Topics of Principal-Agent Contracts: Contract Analysis and Pooling Principals

Item Type text; Electronic Dissertation

Authors Zeng, Shuo

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.

Download date 29/09/2021 22:40:31

Link to Item http://hdl.handle.net/10150/577498 Topics of Principal-Agent Contracts: Contract Analysis and Pooling Principals

by Shuo Zeng

—————————————

A Dissertation Submitted to the Faculty of the

Department of Management Information Systems

In Partial Fulﬁllment of the Requirements For the Degree of

Doctor of Philosophy

In the Graduate College

The University of Arizona 2015 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Shuo Zeng, titled Topics of Principal-Agent Contracts: Contract Analysis and Pooling Principals, and recommend that it be accepted as fulﬁlling the dissertation requirement for the Degree of Doctor of Philosophy.

Date: 8/7/2015 Professor Moshe Dror

Date: 8/7/2015 Professor Paulo Goes

Date: 8/7/2015 Professor Stanley Reynolds

Final approval and acceptance of this dissertation is contingent upon the candidates submis- sion of the ﬁnal copies of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulﬁlling the dissertation requirement.

Date: 8/7/2015 Dissertation Director: Professor Moshe Dror

2 STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulﬁllment of the requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that an accurate acknowledgement of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: Shuo Zeng

3 ACKNOWLEDGEMENTS

I would never have been able to ﬁnish this dissertation without the guidance from many great scholars, help from friends, and understanding and support from my family. It is my pleasure to thank all those who make this dissertation possible.

Foremost, I would like to express my most profound thanks to my advisor, Dr. Moshe Dror, for his continuous encouragement, excellent guidance, and tremendous support throughout my doctoral study. I deeply appreciate his decision to accept me as his doctoral student, his generosity in spending time on me, and his optimism and confidence in me. I will always be indebted to him for his comments on countless revisions of my dissertation, and for his advice on the rehearsal for all my research presentations. He is one of my best mentors and I will definitely benefit a lot from his training in my future career.

I am grateful to my dissertation committee members, Dr. Paulo Goes and Dr. Stanley Reynolds, for their contributions and their helpful comments. I am also grateful to the late Dr. Moshe Shaked, who untimely passed away in October 2014, for his generosity in sharing his time and knowledge with me during the creation of the ﬁrst part of my dissertation.

I would like to thank the following faculty members and staﬀs from the Eller College of Management at the University of Arizona for their various forms of support and advice during the years of my study: Dr. Hsinchun Chen, Dr. Mingfeng Lin, Dr. Nichalin Suakkaphong Summerﬁeld, Dr. Mark Patton, Dr. William Neumann, Dr. Mark Stegeman, Cinda Van Winkle, Mary Lambert, and Cathy Larson. I would also like to thank Dr. Xin Li and Dr. Yida Chen for their emotional support.

Finally and most importantly, my career would never have reached this new height without the love and support from my family. I am deeply grateful to my parents for their unconditional love and care, and for being there whenever I need them. I am also deeply grateful to my wife, Michelle, for her love, understanding, support, and patience. I am lucky to have all of you in my life.

4 DEDICATION

I dedicate my doctoral dissertation to my mother Xiumin Du and my father Ming Zeng, who have always taught me to never give up and have been supporting me unconditionally in all possible ways, especially during the most diﬃcult times of this long journey. I also dedicate my work to my wife, Michelle, whose love and support sustained me throughout my doctoral study.

5 Contents

List of Figures ...... 9

List of Tables ...... 11

Abstract ...... 12

1 Introduction ...... 14

2 Formulating Principal-Agent Service Contracts for a Revenue Generat- ing Unit ...... 15

3 The Basic Principal-Agent ...... 17 3.1 Contractual relationship between a principal and an agent ...... 21

4 Risk-Neutral Agent ...... 24 4.1 Optimal strategies for risk-neutral agent ...... 25 4.1.1 Sensitivity analysis of the optimal strategy ...... 31 4.1.2 The second-best solution ...... 32 4.1.3 Our principal-agent game ...... 34

5 Risk-Averse Agent ...... 35 5.1 Optimal strategies with a weakly risk-averse agent ...... 38 5.1.1 Sensitivity analysis of a weakly risk-averse agent’s optimal strategy . . 53 5.1.2 Principal’s optimal strategy ...... 57 5.2 Optimal strategies given a strongly risk-averse agent ...... 70 5.2.1 Sensitivity analysis of a strongly risk-averse agent’s optimal strategy . 80 5.2.2 Principal’s optimal strategy ...... 83 5.3 Risk-averse agent – a summary ...... 88 5.3.1 Sensitivity analysis of optimal strategies in high revenue industry . . . 90 5.3.2 The second-best solution in high revenue industry ...... 92

6 Risk-Seeking Agent ...... 93

6 6.1 Optimal strategies for the weakly risk-seeking agent ...... 97 6.1.1 Sensitivity analysis of a weakly risk-seeking agent’s optimal strategy . 113 6.1.2 Principal’s optimal strategy ...... 116 6.2 Optimal strategies for the moderately risk-seeking agent ...... 125 6.2.1 Sensitivity analysis of a moderately risk-seeking agent’s optimal strategy135 6.2.2 Principal’s optimal strategy ...... 137 6.3 Optimal strategies for the strongly risk-seeking agent ...... 141 6.4 Risk-seeking agent – a summary ...... 143

7 Summary: Formulating Principal-Agent Service Contracts for a Revenue Generating Unit ...... 145 7.1 Interpreting Table 7.1 ...... 148

8 Pooling Principals ...... 150 8.1 An outline ...... 152

9 Principal-Agent Model with Multiple Principals ...... 153 9.1 The principal-agent model ...... 156 9.2 Performance based contract and the agent’s proﬁt ...... 158 9.3 Principal’s proﬁt ...... 159

10 Selecting Principals Set N ...... 160 10.1 Selecting a ﬁnite set of homogeneous principals ...... 160 10.1.1 Selecting two principals under FCFS queuing discipline ...... 161 10.1.2 Selecting two principals under HOLP queuing discipline ...... 164 10.2 Selecting a ﬁnite set of heterogeneous principals ...... 166 10.3 Simulation results ...... 167 10.3.1 Simulation results with homogeneous principals ...... 169 10.3.2 Simulation results with heterogeneous principals ...... 174 10.4 When to stop pooling principals? ...... 177

11 Serving Principals Set N ...... 179

7 11.1 Principal-agent model with perfectly negative dependencies ...... 179

12 Principal-Agent: The Cooperative Game Perspective ...... 182 12.1 Each principal’s ‘fair contribution’ ...... 189

13 Summary: Pooling Principals ...... 194

14 A Comparison of Performance Based Service Contracts ...... 196

References ...... 199

8 List of Figures

4.1 Illustration of the forms of u(µ) ...... 27 4.2 Structure of the proof for Proposition 4.3 ...... 27 4.3 Conditions when a risk-neutral agent accepts the contract ...... 29 4.4 Structure of the proof for Theorem 4.4 ...... 30 4.5 Structure of the principal-agent extensive form game ...... 34 5.1 π(µ, w, p) as a function of P (1) when η =1...... 37 5.2 Illustration of the forms of u(µ) when η ∈ (0, 3/5] ...... 46 5.3 Structure of the proof for Proposition 5.9 when η ∈ (0, 3/5] ...... 47 5.4 Illustration of the forms of u(µ) when η ∈ (3/5, 4/5) ...... 50 5.5 Structure of the proof for Proposition 5.9 when η ∈ (3/5, 4/5) ...... 51 5.6 Conditions when a weakly risk-averse agent accepts the contract with η = 0.6 54 5.7 Structure of the proof for Proposition 5.18 ...... 64 5.8 Structure of the proof for Theorem 5.19 and Proposition 5.20 ...... 69 ˜ ∗ ∗ ∗ ∗ ∗ ∗ 5.9 The value of ΠP ≡ ΠP (w , p = pcu; µ ) − ΠP (w , p = p3; µ ) for r ∈ (p3, r2) . 71 5.10 Illustration of the forms of u(µ) when η ∈ [4/5, 1) ...... 74 5.11 Structure of the proof for Proposition 5.23 when η ∈ [4/5, 1) ...... 75 5.12 Illustration of the forms of u(µ) when η ≥ 1...... 78 5.13 Structure of the proof for Proposition 5.23 when η ≥ 1 ...... 79 5.14 Conditions when a strongly risk-averse agent accepts the contract with η = 2 81 5.15 Structure of the proof for Theorem 5.27 ...... 86 5.16 Conditions when a risk-neutral principal makes oﬀers to a risk-averse agent . 89 6.1 π(µ, w, p) as a function of P (1) when η = −1...... 94 6.2 Illustration of the forms of u(µ) when η ∈ (0, 3/4] ...... 104 6.3 Structure of the proof for Proposition 6.10 when η ∈ (0, 3/4] ...... 105 6.4 Illustration of the forms of u(µ) when η ∈ (3/4, 8/9) ...... 108 6.5 Structure of the proof for Proposition 6.10 when η ∈ (3/4, 8/9) ...... 109 6.6 Conditions when a weakly risk-seeking agent accepts the contract with η = 0.5 112 6.7 Structure of the proof for Theorem 6.17 ...... 121

9 6.8 Illustration of the forms of u(µ) when η ∈ [8/9, 1) ...... 128 6.9 Structure of the proof for Proposition 6.19 when η ∈ [8/9, 1) ...... 129 6.10 Illustration of the forms of u(µ) when η ∈ [1, 2) ...... 132 6.11 Structure of the proof for Proposition 6.19 when η ∈ [1, 2) ...... 133 6.12 Conditions when a moderately risk-seeking agent accepts the contract with η = 1135 6.13 Structure of the proof for Theorem 6.22 ...... 139 6.14 Illustration of the forms of u(µ) when η ≥ 2 ...... 142 6.15 Structure of the proof for Proposition 6.24 when η ≥ 2 ...... 142 6.16 Conditions when a principal makes contract oﬀers to a risk-seeking agent . . 144 10.1 State transition diagram under FCFS when |N| = 2 ...... 161 10.2 State transition diagram under HOLP when |N| = 2 ...... 164 11.1 State transition diagram with perfectly negative interdependencies ...... 180

10 List of Tables

3.1 The variables of the model ...... 20 5.1 Indicators of the monotonicity and the concavity/convexity of function u(µ) in (5.3) ...... 39 6.1 Indicators of the monotonicity and the concavity/convexity of function u(µ) in (6.3) ...... 97 7.1 Summary of the optimal principal-agent contract formulas under exogenous conditions ...... 149 10.1 Simulation results with homogeneous principals (λ = 0.8, FCFS queuing discipline) ...... 170 10.2 Simulation results with homogeneous principals (λ = 1, FCFS queuing discipline)170 10.3 Simulation results with homogeneous principals (λ = 0.8, HOLP queuing discipline) ...... 171 10.4 Simulation results with homogeneous principals (λ = 1, HOLP queuing discipline) ...... 171

10.5 Simulation results with heterogeneous principals (r1 = 12, λ1 = 0.9, r2 = 10,

λ2 = 0.8, r3 = 11, λ3 = 1.1, r4 = 8, λ4 = 1.2, FCFS queuing discipline) . . . . 175

10.6 Simulation results with heterogeneous principals (r1 = 12, λ1 = 0.9, r2 = 10,

λ2 = 0.8, r3 = 11, λ3 = 1.1, r4 = 8, λ4 = 1.2, HOLP queuing discipline) . . . 176 10.7 Simulation results with homogeneous principals (r = 10 and λ = 1) and

positive interdependencies (ψi4 = ψ4i = 5 for all i ∈ {1, 2, 3}) ...... 178

12.1 A summary of the cases in the proof of Theorem 12.3 where i ∈ NA and

S ⊂ T ⊆ NA \{i} ...... 184

L 12.2 The frequency of a ∈ C(NA, v) for diﬀerent values of |N| ...... 194

11 ABSTRACT

Consider companies who rely on revenue generating equipment that fails from time to time. Assume that a company owns one unit of equipment, whose maintenance and repair services are outsourced to a qualified service provider. We assume that the company (the principal) outsources the maintenance and repair services using performance based contracts. Such contractual relationships fall into economics’s principal-agent framework. The owners of the revenue generating units are referred to as principals, and the service provider as the agent. We address the following questions: What are the optimal contracting strategies for a principal and an agent? Can the agent benefit from pooling the service demands from multiple principals? This dissertation contains two main bodies of work contained in chapters 2∼7 and chapters 8∼13 respectively. In the first part of this dissertation (chapters 2∼7) we examine the contractual options between a single principal and a single agent. The contractual options of a principal and an agent are modeled as a Markov process with an undetermined time horizon. For a risk neutral principal we identify the conditions under which a principal contracts with a risk-neutral, risk-averse, or risk-seeking agent and derive the principal’s optimal offer and the agent’s service capacity response. In essence, we provide an extensive formulating analysis of principal-agent contracts given any exogenous parameter values. That is, we derive mathematical formulas for the optimal contract offers and the agent’s optimal service capacity. It turns out that a small number of formulas cover a large spec- trum of principal-agent conditions. In the second part of this dissertation (chapters 8∼13), in a counter distinction to the vast literature in economics on principal-agent contractual interplay and its predominant concern with the principal, here we focus on the agent. In the case of performance based service contracts it is known that the principal extracts all the economic surplus and the agent breaks even. But this is not the case for an agent of good standing contracting with multiple principals. We show that an agent who contracts a collection of principals with interdependent failure characteristics does better than break- even – such an agent realizes a profit rate that is convexly increasing in the number of principals. The corresponding cooperative game assessing each principal’s contribution to the agent’s profit is convex and its easily computable Louderback’s value seems always to

12 be in its core. In chapter 14 we present the outline of a future study that compares several different options of contract structure faced by the principal and the agent, because the optimal contracting strategies for the principal and the agent may not necessarily be the same under different contract structures. We discuss briefly the agent’s and the principal’s behavior under different forms of performance based contract, which serves as a starting point for future extensions of this dissertation. To summarize, this dissertation provides practical mathematical results and important managerial insights into the principal-agent contract in equipment repair services industry.

13 1 Introduction

Consider a company that owns a revenue generating equipment unit. The unit fails from time to time. When the unit fails no revenue is generated. We assume that the company outsources the equipment unit’s maintenance and repair function to an expert repair service provider. The dissertation examines the contractual relationships between such a company and the service provider. This setting is known in economics as the principal-agent framework where the owner of revenue generating unit is referred to as the principal and the service provider as the agent. The contracting procedure is straight forward: a principal proposes a contract to an agent for repair services. Once the agent accepts the contract, he acquires a level of service capacity (such as purchasing specialized tools, ordering spare parts and hiring well-trained technicians, etc.) to repair the principal’s unit whenever it fails. We assume that the expected repair time during the contract period is inversely proportional to the service provider’s service capacity. The agent is compensated as prescribed by the contract.

Since a failed unit does not generate any revenue, therefore it stands to reason that the principal prefers shorter repair times, which implies a higher service capacity. On the other hand, due to a positive marginal cost of service capacity, the agent has an incentive to install a lower service capacity. We assume that the principal does not contract directly on the agent’s service capacity (unobservable by the principal) for two reasons. First, the agent may acquire the service capacity after signing the contract. Second and more important, the agent’s service capacity is reﬂected by the average repair times, which can only be observed by the end of the contract period. Due to unobservability, the agent’s capacity decision may deviate from the principal’s desired service capacity, and this is known in economics as moral hazard. In this dissertation we focus on performance based contracts used to alleviate moral hazard. By using a performance based contract, the principal can transfer part of her revenue risk to the agent’s revenue risk thus provide an incentive for the agent to install service capacity desired by the principal. We address two questions in this dissertation: (1) What are the optimal contracting strategies for a single principal and a single agent? (2)

14 Can an agent proﬁt from pooling the service demands of multiple principals?

This dissertation is organized as follows. In the first part (chapter 2∼7), we extensively analyze the principal-agent repair service contracts in a single principal and single agent setting. We assume that the principal is risk-neutral and the agent can be either risk- neutral, risk-averse or risk-seeking. We derive a small set of mathematical formulas that represent the optimal contracting strategies for both the principal and the agent for the whole value space of exogenous parameters. In the second part (chapter 8∼13), we examine how an agent of good standing chooses to contract with a subset of principals that benefits him the most. Due to the complexity of an agent’s optimization problem of pooling the ‘best’ subset of principals, we resort to numerical simulations to understand the agent’s optimal pooling strategy. Based on the simulation results, we then propose a theoretical model of the agent’s optimal profit and examine several value allocation schemes to assess each principal’s contribution to the agent’s profit. Finally in chapter 14 we present an outline of an ongoing study that compares the optimal strategies of a principal and an agent under different performance based contract structures.

2 Formulating Principal-Agent Service Contracts for a Revenue

Generating Unit

With the ongoing technological advancements in manufacturing, health delivery systems, information technologies etc., numerous industrial entities become reliant on sophisticated product delivery systems for provision of revenue generating operations. For example, fuel- efficient aircraft engines are essential for airlines to provide affordable transportation services; mining companies operate large interdependent mining equipment units for extraction of hun- dreds truck loads of ores everyday; oil refineries construct groups of fractionating columns to produce various crude oil products; sophisticated flexible manufacturing systems enable manufacturing companies to machine different types of parts with high efficiency at low costs; data server arrays are the backbone of real-time electronic transaction systems oper-

15 ated by banks and credit card companies; advanced oﬃce printing and scanning equipment is indispensable for eﬃcient information collection and dissemination in large companies, universities and government agencies.

There are common characteristics shared by the equipment units of sophisticated product delivery systems. First, the equipment units are mission-critical such that no revenue is generated when the equipment units fail. Second, these units are assumed to operate in a reliable mode with short downtimes relatively to their uptimes. Third, the units are usually of a specialized nature that requires expert maintenance/service providers. It is common for the owner of such systems to outsource the maintenance and repair of her equipment units to an independent supplier of specialized repair services. Therefore the main topic of this work – the analysis of the contractual details that have to be addressed in the agreement between the system’s owner and the supplier of maintenance and repair services.

In this work we examine the contractual options between the owner (principal, she) of a revenue generating unit and a service provider (agent, he) in a framework of principal-agent economic model. Although our initial framing of the principal-agent problem follows Kim et al. (2010), our analysis is signiﬁcantly diﬀerent from Kim et al. (2010) and is much more extensive than their analysis. First, the agent is assumed to be risk-neutral or risk-averse in Kim et al. (2010) while our analysis also include risk-seeking agent. Second, our analysis of the principal-agent contract covers the value of exogenous parameters exhaustively, while Kim et al. (2010)’s assumptions of a reliable equipment unit and negligible downtimes (compared to uptimes) require that the values of certain exogenous parameters fall into a narrow range. Finally and the most important, we derive explicit formulas for optimal principal- agent contract under any market and industry conditions without imposing any additional constraint.

In a counter-distinction to Kim et al. (2010) we model the principal-agent system of a risk-neutral principal with risk-neutral, risk-averse, or risk-seeking agent as a Markov process

16 with an undetermined time horizon instead of a contract for a ﬁnite horizon normalized to 1. In addition, we replace Kim et al. (2010)’s representation of agent’s risk as variance of his revenue stream with a piece wise linear function in a steady state probability of failure as a proxy for a measure of agent’s revenue risk.

Our analysis assumes a single risk-neutral principal who owns one unit of revenue generating entity and a single agent.

3 The Basic Principal-Agent

In a basic principal-agent setting, the principal contracts an agent to perform a service function and the agent chooses the level of his capacity (his ‘effort’) in response to the contract offer and subsequently its effect on the principal’s revenue stream. We assume that the principal’s equipment unit generates revenue at an expected rate of r > 0 $ per unit of uptime. The unit runs for a random period of time before failing, and remains in the failed state until it is repaired. To address the recurring maintenance and equipment failures the principal contracts an agent who subsequently installs a repair capacity and repairs the principal’s equipment when it fails. The contract structure considered is rather simple: the principal proposes to pay the agent w > 0 $ per unit of time during the duration of the contract but the agent pays the principal p > 0 $ per unit of time during the unit’s failure duration. The agent’s capacity decision is unobservable by the principal. Each party is presumed to choose the values that maximize his/her utilities. We assume that the parties are rational and each knows that the other is rational, etc. till infinitum. It includes their individual computational ability to anticipate (compute) the other party’s best response to any offer. Therefore, with some abuse of timing we presume that both, the contract offer and the service capacity decision, occur at the same time with full knowledge of the two parties.

In general, if the agent’s action is observable and contractible, then the principal would contract directly on agent’s service capacity that maximizes the principal’s proﬁt leaving zero

17 surplus to the agent – enough to ensure agent’s participation. Such a scenario is referred to as the first-best solution (Hölmstrom,1979). If the agent’s action is unobservable and therefore uncontractible, then the agent’s response may deviate from the one prescribed by the principal in the first-best solution, and the principal risks realizing lower profits. The likelihood and the degree of agent’s deviation from the desired action is referred to as moral hazard (Luenberger, 1995). When moral hazard is present the principal uses the available information about the agent’s action to alleviate the moral hazard (Hölmstrom, 1979) and proposes a contract with incentives that aim the agent to maximize her profit.

Principal’s main information about the agent’s capacity is deduced from her revenue stream. The revenue consequences of agent’s action are referred to as the service performance characteristics, and quantified service performance metrics are referred to as performance measures. The contracts that use performance measures are called performance based contracts. By offering an agent performance based contract, the principal transfers part of her risk regarding revenue to the agent’s revenue risk, thus providing incentives for the agent to choose the action desired by the principal. If the performance measure is positively correlated with principal’s revenue, a rate of award for each unit of the performance measure, known as the piece rate b, is specified in the contract. If the performance measure is negatively correlated with principal’s revenue, a penalty rate for each unit of the performance measure, denoted by p, is specified in the contract.

Under performance based contracts, the agent maximizes his utility in response to the scheme proposed by the principal, and the principal maximizes her profit while anticipating the agent’s optimizing decision. This scenario is referred to as the second-best solution (Hölmstrom,1979). Given a compensation scheme, if the agent’s utility is globally concave, the second-best solution can be derived using first order condition of the agent’s utility, referred to as the first-order approach. If the agent’s utility is not globally concave, the first-order approach is generally invalid and alternative approaches have to be used such as converting the agent’s utility optimization problem into a convex programming problem

18 (Grossman and Hart, 1983).

In our case short unit’s downtimes (relative to uptimes) imply a higher revenue for the principal, thus the downtimes and their frequency infer the agent’s service performance. The service capacity can only be inferred to by the nature of downtimes, which are unobservable before signing the contract. Therefore moral hazard is of concern with performance based contracts. The performance measure adopted here is based on the unit’s downtimes. The downtimes are negatively correlated with principal’s revenue, and the agent is charged a penalty p $ for each unit (seconds, minutes, hours or days) of the performance measure.

In Kim et al. (2010) the profit function of the principal and the utility function of the agent are based on three assumptions. First, the unit is mission-critical and the principal owns one unit. Second, the unit is highly reliable such that the service times are relatively short as compared to the uptimes. Third, the service times are independently and identically distributed, and the distribution has no upper bound on the realization of the service times. This model has two pitfalls: (i) Kim et al. (2010) assume the failures as a Poisson arrival process independent of the service times. It allows for a new failure to occur while the unit is still in a failed state, contradicting that no new failure can occur when in a failed state. (ii) The profit/utility functions describe the total profit/utility during a single contract period assumed finite and normalized to 1. Although the contract period is finite, it contradicts their assumption about the service time distribution with no upper bound on duration of the service time.

To repeat, the failure rate of the equipment unit is a constant λ, the repair time is exponential with a constant repair rate µ (the service capacity is the repair rate), yielding a less general model than Kim et al. (2010). Furthermore, we do not restrict the contract to a period of time, rather, the contract can be dynamic and and can be oﬀered and accepted/rejected continuously in time.

19 The unit’s failure rate λ > 0, the principal’s expected revenue rate r > 0, and the marginal capacity cost c > 0, are exogenous variables. The payment rate w and the penalty rate p are determined by the principal, whereas the service capacity µ ≥ 0 is determined by the agent. We denote an exogenous scalar parameter η as preference and intensity indicator for agent’s risk attitude: η = 0 for risk-neutral, η > 0 for risk-averse, and η < 0 for risk- seeking.

The seven variables that appear in our model are listed in Table 3.1.

Table 3.1: The variables of the model

Variable Description Type η agent’s risk attitude exogenous r unit’s revenue rate exogenous λ unit’s failure rate exogenous c marginal rate of capacity cost exogenous w agent’s compensation rate determined by the principal p agent’s penalty rate determined by the principal µ service capacity determined by the agent

Two performance measures are considered in Kim et al. (2010). The first one is cumulative downtime – the sum of downtimes during a finite contract period. The second one is the average downtime, which uses the sample average of downtimes during a finite contract period as the performance measure. The two measures provide different incentives for the agent’s capacity decisions. In essence, the agent’s optimal service capacity behaves non- monotonically with the failure rate when using average downtime, while it is monotonically increasing when using cumulative downtime. This is because average downtime reflects the risk differently compared to cumulative downtime. When the failure rate is higher, the expected number of failures is higher during the finite contract period. For a higher number of failures and the same service capacity, average downtime dilutes the agent’s risk by a factor proportional to the square of the number of failures as compared to cumulative downtime, thus provides an incentive for the agent to choose a lower service capacity, leading to reduced service performance. We adopt the steady state probability of the failed state as the sole

20 performance measure, which is equivalent to cumulative downtime in our undetermined time horizon setting.

The literature on principal-agent setting is extensive in economics since the topic is fundamental to the economic analysis of firms’ interdependence via contractual agreements that impact their output. We do not survey here the principal-agent literature. This has been done very well by numerous authors. A partial list includes Ross (1973), Hölmstrom (1979), Stiglitz (1974), Stiglitz (1979), Myerson (1983), Hölmstrom and Milgrom (1987), Fudenberg and Tirole (1990), Maskin and Tirole (1990), Maskin and Tirole (1992), and Bolton and Dewatripont (2005). For analytic and numerical solutions to principal-agent problems see Grossman and Hart (1983) and Guesnerie and Laffont (1984).

3.1 Contractual relationship between a principal and an agent

When an agent contracts a single principal, the agent is always available when the unit fails, therefore the unit’s downtimes are the same as the service times. To mitigate the pitfalls in Kim et al. (2010) we recast this system a Markov process. The state of the Markov process is defined as the state of the principal’s unit: in state 0 when the unit is operational, and in state 1 when the unit is not operational. We assume that the uptimes of the unit are independently and identically distributed following an exponential distribution that is governed by the unit’s failure rate, and the service times of the unit are independently and identically distributed, following an exponential distribution governed by the agent’s service capacity. For a risk-neutral agent we propose an objective function that describes his expected utility rate for each unit of time in an infinite time contract assuming the Markov process is in steady state. Similarly we propose an objective function that describes a risk-neutral principal’s expected profit rate. Both the principal’s and the agent’s objective functions depend on the compensation rate w > 0 paid by the principal to the agent and the penalty rate p > 0 charged by the principal for each unit of downtime. Furthermore, the principal’s expected profit rate also depends on the revenue rate r > 0, and the agent’s expected utility rate also depends on the marginal cost c > 0 of the service capacity for each unit of time. In

21 our principal-agent contractual relationship, the principal controls w and p, and the agent controls µ, therefore we call vector ((w, p), µ) a strategy. The c is exogenously determined by the market and in this work it is normalized as a monetary unit ⇒ c ≡ 1. Observation 4.1 (below) points out that a contract with compensation rate w paid only for each unit of uptime and penalty rate charged for each unit of downtime is equivalent to our setting of principal-agent contract.

Notation: Denote the principal’s expected proﬁt rate by ΠP (w, p; µ) and the agent’s ex-

pected utility rate by uA(µ; w, p), omitting the exogenous parameters.

When the agent does not accept the contract oﬀer he commits no service capacity and

receives no compensation. uA(µ = 0) = 0 is referred to as the agent’s reservation utility rate. An agent accepts the contract only if his expected utility rate is greater than or equal to his reservation utility rate, referred to as the individual rationality (IR) constraints. When the principal does not contract an agent for the repair service, then since an equipment failure will occur after some finite time with probability 1, therefore in the long run the principal’s expected profit rate equals zero, which is referred to as the principal’s reservation profit rate

(ΠP = 0). Individual rationality principal dictates that the principal offers a contract only if her expected profit rate is strictly greater than her reservation profit rate.

When a principal-agent contract exists, the agent’s average utility over a finite period of time converges to his expected utility rate as the period approaches infinity. However with positive probability the agent receives negative revenue stream over some finite period of time, such that his cumulative revenue (utility) drops below a certain threshold and triggers bankruptcy preference claim against the agent. In our work, we presume that the likelihood of such bankruptcy condition to occur is negligible.

The above principal-agent problem is characterized by expression of the principal’s and agent’s expected proﬁt/utility rates and the values of the exogenous parameters. Denote a

22 principal-agent problem by P(ΠP , uA, η, λ, r) or for short P.

Deﬁnition 3.1 (Strategy Set). The strategy set of a principal-agent problem P is deﬁned as a vector S(P) ≡ {((w, p), µ) |w > 0, p > 0, µ ≥ 0}.

Deﬁnition 3.2 (Weak Domination). Consider two strategies ((w, p), µ),((w0, p0), µ0) ∈ S(P). ((w, p), µ) is said to weakly dominates ((w0, p0), µ0), denoted by ((w, p), µ) ((w0, p0), µ0),

0 0 0 0 0 0 if the two strategies result in ΠP (w, p; µ) ≥ ΠP (w , p ; µ ) and uA(µ; w, p) ≥ uA(µ ; w , p ) with at least one strict inequality.

Deﬁnition 3.3 (Set of Admissible Solutions). The set of admissible solutions (also known as the set of Pareto optimal solutions) for the principal-agent problem P is the set s(P) of all strategies ((w, p), µ) ∈ S(P) for which:

(a) @ ((w0, p0), µ0) ∈ S(P) such that ((w0, p0), µ0) ((w, p), µ) – there is no other strategy that weakly dominates ((w, p), µ).

(b)Π P (w, p; µ) > ΠP and uA(µ; w, p) ≥ uA.

Pareto optimality implies that the principal cannot increase her expected proﬁt rate without lowering the agent’s expected utility rate and vice versa (Luenberger, 1995), and it has been proven that generally both the principal and the agent achieve Pareto optimality as a subset of the second-best solutions (Ross, 1973). Since the agent’s IR is always binding, condition (a) in Deﬁnition 3.3 guarantees that all admissible solutions are Pareto optimal. We require that all the solutions proposed in this work be Admissible Solutions.

This work is presented as follows: In Section 4, we describe the basic model with a risk-neutral principal and a risk-neutral agent, and we state the exogenous conditions that guarantee the existence of a contract and the optimal contract terms. In Section 5 we analyze risk-averse agent. Section 6 is dedicated to the analysis of a risk-seeking agent. In Section 7 we summarize our ﬁndings and conclusions. Notation is introduced as needed.

23 4 Risk-Neutral Agent

When a risk-neutral agent accepts a contract oﬀer (w, p), his expected utility rate is composed of the expected value of the compensation rate from the principal and a deterministic cost rate of the service capacity which can be expressed as w−pP (1)−µ, where P (1) denotes the steady state probability of the unit being in the failed state. Similarly denote the steady state probability of the unit being operational by P (0) = 1 − P (1).

Notation: (x)+ = x when x ≥ 0 and (x)+ = 0 when x < 0.

A risk-neutral agent’s expected utility rate is:

uA(µ; w, p) = (w − pP (1) − µ)+ for w > 0, p > 0, µ ≥ 0 (4.1)

P (0) and P (1) (functions of λ and µ), represent the proportion of time in the steady state the Markov process is in state 0 and state 1 respectively (Ross, 2006). They satisfy the balance equations of the Markov process and sum up to 1, thus P (0) = µ/(λ+µ),P (1) = λ/(λ+µ):

pλ uA(µ; w, p) = w − − µ for w > 0, p > 0, µ ≥ 0 (4.2) λ + µ +

Since the principal determines w and p she can always entice the agent to accept the contract.

For r > 0 (determined exogenously by the market), the principal’s expected proﬁt rate is composed of the expected revenue rate generated by her unit, the expected penalty rate collected from the agent and the compensation rate paid to the agent:

rµ pλ Π (w, p; µ) = rP (0) − w + pP (1) = − w + for w > 0, p > 0, µ ≥ 0 (4.3) P λ + µ λ + µ

Observation 4.1. We note that under another type of contract, where the principal compensates the agent only for each unit of uptime (instead of each unit of time), the agent’s expected utility rate is equivalent to (4.2), and the principal’s expected proﬁt rate is equivalent

24 to (4.3): Under the new type of contract, denote the compensation rate by w˜ and the penalty rate by p˜, therefore the agent’s expected utility rate becomes:

wµ˜ pλ˜ uA(µ;w, ˜ p˜) = (wP ˜ (0) − pP˜ (1) − µ)+ = − − µ forw ˜ > 0, p˜ > 0, µ ≥ 0 λ + µ λ + µ + (4.4)

and the principal’s expected proﬁt rate becomes:

rµ wµ˜ pλ˜ Π (w, ˜ p˜; µ) = rP (0) − wP˜ (0) +pP ˜ (1) = − + forw ˜ > 0, p˜ > 0, µ ≥ 0 P λ + µ λ + µ λ + µ (4.5)

Replacing w˜ by w and p˜ by (p − w) in (4.4) and (4.5) we obtain (4.2) and (4.3) respectively.

Note that a performance based contract can even take the form such that a compensation rate is speciﬁed for each unit of uptime (instead of each unit of time) and no penalty rate is charged whatsoever. That is, the principal controls only one variable (the compensation rate) instead of two (the compensation rate and the penalty rate). However this form of performance based contract is not discussed in this work.

Returning to the agent as in (4.2) we deﬁne the part inside the brackets by

pλ u(µ) ≡ w − − µ (4.6) λ + µ

i.e., for µ ≥ 0, u(µ) is continuous and diﬀerentiable everywhere:

du(µ) pλ d2u(µ) 2pλ = − 1 and = − < 0 dµ (λ + µ)2 dµ2 (λ + µ)3

du(µ) p du(µ) u(0) = w − p, = − 1 and lim = −1 µ→+∞ dµ µ=0 λ dµ

4.1 Optimal strategies for risk-neutral agent

Note that u(µ) in (4.6) increases and ΠP (w, p; µ) in (4.3) decreases in w, therefore for any value of penalty rate p, the principal can raise her expected proﬁt rate by adjusting the rate

25 w low enough while ensuring the agent’s participation by setting the agent’s expected utility rate equal to his reservation utility rate. Although the principal cannot contract directly on the agent’s capacity, she presumes the agent will optimize his expected utility rate. That is, for any compensation rate w and penalty rate p proposed by the principal, the agent computes the value of µ that maximizes his expected utility rate and decides whether to accept the contract or not by solving the following optimization problem:

pλ max u(µ) = max w − − µ (4.7) µ≥0 µ≥0 λ + µ

∗ with agent’s optimal service capacity denoted by µ (w, p) = argmaxµ≥0 u(µ).

2 We describe the agent’s optimal response to any possible contract oﬀer (w, p) ∈ R+ in Proposition 4.3, but we start with a simple technical lemma – one of many. √ Lemma 4.2. If p > λ > 0, then p > 2 pλ − λ > 0. √ √ √ √ 2 Proof. If p > λ > 0, then 2 pλ−λ > 2λ−λ = λ > 0 and p−2 pλ+λ = p − λ > 0, √ where the latter inequality indicates p > 2 pλ − λ.

Proposition 4.3. Consider a risk-neutral agent with uA(µ; w, p) given in (4.2).

(a) Given p ∈ (0, λ], then the agent accepts the contract only when w ≥ p and does not commit any service capacity (µ∗(w, p) = 0), which results in expected utility rate

∗ uA(µ (w, p); w, p) = w − p ≥ 0. √ (b) Given p > λ, then the agent accepts the contract only when w ≥ 2 pλ − λ and √ installs service capacity µ∗(w, p) = pλ − λ > 0 resulting in expected utility rate

∗ √ uA(µ (w, p); w, p) = w − 2 pλ + λ ≥ 0.

Proof. Figure 4.1 illustrates the form of u(µ) when the value of p falls in diﬀerent ranges. The structure of the proof for Proposition 4.3 is depicted in Figure 4.2.

Case p ∈ (0, λ]: u(µ) is decreasing for µ ≥ 0, therefore the optimal service capacity is set at µ∗(w, p) = 0 and u(µ∗(w, p)) = w − p.

26 Case u ( µ )

0.95 0.96 0.97 0.98 0.99 w condition order Subcase Subcase Subcase Subcase λ > p 0 − 2 rc foﬀered. if tract tract. √ : pλ λ h evc aaiyta maximizes that capacity service The w w w w + (a) ∈ ≥ ∈ ≥ λ = λ Risk-Neutral codn oLma42w aet eov h olwn subcases: following the resolve to have we 4.2 Lemma to According . 2 0.01 p (0 λ iue42 tutr ftepoffrPooiin4.3 Proposition for proof the of Structure 4.2: Figure 2 p Agent du 0 ∈ √ :

p , , w (0 µ ( u 2 = pλ µ 1 iue41 lutaino h om of forms the of Illustration 4.1: Figure ( √ ): λ ,

) µ p 3 /dµ = λ ] ∗ pλ − 0.5 u ( ,p w, ( λ µ λ | − p µ ∗ : = λ > p ∈ ( 4 )) ,p w, µ λ λ (0 u ∗ λ , ( ( ≥ w,p µ : ] )) ∗ ,tu h gn ol cettecnrc foﬀered. if contract the accept would agent the thus 0, u ( ) ,p w, ( 5 < 0 = λ µ ∗ ,teeoeteaetrjcstecontract. the rejects agent the therefore 0, ( )) ,p w, 27 ⇒ w ≥ w ∈ u ( µ ) )) ≥ w µ ,teeoeteaetwudacp h con- the accept would agent the therefore 0, 0 w ∗ , ∈ 2 2 √ ( < ≥ √ (0 0.950 0.955 0.960 0.965 0.970 0.975 0.980 ,p w, pλ pλ p , p u ,teeoeteaetrjcstecon- the rejects agent the therefore 0, − 0 ) − ( λ = ) µ λ spstv sse rmteﬁrst the from seen as positive is ) √ λ pλ u µ ( µ ∗ − (b) = λ = ) Reject. Reject. µ 2 ∗ > λ √ 0.01 λ 0 = pλ λ > p

w − µ = λ and 0 1

3 p λ = 2 λ u ( 4 λ µ ∗ ( ,p w, 5 λ )= )) In summary, given exogenous market conditions such that there exists a contract bene- ﬁting both the agent and principal (see Theorem 4.4 later), only one formula is necessary √ for the agent to determine his service capacity: µ∗(w, p) = pλ − λ > 0.

The conditions when the agent accepts the contract are depicted by the shaded areas in Figure 4.3. The two shaded areas with diﬀerent grey scales represent conditions √ {(w, p): p ∈ (0, λ] , w ≥ p} and (w, p): p > λ, w ≥ 2 pλ − λ under which the agent accepts the contract but responds diﬀerently. The lower bound function of the shaded areas

(denoted by w0(p)) represents the contract oﬀers that result in agent zero expected utility rate. w0(p) is deﬁned as follows:

  p for p ∈ (0, λ] w0(p) = √  2 pλ − λ for p > λ

Since limp→λ− w0(p) = limp→λ+ w0(p) = λ, limp→λ− dw0(p)/dp = limp→λ+ dw0(p)/dp = 1,

w0(p) is continuous and diﬀerentiable everywhere for p ∈ R+.

Anticipating (calculating) the agent’s optimal response µ∗(w, p) the principal chooses w and p that maximize her expected proﬁt rate by solving the optimization problem:

∗ ∗ rµ (w, p) pλ max ΠP (w, p; µ (w, p)) = max − w + (4.8) w>0,p>0 w>0,p>0 λ + µ∗(w, p) λ + µ∗(w, p)

∗ ∗ ∗ with the optimal rates (w , p ) = argmaxw>0,p>0 ΠP (w, p; µ (w, p)). We only consider pairs 2 (w, p) ∈ R+ such that uA(µ; w, p) ≥ 0.

n p o Deﬁne: DRN ≡ {(w, p): p ∈ (0, λ] , w ≥ p} ∪ (w, p): p > λ, w ≥ 2 pλ − λ (4.9)

Theorem 4.4. Given a risk-neutral agent as in (4.2) and a principal as in (4.3) and suppose that (w, p) ∈ DRN .

28 w

∗ ∗ µ =0 µ = p λ − λ

w=2 p λ − λ w=p 0 p 0 λ

Figure 4.3: Conditions when a risk-neutral agent accepts the contract

(a) If r ∈ (0, λ], then the principal does not propose a contract.

(b) If r > λ, then the principal’s oﬀer and the agent’s capacity are respectively

√ √ (w∗, p∗) = 2 rλ − λ, r and µ∗(w∗, p∗) = rλ − λ (4.10)

√ ∗ ∗ ∗ ∗ ∗ resulting in principal’s expected proﬁt rate ΠP (w , p ; µ (w , p )) = r − 2 rλ + λ.

Proof. The structure of the proof for Theorem 4.4 is depicted in Figure 4.4.

Case p ∈ (0, λ] and w ≥ p: According to Proposition 4.3 part (a), the agent would accept

the contract without installing any service capacity. Since ∂ΠP /∂w = −1 < 0, the

∗ ∗ ∗ ∗ principal chooses w = p and ΠP (w , p; µ (w , p)) = −p + p = 0. Left with zero expected proﬁt rate, the principal does not propose a contract.

29 p ∈ (0, λ] and w ≥ p No contract oﬀered.

Risk-Neutral Principal with Risk-Neutral r ∈ (0, λ] No contract oﬀered. Agent √ p > λ and w ≥ 2 pλ − λ

√ w∗, p∗ = 2 rλ − λ, r r > λ √ µ∗(w∗, p∗) = rλ − λ

Figure 4.4: Structure of the proof for Theorem 4.4

√ Case p > λ and w ≥ 2 pλ − λ: According to Proposition 4.3 part (b), the agent ac- √ cepts the contract and installs capacity pλ − λ. Since ∂ΠP /∂w = −1 < 0, thus

∗ √ ∗ ∗ ∗ w = 2 pλ−λ and the principal’s optimization problem is maxp>λ ΠP (w , p; µ (w , p)) where:

√ √ r Π (w∗, p; µ∗(w∗, p)) = r + λ − λ p + √ (4.11) P p

√ √ Deﬁne x ≡ p, a ≡ λ. The principal’s expected proﬁt rate, denoted by f(x), can be restated as f(x) = r + a2 − a (x + r/x) for x > 0 and a > 0. Maximizing f(x) with

∗ ∗ ∗ respect to x > 0 is equivalent to maximizing ΠP (w , p; µ (w , p)) with respect to p > 0 in the sense that

2 ∗ ∗ ∗ argmax ΠP (w , p; µ (w , p)) = argmax f(x) p>0 x>0

? ∗ ∗ ∗ 2 2 3 Denote p ≡ argmaxp>0 ΠP (w , p; µ (w , p)). Since d f(x)/dx = −2ar/x < 0, thus

f(x) is concave with respect to x > 0 and from the ﬁrst order condition df(x)/dx|x=x∗ = √ ar/ (x∗)2 − a = 0 ⇒ x∗ = r. Therefore p? = (x∗)2 = r. However p? = r is not necessarily the optimal solution because the principal maximizes p for p > λ. Thus p∗ = max{r, λ}.

Subcase r ∈ (0, λ]: p∗ = λ; the principal does not propose a contract since her expected proﬁt rate is zero. √ ∗ ∗ ∗ ∗ ∗ ∗ Subcase r > λ: p = r; the principal receives ΠP (w , p ; µ (w , p )) = r−2 rλ+λ = √ √ 2 √ r − λ > 0 and proposes a contract (w∗, p∗) = (2 rλ − λ, r) that induces

30 √ the agent to install service capacity µ∗(w∗, p∗) = rλ − λ.

In summary, if r ∈ (0, λ], then the principal does not propose a contract (Theorem 4.4 √ (a)). If r > λ, then the principal oﬀers (w∗, p∗) = 2 rλ − λ, r and the agent installs √ capacity µ∗(w∗, p∗) = rλ − λ (Theorem 4.4 (b)), which is an admissible solution according to Deﬁnition 3.3.

Note that in an optimal contract conﬁguration the agent compensates fully the principal for lost revenue during the unit’s fail duration.

4.1.1 Sensitivity analysis of the optimal strategy

The principal-agent rationality assumption are odds with the agent accepting a contract oﬀer and responding with µ∗ = 0. Therefore the only viable case is when the agent accepts √ the contract and installs µ∗(w, p) = pλ − λ. In this case the rate w is bounded below by √ 2 pλ − λ = pP (1) + µ∗(w, p), with pP (1) representing the expected penalty rate charged by the principal when the optimal capacity is installed. It implies that the agent should at least be reimbursed for the expected penalty rate and the cost of the optimal service capacity in exchange for his repair service.

The optimal service capacity itself depends only on p and λ. Note that ∂µ∗/∂p = pλ/4p > 0 and ∂µ∗/∂λ = pp/4λ − 1. It indicates that given a λ the agent will increase the µ when the p increases. However, given a p the change in µ∗ with respect to the √ failure rate is not monotonic. The pλ − λ, as a function of λ, increases when λ ∈ (0, p/4) and decreases when λ ∈ (p/4, p). If the principal’s unit is reliable (λ ∈ (0, p/4)), then the agent increases the µ when λ increases. If the principal’s unit is less reliable (λ ∈ (p/4, p)), then the savings from reducing the µ are greater than the increase in p, therefore the agent will reduce µ∗ when the λ increases.

√ The agent’s optimal expected utility rate when installing capacity µ∗(w, p) = pλ − λ

∗ ∗ √ is uA ≡ uA(µ (w, p); w, p) = w − 2 pλ + λ, and it depends on w, p and λ. Note that

31 ∗ ∗ p ∂uA/∂w = −1 < 0, ∂uA/∂p = − λ/p < 0, indicating that the agent’s optimal expected utility rate decreases with the compensation rate and the penalty rate. Note that

∗ p p ∂uA/∂λ = − p/λ + 1, and from Proposition 4.3 p > λ ⇒ − p/λ + 1 < 0, therefore the agent’s optimal expected utility rate also decreases with the failure rate.

According to Theorem 4.4, a principal offers a contract to a risk-neutral agent only if √ ∗ ∗ ∗ r > λ and her offer is (w , p ) = 2 rλ − λ, r resulting in expected profit rate ΠP ≡ √ √ 2 ∗ ∗ ∗ ∗ ∗ √ ΠP (w , p ; µ (w , p )) = r − 2 rλ + λ = r − λ . The compensation rate and the expected profit rate depend on r and λ, and the penalty rate equals r. Note that ∂w∗/∂r = pλ/r > 0 and ∂w∗/∂λ = pr/λ − 1 > 0 implying that given the λ, the principal will increase w when the revenue rate increases, and given the revenue rate, the principal will √ ∗ √ √ ∗ increase w when λ increases. Note that ∂ΠP /∂r = r − λ / r > 0 and ∂ΠP /∂λ = √ √ √ − r − λ / λ < 0. These results imply that given λ, principal’s expected profit rate will increase when the revenue rate increases, and given the revenue rate, principal’s expected profit rate will decrease when her equipment unit becomes less reliable.

4.1.2 The second-best solution √ √ According to Theorem 4.4, ((w∗, p∗) = 2 rλ − λ, r , µ∗(w∗, p∗) = rλ − λ) is the second- best solution. When the principal can contract directly on µ there is no moral hazard.

FB Therefore in ﬁrst-best setting, the agent’s expected utility rate, denoted by uA (w, µ), is FB simply uA (w, µ) = (w − µ)+ for w > 0 and µ > 0. Since the principal determines w and µ, her optimization problem is:

FB rµ max ΠP (w, µ) = max {rP (0) − w} = max − w (4.12) w>0,µ>0 w>0,µ>0 w>0,µ>0 λ + µ

FB FB FB Denote w and µ the corresponding solution. Since ∂ΠP /∂w = −1 < 0, therefore the principal chooses wFB = µ to ensure the agent’s participation and her optimization problem

32 becomes:

FB rµ max ΠP (µ) = max − µ (4.13) µ>0 µ>0 λ + µ

2 FB 2 3 Since d ΠP (µ)/dµ = −2rλ/(λ+µ) < 0, the principal’s expected profit rate is concave with FB FB respect to µ > 0 and µ can be derived from the first order condition dΠP (µ)/dµ µ=µFB = √ √ rλ/ λ + µFB2 − 1 = 0 ⇒ µFB = rλ − λ. However µFB = rλ − λ may not necessarily be the optimal solution because the principal requires µ > 0. Note that µFB = √ √ √ λ r − λ > 0 only if r > λ. Therefore the first-best solution is:

√ wFB = µFB = rλ − λ for r > λ (4.14)

By comparing the second-best solution (4.10) to the ﬁrst-best solution (4.14), we conclude:

1. The principal oﬀers a contract only when r > λ indicating that the existence of a beneﬁcial contract for risk-neutral agent is determined exogenously by the market (the revenue rate r) and the nature of the equipment (the failure rate λ), which is consistent with Proposition 2 in Harris and Raviv (1978).

2. The proposed w in the second-best solution is higher than that in the first-best solution √ √ (w∗ = 2 rλ − λ > rλ − λ = wFB), because the principal has to compensate for the p when the agent’s µ is not observable. Nevertheless, the second-best contract is efficient (as the first-best contract) because of point 3 below.

3. The optimal capacity in the ﬁrst-best solution and the second-best solution are the same √ (µFB = µ∗(w∗, p∗) = rλ − λ), indicating that the principal can induce a risk-neutral agent to install the desired capacity without contracting on it directly. Furthermore, the principal receives the same expected proﬁt rate no matter if the agent’s action is observable (thus contractible) or not. This is consistent with Proposition 3 part (i) in Harris and Raviv (1978).

4. Finally when the agent is risk-neutral, the principal is guaranteed getting the revenue rate r at all times regardless of the state of the equipment unit (because p∗ = r). This

33 √ comes at the cost of the contract (w∗ = 2 rλ − λ). In other words, the principal’s proﬁt rate appears as if it is deterministic. However this is not true for a risk-averse agent, as seen in Section 5.

4.1.3 Our principal-agent game

To clarify the interplay of decisions by the principal and the agent, we cast the principal- agent problem in an extensive form game depicted in Figure 4.5 below, where “P” represents the principal and “A” the agent.

O1 O2 O3 O4

A A A A

µ∗ µ∗ R µ∗ R µ∗ R R

s s λ λ 0 −w + p 0 −w + p 0 r − w + (p − r) 0 r − w + (p − r) 0 w − p 0 w − p 0 √ p 0 √ p w − 2 pλ + λ w − 2 pλ + λ

Figure 4.5: Structure of the principal-agent extensive form game

There are four possible strategies the principal can choose from:

O1: Oﬀer a contract with p ∈ (0, λ] and w ∈ (0, p).

O2: Oﬀer a contract with p ∈ (0, λ] and w ≥ p.

√ O3: Oﬀer a contract with p > λ and w ∈ 0, 2 pλ − λ . √ O4: Oﬀer a contract with p > λ and w ≥ 2 pλ − λ.

For any contract offer by the principal, there are two strategies for the agent to choose from: “R” when rejecting the contract, and µ∗ for accepting the contract and installing the service capacity that maximizes the agent’s expected profit rate. If the principal offers O1

∗ or O2 and the agent accepts the contract, then µ = 0. If the principal oﬀers O3 and O4 and

34 √ the agent accepts the contract, then µ∗ = pλ − λ.

The principal’s expected proﬁt rate and the agent’s expected utility rate are presented in the leaves of the tree in Figure 4.5. The element above and below are the principal’s and the agent’s values respectively.

The agent would accept the contract only if his maximized expected utility rate is no

less than his reservation utility rate uA = 0, therefore the agent accepts the contract when

the principal offers O2 and O4, and rejects the contract when the principal offers O1 and O3. The principal always prefers the agent to accept the contract and install a positive service capacity. Therefore the principal would choose O4 to all other options. Thus there is only one (subgame perfect) Nash equilibrium: the principal offers a contract with p > λ and √ √ w ≥ 2 pλ − λ and the agent accepts the contract and installs µ∗ = pλ − λ > 0.

5 Risk-Averse Agent

What if the agent is risk-averse. Fluctuations of the agent’s revenue stream occur because the principal’s equipment unit can be either in state 0 (‘operational’) or in state 1 (‘down’). In the operational state the penalty rate is 0, whereas in the down state the penalty rate is p. In other words, the penalty rate at any point of time can be modeled as pB where B is a Bernoulli random variable of value 0 with probability P (0) = µ/(λ + µ) and value 1 with probability P (1) = λ/(λ + µ). The dispersion of B decreases as P (1) moves away from 1/2 in either direction. Denote momentarily a ≡ P (1).

The risk of a random variable is often expressed by the dispersion of the underlying random ﬂuctuation. Standard deviation is commonly used to measure the dispersion of revenue in risk sharing contracts because it is conveniently additive with the revenue stream (Stiglitz 1974, Fukunaga and Huﬀman 2009 and Lewis and Bajari 2014). The standard

35 deviation of pB as a function of a, is denoted by

p s(a) ≡ σpB = p a(1 − a) for a ∈ [0, 1]

We have modiﬁed the above risk measure somewhat. Since s(a) strictly decreases as a moves away from 1/2 in either direction so any other dispersion measure of pB that has this property is a monotone increasing function of the standard deviation s(a). We choose to adopt the dispersion measure:

1 1 r(a) ≡ p − − a for a ∈ [0, 1] 2 2

The r(a) above is strictly decreasing as a gets away from 1/2 in either direction and r(a) has the property that for any a, a0 ∈ [0, 1], we have

r(a) ≤ r(a0) ⇔ s(a) ≤ s(a0)

Note that r(a) increases (decreases) if and only if the standard deviation s(a) increases (decreases).

Risk premium of a risk-averse agent is the $ value he is willing to forfeit to avoid un- certainties (ﬂuctuations) in his revenue stream and as a consequence the risk premium is deﬁned as follows:

1 1 1 1 1 1 λ π(µ, w, p) = ηp − − a = ηp − − P (1) = ηp − − (5.1) 2 2 2 2 2 2 λ + µ

Figure 5.1 is an example that depicts the shape of π(µ, w, p) as a function of P (1) when η = 1. π(µ, w, p) reaches its peak when the equipment has equal likelihood of being operational and being failed. In such case the agent can hardly infer anything from the state of the equipment in order to predict his revenue stream and therefore it is considered the most risky. When the likelihood of the equipment being operational is close to 1, the agent can predict his revenue stream more precisely (less risky). Similarly when the likelihood of the equipment

36 being failed is close to 1, the agent can also predict his revenue stream more precisely.

π ( µ , w , p ) p 2

0 P(1) 0 1 1 2

Figure 5.1: π(µ, w, p) as a function of P (1) when η = 1

The real parameter η indicates the preference and intensity of the agent’s risk attitude. When η > 0 the agent is risk-averse, when η = 0 the agent is risk-neutral (and the model reduces to the model of Section 4), and η < 0 indicates that the agent is risk-seeking (see Section 6). In the analysis below, the value η plays the role of an exogenous variable.

Modifying (4.2), the risk-averse agent’s expected utility rate in this section is:

pλ 1 1 λ uA(µ; w, p) = w − − µ − ηp − − for w > 0, p > 0, µ ≥ 0 (5.2) λ + µ 2 2 λ + µ +

Note that η > 0 ⇒ π(µ, w, p) ≥ 0, and such a risk premium being subtracted from a risk-neutral agent’s expected utility rate (as in (5.2)) implies risk-aversion. The analysis is diﬀerent for η ∈ (0, 4/5) compared to η ≥ 4/5. Thus, for convenience, when η ∈ (0, 4/5) we describe the agent as weakly risk-averse, and when η ≥ 4/5 we describe the agent as strongly risk-averse. We assume, say for historical reasons, that both the agent and the principal know not only the type of the risk-averse agent, but also the value of η.

The principal is always risk-neutral and her expression of expected proﬁt rate ΠP (w, p; µ) is the same as (4.3).

37 Deﬁne the part inside the brackets in (5.2) as

 (1 − η)pλ  w − ηp − − µ, µ ∈ [0, λ] pλ 1 1 λ  λ + µ u(µ) ≡ w − − µ − ηp − − = λ + µ 2 2 λ + µ (1 + η)pλ  w − − µ, µ > λ λ + µ (5.3)

The behavior of the utility function u(µ) for µ ≥ 0 is of prime technical interest. Note that u(µ) is diﬀerentiable everywhere on µ ≥ 0 except at µ = λ. When µ ∈ [0, λ):

du(µ) (1 − η)pλ du(µ) 1 − η λ = − 1, lim = p − dµ (λ + µ)2 µ→0+ dµ λ 1 − η du(µ) 1 − η 4λ d2u(µ) 2(1 − η)pλ lim = p − and = − µ→λ− dµ 4λ 1 − η dµ2 (λ + µ)3 and when µ > λ:

du(µ) (1 + η)pλ du(µ) 1 + η 4λ = − 1, lim = p − dµ (λ + µ)2 µ→λ+ dµ 4λ 1 + η du(µ) d2u(µ) 2(1 + η)pλ lim = −1 and = − < 0 µ→+∞ dµ dµ2 (λ + µ)3

The above derivatives indicate the direction of monotonicity and the concavity/convexity of function u(µ) over [0, λ) and (λ, +∞). Table 5.1 summarizes these indicators for various

2 + regions of the space R+ of the pairs (η, p). In the table uµ(·) = limµ→(·) du/dµ, and uµ(· ) − represents the limit of uµ(µ) as µ approaches (·) from above, and similar for uµ(· ).

5.1 Optimal strategies with a weakly risk-averse agent

Similarly to the risk-neutral agent case, agent’s expected utility rate increases and principal’s expected proﬁt rate decreases in w, therefore for any value of p the principal maximizes her expected proﬁt rate by lowering the compensation rate w yet maintaining the agent’s participation by setting the agent’s expected utility rate equal to his reservation utility rate. Although the principal cannot contract directly on the agent’s service capacity, she

38 Table 5.1: Indicators of the monotonicity and the concavity/convexity of function u(µ) in (5.3)

over [0, λ) over (λ, +∞) Case u (0+) u (λ−) u (λ+) u (+∞) µ u(µ) is µ µ u(µ) is µ λ p ∈ 0, ≤ 0 Concave < 0 < 0 Concave < 0 1 − η λ 4λ † 3 p ∈ , > 0 Concave < 0 ≤ 0 Concave < 0 η ∈ 0, 1 − η 1 + η 5 4λ 4λ p ∈ , > 0 Concave ≤ 0 > 0 Concave < 0 1 + η 1 − η 4λ p ∈ , +∞ > 0 Concave > 0 > 0 Concave < 0 1 − η 4λ p ∈ 0, < 0 Concave < 0 ≤ 0 Concave < 0 1 + η 4λ λ ‡ 3 p ∈ , ≤ 0 Concave < 0 > 0 Concave < 0 η ∈ , 1 1 + η 1 − η 5 λ 4λ p ∈ , > 0 Concave ≤ 0 > 0 Concave < 0 1 − η 1 − η 4λ p ∈ , +∞ > 0 Concave > 0 > 0 Concave < 0 1 − η 4λ p ∈ 0, < 0 Convex < 0 ≤ 0 Concave < 0 η ∈ [1, +∞) 1 + η 4λ p ∈ , +∞ < 0 Convex < 0 > 0 Concave < 0 1 + η †Note that η ∈ (0, 3/5] ⇒ 4λ/(1 + η) ≥ λ/(1 − η).

‡Note that η ∈ (3/5, 1) ⇒ λ/(1 − η) > 4λ/(1 + η). anticipates the agent to optimize his expected utility rate when oﬀered a contract. That is, for any w and p proposed by the principal, the agent computes his value of µ that maximizes his expected utility rate and subsequently decides whether to accept the contract or not, by solving the following optimization problem:

pλ 1 1 λ max u(µ) = max w − − µ − ηp − − (5.4) µ≥0 µ≥0 λ + µ 2 2 λ + µ

∗ The agent’s optimal service capacity is denoted by µ (w, p) = argmaxµ≥0 u(µ).

Notation: p 2 λ λ 8 1 − 1 − η λ p ≡ , p ≡ , and p ≡ (5.5) 1 1 + η 2 1 − η 3 η2

39 and the following identity is easily veriﬁed using the deﬁnition of p3:

p p w3 ≡ ηp3 + 2 (1 − η)p3λ − λ = 2 (1 + η)p3λ − λ (5.6)

p1, p2, p3 and w3 are functions of λ and η. However we suppress the parameters (λ, η).

Next we state a number of technical lemmas used in later proofs.

Lemma 5.1. Let 1 > η > 0 and λ > 0. If p ≥ λ/(1−η), then p ≥ ηp+2p(1 − η)pλ−λ > 0.

Proof. Let 1 > η > 0 and λ > 0, then ηp + 2p(1 − η)pλ − λ increases with respect to p > 0. Further more, if p = λ/(1 − η), then ηp + 2p(1 − η)pλ − λ = λ/(1 − η). Therefore if p ≥ λ/(1 − η), then ηp + 2p(1 − η)pλ − λ ≥ λ/(1 − η) > 0. On the other hand if √ 2 p ≥ λ/(1 − η), then p − ηp + 2p(1 − η)pλ − λ = p(1 − η)p − λ ≥ 0, therefore p ≥ ηp + 2p(1 − η)pλ − λ.

Lemma 5.2. Let 1 > η > 0 and λ > 0. √ √ √ (a) If p > 8 1 − p1 − η2 λ/η2, then ηp − 2 1 + η − 1 − η pλ > 0. √ √ √ (b) If 8 1 − p1 − η2 λ/η2 > p > 0, then 0 > ηp − 2 1 + η − 1 − η pλ. √ √ √ (c) If p = 8 1 − p1 − η2 λ/η2, then ηp − 2 1 + η − 1 − η pλ = 0. √ √ Proof. Let 1 > η > 0 and λ > 0. Deﬁne x ≡ p and a ≡ λ and restate the expression √ √ √ √ √ ηp − 2 1 + η − 1 − η pλ as ηx2 − 2 1 + η − 1 − η ax with x > 0 and a > 0. √ √ The solutions to the quadratic equation ηx2 − 2 1 + η − 1 − η ax = 0 for x are 0 √ √ √ √ and 2 1 + η − 1 − η a/η. Therefore if x > 2 1 + η − 1 − η a/η, or equivalently, √ √ √ x2 > 8 1 − p1 − η2 a2/η2, then ηx2 − 2 1 + η − 1 − η ax > 0. Replacing x by p √ and a by λ we obtain (a). The proofs for (b) and (c) are similar. Lemma 5.3. Let 1 > η > 0 and λ > 0, then 4λ/(1 − η) > 8 1 − p1 − η2 λ/η2 > 4λ/(1 + η).

40 Proof. Let 1 > η > 0 and λ > 0, then we have

2 p1 + η − p1 − η > 0 ⇔ 1 − p1 + ηp1 − η > 0

⇔ 1 + η − p1 + ηp1 − η > η 2 ⇔ (1 + η) p1 + η − p1 − η > η2 2 ⇔ p1 + η − p1 − η /η2 > 1/(1 + η) p ⇔ 8 1 − 1 − η2 λ/η2 > 4λ/(1 + η)

Also we have

2 p1 + η − p1 − η > 0 ⇔ 0 > p1 + ηp1 − η − 1

⇔ η > p1 + ηp1 − η − (1 − η) 2 ⇔ η2 > (1 − η) p1 + η − p1 − η 2 ⇔ 1/(1 − η) > p1 + η − p1 − η /η2 p ⇔ 4λ/(1 − η) > 8 1 − 1 − η2 λ/η2

Lemma 5.4. Let η > 0 and λ > 0. If p > 4λ/(1 + η), then 2p(1 + η)pλ − λ > 0.

Proof. Let η > 0 and λ > 0. If p > 4λ/(1+η), then 2p(1 + η)pλ−λ > 4λ−λ = 3λ > 0.

Lemma 5.5. Let η > 0 and λ > 0. (a) If 1 + 2η + 2pη(1 + η) λ > p > 1 + 2η − 2pη(1 + η) λ, then we have 0 > p − 2p(1 + η)pλ + λ. (b) If 1 + 2η − 2pη(1 + η) λ > p > 0 or p > 1 + 2η + 2pη(1 + η) λ, then p − 2p(1 + η)pλ + λ > 0. (c) If p = 1 + 2η − 2pη(1 + η) λ or 1 + 2η + 2pη(1 + η) λ, then p − 2p(1 + η)pλ + λ = 0.

41 √ √ Proof. Let η > 0 and λ > 0. Deﬁne x ≡ p and a ≡ λ and restate expression p − √ 2p(1 + η)pλ+λ as x2−2ax 1 + η+a2 where x > 0 and a > 0. The solutions to the quadratic √ √ √ √ √ equation x2−2ax 1 + η+a2 = 0 for x are 1 + η − η a and 1 + η + η a. Therefore √ √ √ √ if 1 + η + η a > x > 1 + η − η a, or equivalently, 1 + 2η + 2pη(1 + η) a2 > √ √ x2 > 1 + 2η − 2pη(1 + η) a2, then 0 > x2 − 2ax 1 + η + a2. Replacing x by p and a √ by λ we obtain (a). The proofs for (b) and (c) are similar. Lemma 5.6. Let η > 0 and λ > 0, then 4λ/(1 + η) > 1 + 2η − 2pη(1 + η) λ.

Proof. Let η > 0 and λ > 0. Note that

√ √ p1 + η + η > p1 + η ⇒ 2/p1 + η > 1/p1 + η > 1/ p1 + η + η √ ⇔ 2/(1 + η) > p1 + η − η √ 2 ⇔ 4/(1 + η) > p1 + η − η ⇔ 4λ/(1 + η) > 1 + 2η − 2pη(1 + η) λ

Lemma 5.7. Let λ > 0. (a) If 4/5 > η > 0, then 1 + 2η + 2pη(1 + η) λ > λ/(1 − η). (b) If 1 > η > 4/5, then λ/(1 − η) > 1 + 2η + 2pη(1 + η) . (c) If η = 4/5, then 1 + 2η + 2pη(1 + η) λ = λ/(1 − η).

Proof. Let 4/5 > η > 0 and λ > 0, then we have

4η − 5η2 > 0 ⇔ 2pη(1 − η) > η

⇔ 1 + 2pη(1 − η) > 1 + η √ 2 2 ⇔ p1 − η + η > p1 + η √ ⇔ p1 − η + η > p1 + η √ ⇔ p1 − η > p1 + η − η

42 √ ⇔ p1 − η p1 + η + η > 1 ⇔ 1 + 2η + 2pη(1 + η) λ > λ/(1 − η)

and we obtain (a). The proofs for (b) and (c) are similar.

Lemma 5.8. Let λ > 0. (a) If 4/5 > η > 0, then 8 1 − p1 − η2 λ/η2 > λ/(1 − η). (b) If 1 > η > 4/5, then λ/(1 − η) > 8 1 − p1 − η2 λ/η2. (c) If η = 4/5, then 8 1 − p1 − η2 λ/η2 = λ/(1 − η).

Proof. Let 4/5 > η > 0 and λ > 0, then we have

9(1 − η) > 1 + η ⇔ 3p1 − η > p1 + η

⇔ 4p1 − η > p1 + η + p1 − η ⇔ 2p1 − η p1 + η − p1 − η > η 2 ⇔ 4(1 − η) p1 + η − p1 − η > η2 p ⇔ 8 1 − 1 − η2 λ/η2 > λ/(1 − η)

and we obtain (a). The proofs for (b) and (c) are similar.

Lemma 5.8 part (a) implies η ∈ (0, 4/5) ⇒ p3 > p2.

We identify the optimal response of a weakly risk-averse agent to any contract oﬀer

2 (w, p) ∈ R+ in Proposition 5.9.

Proposition 5.9. Consider a weakly risk-averse agent (η ∈ (0, 4/5)).

(a) Given

w ≥ p ∈ (0, p2] (5.7)

43 then the agent accepts the contract and installs µ∗(w, p) = 0 resulting in expected utility

∗ rate uA(µ (w, p); w, p) = w − p ≥ 0. The agent rejects the contract if p ∈ (0, p2] and w ∈ (0, p).

(b) Given

p p ∈ (p2, p3) and w ≥ ηp + 2 (1 − η)pλ − λ (5.8)

then the agent accepts the contract and installs µ∗(w, p) = p(1 − η)pλ−λ > 0 resulting

∗ p in expected utility rate uA(µ (w, p); w, p) = w − ηp − 2 (1 − η)pλ + λ ≥ 0. The agent p rejects the contract if p ∈ (p2, p3) and w ∈ 0, ηp + 2 (1 − η)pλ − λ .

p = p3 and w ≥ w3 (5.9)

then the agent accepts the contract and is indiﬀerent about installing either µ∗(w, p) =

p ∗ p (1 − η)p3λ − λ or µ (w, p) = (1 + η)p3λ − λ. In both cases the agent receives

∗ expected utility rate uA(µ (w, p); w, p) = w − w3 ≥ 0. If r ∈ (0, p3), then there ex-

∗ ∗ p ists a w such that (w , p3), (1 − η)p3λ − λ is the unique admissible solution (see

∗ ∗ p Deﬁnition 3.3). If r = p3, then there exists w such that (w , p3), (1 − η)p3λ − λ

∗ p and (w , p3), (1 + η)p3λ − λ are both admissible solutions (see Deﬁnition 3.3). If

∗ ∗ p r > p3, then there exists a w such that (w , p3), (1 + η)p3λ − λ is the unique ad-

missible solution (for proof see Proposition 5.12). He rejects the contract if p = p3 and

w ∈ (0, w3).

(d) Given

p p > p3 and w ≥ 2 (1 + η)pλ − λ (5.10)

then the agent accepts the contract and installs µ∗(w, p) = p(1 + η)pλ−λ > 0 resulting

∗ p in expected utility rate uA(µ (w, p); w, p) = w −2 (1 + η)pλ+λ ≥ 0. The agent rejects

44 p the contract if p > p3 and w ∈ 0, 2 (1 + η)pλ − λ .

Proof. According to Table 5.1, the optimization of u(µ) when η ∈ (0, 3/5] versus η ∈ (3/5, 4/5) is diﬀerent. Therefore we prove the proposition separately for η ∈ (0, 3/5] and η ∈ (3/5, 4/5).

Case η ∈ (0, 3/5]: Note that 4p2 > 4p1 ≥ p2 and according to Lemma 5.3, 4p2 > p3 > 4p1.

Therefore we have 4p2 > p3 > 4p1 ≥ p2. Figure 5.2 shows the shape of u(µ) when η ∈ (0, 3/5] and the value of p falls in diﬀerent ranges. The structure of the proof when η ∈ (0, 3/5] is depicted in Figure 5.3.

Case p ∈ (0, p2]: According to Table 5.1, u(µ) is decreasing with respect to µ ≥ 0. Thus the agent’s optimal service capacity is µ∗(w, p) = 0 and from (5.3) u(µ∗(w, p)) = w − p.

Subcase w ∈ (0, p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (p2, 4p1]: According to Table 5.1, the service capacity that maximizes u(µ) lies

∗ in (0, λ). µ (w, p) is computed from ﬁrst order condition du(µ)/dµ|µ=µ∗(w,p) = 0 ⇒ µ∗(w, p) = p(1 − η)pλ − λ > 0 and from equation (5.3) u(µ∗(w, p)) = w − ηp − p p 2 (1 − η)pλ + λ. According to Lemma 5.1, p > p2 ⇒ ηp + 2 (1 − η)pλ − λ > 0.

Subcase w ∈ 0, ηp + 2p(1 − η)pλ − λ : u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ ηp + 2p(1 − η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

Case p ∈ (4p1, 4p2]: According to Table 5.1, there is a service capacity that maximizes u(µ) for µ ∈ (0, λ] and a service capacity that maximizes u(µ) for µ > λ. Denote the op-

∗ timal service capacity in (0, λ] by µ(0,λ](w, p). From the ﬁrst order condition the optimal ∗ p ∗ service capacity is µ(0,λ](w, p) = (1 − η)pλ − λ and from (5.3) u µ(0,λ](w, p) = w −

45 u( µ ) u( µ )

0.940 0.945 0.950 0.955 0.960 0.95 0.96 0.97 0.98 0 0 iue52 lutaino h om of forms the of Illustration 5.2: Figure λ λ (c) (a) = η = η p 0.5 0.5 p

∈ ,

λ= λ , λ= λ ∈ (4 0.01 0.01 (0 µ µ µ p u( ) ,

, w

1 w p , = p , = 1 1 0.900 0.905 0.910 0.915 0.920 0.925 0.930 , 2

, p

3 p ] = = ) 1.5 0 4 λ λ λ = η 5 5 (e) λ λ 0.5

, λ= λ > p 46 0.01

µ u( µ ) u( µ )

, 4

w p = 1 2

0.935 0.940 0.945 0.950 0.945 0.950 0.955 0.960 0.965 0.970 0.975 ,

p = 10 0 0 λ u ( µ when ) λ λ (b) (d) = η 5 = η λ 0.5 p p 0.5

, ∈ ∈

λ= λ , λ= λ η 0.01 ( ( 0.01 p p µ µ ∈

2 3 ,

, w

w , , = = 4 4 (0 1 1

, p p

, p

p , 1 2 = = 2.5 ] ] 3 6 λ / λ 5] 5 5 λ λ w ∈ (0, p) Reject.

p ∈ (0, p2]

w ≥ p µ∗ = 0

w ∈ 0, ηp + 2p(1 − η)pλ − λ Reject.

p ∈ (p2, 4p1]

w ≥ ηp + 2p(1 − η)pλ − λ µ∗ = p(1 − η)pλ − λ

w ∈ 0, ηp + 2p(1 − η)pλ − λ Reject. 3 η ∈ 0, 5 p ∈ (4p1, p3)

w ≥ ηp + 2p(1 − η)pλ − λ µ∗ = p(1 − η)pλ − λ

w ∈ (0, w3) Reject.

p ∈ (4p1, 4p2] p = p3 ∗ p µ = (1 − η)p3λ − λ w ≥ w3 ∗ p or µ = (1 + η)p3λ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

p ∈ (p3, 4p2]

w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

p > 4p2

w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

Figure 5.3: Structure of the proof for Proposition 5.9 when η ∈ (0, 3/5]

p ∗ ηp−2 (1 − η)pλ+λ. Denote the optimal service capacity for µ > λ by µλ(w, p), which ∗ p is solved from ﬁrst order condition du(µ)/dµ| ∗ = 0 ⇒ µ (w, p) = (1 + η)pλ−λ µ=µλ(w,p) λ ∗ p and from (5.3) u (µλ(w, p)) = w − 2 (1 + η)pλ + λ. The agent has a choice of two service capacities and he installs the one that generates a higher expected utility rate. √ ∗ ∗ √ √ Note that u (µλ(w, p)) − u µ(0,λ](w, p) = ηp − 2 1 + η − 1 − η pλ. According

to Lemma 5.3, 4p2 > p3 > 4p1, therefore we examine the following subcases.

∗ ∗ Subcase p ∈ (4p1, p3): From Lemma 5.2 part (b), u µ(0,λ](w, p) > u (µλ(w, p)), thus the agent’s optimal service capacity is µ∗(w, p) = p(1 − η)pλ − λ and

∗ p u(µ (w, p)) = w − ηp − 2 (1 − η)pλ + λ. From Lemma 5.1, p > 4p1 ≥ p2 ⇒ ηp + 2p(1 − η)pλ − λ > 0. Subsubcase w ∈ 0, ηp + 2p(1 − η)pλ − λ : u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subsubcase w ≥ ηp + 2p(1 − η)pλ − λ: u(µ∗(w, p)) ≥ 0, so the agent would

47 accept the contract if oﬀered.

∗ ∗ Subcase p = p3: According to Lemma 5.2 part (c), u µ(0,λ](w, p3) = u (µλ(w, p3)), ∗ ∗ indicating that installing µ(0,λ](w, p3) or µλ(w, p3) leads to the same agent’s expected utility rate. Thus the agent is indiﬀerent about installing either µ∗(w, p) =

p ∗ p (1 − η)p3λ − λ or µ (w, p) = (1 + η)p3λ − λ. Still, the capacity value has to

lead to admissible solutions (see Proposition 5.12). Recall the deﬁnition of w3 from p (5.6). According to Lemma 5.1, p3 > 4p1 ≥ p2 ⇒ w3 = ηp3+2 (1 − η)p3λ−λ > 0.

∗ Subsubcase w ∈ (0, w3): u(µ (w, p)) < 0, thus the agent rejects the contract.

∗ Subsubcase w ≥ w3: u(µ (w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

∗ ∗ Subcase p ∈ (p3, 4p2]: By Lemma 5.2 part (a), u (µλ(w, p)) > u µ(0,λ](w, p) , thus the agent’s optimal service capacity is µ∗(w, p) = p(1 + η)pλ−λ and u(µ∗(w, p)) = p p w − 2 (1 + η)pλ + λ. From Lemma 5.4, p > p3 > 4p1 ⇒ 2 (1 + η)pλ − λ > 0. Subsubcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subsubcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p > 4p2: According to Table 5.1, the service capacity that maximizes u(µ) satisﬁes µ > λ. From the ﬁrst order condition the agent’s optimal service capacity is µ∗(w, p) = p(1 + η)pλ−λ and from equation (5.3) u(µ∗(w, p)) = w−2p(1 + η)pλ+λ. According p to Lemma 5.4, p > 4p2 > 4p1 ⇒ 2 (1 + η)pλ − λ > 0.

Subcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, thus the agent rejects the contract.

Subcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

This completes the proof for Proposition 5.9 when η ∈ (0, 3/5].

48 Case η ∈ (3/5, 4/5): Note that 4p2 > p2 > 4p1 and according to Lemma 5.3 and 5.8 part

(a), 4p2 > p3 > p2. Therefore we have 4p2 > p3 > p2 > 4p1. Figure 5.4 shows the shape of u(µ) when η ∈ (3/5, 4/5) and the value of p falls in diﬀerent ranges. The structure of the proof when η ∈ (3/5, 4/5) is depicted in Figure 5.5.

Case p ∈ (0, 4p1]: According to Table 5.1, u(µ) is decreasing with respect to µ ≥ 0. Thus the agent’s optimal service capacity is µ∗(w, p) = 0 and from (5.3) u(µ∗(w, p)) = w − p.

Subcase w ∈ (0, p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (4p1, p2]: According to Table 5.1, there is a service capacity that maximizes u(µ) for µ ∈ [0, λ) and a service capacity that maximizes u(µ) for µ > λ. Denote

∗ the optimal service capacity in [0, λ) by µ[0,λ)(w, p). Note that u(µ) is decreasing with ∗ respect to µ over [0, λ), therefore the agent’s optimal service capacity is µ[0,λ)(w, p) = 0 ∗ and from (5.3) u µ[0,λ)(w, p) = w − p. Denote the optimal service capacity for ∗ ∗ p µ > λ by µλ(w, p). From the ﬁrst order condition µλ(w, p) = (1 + η)pλ − λ and ∗ p from equation (5.3) u (µλ(w, p)) = w − 2 (1 + η)pλ + λ. The agent has to choose one of the two service capacities and installs the one with higher expected utility

∗ ∗ p rate. Note that u (µλ(w, p)) − u µ[0,λ)(w, p) = p − 2 (1 + η)pλ + λ. Accord- p ing to Lemma 5.6, 4p1 > 1 + 2η − 2 η(1 + η) λ and according to Lemma 5.7 p part (a), 1 + 2η + 2 η(1 + η) λ > p2. Thus according to Lemma 5.5 part (a),

∗ ∗ ∗ u µ[0,λ)(w, p) > u (µλ(w, p)), the agent’s optimal service capacity is µ (w, p) = 0 and u(µ∗(w, p)) = w − p.

Subcase w ∈ (0, p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (p2, 4p2]: According to Table 5.1, there is a service capacity that maximizes u(µ) for µ ∈ (0, λ] and a service capacity that maximizes u(µ) for µ > λ. Denote

49 u( µ ) u( µ )

0.940 0.945 0.950 0.955 0.960 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0 0 iue54 lutaino h om of forms the of Illustration 5.4: Figure λ λ (c) (a) = η = η p 0.7 0.7 p

, , ∈ ∈ λ= λ = λ 0.01 0.01 ( (0 p µ u( µ ) µ

2 , , ,

w w p , 4 = = p 0.85 0.86 0.87 0.88 0.89 0.90 0.91 1 1

3 , , 1

p p ) ] = = 0 4 2 λ λ λ = η 5 5 (e) λ λ 0.7

, λ= λ > p 50 0.01

µ u( µ ) u( µ )

, 4

w p = 1 2

0.900 0.905 0.910 0.915 0.920 0.925 0.945 0.950 0.955 0.960 0.965 0.970 ,

p = 15 0 0 u λ ( µ when ) λ λ (b) (d) = η = η 5 λ 0.7 p p 0.7

, , ∈ ∈ λ= λ λ= λ η 0.01 0.01 (4 ( ∈ p µ µ

p 3 , ,

w (3 w , 1 = = 4 p , 1 1

, p ,

p p 5 2 2 = = 2.7 ] ] , 10 4 λ λ / 5) 5 5 λ λ w ∈ (0, p) Reject.

p ∈ (0, 4p1]

w ≥ p µ∗ = 0

w ∈ (0, p) Reject.

p ∈ (4p1, p2]

w ≥ p µ∗ = 0

w ∈ 0, ηp + 2p(1 − η)pλ − λ Reject. 3 4 η ∈ , 5 5 p ∈ (p2, p3)

w ≥ ηp + 2p(1 − η)pλ − λ µ∗ = p(1 − η)pλ − λ

w ∈ (0, w3) Reject.

p ∈ (p2, 4p2] p = p3 ∗ p µ = (1 − η)p3λ − λ w ≥ w3 ∗ p or µ = (1 + η)p3λ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

p ∈ (p3, 4p2]

w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

p > 4p2

w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

Figure 5.5: Structure of the proof for Proposition 5.9 when η ∈ (3/5, 4/5)

∗ the optimal service capacity in (0, λ] by µ(0,λ](w, p). From the ﬁrst order condition ∗ p ∗ µ(0,λ](w, p) = (1 − η)pλ − λ and from equation (5.3) u µ(0,λ](w, p) = w − ηp − p ∗ 2 (1 − η)pλ + λ. Denote the optimal service capacity for µ > λ by µλ(w, p). From the ∗ p ∗ ﬁrst order condition µλ(w, p) = (1 + η)pλ − λ and from equation (5.3) u (µλ(w, p)) = w − 2p(1 + η)pλ + λ. The agent has to choose one of the two service capacities and

∗ installs the one that generates a higher expected utility rate. Note that u (µλ(w, p)) − √ ∗ √ √ u µ(0,λ](w, p) = ηp − 2 1 + η − 1 − η pλ. According to Lemma 5.3 and 5.8

part (a), 4p2 > p3 > p2, therefore we examine the following subcases.

∗ ∗ Subcase p ∈ (p2, p3): By Lemma 5.2 part (b), u µ(0,λ](w, p) > u (µλ(w, p)), thus the agent’s optimal service capacity is µ∗(w, p) = p(1 − η)pλ−λ and u(µ∗(w, p)) = p p w−ηp−2 (1 − η)pλ+λ. According to Lemma 5.1, p > p2 ⇒ ηp+2 (1 − η)pλ− λ > 0. Subsubcase w ∈ 0, ηp + 2p(1 − η)pλ − λ : u(µ∗(w, p)) < 0, therefore the

51 agent rejects the contract.

Subsubcase w ≥ ηp + 2p(1 − η)pλ − λ: u(µ∗(w, p)) ≥ 0, so the agent would accept the contract if oﬀered.

∗ ∗ Subcase p = p3: According to Lemma 5.2 part (c), u µ(0,λ](w, p3) = u (µλ(w, p3)), ∗ ∗ indicating that installing µ(0,λ](w, p) or µλ(w, p) leads to the same agent’s expected utility rate. Therefore the agent is indiﬀerent about installing µ∗(w, p) =

p ∗ p (1 − η)p3λ − λ or µ (w, p) = (1 + η)p3λ − λ. Again, the service capacity has

to lead to admissible solutions (see Proposition 5.12). Recall the deﬁnition of w3 p from (5.6). According to Lemma 5.1, p3 > p2 ⇒ w3 = ηp3 +2 (1 − η)p3λ−λ > 0.

∗ Subsubcase w ∈ (0, w3): u(µ (w, p)) < 0, thus the agent rejects the contract.

∗ Subsubcase w ≥ w3: u(µ (w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

∗ ∗ Subcase p ∈ (p3, 4p2]: By Lemma 5.2 part (a), u (µλ(w, p)) > u µ(0,λ](w, p) , thus the agent’s optimal service capacity is µ∗(w, p) = p(1 + η)pλ−λ and u(µ∗(w, p)) = p p w−2 (1 + η)pλ+λ. From Lemma 5.4, p > p3 > p2 > 4p1 ⇒ 2 (1 + η)pλ−λ > 0. Subsubcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subsubcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Subcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, thus the agent rejects the contract.

Subcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

52 This completes the proof for Proposition 5.9 when η ∈ (3/5, 4/5).

In summary, given exogenous market conditions such that a contract oﬀer satisfying the reservation value constraints for both the principal and a weakly risk-averse agent exists (see Theorem 5.19 and Proposition 5.20 later), the agent determines his optimal capacity using one of two formulas:

µ∗(w, p) = p(1 − η)pλ − λ > 0 or µ∗(w, p) = p(1 + η)pλ − λ > 0

The conditions when a weakly risk-averse agent accepts the contract can be depicted by the shaded areas in Figure 5.6, where η = 0.6. The three shaded areas with diﬀerent grey scales represent conditions (5.7), (5.8) and (5.10) under which the agent accepts the contract but responds diﬀerently. The lower bound function of the shaded areas (denoted by w0(p))

represents the set of oﬀers with agent’s zero expected utility rate. w0(p) is deﬁned as follows:

  p when p ∈ (0, p2]   p w0(p) = ηp + 2 (1 − η)pλ − λ when p ∈ (p2, p3]   p  2 (1 + η)pλ − λ when p > p3

Note that since lim − w0(p) = lim + w0(p) = p2 and lim − w0(p) = lim + = p→p2 p→p2 p→p3 p→p3 p ηp3 + 2 (1 − η)p3λ − λ, w0(p) is continuous everywhere over interval p ∈ R+. Since

lim − dw0(p)/dp = lim + dw0(p)/dp = 1, w0(p) is diﬀerentiable at p = p2. However p→p2 p→p2 p p since lim − dw0(p)/dp = η + (1 − η)λ/p3 6= (1 + η)λ/p3 = lim + dw0(p)/dp, w0(p) p→p3 p→p3

is not diﬀerentiable at p = p3.

5.1.1 Sensitivity analysis of a weakly risk-averse agent’s optimal strategy

∗ The principal would not propose an acceptable contract that results in uA(µ = 0) ≥ uA = 0. Therefore the only viable cases to consider are when the agent accepts the contract and installs positive service capacities: µ∗(w, p) = p(1 − η)pλ − λ or µ∗(w, p) = p(1 + η)pλ − λ. We examine the two viable contracts with positive service capacities.

53 w

∗ ∗ ∗ µ =0 µ = (1− η )p λ − λ µ = (1+ η )p λ − λ λ − λ 3 p

) η η (1+

2 w=2 (1+ η )p λ − λ 2 p

w= η p+2 (1− η )p λ − λ

w=p

0 p 0 p2 p3

Figure 5.6: Conditions when a weakly risk-averse agent accepts the contract with η = 0.6

First the case µ∗(w, p) = p(1 − η)pλ − λ. According to (5.8) the compensation rate w is bounded below by ηp+2p(1 − η)pλ−λ = ηpP (0)+pP (1)+µ∗(w, p), with the term ηpP (0) representing the expected risk rate perceived by the agent and the term pP (1) representing the expected penalty rate charged by the principal when the optimal capacity is installed. It indicates that the agent ought to be reimbursed for the expected risk rate, the expected penalty rate and the cost of the optimal service capacity.

The optimal service capacity p(1 − η)pλ − λ depends on p, λ, and η. Its derivatives are:

s r s ∂µ∗ (1 − η)λ ∂µ∗ (1 − η)p ∂µ∗ pλ = > 0, = − 1 and = − < 0 ∂p 4p ∂λ 4λ ∂η 4(1 − η)

These derivatives indicate that given a λ and η, the agent will increase his service capacity

54 when the penalty rate increases. Note that p(1 − η)pλ − λ, as a function of λ, decreases when λ > (1 − η)p/4. From conditions (5.8) and (5.9) the agent installs service capacity p (1 − η)pλ − λ when p ∈ (p2, p3] and from Lemma 5.3 we have 4p2 > p3. Therefore we have

∗ 4λ/(1 − η) = 4p2 > p ⇒ λ > (1 − η)p/4 ⇒ ∂µ /∂λ < 0. Thus, given a p and η, the savings from reducing the service capacity are greater than the increase in the penalty charge and in the risk rate, and the agent will reduce µ when λ increases. Given a p and λ, the agent will reduce the µ when he is more risk-averse.

The agent’s optimal expected utility rate when installing capacity µ∗(w, p) = p(1 − η)pλ−λ

∗ ∗ p is uA ≡ uA(µ (w, p); w, p) = w−ηp−2 (1 − η)pλ+λ, and it depends on w, p, η and λ. Note ∗ ∗ p that ∂uA/∂w = −1 < 0, ∂uA/∂p = −η − (1 − η)λ/p < 0, indicating that the agent’s optimal expected utility rate decreases with the compensation rate and the penalty rate. Note

∗ √ √ √ ∗ √ √ √ that ∂uA/∂η = − p p − p2 and ∂uA/∂λ = − p − p2 / p2, and from Proposition √ √ 5.9 p > p2 ⇒ p− p2 > 0, therefore the agent’s optimal expected utility rate also decreases with his risk intensity and the failure rate.

Next we examine the case µ∗(w, p) = p(1 + η)pλ − λ. According to (5.10) the compensation rate w is bounded below by 2p(1 + η)pλ − λ = ηpP (1) + pP (1) + µ∗(w, p), with the term ηpP (1) representing the expected risk rate perceived by the agent and pP (1) representing the expected penalty rate charged by the principal when the optimal capacity is installed. It indicates that the agent should at least be reimbursed for the expected risk rate, the expected penalty rate and the cost of the optimal service capacity.

The optimal service capacity p(1 + η)pλ − λ depends on p, λ, and η. Its derivatives are:

s r s ∂µ∗ (1 + η)λ ∂µ∗ (1 + η)p ∂µ∗ pλ = > 0, = − 1 and = > 0 ∂p 4p ∂λ 4λ ∂η 4(1 + η)

The derivatives indicate that given λ and η, the agent will increase the µ when the penalty rate increases. Note that p(1 + η)pλ − λ, as a function of λ, increases when (1 + η)p/4 > λ.

55 p From (5.9) and (5.10) the agent installs service capacity (1 + η)pλ − λ when p ≥ p3, and

from Lemma 5.3 we have p3 > 4p1. Therefore we have p > 4p1 = 4λ/(1 + η) ⇒ (1 + η)p/4 > λ ⇒ ∂µ∗/∂λ > 0. Thus, given p and η, the agent will increase µ when λ increases. Given p and λ, the agent will increase his µ when he is more risk-averse.

The agent’s optimal expected utility rate when installing capacity µ∗(w, p) = p(1 + η)pλ−λ

∗ ∗ p is uA ≡ uA(µ (w, p); w, p) = w − 2 (1 + η)pλ + λ, and it depends on w, p, η and λ. Note ∗ ∗ p ∗ p that ∂uA/∂w = −1 < 0, ∂uA/∂p = − (1 + η)λ/p < 0 and ∂uA/∂η = − pλ/(1 + η) < 0, indicating that the agent’s optimal expected utility rate decreases with the compensation

∗ √ √ √ rate, the penalty rate and his risk intensity. Note that ∂uA/∂λ = − p − p1 / p1, and √ √ from Proposition 5.9 p ≥ p3 > p1 ⇒ p − p1 > 0, therefore the agent’s optimal expected utility rate also decreases with the failure rate.

Summary: Recall that given the set of oﬀers {(w, p): p ∈ (0, λ], w ≥ p} a risk-neutral agent would accept the contract, install µ∗(w, p) = 0 and receive expected utility rate √ u(µ∗(w, p); w, p) = w − p. Given the set of oﬀers (w, p): p > λ, w ≥ 2 pλ − λ he would √ accept the contract, install service capacity µ∗(w, p) = pλ − λ and receive expected utility √ rate u(µ∗(w, p); w, p) = w − 2 pλ + λ. By comparing the optimal capacities of a weakly risk-averse agent to that of a risk-neutral agent, three conclusions are drawn:

1. Given a λ, the principal has to set a higher p in order to induce a weakly risk-averse agent to install a positive service capacity versus a risk-neutral agent (p > λ for risk- neutral agent, p > λ/(1 − η) for weakly risk-averse agent).

2. Given a λ, when p is relatively low, the µ value plays a more prominent role in the utility of a weakly risk-averse agent who therefore installs a service capacity lower than √ a risk-neutral agent ( pλ − λ > p(1 − η)pλ − λ). As the p increases, the penalty charge and the risk become of greater concern, therefore the weakly risk-averse agent √ installs a µ∗ higher than a risk-neutral agent (p(1 + η)pλ − λ > pλ − λ).

3. In essence, weakly risk-averse attitude makes an agent worse oﬀ. We state this conclu-

56 sion formally in Proposition 5.10.

Proposition 5.10. Given w and p, an agent who accepts the contract and installs a positive service capacity has a decreasing expected utility rate in η ∈ [0, 4/5).

Proof. Recall that when w and p satisfy conditions (5.8) and (5.9), the agent installs capacity µ∗(w, p) = p(1 − η)pλ − λ > 0, and the agent’s expected utility rate is u (µ∗(w, p)) = w − ηp − 2p(1 − η)pλ + λ. Note that ∂u/∂η = −p + pλ/p(1 − η)pλ = p √ √ √ √ − p − λ/(1 − η) p = − p p − p2 . Since p > p2, therefore ∂u/∂η < 0. When the compensation rate w and the penalty rate p satisfy conditions (5.9) and (5.10), the agent installs µ∗(w, p) = p(1 + η)pλ − λ > 0, and the agent’s expected utility rate is u (µ∗(w, p)) = w − 2p(1 + η)pλ + λ, thus ∂u/∂η = −ppλ/(1 + η) < 0.

The following Corollary follows from Proposition 5.10.

Corollary 5.11. Given w and p, an agent that accepts the contract and subsequently installs a positive service capacity is always worse oﬀ when he is weakly risk-averse (η ∈ (0, 4/5)) than risk-neutral (η = 0).

We discuss the case for η ≥ 4/5 in subsection 5.2.1.

5.1.2 Principal’s optimal strategy

Anticipating the agent’s optimal µ∗(w, p) the principal chooses the w and p that maximize her expected proﬁt rate by solving the optimization problem (5.11).

∗ ∗ rµ (w, p) pλ max ΠP (w, p; µ (w, p)) = max − w + (5.11) w>0,p>0 w>0,p>0 λ + µ∗(w, p) λ + µ∗(w, p)

∗ ∗ ∗ Denote (w , p ) = argmaxw>0,p>0 ΠP (w, p; µ (w, p)).

Before deriving the principal’s optimal strategy, we examine the case when the principal’s

contract offer satisfies p = p3 and w ≥ w3, in which case the agent is indifferent with respect to installing two different service capacities. Nevertheless, the corresponding solutions

57 ((w, p), µ) have to be admissible solutions (see Deﬁnition 3.3). We state this case formally in Proposition 5.12.

Proposition 5.12. Suppose a weakly risk-averse agent. Assume that the principal’s potential oﬀers are in the set {(w, p): p = p3, w ≥ w3}.

∗ p (a) If r ∈ (0, p3), the agent installs µ = (1 − η)p3λ − λ if oﬀered a contract.

∗ p ∗ p (b) If r = p3, both µ = (1 − η)p3λ − λ and µ = (1 + η)p3λ − λ lead to admissible p p solutions. Therefore the agent installs either (1 − η)p3λ − λ or (1 + η)p3λ − λ if oﬀered a contract.

∗ p (c) If r > p3, the agent installs µ = (1 + η)p3λ − λ if oﬀered a contract.

2 Proof. Note that for w ≥ w3 we have ∂ΠP (w, p3; µ)/∂µ = (r − p3)λ/(λ + µ) . Define p p µL ≡ (1 − η)p3λ − λ and µH ≡ (1 + η)p3λ − λ. Note that µH > µL. If r ∈ (0, p3), then ∂ΠP /∂µ < 0, therefore ((w, p3), µL) ((w, p3), µH ). If the principal offers a contract (the conditions are discussed in Proposition 5.18 that follows), then by Definition 3.3 only

µL leads to admissible solutions. Thus we obtain (a). If r > p3, then ∂ΠP /∂µ > 0, therefore

((w, p3), µH ) ((w, p3), µL). If the principal offers a contract (see Proposition 5.18), then only µH leads to admissible solutions. Therefore we obtain (c). If r = p3, then ∂ΠP /∂µ = 0, indicating that the principal receives the same expected profit rate when the agent installs capacity µL or µH . If the principal offers a contract (see Proposition 5.18), then both µL and µH lead to admissible solutions and we obtain (b).

Notation:

√ √ √ p3 − p2 r1 ≡ ηp3 + (1 − η) p2p3 , r2 ≡ 1 + 2η √ p3 , and r3 ≡ (1 + 2η)p3 (5.12) p2

Note that r1, r2 and r3 are functions of λ and η. However we suppress the parameters (λ, η).

58 1 Deﬁne pcu as follows :

1 2 p ≡ b + C + C (5.13) cu 9a2

√ √ where a ≡ 2η, b ≡ (1 − 2η) p2, and d ≡ −r p2 and

s s p 2 3 p 2 3 3 ∆ + ∆ − 4∆ 3 ∆ − ∆ − 4∆ C ≡ 1 1 0 , C ≡ 1 1 0 , where ∆ ≡ b2, ∆ ≡ 2b3 + 27a2d 2 2 0 1

Replacing ∆0 and ∆1 by the expressions of a, b and d we have

s 3p 3 2 √ p 2 3 2 4 2 3 2(1 − 2η) p − 108η r p + −432η r(1 − 2η) p + 11664η r p C = 2 2 2 2 and 2 s 3p 3 2 √ p 2 3 2 4 2 3 2(1 − 2η) p − 108η r p − −432η r(1 − 2η) p + 11664η r p C = 2 2 2 2 2

We introduce several technical lemmas.

Lemma 5.13. Let 4/5 > η > 0 and λ > 0. √ (a) ηp3 + (1 − η) p2p3 > p2 > 0. √ (b) p3 > ηp3 + (1 − η) p2p3.

√ √ √ (c) 1 + 2η p3 − p2 / p2 p3 > p3.

√ √ √ (d) (1 + 2η)p3 > 1 + 2η p3 − p2 / p2 p3.

Proof. Let 4/5 > η > 0 and λ > 0. First we prove (a) and (b) together. According to √ Lemma 5.8 part (a), p3 > p2, therefore p3 = ηp3 + (1 − η)p3 > ηp3 + (1 − η) p2p3 >

ηp2 +(1−η)p2 = p2. Next we prove (c). According to Lemma 5.8 part (a), p3 > p2, therefore √ √ √ √ √ √ 2η p3 − p2 / p2 > 0 ⇒ 1 + 2η p3 − p2 / p2 p3 > p3. Finally we prove (d). √ √ √ √ √ According to Lemma 5.3, we have 4p2 > p3, therefore 2 p2 > p3 ⇔ p2 > p3 − p2 ⇔ √ √ √ √ √ √ 1 > p3 − p2 / p2. Therefore (1 + 2η)p3 > 1 + 2η p3 − p2 / p2 p3.

1The subscript “cu” stands for “cubic” because (5.13) is the square of the solution to equation (5.14), which is a cubic equation that is introduced later in the proof for Lemma 5.16.

59 Lemma 5.13 implies that for η ∈ (0, 4/5) we have r3 > r2 > p3 > r1 > p2.

Lemma 5.14. Let η > 0 and λ > 0. (a) If 1 + 3η + 2pη(1 + 2η) λ/(1 + η) > r > 1 + 3η − 2pη(1 + 2η) λ/(1 + η) then 0 > r − 2p(1 + 2η)rλ/(1 + η) + λ. (b) If 1 + 3η − 2pη(1 + 2η) λ/(1+η) > r > 0 or r > 1 + 3η + 2pη(1 + 2η) λ/(1+η) then r − 2p(1 + 2η)rλ/(1 + η) + λ > 0. (c) If r = 1 + 3η − 2pη(1 + 2η) λ/(1 + η) or r = 1 + 3η + 2pη(1 + 2η) λ/(1 + η) then r − 2p(1 + 2η)rλ/(1 + η) + λ = 0. √ √ Proof. Let η > 0 and λ > 0. Deﬁne x ≡ r and a ≡ λ and the expression r − 2p(1 + 2η)rλ/(1 + η)+λ can be restated as x2 −2axp(1 + 2η)/(1 + η)+a2 with x > 0 and a > 0. The solutions to the quadratic equation x2 −2axp(1 + 2η)/(1 + η)+a2 = 0 for x are √ √ √ √ √ √ √ √ √ 1 + 2η − η a/ 1 + η and 1 + 2η + η a/ 1 + η. If 1 + 2η + η a/ 1 + η > √ √ √ x > 1 + 2η − η a/ 1 + η, or equivalently, if 1 + 3η + 2pη(1 + 2η) a/(1 + η) > x2 > 1 + 3η − 2pη(1 + 2η) a/(1 + η), then 0 > x2 − 2axp(1 + 2η)/(1 + η) + a2. Replacing x √ √ by r and a by λ we obtain (a). The proofs for (b) and (c) are similar.

p Lemma 5.15. Given 1 > η > 0 and λ > 0, then (1+2η)p3 > 1 + 3η + 2 η(1 + 2η) λ/(1+ η). √ √ √ √ Proof. Let 1 > η > 0 and λ > 0, and we have 1 + η > 1 − η and 1 + 2η > η, there- √ √ √ √ √ √ fore we have 2 1 + η > 1 + η + 1 − η and 2 1 + 2η > 1 + 2η + η. By multiplying √ √ √ √ √ √ these inequalities 4 1 + 2η 1 + η > 1 + 2η + η 1 + η + 1 − η. By multiplying √ √ √ √ √ √ 1 + η − 1 − η and dividing 2η 1 + η on both sides, 2 1 + 2η 1 + η − 1 − η /η > √ √ p 1 + 2η + η . By squaring both sides (1+2η)p3 > 1 + 3η + 2 η(1 + 2η) λ/(1+η).

2 √ Lemma 5.16. Consider max √ √ f(x) where f(x) = r+λ−ηx − p2 ((1 − 2η)x + r/x) x∈[ p2, p3] and denote x∗ = argmax √ √ f(x). The solutions to this optimization problem are x∈[ p2, p3]

∗ √ (a) x = p2 if r ∈ (0, p2].

∗ √ √ √ (b) x = pcu ∈ p2, p3 if r ∈ (p2, r2).

60 ∗ √ (c) x = p3 if r ≥ r2. √ √ Proof. Note that f(x) is continuous and diﬀerentiable over interval p2, p3 : √ df(x) √ r d2f(x) 2r p = −2ηx − p (1 − 2η) − and = −2η − 2 < 0 dx 2 x2 dx2 x3

df(x) √ r 1 = − p2 + √ = √ (r − p2) dx √ p p x= p2 2 2 √ √ df(x) √ √ √ r p2 p2 = − p2 − 2η( p3 − p2) + = (r − r2) dx √ p p x= p3 3 3

√ √ ∗ √ If r ∈ (0, p2], then f(x) is decreasing over p2, p3 , therefore x = p2. If r ≥ r2, √ √ ∗ √ then f(x) is increasing over p2, p3 , therefore x = p3. If r ∈ (p2, r2), then f(x) is √ increasing in the neighborhood above x = p2 and decreasing in the neighborhood below √ 2 2 ∗ √ √ x = p3. Also note that d f(x)/dx < 0, therefore x ∈ p2, p3 that satisﬁes the ﬁrst order condition

df(x) ∗ 3 √ ∗ 2 √ = 0 ⇒ 2η (x ) + (1 − 2η) p2 (x ) − r p2 = 0 (5.14) dx x=x∗

which is a cubic equation. According to the general formula for roots of cubic equation,

∗ √ √ √ x = pcu ∈ p2, p3 .

√ √ Lemma 5.17. Consider max x≥ p3 f(x) where f(x) = r + λ − p1 ((1 + 2η)x + r/x) and

∗ √ denote x = argmax x≥ p3 f(x). The solutions to this optimization problem are

∗ √ (a) x = p3 if r ∈ (0, r3].

∗ p (b) x = r/(1 + 2η) if r > r3. √ Proof. Note that f(x) is continuous and diﬀerentiable for x ≥ p3: √ df(x) √ r d2f(x) 2r p = − p 1 + 2η − and = − 1 < 0 dx 1 x2 dx2 x3 √ df(x) p1 df(x) √ = (r − r3) and lim = −(1 + 2η) p1 < 0 dx √ p x→+∞ dx x= p3 3

√ ∗ √ If r ∈ (0, r3], then f(x) is decreasing for x ≥ p3, therefore x = p3. If r > r3, then √ f(x) is increasing in the neighborhood above x = p3 and decreasing when x approaches

61 2 2 ∗ √ +∞. Also note that d f(x)/dx < 0, therefore x > p3 that satisﬁes ﬁrst order condition

∗ p df(x)/dx|x=x∗ = 0 ⇒ x = r/(1 + 2η).

Now we state Proposition 5.18, which serves as a stepping stone towards the main results Theorem 5.19 and Proposition 5.20 that follow later. Proposition 5.18 provides the optimal w∗ and optimal p∗ under some restrictions. These restrictions are later removed in the main results Theorem 5.19 and Proposition 5.20.

Recall that Proposition 5.9 describes the agent’s optimal response to each contract oﬀer (w, p). Since the principal will not propose a contract that is going to be rejected by a weakly risk-averse (WRA) agent, therefore Proposition 5.18 only considers pairs (w, p) that result in agent’s non-negative expected utility rate. Deﬁne:

D(5.7) ≡ {(w, p) that satisﬁes (5.7) when η ∈ (0, 4/5)}

D(5.8) ≡ {(w, p) that satisﬁes (5.8) when η ∈ (0, 4/5)}

D(5.9) ≡ {(w, p) that satisﬁes (5.9) when η ∈ (0, 4/5)} (5.15)

D(5.10) ≡ {(w, p) that satisﬁes (5.10) when η ∈ (0, 4/5)}

DWRA ≡ D(5.7) ∪ D(5.8) ∪ D(5.9) ∪ D(5.10)

Proposition 5.18. Given a weakly risk-averse agent;

(a) If (w, p) ∈ D(5.7), then the principal does not propose a contract.

(b) Consider oﬀers (w, p) ∈ D(5.8) ∪ D(5.9).

(b1) If r ∈ (0, p2], then the principal does not propose a contract.

∗ ∗ p (b2) If r ∈ (p2, p3], then the principal oﬀers (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu

∗ ∗ ∗ p and the agent installs service capacity µ (w , p ) = (1 − η)pcuλ − λ.

∗ ∗ (b3) If r ∈ (p3, r2), then the principal either oﬀers (w , p ) = (w3, p3) resulting in

∗ p ∗ ∗ the agent installing capacity µ (w, p) = (1 + η)p3λ − λ, or oﬀers (w , p ) = p ηpcu + 2 (1 − η)pcuλ − λ, pcu resulting in the agent installing service capacity

∗ ∗ ∗ p µ (w , p ) = (1 − η)pcuλ − λ.

62 ∗ ∗ (b4) If r ≥ r2, then the principal’s oﬀer is (w , p ) = (w3, p3) and the agent installs

∗ ∗ ∗ p service capacity µ (w , p ) = (1 + η)p3λ − λ.

(c1) If r ∈ (0, r1], then the principal does not propose a contract.

∗ ∗ (c2) If r ∈ (r1, p3], the principal oﬀers a contract with (w , p ) = (w3, p3) and the agent

∗ ∗ ∗ p installs service capacity µ (w , p ) = (1 − η)p3λ − λ.

∗ ∗ (c3) If r ∈ (p3, r3], the principal oﬀers a contract with (w , p ) = (w3, p3) and the agent

∗ ∗ ∗ p installs service capacity µ (w , p ) = (1 + η)p3λ − λ.

∗ ∗ p (c4) If r > r3, the principal oﬀers (w , p ) = 2 (1 + η)rλ/(1 + 2η) − λ, r/(1 + 2η) and the agent installs service capacity µ∗(w∗, p∗) = p(1 + η)rλ/(1 + 2η) − λ.

Proof. The structure of the proof for Proposition 5.18 is depicted in Figure 5.7.

Case (w, p) ∈ D(5.7): According to Proposition 5.9 part (a), in case the principal makes an oﬀer, the agent accepts the contract but does not install any service capacity. Since

∗ ∗ ∗ ∗ ∂ΠP /∂w = −1 < 0, thus w = p and from equation (4.3) ΠP (w , p; µ (w , p)) = −w∗ + p = −p + p = 0. Therefore the principal does not propose a contract.

Case (w, p) ∈ D(5.8) ∪ D(5.9): According to Proposition 5.9 part (b), if (w, p) ∈ D(5.8), then in case the principal makes an oﬀer, the agent accepts the contract and installs

p ∗ p (1 − η)pλ − λ. Since ∂ΠP /∂w = −1 < 0, therefore w = ηp + 2 (1 − η)pλ − λ.

According to Proposition 5.9 part (c) and Proposition 5.12, if (w, p) ∈ D(5.9) (which

implies p = p3), then in case the principal makes an oﬀer, the agent accepts the p p contract and installs (1 − η)p3λ − λ if r ∈ (0, p3), installs either (1 − η)p3λ − λ or p p (1 + η)p3λ−λ if r = p3, or installs (1 + η)p3λ−λ if r > p3. Since ∂ΠP /∂w = −1 <

∗ 0, therefore w = w3. Denote the principal’s expected proﬁt rate when (w, p) = (w3, p3)

p L and µ = (1 − η)p3λ − λ by ΠP (p3), and denote the principal’s expected proﬁt rate p H when (w, p) = (w3, p3) and µ = (1 + η)p3λ − λ by ΠP (p3). By plugging the value of

63 (w, p) ∈ D(5.7) No contract oﬀered.

r ∈ (0, p2] No contract oﬀered.

∗ p w = ηpcu + 2 (1 − η)pcuλ − λ r ∈ (p2, p3] ∗ ∗ p and p = pcu and µ = (1 − η)pcuλ − λ Risk-Neutral Principal with Weakly (w, p) ∈ D(5.8) ∪ D(5.9) Risk-Averse Agent ∗ p w = ηpcu + 2 (1 − η)pcuλ − λ ∗ ∗ p r ∈ (p3, r2) and p = pcu and µ = (1 − η)pcuλ − λ ∗ ∗ ∗ p or w = w3 and p = p3 and µ = (1 + η)p3λ − λ

∗ ∗ ∗ p r ≥ r2 w = w3 and p = p3 and µ = (1 + η)p3λ − λ

r ∈ (0, r1] No contract oﬀered.

∗ ∗ ∗ p r ∈ (r1, p3] w = w3 and p = p3 and µ = (1 − η)p3λ − λ

(w, p) ∈ D(5.9) ∪ D(5.10)

∗ ∗ ∗ p r ∈ (p3, r3] w = w3 and p = p3 and µ = (1 + η)p3λ − λ

s (1 + η)rλ r w∗ = 2 − λ and p∗ = 1 + 2η 1 + 2η r > r3 s (1 + η)rλ and µ∗ = − λ 1 + 2η

Figure 5.7: Structure of the proof for Proposition 5.18

w, p and µ into equation (4.3):

√ √ L √ √ r p3 − p2 ΠP (p3) =r + λ − ηp3 − p2 (1 − 2η) p3 + √ = √ (r − r1) (5.16) p3 p3 H √ √ r ΠP (p3) =r + λ − p1 (1 + 2η) p3 + √ (5.17) p3

∗ ∗ ∗ and the principal’s optimization problem is maxp∈[p2,p3] ΠP (w , p; µ (w , p)) where:

 √ √ r  r + λ − ηp − p2 (1 − 2η) p + √ , for p ∈ [p2, p3) ∗ ∗ ∗ p ΠP (w , p; µ (w , p)) =  L H  max ΠP (p3), ΠP (p3) , for p = p3

√ √ √ √ Deﬁne x ≡ p, the expression r + λ − ηp − p2 (1 − 2η) p + r/ p can be restated

2 √ as f(x) = r + λ − ηx − p2 ((1 − 2η)x + r/x). Maximizing f(x) with respect to x over

64 √ √ √ √ √ p2, p3 is equivalent to maximizing r + λ − ηp − p2 (1 − 2η) p + r/ p with

respect to p over the interval [p2, p3] in the sense that

 2 √ √ r argmax r + λ − ηp − p2 (1 − 2η) p + √ =  argmax f(x) p∈[p ,p ] p √ √ 2 3 x∈[ p2, p3]

According to Lemma 5.13, r2 > p3 > p2, therefore we examine the following subcases.

∗ Subcase r ∈ (0, p2]: According to Lemma 5.16 part (a), p = p2; this case is taken

care of in the case when (w, p) ∈ D(5.7) and the principal does not propose a contract.

Subcase r ∈ (p2, p3]: From Lemma 5.16 part (b) and Proposition 5.12 part (a) and

∗ ∗ ∗ ∗ ∗ ∗ (b), p = pcu and the principal’s expected proﬁt rate is ΠP (w , p ; µ (w , p )) >

∗ p ΠP (p2, p2; 0) = 0. Thus the principal proposes w = ηpcu + 2 (1 − η)pcuλ − λ and

∗ ∗ ∗ ∗ p p = pcu that induces the agent to install µ (w , p ) = (1 − η)pcuλ − λ.

Subcase r ∈ (p3, r2): From Lemma 5.16 part (b) and Proposition 5.12 part (c), the

∗ ∗ ∗ ∗ ∗ ∗ principal chooses either p = pcu with expected proﬁt rate ΠP (w , p ; µ (w , p )) = √ √ √ r + λ − ηpcu − p2 (1 − 2η) pcu + r/ pcu > ΠP (p2, p2; 0) = 0, or chooses

∗ ∗ ∗ ∗ ∗ ∗ H L p = p3 with expected profit rate ΠP (w , p ; µ (w , p )) = ΠP (p3) > ΠP (p3) = √ √ √ p3 − p2 (r − r1)/ p3 > 0. However due to the difficulty of computing pcu we do not explicitly identify the principal’s optimal offer.

∗ Subcase r ≥ r2: According to Lemma 5.16 part (c), p = p3. According to Proposi- p tion 5.12 part (c) the agent installs capacity (1 + η)p3λ−λ and the principal’s ex- ∗ ∗ ∗ ∗ ∗ H L √ √ pected proﬁt rate is ΠP (w , p ; µ (w , p )) = ΠP (p3) > ΠP (p3) = p3 − p2 (r− √ ∗ p ∗ r1)/ p3 > 0. Therefore the principal proposes w = 2 (1 + η)p3λ−λ and p = p3

∗ ∗ ∗ p that induces the agent to install µ (w , p ) = (1 + η)p3λ − λ.

Case (w, p) ∈ D(5.9) ∪ D(5.10): According to Proposition 5.9 part (d), if (w, p) ∈ D(5.10), then in case the principal makes an oﬀer, the agent accepts the contract and installs

p ∗ p (1 + η)pλ−λ. Since ∂ΠP /∂w = −1 < 0, therefore w = 2 (1 + η)pλ−λ. According

to Proposition 5.9 part (c) and Proposition 5.12, if (w, p) ∈ D(5.9) (which implies

65 p = p3), then in case the principal makes an oﬀer, the agent accepts the contract and p p p installs (1 − η)p3λ−λ if r ∈ (0, p3), installs either (1 − η)p3λ−λ or (1 + η)p3λ−λ p if r = p3, or installs (1 + η)p3λ − λ if r > p3. Since ∂ΠP /∂w = −1 < 0, therefore

∗ L H w = w3. Recall the deﬁnition of ΠP (p3) and ΠP (p3) (see equations (5.16) and (5.17)). ∗ ∗ ∗ The principal’s optimization problem is maxp≥p3 ΠP (w , p; µ (w , p)) where:

 L H  max ΠP (p3), ΠP (p3) , for p = p3 ∗ ∗ ∗  ΠP (w , p; µ (w , p)) = √ √ r  r + λ − p1 (1 + 2η) p + √ , for p > p3  p

√ √ √ √ Deﬁne x ≡ p, the expression r + λ − p1 (1 + 2η) p + r/ p can be restated as √ √ f(x) = r + λ − p1 ((1 + 2η)x + r/x). Maximizing f(x) for x ≥ p3 is equivalent to √ √ √ maximizing r + λ − p1 (1 + 2η) p + r/ p for p ≥ p3 in the sense that

!2 √ √ r argmax r + λ − p (1 + 2η) p + = argmax f(x) 1 √ √ p≥p3 p x≥ p3

According to Lemma 5.13, r3 > p3 > r1, therefore we examine the following subcases.

∗ Subcase r ∈ (0, r1]: According to Lemma 5.17 part (a), p = p3. By Proposition ∗ ∗ ∗ ∗ ∗ L √ √ √ 5.12 part (a), ΠP (w , p ; µ (w , p )) = ΠP (p3) = p3 − p2 (r − r1)/ p3 and ∗ ∗ ∗ ∗ ∗ note that ΠP (w , p ; µ (w , p )) ≤ 0, therefore the principal does not propose a contract.

∗ Subcase r ∈ (r1, p3]: From Lemma 5.17 part (a), p = p3. According to Proposition

∗ ∗ ∗ ∗ ∗ 5.12 part (a) and (b), the principal’s expected proﬁt rate is ΠP (w , p ; µ (w , p )) = L √ √ √ ΠP (p3) = p3 − p2 (r − r1)/ p3 > 0, therefore the principal proposes a con- ∗ ∗ ∗ ∗ ∗ tract with w = w3 and p = p3 that induces the agent to install µ (w , p ) = p (1 − η)p3λ − λ.

∗ Subcase r ∈ (p3, r3]: From Lemma 5.17 part (a), p = p3. According to Proposi-

∗ ∗ ∗ ∗ ∗ tion 5.12 part (c), the principal’s expected proﬁt rate is ΠP (w , p ; µ (w , p )) = H L √ √ √ ΠP (p3) > ΠP (p3) = p3 − p2 (r − r1)/ p3 > 0, therefore the principal pro- ∗ ∗ poses a contract with w = w3 and p = p3 that induces the agent to install

66 ∗ ∗ ∗ p µ (w , p ) = (1 + η)p3λ − λ.

∗ Subcase r > r3: According to Lemma 5.17 part (b), p = r/(1 + 2η) and the princi-

∗ ∗ ∗ ∗ ∗ p pal’s expected proﬁt rate is ΠP (w , p ; µ (w , p )) = r−2 (1 + 2η)rλ/(1 + η)+λ.

∗ ∗ ∗ ∗ ∗ According to Lemma 5.14 and 5.15, ΠP (w , p ; µ (w , p )) > 0, therefore the principal proposes a contract with w∗ = 2p(1 + η)rλ/(1 + 2η)−λ and p∗ = r/(1+2η) that induces the agent to install capacity µ∗(w∗, p∗) = p(1 + η)rλ/(1 + 2η) − λ.

We describe the principal’s optimal strategy in Theorem 5.19 and Proposition 5.20. We identify the principal’s optimal oﬀer only when r ∈ (0, p3] or r ≥ r2, (see Theorem 5.19).

The cases when r ∈ (p3, r2) are discussed in Proposition 5.20. We prove Theorem 5.19 and Proposition 5.20 together.

Theorem 5.19. Consider a weakly risk-averse agent and (w, p) ∈ DWRA.

(a) If r ∈ (0, p2], then the principal does not propose a contract.

(b) If r ∈ (p2, p3], then the principal’s oﬀer and the capacity installed by the agent are

∗ ∗ p ∗ ∗ ∗ p (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu and µ (w , p ) = (1 − η)pcuλ − λ (5.18)

and the principal’s expected proﬁt rate is

∗ ∗ ∗ ∗ ∗ √ √ r ΠP (w , p ; µ (w , p )) = r + λ − ηpcu − p2 (1 − 2η) pcu + √ (5.19) pcu

∗ ∗ ∗ ∗ ∗ p (w , p ) = (w3, p3) and µ (w , p ) = (1 + η)p3λ − λ (5.20)

and the principal’s expected proﬁt rate is

∗ ∗ ∗ ∗ ∗ √ √ r ΠP (w , p ; µ (w , p )) = r + λ − p1 (1 + 2η) p3 + √ (5.21) p3

67 (d) If r > r3, then the principal’s oﬀer and the capacity installed by the agent are

s ! s (1 + η)rλ r (1 + η)rλ (w∗, p∗) = 2 − λ, and µ∗(w∗, p∗) = − λ (5.22) 1 + 2η 1 + 2η 1 + 2η

and the principal’s expected proﬁt rate is

s (1 + 2η)rλ Π (w∗, p∗; µ∗(w∗, p∗)) = r − 2 + λ (5.23) P 1 + η

Proposition 5.20. Given a weakly risk-averse agent and (w, p) ∈ DWRA. If r ∈ (p3, r2), then either

∗ ∗ p ∗ ∗ ∗ p (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu and µ (w , p ) = (1 − η)pcuλ − λ resulting in principal’s expected proﬁt rate

∗ ∗ ∗ ∗ ∗ √ √ √ ΠP (w , p ; µ (w , p )) = r + λ − ηpcu − p2 ((1 − 2η) pcu + r/ pcu) or the principal oﬀers and the agent installs

∗ ∗ ∗ ∗ ∗ p (w , p ) = (w3, p3) and µ (w , p ) = (1 + η)p3λ − λ resulting in principal’s expected utility rate

∗ ∗ ∗ ∗ ∗ √ √ √ ΠP (w , p ; µ (w , p )) = r + λ − p1 ((1 + 2η) p3 + r/ p3)

Proof. In part (b) of Proposition 5.18, we solved the problem (5.11) for (w∗, p∗) by restricting r to be in (0, p2], or in (p2, p3], or in (p3, r2), or in [r2, +∞). In part (c) of Proposition 5.18,

∗ ∗ we solved the problem (5.11) for (w , p ) by restricting r to be in (0, r1], or in (r1, p3], or in (p3, r3], or in (r3, +∞). The principal maximizes her expected profit rate by offering contract that lead to admissible solutions (Definition 3.3) for any given value of r, η and λ. The structure of the proof for Theorem 5.19 and Proposition 5.20 is depicted in Figure 5.8.

68 r ∈ (0, p2] No contract.

∗ p w = ηpcu + 2 (1 − η)pcuλ − λ r ∈ (p2, p3] ∗ ∗ p and p = pcu and µ = (1 − η)pcuλ − λ

Risk-Neutral ∗ p Principal Either w = ηpcu + 2 (1 − η)pcuλ − λ ∗ ∗ p with Weakly r ∈ (p3, r2) and p = pcu and µ = (1 − η)pcuλ − λ ∗ ∗ ∗ p Risk-Averse or w = w3 and p = p3 and µ = (1 + η)p3λ − λ Agent

∗ ∗ ∗ p r ∈ [r2, r3] w = w3 and p = p3 and µ = (1 + η)p3λ − λ

s (1 + η)rλ r w∗ = 2 − λ and p∗ = 1 + 2η 1 + 2η r > r3 s (1 + η)rλ and µ∗ = − λ 1 + 2η

Figure 5.8: Structure of the proof for Theorem 5.19 and Proposition 5.20

Case r ∈ (0, p2]: According to Proposition 5.18 part (a), (b1) and (c1), the principal does not propose a contract. This case is covered by Theorem 5.19 (a).

Case r ∈ (p2, p3]: If r ∈ (p2, r1], then according to Proposition 5.18 part (a), (b2) and

∗ ∗ p (c1), the principal oﬀers (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu and the agent

∗ ∗ ∗ p installs µ (w , p ) = (1 − η)pcuλ − λ. If r ∈ (r1, p3], then according to Proposition 5.18 part (a), (b2) and (c2) and Lemma 5.16 part (b) the principal oﬀers (w∗, p∗) =

p ∗ ∗ ∗ p ηpcu + 2 (1 − η)pcuλ − λ, pcu and the agent installs µ (w , p ) = (1 − η)pcuλ − λ. This case is covered by Theorem 5.19 (b).

Case r ∈ (p3, r2): According to Proposition 5.18 part (a), (b3) and (c3), the principal

∗ ∗ p either oﬀers a contract (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu and the agent in-

∗ ∗ ∗ p ∗ ∗ stalls µ (w , p ) = (1 − η)pλ − λ, or oﬀers (w , p ) = (w3, p3) and the agent installs

∗ ∗ ∗ p µ (w , p ) = (1 + η)p3λ − λ. This case is covered by Proposition 5.20.

Case r ∈ [r2, r3]: According to Proposition 5.18 part (a), (b4) and (c3), the principal oﬀers

∗ ∗ ∗ ∗ ∗ p a contract with (w , p ) = (w3, p3) and the agent installs µ (w , p ) = (1 + η)p3λ−λ. This case is covered by Theorem 5.19 (c).

69 Case r > r3: From Proposition 5.18 part (a), (b4) and (c4) and Lemma 5.17 part (b), the principal oﬀers a contract with (w∗, p∗) = 2p(1 + η)rλ/(1 + 2η) − λ, r/(1 + 2η) and the agent installs service capacity µ∗(w∗, p∗) = p(1 + η)rλ/(1 + 2η) − λ. This case is covered by Theorem 5.19 (d).

Theorem 5.19 and Proposition 5.20 indicate that the existence of a beneﬁcial contract with a weakly risk-averse agent is determined exogenously by the revenue rate r, the failure rate λ, and the risk coeﬃcient η.

Since it is diﬃcult to identify the principal’s optimal oﬀer when r ∈ (p3, r2) due to

the diﬃculty of computing pcu we resort to simulation to better understand the principal’s choices.

Remark 5.21. Figure 5.9 shows that when η = 0.1 and η = 0.5 there exists an r0 ∈ (p3, r2)

∗ ∗ p such that when r ∈ (p3, r0), the principal oﬀers (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu ,

∗ ∗ when r ∈ (r0, r2), she oﬀers (w , p ) = (w3, p3) and when r = r0, the principal is indiﬀerent

about the two alternative oﬀers. However due to the diﬃculty of computing pcu (equation

(5.13)), it is not clear how to determine analytically the value of r0 for all η ∈ (0, 4/5) as a function of λ and η.

5.2 Optimal strategies given a strongly risk-averse agent

For the strongly risk-averse (SRA) agent we ﬁrst derive the agent’s optimal strategy. The agent’s optimization problem is stated in (5.4).

Notation:

p w4 = p4 ≡ 1 + 2η + 2 η(1 + η) λ (5.24)

70 ~ ∗ ∗ ∗ ∗ ∗ ∗ ~ ∗ ∗ ∗ ∗ ∗ ∗ ΠP = ΠP ( w , p = pcu ; µ ) − ΠP ( w , p = p3 ; µ ) ΠP = ΠP ( w , p = pcu ; µ ) − ΠP ( w , p = p3 ; µ )

~ η = 0.1 and λ = 0.01 ~ η = 0.5 and λ = 0.01 ΠP ΠP

r r

p3 r0 ( λ , η ) r2 ( λ , η ) p3 r0 ( λ , η ) r2 ( λ , η )

(a) η = 0.1 (b) η = 0.5

∗ ∗ ∗ ∗ ∗ ∗ Figure 5.9: The value of Π˜ P ≡ ΠP (w , p = pcu; µ ) − ΠP (w , p = p3; µ ) for r ∈ (p3, r2)

Note that w4 and p4 are functions of λ and η which are suppressed in our notation.

A technical lemma used later is introduced next. Lemma 5.22. Let η > 1/3 and λ > 0, then 1 + 2η + 2pη(1 + η) λ > 4λ/(1 + η).

Proof. Let η > 1/3 and λ > 0. Note that

√ 3η > 1 ⇔ 2 η > p1 + η √ ⇔ p1 + η > 2 p1 + η − η √ 2 ⇔ (1 + η) p1 + η + η > 4 ⇔ 1 + 2η + 2pη(1 + η) λ > 4λ/(1 + η)

We describe a strongly risk-averse agent’s optimal response to any possible contract

2 (w, p) ∈ R+ in Proposition 5.23.

Proposition 5.23. Consider a strongly risk-averse agent (η ≥ 4/5).

71 (a) Given

w ≥ p ∈ (0, p4) (5.25)

then the agent would accept the contract if oﬀered and install µ∗(w, p) = 0 with resulting

∗ expected utility rate uA(µ (w, p); w, p) = w − p ≥ 0. The agent rejects the contract if

p ∈ (0, p4) and w ∈ (0, p).

(b) Given

p = p4 and w ≥ w4 (5.26)

then the agent would accept the contract if oﬀered and is indiﬀerent about installing

∗ ∗ p either µ (w, p) = 0 or µ (w, p) = (1 + η)p4λ − λ. In both cases the agent’s expected

∗ ∗ utility rate is uA(µ (w, p); w, p) = w − p4 ≥ 0. If r ∈ (0, p4], then neither µ = 0 nor

∗ p µ = (1 + η)p4λ−λ leads to admissible solutions (see Deﬁnition 3.3). If r > p4, then

∗ ∗ ∗ p there exists w such that (w , p4), µ = (1 + η)p4λ − λ is the unique admissible

solution (for proof see Proposition 5.24). The agent rejects the contract if p = p4 and

w ∈ (0, w4).

p p > p4 and w ≥ 2 (1 + η)pλ − λ (5.27)

then the agent would accept the contract if oﬀered and install µ∗(w, p) = p(1 + η)pλ−λ

∗ p with resulting expected utility rate uA(µ (w, p); w, p) = w − 2 (1 + η)pλ + λ ≥ 0. The p agent rejects the contract if p > p4 and w ∈ 0, 2 (1 + η)pλ − λ .

Proof. According to Table 5.1, the optimization of u(µ) when η ∈ [4/5, 1) versus η ≥ 1 is diﬀerent. Therefore we prove the proposition separately for η ∈ [4/5, 1) and η ≥ 1.

Case η ∈ [4/5, 1): Recall the deﬁnition of p1 and p2 in (5.5). Note that 4p2 > p2 > 4p1 and according to Lemmas 5.7 part (b) and (c) and 5.22, p2 ≥ p4 > 4p1. Therefore we have

72 4p2 > p2 ≥ p4 > 4p1. Figure 5.10 depicts the shape of u(µ) when η ∈ [4/5, 1) and the value of p falls in diﬀerent ranges. The structure of the proof when η ∈ [4/5, 1) is depicted in Figure 5.11.

Case p ∈ (0, 4p1]: According to Table 5.1, u(µ) is decreasing for µ ≥ 0. Thus the agent’s optimal service capacity is µ∗(w, p) = 0 and from (5.3) u(µ∗(w, p)) = w − p.

Subcase w ∈ (0, p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (4p1, p2]: According to Table 5.1, there is a service capacity that maximizes u(µ) for µ ∈ [0, λ) and a service capacity that maximizes u(µ) for µ > λ. Denote

∗ the optimal service capacity in [0, λ) by µ[0,λ)(w, p). Note that u(µ) is decreasing with ∗ respect to µ over [0, λ), therefore the agent’s optimal service capacity is µ[0,λ)(w, p) = 0 ∗ and from (5.3) u µ[0,λ)(w, p) = w − p. Denote the optimal service capacity for µ > λ ∗ ∗ p by µλ(w, p). From the ﬁrst order condition µλ(w, p) = (1 + η)pλ − λ and from ∗ p (5.3) u (µλ(w, p)) = w − 2 (1 + η)pλ + λ. The agent has a choice of two service capacities and he installs the one that generates a higher expected utility rate. Note

∗ ∗ p that u (µλ(w, p)) − u µ[0,λ)(w, p) = p − 2 (1 + η)pλ + λ. According to Lemmas 5.7

part (b) and (c) and 5.22, p2 ≥ p4 > 4p1, therefore we examine the following subcases.

p Subcase p ∈ (4p1, p4): According to Lemma 5.6, 4p1 > 1 + 2η − 2 η(1 + η) λ

∗ ∗ and according to Lemma 5.5 part (a), u µ[0,λ)(w, p) > u (µλ(w, p)), thus the agent’s optimal service capacity is µ∗(w, p) = 0 and u(µ∗(w, p)) = w − p.

Subsubcase w ∈ (0, p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subsubcase w ≥ p: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

∗ ∗ Subcase p = p4: According to Lemma 5.5 part (c), u µ[0,λ)(w, p) = u (µλ(w, p)), in- ∗ ∗ dicating that installing µ[0,λ)(w, p) or µλ(w, p) results in the same agent’s expected

73 u( µ ) u( µ )

0.920 0.925 0.930 0.935 0.95 0.96 0.97 0.98 0 0 iue51:Ilsrto ftefrsof forms the of Illustration 5.10: Figure λ λ (a) (c) = η = η 0.9 0.9 p p

, , λ= λ λ= λ ∈ ∈ 0.01 0.01 ( (0 µ µ µ p

u( ) , ,

4 , w w 4 p , = = 1 1 p

0.50 0.55 0.60 0.65 0.70 0.75 0.80 , ,

2 p p 1 = = ] ] 1.5 7.5 0 λ λ λ = η 5 5 (e) λ λ 0.9

, λ= λ > p 74 0.01

µ u( µ ) u( µ )

, 4

w p = 1 2

0.85 0.86 0.87 0.88 0.89 0.90 0.940 0.945 0.950 0.955 0.960 ,

p = 50 0 0 λ u ( µ λ λ when ) (b) (d) = η 5 = η λ p p 0.9 0.9

∈ , ∈

, λ= λ λ= λ (4 0.01 ( 0.01 η p µ µ p

2 ,

∈ ,

w 1 , = = p , 4 1 [4 1

p ,

p 4 2 = / = ) ] 15 4 5 λ λ , 1) 5 5 λ λ w ∈ (0, p) Reject.

p ∈ (0, 4p1]

w ≥ p µ∗ = 0

w ∈ (0, p) Reject.

p ∈ (4p1, p4)

w ≥ p µ∗ = 0

w ∈ (0, w4) Reject.

p ∈ (4p1, p2] p = p4

∗ ∗ p w ≥ w4 µ = 0 or µ = (1 + η)p4λ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

4 p ∈ (p4, p2] η ∈ , 1 5 w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

p ∈ (p2, 4p2]

w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

p > 4p2

w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

Figure 5.11: Structure of the proof for Proposition 5.23 when η ∈ [4/5, 1)

utility rate. Therefore the agent is indiﬀerent about installing µ∗(w, p) = 0 or

∗ p ∗ µ (w, p) = (1 + η)p4λ − λ with expected utility rate u (µ (w, p); w, p) = w − w4. However the principal would not propose a contract in this case because none of these capacities leads to admissible solutions (see Deﬁnition 3.3). For proof see p Proposition 5.24. According to Lemma 5.4, p4 > 4p1 ⇒ w4 = 2 (1 + η)p4λ − λ > 0.

∗ Subsubcase w ∈ (0, w4): u(µ (w, p)) < 0, therefore the agent rejects the contract.

∗ Subsubcase w ≥ w4: u(µ (w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

∗ ∗ Subcase p ∈ (p4, p2]: From Lemma 5.5 part (b), u (µλ(w, p)) > u µ[0,λ)(w, p) , thus the agent’s optimal service capacity is µ∗(w, p) = p(1 + η)pλ−λ and u(µ∗(w, p)) = p p w−2 (1 + η)pλ+λ. According to Lemma 5.4, p > p4 > 4p1 ⇒ 2 (1 + η)pλ−λ >

75 0, therefore we further examine the following subcases. Subsubcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subsubcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

Case p ∈ (p2, 4p2]: According to Table 5.1, there is a service capacity that maximizes u(µ) for µ ∈ (0, λ] and a service capacity that maximizes u(µ) for µ > λ. Denote

∗ the optimal service capacity in (0, λ] by µ(0,λ](w, p). From the ﬁrst order condition ∗ p ∗ p µ(0,λ](w, p) = (1 − η)pλ−λ and from (5.3) u µ(0,λ](w, p) = w−ηp−2 (1 − η)pλ+λ. ∗ Denote the optimal service capacity for µ > λ by µλ(w, p). From the ﬁrst order ∗ p ∗ p condition µλ(w, p) = (1 + η)pλ−λ and from (5.3) u (µλ(w, p)) = w−2 (1 + η)pλ+λ. The agent has to decide which of the two service capacities he installs and he chooses

∗ ∗ the one with higher expected utility rate. Note that u (µλ(w, p)) − u µ(0,λ](w, p) = √ √ √ ηp−2 1 + η − 1 − η pλ. According to Lemma 5.8 part (b) and (c), p > p2 ≥ p3,

∗ ∗ and according to Lemma 5.2 part (a) u (µλ(w, p)) > u µ(0,λ](w, p) , therefore the agent’s optimal service capacity is µ∗(w, p) = p(1 + η)pλ − λ and u(µ∗(w, p)) = w − p p 2 (1 + η)pλ + λ. According to Lemma 5.4, p > p2 > 4p1 ⇒ 2 (1 + η)pλ − λ > 0, therefore we examine the following subcases.

Subcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, thus the agent rejects the contract.

Subcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

Case p > 4p2: According to Table 5.1, the service capacity that maximizes u(µ) must satisfy µ > λ. From the ﬁrst order condition µ∗(w, p) = p(1 + η)pλ−λ and u(µ∗(w, p)) = p p w−2 (1 + η)pλ+λ. According to Lemma 5.4, p > 4p2 > 4p1 ⇒ 2 (1 + η)pλ−λ > 0, therefore we examine the following subcases.

Subcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, thus the agent rejects the

76 contract.

Subcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

This completes the proof for Proposition 5.23 when η ∈ [4/5, 1).

Case η ≥ 1: According to Lemma 5.22, p4 > 4p1. Figure 5.12 depicts the shape of u(µ) when η ≥ 1 and the value of p falls in diﬀerent ranges. The proof when η ≥ 1 is depicted in Figure 5.13.

Case p ∈ (0, 4p1]: According to Table 5.1, u(µ) is decreasing with respect to µ ≥ 0. Thus the agent’s optimal service capacity is µ∗(w, p) = 0 and from (5.3) u(µ∗(w, p)) = w − p.

Subcase w ∈ (0, p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p > 4p1: According to Table 5.1, there is a service capacity that maximizes u(µ) for µ ∈ [0, λ) and a service capacity that maximizes u(µ) for µ > λ. Denote the optimal

∗ service capacity in [0, λ) by µ[0,λ)(w, p). Note that u(µ) is decreasing with respect to ∗ µ over [0, λ), therefore the agent’s optimal service capacity is µ[0,λ)(w, p) = 0 and from ∗ (5.3) u µ[0,λ)(w, p) = w − p. Denote the optimal service capacity for µ > λ by ∗ ∗ p µλ(w, p). From the ﬁrst order condition µλ(w, p) = (1 + η)pλ − λ and from (5.3) ∗ p u (µλ(w, p)) = w − 2 (1 + η)pλ + λ. The agent must decide which of the two service capacities he is going to install and he chooses the one that generates a higher expected

∗ ∗ p utility rate. Note that u (µλ(w, p))−u µ[0,λ)(w, p) = p−2 (1 + η)pλ+λ. According

to Lemma 5.22, p4 > 4p1 and we need to examine the following subcases.

p Subcase p ∈ (4p1, p4): According to Lemma 5.6, 4p1 > 1 + 2η − 2 η(1 + η) λ

∗ ∗ and according to Lemma 5.5 part (a), u µ[0,λ)(w, p) > u (µλ(w, p)), therefore the agent’s optimal service capacity is µ∗(w, p) = 0 and u(µ∗(w, p)) = w − p.

77 Subcase

u( µ ) Subsubcase Subsubcase h rnia ol o rps otati hscs,bcuenn fteeca- these of none because case, this in contract a propose not would principal installing the about indiﬀerent is agent µ the Therefore rate. utility installing that dicating ∗

( 0.95 0.96 0.97 0.98 0.99 ,p w, tract. rc foﬀered. if tract 0 p = ) = iue51:Ilsrto ftefrsof forms the of Illustration 5.12: Figure λ p (a) p 4 = η : 1+ (1 2 w w p

codn oLma55pr (c), part 5.5 Lemma to According , λ= λ ∈ 0.01 ≥ ∈ (0 µ

η u( µ ) ,

w , (0 ) p = 4 1 p

p : ,

0.84 0.85 0.86 0.87 0.88 0.89 0.90 p 4 p , 1 = λ ] u 1.1 0 µ λ ( − ): µ [0 ∗ λ ∗ ,λ u ( ) ,p w, n nsc case such in and ( ( λ µ ,p w, ∗ 5 = η (c) ( λ )) ,p w, 2

or ) , λ= λ ≥ p > p 78 0.01 )) µ

µ ,teeoeteaetwudacp h con- the accept would agent the therefore 0, , u( )

w µ = 1 4 λ ∗ <

, 0.915 0.920 0.925 0.930 0.935 0.940 0.945 0.950

p ( = ,p w, 10 0 ,teeoeteaetrjcstecon- the rejects agent the therefore 0, λ ed otesm gn’ expected agent’s same the to leads ) u u ( λ (b) µ u ( µ ∗ 5 ( when ) = η λ p ,p w, µ 2

, ∈ [0 ∗ λ= λ ,λ 0.01 (4 ); ) µ

( p ,

w ,p w, ,p w, 1 η = p , 1

≥ p 4 = ) 5 = ) ) λ 1 = w µ u − ∗ ( ( 5 ,p w, µ w λ λ ∗ 4 ( However . ,p w, or 0 = ) ) in- )), w ∈ (0, p) Reject.

p ∈ (0, 4p1]

w ≥ p µ∗ = 0

w ∈ (0, p) Reject.

η ≥ 1 p ∈ (4p1, p4)

w ≥ p µ∗ = 0

w ∈ (0, w4) Reject.

p > 4p1 p = p4

∗ ∗ p w ≥ w4 µ = 0 or µ = (1 + η)p4λ − λ

w ∈ 0, 2p(1 + η)pλ − λ Reject.

p > p4

w ≥ 2p(1 + η)pλ − λ µ∗ = p(1 + η)pλ − λ

Figure 5.13: Structure of the proof for Proposition 5.23 when η ≥ 1

pacities leads to admissible solutions (see Deﬁnition 3.3). For proof see Proposition p 5.24. According to Lemma 5.4, p4 > 4p1 ⇒ w4 = 2 (1 + η)p4λ − λ > 0.

∗ Subsubcase w ∈ (0, w4): u(µ (w, p)) < 0, therefore the agent rejects the contract.

∗ Subsubcase w ≥ w4: u(µ (w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

∗ ∗ Subcase p > p4: From Lemma 5.5 part (b), u (µλ(w, p)) > u µ[0,λ)(w, p) , thus the agent’s optimal capacity is µ∗(w, p) = p(1 + η)pλ − λ and u(µ∗(w, p)) = w − p p 2 (1 + η)pλ+λ. According to Lemma 5.4, p > p4 > 4p1 ⇒ 2 (1 + η)pλ−λ > 0, therefore we further examine the following subcases. Subsubcase w ∈ 0, 2p(1 + η)pλ − λ : u(µ∗(w, p)) < 0, thus the agent rejects the contract.

Subsubcase w ≥ 2p(1 + η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

In summary, given exogenous market conditions such that a mutually beneﬁcial contract

79 with a strongly risk-averse agent exists (see Theorem 5.27 later), only one formula is needed for the agent to compute his optimal service capacity: µ∗(w, p) = p(1 + η)pλ − λ > 0.

The conditions when a strongly risk-averse agent accepts the contract can be depicted by the shaded areas in Figure 5.14, where η = 2. The two shaded areas with diﬀerent grey scales represent conditions (5.25) and (5.27) under which the agent accepts the contract

but responds diﬀerently. The lower bound function of the shaded areas (denoted by w0(p))

represents the set of oﬀers that give the agent zero expected utility rate. w0(p) is deﬁned as follows:

  p when p ∈ (0, p4] w (p) = 0 p  2 (1 + η)pλ − λ when p > p4

Since lim − w0(p) = lim + w0(p) = p4, w0(p) is continuous everywhere over interval p ∈ p→p4 p→p4 √ √ √ +. However since lim − dw0(p)/dp = 1 6= 1 + η 1 + η + η = lim + dw0(p)/dp, R p→p4 p→p4

w0(p) is not diﬀerentiable at p = p4.

5.2.1 Sensitivity analysis of a strongly risk-averse agent’s optimal strategy

Since the principal does not propose a contract that will be responded to with zero service capacity, therefore the only viable case is when the agent in response installs positive service capacity: µ∗(w, p) = p(1 + η)pλ − λ. The w is bounded below by 2p(1 + η)pλ − λ = ηpP (1) + pP (1) + µ∗(w, p) (see (5.27)), with ηpP (1) representing the expected risk rate perceived by the agent and pP (1) representing the expected penalty rate charged by the principal. It indicates that the agent has to be reimbursed for the expected risk rate, the expected penalty rate, and the cost of the optimal service capacity.

The optimal service capacity µ∗(w, p) = p(1 + η)pλ − λ depends on the penalty rate p, the

80 w

∗ ∗ µ =0 µ = (1+ η )p λ − λ

w=2 (1+ η )p λ − λ

w=p

0 p 0 p4

Figure 5.14: Conditions when a strongly risk-averse agent accepts the contract with η = 2

failure rate λ and the risk coeﬃcient η. Its derivatives are:

s r s ∂µ∗ (1 + η)λ ∂µ∗ (1 + η)p ∂µ∗ pλ = > 0, = − 1 and = > 0 ∂p 4p ∂λ 4λ ∂η 4(1 + η)

The derivatives indicate that given λ and η the agent will increase the service capacity when the penalty rate increases. Note that p(1 + η)pλ − λ, as a function of λ, increases when (1 + η)p/4 > λ. From conditions (5.26) and (5.27) the agent installs service capac- p ity (1 + η)pλ − λ when p ≥ p4 and from Lemma 5.22 we have p4 > 4p1. Therefore we

∗ have p > 4p1 = 4λ/(1 + η) ⇒ (1 + η)p/4 > λ ⇒ ∂µ /∂λ > 0. Thus, given p and η, the agent will increase the service capacity when the failure rate increases. Given the penalty rate and the failure rate, the agent will increase the service capacity as his risk-aversion increases.

The agent’s optimal expected utility rate when installing capacity µ∗(w, p) = p(1 + η)pλ−λ

81 ∗ ∗ p is uA ≡ uA(µ (w, p); w, p) = w − 2 (1 + η)pλ + λ, and it depends on w, p, η and λ. Note ∗ ∗ p ∗ p that ∂uA/∂w = −1 < 0, ∂uA/∂p = − (1 + η)λ/p < 0 and ∂uA/∂η = − pλ/(1 + η) < 0, indicating that the agent’s optimal expected utility rate decreases with the compensation

∗ √ √ √ rate, the penalty rate and his risk intensity. Note that ∂uA/∂λ = − p − p1 / p1, and √ √ from Proposition 5.23 p ≥ p4 > 4p1 ⇒ p− p1 > 0, therefore the agent’s optimal expected utility rate also decreases with the failure rate.

Summary: Recall that a risk-neutral agent would accept a contract, install µ∗(w, p) = 0 and receive u(µ∗(w, p); w, p) = w − p given the set of oﬀers {(w, p): p ∈ (0, λ], w ≥ p}. √ Given the set of oﬀers (w, p): p > λ, w ≥ 2 pλ − λ he would accept the contract, install √ √ µ∗(w, p) = pλ − λ and receive expected utility rate u(µ∗(w, p); w, p) = w − 2 pλ + λ. By comparing the optimal solutions of a strongly risk-averse agent with that of a risk-neutral agent, two conclusions are drawn:

1. Given a λ, the principal must set a higher p in order to induce a strongly risk-averse agent to install a positive service capacity versus a risk-neutral agent (p > λ for risk- neutral agent, p > 1 + 2η + 2pη(1 + η) λ for strongly risk-averse agent).

2. With the same w and p, given that the agent accepts the contract and installs a positive service capacity, the expected utility rate of a strongly risk-averse agent decreases with respect to η since

s ∂u pλ u µ∗(w, p) = p(1 + η)pλ − λ = w − 2p(1 + η)pλ + λ ⇒ = − < 0 ∂η 1 + η

Therefore a strongly risk-averse agent is always strictly worse oﬀ than a risk-neutral agent.

Compared to a weakly risk-averse agent, a strongly risk-averse agent has fewer options of positive optimal service capacities (he will never install µ∗(w, p) = p(1 − η)pλ − λ when η ∈ [4/5, 1)) because the perceived risk rate is high enough such that the only reasonable choice is to invest more in service capacity to compensate for the risk.

82 5.2.2 Principal’s optimal strategy

We now derive the principal’s optimal strategy while anticipating the agent’s optimal response µ∗(w, p). For that the principal solves the optimization problem:

∗ ∗ rµ (w, p) pλ max ΠP (w, p; µ (w, p)) = max − w + (5.28) w>0,p>0 w>0,p>0 λ + µ∗(w, p) λ + µ∗(w, p)

∗ ∗ ∗ and recovers the optimizing values: (w , p ) = argmaxw>0,p>0 ΠP (w, p; µ (w, p)).

Before deriving the principal’s optimal strategy, we reexamine the case when the principal offers p = p4 and w ≥ w4, under which the agent is indifferent regarding two different service capacities which however effect the principal differently. Since any selected solution ((w, p), µ) has to be an admissible solution (see Definition 3.3) we test the solutions’ membership in Proposition 5.24.

Proposition 5.24. Suppose a strongly risk-averse agent. Assume that the principal’s potential oﬀers are constrained to set {(w, p): p = p4, w ≥ w4}.

(a) If r ∈ (0, p4], then the principal does not propose a contract.

p (b) If r > p4, the agent installs (1 + η)p4λ − λ if oﬀered a contract.

2 Proof. For w ≥ w4 we have ∂ΠP (w, p4; µ)/∂µ = (r − p4)λ/(λ + µ) . Define µL ≡ 0 and p µH ≡ (1 + η)p4λ − λ and note that µH > µL. If r ∈ (0, p4), then ∂ΠP /∂µ < 0, therefore ((w, p4), µL) ((w, p4), µH ) and the agent would install µL if offered a contract since ((w, p4), µH ) is not an admissible solution. However in such case the principal’s expected profit rate is ΠP (w, p4; µL) = −w + p4 ≤ 0, therefore the principal would not propose a contract. If r = p4, then ∂ΠP /∂µ = 0, therefore the agent installs either µL or µH if offered a contract. However in such case the principal’s expected profit rate is

ΠP (w, p4; µL) = ΠP (w, p4; µH ) = −w + p4 ≤ 0, therefore the principal would not propose a contract. If r > p4, then ∂ΠP /∂µ > 0 and ((w, p4), µH ) ((w, p4), µL). If the principal oﬀers a contract (where the conditions will be discussed in detail in Theorem 5.27 that follows), then by Deﬁnition 3.3 only µH leads to admissible solutions.

83 Notation:

p r4 ≡ 1 + 2η + 2 η(1 + η) (1 + 2η)λ = (1 + 2η)p4 (5.29)

r4 is a function of λ and η however we suppress the parameters (λ, η).

Next we state several technical lemmas.

√ √ Lemma 5.25. Consider maxx≥ p4 f(x) where f(x) = r + λ − p1 ((1 + 2η)x + r/x) and

∗ √ denote x = argmaxx≥ p4 f(x). The solutions to this optimization problem are

∗ √ (a) x = p4 if r ∈ (0, r4].

∗ p (b) x = r/(1 + 2η) if r > r4. √ Proof. Note that f(x) is continuous and diﬀerentiable for x ≥ p4: √ df(x) √ r d2f(x) 2r p = − p 1 + 2η − and = − 1 < 0 dx 1 x2 dx2 x3 √ df(x) p1 df(x) √ = (r − r4) and lim = −(1 + 2η) p1 < 0 dx √ p x→+∞ dx x= p4 4

√ ∗ √ If r ∈ (0, r4], then f(x) is decreasing for x ≥ p4, therefore x = p4. If r > r4, then f(x) √ is increasing in the neighborhood above x = p4 and decreasing when x approaches +∞.

2 2 ∗ √ Note that d f(x)/dx < 0, therefore x > p4 satisﬁes ﬁrst order condition df(x)/dx|x=x∗ = 0 ⇒ x∗ = pr/(1 + 2η).

p Lemma 5.26. Let η > 0 and λ > 0, then (1 + 2η)p4 > 1 + 3η + 2 η(1 + 2η) λ/(1 + η).

Proof. Let η > 0 and λ > 0, then we have

√ √ √ η p1 + 2η − p1 + η + η > 0 ⇔ 0 > − ηp1 + 2η + ηp1 + η − η √ √ ⇔ p1 + 2ηp1 + η > p1 + 2η + η p1 + η − η √ √ √ 1 + 2η + η ⇔ p1 + η + η p1 + 2η > √ 1 + η

p by squaring both sides we have (1 + 2η)p4 > 1 + 3η + 2 η(1 + 2η) λ/(1 + η).

84 The principal’s optimal strategy is derived in Theorem 5.27. Recall that Proposition 5.23 describes the agent’s optimal response to each pair of compensation rate and penalty rate

2 (w, p) ∈ R+. Since the principal will not propose a contract that is going to be rejected by 2 a strongly risk-averse (SRA) agent, therefore Theorem 5.27 only considers pairs (w, p) ∈ R+ such that the agent receives a non-negative expected utility rate. Deﬁne

D(5.25) ≡ {(w, p) that satisﬁes (5.25) when η ≥ 4/5}

D(5.26) ≡ {(w, p) that satisﬁes (5.26) when η ≥ 4/5} (5.30) D(5.27) ≡ {(w, p) that satisﬁes (5.27) when η ≥ 4/5}

DSRA ≡ D(5.25) ∪ D(5.26) ∪ D(5.27)

Theorem 5.27. Given a strongly risk-averse agent and (w, p) ∈ DSRA.

(a) If r ∈ (0, p4], then the principal does not propose a contract.

(b) If r ∈ (p4, r4], then the principal’s oﬀer and the capacity installed by the agent are

∗ ∗ ∗ ∗ ∗ p (w , p ) = (w4, p4) and µ (w , p ) = (1 + η)p4λ − λ (5.31)

and the principal’s expected proﬁt rate is

∗ ∗ ∗ ∗ ∗ √ √ r ΠP (w , p ; µ (w , p )) = r + λ − p1 (1 + 2η) p4 + √ (5.32) p4

s ! s (1 + η)rλ r (1 + η)rλ (w∗, p∗) = 2 − λ, and µ∗(w∗, p∗) = − λ (5.33) 1 + 2η 1 + 2η 1 + 2η

and the principal’s expected proﬁt rate is

s (1 + 2η)rλ Π (w∗, p∗; µ∗(w∗, p∗)) = r − 2 + λ (5.34) P 1 + η

Proof. The structure of the proof is depicted in Figure 5.15.

85 r ∈ (0, p4] No contract oﬀered.

(w, p) ∈ D(5.25) ∪ D(5.26) w∗ = w and p∗ = p r > p 4 4 4 and µ∗ = p(1 + η)p λ − λ Risk-Neutral 4 Principal with Strongly Risk-Averse r ∈ (0, p4] No contract oﬀered. Agent

∗ ∗ w = w4 and p = p4 (w, p) ∈ D(5.26) ∪ D(5.27) r ∈ (p4, r4] ∗ p and µ = (1 + η)p4λ − λ

s (1 + η)rλ r w∗ = 2 − λ and p∗ = 1 + 2η 1 + 2η r > r4 s (1 + η)rλ and µ∗ = − λ 1 + 2η

Figure 5.15: Structure of the proof for Theorem 5.27

Case (w, p) ∈ D(5.25) ∪ D(5.26): According to Proposition 5.23 part (a), if (w, p) ∈ D(5.25), then in case the principal makes an oﬀer, the agent accepts the contract but does not

∗ install any service capacity. Since ∂ΠP /∂w = −1 < 0, therefore w = p and from

∗ ∗ ∗ ∗ equation (4.3) ΠP (w , p; µ (w , p)) = −w + p = −p + p = 0. According to Proposition

5.23 part (b) and Proposition 5.24, if (w, p) ∈ D(5.26) (which implies p = p4), then the p principal does not propose a contract if r ∈ (0, p4], or installs (1 + η)p4λ − λ in case

∗ the principal makes an oﬀer when r > p4. Since ∂ΠP /∂w = −1 < 0, therefore w = w4.

From Proposition 5.23 part (b), if the principal oﬀers a contract with (w, p) = (w4, p4),

∗ ∗ p then the agent installs either µ (w4, p4) = 0 or µ (w4, p4) = (1 + η)p4λ − λ. Denote

∗ L the principal’s expected proﬁt rate when (w, p) = (w4, p4) and µ (w, p) = 0 by ΠP (p4), ∗ and denote the principal’s expected proﬁt rate when (w, p) = (w4, p4) and µ (w, p) =

p H (1 + η)p4λ − λ by ΠP (p4). By plugging the value of w, p and µ into equation (4.3):

L ΠP (p4) = −w4 + p4 = 0 (5.35) √ √ H √ √ r p4 − p1 ΠP (p4) = r + λ − p1 (1 + 2η) p4 + √ = √ (r − p4) (5.36) p4 p4

∗ ∗ ∗ In such case the principal’s optimization problem is maxp∈(0,p4] ΠP (w , p; µ (w , p))

86 where:

 ∗ ∗ ∗  0 for p ∈ (0, p4) ΠP (w , p; µ (w , p)) = L H  max ΠP (p4), ΠP (p4) for p = p4

Subcase r ∈ (0, p4]: By Proposition 5.24 part (a), the principal does not oﬀer a contract.

∗ Subcase r > p4: According to Lemma 5.25 part (b), p = p4 and according to Propo-

∗ ∗ ∗ ∗ ∗ sition 5.24 part (b) the principal’s expected proﬁt rate ΠP (w , p ; µ (w , p )) =

H L ∗ ΠP (p4) > ΠP (p4) = 0. Thus the principal proposes a contract with w = w4 and ∗ ∗ ∗ ∗ p p = p4 that induces the agent to install µ (w , p ) = (1 + η)p4λ − λ.

Case (w, p) ∈ D(5.26) ∪ D(5.27): According to Proposition 5.23 part (c), if (w, p) ∈ D(5.27), then in case the principal makes an oﬀer, the agent accepts the contract and installs

p ∗ p (1 + η)pλ − λ. Since ∂ΠP /∂w = −1 < 0, therefore w = 2 (1 + η)pλ − λ. Ac-

cording to Proposition 5.23 part (b) and Proposition 5.24, if (w, p) ∈ D(5.26) (which

implies p = p4), then the principal does not propose a contract if r ∈ (0, p4], or installs p (1 + η)p4λ − λ in case the principal makes an oﬀer when r > p4. Since ∂ΠP /∂w =

∗ −1 < 0, therefore w = w4. From Proposition 5.23 part (b), if the principal oﬀers a con-

∗ ∗ tract with (w, p) = (w4, p4), then the agent installs either µ (w4, p4) = 0 or µ (w4, p4) =

p ∗ ∗ ∗ (1 + η)p4λ − λ. The principal’s optimization problem is maxp≥p4 ΠP (w , p; µ (w , p)) where:

 L H  max ΠP (p4), ΠP (p4) , for p = p4 ∗ ∗ ∗  ΠP (w , p; µ (w , p)) = √ √ r  r + λ − p1 (1 + 2η) p + √ , for p > p4  p

√ √ √ √ Deﬁne x ≡ p, the expression r + λ − p1 (1 + 2η) p + r/ p can be restated as √ √ f(x) = r + λ − p1 ((1 + 2η)x + r/x). Maximizing f(x) with respect for x ≥ p4 is √ √ √ equivalent to maximizing r +λ− p1 (1 + 2η) p + r/ p for p ≥ p4 in the sense that

!2 √ √ r argmax r + λ − p (1 + 2η) p + = argmax f(x) 1 √ √ p≥p4 p x≥ p4

87 Since r4 = (1 + 2η)p4 > p4, we examine the following subcases.

Subcase r ∈ (0, p4]: By Proposition 5.24 part (a), the principal does not propose a contract.

∗ Subcase r ∈ (p4, r4]: According to Lemma 5.25 part (a), p = p4. According to

∗ ∗ ∗ ∗ ∗ H L Proposition 5.24 part (b), ΠP (w , p ; µ (w , p )) = ΠP (p4) > ΠP (p4) = 0. There- ∗ ∗ fore the principal proposes a contract with w = w4 and p = p4 that induces the

∗ ∗ ∗ p agent to install µ (w , p ) = (1 + η)p4λ − λ.

∗ Subcase r > r4: According to Lemma 5.25 part (b), p = r/(1 + 2η) and the princi-

∗ ∗ ∗ ∗ ∗ p pal’s expected proﬁt rate is ΠP (w , p ; µ (w , p )) = r−2 (1 + 2η)rλ/(1 + η)+λ.

∗ ∗ ∗ ∗ ∗ According to Lemmas 5.14 and 5.26 ΠP (w , p ; µ (w , p )) > 0, therefore the principal proposes a contract with w∗ = 2p(1 + η)rλ/(1 + 2η)−λ and p∗ = r/(1+2η) that induces the agent to install capacity µ∗(w∗, p∗) = p(1 + η)rλ/(1 + 2η) − λ.

To summarize, when r ∈ (0, p4], the principal does not propose a contract. This case is

∗ ∗ covered by Theorem 5.27 (a). If r ∈ (p4, r4], then the principal oﬀers (w , p ) = (w4, p4)

∗ ∗ ∗ p and the agent installs capacity µ (w , p ) = (1 + η)p4λ − λ. This case is covered by

Theorem 5.27 (b). Finally if r > r4, then according to Lemma 5.25 part (b), the principal oﬀers (w∗, p∗) = 2p(1 + η)rλ/(1 + 2η) − λ, r/(1 + 2η) and the agent installs capacity µ∗(w∗, p∗) = p(1 + η)rλ/(1 + 2η) − λ. This case is covered by Theorem 5.27 (c).

Theorem 5.27 indicates that the existence of a beneﬁcial contract for strongly risk-averse agent is determined exogenously by the market (the revenue rate r), the nature of the equipment (the failure rate λ) and the nature of the agent (the risk coeﬃcient η).

5.3 Risk-averse agent – a summary

Recall the definition of p2, p3, p4, r2, r3 and r4 from (5.5), (5.12), (5.24) and (5.29). The conditions that a principal makes an offer to a risk-averse agent is depicted by the shaded areas in Figure 5.16. The horizontal axis represents the agent’s risk coefficient, and the vertical axis represents the revenue rate generated by the principal’s unit, which is exogenously

88 determined by the market. The principal makes different offers to the agent when (r, η) is in the five shaded areas with different gray scales. We define

   p3 for η ∈ (0, 4/5)  r3 for η ∈ (0, 4/5) p34 ≡ and r34 ≡  p4 for η ≥ 4/5  r4 for η ≥ 4/5

Note that limη→(4/5)− r34 = 13λ = limη→(4/5)+ r34, and note that limη→(4/5)− ∂r34/∂η =

125λ/6 = limη→(4/5)+ ∂r34/∂η, therefore r34 is continuous and diﬀerentiable everywhere over R+. Since limη→(4/5)− p34 = 5λ = limη→(4/5)+ p34 and limη→(4/5)− ∂p34/∂η = 25λ/6 = limη→(4/5)+ ∂p34/∂η, therefore p34 is continuous and diﬀerentiable everywhere over R+ as well.

Furthermore, note that limη→0+ p3 = limη→0+ r2 = limη→0+ r3 = 4λ and limη→(4/5)− r2 = limη→(4/5)− p2 = 5λ.

∗ r r p = 4 1+2 η

13 λ

∗ p = p4

∗ r3 p = p3

r2 p4 ∗ p = p or p p3 cu 3 5 λ 4 λ ∗ p = pcu p2 λ η 4 0 Weakly Risk−Averse Strongly Risk−Averse 5

Figure 5.16: Conditions when a risk-neutral principal makes oﬀers to a risk-averse agent

89 5.3.1 Sensitivity analysis of optimal strategies in high revenue industry

The revenue rate r is determined exogenously by the market, and consider r > r34 (high revenue rate). (5.22) and (5.33) are the second-best solutions when the agent is weakly and strongly risk-averse respectively, and they have the same functional form.

The risk-averse agent’s optimal strategy is examined ﬁrst. The optimal service capacity of a risk-averse agent (µ∗ = p(1 + η)rλ/(1 + 2η)−λ) is a function of r, λ, and η. The derivatives of µ∗ with respect to the parameters are ∂µ∗/∂r = p(1 + η)λ/ 2p(1 + 2η)r > 0, √ ∂µ∗/∂λ = p(1 + η)r/ 2p(1 + 2η)λ −1 and ∂µ∗/∂η = − rλ/ 2p(1 + η)(1 + 2η)3 < 0. The derivatives indicate that given λ and η, the optimal capacity increases when the revenue rate increases, and therefore the average downtime of the principal’s unit decreases. Given the revenue rate and the failure rate, the average downtime of the principal’s equipment will increase as the agent becomes more risk-averse. Note that µ∗ = p(1 + η)rλ/(1 + 2η) − λ, as a function of λ, increases when (1 + η)r/4(1 + 2η) > λ. According to Lemma 5.3 we have p3 > 4p1 and according to Lemma 5.22 we have p4 > 4p1. Furthermore, since we assume that r > r34, then r > r3 = (1 + 2η)p3 ⇒ r/(1 + 2η) > p3 ⇒ r/(1 + 2η) > 4p1 = 4λ/(1 + η) ⇒

(1 + η)r/4(1 + 2η) > λ if η ∈ (0, 4/5) and r > r4 = (1 + 2η)p4 ⇒ r/(1 + 2η) > p4 ⇒ r/(1 + 2η) > 4p1 = 4λ/(1 + η) ⇒ (1 + η)r/4(1 + 2η) > λ if η ≥ 4/5. Thus, given the revenue rate and the risk coeﬃcient, the failure rate is low compared to the revenue rate, and the average downtime of the principal’s equipment will decrease when the failure rate increases.

Next we examine the principal’s optimal strategy. Note that the optimal compensation rate of a principal with a risk-averse agent (w∗ = 2p(1 + η)rλ/(1 + 2η) − λ) is a function of r, λ, and η. The derivatives of w∗ with respect to the parameters are ∂w∗/∂r = p(1 + η)λ/(1 + 2η)r > 0, ∂w∗/∂λ = p(1 + η)r/(1 + 2η)λ − 1 and ﬁnally ∂w∗/∂η = −prλ/(1 + η)(1 + 2η)3 < 0. The derivatives indicate that given the λ and η, the optimal compensation rate increases with respect to r. Given the r and the λ, the optimal compensation rate decreases as the agent becomes more risk-averse. Note that w∗ = 2p(1 + η)rλ/(1 + 2η) − λ, as a function of λ, increases when (1 + η)r/(1 + 2η) > λ.

90 According to Lemma 5.3 we have p3 > 4p1 > p1 and according to Lemma 5.22 we have

p4 > 4p1 > p1. Furthermore, since we assume that r > r34, then r > r3 = (1 + 2η)p3 ⇒ r/(1 + 2η) > p3 ⇒ r/(1 + 2η) > p1 = λ/(1 + η) ⇒ (1 + η)r/(1 + 2η) > λ if η ∈ (0, 4/5) and

r > r4 = (1+2η)p4 ⇒ r/(1+2η) > p4 ⇒ r/(1+2η) > p1 = λ/(1+η) ⇒ (1+η)r/(1+2η) > λ if η ≥ 4/5. Therefore the failure rate is low compared to the revenue rate ((1+η)r/(1+2η) > λ ⇒ ∂w∗/∂λ > 0), indicating that the w∗ increases with respect to the failure rate.

The principal’s optimal p∗ given a risk-averse agent (p∗ = r/(1 + 2η)) is a function of r and η. Note that p∗ is independent of the failure rate λ under the assumption that the revenue rate is suﬃciently high compared to the failure rate. The derivatives of p∗ with respect to the parameters are ∂p∗/∂r = 1/(1 + 2η) > 0 and ∂p∗/∂η = −2r/(1 + 2η)2 < 0. The derivatives indicate that given the risk η, the optimal penalty p∗ increases with respect to r, and given r, the p∗ decreases with respect to η.

The principal’s optimal expected proﬁt rate given a risk-averse agent

∗ ∗ ∗ ∗ ∗ ∗ p ΠP ≡ ΠP (w , p ; µ (w , p )) = r − 2 (1 + 2η)rλ/(1 + η) + λ

∗ is a function of r, λ, and η. The derivatives of ΠP with respect to these parameters are ∗ p ∗ p ∗ ∂ΠP /∂r = 1 − (1 + 2η)λ/(1 + η)r, ∂ΠP /∂λ = 1 − (1 + 2η)r/(1 + η)λ, ∂ΠP /∂η = −prλ/(1 + 2η)(1 + η)3 < 0. The derivatives indicate that given r and λ, the principal’s optimal expected proﬁt rate decreases as the agent becomes more risk-averse. Note that

∗ p ΠP = r−2 (1 + 2η)rλ/(1 + η)+λ, as a function of λ, decreases when (1+2η)r/(1+η) > λ, and as a function of r, increases when r > (1+2η)λ/(1+η). According to Lemma 5.3 we have

p3 > 4p1 > p1 and according to Lemma 5.22 we have p4 > 4p1 > p1. Furthermore, since we

assume that r > r34, then r > r3 = (1+2η)p3 ⇒ r/(1+2η) > p3 ⇒ r/(1+2η) > p1 = λ/(1+η)

if η ∈ (0, 4/5) and r > r4 = (1 + 2η)p4 ⇒ r/(1 + 2η) > p4 ⇒ r/(1 + 2η) > p1 = λ/(1 + η) if η ≥ 4/5. Therefore given a λ and an η, the revenue rate is high compared to the fail-

∗ ure rate (r > (1 + 2η)λ/(1 + η) ⇒ ∂ΠP /∂r > 0), thus the principal’s optimal expected

91 proﬁt rate increases with respect to the revenue rate. Note that since η > 0, therefore r > (1 + 2η)λ/(1 + η) > (1 + η)λ/(1 + 2η) ⇒ (1 + 2η)r/(1 + η) > λ, which implies that given an r and η, the failure rate is low compared to the revenue rate ((1 + 2η)r/(1 + η) > λ ⇒

∗ ∂ΠP /∂λ < 0), therefore the principal’s optimal expected proﬁt rate decreases with respect to λ.

5.3.2 The second-best solution in high revenue industry

By comparing the second-best solution given a risk-averse agent ((5.22) and (5.33)) with the second-best given a risk-neutral agent when r > r34, four conclusions are drawn.

1. The optimal w∗ and the optimal p∗ decrease when the agent is risk-averse versus risk- √ neutral agent (w∗ : 2 rλ − λ > 2p(1 + η)rλ/(1 + 2η) − λ and p∗ : r > r/(1 + 2η)). It indicates that the risk adds an incentive for the agent to install a higher service capacity by coupling it to the penalty charge collected by the principal. √ 2. The principal is worse off with a risk-averse agent than a risk-neutral agent (r−2 rλ+ λ > r−2p(1 + 2η)rλ/(1 + η)+λ), as well as with an agent whose action is contractible (recall that the principal receives the same expected profit rate with a risk-neutral agent in first-best and second-best setting). This conclusion is consistent with Proposition 3 part (ii) in Harris and Raviv (1978). The principal’s loss can be explained as follows: On one hand, the decrease in the agent’s optimal capacity when risk-averse reduces the revenue performance of the principal’s unit. At the same time, the monetary equivalency of the risk perceived by the agent is not channeled to the principal, although from the agent’s perspective it serves as part of the penalty charge. √ 3. The µ∗ of a risk-averse agent is strictly less than that of a risk-neutral agent ( rλ−λ > p(1 + η)rλ/(1 + 2η) − λ). Recall that when the agent is risk-neutral, the µ∗ in the second-best solution is the same as that in the first-best solution, indicating that the unobservability of the agent’s service capacity does not contribute to the decrease of the optimal service capacity. When the agent is risk-averse, he compensates for his risk by reducing µ.

92 4. Given the compensation rate and penalty rate, both weakly and strongly risk-averse agents are worse oﬀ compared to a risk-neutral agent.

To summarize, for a principal with high revenue generating unit, agent’s risk-aversion reduces the eﬃciency of the contract (compared to the ﬁrst-best contract), and therefore it reduces the social welfare.

6 Risk-Seeking Agent

In previous section we represented agent’s perceived risk by a measure that reflects the dispersion of his revenue stream. Although the dispersion of possible outcomes has been widely used as the measure of risk (Pratt 1964, Rothschild and Stiglitz 1970, Stiglitz 1974, Levy 1992, Fukunaga and Huffman 2009 and Lewis and Bajari 2014) it fails to capture observable behavior in risky settings. In this section we extend our principal-agent analysis to risk- seeking agent. We note that there is an ongoing evaluation of risk attitudes in an attempt to explain peoples’ behavior when faced with risky choices. For instance Prospect Theory claims to offer a better model that covers discrepancies observed elsewhere (Kahneman and Tversky 1979, Tversky and Kahneman 1992). Prospect Theory claims that people are less sensitive to the variation of the probability of outcomes compared to the expectation, and losses loom larger than gains. Furthermore, empirical evidences indicates that decision mak- ers prefer expressions of risk in terms of the expected value at stake, and they appear to be risk-averse when dealing with a risky alternative whose possible outcomes are generally good and tend to be risk-seeking when dealing with a risky alternative whose possible outcomes are generally poor (March and Shapira 1987, Filiz-Ozbay et al. 2013, Page et al. 2014).

In our principal-agent setting with a risk-seeking agent we propose that an agent perceives a greater loss when he is charged a larger penalty rate for each unit of downtime and also when the probability of being in the failed state goes up. The agent’s penalty rate at any point of time can be modeled as pB where B is a Bernoulli random variable that takes value 0 with probability P (0) = µ/(λ + µ) and value 1 with probability P (1) = λ/(λ + µ). For

93 simplicity denote momentarily a ≡ P (1). In this section we adopt the following risk measure:

1 r(a) ≡ p a − for a ∈ [0, 1] 2 +

We note that R(pB) ≡ r(a) satisﬁes the properties of monotonicity, sub-additivity and positive homogeneity of a coherent risk measure but fails to satisfy the property of trans- lation invariance, since R(pB) is independent of the expectation of pB (Artzner et al., 1999).

Risk premium of a risk-seeking agent is the $ value considered by the agent as extra gains to his revenue stream. As a consequence, just for the risk-seeking agent we modify the risk premium deﬁned earlier in (5.1), in a manner that reﬂects the expected amount at stake instead of the dispersion of the revenue stream:

1 λ 1 π(µ, w, p) ≡ −ηp P (1) − = −ηp − (6.1) 2 + λ + µ 2 +

Note that for risk-seeking agent η < 0 ⇒ π(µ, w, p) ≥ 0, and adding such a risk premium to a risk-neutral agent’s expected utility rate (as in (6.2)) implies risk-seeking. Figure 6.1 depicts π(µ, w, p) as a function of P (1) for η = −1.

π ( µ , w , p ) p 2

0 P(1) 0 1 1 2

Figure 6.1: π(µ, w, p) as a function of P (1) when η = −1

The representation of the risk premium in (6.1) is consistent with the properties of risk in the Prospect Theory (Kahneman and Tversky 1979, Tversky and Kahneman 1992) and the empirical ﬁndings in (March and Shapira, 1987): The risk premium is zero when P (1) is

94 lower than 1/2. The risk premium increases with P (1) linearly when P (1) exceeds one half, and reaches its peak when P (1) = 1.

Denote η ≡ −η > 0. Modifying (4.2), the risk-seeking agent’s expected utility rate is:

pλ λ 1 u (µ; w, p) = w − − µ + ηp − for w > 0, p > 0, µ ≥ 0 (6.2) A λ + µ λ + µ 2 + +

Since the analysis is diﬀerent for η ∈ (0, 8/9), η ∈ [8/9, 2) and η ≥ 2, therefore, when η ∈ (0, 8/9) we consider the agent as weakly risk-seeking, when η ∈ [8/9, 2) we consider the agent as moderately risk-seeking, and when η ≥ 2 we consider the agent as strongly risk- seeking. We assume, say for historic reasons, that both the agent and the principal know not only the agent’s type as risk-seeking but also the value of η.

The expression for the principal’s expected revenue rate ΠP (w, p; µ) remains the same as in (4.3).

Before examining the details of the optimal contracts we discuss a potential case of the agent compensating the principal at times during the contract. Such occurrence of utility transfer from an risk-seeking agent to the principal can have one of two forms: either the compensation rate is non-positive (w ≤ 0), or the principal is guaranteed a positive expected revenue rate even with her unit in the failed state forever (−w + p > 0 if µ = 0). Under our setting of undetermined contract horizon it is unrealistic to accept that the agent might compensate the principal when the unit is forever in the failed state. Therefore the likelihood of a non-positive compensation rate (w ≤ 0) has to be ruled out in the definition of the Strategy Set (Definition 3.1). Nevertheless, the possibility of the principal receiving a positive expected revenue rate with a failed unit has to be considered. Therefore we extend the definition of the Set of Admissible Solutions (Definition 3.3) as follows.

Deﬁnition 6.1 (Set of Admissible Solutions). The set of admissible solutions for the principal-agent problem P is the set s(P) of all strategies ((w, p), µ) ∈ S(P) for which:

95 (a) @ ((w0, p0), µ0) ∈ S(P) such that ((w0, p0), µ0) ((w, p), µ) – there is no other strategy that weakly dominates ((w, p), µ).

(b)Π P (w, p; µ) > ΠP and uA(µ; w, p) ≥ uA.

We denote the part inside the brackets in equation (6.2) as

 ηp (1 − η)pλ  w − − − µ, µ ∈ [0, λ]  2 λ + µ u(µ) ≡ pλ (6.3)  w − − µ, µ > λ  λ + µ

Note that u(µ) is diﬀerentiable everywhere for µ ≥ 0 except at µ = λ. When µ ∈ [0, λ):

du(µ) (1 − η)pλ du(µ) 1 − η λ = − 1, lim = p − dµ (λ + µ)2 µ→0+ dµ λ 1 − η du(µ) 1 − η 4λ d2u(µ) 2(1 − η)pλ lim = p − , = − µ→λ− dµ 4λ 1 − η dµ2 (λ + µ)3 and when µ > λ:

du(µ) pλ du(µ) p − 4λ = − 1, lim = dµ (λ + µ)2 µ→λ+ dµ 4λ du(µ) d2u(µ) 2pλ lim = −1 < 0 and = − < 0 µ→+∞ dµ dµ2 (λ + µ)3

The positivity or negativity of the above derivatives indicate the direction of monotonicity and the concavity/convexity of the function u(µ) over [0, λ) and (λ, +∞). Table 6.1 sum-

2 marizes these indicators for various regions of the space R+ for pairs of (η, p). In the table + uµ(·) = limµ→· du/dµ, and uµ(· ) represents the limit of uµ(µ) as µ approaches (·) from

− above, and similarly uµ(· ) represents the limit of uµ(µ) as µ approaches (·) from below.

96 Table 6.1: Indicators of the monotonicity and the concavity/convexity of function u(µ) in (6.3)

over [0, λ) over (λ, +∞) Case u (0+) u (λ−) u (λ+) u (+∞) µ u(µ) is µ µ u(µ) is µ λ p ∈ 0, ≤ 0 Concave < 0 < 0 Concave < 0 1 − η λ † 3 p ∈ , 4λ > 0 Concave < 0 ≤ 0 Concave < 0 η ∈ 0, 1 − η 4 † 4λ p ∈ 4λ, > 0 Concave ≤ 0 > 0 Concave < 0 1 − η 4λ p ∈ , +∞ > 0 Concave > 0 > 0 Concave < 0 1 − η p ∈ (0, 4λ] < 0 Concave < 0 ≤ 0 Concave < 0 λ ‡ 3 p ∈ 4λ, ≤ 0 Concave < 0 > 0 Concave < 0 η ∈ , 1 1 − η 4 λ 4λ ‡ p ∈ , > 0 Concave ≤ 0 > 0 Concave < 0 1 − η 1 − η 4λ p ∈ , +∞ > 0 Concave > 0 > 0 Concave < 0 1 − η p ∈ (0, 4λ]§ < 0 Convex < 0 ≤ 0 Concave < 0 η ∈ [1, +∞) p ∈ (4λ, +∞) < 0 Convex < 0 > 0 Concave < 0

†Note that η ∈ (0, 3/4] ⇒ 4λ/(1 − η) > 4λ ≥ λ/(1 − η).

‡Note that η ∈ (3/4, 1) ⇒ 4λ/(1 − η) > λ/(1 − η) > 4λ.

§Note that η > 1 ⇒ 4λ > 0 > λ/(1 − η) > 4λ/(1 − η).

6.1 Optimal strategies for the weakly risk-seeking agent

Note that agent’s expected utility rate (see (6.2)) increases and principal’s expected profit rate (see (4.3)) decreases in w, therefore for any value of p the principal can maximize her expected profit rate by lowering w yet safeguarding agent’s participation by setting the agent’s expected utility rate equal to his reservation utility rate. Although the principal cannot contract directly on the agent’s service capacity, she anticipates the agent optimizing his expected utility rate when offered a contract. That is, for any w and p values proposed by the principal, the agent computes the µ that maximizes his expected utility rate and subsequently decides whether to accept the contract or not, by solving the following optimization

97 problem:

pλ λ 1 max u(µ) = max w − − µ + ηp − (6.4) µ≥0 µ≥0 λ + µ λ + µ 2 +

∗ The agent’s optimal service capacity is denoted by µ (w, p) = argmaxµ≥0 u(µ).

Before proceeding to derive the agent’s optimal strategy we introduce some notation: √ λ 16 2 − η − 2 1 − η λ p ≡ , p ≡ (6.5) 1 1 − η 2 η2

and the following identity is veriﬁed using the deﬁnition of p2:

ηp w ≡ 2 + 2p(1 − η)p λ − λ = 2pp λ − λ (6.6) 2 2 2 2

Note that p1, p2 and w2 are functions of λ and η. However we suppress (λ, η).

Next we introduce a number of technical lemmas.

Lemma 6.2. Let 1 > η > 0 and λ > 0. p √ 2 4 1 − η/2 − 1 − η λ ηp (a) If > p > 0, then 0 > + 2p(1 − η)pλ − λ. η2 2 p √ 2 4 1 − η/2 − 1 − η λ ηp (b) If p > , then + 2p(1 − η)pλ − λ > 0. η2 2 p √ 2 4 1 − η/2 − 1 − η λ ηp (c) If p = , then + 2p(1 − η)pλ − λ = 0. η2 2 √ √ Proof. Let 1 > η > 0 and λ > 0. Deﬁne x ≡ p and a ≡ λ and restate the expression ηp/2 + 2p(1 − η)pλ − λ as ηx2/2 + 2axp(1 − η) − a2 with x > 0 and a > 0. The solutions to the quadratic equation ηx2/2 + 2axp(1 − η) − a2 = 0 for x are √ √ 0 > −2 p1 − η/2 + 1 − η a/η and 2 p1 − η/2 − 1 − η a/η > 0. Therefore, if √ √ 2 2 p1 − η/2 − 1 − η a/η > x > 0, or equivalently, 4 p1 − η/2 − 1 − η a2/η2 >

98 √ √ x2 > 0, then 0 > ηx2/2 + 2axp(1 − η) − a2. Replacing x by p and a by λ we obtain (a). The proofs for (b) and (c) are similar.

√ 2 Lemma 6.3. Let 1 > η > 0 and λ > 0, then λ/(1 − η) > 4 p1 − η/2 − 1 − η λ/η2.

Proof. Let 1 > η > 0 and λ > 0, then we have

0 > η2 − 2η ⇔ η2 − 4η + 4 > 2η2 − 6η + 4

⇔ (2 − η)2 > 4(1 − η/2)(1 − η)

⇔ 2 − η > 2p1 − η/2p1 − η

⇔ η > 2p1 − η/2p1 − η − 2(1 − η) ⇔ 1/p1 − η > 2 p1 − η/2 − p1 − η /η 2 ⇔ λ/(1 − η) > 4 p1 − η/2 − p1 − η λ/η2

Lemma 6.4. Let 2 > η > 0 and λ > 0. 2λ 2λ η √ (a) If √ > p > √ , then 0 > 1 − p − 2 pλ + λ. 2 + η − 2 2η 2 + η + 2 2η 2 2λ 2λ η √ (b) If √ > p > 0 or p > √ , then 1 − p − 2 pλ + λ > 0. 2 + η + 2 2η 2 + η − 2 2η 2 2λ 2λ η √ (c) If p = √ or p = √ , then 1 − p − 2 pλ + λ = 0. 2 + η + 2 2η 2 + η − 2 2η 2 √ √ Proof. Let 2 > η > 0 and λ > 0. Deﬁne x ≡ p and a ≡ λ and restate the expression √ (1 − η/2)p − 2 pλ + λ as (1 − η/2)x2 − 2ax + a2 with x > 0 and a > 0. The solutions √ √ √ to the quadratic equation (1 − η/2)x2 − 2ax + a2 = 0 for x are 2a/ 2 + η > 0 √ √ √ √ √ √ √ √ √ and 2a/ 2 − η > 0. Therefore if 2a/ 2 − η > x > 2a/ 2 + η, or √ √ equivalently, 2a2/ 2 + η − 2 2η > x2 > 2a2/ 2 + η + 2 2η, then 0 > (1 − η/2)x2 − √ √ 2ax + a2. Replacing x by p and a by λ we obtain (a). The proofs for (b) and (c) are similar. √ Lemma 6.5. Let η > 0 and λ > 0, then 4λ > 2λ/ 2 + η + 2 2η.

99 √ √ √ √ √ Proof. Let η > 0 and λ > 0, then 1 + 2η > 0 ⇔ 2 + 2η > 1 ⇔ 2 > 1/ 2 + η ⇔ √ 4λ > 2λ/ 2 + η + 2 2η.

Lemma 6.6. Let λ > 0. 8 2λ λ (a) If > η > 0, then √ > . 9 2 + η − 2 2η 1 − η 8 λ 2λ (b) If 1 > η > , then > √ . 9 1 − η 2 + η − 2 2η 8 2λ λ (c) If η = , then √ = . 9 2 + η − 2 2η 1 − η Proof. Let 8/9 > η > 0 and λ > 0, then we have

8η − 9η2 > 0 ⇔ 2p2η > 3η

⇔ 2 − 2η > 2 + η − 2p2η √ √ ⇔ 2p1 − η > 2 − pη √ √ ⇔ 2/ 2 − pη > 1/p1 − η ⇔ 2λ/ 2 + η − 2p2η > λ/(1 − η)

and we obtain (a). The proofs for (b) and (c) are similar.

Lemma 6.7. Let 1 > η > 0 and λ > 0. √ 16 2 − η − 2 1 − η λ ηp √ √ (a) If > p > 0, then 0 > − 2 1 − 1 − η pλ. η2 2 √ 16 2 − η − 2 1 − η λ ηp √ √ (b) If p > , then − 2 1 − 1 − η pλ > 0. η2 2 √ 16 2 − η − 2 1 − η λ ηp √ √ (c) If p = 0 or p = , then − 2 1 − 1 − η pλ = 0. η2 2 √ √ Proof. Let 1 > η > 0 and λ > 0. Deﬁne x ≡ p and a ≡ λ and restate expression √ √ √ ηp/2−2 1 − 1 − η pλ as ηx2/2−2 1 − 1 − η ax with x > 0 and a > 0. The solution √ √ to the quadratic equation ηx2/2−2 1 − 1 − η ax = 0 for x are 0 and 4 1 − 1 − η a/η > √ √ 0. Therefore if 4 1 − 1 − η a/η > x > 0, or equivalently, 16 2 − η − 2 1 − η a2/η2 > √ √ √ x2 > 0, then 0 > ηx2/2 − 2 1 − 1 − η ax. Replacing x by p and a by λ we obtain (a). The proofs for (b) and (c) are similar.

100 Lemma 6.8. Let λ > 0. √ 8 16 2 − η − 2 1 − η λ λ (a) If > η > 0, then > . 9 η2 1 − η √ 8 λ 16 2 − η − 2 1 − η λ (b) If 1 > η > , then > . 9 1 − η η2 √ 8 16 2 − η − 2 1 − η λ λ (c) If η = , then = . 9 η2 1 − η Proof. Let 8/9 > η > 0 and λ > 0, then we have

0 > 9η2 − 8η ⇔ 16(1 − η) > 9η2 − 24η + 16

⇔ 4p1 − η > 4 − 3η ⇔ 4 1 − p1 − η /η > 1/p1 − η 2 ⇔ 16 2 − η − 2p1 − η λ/η2 > λ/(1 − η)

and we obtain (a). The proofs for (b) and (c) are similar. √ Lemma 6.9. Let 1 > η > 0 and λ > 0, then 4λ/(1 − η) > 16 2 − η − 2 1 − η λ/η2 > 4λ. √ √ Proof. Let 1 > η > 0 and λ > 0, then 1 > 1 − η ⇔ η > 2 1 − η − 2 + 2η ⇔ 4λ/(1 − √ √ √ η) > 16 2 − η − 2 1 − η λ/η2. Also note that 1 > 1 − η ⇔ 2 − η − 2 1 − η > 0 ⇔ √ √ 2 1 − 1 − η > η ⇔ 16 2 − η − 2 1 − η λ/η2 > 4λ.

Lemma 6.8 and 6.9 imply η ∈ (0, 3/4) ⇒ 4p1 > p2 > 4λ ≥ p1 > 0 and η ∈ (3/4, 8/9) ⇒

4p1 > p2 > p1 > 4λ > 0.

2 We present weakly risk-seeking agent’s optimal response to any contract oﬀers (w, p) ∈ R+ in Proposition 6.10.

Proposition 6.10. Consider a weakly risk-seeking agent (η ∈ (0, 8/9)).

(a) Given

η p ∈ (0, p ] and w ≥ 1 − p (6.7) 1 2

101 then the agent accepts the contract and installs µ∗(w, p) = 0 with resulting expected

∗ utility rate uA(µ (w, p); w, p) = w − (1 − η/2)p ≥ 0. The agent rejects the contract if

both p ∈ (0, p1] and w ∈ (0, (1 − η/2)p).

(b) Given

ηp p ∈ (p , p ) and w ≥ + 2p(1 − η)pλ − λ (6.8) 1 2 2

then the agent accepts the contract and installs µ∗(w, p) = p(1 − η)pλ−λ with resulting

∗ p expected utility rate uA(µ (w, p); w, p) = w − ηp/2 − 2 (1 − η)pλ + λ ≥ 0. The agent p rejects the contract if both p ∈ (p1, p2) and w ∈ 0, ηp/2 + 2 (1 − η)pλ − λ .

p = p2 and w ≥ w2 (6.9)

then the agent accepts the contract and is indiﬀerent about installing either µ∗(w, p) =

p ∗ p (1 − η)p2λ − λ or µ (w, p) = p2λ − λ. In both cases the agent receives expected ∗ ∗ utility rate uA(µ (w, p); w, p) = w − w2 ≥ 0. If r ∈ (0, p2), then there exists w ∗ ∗ p such that (w , p2), µ = (1 − η)p2λ − λ is the unique admissible solution (see Def- ∗ ∗ ∗ p inition 6.1). If r = p2, then there exists w such that (w , p2), µ = p2λ − λ ∗ ∗ p and (w , p2), µ = (1 − η)p2λ − λ are both admissible solutions. If r > p2, then ∗ ∗ ∗ p there exists w such that (w , p2), µ = p2λ − λ is the unique admissible solution

(for proof see Proposition 6.13). The agent rejects the contract if both p = p2 and

w ∈ (0, w2).

(d) Given

p p > p2 and w ≥ 2 pλ − λ (6.10)

√ then the agent accepts the contract and installs µ∗(w, p) = pλ − λ with resulting

∗ √ expected utility rate uA(µ (w, p); w, p) = w − 2 pλ + λ ≥ 0. The agent rejects the

102 √ contract if both p > p2 and w ∈ 0, 2 pλ − λ .

Proof. According to Table 6.1, the behavior of u(µ) when η ∈ (0, 3/4] versus η ∈ (3/4, 8/9) is diﬀerent. Therefore we prove the proposition separately for η ∈ (0, 3/4] and η ∈ (3/4, 8/9).

Case η ∈ (0, 3/4]: According to Lemma 6.8 part (a) and 6.9, 4p1 > p2 > 4λ ≥ p1 > 0. Figure 6.2 depicts the shape of u(µ) when η ∈ (0, 3/4] and the value of p falls in diﬀerent ranges. The structure of the proof when η ∈ (0, 3/4] is depicted in Figure 6.3.

Case p ∈ (0, p1]: According to Table 6.1, u(µ) is decreasing with respect to µ ≥ 0. Thus the agent’s optimal service capacity is µ∗(w, p) = 0 and from (6.3) u(µ∗(w, p)) = w − (1 − η/2)p. Note that 1 − η/2 > 0.

Subcase w ∈ (0, (1 − η/2)p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ (1 − η/2)p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (p1, 4λ]: According to Table 6.1, the service capacity that maximizes u(µ) lies in ∗ ∗ (0, λ). µ is computed from ﬁrst order condition du(µ)/dµ|µ=µ∗(w,p) = 0 ⇒ µ (w, p) = p(1 − η)pλ − λ > 0 and from (6.3) u(µ∗(w, p)) = w − ηp/2 − 2p(1 − η)pλ + λ. According to Lemma 6.2 part (b) and 6.3, ηp/2 + 2p(1 − η)pλ − λ > 0, therefore we examine the following subcases.

Subcase w ∈ 0, ηp/2 + 2p(1 − η)pλ − λ : u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ ηp/2 + 2p(1 − η)pλ − λ: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (4λ, 4p1]: According to Table 6.1, there is a service capacity that maximizes u(µ) for µ ∈ (0, λ] and a service capacity that maximizes u(µ) for µ > λ. Denote

103 u( µ ) u( µ )

0.945 0.950 0.955 0.960 0.965 0.95 0.96 0.97 0.98 0.99 0 0 iue62 lutaino h om of forms the of Illustration 6.2: Figure λ λ (c) (a) η η = = 0.5 p 0.5 p

λ= λ ∈ , λ= λ ∈ 0.01 (4 0.01 (0 µ µ µ

u( ) , λ,

, w

w , = = p 1 p 1 0.925 0.930 0.935 0.940 0.945 , 1

, 2 p

p ] = ) = 1.5 0 5 λ λ λ η (e) = 0.5

, λ= λ > p 104 0.01

µ u( µ ) u( µ )

, 4

w p = 1 1

0.940 0.945 0.950 0.955 0.945 0.950 0.955 0.960 0.965 0.970 0.975 ,

p = 10 0 0 λ u ( µ when ) λ λ (d) (b) η η = = p 0.5 0.5 p ∈

, , ∈ λ= λ λ= λ η ( 0.01 0.01 ( p µ µ p ∈ 2

1 , ,

w w , , = = 4 (0 4 1 1 p

, , λ

p p , 1 = = ] ] 3 3 7 λ λ / 4] η w ∈ 0, 1 − p Reject. 2

p ∈ (0, p1] η w ≥ 1 − p µ∗ = 0 2

ηp w ∈ 0, + 2p(1 − η)pλ − λ Reject. 2

p ∈ (p1, 4λ] ηp w ≥ + 2p(1 − η)pλ − λ µ∗ = p(1 − η)pλ − λ 2

ηp w ∈ 0, + 2p(1 − η)pλ − λ Reject. 2 3 η ∈ 0, p ∈ (4λ, p2) 4 ηp w ≥ + 2p(1 − η)pλ − λ µ∗ = p(1 − η)pλ − λ 2

w ∈ (0, w2) Reject.

p ∈ (4λ, 4p1] p = p2

∗ p µ = (1 − η)p2λ − λ w ≥ w2 ∗ p or µ = p2λ − λ

√ w ∈ 0, 2 pλ − λ Reject.

p ∈ (p2, 4p1]

√ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

√ w ∈ 0, 2 pλ − λ Reject.

p > 4p1 √ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

Figure 6.3: Structure of the proof for Proposition 6.10 when η ∈ (0, 3/4]

∗ the optimal service capacity in (0, λ] by µ(0,λ](w, p). From the ﬁrst order condition ∗ p the optimal service capacity is µ(0,λ](w, p) = (1 − η)pλ − λ and from equation (6.3) ∗ p u µ(0,λ](w, p) = w −ηp/2−2 (1 − η)pλ+λ. Denote the optimal service capacity for ∗ µ > λ by µ (w, p), which is obtained from ﬁrst order condition du(µ)/dµ| ∗ = λ µ=µλ(w,p) ∗ √ ∗ √ 0 ⇒ µλ(w, p) = pλ − λ and from equation (6.3) u (µλ(w, p)) = w − 2 pλ + λ. The agent has a choice of two service capacities and he installs the one that generates

∗ ∗ a higher expected utility rate. Note that u (µλ(w, p)) − u µ(0,λ](w, p) = ηp/2 − √ √ 2 1 − 1 − η pλ. According to Lemma 6.9, 4p1 > p2 > 4λ, therefore we examine the following subcases.

∗ ∗ Subcase p ∈ (4λ, p2): By Lemma 6.7 part (a), u µ(0,λ](w, p) > u (µλ(w, p)), thus the agent’s optimal service capacity is µ∗(w, p) = p(1 − η)pλ−λ and u(µ∗(w, p)) =

105 w − ηp/2 − 2p(1 − η)pλ + λ. According to Lemma 6.2 part (a) and 6.3, ηp/2 + 2p(1 − η)pλ − λ > 0, therefore we examine the following subcases. Subsubcase w ∈ 0, ηp/2 + 2p(1 − η)pλ − λ : u(µ∗(w, p)) < 0, thus the agent rejects the contract.

Subsubcase w ≥ ηp/2 + 2p(1 − η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

∗ ∗ Subcase p = p2: According to Lemma 6.7 part (c), u µ(0,λ](w, p) = u (µλ(w, p)), ∗ ∗ indicating that installing either µ(0,λ](w, p2) or µλ(w, p2) leads to the same agent’s expected utility rate. Therefore the agent is indiﬀerent about installing µ∗(w, p) = √ p(1 − η)pλ−λ or µ∗(w, p) = pλ−λ. Still, the capacity value leads to admissible

solutions (see Proposition 6.13). Recall the deﬁnition of w2 from (6.6). By Lemma p 6.2, p2 > 4λ > p1 ⇒ w2 = ηp2/2 + 2 (1 − η)p2λ − λ > 0, therefore we examine the following subcases.

∗ Subsubcase w ∈ (0, w2): u(µ (w, p)) < 0, therefore the agent rejects the contract.

∗ Subsubcase w ≥ w2: u(µ (w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

∗ ∗ Subcase p ∈ (p2, 4p1]: From Lemma 6.7 part (b), u (µλ(w, p)) > u µ(0,λ](w, p) , √ thus the agent’s optimal service capacity is µ∗(w, p) = pλ − λ and u(µ∗(w, p)) = √ √ w − 2 pλ + λ. Since p > p2 > 4λ ⇒ 2 pλ − λ > 3λ > 0, therefore we examine the following subcases. √ Subsubcase w ∈ 0, 2 pλ − λ: u(µ∗(w, p)) < 0, thus the agent rejects the contract. √ Subsubcase w ≥ 2 pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

Case p > 4p1: According to Table 6.1, the service capacity that maximizes u(µ) satisﬁes µ > λ. From the ﬁrst order condition the agent’s optimal service capacity is µ∗(w, p) =

106 √ ∗ √ pλ − λ and from equation (6.3) u(µ (w, p)) = w − 2 pλ + λ. Since p > 4p1 > 4λ, √ therefore 2 pλ − λ > 3λ > 0 and we examine the following subcases.

√ Subcase w ∈ 0, 2 pλ − λ: u(µ∗(w, p)) < 0, thus the agent rejects the contract. √ Subcase w ≥ 2 pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

This completes the proof for Proposition 6.10 when η ∈ (0, 3/4].

Case η ∈ (3/4, 8/9): According to Lemma 6.8 and 6.9, 4p1 > p2 > p1 > 4λ > 0. Figure 6.4 depicts the shape of u(µ) when η ∈ (3/4, 8/9) and the value of p falls in diﬀerent ranges. The structure of the proof when η ∈ (3/4, 8/9) is depicted in Figure 6.5.

Case p ∈ (0, 4λ]: According to Table 6.1, u(µ) is decreasing with respect to µ ≥ 0. There- fore the agent’s optimal service capacity is µ∗(w, p) = 0 and from equation (6.3) u(µ∗(w, p)) = w − (1 − η/2)p. Note that 1 − η/2 > 0.

Subcase w ∈ (0, (1 − η/2)p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ (1 − η/2)p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (4λ, p1]: According to Table 6.1, there is a service capacity that maximizes u(µ) for µ ∈ [0, λ) and a service capacity that maximizes u(µ) for µ > λ. Denote the optimal

∗ service capacity in [0, λ) by µ[0,λ)(w, p). Since u(µ) is decreasing with respect to µ over ∗ ∗ [0, λ), therefore µ[0,λ)(w, p) = 0 and from (6.3) u µ[0,λ)(w, p) = w − (1 − η/2)p. ∗ Denote the optimal service capacity for µ > λ by µλ(w, p). From ﬁrst order condition ∗ √ ∗ √ µλ(w, p) = pλ−λ and from (6.3) u (µλ(w, p)) = w−2 pλ+λ. The agent has to choose one of the two service capacities and he installs the one with higher expected utility rate.

∗ ∗ √ Note that u (µλ(w, p))−u µ[0,λ)(w, p) = (1−η/2)p−2 pλ+λ. According to Lemma

107 u( µ ) u( µ )

0.940 0.945 0.950 0.955 0.960 0.965 0.95 0.96 0.97 0.98 0 0 iue64 lutaino h om of forms the of Illustration 6.4: Figure λ λ (c) (a) η η = = p 0.8 0.8 p

, , ∈ λ= λ λ= λ ∈ 0.01 0.01 ( (0 p µ u( µ ) µ

1 , ,

, w w , 4 = = p 1 1

0.85 0.86 0.87 0.88 0.89 0.90 0.91 λ

2 , ,

p p ] ) = = 0 2 6 λ λ λ η (e) = 0.8

, λ= λ > p 108 0.01

µ u( µ ) u( µ )

, 4

w p = 1 1

0.934 0.936 0.938 0.940 0.942 0.944 0.946 0.945 0.950 0.955 0.960 0.965 0.970 ,

p = 25 0 0 u λ ( µ when ) λ λ (d) (b) η η = = 0.8 p 0.8 p

, , ∈ λ= λ λ= λ ∈ η 0.01 0.01 ( ∈ (4 p µ µ

2 , ,

λ,

w (3 w , = = 4 1 1 p

, p ,

p p 1 4 1 = = ] 4.8 ] , 10 8 λ λ / 9) η w ∈ 0, 1 − p Reject. 2 p ∈ (0, 4λ] η w ≥ 1 − p µ∗ = 0 2

η w ∈ 0, 1 − p Reject. 2

p ∈ (4λ, p1] η w ≥ 1 − p µ∗ = 0 2

ηp w ∈ 0, + 2p(1 − η)pλ − λ Reject. 2 3 8 η ∈ , p ∈ (p1, p2) 4 9 ηp w ≥ + 2p(1 − η)pλ − λ µ∗ = p(1 − η)pλ − λ 2

w ∈ (0, w2) Reject.

p ∈ (p1, 4p1] p = p2

∗ p µ = (1 − η)p2λ − λ w ≥ w2 ∗ p or µ = p2λ − λ

√ w ∈ 0, 2 pλ − λ Reject.

p ∈ (p2, 4p1]

√ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

√ w ∈ 0, 2 pλ − λ Reject.

p > 4p1 √ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

Figure 6.5: Structure of the proof for Proposition 6.10 when η ∈ (3/4, 8/9)

√ √ 6.5, 4λ > 2λ/ 2 + η + 2 2η and according to Lemma 6.6, 2λ/ 2 + η − 2 2η > p1. ∗ ∗ Therefore according to Lemma 6.4 part (a), u µ[0,λ)(w, p) > u (µλ(w, p)), the agent’s optimal service capacity is µ∗(w, p) = 0 and u(µ∗(w, p)) = w − (1 − η/2)p. Note that 1 − η/2 > 0, therefore we examine the following subcases.

Subcase w ∈ (0, (1 − η/2)p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ (1 − η/2)p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (p1, 4p1]: According to Table 6.1, there is a service capacity that maximizes u(µ) for µ ∈ (0, λ] and a service capacity that maximizes u(µ) for µ > λ. Denote the

109 ∗ optimal service capacity in (0, λ] by µ(0,λ](w, p). From ﬁrst order condition the optimal ∗ p ∗ service capacity is µ(0,λ](w, p) = (1 − η)pλ − λ and from (6.3) u µ(0,λ](w, p) = w − ηp/2 − 2p(1 − η)pλ + λ. Denote the optimal service capacity for µ > λ by

∗ ∗ √ ∗ µλ(w, p). From ﬁrst order condition µλ(w, p) = pλ − λ and from (6.3) u (µλ(w, p)) = √ w − 2 pλ + λ. The agent has a choice of two service capacities and he installs the one

∗ ∗ that generates a higher expected utility rate. Note that u (µλ(w, p))−u µ(0,λ](w, p) = √ √ ηp/2 − 2 1 − 1 − η pλ. According to Lemma 6.8 and Lemma 6.9, 4p1 > p2 > p1, therefore we examine the following subcases.

∗ ∗ Subcase p ∈ (p1, p2): By Lemma 6.7 part (a), u µ(0,λ](w, p) > u (µλ(w, p)), thus the agent’s optimal service capacity is µ∗(w, p) = p(1 − η)pλ−λ and u(µ∗(w, p)) = p w − ηp/2 − 2 (1 − η)pλ + λ. According to Lemma 6.2 and 6.3, p > p1 ⇒ ηp/2 + 2p(1 − η)pλ − λ > 0, therefore we examine the following subcases. Subsubcase w ∈ 0, ηp/2 + 2p(1 − η)pλ − λ : u(µ∗(w, p)) < 0, thus the agent rejects the contract.

Subsubcase w ≥ ηp/2 + 2p(1 − η)pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

∗ ∗ Subcase p = p2: According to Lemma 6.7 part (c), u µ(0,λ](w, p) = u (µλ(w, p)), ∗ ∗ indicating that installing µ(0,λ](w, p2) or µλ(w, p2) leads to the same agent’s expected utility rate. Therefore the agent is indiﬀerent about installing µ∗(w, p) = √ p(1 − η)pλ − λ or µ∗(w, p) = pλ − λ. Still, the capacity value has to lead to

admissible solutions (see Proposition 6.13). Recall the deﬁnition of w2 in (6.6). p According to Lemma 6.2, p2 > p1 ⇒ w2 = ηp2/2+2 (1 − η)p2λ−λ > 0, therefore we examine the following subcases.

∗ Subsubcase w ∈ (0, w2): u(µ (w, p)) < 0, therefore the agent rejects the contract.

∗ Subsubcase w ≥ w2: u(µ (w, p)) ≥ 0, so the agent would accept the contract if oﬀered.

∗ ∗ Subcase p ∈ (p2, 4p1]: By Lemma 6.7 part (b), u (µλ(w, p)) > u µ(0,λ](w, p) , there-

110 √ fore the agent’s optimal service capacity is µ∗(w, p) = pλ − λ and u(µ∗(w, p)) = √ √ w − 2 pλ + λ. Since p > p2 > 4λ ⇒ 2 pλ − λ > 3λ > 0, therefore we examine the following subcases. √ Subsubcase w ∈ 0, 2 pλ − λ: u(µ∗(w, p)) < 0, thus the agent rejects the contract. √ Subsubcase w ≥ 2 pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

Case p > 4p1: According to Table 6.1, the service capacity that maximizes u(µ) satisﬁes µ > λ. From the ﬁrst order condition the agent’s optimal service capacity is µ∗(w, p) =

√ ∗ √ pλ − λ and from (6.3) u(µ (w, p)) = w − 2 pλ + λ. Since p > 4p1 > 4λ, therefore √ 2 pλ − λ > 3λ > 0 and we examine the following subcases.

This complete the proof for Proposition 6.10 when η ∈ (3/4, 8/9).

To summarize: Given exogenous market conditions that enable a mutually beneﬁcial contract between a principal and weakly risk-seeking agent (see Theorem 6.17 later), the agent determines his service capacity by using one of only two formulas:

µ∗ = p(1 − η)pλ − λ > 0 or µ∗(w, p) = ppλ − λ > 0

The conditions when a weakly risk-seeking agent accepts the contract can be depicted by the shaded areas in Figure 6.6, where η = 0.5. The three shaded areas with different grey scales represent conditions (6.7), (6.8) and (6.10) under which the agent accepts the contract but responds differently. The lower bound function of the shaded area (denoted by w0(p)) represents the set of offers of zero expected utility rate for the agent. The w0(p) line is

111 deﬁned as follows:

 η  1 − p when p ∈ (0, p1]  2  ηp w0(p) = p + 2 (1 − η)pλ − λ when p ∈ (p1, p2]  2  √  2 pλ − λ when p > p2

Since lim − w0(p) = lim + w0(p) = (1 − η/2)p and lim − w0(p) = lim + w0(p) = p→p1 p→p1 1 p→p2 p→p2 p ηp2/2 + 2 (1 − η)p2λ − λ, therefore w0(p) is continuous everywhere over interval p ∈

+. Since lim − dw0(p)/dp = lim + dw0(p)/dp = 1 − η/2, therefore w0(p) is diﬀer- R p→p1 p→p1 √ √ entiable at p = p . However since lim − dw0(p)/dp = η 2 − 1 − η /4 1 − 1 − η 6= 1 p→p2 √ η/4 1 − 1 − η = lim + dw0(p)/dp, therefore w0(p) is not diﬀerentiable at p = p . p→p2 2

∗ ∗ ∗ µ =0 µ = (1− η )p λ − λ µ = p λ − λ λ − λ 2 p

w=2 p λ − λ 2

1 η p p w= +2 (1− η )p λ − λ 2 /2)

η w=(1− η /2)p (1−

0 p 0 p1 p2

Figure 6.6: Conditions when a weakly risk-seeking agent accepts the contract with η = 0.5

112 6.1.1 Sensitivity analysis of a weakly risk-seeking agent’s optimal strategy

A principal does not propose a contract that will be accepted by the agent but results in zero service capacity. Therefore the only viable cases when the agent accepts the contract √ and installs positive service capacities are: µ∗(w, p) = p(1 − η)pλ−λ or µ∗(w, p) = pλ−λ.

First the case when a weakly risk-seeking agent installs µ∗(w, p) = p(1 − η)pλ − λ. Ac- cording to (6.8) the compensation rate w is bounded below by ηp/2 + 2p(1 − η)pλ − λ = pP (1) − ηp (P (1) − 1/2) + µ∗(w, p), with the term pP (1) representing the expected penalty rate charged by the principal and the term ηp (P (1) − 1/2) representing the expected risk rate perceived by the agent when the optimal capacity is installed. It dictates that the agent be reimbursed for the expected penalty rate and the cost of the optimal service capacity discounted by his perceived risk rate in exchange.

The optimal service capacity p(1 − η)pλ − λ depends on p, λ, and η. Its derivatives are:

s r s ∂µ∗ (1 − η)λ ∂µ∗ (1 − η)p ∂µ∗ pλ = > 0, = − 1 and = − < 0 ∂p 4p ∂λ 4λ ∂η 4(1 − η)

The above derivatives indicate that given a λ and η the agent will increase the service capacity when the penalty rate increases. Note that p(1 − η)pλ − λ, as a function of λ, decreases when λ > (1 − η)p/4. From conditions (6.8) and (6.9) the agent installs service capacity p (1 − η)pλ − λ when p ∈ (p1, p2], and according to Lemma 6.9 we have 4p1 > p2. Therefore ∗ we have 4λ/(1 − η) = 4p1 > p ⇒ λ > (1 − η)p/4 ⇒ 0 > ∂µ /∂λ. Thus, given the penalty rate and the risk coeﬃcient, the agent will decrease the service capacity when the failure rate increases. Given a penalty rate and a failure rate, the agent will reduce the service capacity when he is more risk-seeking.

The agent’s optimal expected utility rate when installing capacity µ∗(w, p) = p(1 − η)pλ−λ

∗ ∗ p is uA ≡ uA(µ (w, p); w, p) = w − ηp/2 − 2 (1 − η)pλ + λ, and it depends on w, p, η ∗ ∗ p and λ. Note that ∂uA/∂w = −1 < 0, ∂uA/∂p = −η/2 − (1 − η)λ/p < 0, indicat-

113 ing that the agent’s optimal expected utility rate decreases with the compensation rate

∗ √ √ √ and the penalty rate. Note that ∂uA/∂η = − p p − 4p1 /2. From Proposition 6.10 √ √ p < p2 < 4p1 ⇒ p < 4p1, therefore the agent’s optimal expected utility rate increases ∗ √ √ √ with his risk intensity. Note that ∂uA/∂λ = − p − p1 / p1, and from Proposition 5.23 √ √ p > p1 ⇒ p − p1 > 0, therefore the agent’s optimal expected utility rate decreases with the failure rate.

√ Then the case when a weakly risk-seeking agent installs µ∗(w, p) = pλ − λ. In this case the agent’s optimal strategy is identical to the optimal strategy when he is risk-neutral. √ According to (6.10) the w is bounded below by 2 pλ − λ = pP (1) + µ∗(w, p), with the term pP (1) representing the expected penalty rate charged by the principal. It indicates that the agent will have to be reimbursed for the expected penalty rate and the cost of the optimal service capacity.

√ The optimal service capacity pλ − λ depends on the penalty rate p and the failure rate λ. Its derivatives are ∂µ∗/∂p = pλ/4p > 0 and ∂µ∗/∂λ = pp/4λ − 1. These derivatives imply that given λ, the agent will increase the service capacity when the penalty rate increases. √ Note that pλ − λ, as a function of λ, increases when p/4 > λ. From conditions (6.9) and √ (6.10) the agent installs service capacity pλ − λ when p ≥ p2, and according to Lemma 6.9 ∗ we have p2 > 4λ. Therefore we have p > 4λ ⇒ p/4 > λ ⇒ ∂µ /∂λ > 0. Thus, given p, an agent will increase µ when λ increases.

√ The agent’s optimal expected utility rate when installing capacity µ∗(w, p) = pλ − λ

∗ ∗ √ is uA ≡ uA(µ (w, p); w, p) = w − 2 pλ + λ, and it depends on w, p and λ only. Note ∗ ∗ p that ∂uA/∂w = −1 < 0, ∂uA/∂p = − λ/p < 0, indicating that the agent’s optimal expected utility rate decreases with the compensation rate and the penalty rate. Note that

∗ p p ∂uA/∂λ = − p/λ + 1, and from Proposition 6.10 p ≥ p2 > 4λ ⇒ − p/λ + 1 < 0, therefore the agent’s optimal expected utility rate also decreases with the failure rate.

114 Summary: Recall that given the set of contract oﬀers {(w, p): p ∈ (0, λ], w ≥ p} a risk- neutral agent would accept the contract, install µ∗(w, p) = 0 and receive expected utility √ rate u(µ∗(w, p); w, p) = w − p. Given the set of oﬀers (w, p): p > λ, w ≥ 2 pλ − λ he √ would accept the contract, install µ∗(w, p) = pλ − λ and receive expected utility rate √ u(µ∗(w, p); w, p) = w − 2 pλ + λ. By comparing the optimal capacities of a weakly risk- seeking agent to that of a risk-neutral agent, three conclusions are drawn.

1. The principal has to set a higher penalty rate p in order to induce a weakly risk- seeking agent to install a positive service capacity versus a risk-neutral agent (p > λ for risk-neutral agent, p > λ/(1 − η) for weakly risk-seeking agent).

2. When p is relatively low, µ plays a more prominent role in the utility of a weakly risk-seeking agent who therefore installs a µ lower than that when he is risk-neutral √ ( pλ − λ > p(1 − η)pλ − λ). As p increases, the weakly risk-seeking agent installs µ √ that is identical to the one for risk-neutral agent ( pλ − λ).

3. Weakly risk-seeking agent is not worse oﬀ. This conclusion is restated in Proposition 6.11.

Proposition 6.11. Given w and p, an agent who accepts the contract and installs a positive service capacity has a non-decreasing expected utility rate with η for η ∈ [0, 8/9).

Proof. Recall that when the compensation rate w and the penalty rate p satisfy conditions (6.8) and (6.9), the agent installs service capacity µ∗(w, p) = p(1 − η)pλ−λ > 0, and the agent’s expected utility rate is u (µ∗(w, p)) = w−ηp/2−2p(1 − η)pλ+λ. Note p √ √ √ that ∂u/∂η = −p/2 + pλ/(1 − η) = − p p − 4p1 /2. According to Lemma 6.9,

4p1 > p2 ≥ p, therefore ∂u/∂η > 0. When the compensation rate w and the penalty rate p satisfy conditions (6.9) and (6.10), the agent installs service capacity µ∗(w, p) = √ √ pλ − λ > 0, and the agent’s expected utility rate is u (µ∗(w, p)) = w − 2 pλ + λ, therefore ∂u/∂η = 0.

Corollary 6.12. Given w and p, an agent who accepts the contract and subsequently installs a positive service capacity will not be worse oﬀ when he is weakly risk-seeking

115 (η ∈ (0, 8/9)) comparing to risk-neutral (η = 0).

We return to the case of η ≥ 8/9 in subsection 6.2.1 and 6.3.

6.1.2 Principal’s optimal strategy

We now proceed to derive the principal’s optimal strategy. Anticipating the agent’s optimal selection of µ∗(w, p) the principal chooses w and p that maximize her expected proﬁt rate by solving the optimization problem

∗ ∗ rµ (w, p) pλ max ΠP (w, p; µ (w, p)) = max − w + (6.11) w>0,p>0 w>0,p>0 λ + µ∗(w, p) λ + µ∗(w, p)

∗ ∗ ∗ Denote (w , p ) = argmaxw>0,p>0 ΠP (w, p; µ (w, p)).

Before deriving the principal’s optimal strategy, we examine the case when the principal

offers p = p2 and w ≥ w2, under which the agent is indifferent between two different service capacities. In such a case, the solution ((w, p), µ) has to be an admissible solution (see Definition 6.1). We state this case formally in Proposition 6.13.

Proposition 6.13. Suppose a weakly risk-seeking agent. Assume that the principal’s possible

oﬀers are constrained to set {(w, p): p = p2, w ≥ w2}.

∗ p (a) If r ∈ (0, p2), then the agent installs µ = (1 − η)p2λ − λ if oﬀered a contract.

∗ p ∗ p (b) If r = p2, then both µ = (1 − η)p2λ − λ and µ = p2λ − λ lead to admissible p p solutions and the agent installs either (1 − η)p2λ−λ or p2λ−λ if oﬀered a contract.

∗ p (c) If r > p2, then the agent installs µ = p2λ − λ if oﬀered a contract.

2 Proof. Note that for w ≥ w2 we have ∂ΠP (w, p2; µ)/∂µ = (r − p2)λ/(λ + µ) . Deﬁne p p µL ≡ (1 − η)p2λ − λ and µH ≡ p2λ − λ. Note that µH > µL. If r ∈ (0, p2), then

∂ΠP /∂µ < 0, therefore ((w, p2), µL) ((w, p2), µH ). If the principal oﬀers a contract (the conditions are discussed in detail in Theorem 6.17 that follows), then by Deﬁnition 6.1 only

µL leads to admissible solutions and we obtain (a). If r > p2, then ∂ΠP /∂µ > 0, therefore

116 ((w, p2), µH ) ((w, p2), µL). If the principal oﬀers a contract (the conditions are discussed

in Theorem 6.17 that follows), then by Deﬁnition 6.1 only µH leads to admissible solutions

and we obtain (c). If r = p2, then ∂ΠP /∂µ = 0, indicating that the principal receives the same expected proﬁt rate when the agent installs capacity µL or µH . If the principal oﬀers a contract (the conditions are discussed in Theorem 6.17 that follows), then both µL and µH lead to admissible solutions. Therefore we obtain (b).

Notation:

√ √ √ p ηp2 p2 p2 − p1 r1 ≡ ηp2 + (1 − η) p1p2 − √ √ , r2 ≡ (1 − η)p2 + ηp2 √ 2 p2 − p1 p1 (6.12)

Note that r1 and r2 are functions of λ and η. However we suppress the parameters (λ, η).

2 We deﬁne pcu as follows :

1 2 p ≡ b + C + C (6.13) cu 9a2 √ √ where a ≡ η, b ≡ (1 − 2η) p1, and d ≡ −r p1 and

s s p 2 3 p 2 3 3 ∆ + ∆ − 4∆ 3 ∆ − ∆ − 4∆ C ≡ 1 1 0 , C ≡ 1 1 0 , where ∆ ≡ b2, ∆ ≡ 2b3 + 27a2d 2 2 0 1

Replacing ∆0 and ∆1 by the expressions of a, b and d we have

s √ 3p 3 2 p 2 3 2 4 2 3 2(1 − 2η) p − 27η r p + −108η r(1 − 2η) p + 729η r p C = 1 1 1 1 and 2 s √ 3p 3 2 p 2 3 2 4 2 3 2(1 − 2η) p − 27η r p − −108η r(1 − 2η) p + 729η r p C = 1 1 1 1 2

Next we state a number of technical lemmas.

2The subscript “cu” stands for “cubic” because (6.13) is the square of the solution to equation (6.14), which is a cubic equation that is introduced later in the proof for Lemma 6.15.

117 Lemma 6.14. Let 8/9 > η > 0 and λ > 0, then √ √ √ p2 − p1 √ ηp2 p2 (a) p2 > (1 − η)p2 + ηp2 √ > ηp2 + (1 − η) p1p2 − √ √ . p1 2 p2 − p1 √ √ p2 − p1 (b) (1 − η)p2 + ηp2 √ > λ. p1

Proof. Let 8/9 > η > 0 and λ > 0. According to Lemma 6.8 and 6.9, 4p1 > p2 > p1, therefore:

p p p p p 2 p1 > p2 ⇔ p1 > p2 − p1 > 0 p p p ⇔ 1 > p2 − p1 / p1 p p p ⇔ η > η p2 − p1 / p1 p p p ⇔ 1 > (1 − η) + η p2 − p1 / p1 p p p ⇔ p2 > (1 − η)p2 + ηp2 p2 − p1 / p1

Note that √ √ √ p ηp2 p2 p ηp2 ( p2 − 4p1) ηp2 + (1 − η) p1p2 − √ √ = (1 − η) p1p2 + √ √ 2 ( p2 − p1) 2 ( p2 − p1) √ √ √ √ Since 4p1 > p2 > p1, therefore we have (1−η)p2 > (1−η) p1p2 and ηp2 ( p2 − p1) / p1 > √ √ √ √ √ √ √ 0 > ηp2 ( p2 − 4p1) /2 ( p2 − p1). Thus (1 − η)p2 + ηp2 ( p2 − p1) / p1 > ηp2 + √ √ √ √ (1 − η) p1p2 − ηp2 p2/2 ( p2 − p1). Therefore we obtain (a). According to Lemma √ √ √ 6.8, p2 > p1, therefore (1 − η)(p2 − p1) + ηp2 ( p2 − p1) / p1 > 0 ⇔ (1 − η)p2 + √ √ √ ηp2 ( p2 − p1) / p1 > (1 − η)p1 = λ. Therefore we obtain (b).

Lemma 6.15. Consider optimization problem max √ √ f(x) in which f(x) = r + λ − x∈[ p1, p2] √ 2 ∗ √ √ ηx /2 − p1 ((1 − 2η)x + r/x) and denote x = argmax f(x). The solutions to x∈[ p1, p2] this optimization problem are:

∗ √ (a) x = p1 if r ∈ (0, λ].

∗ √ √ √ (b) x = pcu ∈ ( p1, p2) if r ∈ (λ, r2).

∗ √ (c) x = p2 if r ≥ r2.

118 √ √ Proof. Note that f(x) is continuous and diﬀerentiable over interval [ p1, p2]: √ df(x) r d2f(x) 2r p = −ηx − pp (1 − 2η) − and = −η − 1 < 0 dx 1 x2 dx2 x3

df(x) p r 1 = −(1 − η) p + √ = √ (r − λ) dx √ 1 p p x= p1 1 1 √ √ df(x) p p r p1 p1 = −η p − (1 − 2η) p + = (r − r2) dx √ 2 1 p p x= p2 2 2

√ √ ∗ √ If r ∈ (0, λ], then f(x) is decreasing over [ p1, p2], therefore x = p1. If r ≥ r2, then f(x) √ √ ∗ √ is increasing over [ p1, p2], therefore x = p2. If r ∈ (λ, r2), then f(x) is increasing in the √ √ neighborhood above x = p1 and is decreasing in the neighborhood below x = p2. Also 2 2 ∗ √ √ note that d f(x)/dx < 0, therefore x ∈ ( p1, p2) that satisﬁes the ﬁrst order condition

df(x) ∗ 3 p ∗ 2 p = 0 ⇒ η (x ) + (1 − 2η) p1 (x ) − r p1 = 0 (6.14) dx x=x∗ which is a cubic equation. According to the general formula for roots of cubic equation,

∗ √ √ √ x = pcu ∈ ( p1, p2). √ Lemma 6.16. Consider max √ f(x) where f(x) = r + λ − λ (x + r/x) and denote x≥ p2 x∗ = argmax √ f(x). Solutions to this optimization problem are x≥ p2

∗ √ (a) x = p2 if r ∈ (0, p2].

∗ √ (b) x = r if r > p2. √ Proof. Note that f(x) is continuous and diﬀerentiable for x ≥ p2: √ df(x) √ r d2f(x) 2r λ = − λ 1 − and = − < 0 dx x2 dx2 x3 √ df(x) λ df(x) √ = (r − p ) and = − λ < 0 dx √ p 2 dx x= p2 2 x→+∞

√ ∗ √ √ If r ∈ (0, p2], then f(x) is decreasing for r ≥ p2, therefore x = p2. If r > p2, then √ f(x) is increasing in the neighborhood above x = p2. Since f(x) is decreasing as x → +∞ 2 2 ∗ √ and since f(x) is concave (d f(x)/dx < 0), therefore x > p2 is solved from ﬁrst order ∗ √ condition df(x)/dx|x=x∗ = 0 ⇒ x = r.

119 Lemma 6.14 implies p2 > r2 > r1 and r2 > λ.

Recall that Proposition 6.10 describes the agent’s optimal response to each pair (w, p) ∈

2 R+. Since the principal will not propose a contract knowing ex ante that it will be rejected by 2 a weakly risk-seeking (WRS) agent, therefore Theorem 6.17 only considers pairs (w, p) ∈ R+ that result in agent’s non-negative expected utility rate. Deﬁne

D(6.7) ≡ {(w, p) that satisﬁes (6.7) when η ∈ (0, 8/9)}

D(6.8) ≡ {(w, p) that satisﬁes (6.8) when η ∈ (0, 8/9)}

D(6.9) ≡ {(w, p) that satisﬁes (6.9) when η ∈ (0, 8/9)} (6.15)

D(6.10) ≡ {(w, p) that satisﬁes (6.10) when η ∈ (0, 8/9)}

DWRS ≡ D(6.7) ∪ D(6.8) ∪ D(6.9) ∪ D(6.10)

Theorem 6.17. Given a weakly risk-seeking agent and (w, p) ∈ DWRS.

(a) If r ∈ (0, λ], then the principal does not propose a contract.

(b) If r ∈ (λ, r2), then the principal’s oﬀer and the capacity installed by the agent are:

ηp (w∗, p∗) = cu + 2p(1 − η)p λ − λ, p and µ∗(w∗, p∗) = p(1 − η)p λ − λ 2 cu cu cu (6.16)

and the principal’s expected proﬁt rate is:

∗ ∗ ∗ ∗ ∗ ηpcu p p r ΠP (w , p ; µ (w , p )) = r + λ − − p1 (1 − 2η) pcu + √ (6.17) 2 pcu

∗ ∗ ∗ ∗ ∗ p (w , p ) = (w2, p2) and µ (w , p ) = (1 − η)p2λ − λ (6.18)

120 and the principal’s expected proﬁt rate is:

∗ ∗ ∗ ∗ ∗ ηp2 p p r ΠP (w , p ; µ (w , p )) = r + λ − − p1 (1 − 2η) p2 + √ (6.19) 2 p2

(d) If r > p2, then the principal’s oﬀer and the capacity installed by the agent are

√ √ (w∗, p∗) = 2 rλ − λ, r and µ∗(w∗, p∗) = rλ − λ (6.20)

and the principal’s expected proﬁt rate is:

√ ∗ ∗ ∗ ∗ ∗ ΠP (w , p ; µ (w , p )) = r − 2 rλ + λ (6.21)

Proof. The structure of the proof for Theorem 6.17 is depicted in Figure 6.7.

(w, p) ∈ D(6.7) No contract oﬀered.

r ∈ (0, λ] No contract oﬀered.

∗ ηpcu p w = + 2 (1 − η)pcuλ − λ r ∈ (λ, r2) 2 ∗ ∗ p Risk-Neutral and p = pcu and µ = (1 − η)pcuλ − λ Principal with Weakly (w, p) ∈ D(6.8) ∪ D(6.9) Risk-Seeking Agent ∗ ∗ ∗ p r ∈ [r2, p2] w = w2 and p = p2 and µ = (1 − η)p2λ − λ

∗ ∗ ∗ p r > p2 w = w2 and p = p2 and µ = p2λ − λ

r ∈ (0, max {0, r1}] No contract oﬀered.

(w, p) ∈ D ∪ D ∗ ∗ ∗ p (6.9) (6.10) r ∈ (0, max {0, r1} , p2] w = w2 and p = p2 and µ = (1 − η)p2λ − λ

√ w∗ = 2 rλ − λ and p∗ = r r > p √ 2 and µ∗ = rλ − λ

Figure 6.7: Structure of the proof for Theorem 6.17

121 Case (w, p) ∈ D(6.7): According to Proposition 6.10 part (a), in case the principal makes an oﬀer, the agent accepts the contract but does not install any service capacity. Since

∗ ∗ ∗ ∗ ∂ΠP /∂w = −1 < 0, thus we have w = (1−η/2)p and from (4.3) ΠP (w , p; µ (w , p)) = −w∗+p = ηp/2 > 0. However in such case p > w∗ = (1−η/2)p, which violates condition (c) in Deﬁnition 6.1, therefore ((w∗ = (1−η/2)p, p), µ∗ = 0) is not an admissible solution and the principal does not propose a contract.

Case (w, p) ∈ D(6.8) ∪ D(6.9): According to Proposition 6.10 part (b), if (w, p) ∈ D(6.8), then in case the principal makes an oﬀer, the agent accepts the contract and installs

p ∗ p (1 − η)pλ − λ. Since ∂ΠP /∂w = −1 < 0, therefore w = ηp/2 + 2 (1 − η)pλ − λ.

According to Proposition 6.10 part (c) and Proposition 6.13, if (w, p) ∈ D(6.9) (which

implies p = p2), then in case the principal makes an oﬀer, the agent accepts the contract p p p and installs (1 − η)p2λ − λ if r ∈ (0, p2), installs either (1 − η)p2λ − λ or p2λ − λ p ∗ if r = p2, or installs p2λ − λ if r > p2. Since ∂ΠP /∂w = −1 < 0, therefore w = w2.

For convenience denote the principal’s expected proﬁt rate when (w, p) = (w2, p2) and ∗ p L µ (w, p) = (1 − η)p2λ − λ by ΠP (p2), and denote the principal’s expected proﬁt rate ∗ p H when (w, p) = (w2, p2) and µ (w, p) = p2λ − λ by ΠP (p2). By plugging the value of w, p and µ into (4.3):

√ √ L ηp2 p p r p2 − p1 ΠP (p2) =r + λ − − p1 (1 − 2η) p2 + √ = √ (r − r1) 2 p2 p2 (6.22) √ H p r ΠP (p2) =r + λ − λ p2 + √ (6.23) p2

∗ ∗ ∗ In such case the principal’s optimization problem is: maxp∈[p1,p2] ΠP (w , p; µ (w , p)) where:

 ηp √ √ r  r + λ − − p1 (1 − 2η) p + √ , for p ∈ [p1, p2) ∗ ∗ ∗ 2 p ΠP (w , p; µ (w , p)) =  L H  max ΠP (p2), ΠP (p2) , for p = p2

√ √ √ √ Deﬁne x ≡ p, the expression r +λ−ηp/2− p1 (1 − 2η) p + r/ p can be restated

122 2 √ as f(x) = r + λ − ηx /2 − p1 ((1 − 2η)x + r/x). Maximizing f(x) with respect to x √ √ √ √ √ over [ p1, p2] is equivalent to maximizing r + λ − ηp/2 − p1 (1 − 2η) p + r/ p

with respect to p over [p1, p2] in the sense that

 2 ηp p √ r argmax r + λ − − p1 (1 − 2η) p + √ =  argmax√ √ f(x) p∈[p ,p ] 2 p 1 2 x∈[ p1, p2]

From Lemma 6.14, p2 > r2 > λ and we examine the following subcases:

∗ Subcase r ∈ (0, λ]: According to Lemma 6.15 part (a), p = p1. It is covered by the

case (w, p) ∈ D(6.7) and the principal does not propose a contract.

∗ Subcase r ∈ (λ, r2): According to Lemma 6.15 part (b), p = pcu and the principal’s ∗ ∗ ∗ ∗ ∗ expected proﬁt rate is ΠP (w , p ; µ (w , p )) > ΠP ((1 − η/2)p1, p1; 0) = ηp1/2 > 0. ∗ p Therefore the principal proposes a contract with w = ηpcu/2+2 (1 − η)pcuλ−λ ∗ ∗ ∗ ∗ p and p = pcu that induces the agent to install µ (w , p ) = (1 − η)pcuλ − λ.

∗ Subcase r ∈ [r2, p2]: From Lemma 6.15 part (c), p = p2 and from Proposition 6.13 ∗ ∗ ∗ ∗ ∗ part (a) and (b) the principal’s expected proﬁt rate is ΠP (w , p ; µ (w , p )) =

L ΠP (p2) > ΠP ((1 − η/2)p1, p1; 0) = ηp1/2 > 0. Therefore the principal proposes a ∗ ∗ ∗ ∗ ∗ contract with w = w2 and p = p2 that induces the agent to install µ (w , p ) = p (1 − η)p2λ − λ.

∗ Subcase r > p2: According to Lemma 6.15 part (c), p = p2 and according to Propo- ∗ ∗ ∗ ∗ ∗ H sition 6.13 part (c) her expected proﬁt rate is ΠP (w , p ; µ (w , p )) = ΠP (p2) > L ΠP (p2) > ΠP ((1 − η/2)p1, p1; 0) = ηp1/2 > 0. Therefore the principal pro- ∗ ∗ poses a contract with w = w2 and p = p2 that induces the agent to install ∗ ∗ ∗ p µ (w , p ) = p2λ − λ.

Case (w, p) ∈ D(6.9) ∪ D(6.10): According to Proposition 6.10 part (d), if (w, p) ∈ D(6.10), then in case the principal makes an oﬀer, the agent accepts the contract and installs

√ ∗ √ pλ−λ. Since ∂ΠP /∂w = −1 < 0, therefore w = 2 pλ−λ. According to Proposition

6.10 part (c) and Proposition 6.13, if (w, p) ∈ D(6.9) (which implies p = p2), then in case p the principal makes an oﬀer, the agent accepts the contract and installs (1 − η)p2λ−λ

123 p p p if r ∈ (0, p2), installs either (1 − η)p2λ−λ or p2λ−λ if r = p2, or installs p2λ−λ ∗ L if r > p2. Since ∂ΠP /∂w = −1 < 0, therefore w = w2. Recall the deﬁnition of ΠP (p2) H and ΠP (p2) (see equation (6.22) and (6.23)). Thus the principal’s optimization problem ∗ ∗ ∗ is maxp≥p2 ΠP (w , p; µ (w , p)) where:

 L H  max ΠP (p2), ΠP (p2) , for p = p2 ∗ ∗ ∗  ΠP (w , p; µ (w , p)) = √ √ r  r + λ − λ p + √ , for p > p  p 2

√ √ √ √ Deﬁne x ≡ p, the expression r + λ − λ p + r/ p can be restated as f(x) = √ √ r + λ − λ (x + r/x). Maximizing f(x) with respect to x ≥ p2 is equivalent to √ √ √ maximizing r + λ − λ p + r/ p with respect to p ≥ p2 in the sense that

 2 √ √ r argmax r + λ − λ p + √ = argmax√ f(x) p≥p2 p x≥ p2

According to Lemma 6.14, p2 > r1. Also note that limη→0+ r1 = 2λ and according to

Lemma 6.8 limη→8/9− r1 = −∞. Therefore we examine the following subcases:

∗ Subcase r ∈ (0, max {0, r1}]: According to Lemma 6.16 part (a), p = p2. Accord- ∗ ∗ ∗ ∗ ∗ L ing to Proposition 6.13 part (a), ΠP (w , p ; µ (w , p )) = ΠP (p2) ≤ 0, therefore the principal does not propose a contract.

∗ Subcase r ∈ (max {0, r1} , p2]: According to Lemma 6.16 part (a), p = p2. Ac- ∗ ∗ ∗ ∗ ∗ L cording to Proposition 6.13 part (a) and (b), ΠP (w , p ; µ (w , p )) = ΠP (p2) > 0, ∗ ∗ therefore the principal proposes a contract with w = w2 and p = p2 that induces ∗ ∗ ∗ p the agent to install µ (w , p ) = (1 − η)p2λ − λ.

∗ Subcase r > p2: According to Proposition 6.16 part (b), p = r and the principal’s √ ∗ ∗ ∗ ∗ ∗ expected proﬁt rate is ΠP (w , p ; µ (w , p )) = r − 2 rλ + λ > 0. Thus the √ principal proposes a contract with w∗ = 2 rλ − λ and p∗ = r that induces the √ agent to install µ∗(w∗, p∗) = rλ − λ.

To summarize, if r ∈ (0, λ], then the principal does not propose a contract. If r ∈ (λ, r2),

124 ∗ ∗ p then the principal offers (w , p ) = ηpcu/2 + 2 (1 − η)pcuλ − λ, pcu and the agent installs ∗ ∗ ∗ p ∗ ∗ capacity µ (w , p ) = (1 − η)pcuλ − λ. If r ∈ [r2, p2], then the principal offers (w , p ) = ∗ ∗ ∗ p (w2, p2) and the agent installs capacity µ (w , p ) = (1 − η)p2λ − λ. If r > p2, then √ the principal offers (w∗, p∗) = 2 rλ − λ, r and the agent installs capacity µ∗(w∗, p∗) = √ rλ − λ.

Theorem 6.17 indicates that the existence of a contract acceptable by weakly risk-seeking agent is determined exogenously by r, λ, and η.

6.2 Optimal strategies for the moderately risk-seeking agent

For the moderately risk-seeking agent we ﬁrst derive the agent’s optimal strategy. The agent’s optimization problem is deﬁned in (6.3).

Notation:

2λ p ≡ √ (6.24) 3 2 + η − 2 2η

and the following identity is veriﬁed using the deﬁnition of p3:

η w ≡ 1 − p = 2pp λ − λ (6.25) 3 2 3 3

Note that p3 and w3 are functions of λ and η. However we suppress the exogenous parameters (λ, η).

√ Lemma 6.18. Let 2 > η ≥ 8/9 and λ > 0, then 2λ/ 2 + η − 2 2η > 4λ.

Proof. Let 2 > η ≥ 8/9 and λ > 0, then we have

2η ≥ 16/9 ⇔ p2η ≥ 4/3 > 1

⇔ 1 > 2 − p2η √ √ ⇔ 2/ 2 − pη > 2

125 ⇔ 2λ/ 2 + η − 2p2η > 4λ

2 We describe a moderately risk-seeking agent’s optimal response to any (w, p) ∈ R+ in Proposition 6.19.

Proposition 6.19. Consider a moderately risk-seeking agent (η ∈ [8/9, 2)).

(a) Given

η p ∈ (0, p ) and w ≥ 1 − p (6.26) 3 2

then the agent accepts the contract and installs µ∗(w, p) = 0 with resulting expected

∗ utility rate uA(µ (w, p); w, p) = w − (1 − η/2)p ≥ 0. The agent rejects the contract if

p ∈ (0, p3] and w ∈ (0, (1 − η/2)p).

(b) Given

p = p3 and w ≥ w3 (6.27)

then the agent accepts the contract and is indiﬀerent installing either µ∗(w, p) = 0 or

∗ p ∗ µ (w, p) = p3λ−λ. In both cases the agent’s expected utility rate is uA(µ (w, p); w, p) = p ∗ w − (1 − η/2)p3 = w − 2 p3λ + λ ≥ 0. If r ∈ (0, p3], then neither µ = 0 nor ∗ p µ = p3λ − λ leads to admissible solutions (see Deﬁnition 6.1). If r > p3, then there ∗ ∗ ∗ p exists w such that (w , p3), µ = p3λ − λ is the only admissible solution (for proof

see Proposition 6.20). He rejects the contract if p = p3 and w ∈ (0, w3).

p p > p3 and w ≥ 2 pλ − λ (6.28)

√ then the agent accepts the contract and installs µ∗(w, p) = pλ−λ resulting in expected

∗ √ utility rate uA(µ (w, p); w, p) = w − 2 pλ + λ ≥ 0. The agent rejects the contract if

126 √ p > p3 and w ∈ 0, 2 pλ − λ .

Proof. According to Table 6.1, the optimization of u(µ) when η ∈ [8/9, 1) versus η ∈ [1, 2) is diﬀerent. Therefore we prove the proposition separately for η ∈ [8/9, 1) and η ∈ [1, 2).

Case η ∈ [8/9, 1): According to Lemma 6.6 and 6.18, 4p1 > p1 ≥ p3 > 4λ. Figure 6.8 shows the shape of u(µ) when η ∈ [8/9, 1) and the value of p falls in diﬀerent ranges. The structure of the proof when η ∈ [8/9, 1) is depicted in Figure 6.9.

Case w ∈ (0, (1 − η/2)p) : u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Case w ≥ (1 − η/2)p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p ∈ (4λ, p1]: According to Table 6.1, there is a service capacity that maximizes u(µ) for µ ∈ [0, λ) and a service capacity that maximizes u(µ) for µ > λ. Denote the optimal

∗ ∗ √ Note that u (µλ(w, p))−u µ[0,λ)(w, p) = (1−η/2)p−2 pλ+λ. According to Lemma √ 6.5, 4λ > 2λ/ 2 + η + 2 2η . According to Lemma 6.6 and 6.18, p1 ≥ p3 > 4λ, therefore we examine the following subcases.

∗ ∗ Subcase p ∈ (4λ, p3): By Lemma 6.4 part (a), u µ[0,λ)(w, p) > u (µλ(w, p)), therefore the agent’s optimal service capacity is µ∗(w, p) = 0 and u(µ∗(w, p)) = w − (1 − η/2)p. Note that 1 − η/2 > 0.

127 u( µ ) u( µ )

0.934 0.936 0.938 0.940 0.942 0.944 0.946 0.948 0.95 0.96 0.97 0.98 0.99 0 0 iue68 lutaino h om of forms the of Illustration 6.8: Figure λ λ (c) (a) η η = = 0.9 0.9 p p

λ= λ , ∈ λ= λ ∈ 0.01 0.01 ( (0 µ µ µ p

u( ) ,

, 3 w

, w , = 4 = 1 p 1

0.76 0.78 0.80 0.82 0.84 0.86 λ ,

, 1 p

p ] = ] = 9.7 0 1 λ λ λ η (e) = 0.9

, λ= λ > p 128 0.01

µ u( µ ) u( µ )

, 4

w p = 1 1

0.87 0.88 0.89 0.90 0.91 0.940 0.945 0.950 0.955 0.960 ,

p = 45 0 0 λ u ( µ when ) λ λ (d) (b) η η = = p 0.9 p 0.9

, ∈

, λ= λ ∈ λ= λ η 0.01 ( 0.01 (4 p µ µ ∈

1 ,

λ, ,

w , = = 4 [8 1 p 1

p ,

3 p

/ p 1 = ) = 9 ] 25 7 λ , λ 1) η w ∈ 0, 1 − p Reject. 2 p ∈ (0, 4λ] η w ≥ 1 − p µ∗ = 0 2

η w ∈ 0, 1 − p Reject. 2

p ∈ (4λ, p3) η w ≥ 1 − p µ∗ = 0 2

w ∈ (0, w3) Reject.

p ∈ (4λ, p1] p = p3

∗ ∗ p w ≥ w3 µ = 0 or µ = p3λ − λ

√ w ∈ 0, 2 pλ − λ Reject.

8 η ∈ , 1 p ∈ (p3, p1] 9 √ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

√ w ∈ 0, 2 pλ − λ Reject.

p ∈ (p1, 4p1] √ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

√ w ∈ 0, 2 pλ − λ Reject.

p > 4p1 √ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

Figure 6.9: Structure of the proof for Proposition 6.19 when η ∈ [8/9, 1)

Subsubcase w ∈ (0, (1 − η/2)p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract. Subsubcase w ≥ (1 − η/2)p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

∗ ∗ Subcase p = p3: According to Lemma 6.4 part (c), u µ[0,λ)(w, p) = u (µλ(w, p)), ∗ ∗ indicating that installing µ[0,λ)(w, p3) or µλ(w, p3) leads to the same agent’s expected utility rate. Therefore the agent is indiﬀerent about installing µ∗(w, p) = 0 √ or µ∗(w, p) = pλ − λ. Still, the capacity value has to lead to admissible solutions

(see Proposition 6.20). Recall that by deﬁnition w3 = (1 − η/2)p3 (see (6.25)). Note that 1 − η/2 > 0.

∗ Subsubcase w ∈ (0, (1 − η/2)p3): u(µ (w, p)) < 0, therefore the agent rejects

129 the contract.

∗ Subsubcase w ≥ (1 − η/2)p3: u(µ (w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

∗ ∗ Subcase p ∈ (p3, p1]: From Lemma 6.4 part (b), u (µλ(w, p)) > u µ[0,λ)(w, p) , thus √ the agent’s optimal service capacity is µ∗(w, p) = pλ − λ and u(µ∗(w, p)) = √ √ w − 2 pλ + λ. Since p > p3 > 4λ ⇒ 2 pλ − λ > 3λ > 0, therefore we examine the following subcases. √ Subsubcase w ∈ 0, 2 pλ − λ: u(µ∗(w, p)) < 0, thus the agent rejects the contract. √ Subsubcase w ≥ 2 pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

Case p ∈ (p1, 4p1]: According to Table 6.1, there is a service capacity that maximizes u(µ) for µ ∈ (0, λ] and a service capacity that maximizes u(µ) for µ > λ. Denote the

∗ optimal service capacity in (0, λ] by µ(0,λ](w, p). From ﬁrst order condition the optimal ∗ p ∗ service capacity is µ(0,λ](w, p) = (1 − η)pλ − λ and from (6.3) u µ(0,λ](w, p) = w − ηp/2 − 2p(1 − η)pλ + λ. Denote the optimal service capacity for µ > λ by

∗ ∗ that generates a higher expected utility rate. Note that u (µλ(w, p))−u µ(0,λ](w, p) = √ √ ηp/2 − 2 1 − 1 − η pλ. According to Lemma 6.8, p1 ≥ p2, therefore according to ∗ ∗ Lemma 6.7 part (b), u (µλ(w, p)) > u µ(0,λ](w, p) , the agent’s optimal service capacity ∗ √ ∗ √ √ is µ (w, p) = pλ−λ and u(µ (w, p)) = w−2 pλ+λ. Since p > p1 > 4λ ⇒ 2 pλ−λ > 3λ > 0, therefore we examine the following subcases.

√ Subcase w ∈ 0, 2 pλ − λ: u(µ∗(w, p)) < 0, therefore the agent rejects the contract. √ Subcase w ≥ 2 pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

130 Case p > 4p1: According to Table 6.1, the service capacity that maximizes u(µ) satisﬁes µ > λ. From the ﬁrst order condition the agent’s optimal service capacity is µ∗(w, p) =

√ ∗ √ √ pλ − λ and u(µ (w, p)) = w − 2 pλ + λ. Since p > 4p1 > 4λ ⇒ 2 pλ − λ > 3λ > 0, therefore we examine the following subcases.

This complete the proof for Proposition 6.19 when η ∈ [8/9, 1).

Case η ∈ [1, 2): Note that 4λ > 0 > p1 > 4p1 and according to Lemma 6.18, p3 > 4λ.

Therefore p3 > 4λ > 0 > p1 > 4p1. Figure 6.10 depicts the shape of u(µ) when η ∈ [1, 2) and the value of p falls in diﬀerent ranges. The structure of the proof when η ∈ [1, 2) is depicted in Figure 6.11.

Subcase w ∈ (0, (1 − η/2)p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subcase w ≥ (1 − η/2)p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

Case p > 4λ: According to Table 6.1, there is a service capacity that maximizes u(µ) for µ ∈ [0, λ) and a service capacity that maximizes u(µ) for µ > λ. Denote the optimal

∗ service capacity in [0, λ) by µ[0,λ)(w, p). Since u(µ) is decreasing with respect to µ ∗ ∗ over [0, λ), therefore µ[0,λ)(w, p) = 0 and from (6.3) u µ[0,λ)(w, p) = w − (1 − η/2)p. ∗ Denote the optimal service capacity for µ > λ by µλ(w, p). From ﬁrst order condition

131 Subcase subcases. following the examine we utility expected higher 4 with 6.5, one Lemma the installs he that Note and capacities rate. service two the of one µ λ ∗ ( ,p w, u( µ ) oeteaetsotmlsriecpct is capacity (1 service optimal agent’s the fore = ) − 0.95 0.96 0.97 0.98 0.99 0 η/ p √ 2) ∈ iue61:Ilsrto ftefrsof forms the of Illustration 6.10: Figure pλ > λ p (4 λ oeta 1 that Note . − (a) u λ, η λ 2 ( = 1.5 µ λ/ p n rm(6.3) from and p

, λ ∗ λ= λ ∈ ( 3 0.01 ,p w, ): (0 µ + 2 u( µ )

, w 4 = yLma64pr (a), part 6.4 Lemma By 1 ))

0.70 0.75 0.80 0.85 λ

η p ] = 0 − 2 2 + λ − u η/ √ λ µ 2 2 u [0 ∗ η > η ( (c) ,λ = µ 1.5 ) 0. n codn oLma6.18, Lemma to according and λ ∗

( , λ= λ 132 ( > p ,p w, ,p w, 0.01

µ u( µ )

w p ) = 3 )= )) 1

0.89 0.90 0.91 0.92 0.93 0.94 0.95 ,

p = (1 = 60 0 λ w µ u u ∗ − ( − ( µ λ ,p w, 2 µ (b) when ) η/ √ [0 ∗ η = ,λ pλ 1.5 2) p and 0 = ) )

, ( λ= λ ∈ p + ,p w, 0.01 (4 − λ µ η

, λ,

h gn a ochoose to has agent The . w 2 ∈ ) = 1 p √

3 p [1 = ) pλ u > 20 , λ 2) u + p ( ( µ 3 µ λ ∗ > λ ∗ codn to According . ( ( ,p w, ,p w, 4 λ therefore , )= )) ) there- )), w − η w ∈ 0, 1 − p Reject. 2 p ∈ (0, 4λ] η w ≥ 1 − p µ∗ = 0 2

η w ∈ 0, 1 − p Reject. 2

η ∈ [1, 2) p ∈ (4λ, p3) η w ≥ 1 − p µ∗ = 0 2

w ∈ (0, w3) Reject.

p > 4λ p = p3

∗ ∗ p w ≥ w3 µ = 0 or µ = p3λ − λ

√ w ∈ 0, 2 pλ − λ Reject.

p > p3

√ √ w ≥ 2 pλ − λ µ∗ = pλ − λ

Figure 6.11: Structure of the proof for Proposition 6.19 when η ∈ [1, 2)

Subsubcase w ∈ (0, (1 − η/2)p): u(µ∗(w, p)) < 0, therefore the agent rejects the contract.

Subsubcase w ≥ (1 − η/2)p: u(µ∗(w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

(see Proposition 6.20). Recall that by deﬁnition of w3 = (1 − η/2)p3 (see (6.25))

and 1 − η/2 > 0 ⇒ w3 > 0.

∗ Subsubcase w ∈ (0, w3): u(µ (w, p)) < 0, therefore the agent rejects the contract.

∗ Subsubcase w ≥ w3: u(µ (w, p)) ≥ 0, thus the agent would accept the contract if oﬀered.

∗ ∗ Subcase p > p3: From Lemma 6.4 part (b), u (µλ(w, p)) > u µ[0,λ)(w, p) , therefore √ the agent’s optimal service capacity is µ∗(w, p) = pλ − λ and u(µ∗(w, p)) =

133 √ √ w − 2 pλ + λ. Since p > p3 > 4λ ⇒ 2 pλ − λ > 3λ > 0, therefore we examine the following subcases. √ Subsubcase w ∈ 0, 2 pλ − λ: u(µ∗(w, p)) < 0, thus the agent rejects the contract. √ Subsubcase w ≥ 2 pλ − λ: u(µ∗(w, p)) ≥ 0, therefore the agent would accept the contract if oﬀered.

This completes the proof for Proposition 6.19 when η ∈ [1, 2).

In summary, under the exogenous market conditions such that a contract between the principal and a moderately risk-seeking agent is feasible (see Theorem 6.22 later), only one formula is needed for the agent to compute his optimal service capacity: µ∗(w, p) = √ pλ − λ > 0.

The conditions when a moderately risk-seeking agent accepts the contract can be depicted by the shaded areas in Figure 6.12, where η = 1. The two shaded areas with diﬀerent grey scales represent conditions (6.26) and (6.28) under which the agent accepts the contract

but responds diﬀerently. The lower bound function of the shaded area (denoted by w0(p))

represents the set of oﬀers that give the agent zero expected utility rate. w0(p) is deﬁned as follows:

 η  1 − p when p ∈ (0, p3] w0(p) = 2  √  2 pλ − λ when p > p3

√ √ √ √ Since lim − w0(p) = lim + w0(p) = 2 + η λ/ 2 − η , therefore w0(p) is con- p→p3 p→p3

tinuous everywhere over interval p ∈ +. However since lim − dw0(p)/dp = 1 − η/2 6= R p→p3 p 1 − η/2 = lim + dw0(p)/dp, therefore w0(p) is not diﬀerentiable at p = p . p→p3 3

134 w

∗ ∗ µ =0 µ = p λ − λ 3 p

/2)

η (1−

w=(1− η /2)p w=2 p λ − λ 0 p 0 p3

Figure 6.12: Conditions when a moderately risk-seeking agent accepts the contract with η = 1

6.2.1 Sensitivity analysis of a moderately risk-seeking agent’s optimal strategy

Since the principal does not propose a contract that even if accepted will result in zero service capacity, therefore the only viable case is when the agent accepts the contract and √ installs positive service capacity: µ∗(w, p) = pλ − λ. In such a case the agent’s optimal strategy is identical to the optimal strategy for risk-neutral agent. According to (6.10) the √ compensation rate w is bounded below by 2 pλ−λ = pP (1)+µ∗(w, p), with the term pP (1) representing the expected penalty rate charged by the principal when the optimal capacity is installed. It indicates that the agent must be reimbursed for the expected penalty rate and the cost of service capacity.

√ The optimal service capacity pλ−λ depends on the penalty rate p and the failure rate λ. Its derivatives are ∂µ∗/∂p = pλ/4p > 0 and ∂µ∗/∂λ = pp/4λ − 1. These derivatives suggest that given the failure rate, the agent will increase the service capacity when the penalty rate

135 √ increases. Note that pλ − λ, as a function of λ, increases when p/4 > λ. From conditions √ (6.27) and (6.28) the agent installs service capacity pλ − λ when p ≥ p3, and according ∗ to Lemma 6.18 we have p3 > 4λ. Therefore we have p > 4λ ⇒ p/4 > λ ⇒ ∂µ /∂λ > 0. Thus, given the penalty rate, the agent will increase the service capacity when the failure rate increases.

√ The agent’s optimal expected utility rate when installing capacity µ∗(w, p) = pλ − λ

∗ p p ∂uA/∂λ = − p/λ + 1, and from Proposition 6.10 p ≥ p3 > 4λ ⇒ − p/λ + 1 < 0, therefore the agent’s optimal expected utility rate also decreases with the failure rate.

Summary: Recall that given the set of oﬀers {(w, p): p ∈ (0, λ], w ≥ p} a risk-neutral agent √ would accept the contract, install µ∗(w, p) = 0. When (w, p): p > λ, w ≥ 2 pλ − λ he √ would accept the contract, install µ∗(w, p) = pλ − λ and realize an expected utility rate √ u(µ∗(w, p); w, p) = w − 2 pλ + λ. By comparing the optimal capacities of a moderately risk-seeking agent to that of a risk-neutral agent, three conclusions are drawn.

1. The principal has to set a higher p in order to induce a moderately risk-seeking agent to install a positive service capacity versus a risk-neutral agent (p > λ for risk-neutral √ √ 2 agent, p > p3 = 2λ/ 2 − η > λ for moderately risk-seeking agent).

2. A moderately risk-seeking agent would install the same positive service capacity as a √ risk-neutral agent ( pλ − λ).

3. Given w and p, an agent who accepts the contract and subsequently installs a positive service capacity will receive the same expected utility rate when he is moderately risk- seeking (η ∈ [8/9, 2)) as risk-neutral (η = 0). This is because the positive service capacity installed by a moderately risk-seeking agent is high enough such that the limiting probability of the principal’s unit being operational is no less than 1/2 and the

136 agent perceives zero risk (see (6.1)).

6.2.2 Principal’s optimal strategy

We now proceed to derive the principal’s optimal strategy. Anticipating the agent’s optimal selection of µ∗(w, p) the principal chooses w and p to maximize her expected proﬁt rate by solving the optimization problem

∗ ∗ rµ (w, p) pλ max ΠP (w, p; µ (w, p)) = max − w + (6.29) w>0,p>0 w>0,p>0 λ + µ∗(w, p) λ + µ∗(w, p)

∗ ∗ ∗ where the principal’s optimal solution values are (w , p ) = argmaxw>0,p>0 ΠP (w, p; µ (w, p)).

Before we describe the principal’s optimal strategy, we reexamine the case when the

principal offers p = p3 and w ≥ w3, under which the agent is indifferent about installing two different service capacities. The selected solutions ((w, p), µ) have to be admissible solutions (see Definition 6.1). We state this case formally in Proposition 6.20.

Proposition 6.20. Suppose a moderately risk-seeking agent and principal. Assume that the principal’s oﬀers are constrained to {(w, p): p = p3, w ≥ w3}.

(a) If r ∈ (0, p3], then the principal does not propose a contract.

∗ p (b) If r > p3, then the agent installs µ = p3λ − λ if oﬀered a contract.

2 Proof. Note that for w ≥ w3 we have ∂ΠP (w, p3; µ)/∂µ = (r − p3)λ/(λ + µ) . Deﬁne µL ≡ 0 p and µH ≡ p3λ − λ and note that µH > µL. If r ∈ (0, p3), then ∂ΠP /∂µ < 0, therefore

((w, p3), µL) ((w, p3), µH ) and the agent would install µL if oﬀered a contract. However

condition (c) in Deﬁnition 6.1 requires that w ≥ p3, therefore ΠP (w, p3; µL) = −w + p3 ≤ 0 and the principal would not propose a contract. If r = p3, then ∂ΠP /∂µ = 0, therefore

the agent installs either µL or µH if oﬀered a contract. However in such case the principal’s

expected proﬁt rate is ΠP (w, p3; µL) = ΠP (w, p3; µH ) = −w + p3, which is non-positive due

to condition (c) in Deﬁnition 6.1, thus the principal would not propose a contract. If r > p3, then ∂ΠP /∂µ > 0 and ((w, p3), µH ) ((w, p3), µL). If the principal oﬀers a contract (where

137 the conditions will be discussed in detail in Theorem 6.22 that follows), then by Deﬁnition

6.1 only µH leads to admissible solutions. √ Lemma 6.21. Consider max √ f(x) where f(x) = r + λ − λ (x + r/x) and denote x≥ p3 x∗ = argmax √ f(x). The solutions to this optimization problem are: x≥ p3

∗ √ (a) x = p3 if r ∈ (0, p3].

∗ √ (b) x = r if r > p3. √ Proof. Note that f(x) is continuous and diﬀerentiable for x ≥ p3: √ df(x) √ r d2f(x) 2r λ = − λ 1 − and = − < 0 dx x2 dx2 x3 √ df(x) λ df(x) √ = (r − p ) and = − λ < 0 dx √ p 3 dx x= p3 3 x→+∞

√ ∗ √ √ If r ∈ (0, p3], then f(x) is decreasing for x ≥ p3, therefore x = p3. If r > p3, then √ f(x) is increasing in the neighborhood above x = p3. Since f(x) is decreasing as x → +∞ 2 2 ∗ √ and since f(x) is concave (d f(x)/dx < 0), therefore x > p3 is solved from ﬁrst order ∗ √ condition df(x)/dx|x=x∗ = 0 ⇒ x = r.

The principal’s optimal strategy is described in Theorem 6.22. Recall that Proposition

2 6.19 describes the agent’s optimal response to each pair (w, p) ∈ R+. Since the principal will not propose a contract that is going to be rejected by a moderately risk-seeking (MRS) agent,

2 therefore Theorem 6.22 only considers pairs (w, p) ∈ R+ that result in agent’s non-negative expected utility rate. Deﬁne

D(6.26) ≡ {(w, p) that satisﬁes (6.26) when η ∈ [8/9, 2)}

D(6.27) ≡ {(w, p) that satisﬁes (6.27) when η ∈ [8/9, 2)} (6.30) D(6.28) ≡ {(w, p) that satisﬁes (6.28) when η ∈ [8/9, 2)}

DMRS ≡ D(6.26) ∪ D(6.27) ∪ D(6.28)

Theorem 6.22. Given a moderately risk-seeking agent and (w, p) ∈ DMRS.

(a) If r ∈ (0, p3], then the principal does not propose a contract.

138 (b) If r > p3, then the principal’s oﬀer and the capacity installed by the agent are

√ √ (w∗, p∗) = 2 rλ − λ, r and µ∗(w∗, p∗) = rλ − λ (6.31)

and the principal’s expected proﬁt rate is

√ ∗ ∗ ∗ ∗ ∗ ΠP (w , p ; µ (w , p )) = r − 2 rλ + λ (6.32)

Proof. The structure of the proof for Theorem 6.22 is depicted in Figure 6.13.

(w, p) ∈ D(6.26) No contract oﬀered.

Risk-Neutral Principal with Moderately Risk-Seeking Agent r ∈ (0, p3] No contract oﬀered.

(w, p) ∈ D(6.27) ∪ D(6.28) √ w∗ = 2 rλ − λ and p∗ = r r > p √ 3 and µ∗ = rλ − λ

Figure 6.13: Structure of the proof for Theorem 6.22

Case (w, p) ∈ D(6.26): According to Proposition 6.19 part (a), in case the principal makes an oﬀer, the agent accepts the contract but does not install any service capacity. Since

∗ ∗ ∗ ∗ ∂ΠP /∂w = −1 < 0, thus we have w = (1−η/2)p and from (4.3) ΠP (w , p; µ (w , p)) = −w∗ + p = ηp/2 > 0. However in such case p > w∗ = (1 − η/2)p, which violates condition (c) in Deﬁnition 6.1, therefore ((w∗ = (1 − η/2)p, p), µ∗ = 0) is not an admissible solution and the principal does not propose a contract.

Case (w, p) ∈ D(6.27) ∪ D(6.28): According to Proposition 6.19 part (c), if (w, p) ∈ D(6.28), then in case the principal makes an oﬀer, the agent accepts the contract and installs

√ ∗ √ pλ−λ. Since ∂ΠP /∂w = −1 < 0, therefore w = 2 pλ−λ. According to Proposition

6.19 part (b) and Proposition 6.20, if (w, p) ∈ D(6.27) (which implies p = p3), then the p principal does not propose a contract if r ∈ (0, p3], or installs p3λ − λ in case the ∗ principal makes an oﬀer when r > p3. Since ∂ΠP /∂w = −1 < 0, therefore w = w3. ∗ Denote the principal’s expected proﬁt rate when (w, p) = (w3, p3) and µ (w, p) = 0

139 L by ΠP (p3), and denote the principal’s expected proﬁt rate when (w, p) = (w3, p3) and ∗ p H µ (w, p) = p3λ − λ by ΠP (p3). By plugging the value of w, p and µ into (4.3):

η ηp ΠL (p ) = − 1 − p + p = 3 (6.33) P 3 2 3 3 2 √ H p r ΠP (p3) = r + λ − λ p3 + √ (6.34) p3

∗ ∗ ∗ In such case the principal’s optimization problem is maxp≥p3 ΠP (w , p; µ (w , p)) where:

 L H  max ΠP (p3), ΠP (p3) , for p = p3 ∗ ∗ ∗  ΠP (w , p; µ (w , p)) = √ √ r  r + λ − λ p + √ , for p > p  p 3

√ √ √ √ Deﬁne x ≡ p, the expression r + λ − λ p + r/ p can be restated as f(x) = √ √ r + λ − λ (x + r/x). Maximizing f(x) with respect to x ≥ p3 is equivalent to √ √ √ maximizing r + λ − λ p + r/ p with respect to p ≥ p3 in the sense that

 2 √ √ r argmax r + λ − λ p + √ = argmax√ f(x) p≥p3 p x≥ p3

Therefore we examine the following subcases.

∗ Subcase r ∈ (0, p3]: According to Lemma 6.21 part (a), p = p3 and according to Proposition 6.20 part (a) the principal does not propose a contract.

∗ Subcase r > p3: According to Lemma 6.21 part (b), p = r and the principal’s ex- √ ∗ ∗ ∗ ∗ ∗ H L pected proﬁt rate is ΠP (w , p ; µ (w , p )) = r − 2 rλ + λ > ΠP (p3) > ΠP (p3) = √ ∗ ∗ ηp3/2 > 0. Thus the principal proposes a contract with w = 2 rλ−λ and p = r √ that induces the agent to install µ∗(w∗, p∗) = rλ − λ.

To summarize, if r ∈ (0, p3], then the principal does not propose a contract. If r > p3, then √ the principal oﬀers (w∗, p∗) = 2 rλ − λ, r and the agent installs capacity µ∗(w∗, p∗) = √ rλ − λ.

Theorem 6.22 indicates that the existence of a contract for moderately risk-seeking agent

140 is determined exogenously by the r, λ, and η.

6.3 Optimal strategies for the strongly risk-seeking agent

We start by deriving the strongly risk-seeking agent’s optimal strategy. The agent’s optimization problem is deﬁned in (6.3).

First a technical lemma.

Lemma 6.23. Let η > 2 and λ > 0. 2λ η √ (a) If √ > p > 0, then 1 − p − 2 pλ + λ > 0. 2 + η + 2 2η 2 2λ η √ (b) If p > √ , then 0 > 1 − p − 2 pλ + λ. 2 + η + 2 2η 2 2λ η √ (c) If p = √ , then 1 − p − 2 pλ + λ = 0. 2 + η + 2 2η 2 √ √ Proof. Let η > 2 and λ > 0. Deﬁne x ≡ p and a ≡ λ and restate the expression (1 − √ η/2)p−2 pλ+λ as (1−η/2)x2−2ax+a2 with x > 0 and a > 0. The solutions to the quadratic √ √ √ √ √ √ equation (1−η/2)x2 −2ax+a2 = 0 for x are 2a/ 2 + η > 0 and 0 > 2a/ 2 − η. √ √ √ √ Therefore if 2a/ 2 + η > x > 0, or equivalently, 2a2/ 2 + η + 2 2η > x2 > 0, then √ √ (1 − η/2)x2 − 2ax + a2 > 0. Replacing x by p and a by λ we obtain (a). The proofs for (b) and (c) are similar.

We describe a strongly risk-seeking agent’s optimal response to any possible oﬀered con-

2 tract (w, p) ∈ R+ in Proposition 6.24.

Proposition 6.24. Consider a strongly risk-seeking agent (η ≥ 2). ∀ w > 0 and p > 0, the agent accepts the contract and installs µ∗(w, p) = 0 with resulting expected utility rate

∗ uA(µ (w, p); w, p) = w − (1 − η/2)p > 0.

Proof. Figure 6.14 shows the shape of u(µ) when η ≥ 2 and the value of p falls in diﬀerent ranges. The structure of the proof when η ≥ 2 is depicted in Figure 6.15.

141 Case Case vr[0 over µ hoeoeo h w evc aaiisadh ntlsteoewt ihrexpected higher with one η the that installs Note he and rate. capacities utility service two the of one choose µ for capacity service optimal the Denote [0 in capacity service is capacity service optimal agent’s the fore gn ol cettecnrc foﬀered. if contract the accept would agent w p > p λ ∗ ,then 2, = ∈ − ( ∈ ,p w, µ

[0 u( ) (1 (0 4 λ , λ , = ) λ 0.96 0.98 1.00 1.02 − , n evc aaiyta maximizes that capacity service a and ) : iue61:Srcueo h ro o rpsto .4when 6.24 Proposition for proof the of Structure 6.15: Figure ,therefore ), 4 0 η/ λ codn oTbe61 hr sasriecpct htmaximizes that capacity service a is there 6.1, Table to According √ ]: u 2) pλ ( iue61:Ilsrto ftefrsof forms the of Illustration 6.14: Figure p codn oTbe6.1, Table to According µ λ oeta 1 that Note . (a) λ ∗ − ( η ,p w, = 4 p λ

, λ= λ ∈ µ n rm(6.3) from and 0.01 λ , )) (0 [0 ∗ µ

,λ w η by ) , − u = 4 ≥ ) 1

( λ , (

p u 2 ,p w, µ ] = 2 λ λ ∗ µ ( µ − ,p w, n rm(6.3) from and 0 = ) [0 ∗ [0 ∗ ,λ η/ ,λ p ) ) )) ( ( > p 2 ∈ ,p w, ,p w, (0 ≤ − u 4 , 4 λ u ( λ 142 .Since ). ) u ,therefore 0, ] µ ( λ > µ µ λ ∗ u( µ ) ( sdcesn ihrsetto respect with decreasing is ) = µ ,p w, [0 ∗ µ 0.94 0.96 0.98 1.00 1.02 1.04 − ,λ ∗ ( ) 2 0 )= )) ,p w, by ( √ ,p w, u pλ u ( µ n rm(6.3) from and 0 = ) µ ( u λ ∗ ∀ ) µ λ w u sdcesn ihrsetto respect with decreasing is ) + µ µ ( ( for ) ∗ ∗ µ ,p w, > w − (b) 0 = 0 = when ) η λ µ (1 = = n since and , 4 [0 ∗

2 , λ= λ .Fo rtodrcondition order ﬁrst From ). > p ,λ √ λ > µ 0.01 0, ) pλ µ (

, −

,p w, w 4 u η = λ 1 +

( , η/

≥ p µ = eoeteoptimal the Denote . ) 5 ∗ λ λ 2) 2 ( h gn a to has agent The . η ,p w, > p p = ≥ − w 2 )) 4 2 µ u − λ √ > ( ≥ µ (1 ⇔ pλ ∗ n the and 0 .There- 0. ( − ,p w, 2 u + √ ( η/ µ λ λ> pλ for ) )= )) If . 2) p µ . √ ∗ ∗ 4λ ⇔ 0 > −3λ > −2 pλ + λ, we have u µ[0,λ)(w, p) > u (µλ(w, p)). If η > 2, √ then according to Lemma 6.5 and 6.23 part (b), p > 4λ > 2λ/ 2 + η + 2 2η ⇒

∗ ∗ ∗ u µ[0,λ)(w, p) > u (µλ(w, p)). Thus the agent’s optimal service capacity is µ (w, p) = 0 and u(µ∗(w, p)) = w − (1 − η/2)p. Note that 1 − η/2 ≤ 0, therefore ∀ w > 0, u(µ∗(w, p)) > 0 and the agent would accept the contract if oﬀered.

Proposition 6.24 indicates that a strongly risk-seeking agent does not commit any capacity, therefore the principal does not propose any contract, which we state formally in Theorem 6.25.

Theorem 6.25. A principal never oﬀers a contract to a strongly risk-seeking agent!

Proof. According to Proposition 6.24, the agent accepts the contract but does not install

2 any service capacity for all (w, p) ∈ R+. In such case the principal’s expected proﬁt rate ∗ is ΠP (w, p; µ (w, p)) = −w + p. Since condition (c) of Deﬁnition 6.1 requires that w ≥ p,

∗ therefore ΠP (w, p; µ (w, p)) ≤ 0 and the principal does not propose a contract to a strongly risk-seeking agent!

6.4 Risk-seeking agent – a summary

Recall the definition of p2, r2 and p3 from (6.5), (6.12) and (6.24). The conditions when a principal makes contract offers to a risk-seeking agent is depicted by the shaded areas in Figure 6.16. The horizontal axis represents the agent’s risk coefficient η, and the vertical axis represents the revenue rate generated by the principal’s equipment unit, which is exogenously determined by the market. The principal makes different offers to the agent when (r, η) is in the three shaded areas with different gray scales. We define

  p2 for η ∈ (0, 8/9) p23 ≡  p3 for η ∈ [8/9, 2)

Since limη→(8/9)− p23 = limη→(8/9)+ p23 = 9λ and limη→(8/9)− ∂p23/∂η = limη→(8/9)+ ∂p23/∂η =

81λ/4, and note that limη→2− p23 = limη→2− p3 = +∞, therefore p23 is continuous and

143 differentiable everywhere over (0, 2). In Figure 6.16 we only describe the conditions of a risk- neutral principal making offers to a weakly and moderately risk-seeking agent (η ∈ (0, 2)), because the principal never makes a contract offer to a strongly risk-seeking agent (η ≥ 2).

+ ∞

∗ p = r

9 λ

p2 ∗ p = p2 4 λ r2

∗ p = pcu λ λ 0 η 8 0 Weakly Risk−Seeking Moderately Risk−Seeking 9

Figure 6.16: Conditions when a principal makes contract oﬀers to a risk-seeking agent

The revenue rate parameter r is determined exogenously by the market, and we assume that the principal is only interested in operating the equipment when the revenue rate is sufficiently high, specifically r > p23. In such case weakly and moderately risk-seeking agents would behave exactly the same as a risk-neutral agent, and a strongly risk-seeking agent will never be offered a contract.

144 7 Summary: Formulating Principal-Agent Service Contracts for a Revenue Generating Unit

In this paper we examine a basic principal-agent arrangement for contracting an exclusive equipment repair service supplier. The system setting consists of one principal, one agent, and one revenue generating unit that breaks down from time to time and needs to be repaired when a failure occurs. Our assumptions are that the risk-neutral principal maximizes her expected proﬁt rate given market driven revenue rate r collected during the unit’s uptime, the unit’s failure rate λ, and the agent’s risk attitude η. We consider diﬀerent agent types – risk neutral, weakly risk-averse, strongly risk-averse, weakly risk-seeking, moderate risk- seeking, and strongly risk-seeking. As is common in a principal-agent context the principal cannot contract directly for the agent’s service capacity µ. The nature of the principal-agent contract is that the principal supports the agent at a compensation rate w > 0 but imposes on the agent a penalty rate p > 0 during the time the unit is down. We note that the nature of the contract does not change if the w is paid to the agent only during the unit’s uptime. In fact, the two contract versions are equivalent (see Observation 4.1).

The main contribution of this paper is in the complete analysis of the contractual details that have to be addressed in the agreement between the unit’s owner and the supplier of repair services. Our pedestrian assumptions are that the failure rate of the equipment unit is a constant λ, the repair time duration has an exponential distribution with a constant repair rate µ. Furthermore, we do not restrict the contract to a specific period of time, rather the contract can be for undetermined time. With the assumption that both the principal and the agent are infinitely rational the surprising outcome is that calculating the optimal strategies for the two parties in all circumstances can be accomplished with an aid of small number of formulas – 7 sets in total. That is, given exogenously determined values of market driven revenue rate, equipment’s failure rate, repair capacity marginal cost, and the type of a repair agent, it is straight forward to calculate principal’s optimal contract offer if one exists, together with agent’s optimal service capacity decision. An optimal contract consists of

145 compensation rate w and penalty rate p, both determined by the principal, and the capacity value of µ determined by the agent.

Our analysis of the above principal-agent cooperation is divided into three main parts based on agent’s type starting with risk-neutral agent. The second part examines the case of a contracting a risk-averse agent followed by the analysis of a contract given a risk-seeking agent. To our knowledge analysis of principal-agent with risk-seeking agent has not received much coverage in the literature.

As for the analysis of principal-agent construct given a risk-neutral agent, for the entire range of exogenous parameters’ values, it can be summarized for the principal by one set of √ formulas calculating optimal compensation rate w∗ = 2 rλ − λ and optimal penalty rate √ p∗ = r. The agent’s optimal capacity rate formula is µ∗(w∗, p∗) = rλ − λ. We note that this case has the property that without checking if the given market conditions guarantee the existence of a contract, by calculating principal’s optimal contract terms w∗ and p∗ and agent’s optimal capacity value µ∗(w∗, p∗), we simultaneously verify contract existence if the resulting µ∗(w∗, p∗) is positive. If the optimal capacity value is zero or negative, then it means that the given market conditions do not support a service contract. It also important to note that, for our principal-agent given a risk-neutral agent, if an optimal contract is feasible then it is also eﬃcient.

When considering a risk-averse agent the ﬁrst task is to decide on the appropriate mathematical expression that captures the agent’s disutility with regard to his revenue dispersion. After examining risk premium expressions in the literature we opted for a new risk expression not yet seen in the literature. We express agent’s disutility as ηp(1/2 − |1/2 − λ/(λ + µ)|). This measure of agent’s utility value due to his revenue ﬂuctuation is introduced and discussed in Section 5. The main points are that the risk expression acts like standard deviation and is unit-wise compatible with other terms of agent’s utility. In high revenue industry, if the principal contracts with a risk-averse agent with the risk disutility measured by the

146 dispersion of the agent’s revenue stream, then agent’s risk-aversion reduces the principal’s optimal penalty rate and leads to deterioration of the equipment unit’s performance. Fur- thermore, with risk-averse agent the principal is strictly worse oﬀ in relation to risk-neutral agent and the social welfare is reduced as the agent’s risk-aversion increases.

We divided risk-averse agents into two types based on risk intensity parameter η. That is, for η ∈ (0, 4/5) we refer to the agent as weakly risk-averse (Subsection 5.1) and for η ≥ 4/5 we refer to the agent as strongly risk-averse (Subsection 5.2). A weakly risk-averse agent has only two formulas to consider: (i) µ∗(w, p) = p(1 − η)pλ − λ or (ii) µ∗(w, p) = p(1 + η)pλ − λ. Only one formula, the same as (ii), is suﬃcient given a strongly risk-averse agent. Formula (i) exists only for WRA agent because when the penalty rate is low, the savings from reducing the service capacity is more prominent than the increase in the penalty charge, providing an incentive for the agent to reduce the optimal service capacity, which deteriorates the performance of the principal’s equipment unit. When the penalty rate becomes high, WRA agent increases his service capacity to reduce the penalty charge, which results in formula (ii).

For a risk-seeking agent we adopt a risk premium expression that reﬂects the expected amount at stake instead of the dispersion of his revenue stream. Our new risk premium expression is consistent with the theoretical developments and empirical evidences regarding the properties of risk in recent literature. We express the agent’s risk premium as

−ηp (λ/(λ + µ) − 1/2)+, which is unit-wise compatible with other terms of agent’s utility (see Section 6). If the principal contracts with a risk-seeking agent with low penalty rate, then the agent’s risk-seeking deteriorates the performance of the principal’s equipment unit. If the principal contract with a risk-seeking agent with high penalty rate, then she can achieve the same equipment performance and contract eﬃciency as with a risk-neutral agent. How- ever a principal never contracts with a strongly risk-seeking agent.

We categorize risk-seeking agents into three types based on η – risk intensity parameter. That is, for η ∈ (0, 8/9) we refer to the agent as weakly risk-seeking (Subsection 6.1), for

147 η ∈ [8/9, 2) we refer to the agent as moderately risk-seeking (Subsection 6.2) and the agent as strongly risk-seeking (Subsection 6.3). A weakly risk-seeking agent has only two formulas √ to consider: (i) µ∗(w, p) = p(1 − η)pλ − λ or (ii) µ∗(w, p) = pλ − λ. Only one formula, the same as (ii), is suﬃcient given a moderately risk-seeking agent. A strongly risk-seeking agent never commits any service capacity. Formula (i) exists only for WRS agent because when the penalty rate is low, the risk premium covers the penalty charge thus provides an incentive for the agent to reduce the optimal service capacity compared to risk-neutral. When the penalty rate increases, WRS agent increases his service capacity to reduce the penalty charge that cannot be covered by risk premium, which results in formula (ii).

7.1 Interpreting Table 7.1

Table 7.1 summarizes the formulas for calculating the principal’s optimal contract terms and the agent’s optimal service capacity when a contract is supported by exogenous market and industry conditions. Mutually exclusive exogenous conditions that support a contract are listed in the column labeled “Exogenous Condition”, and the formulas of the principal’s optimal contract terms and the agent’s optimal capacities are listed in the column labeled “Principal’s Formula” and “Agent’s Formula” respectively.

If a set of specific market and industry values are observed, namely the value of the agent’s risk coefficient η (or η), the revenue rate r, and the failure rate λ, then these values can be validated against the exogenous conditions listed in the table. If the set of values satisfies a certain condition, then the principal’s formula and the agent’s formula corresponding to that condition can be used to calculate the optimal contract terms and the optimal capacity. No contract is supported if the set of values does not satisfy any condition listed in the table.

To verify that whether the observed values of η, r, and λ satisfy a certain condition, one

has to calculate the values that separate the range of r into diﬀerent intervals, including p2,

p3, r2, r3, p4, r4, r2, p2, and p3. Recall that p2 and p3 are defined in (5.5), r2 and r3 are defined in (5.12), p4 and w4 are defined in (5.24), r4 is defined in (5.29), p2 is defined in (6.5),

148 Table 7.1: Summary of the optimal principal-agent contract formulas under exogenous conditions

Exogenous Condition Principal’s Formula Agent’s Formula Agent’s Type Revenue

η = 0 √ √ r > λ (w∗, p∗) = 2 rλ − λ, r µ∗(w∗, p∗) = rλ − λ (RN)

∗ ∗ p ∗ ∗ ∗ p r ∈ (p2, p3] (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu µ (w , p ) = (1 − η)pcuλ − λ

∗ ∗ p ∗ ∗ ∗ p (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu µ (w , p ) = (1 − η)pcuλ − λ 4 r ∈ (p3, r2) η ∈ 0, 5 ∗ ∗ ∗ ∗ ∗ p (w , p ) = (w3, p3) µ (w , p ) = (1 + η)p3λ − λ (WRA)

∗ ∗ ∗ ∗ ∗ p r ∈ [r2, r3] (w , p ) = (w3, p3) µ (w , p ) = (1 + η)p3λ − λ

r(1 + η)rλ r r(1 + η)rλ r > r (w∗, p∗) = 2 − λ, µ∗(w∗, p∗) = − λ 3 1 + 2η 1 + 2η 1 + 2η

4 r ∈ (p , r ] (w∗, p∗) = (w , p ) µ∗(w∗, p∗) = p(1 + η)p λ − λ η ≥ 4 4 4 4 4 5 (SRA) r(1 + η)rλ r r(1 + η)rλ r > r (w∗, p∗) = 2 − λ, µ∗(w∗, p∗) = − λ 4 1 + 2η 1 + 2η 1 + 2η ηp r ∈ (λ, r ) (w∗, p∗) = cu + 2p(1 − η)p λ − λ, p µ∗(w∗, p∗) = p(1 − η)p λ − λ 2 2 cu cu cu 8 η ∈ 0, 9 ∗ ∗ ∗ ∗ ∗ p r ∈ [r2, p ] (w , p ) = (w2, p ) µ (w , p ) = (1 − η)p λ − λ (WRS) 2 2 2 √ √ ∗ ∗ ∗ ∗ ∗ r > p2 (w , p ) = 2 rλ − λ, r µ (w , p ) = rλ − λ 8 √ √ η ∈ , 2 ∗ ∗ ∗ ∗ ∗ 9 r > p3 (w , p ) = 2 rλ − λ, r µ (w , p ) = rλ − λ (MRS)

r2 is deﬁned in (6.12), and p3 is deﬁned in (6.24). Furthermore, to calculate the principal’s optimal contract terms and the agent’s optimal capacity, one may need to calculate the val-

ues of pcu, w3, pcu, and w2. Recall that pcu can be calculated using (5.13), w3 is deﬁned in

(5.6), pcu can be calculated using (6.13), and w2 is deﬁned in (6.6).

Speciﬁcally, note that when the revenue rate r ∈ (p3, r2), there are two sets of formulas listed in the table to calculate the principal’s optimal contract terms and the agent’s optimal

149 capacity:

∗ ∗ p ∗ ∗ ∗ p (w , p ) = ηpcu + 2 (1 − η)pcuλ − λ, pcu , µ (w , p ) = (1 − η)pcuλ − λ

∗ ∗ ∗ ∗ ∗ p (w , p ) = (w3, p3) , µ (w , p ) = (1 + η)p3λ − λ

According to Proposition 5.20 it is difficult to identify the principal’s optimal offer when r ∈ (p3, r2) due to the difficulty of computing pcu (see equation (5.13)). However, given the value of η, r, and λ, the principal’s expected profit rate of both offers can be calculated (see the formulas for calculating the principal’s expected profit rate in Proposition 5.20), and the offer with higher expected profit rate should be selected by the principal.

In summary, this work provides a small set of formulas that exhaustively covers the computing of Pareto optimal principal-agent contract oﬀer and corresponding service capacity for any values of market and industry parameters.

8 Pooling Principals

Suppose a number of technologically similar commercial entities each operating a revenue generating unit. It is reasonable to presuppose that the units fail from time to time and require repair services. We refer to the commercial entities as principals and assume a large population of principals who outsource their units’ repair function with each principal independently proposing a service contract to a qualiﬁed service provider. The service providers are referred to as agents. Just consider an academic department with a single high cost – high volume sophisticated copier that breaks down from time to time and requires expert service. In such cases it is common for a single repair agent to pool multiple principals’ service demands and sign an individual service contract with each of the pooled principals (Haque and Armstrong, 2007). In this chapter we examine the determination of contract parameters controlled by the principals and the agent in a setting in which multiple principals independently contract with an agent for the provision of repair services. We assume that the agent is of good standing and can choose the subset of principals to contract with.

150 A question of interest is how are the contracting principals compared? Whose contract oﬀer is accepted by the agent given that both the agent and each principal are rational proﬁt maximizers?

An example from insurance industry is described in Luenberger (1995), with insurance companies as principals and an insurance seeking individual as the sole agent. The contracting time sequence in the insurance market is straight forward: The insurance company offers a premium schedule as a function of the coverage level, and the insurance seeking individual subsequently chooses the coverage level and determines his effort to maximize his contract utility. An insurance company anticipates the insurance seeking individual’s self-interest optimizing response and therefore designs a premium schedule that maximizes the company’s expected profit. As is common in insurance industry the choice of the insurance seeking individual’s effort is not observed thus the insurance company is exposed to moral hazard. If there are multiple insurance companies in the insurance market, and if the premium schedule is a convex increasing function of the coverage level, then the insurance seeking individual has an incentive to purchase a number of policies with relatively small coverage levels from different insurance companies to obtain a large aggregate coverage level. The agent does not reveal the policies he has purchased from multiple insurance companies. This way the insurance seeking individual (the agent) takes advantage of the superadditivity in coverage level and the information asymmetry between him and the insurance companies (the principals) to reduce the total premium for the same aggregate coverage level by purchasing several insurance policies with small coverage level from multiple insurance companies.

This work is a follow up analysis on principal-agent performance based service contracts of the type introduced in Kim et al. (2010) and comprehensively analyzed in Zeng and Dror (2015a). We assume that the agent has a good reputation and therefore is in high demand and can choose whose contract oﬀer to accept. We assume that each of the principals acts independently anticipating the agent’s optimal response to its contract proposal without considering the existence of similar principals in the market place. In such a setting we

151 expect that a rational agent would accept the contract offers that collectively benefit him the most and adjusts his service capacity to maximize his expected profit rate. Alternatively the agent can consider installing a dedicated capacity for each principal and a setup of multiple replications of a single principal setting as described in Zeng and Dror (2015a). In this paper our focus is on the opportunities and conditions that exist in a multiple principals single agent contract configurations. We address two questions: (1) When does it ‘pay’ for the agent to contract with multiple principals? and (2) how to assess each principal’s fair contribution to the agent’s profit?

8.1 An outline

In Section 9 we present a model that describes the contractual interplay between an agent and multiple principals, where the failures of different principals’ units are assumed to be interdependent. In Section 10 we examine how an agent determines a subset of two principals to contract with that collectively benefits him the most. We then resort to numerical simulations to study how the agent determines the optimal subset of more than two principals to contract with. The simulation study is time consuming because not only Monte Carlo simulation is required to learn the interplay between the agent and the principals, but also a numerical optimization is necessary in order to determine the agent’s expected profit rate. Based on the implications of the simulation results from Section 10 we refined our model in Section 11 and describe the contractual relationship between an agent and multiple principals given that the agent is able to identify the best subset of principals. In order to assess each principal’s fair contribution to the agent’s expected profit rate, in Section 12 we restate the principal-agent options of a contractual relationship in the form of a cooperative game and suggest two allocation schemes that assign to each principal a contribution to agent’s expected profit rate. Section 13 provides a summary of the results.

152 9 Principal-Agent Model with Multiple Principals

Assume that each of the principals and the agent are risk-neutral inﬁnitely rational proﬁt maximizers. A principal owns and operates a single revenue generating unit that experiences a random uptime before failing. Once a unit fails, it remains inoperative until it is repaired (brought back to its operational state). We refer to the principal as ‘she’ and the agent as ‘he’.

Consider a finite set of principals who independently contract an agent for indefinite length of time (the contract can be terminated/renewed any time in an infinite time horizon) to oversee and repair her revenue generating unit whenever it fails. In response the agent installs a service capacity and repairs the contracted principals’ failed units one at a time subject to a queuing discipline for principals’ units in a failed state. In terms of the queuing disciplines we restrict the agent’s choice to be either First-Come-First-Serve (FCFS) or Head-Of-the-Line-Preemption (HOLP) based on which queuing discipline results in a higher expected profit rate. The reason of restricting the queuing disciplines to FCFS and HOLP is that they represent two extremes in the set of service rules, because the average waiting times of failed units is completely independent of the units’ priorities under FCFS while the average waiting times of failed units increase with decreasing priority of the units (Chandra, 1986). With an agent committed to repair services with multiple principals the downtimes of the corresponding principals’ units have a different pattern compared to that of a single principal and a single agent, because the agent, at times, may be engaged in repairing a unit of one principal when another principal’s unit fails. In essence, when the agent serves multiple principals, the downtimes of any principal’s unit consist of the unit’s waiting times and service times.

We are not the first to pool principals. The setting that considers pooling multiple principals by a single agent is discussed in Jackson and Pascual (2008), who proposed a tractable model under the assumption that the principals’ revenue generating units are identical (the same failure rate and the same revenue rate). This simplifying assumption implies that there is no difference between different principals. In contrast, our model allows the principals’

153 units to have diﬀerent failure rates and diﬀerent revenue rates.

An agent pooling the repair services for multiple principals repairs all the failed units in question. A similar problem, referred to as the resource pooling game, has been examined in several studies (Anily and Haviv 2010, Karsten et al. 2011 and Karsten et al. 2015). Their resource pooling scheme consists of multiple service providers, each serving a stream of customers that is essentially equivalent to the sequence of failures of a principal’s unit in our principal-agent framework. Their goal is to examine whether service providers can improve the service quality (represented by the customers’ average waiting times) by pooling their service capacities to serve the union of their customer streams. Anily and Haviv (2010) assumed that the service providers can pool their service capacities into a single server and model their system as a M/M/1/∞ queue. Karsten et al. (2011) extended the single server system to a multiple-servers system with blocking, where the service provider (the agent) is charged a ﬁxed penalty for each of the blocked customers (a customer is blocked if he ﬁnds no available server upon arrival and is rejected immediately). Such a system can be modeled as a M/G/s/s queue. Karsten et al. (2015) extended the analysis to a multiple-servers system modeled as a M/G/s/∞ queue in which a customer waits for service if no server is available upon her arrival, and the service provider is charged a penalty that is linear in the waiting times of each of the waiting customers.

An important assumption shared by the models of Anily and Haviv (2010), Karsten et al. (2011) and Karsten et al. (2015) is that the customer streams are independent Poisson pro- cesses such that the union of customer streams is still a Poisson process with the arrival rate being the sum of the arrival rates of all individual customer streams. We do not impose such an assumption for two reasons. First, the assumption of Poisson failure arrivals does not hold in our setting because a failure cannot occur to an inoperative unit. Second, and more important we relax the assumption of independence between failures of diﬀerent principals’ units.

154 Interdependence of failures: In a number of industries the revenue generating units are owned by a few oligopolistic suppliers (such as commercial aircraft engines, power generating gas turbines, nuclear reactors, etc.) and the principals have a limited number of brands and models to choose from. Therefore different principals in such industry may operate technically the same or similar units. Although the principals operate their units independently, we cannot exclude the fact that commonality of components, designs, and operational procedures introduces dependencies of failures across these units (Sampson and Smith, 1982), meaning that if one principal’s unit experiences specific component failures, then it is more/less likely to happen to other principals’ units. For example a company that provides repair and maintenance services for gas turbines may serve different power plants who operate the same model of gas turbines to generate electricity. If at a certain moment one of these gas turbines fails due to metal fatigue of the turbine blades, then the other gas turbines would have a higher likelihood of a similar failure occurring soon because they share the same turbine design. This is an example of positive dependencies of failures. On the other hand, if one of these gas turbines fails due to a bug in the control software, and the manufacturer quickly fixes the problem and updates the software on all other gas turbines of the same model, the remaining operational gas turbines would have a lower chance experi- encing a similar failure. This is an example of negative dependencies of failures. Our model accounts for the dependencies between failures and the agent leverages these dependencies when choosing the principals to contract with. We assume that the agent is in such a high demand that he can select (pick and choose) to contract with a set of principals who have any desired interdependencies. Although from theoretical point of view an agent of good standing can pool as many principals with the desired interdependencies, as he wants, in practice the number of principals who seeks the agent’s service is finite. Moreover, once the agent depletes the pool of principals with the desired interdependencies adding one more principal may in fact reduce the agent’s expected profit rate, providing a disincentive for the agent to pool more principals.

155 9.1 The principal-agent model

Denote the agent by A and the subpopulation of principals who seek to contract agent A’s repair services by P. We assume that agent A chooses to contract with some nonempty ﬁnite set of principals N ⊆ P. In the analysis of contractual setting for a nonempty S ⊆ N we model the service interactions of any S ⊆ N with A as a continuous time Markov process.

Assume that principal i’s unit generates an expected revenue of ri (ri > 0) $ per unit of uptime with ri determined exogenously by the market, and the unit’s uptimes follow an exponential distribution governed by the unit’s failure rate λi (λi > 0). Denote the agent’s service capacity when serving principal subset S ⊆ N (|S| ≥ 1) by µS (µS ≥ 0). The definition of system state is different under different queuing disciplines. We consider two queuing disciplines: FCFS and HOLP. Under FCFS queuing discipline, the agent begins to repair a failed unit immediately if he is idle at the instant the unit fails. Otherwise the failed unit joins the queue of failed units at the tail and waits until its turn to be repaired. Under HOLP queuing discipline, a priority order is defined by the agent over the units of N. When a unit fails, an idle agent begins to repair this unit immediately, or if all other failed units (including the one being repaired by the agent) are of lower priority. Once the agent completes the repair of a unit, he starts to repair the next failed unit with the highest priority, or he idles if the repair queue is empty. The HOLP queuing discipline implies that at any moment the agent is working on the failed unit with the highest priority.

Under FCFS queuing discipline, we deﬁne the state of the system with nonempty principals set S ⊆ N and agent A as the queue of failed units from S in their FCFS order including the one being repaired by the agent. Denote the state by sequence (i1, ··· , in) where i1 represents the head of the queue and n, (0 < n ≤ |S|) represents the length of the queue. If all units are operational then the state is denoted by ∅. In total the number of

P|S| |S| P|S| |S| states under FCFS queuing discipline is k=0 k k! = k=0 |S|!/(|S| − k)! > 2 .

Under HOLP queuing discipline, we deﬁne the state of the system for nonempty set

156 S ⊆ N and agent A as the queue of failed units from S in their decreasing HOLP priority or-

der, including the one being repaired by the agent. Denote the state by sequence (i1, ··· , in)

where i1 represents the head of the queue, which has the highest HOLP priority in the queue, and n, (0 < n ≤ |S|) represents the length of the queue. For clarity we assume that the

HOLP priority order is strict with j < k implying that ij has higher priority than ik. In total the number of states under HOLP queuing discipline is 2|S|.

We relax the assumption shared by Anily and Haviv (2010), Karsten et al. (2011), and Karsten et al. (2015) models and assume that the likelihood of failures of diﬀerent units may be pairwise interdependent. Speciﬁcally we assume that the failure rate of an operational

unit i not only depends on its own failure rate λi, but also depends on the set of failed units in the repair queue.

We model the pairwise dependencies of units’ failures as follows: Given any failed unit i and any operational unit j, we represent the dependency of unit j’s likelihood to fail

based on the state of unit i by multiplying j’s failure rate λj by a positive factor ψij. We

refer to ψij, i, j ∈ N as dependency coeﬃcients assumed to be exogenously determined. If

the failure rate of unit j decreases given unit i’s failure status (λj > ψijλj > 0 ⇒ 1 >

ψij > 0), then unit j’s failures are negatively dependent on unit i’s failures; if unit j’s

failure rate increases when unit i is in the failed state (ψijλj > λj > 0 ⇒ ψij > 1), then unit j’s failures are positively dependent on unit i’s failures; if unit j’s failure rate remains

unchanged when unit i is inoperative (ψijλj = λj > 0 ⇒ ψij = 1), then unit j’s failures are independent of unit i’s failures. Element ψij captures the direction and the strength of the interdependencies: elements ψij ∈ (0, 1) for negative dependencies, and elements ψij > 1

for positive dependencies, with ψij = 1 for no dependency between the failures. Note that

ψii = 1. In a steady state we assume that the dependencies are time homogeneous and

symmetric: ψij(t) = ψij, ∀i, j ∈ N and ∀t ∈ R+, and ψij = ψji, ∀i, j ∈ N. We represent the dependency coeﬃcients of principals set S ⊆ N in the form of a |S| × |S| square matrix

ΨS ≡ {ψij, i, j ∈ S}. If our Markov process reached a state with failed units F ⊆ S, then

157 Q the failure rate of an operational unit j ∈ S \ F in this state is i∈F ψij λj. Note that the role of ΨS is similar to the covariance matrices that describe the interdependencies of demands between the diﬀerent outlets as in Hartman and Dror (2003) and Burer and Dror (2012), where the stochastic demands of each outlet is assumed to be normally distributed with ﬁxed standard deviation.

9.2 Performance based contract and the agent’s proﬁt

Performance based principal-agent contracts codify agreements between the two rational profit maximizing parties (Straub 2009 and Kim et al. 2010). To derive such contracts we begin by examining a profit function of a risk-neutral agent given risk-neutral principals. The agent’s profit function describes the agent’s expected profit rate in an indefinite time contract assuming the corresponding Markov process is in steady state. Principal i’s proposed compensation rate, is denoted by wi (wi > 0) paid to the agent by i for each unit of contract time irrespective of the state of principal i’s unit. We denote principal i’s penalty

rate by pi (pi > 0), collected by principal i from the agent for each unit of downtime of principal i’s unit. We denote the limiting probability of principal i’s unit being inoperative independent of the state of other units by P(i), i ∈ S, with P(i) equal the sum of the limiting probabilities of states in which principal i’s unit is inoperative. Recall that we deﬁne the

state of the Markov process as the queue of failed units (i1, ··· , in), and denote the limiting

probability of state (i1, ··· , in) when the Markov process is in steady state by P (i1, ··· , in). Therefore the summation is over the limiting probabilities of all the states in which i is a member: X P(i) = P (i1, ··· , in)

{(i1,··· ,in)3i}

Therefore the expected penalty rate collected by principal i can be calculated by piP(i).

The agent’s expected proﬁt rate serving principals set S ⊆ N is a function of the agent’s service capacity µS, principals’ compensation rates wS ≡ {wi, i ∈ S}, principals’ penalty rates pS ≡ {pi, i ∈ S} while accounting for the dependency coeﬃcients between the princi-

158 pals by ΨS = {ψij, i, j ∈ S}. Furthermore, the agent’s expected proﬁt rate is also a function

of the revenue rate rS ≡ {ri, i ∈ S} and the failure rate ΛS ≡ {λi, i ∈ S}.

We denote the agent’s proﬁt function by Π(µS; wS, pS, ΨS) when the agent contracts

individually the principals in set S ⊆ N. Exogenous parameters rS and ΛS are omitted for simplicity. Similarly to the agent’s expected proﬁt rate deﬁned in Zeng and Dror (2015a):

X X Π(µS; wS, pS, ΨS) = wi − piP(i) − µS for wi > 0, pi > 0, µS ≥ 0, i ∈ S (9.1) i∈S i∈S

The marginal cost of service capacity is normalized to 1 as in Zeng and Dror (2015a).

Since the principals are assumed to be rational and risk-neutral, they anticipate the agent’s optimal response to the individual principal’s contract oﬀers. Therefore according to Zeng and Dror (2015a), the individual contract oﬀers of principals in set S are √ {(wi, pi) = 2 riλi − λi, ri , i ∈ S}.

Therefore the agent’s expected proﬁt rate (9.1) is:

X p X Π(µS; wS, pS, ΨS) = 2 riλi − λi − riP(i) − µS (9.2) i∈S i∈S

The agent, given the contracts with principals set S ⊆ N, chooses the service capacity that maximizes his expected proﬁt rate by solving the following optimization problem:

( ) X p X max Π(µS; wS, pS, ΨS) = max 2 riλi − λi − riP(i) − µS (9.3) µS ≥0 µS ≥0 i∈S i∈S

Denote the corresponding agent’s optimal capacity by µ∗ = argmax Π(µ ; w , p , Ψ ). S µS ≥0 S S S S

9.3 Principal’s proﬁt

Since we assume that the agent is risk-neutral, then principal i’s optimal contract oﬀer is √ ∗ ∗ (wi , pi ) = 2 riλi − λi, ri (Zeng and Dror, 2015a). If a contract between a single risk-

159 neutral principal and a single risk-neutral agent is supported by exogenous condition (i.e.,

ri > λi), then given that the agent accepts principal i’s contract oﬀer represented by (wi, pi) = √ 2 riλi − λi, ri , the principal’s expected proﬁt rate is

ΠPi (wi, pi; µS) =riP(0) − wi + piP(1) p =riP(0) + riP(1) − 2 riλi + λi √ p 2 = ri − λi > 0 (9.4)

Compared to the principal’s optimal expected profit rate with a risk-neutral agent, the expected profit rate of each of the principals in S remains the same as if she is the only principal contracting for the agent’s repair services, and all surplus profit generated by the set S ∪ {A} (if any) goes to the agent.

10 Selecting Principals Set N

The agent pools the service demands of multiple principals as long as doing so allows him to realize a superadditive expected profit rate with respect to the principals set he contracts, where superadditivity means that the agent’s expected profit rate with respect to any principals set S he contracts with is no less than the sum of his expected profit rates when he contracts individually with all the principal subsets in any partition of S (Peleg and Sudhölter,2007). To justify the option of contracting the agent’s repair services to multiple principals we assume that the agent is in high demand and therefore can choose any finite principals set N ⊆ P with the desired interdependencies between the principals in N (the desired values of ψij, i, j ∈ N). Therefore the agent’s objective is to select the principals set N that collectively benefit him the most, and at the same time installing the service capacity that maximizes his expected profit rate serving the set N.

10.1 Selecting a ﬁnite set of homogeneous principals

For simplicity we ﬁrst examine the case of an agent contracting a set of homogeneous principals N ⊆ P, where homogeneity means the principals’ units have the same revenue rate

160 and initially experience the same failure rate (for i ∈ N, ri = r > 0 and λi = λ > 0). This is a special case of the model described in Section 9, where the principals are assumed to be heterogeneous with the revenue rates and the failure rates of different units not necessarily the same. In the following subsections, we first formulate the agent’s optimization problem of selecting and serving a finite set of principals under FCFS and HOLP queuing disciplines, then we resort to numerical simulations to demonstrate how the agent selects a set of principals that benefit him the most.

10.1.1 Selecting two principals under FCFS queuing discipline

Consider principals set N = {1, 2}. Recall that under FCFS queuing discipline the system is in state ∅ when there is no failed unit, and is represented by the queue of failed units 2 2 2 otherwise. Therefore in total there are 0 0! + 1 1! + 2 2! = 5 states: ∅, (1), (2), (1, 2) and (2, 1), with the limiting probabilities of these states are denoted by P (∅), P (1), P (2), P (1, 2) and P (2, 1) respectively. The state transition diagram is illustrated by Figure 10.1.

ψ12λ 1 1,2 λ µ µ N N µN ∅ λ

µN ψ12λ 2 2,1

Figure 10.1: State transition diagram under FCFS when |N| = 2

The corresponding balance equations are:

2λP (∅) = µN (P (1) + P (2))

(µN + ψ12λ)P (1) = λP (∅) + µN P (2, 1)

(µN + ψ12λ)P (2) = λP (∅) + µN P (1, 2)

µN P (1, 2) = ψ12λP (1)

161 µN P (2, 1) = ψ12λP (2)

Therefore limiting probabilities P (1), P (2), P (1, 2) and P (2, 1) can be expressed in terms of P (∅):

2 λ ψ12λ P (1) = P (2) = P (∅) and P (1, 2) = P (2, 1) = 2 P (∅) µN µN

Since the limiting probabilities sum up to 1, therefore

2 µN P (∅) = 2 2 µN + 2λµN + 2ψ12λ λµN P (1) =P (2) = 2 2 µN + 2λµN + 2ψ12λ 2 ψ12λ P (1, 2) =P (2, 1) = 2 2 µN + 2λµN + 2ψ12λ

Recall that P(i) denotes the limiting probability of principal i’s unit being inoperative independent of the state of other units, thus P(i) equals the sum of the limiting probabilities of states in which principal i’s unit is inoperative. For example, P(1) represents the limiting probability of principal 1’s unit being inoperative regardless of the state of principal 2’s unit, therefore P(1) equals the sum of the limiting probabilities of all the states that include principal 1’s unit. Therefore

λ(µN + 2ψ12λ) P(1) =P (1) + P (1, 2) + P (2, 1) = 2 2 µN + 2λµN + 2ψ12λ λ(µN + 2ψ12λ) P(2) =P (2) + P (1, 2) + P (2, 1) = 2 2 µN + 2λµN + 2ψ12λ

According to (9.2) the agent’s expected proﬁt rate is:

√ 2rλ(µN + 2ψ12λ) Π(µN ; wN , pN , ΨN ) = 2 2 rλ − λ − 2 2 − µN (10.1) µN + 2λµN + 2ψ12λ

162 Note that

dΠ(µ ) 2rλ (µ2 + 4ψ λµ + 2ψ λ2) N = N 12 N 12 − 1 dµ 2 2 2 N (µN + 2λµN + 2ψ12λ )

dΠ(µN ) r dΠ(µN ) = − 1 and = −1 dµ ψ λ dµ N µN =0 12 N µN →+∞

If the units’ revenue rate is suﬃciently high compared to the value of the failure rate:

r > ψ12λ (10.2) then dΠ(µ )/dµ | > 0 and there exists a µ∗ > 0 that satisﬁes the ﬁrst order condition N N µN =0 N

dΠ(µN )/dµN | ∗ = 0. The ﬁrst order condition can be rewritten as a quartic equation of µN =µN ∗ µN :

∗ 4 ∗ 3 ∗ 2 2 ∗ 3 µN + 4λµN + 2λ (2(1 + ψ12)λ − r) µN + 8ψ12λ (λ − r) µN + 4ψ12λ (ψ12λ − r) = 0 (10.3)

∗ Although formula for calculating the root of a quartic equation (an explicit expression of µN

as a function of λ, r and ψ12) exists (see Cardano and Witmer (1993)), its complex functional

∗ form makes it diﬃcult to analyze the agent’s optimal capacity µN and his optimal expected proﬁt rate, which equals:

√ 2rλ(µ∗ + 2ψ λ) Π(µ∗ ; w , p , Ψ ) = 2 2 rλ − λ − N 12 − µ∗ (10.4) N N N N ∗ 2 ∗ 2 N µN + 2λµN + 2ψ12λ

Based on (10.4) the agent’s problem of selecting a set of two principals N under FCFS queuing discipline can be formulated as the following optimization problem with respect to

ψ12:

√ ∗ ∗ 2rλ(µN + 2ψ12λ) ∗ max Π(µN ; wN , pN , ΨN ) = max 2 2 rλ − λ − − µN (10.5) ψ >0 ψ >0 ∗ 2 ∗ 2 12 12 µN + 2λµN + 2ψ12λ

163 10.1.2 Selecting two principals under HOLP queuing discipline

Consider the set of two principals N = {1, 2}. Without loss of generality, we assume that principal i’s unit has higher priority than principal j’s unit if i < j. Recall that under HOLP queuing discipline the state is denoted by ∅ when there is no failed unit, and the state is represented by the queue of failed units otherwise. Note that the failed units in the queue are always ordered in descending order based on their priority. Therefore in total there are 22 = 4 states: ∅, (1), (2), (1,2), with their limiting probabilities denoted by P (∅), P (1), P (2) and P (1, 2). The state transition diagram is illustrated in Figure 10.2.

1 λ ψ12λ

µN ∅ 1,2 λ λ ψ 12

µN µN 2

Figure 10.2: State transition diagram under HOLP when |N| = 2

The balanced equations are:

2λP (∅) =µN (P (1) + P (2))

(µN + ψ12λ)P (1) =λP (∅)

(µN + ψ12λ)P (2) =λP (∅) + µN P (1, 2)

µN P (1, 2) =ψ12λP (1) + ψ12λP (2)

Therefore limiting probabilities P (1), P (2) and P (1, 2) can be expressed in terms of P (∅):

2 λ λ(µN + 2ψ12λ) 2ψ12λ P (1) = P (∅), P (2) = P (∅) and P (1, 2) = 2 P (∅) µN + ψ12λ µN (µN + ψ12λ) µN

164 Since the limiting probabilities sum up to 1, therefore

2 2 µN λµN P (∅) = 2 2 ,P (1) = 2 2 µN + 2λµN + 2ψ12λ (µN + ψ12λ)(µN + 2λµN + 2ψ12λ ) 2 λ (µN + 2ψ12λ) µN 2ψ12λ P (2) = 2 2 ,P (1, 2) = 2 2 (µN + ψ12λ)(µN + 2λµN + 2ψ12λ ) µN + 2λµN + 2ψ12λ

Recall that P(i) denotes the limiting probability of principal i’s unit being inoperative independent of the state of other units, and is equal to the sum of the limiting probabilities of states in which principal i’s unit is inoperative. Therefore

2 2 2 3 λµN + 2ψ12λ µN + 2ψ12λ P(1) =P (1) + P (1, 2) = 2 2 (µN + ψ12λ)(µN + 2λµN + 2ψ12λ ) 2 2 2 3 λµN + 4ψ12λ µN + 2ψ12λ P(2) =P (2) + P (1, 2) = 2 2 (µN + ψ12λ)(µN + 2λµN + 2ψ12λ )

2 2 2 Note that P(2) − P(1) = 2ψ12λ µN / ((µN + ψ12λ)(µN + 2λµN + 2ψ12λ )) > 0 ⇒ P(2) > P(1), which implies that a unit with lower priority has higher probability of being inoperative than a unit with higher priority when the failure rates of all units are the same, since a failed unit with lower priority has to wait when a unit with higher priority is inoperative and being repaired. This result is consistent with the statement in Chandra (1986).

According to (9.2) the agent’s expected proﬁt rate is:

√ 2rλ (µN + 2ψ12λ) Π(µN ; wN , pN , ΨN ) = 2 2 rλ − λ − 2 2 − µN (10.6) µN + 2λµN + 2ψ12λ

Note that (10.6) is the same as (10.1) because when the principals are homogeneous, the agent does not distinguish between the units of different principals and the agent’s profit rate at any time instance only depends on the total number of units that are inoperative. There- fore under HOLP queuing discipline, if the units’ revenue rate is sufficiently high compared to the failure rate (see condition (10.2)), then the agent’s optimal service capacity satisfies first order condition (10.3), and the agent’s problem of selecting a set of two principals N is the same as (10.5).

165 Due to the complexity of optimization problem (10.5), it is diﬃcult to select a set of two homogeneous principals analytically. Since selecting a set of homogeneous principals is a special case of selecting a set of heterogeneous principals, therefore performing the latter task cannot be less diﬃcult than the former task.

10.2 Selecting a ﬁnite set of heterogeneous principals

Due to the analytic difficulty of selecting a set of heterogeneous principals, we resort to numerical simulations to understand how the agent selects a heterogeneous principals set N ⊆ P that collectively benefits him the most. Since the dependency coefficient between any pair of principals, say i and j, is assumed to be symmetric (i.e. ψij = ψji), therefore an agent’s problem of selecting the ‘best’ principals set N ⊆ P is equivalent to the following optimization problem with respect to all the dependency coefficients of the principal pairs in N:

∗ max Π(µN ; wN , pN , ΨN ) (10.7) {ψij >0|i

∗ ∗ ∗ Denote the optimal dependency matrix by ΨN ≡ {ψij|i, j ∈ N} where ψij = 1 for i = j, ∗ ∗ ∗ ∗ ψij ∈ argmax{ψij >0|i j. The total number of variables in (10.7) is |N|(|N| − 1)/2 increasing quadratically with |N|.

For example, say the agent considers contracts with principals set N, |N| = 3 under FCFS queuing discipline. By deﬁnition the dependency matrix is:

  1 ψ12 ψ13     ΨN = ψ12 1 ψ23   ψ13 ψ23 1

166 Then the agent’s problem of selecting principals set N can be formulated as:

∗ max Π(µN ; wN , pN , ΨN ) ψ12,ψ13,ψ23>0

∗ where Π(µN ; wN , pN , ΨN ) represents the agent’s optimal expected proﬁt rate under FCFS queuing discipline given the dependency matrix ΨN , and the optimal dependency matrix is:

 ∗ ∗  1 ψ12 ψ13   Ψ∗ =  ∗ ∗  N ψ12 1 ψ23   ∗ ∗ ψ13 ψ23 1

∗ ∗ ∗ ∗ where {ψ12, ψ13, ψ23} = argmaxψ12,ψ13,ψ23>0 Π(µN ; wN , pN , ΨN ).

Although we assume in this work that the dependency matrix is symmetric, it is probable in practice that the dependency matrix is asymmetric. An asymmetric dependency matrix makes the maximization problem (10.7) signiﬁcantly more diﬃcult to solve because the number of variables of (10.7) is doubled thus the dimension of the search space is doubled.

10.3 Simulation results

As demonstrated by the results for a set of two principals under FCFS and HOLP queuing disciplines (see (10.5)), ‘intractability’ is unavoidable when solving the optimization problem (10.7) with a ﬁnite principals set of any size. Therefore we resort to numerical simulations to better understand how the agent selects the ﬁnite principals set to contract. The simulation program is implemented using Java and it’s objective is to solve optimization problem (10.7) given the choice of queuing discipline, the values of revenue rates ri, failure rates λi and the size of principals set |N|, by means of Monte Carlo simulation.

Including the service capacity µN , (10.7) is a multivariate nonlinear optimization problem with 1+|N|(|N|−1)/2 variables: {µN }∪{ψij|i < j, i, j ∈ N}. Due to analytic diﬃculty issue, explicit expressions of partial derivatives of the objective function in (10.7) with respective

167 to variables {µN } ∪ {ψij|i < j, i, j ∈ N} are hard to derive, therefore we resort to direct search methods to numerically solve this multivariate optimization problem. Direct search methods only require evaluations of the objective function values, they neither compute nor approximate the derivatives of the objective function therefore they are often describes as “derivatives-free” (Lewis et al., 2000). We have implemented two of the most popular direct search methods: the Nelder-Mead simplex method, and the Covariance Matrix Adaptation Evolution Strategy.

Proposed by Nelder and Mead (1965), Nelder-Mead simplex method is a deterministic direct search algorithm for the maximization of a real-valued objective function of n variables, which depends on the comparison of function values at the (n+1) vertices of a general simplex, followed by the replacement of the vertex with the lowest value by another point. The simplex adapts itself to the local landscape in the search space, and contracts on to the ﬁnal maximum. Nelder-Mead simplex method is one of the most popular deterministic direct search methods (Lewis et al., 2000).

Proposed by Hansen and Ostermeier (1996), Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is a stochastic direct search algorithm that maximizes a nonlinear real- valued multivariate objective function, which follows the principles of biological evolution. In each of the search steps, a new population of search points is created by adding normally distributed random vectors with zero mean to an initial search point derived from the previous search step, and a subset of ‘better’ search points are selected out of these new search points to calculate the initial search point for the next step (Hansen and Ostermeier, 2001). CMA-ES is among one of the most powerful evolutionary algorithms for real-valued optimization with many successful applications (Igel et al., 2007).

Although an eﬃcient algorithm, we observe that when the number of principals |N| (and thus the number of variables of the optimization problem (10.7)) increases, Nelder-Mead simplex method becomes less stable and does not converge sometime (Kolda et al., 2003).

168 Therefore in our simulations we choose CMA-ES over Nelder-Mead simplex method as the numerical optimization methodology for solving (10.7).

The steps of our simulation program can be summarized as follows:

Step 1. User chooses the queuing discipline (FCFS or HOLP) and the values of parameters

|N|, ri, and λi for all i ∈ N.

P Step 2. The simulation program initializes µN = 2 i∈N λi and ψij = 1 for i, j ∈ N.

Step 3. The simulation program simulates the Markov process for a preset length of period.

Step 4. The simulation program calculates the downtimes of each principal’s unit and the agent’s proﬁt rate from the results of simulation from Step 3.

Step 5. CMA-ES algorithm evaluates if the agent’s proﬁt rate can be signiﬁcantly improved. If no, the simulation program terminates. Otherwise CMA-ES algorithm adjusts the

values of µN and ψij, i, j ∈ N and return to Step 3.

In following subsections, we ﬁrst present the simulation results for homogeneous principals, then we present the simulation results for heterogeneous principals. As demonstrated by the simulation results, the agent’s optimal expected proﬁt rate is convexly increasing with respect to the set of principals he contracts with.

10.3.1 Simulation results with homogeneous principals

The simulation setting with homogeneous principals is characterized by the values of revenue rate r, failure rate λ, size of principals set |N| and the queuing discipline. We ran simulations for the combinations of diﬀerent values of r, λ and |N| under a choice of queuing discipline from set {FCFS, HOLP}, where r ∈ {10, 15, 20}, λ ∈ {0.8, 1.0} and |N| ∈ {2, 3, 4}. The sample size of each simulation setting is 30, which means for each combination of revenue rate, failure rate, size of principals set and queuing discipline, we solve optimization problem (10.7) independently for 30 times by numerical simulations. The sample mean

169 and standard deviation (in parentheses in tables) of the optimal point (the optimal values of variables {µN } ∪ {ψij|i < j, i, j ∈ N}) and the agent’s optimal expected proﬁt rate

∗ ∗ ∗ ΠN ≡ Π(µN ; wN , pN , ΨN ) are reported in Table 10.1, Table 10.2, Table 10.3 and Table 10.4.

Table 10.1: Simulation results with homogeneous principals (λ = 0.8, FCFS queuing discipline)

r = 10 r = 15 r = 20 ∗ µN 2.3989 (0.1134) 3.354 (0.0742) 4.1157 (0.0959) ∗ |N| = 2 ψ12 0.0021 (0.0034) 0.0015 (0.0021) 0.0017 (0.0018) ∗ ΠN 3.3388 (0.0231) 4.0834 (0.0159) 4.6608 (0.0713) ∗ µN 2.6113 (0.2036) 3.7312 (0.2781) 4.5688 (0.3882) ∗ ψ12 0.002 (0.0069) 0.0017 (0.0022) 0.0033 (0.0083) ∗ |N| = 3 ψ13 0.0012 (0.0015) 0.0032 (0.0032) 0.0034 (0.0056) ∗ ψ23 0.0018 (0.0031) 0.0013 (0.0008) 0.0014 (0.0024) ∗ ΠN 7.1963 (0.0183) 8.793 (0.0119) 10.0731 (0.0132) ∗ µN 3.0361 (0.4242) 4.2611 (0.4235) 5.0669 (0.642) ∗ ψ12 0.0098 (0.0183) 0.0062 (0.0119) 0.0074 (0.0132) ∗ ψ13 0.0067 (0.0141) 0.0046 (0.0126) 0.0073 (0.0098) ∗ ψ14 0.0072 (0.0194) 0.0104 (0.028) 0.0416 (0.1642) |N| = 4 ∗ ψ23 0.0067 (0.0091) 0.0047 (0.0079) 0.0104 (0.0208) ∗ ψ24 0.0053 (0.0082) 0.0085 (0.0288) 0.0096 (0.0294) ∗ ψ34 0.0073 (0.0108) 0.0302 (0.0975) 0.0054 (0.004) ∗ ΠN 11.2236 (0.0978) 13.7766 (0.1503) 15.8563 (0.2027)

Table 10.2: Simulation results with homogeneous principals (λ = 1, FCFS queuing discipline)

r = 10 r = 15 r = 20 ∗ µN 2.4546 (0.201) 3.4238 (0.2427) 4.3279 (0.1893) ∗ |N| = 2 ψ12 0.0013 (0.0018) 0.0007 (0.0008) 0.0022 (0.0039) ∗ ΠN 3.6842 (0.0404) 4.5021 (0.0383) 5.2218 (0.0149) ∗ µN 2.7243 (0.2658) 3.7049 (0.3153) 4.8093 (0.4247) ∗ ψ12 0.0015 (0.0018) 0.0009 (0.0011) 0.0018 (0.0015) ∗ |N| = 3 ψ13 0.0017 (0.0039) 0.0011 (0.0014) 0.0014 (0.0014) ∗ ψ23 0.0006 (0.0009) 0.0021 (0.0041) 0.0012 (0.001) ∗ ΠN 8.0173 (0.0254) 9.8743 (0.0147) 11.3176 (0.0063) ∗ µN 3.1026 (0.5354) 4.1645 (0.5431) 5.154 (0.776) ∗ ψ12 0.0075 (0.0254) 0.0064 (0.0147) 0.0042 (0.0063) ∗ ψ13 0.0029 (0.0052) 0.0062 (0.007) 0.008 (0.0244) ∗ ψ14 0.0047 (0.0147) 0.0067 (0.0148) 0.0053 (0.0088) |N| = 4 ∗ ψ23 0.003 (0.0059) 0.0114 (0.0361) 0.0071 (0.019) ∗ ψ24 0.0038 (0.0079) 0.0023 (0.0033) 0.0023 (0.0045) ∗ ψ34 0.0027 (0.0044) 0.0009 (0.001) 0.0045 (0.0054) ∗ ΠN 12.5215 (0.1254) 15.4347 (0.1156) 17.8035 (0.1275)

170 Table 10.3: Simulation results with homogeneous principals (λ = 0.8, HOLP queuing discipline)

r = 10 r = 15 r = 20 ∗ µN 2.4618 (0.0751) 3.2601 (0.2372) 4.0712 (0.254) ∗ |N| = 2 ψ12 0.0026 (0.004) 0.0035 (0.0099) 0.0033 (0.0049) ∗ ΠN 3.3155 (0.011) 4.0081 (0.0879) 4.6462 (0.065) ∗ µN 2.6549 (0.1298) 3.6443 (0.3822) 4.604 (0.4944) ∗ ψ12 0.0016 (0.0021) 0.0025 (0.0031) 0.0182 (0.0743) ∗ |N| = 3 ψ13 0.0023 (0.0024) 0.0039 (0.0097) 0.0031 (0.0077) ∗ ψ23 0.0023 (0.0025) 0.0019 (0.003) 0.0048 (0.0054) ∗ ΠN 7.1947 (0.0047) 8.7527 (0.0081) 10.1231 (0.0359) ∗ µN 3.0487 (0.4424) 4.2282 (0.6103) 5.0762 (0.6075) ∗ ψ12 0.0029 (0.0047) 0.0066 (0.0081) 0.0151 (0.0359) ∗ ψ13 0.0165 (0.0498) 0.0041 (0.0075) 0.0525 (0.2404) ∗ ψ14 0.0047 (0.0053) 0.0057 (0.0118) 0.0172 (0.0336) |N| = 4 ∗ ψ23 0.0085 (0.026) 0.0513 (0.1753) 0.0138 (0.0433) ∗ ψ24 0.0038 (0.0078) 0.0082 (0.0146) 0.0038 (0.0052) ∗ ψ34 0.0045 (0.0068) 0.039 (0.1853) 0.0189 (0.0793) ∗ ΠN 11.2347 (0.1044) 13.6673 (0.2756) 15.8595 (0.228)

Table 10.4: Simulation results with homogeneous principals (λ = 1, HOLP queuing discipline)

r = 10 r = 15 r = 20 ∗ µN 2.5079 (0.1611) 3.4593 (0.16) 4.3164 (0.1044) ∗ |N| = 2 ψ12 0.001 (0.001) 0.0016 (0.0018) 0.0019 (0.0029) ∗ ΠN 3.6899 (0.0345) 4.5241 (0.022) 5.2512 (0.0279) ∗ µN 2.6975 (0.3243) 3.6569 (0.4374) 4.7314 (0.5882) ∗ ψ12 0.0006 (0.0009) 0.001 (0.0009) 0.0021 (0.0033) ∗ |N| = 3 ψ13 0.0009 (0.0014) 0.0007 (0.0008) 0.0023 (0.0074) ∗ ψ23 0.0017 (0.0015) 0.0006 (0.0007) 0.0006 (0.0008) ∗ ΠN 7.9887 (0.0038) 9.7921 (0.0151) 11.3425 (0.1654) ∗ µN 2.9707 (0.4263) 4.432 (0.5938) 5.0964 (0.8208) ∗ ψ12 0.002 (0.0038) 0.0094 (0.0151) 0.0326 (0.1654) ∗ ψ13 0.0034 (0.0064) 0.0179 (0.0647) 0.0107 (0.0281) ∗ ψ14 0.0018 (0.0053) 0.0052 (0.0115) 0.0061 (0.0173) |N| = 4 ∗ ψ23 0.006 (0.0143) 0.0039 (0.0064) 0.0088 (0.0208) ∗ ψ24 0.0029 (0.0061) 0.0044 (0.0072) 0.0063 (0.0103) ∗ ψ34 0.0028 (0.0067) 0.0043 (0.0054) 0.0025 (0.0037) ∗ ΠN 12.5503 (0.0764) 15.3391 (0.1094) 17.7725 (0.1964)

∗ ∗ ∗ ∗ The optimal values of the dependency coeﬃcients (i.e. {ψ12} for |N| = 2; {ψ12, ψ13, ψ23} ∗ ∗ ∗ ∗ ∗ ∗ for |N| = 3; {ψ12, ψ13, ψ14, ψ23, ψ24, ψ34} for |N| = 4) represent the agent’s desired pairwise interdependencies between all principals, and these values exhibit two important characteristics:

1. the means of all dependency coeﬃcients are close to zero.

171 2. the standard deviations (reported in the parentheses following the means) are large compared to the corresponding means.

These two characteristics imply that the agent’s desired dependency coefficients cannot be significantly distinguished from zero. Recall that by definition the failures of principal i’s and principal j’s units are negatively interdependent if ψij ∈ (0, 1), and the smaller the value of ψij, the stronger the negative interdependency. Therefore the simulation results imply that the agent prefers total negative interdependencies between all pairs of homogeneous principals in set N (i.e. ψij ∼ 0 for all i, j ∈ N and i 6= j). In essence total negative interdependency implies that it is extremely unlikely to observe more than one revenue generating units from set N being inoperative simultaneously at any instant during the infinite contract horizon.

The agent beneﬁts from pooling the service demands of multiple principals if he can achieve a superadditive expected proﬁt rate with respect to the set of principals he contracts with. Due to our assumption of homogeneous revenue rate and failure rate (ri = r > 0 and

λi = λ > 0 for all i ∈ N), all principals in N propose the same compensation rate and √ the same penalty rate (i.e. wi = 2 rλ − λ and pi = r for all i ∈ N). Therefore the agent does not distinguish between diﬀerent principals in N and his optimal expected proﬁt rate

∗ ∗ Π(µN ; wN , pN , ΨN ) only depends on the values of r, λ and |N|. The agent’s expected proﬁt rate is superadditive with respect to the set of principals he contracts with if and only if for

∗ ∗ ∗ ∗ any principal subsets S and T where S ∩ T = ∅, Π(µS; wS, pS, ΨS) + Π(µT ; wT , pT , ΨT ) ≤ ∗ ∗ Π(µS∪T ; wS∪T , pS∪T , ΨS∪T ) (Peleg and Sudhölter 2007). We check the superadditivity of agent’s optimal expected profit rate for all values of r and λ in Table 10.1, Table 10.2, Table 10.3 and Table 10.4. For example, for r = 10 and λ = 0.8, the agent’s optimal expected profit rates with a set of 2, 3, and 4 homogeneous principals under FCFS queuing

∗ ∗ discipline can be found in the 3rd column in Table 10.1: ΠN,|N|=2 = 3.3388, ΠN,|N|=3 = 7.1963 ∗ ∗ and ΠN,|N|=4 = 11.2236. Recall that from Zeng and Dror (2015a) we have ΠN,|N|=1 = 0. Note

172 that the following inequalities hold:

∗ ∗ ∗ ΠN,|N|=1 + ΠN,|N|=1 = 0 <3.3388 = ΠN,|N|=2

∗ ∗ ∗ ΠN,|N|=1 + ΠN,|N|=2 = 3.3388 <7.1963 = ΠN,|N|=3

∗ ∗ ∗ ΠN,|N|=1 + ΠN,|N|=3 = 7.1963 <11.2236 = ΠN,|N|=4

∗ ∗ ∗ ΠN,|N|=2 + ΠN,|N|=2 = 6.6776 <11.2236 = ΠN,|N|=4 indicating that for r = 10 and λ = 0.8, the agent’s optimal expected proﬁt rate is superadditive with respect to the set of principals he contracts with. In a similar way we validate that the agent’s optimal expected proﬁt rate is superadditive for all values of r and λ in Table 10.1, Table 10.2, Table 10.3 and Table 10.4.

Given superadditivity has been determined we check if the agent’s optimal expected proﬁt rate is convex with respect to the set of principals he contracts with. Convexity implies superadditivity and many other desired solution properties. The agent’s expected proﬁt rate is convex with respect to the set of principals he contracts with if and only if for all i ∈ N

∗ ∗ and for all subsets S,T such that S ⊆ T ⊆ N \{i}, we have Π(µS∪{i}; wS∪{i}, pS∪{i}, ΨS∪{i})− ∗ ∗ ∗ ∗ ∗ ∗ Π(µS; wS, pS, ΨS) ≤ Π(µT ∪{i}; wT ∪{i}, pT ∪{i}, ΨT ∪{i})−Π(µT ; wT , pT , ΨT ) (Peleg and Sudhölter 2007). We check the convexity of agent’s optimal expected profit rate for all values of r and λ in Table 10.1, Table 10.2, Table 10.3 and Table 10.4. For example, for r = 10 and λ = 0.8, the agent’s optimal expected profit rate with a set of 2, 3, and 4 homogeneous principals under

∗ ∗ ∗ FCFS queuing discipline are: ΠN,|N|=2 = 3.3388, ΠN,|N|=3 = 7.1963 and ΠN,|N|=4 = 11.2236, and the following inequalities hold:

∗ ∗ ∗ ∗ ΠN,|N|=2 − ΠN,|N|=1 = 3.3388 < 3.8575 = ΠN,|N|=3 − ΠN,|N|=2

∗ ∗ ∗ ∗ ΠN,|N|=3 − ΠN,|N|=2 = 3.8575 < 4.0273 = ΠN,|N|=4 − ΠN,|N|=3 indicating that for r = 10 and λ = 0.8, the agent’s optimal expected proﬁt rate is convex with respect to the set of principals he contracts with. In a similar way we validate that

173 the agent’s optimal expected proﬁt rate is convex for all values of r and λ in Table 10.1, Table 10.2 and Table 10.3 and Table 10.4.

10.3.2 Simulation results with heterogeneous principals

The simulation setting with heterogeneous principals is characterized by the values of revenue

rates ri, failure rates λi and the queuing discipline. We ran simulations for all nonempty

subsets of a set of 4 principals with diﬀerent revenue rates and diﬀerent failure rates (r1 = 12,

λ1 = 0.9, r2 = 10, λ2 = 0.8, r3 = 11, λ3 = 1.1, r4 = 8 and λ4 = 1.2) under a choice of queuing discipline from set {FCFS, HOLP}. When the principals are heterogeneous, the agent distinguishes between diﬀerent principals he contracts with, and subjectively chooses a HOLP priority order over the set of principals that is optimal for him. We assume that

the agent assigns the HOLP priority order in descending order of the value ri/λi, because a principal with higher revenue rate charges the agent more during her unit’s downtime

(recall that pi = ri) under the same failure rate. Recall that in Subsection 9.1 we assume that HOLP priority order is strict with i < j implying that principal i has higher priority than principal j, and it can be verified that the configuration of our simulations satisfies such assumption (r1/λ1 = 13.33 > r2/λ2 = 12.5 > r3/λ3 = 10 > r4/λ4 = 6.67). The sample size of each simulation setting is 30, which means for each of the nonempty subsets and queuing discipline, we solve optimization problem (10.7) independently for 30 times by numerical simulations. The sample mean and standard deviation (in parentheses in tables)

of the optimal point (the optimal values of variables {µN } ∪ {ψij|i < j, i, j ∈ N}) and the

∗ ∗ ∗ agent’s optimal expected proﬁt rate ΠN ≡ Π(µN ; wN , pN , ΨN ) are reported in Table 10.5 and Table 10.6.

∗ The optimal values of dependency coeﬃcients ψij, i < j, i, j ∈ N represent the agent’s desired pairwise interdependencies between all principals in set N. Similar to the simulation results with homogeneous principals, these values exhibit two important characteristics:

1. the means of all dependency coeﬃcients are close to zero.

2. the standard deviations (reported in the parenthesis following the means) are large

174 Table 10.5: Simulation results with heterogeneous principals (r1 = 12, λ1 = 0.9, r2 = 10, λ2 = 0.8, r3 = 11, λ3 = 1.1, r4 = 8, λ4 = 1.2, FCFS queuing discipline)

∗ ∗ µN 2.6621 (0.1157) µN 2.7282 (0.258) ∗ ∗ N = {1, 2} ψ12 0.0009 (0.0012) ψ12 0.0012 (0.0012) ∗ ∗ ΠN 3.5648 (0.0083) N = {1, 2, 4} ψ14 0.0011 (0.0031) ∗ ∗ µN 2.731 (0.2135) ψ24 0.0006 (0.0014) ∗ ∗ N = {1, 3} ψ13 0.0011 (0.0021) ΠN 7.7282 (0.038) ∗ ∗ ΠN 4.0189 (0.0289) µN 2.6277 (0.2602) ∗ ∗ µN 2.4918 (0.1362) ψ13 0.0007 (0.0018) ∗ ∗ N = {1, 4} ψ14 0.0021 (0.003) N = {1, 3, 4} ψ14 0.0009 (0.0007) ∗ ∗ ΠN 3.7287 (0.0137) ψ34 0.0025 (0.002) ∗ ∗ µN 2.5387 (0.1518) ΠN 8.3367 (0.0306) ∗ ∗ N = {2, 3} ψ23 0.0002 (0.0004) µN 2.5325 (0.1948) ∗ ∗ ΠN 3.6707 (0.0092) ψ23 0.0003 (0.0002) ∗ ∗ µN 2.1887 (0.1266) N = {2, 3, 4} ψ24 0.0005 (0.0005) ∗ ∗ N = {2, 4} ψ24 0.0007 (0.0007) ψ34 0.001 (0.0003) ∗ ∗ ΠN 3.4441 (0.0081) ΠN 7.9099 (0.0182) ∗ ∗ µN 2.4472 (0.1663) µN 3.2566 (0.5985) ∗ ∗ N = {3, 4} ψ34 0.0018 (0.0022) ψ12 0.0089 (0.0197) ∗ ∗ ΠN 3.8102 (0.015) ψ13 0.0055 (0.016) ∗ ∗ µN 3.006 (0.199) ψ14 0.004 (0.0146) ∗ N = {1, 2, 3, 4} ∗ ψ12 0.0012 (0.0024) ψ23 0.0042 (0.0137) ∗ ∗ N = {1, 2, 3} ψ13 0.0009 (0.0006) ψ24 0.0064 (0.0178) ∗ ∗ ψ23 0.0011 (0.0011) ψ34 0.0021 (0.0034) ∗ ∗ ΠN 8.0374 (0.0464) ΠN 12.5121 (0.159)

compared to the corresponding means.

These two characteristics imply that the agent’s desired dependency coeﬃcients cannot be signiﬁcantly distinguished from zero thus the simulation results imply that the agent prefers total negative interdependencies between all pairs of heterogeneous principals in set N (i.e.

ψij ∼ 0 for all i, j ∈ N and i 6= j).

We check if the agent’s optimal expected proﬁt rate is convex with respect to the set of principals he contracts with using the deﬁnition in Subsection 10.3.1. It is trivial if

∗ ∗ ∗ ∗ S = T ⊆ N \{i} since ΠS∪{i} − ΠS = ΠT ∪{i} − ΠT , therefore we focus on cases where S ⊂ T ⊆ N \{i}. For example, we check the convexity of agent’s optimal expected proﬁt

175 Table 10.6: Simulation results with heterogeneous principals (r1 = 12, λ1 = 0.9, r2 = 10, λ2 = 0.8, r3 = 11, λ3 = 1.1, r4 = 8, λ4 = 1.2, HOLP queuing discipline)

∗ ∗ µN 2.6668 (0.118) µN 2.4712 (0.2649) ∗ ∗ N = {1, 2} ψ12 0.0003 (0.0003) ψ12 0.0021 (0.0056) ∗ ∗ ΠN 3.6042 (0.0027) N = {1, 2, 4} ψ14 0.0019 (0.0047) ∗ ∗ µN 2.8024 (0.1128) ψ24 0.0015 (0.0022) ∗ ∗ N = {1, 3} ψ13 0.0004 (0.0004) ΠN 7.7971 (0.0232) ∗ ∗ ΠN 3.9797 (0.0079) µN 2.6475 (0.3247) ∗ ∗ µN 2.4603 (0.1609) ψ13 0.0014 (0.0017) ∗ ∗ N = {1, 4} ψ14 0.0014 (0.0015) N = {1, 3, 4} ψ14 0.0006 (0.0003) ∗ ∗ ΠN 3.7159 (0.0121) ψ34 0.0011 (0.0009) ∗ ∗ µN 2.567 (0.1555) ΠN 8.2435 (0.0434) ∗ ∗ N = {2, 3} ψ23 0.0014 (0.0014) µN 2.6244 (0.2679) ∗ ∗ ΠN 3.6601 (0.0176) ψ23 0.0008 (0.0007) ∗ ∗ µN 2.1756 (0.0526) N = {2, 3, 4} ψ24 0.0013 (0.0033) ∗ ∗ N = {2, 4} ψ24 0.0001 (0.0002) ψ34 0.0015 (0.0021) ∗ ∗ ΠN 3.5008 (0.0019) ΠN 7.9071 (0.0364) ∗ ∗ µN 2.3428 (0.1166) µN 3.1511 (0.5022) ∗ ∗ N = {3, 4} ψ34 0.0011 (0.001) ψ12 0.0109 (0.0255) ∗ ∗ ΠN 3.8727 (0.0052) ψ13 0.0061 (0.0233) ∗ ∗ µN 2.8444 (0.3135) ψ14 0.0101 (0.0306) ∗ N = {1, 2, 3, 4} ∗ ψ12 0.0037 (0.005) ψ23 0.0053 (0.0124) ∗ ∗ N = {1, 2, 3} ψ13 0.001 (0.0013) ψ24 0.015 (0.0398) ∗ ∗ ψ23 0.0013 (0.0015) ψ34 0.0039 (0.0125) ∗ ∗ ΠN 8.046 (0.0369) ΠN 12.4862 (0.1464) rate in Table 10.5 when i = 1 as follows:

∗ ∗ ∗ ∗ Π{1,2} − Π{2} = 3.5648 <4.3667 = Π{1,2,3} − Π{2,3}

∗ ∗ ∗ ∗ Π{1,2} − Π{2} = 3.5648 <4.2841 = Π{1,2,4} − Π{2,4}

∗ ∗ ∗ ∗ Π{1,2} − Π{2} = 3.5648 <4.6022 = Π{1,2,3,4} − Π{2,3,4}

∗ ∗ ∗ ∗ Π{1,3} − Π{3} = 4.0189 <4.3667 = Π{1,2,3} − Π{2,3}

∗ ∗ ∗ ∗ Π{1,3} − Π{3} = 4.0189 <4.3178 = Π{1,3,4} − Π{3,4}

∗ ∗ ∗ ∗ Π{1,3} − Π{3} = 4.0189 <4.6022 = Π{1,2,3,4} − Π{2,3,4}

∗ ∗ ∗ ∗ Π{1,4} − Π{4} = 3.7287 <4.2841 = Π{1,2,4} − Π{2,4}

∗ ∗ ∗ ∗ Π{1,4} − Π{4} = 3.7287 <4.5265 = Π{1,3,4} − Π{3,4}

∗ ∗ ∗ ∗ Π{1,4} − Π{4} = 3.7287 <4.6022 = Π{1,2,3,4} − Π{2,3,4}

176 ∗ ∗ ∗ ∗ Π{1,2,3} − Π{2,3} = 4.3667 <4.6022 = Π{1,2,3,4} − Π{2,3,4}

∗ ∗ ∗ ∗ Π{1,2,4} − Π{2,4} = 4.2841 <4.6022 = Π{1,2,3,4} − Π{2,3,4}

∗ ∗ ∗ ∗ Π{1,3,4} − Π{3,4} = 4.5265 <4.6022 = Π{1,2,3,4} − Π{2,3,4}

We check the convexity of agent’s optimal expected proﬁt rate in Table 10.5 and Table 10.6 for i = 1, 2, 3, 4 in the same manner. Based on the simulation results we conclude that the agent’s optimal expected proﬁt rate is convex with respect to the set of principals he contracts with under both FCFS and HOLP queuing disciplines.

10.4 When to stop pooling principals?

Simulation results in Subsection 10.3 imply that the agent requires total negative interdependencies of failures between all pairs of principals in set N (i.e. ψij ∼ 0 for all i, j ∈ N and i 6= j), and in such case the agent’s expected profit rate is convex in the principals set he contracts with. Since we assume that the agent is in high demand, therefore presum- ably the agent can expand indefinitely the set N of principals to contract with. However in practice the pairwise total negative interdependency between failures is likely to hold true only for a small number of principals. In this subsection, we provide a numerical example to demonstrate that if the agent has already contracted with all principals with total negative interdependencies, say N, then adding one more principal j 6∈ N such that ψji = ψij > 0 for all i ∈ N may result in a loss of superadditivity property of the agents expected profit rate with respect to N and thus effectively serve as disincentive to increase the pool of principals beyond a certain N. Correspondingly the agents service capacity is bounded from above by µ∗ . N

In our numerical example we assume that the agent signs contracts with a set of 3 principals: N = {1, 2, 3}, and he wants to expand set N to include another principal ‘4’. We assume that all four principals are homogeneous with revenue rate r = 10 and failure rate λ = 1. Furthermore we assume negative pairwise interdependencies between principals 1, 2

and 3 (e.g. ψij = 0.001 for all i, j ∈ {1, 2, 3}, i 6= j), and we assume that principal 4 has high

177 positive interdependencies with principals 1, 2 and 3 (e.g. ψi4 = ψ4i = 5 for all i ∈ {1, 2, 3}). In other words, the dependency matrix of principals set N is as follows:

  1 0.001 0.001 5     0.001 1 0.001 5 ΨN =     0.001 0.001 1 5   5 5 5 1

Given the revenue rate r, the failure rate λ and the dependency matrix ΨN , the agent’s optimization problem becomes maxµN ≥0 Π(µN ; wN , pN , ΨN ), and we solve this optimization problem independently for 30 times by numerical simulations under FCFS and HOLP queuing disciplines respectively. The sample mean and standard deviation (in parenthe-

∗ ses) of the agent’s optimal service capacity µN and the agent’s optimal expected proﬁt rate ∗ ∗ ΠN ≡ Π(µN ; wN , pN , ΨN ) are reported in Table 10.7.

Table 10.7: Simulation results with homogeneous principals (r = 10 and λ = 1) and positive interdependencies (ψi4 = ψ4i = 5 for all i ∈ {1, 2, 3})

FCFS HOLP ∗ ∗ µN 2.4377 (0.0385) µN 2.3854 (0.0548) N = {1, 2} ∗ N = {1, 2} ∗ ΠN 3.6918 (0.0422) ΠN 3.6988 (0.0174) ∗ ∗ µN 2.4518 (0.0532) µN 2.4178 (0.0913) N = {1, 2, 3} ∗ N = {1, 2, 3} ∗ ΠN 7.9933 (0.0326) ΠN 8.014 (0.029) ∗ ∗ µN 7.1239 (0.2512) µN 7.9654 (0.2007) N = {1, 2, 3, 4} ∗ N = {1, 2, 3, 4} ∗ ΠN 6.0771 (0.0303) ΠN 4.9778 (0.0413)

As shown in Table 10.7, the agent’s optimal expected proﬁt rate loses the superadditivity property when he expands principals set {1, 2, 3} to include principal 4 under both FCFS and

∗ HOLP queuing disciplines (i.e. under FCFS queuing discipline Π{1,2,3} = 7.9933 > 6.0771 = ∗ ∗ ∗ Π{1,2,3,4} and under HOLP queuing discipline Π{1,2,3} = 8.014 > 4.9778 = Π{1,2,3,4}), which serves as threshold for the agent not to pool principals beyond principals set {1, 2, 3}. In this numerical example, N = {1, 2, 3}, µ∗ = 2.4518 under FCFS queuing discipline, and N µ∗ = 2.4178 under HOLP queuing discipline. N

178 11 Serving Principals Set N

The simulation results from Section 10 demonstrate that the agent prefers a principals set with total negative interdependencies between all pairs of principals in that set. In this section we propose a model that describes the interplay between the agent and a ﬁnite set of principals N ⊆ P he has selected by solving the optimization problem (10.7). Based on the simulation results we approximate the agent’s desired total negative interdependencies between all pairs of principals in N by setting the dependency matrix ΨN to be the identity matrix (i.e. ψij = 1 for i = j ∈ N and ψij = 0 for i 6= j, i, j ∈ N), which implies that the units between all pairs of principals in N have perfectly negative interdependencies. The principals are assumed to be heterogeneous, which means that the revenue rates and the failure rates of diﬀerent principals’ units are not necessarily the same.

The assumption of perfectly negative interdependencies between diﬀerent principals’ units imply that at any moment during the inﬁnite contract horizon, whenever a unit is inoperative, no other unit can be inoperative at the same time. Although it is not very likely to observe perfectly negative interdependencies in real life, it serves as an approximation of the agent’s optimizing behavior yet allows tractability, as demonstrated later in this section.

11.1 Principal-agent model with perfectly negative dependencies

In line with Section 9 the system of principals subset S ⊆ N and a single agent A is modeled as a continuous time Markov process. Under the assumption of perfectly negative interdependencies, the state space of the Markov process degenerates to {∅} ∪ {(i), i ∈ S} with (1 + |S|) states in total, compared to the Markov process in Section 9. The limiting probabilities of these states are denoted by P (∅) and P (i), i ∈ S. Note that the maximum length of queue of failed units is 1 because when units have perfectly negative interdependencies, no other unit will fail during the inoperative time of one unit, therefore queuing discipline does not apply here. The state transition diagram is illustrated by Figure 11.1.

179 λ 1 1

µS

λ2 2

µS

∅ λ3 3

µS ··· λ|S|

µS |S|

Figure 11.1: State transition diagram with perfectly negative interdependencies

The balanced equations are:

X X P (∅) λi = µS P (i) and µSP (i) = λiP (∅) for all i ∈ S i∈S i∈S

P Since P (∅) + i∈S P (i) = 1, therefore

µS λi P (∅) = P ,P (i) = P µS + j∈S λj µS + j∈S λj

Recall that P(i) denotes the limiting probability of principal i’s unit being inoperative independent of the state of other units, therefore

λi P(i) = P (i) = P µS + j∈S λj

According to (9.2) the agent’s expected proﬁt rate is:

P X p i∈S riλi Π(µS; wS, pS, ΨS) = 2 riλi − λi − P − µS µS + λi i∈S i∈S

Note that

P 2 P dΠ(µS) i∈S riλi d Π(µS) 2 i∈S riλi = 2 − 1 and 2 = − 3 < 0 dµS P dµ P µS + i∈S λi S µS + i∈S λi

180 ∗ therefore there exists a unique µS that satisﬁes the ﬁrst order condition:

s dΠ(µS) ∗ X X = 0 ⇒ µS = riλi − λi (11.1) dµS ∗ µS =µS i∈S i∈S

Note that agent A accepts the contract oﬀers from principals set S and installs a strictly

∗ positive service capacity (µS > 0) only if the following condition is satisﬁed:

sX X riλi > λi, ∀S ⊆ N,S 6= ∅ (11.2) i∈S i∈S

Condition (11.2) implies that the revenue rates are high enough in relation to the failure P rates. A suﬃcient condition that guarantees (11.2) is that ri > j∈S λj, ∀ i ∈ S. Note √ that setting S = {i}, i ∈ N implies: riλi > λi ⇔ ri > λi, which is consistent with the condition to justify a principal-agent contract with a single risk-neutral principal and a single risk-neutral agent (Zeng and Dror, 2015a). The agent’s optimal expected proﬁt rate is

  s ∗ X p X Π(µS; wS, pS, ΨS) = 2  riλi − riλi (11.3) i∈S i∈S

P P p Note that since i∈S j∈S,j6=i riλirjλj > 0, therefore

  s X X X p X X p X riλi + riλirjλj > riλi ⇒ 2  riλi − riλi > 0 i∈S i∈S j∈S,j6=i i∈S i∈S i∈S

When the agent serves principals set S (|S| ≥ 2), both his optimal capacity (11.1) and his optimal expected proﬁt rate (11.3) are functions of the revenue rates and the failure rates of the units owned by the principals in set S.

Subsequent sensitivity analysis help us understand the eﬀect of the revenue rates and the failure rates on the agent’s optimal service capacity and optimal expected proﬁt rate. We

181 ﬁrst examine the agent’s optimal capacity (11.1). Note that:

∂µ∗ λ ∂µ∗ r S = √ i and S = √ i − 1, ∀ i ∈ S ∂ri 2 riλi ∂λi 2 riλi

∗ Since ri > 0 and λi > 0 for i ∈ S, therefore ∂µS/∂ri > 0 for i ∈ S, indicating that the agent’s optimal capacity increases with respect to all the revenue rates of the principals in

∗ S. According to condition (11.2), λi < ri for i ∈ S. If λi ∈ (0, ri/4], then ∂µS/∂λi ≥ 0 and if ∗ λi ∈ (ri/4, ri), then ∂µS/∂λi < 0. The derivatives indicate that the agent’s optimal capacity is not monotonic with respect to the failure rates of principals in set S. For example, if principal i’s failure rate is relatively small compared to her revenue rate (λi ∈ (0, ri/4]) then a small increase in her unit’s failure rate causes the agent to raise the optimal capacity, because the penalty rate plays a prominent role in this case. However if principal i’s failure rate is relatively large compared to her revenue rate (λi ∈ (ri/4, ri)) then an increase in her unit’s failure rate causes the agent to reduce the optimal capacity.

Now let’s examine the agent’s optimal proﬁt rate (11.3). Note that:

∗ ∗ ∂Π(µS) λi λi ∂Π(µS) ri ri = √ − q > 0, and = √ − q > 0, ∀i ∈ S ∂ri riλi P ∂λi riλi P j∈S rjλj j∈S rjλj

The derivatives indicate that the agent’s optimal proﬁt rate monotonically increases with respect to the revenue rates and the failure rates of the units owned by principals in set S.

12 Principal-Agent: The Cooperative Game Perspective

In order for the agent to assess fairly the contributions of the diﬀerent principals in this service environment, we examine the option of a risk-neutral agent contracting with many principals as a cooperative game with transferable utility. First, we deﬁne the game and we prove that this game is convex.

Given the set NA ≡ N ∪{A} of participants including the agent, we deﬁne principal-agent

182 NA cooperative game (NA, v) with v : 2 → R as the set value function. The game is deﬁned as agent A’s optimal expected proﬁt rate when serving principals set S ⊆ N, normalized at {i}, i ∈ N by v({i} ∪ {A}) = 0. The value function is given by (12.1). It equals 0 if X is empty or if X consists of only the agent or only of principals, since both the agent and the principals are vital players in this cooperative game. That is

  0 if X = ∅ or X = {A} or X ⊆ N v(X) = (12.1) P √ pP  2 i∈S riλi − i∈S riλi if X = S ∪ {A}, ∅= 6 S ⊆ N

Note that by setting X = {i} ∪ {A}, i ∈ N, it is straight forward to verify that v(X) = √ √ 2 riλi − riλi = 0, since in a contract between an agent and a single principal the agent only breaks even.

Say the agent has determined the principals set N ⊆ P to contract with, (that is the revenue rates and the failure rates of any principal subset S ⊆ N, |S| ≥ 2 satisfy condition (11.2)), then the agent accepts the contracts and installs a strictly positive service capacity, and the principals in N have perfectly negative interdependencies. In that case we

prove that the principal-agent cooperative game (NA, v) is convex (Theorem 12.3).

We ﬁrst state a couple of technical lemmas used in the proof of Theorem 12.3. √ √ √ Lemma 12.1. x + y − x + y > 0, ∀ x > 0 and y > 0.

√ √ √ √ 2 Proof. Let x > 0 and y > 0. 2 xy > 0 ⇔ x + 2 xy + y > x + y ⇔ x + y > √ 2 √ √ √ x + y ⇔ x + y − x + y > 0. √ √ √ √ Lemma 12.2. y + a − y > x + a − x, ∀ x > y > 0 and a > 0. √ √ √ Proof. Let a > 0. Deﬁne f(t) = t + a − t for t > 0. Note that f 0(t) = 1/2 t + a − √ √ √ √ √ 1/2 t < 0. Let x > y > 0, therefore y + a − y > x + a − x.

Notation: NA ≡ N ∪ {A}.

Theorem 12.3 (Convexity). Principal-agent cooperative game (NA, v) is convex.

183 Proof. Note that the game (NA, v) is convex if and only if for all i ∈ NA and for all S,T such that S ⊆ T ⊆ NA \{i}, we have v(S ∪ {i}) − v(S) ≤ v(T ∪ {i}) − v(T ). It is easy to see that given i ∈ NA and S = T ⊆ NA \{i}, we have v(S ∪ {i}) − v(S) = v(T ∪ {i}) − v(T ).

Therefore we exhaustively examine the remaining cases where i ∈ NA and S ⊂ T ⊆ NA \{i}. Table 12.1 helps illustrate the proof by highlighting the cardinality of set S and T in diﬀerent cases and subcases.

Table 12.1: A summary of the cases in the proof of Theorem 12.3 where i ∈ NA and S ⊂ T ⊆ NA \{i}

Case 1: i 6= A, S 63 A and T 63 A |T | > |S| ≥ 0 Case 2.1: |S| = 0 and |T | = 1 Case 2.2: |S| = 0 and |T | ≥ 2 Case 2: i = A, S 63 A and T 63 A Case 2.3: |S| = 1 and |T | ≥ 2 Case 2.4: |T | > |S| ≥ 2 Case 3.1: 1 = |T \{A}| > |S| ≥ 0 Case 3: i 6= A, S 63 A and T 3 A Case 3.2: |T \{A}| ≥ 2 and |T \{A}| > |S| ≥ 0 Case 4.1: |S \{A}| = 0 and |T \{A}| = 1 Case 4.2: |S \{A}| = 0 and |T \{A}| ≥ 2 Case 4: i 6= A, S 3 A and T 3 A Case 4.3: |S \{A}| = 1 and |T \{A}| ≥ 2 Case 4.4: |T \{A}| > |S \{A}| ≥ 2

Case 1: i ∈ NA and S ⊂ T ⊆ NA \{i} where i 6= A, S 63 A and T 63 A. This is the case where i is a principal and both set S and T consist of principals only. According to (12.1) v(S ∪ {i}) − v(S) = v(T ∪ {i}) − v(T ) = 0.

Case 2: i ∈ NA and S ⊂ T ⊆ NA \{i} where i = A, S 63 A and T 63 A. This is the case where i is the agent, and set S and set T consist of principals only. We exhaustively examine the following subcases.

Subcase 2.1: S = ∅ and {j} = T ⊆ N. According to (12.1) v(S ∪ {i}) = v(S) = v(T ∪ {i}) = v(T ) = 0, therefore v(S ∪ {i}) − v(S) = v(T ∪ {i}) − v(T ).

Subcase 2.2: S = ∅ and T ⊆ N where |T | ≥ 2. According to (12.1):

  s X p X v(S ∪ {i}) = v(S) = v(T ) = 0, v(T ∪ {i}) = 2  rjλj − rjλj j∈T j∈T

184 Since rj, λj > 0 for j ∈ N, according to Lemma 12.1, v(T ∪ {i}) > 0. Therefore v(S ∪ {i}) − v(S) < v(T ∪ {i}) − v(T ).

Subcase 2.3: {j} = S ⊂ T ⊆ N. According to (12.1):

  X p sX v(S ∪ {i}) = v(S) = v(T ) = 0, v(T ∪ {i}) = 2  rkλk − rkλk k∈T k∈T

Since rk, λk > 0 for k ∈ N, according to Lemma 12.1, v(T ∪ {i}) > 0. Therefore v(S ∪ {i}) − v(S) < v(T ∪ {i}) − v(T ).

Subcase 2.4: S ⊂ T ⊆ N where |S| ≥ 2. According to (12.1):

  s X p X v(S) = v(T ) = 0, v(S ∪ {i}) = 2  rjλj − rjλj j∈S j∈S   s X p X v(T ∪ {i}) = 2  rjλj − rjλj j∈T j∈T

Note that

v(T ∪ {i})−v(T ) − (v(S ∪ {i}) − v(S))   s s X p X X =2  rjλj + rjλj − rjλj j∈T \S j∈S j∈T

Since rj, λj > 0 for j ∈ N, according to Lemma 12.1:

s s X p X X rjλj+ rjλj − rjλj > 0 j∈T \S j∈S j∈T

⇒ v(T ∪ {i}) − v(T ) > v(S ∪ {i}) − v(S)

Case 3: i ∈ NA and S ⊂ T ⊆ NA \{i} where i 6= A, S 63 A and T 3 A. This is the case where i is a principal, set S consists of principals only but set T includes the agent. We exhaustively examine the following subcases.

185 Subcase 3.1: S ⊂ T \{A} = {j} ⊆ N \{i}. According to (12.1):

p p p v(S ∪ {i}) = v(S) = v(T ) = 0, v(T ∪ {i}) = 2 riλi + rjλj − riλi + rjλj

Since ri, λi, rj, λj > 0 for i, j ∈ N, let x = riλi and y = rjλj. According to Lemma 12.1:

p p p riλi + rjλj − riλi + rjλj > 0 ⇒ v(T ∪ {i}) − v(T ) > v(S ∪ {i}) − v(S)

Subcase 3.2: S ⊂ T \{A} ⊆ N \{i} where |T \{A}| ≥ 2. According to (12.1):

v(S ∪ {i}) = v(S) = v(T ) = 0   p X p s X v(T ∪ {i}) = 2  riλi + rjλj − riλi + rjλj j∈T \{A} j∈T \{A}

Since ri, λi, rj, λj > 0 for i, j ∈ N, according to Lemma 12.1:

p X p s X riλi + rjλj − riλi + rjλj > 0 j∈T \{A} j∈T \{A}

⇒ v(T ∪ {i}) − v(T ) > v(S ∪ {i}) − v(S)

Case 4: i ∈ NA and S ⊂ T ⊆ NA \{i} where i 6= A, S 3 A and T 3 A. This is the case where i is a principal, and set S and set T both include the agent. We exhaustively examine the following subcases.

Subcase 4.1: S \{A} = ∅ and {j} = T \{A} ⊆ N \{i}. According to (12.1):

v(S ∪ {i}) = v(S) = v(T ) = 0 and p p p v(T ∪ {i}) = 2 riλi + rjλj − riλi + rjλj

Since ri, λi, rj, λj > 0 for i, j ∈ N, let x = riλi and y = rjλj. According to

186 Lemma 12.1:

p p p riλi + rjλj − riλi + rjλj > 0 ⇒ v(T ∪ {i}) − v(T ) > v(S ∪ {i}) − v(S)

Subcase 4.2: S \{A} = ∅ and T \{A} ⊆ N \{i} where |T \{A}| ≥ 2. From (12.1):

  X p s X v(S ∪ {i}) = v(S) = 0, v(T ) = 2  rjλj − rjλj j∈T \{A} j∈T \{A}   X p s X v(T ∪ {i}) = 2  rjλj − rjλj j∈T ∪{i}\{A} j∈T ∪{i}\{A}

Note that

v(T ∪ {i})−v(T ) − (v(S ∪ {i}) − v(S))   p s X s X =2  riλi + rjλj − riλi + rjλj j∈T \{A} j∈T \{A}

P Since ri, λi, rj, λj > 0 for i, j, ∈ N, let x = riλi and y = j∈T \{A} rjλj. From Lemma 12.1:

p s X s X riλi + rjλj − riλi + rjλj > 0 j∈T \{A} j∈T \{A}

⇒ v(T ∪ {i}) − v(T ) > v(S ∪ {i}) − v(S)

Subcase 4.3: {j} = S \{A} ⊂ T \{A} ⊆ N \{i}. According to (12.1):

  X p s X v(S) = 0, v(T ) = 2  rkλk − rkλk k∈T \{A} k∈T \{A} p p p v(S ∪ {i}) = 2 riλi + rjλj − riλi + rjλj

187   X p s X v(T ∪ {i}) = 2  rkλk − rkλk k∈T ∪{i}\{A} k∈T ∪{i}\{A}

Note that

v(T ∪ {i})−v(T ) − (v(S ∪ {i}) − v(S)) p p =2 riλi + rjλj − rjλj   s X s X − 2  riλi + rkλk − rkλk k∈T \{A} k∈T \{A}

P Since ri, λi, rj, λj > 0 for i, j ∈ N, let x = k∈T \{A} rkλk, y = rjλj and a = riλi. Since {j} = S \{A} ⊂ T \{A}, therefore x > y > 0. According to Lemma 12.2:

p p s X s X riλi + rjλj − rjλj > riλi + rkλk − rkλk k∈T \{A} k∈T \{A}

⇒ v(T ∪ {i}) − v(T ) > v(S ∪ {i}) − v(S)

Subcase 4.4: S \{A} ⊂ T \{A} ⊆ N \{i} where |S \{A}| ≥ 2. According to (12.1):

  X p s X v(S) = 2  rjλj − rjλj j∈S\{A} j∈S\{A}   X p s X v(S ∪ {i}) = 2  rjλj − rjλj j∈S∪{i}\{A} j∈S∪{i}\{A}   X p s X v(T ) = 2  rjλj − rjλj j∈T \{A} j∈T \{A}   X p s X v(T ∪ {i}) = 2  rjλj − rjλj j∈T ∪{i}\{A} j∈T ∪{i}\{A}

188 Note that

v(T ∪ {i})−v(T ) − (v(S ∪ {i}) − v(S))   s X s X =2  riλi + rjλj − rjλj j∈S\{A} j∈S\{A}   s X s X − 2  riλi + rjλj − rjλj j∈T \{A} j∈T \{A}

P P Since ri, λi, rj, λj > 0 for i, j ∈ N, let x = j∈T \{A} rjλj, y = j∈S\{A} rjλj and

a = riλi. Since S \{A} ⊂ T \{A}, therefore x > y > 0. According to Lemma 12.2:

s X s X s X s X riλi + rjλj − rjλj > riλi + rjλj − rjλj j∈S\{A} j∈S\{A} j∈T \{A} j∈T \{A}

⇒ v(T ∪ {i}) − v(T ) > v(S ∪ {i}) − v(S)

To summarize, (NA, v) is convex.

12.1 Each principal’s ‘fair contribution’

The value allocation of game (N , v) is a vector a = (a ) ∈ |N|+1 such that P a = A i i∈NA R i∈NA i v(NA), in which ai represents a ‘fair’ portion of the value v(NA) attributed to player i.

P Denote a(S) ≡ i∈S ai for all S ⊆ NA. The core of the cooperative game (NA, v), |N|+1 denoted by C(NA, v), is the set of all value allocations a ∈ R that satisfy the following conditions (Peleg and Sudh¨olter2007):

1. Eﬃciency (all of the value is distributed): a(NA) = v(NA);

2. Stability: a(S) ≥ v(S), ∀ S ⊆ NA.

If cooperative game (NA, v) is convex, then it is totally balanced and has nonempty core

(Shapley 1971 and Peleg and Sudh¨olter2007). According to Theorem 12.3, (NA, v) is convex therefore C(NA, v) 6= ∅.

189 S Furthermore, (NA, v) being convex implies that the Shapley value (denoted by a ) is in

the core. For cooperative game (NA, v) the Shapley value attributes the portion of v(NA) to each of the players (the agent A and the principals in N) the expected marginal contribution to the agent’s expected proﬁt rate when this player joins the coalition of players in all possible ways with equal chance (Shapley, 1971), and it is the unique value allocation rule that is symmetric (the value allocation is invariant under any renaming of the players), attributes

dummies nothing (a player i is a dummy if v(S ∪ {i}) = v(S) for all S ⊆ NA \{i}), and is additive (a value allocation a is additive if v(S) = v1(S) + v2(S) ⇒ a(S) = a1(S) + a2(S)

for all S ⊆ NA) (Young, 1994). Therefore the Shapley value is an attractive candidate for evaluating each principal’s contribution to the agent’s expected proﬁt rate when the agent

contracting with principals set N. For game (NA, v) the Shapley value is

X |S|!(|N| − |S|)! aS = v(S ∪ {i}) − v(S) for all i ∈ N (12.2) i (|N| + 1)! A S⊆NA\{i}

Note that the agent is not a dummy player because v(S) = 0 but v(S ∪ {A}) > 0 for all

S S P S S ⊆ N, |S| ≥ 2, thus aA > 0 ⇒ a (N) = i∈N ai < v(NA).

Although the Shapley value is often proposed as allocation scheme, simulation results suggest that when a value game is not convex, the Shapley value has a relatively high probability not being in the core (Dror et al., 2012). Furthermore, the time complexity of computing

|NA| the Shapley value is O(2 ) because the values of all player subsets v(S), S ⊆ NA have to be computed. An alternative value allocation scheme is the Louderback’s value, which was introduced by Louderback (1976) and is essentially equivalent to the Alternate Cost Avoided (ACA) method introduced in Young (1994). The time complexity of computing the Loud-

erback’s value is O(|NA|), which is signiﬁcantly better compared to the time complexity of computing the Shapley value, because the Louderback’s value only requires the computation

of each player’s individual value v({i}), i ∈ NA and each player’s marginal contribution to

the value of the grand coalition (v(NA) − v(NA \{i})), i ∈ NA.

190 In order to compute the Louderback’s value of our game (NA, v), we introduce the following notation first. Denoted by ui, the separable profit of player i is defined as the marginal profit rate player i brings to the agent when player i joins the coalition NA \{i}:

 P p qP  2 j∈N rjλj − j∈N rjλj if i = A u ≡ v(N ) − v(N \{i}) = i A A √ qP qP  2 riλi + j∈N\{i} rjλj − j∈N rjλj if i ∈ N (12.3)

Note that by Lemma 12.1 ui > 0 for all i ∈ NA. The alternate profit of player i is the agent’s optimal expected profit rate when player i is the sole player in the game, thus according to (12.1) v({i}) = 0 for all i ∈ NA. The difference between the separable profit and the alternate profit is the alternate profit claimed by player i in a value game, which is similar to the alternate cost avoided in a cost game (Young, 1994). Denoted player i’s alternate profit claimed by di, and note that di = ui − v({i}) = ui. The Louderback’s method attributes

L value ai to player i that equals player i’s separable profit plus a fraction of the remaining value based on the proportion of player i’s alternate profit claimed compared to other players. The Louderback’s value is defined as follows:

L di ui ai = ui + (v(NA) − u(NA)) = v(NA) (12.4) d(NA) u(NA)

In (12.4) u(N ) ≡ P u and d(N ) ≡ P d . Note that for our game (N , v) the A j∈NA j A j∈NA j A Louderback’s value is essentially equivalent to the Incremental value examined in Dror et al.

(2012) because v({i}) = 0 for all i ∈ NA.

Although simulation results in Dror et al. (2012) demonstrated that the Louderback’s value has a high chance to be in the core of a value game with non-empty core, it is hard to prove in general that the Louderback’s value is a member of the core (Young, 1994). However if a value game is superadditive, then the Louderback’s value is proven to be in the value

0 game’s semicore (Young, 1994). Denote the semicore of game (NA, v) by C (NA, v). A value

191 0 allocation a ∈ C (NA, v) if it satisﬁes the following conditions (see Young 1994):

1. Eﬃciency (all of the value is distributed): a(NA) = v(NA);

2. Individual Rationality: a({i}) ≥ v({i}), ∀ i ∈ NA;

3. Stability: a(NA \{i}) ≥ v(NA \{i}), ∀ i ∈ NA.

0 0 By deﬁnition C(NA, v) ⊆ C (NA, v). According to Theorem 12.3 C(NA, v) 6= ∅ ⇒ C (NA, v) 6=

L 0 ∅, therefore a ∈ C (NA, v).

Furthermore, we show in Theorem 12.4 that for homogeneous principals (ri = r and

λi = λ for i ∈ N), the Louderback’s value is in the core.

L Theorem 12.4. a ∈ C(NA, v) if the principals are homogeneous with r > 0 and λ > 0.

L Proof. Consider a set N of homogeneous principals. By deﬁnition a ∈ C(NA, v) only if for

L L any subset ∅ 6= S ⊂ NA, a (S) ≥ v(S) and a (NA) = v(NA). According to Lemma 12.1,

L ui > 0 for all i ∈ NA (see (12.3)), thus ai > 0 = v({i}) for all i ∈ NA. Furthermore, note that aL(S) > 0 = v(S) for all S ⊆ N (which implies that S 63 A). Therefore we focus on the only remaining case where {A} ⊂ S ⊆ NA (which implies that |S| ≥ 2), and in such case the Louderback’s value of S is

p p p |N| − |N| + (|S| − 1) 1 + |N| − 1 − |N| √ L p a (S) = 2 |N| − |N| rλ |N| − p|N| + |N| 1 + p|N| − 1 − p|N|

√ and note that v(S) = 2 (|S| − 1) − p|S| − 1 rλ. Since |N| ≥ |S| ≥ 2, therefore

p|N| + p|S| − 1 ≥ 1 ⇔ |N| − (|S| − 1) ≥ p|N| − p|S| − 1

⇔ |N| − p|N| ≥ (|S| − 1) − p|S| − 1 2 ⇔ |N| − p|N| ≥ |N| − p|N| (|S| − 1) − p|S| − 1

192 Furthermore, note that

p|N| > p|S| − 1 ⇔ p|S| − 1 p|N| − 1 > p|N| p|S| − 1 − 1 ⇔ (|S| − 1) |N| − p|N| > |N| (|S| − 1) − p|S| − 1

According to Lemma 12.1, 1 + p|N| − 1 > p|N| ⇔ 1 + p|N| − 1 − p|N| > 0, therefore

2 |N| − p|N| + (|S| − 1) 1 + p|N| − 1 − p|N| |N| − p|N| > |N| − p|N| (|S| − 1) − p|S| − 1 + |N| 1 + p|N| − 1 − p|N| (|S| − 1) − p|S| − 1 |N| − p|N| + (|S| − 1) 1 + p|N| − 1 − p|N| p √ ⇔ 2 |N| − |N| rλ |N| − p|N| + |N| 1 + p|N| − 1 − p|N| √ > 2 (|S| − 1) − p|S| − 1 rλ

⇔ aL(S) > v(S)

L L Speciﬁcally a (NA) = u(NA)v(NA)/u(NA) = v(NA). Therefore a ∈ C(NA, v) for homogeneous principals.

To show that the Louderback’s value is a core allocation of our game (NA, v) we resort to numerical simulation to demonstrate that the Louderback’s value has a very high chance to be

in the core of (NA, v) when the principals are heterogeneous. By observing the deﬁnitions of the value function (12.1) and the Louderback’s value (12.4) we note that they are functions of the revenue rates {ri, i ∈ N} and the failure rates {λi, i ∈ N} of principals in set N.

Therefore the parameters of our simulations are the player set N, the failure rates {λi, i ∈ N} and the revenue rates {ri, i ∈ N}. We restrict the value of |N| to be the integers in interval [2, 15]. Furthermore note that for any principal i ∈ N, her revenue rate and failure rate

always appear as a single term riλi. Therefore without loss of generality in each round of

our simulation we set λi = 1 and ri ∈ U(50, 500000) for all i ∈ N, where U(α, β) is the uniform distribution with α and β as its minimum and maximum values respectively. For

193 each value of |N| + 1 we run the simulation for 100000 rounds, where in each round we draw

a sample of {ri, i ∈ N} independently from the uniform distribution U(50, 500000), calculate

L L v(S) and a (S) for all S ⊆ NA and check whether a ∈ C(NA, v) by deﬁnition. We report the frequency of the Louderback’s value being in the core of (NA, v) for each value of |N| in Table 12.2.

L Table 12.2: The frequency of a ∈ C(NA, v) for diﬀerent values of |N|

|N| Frequency |N| Frequency 2 100% 9 100% 3 100% 10 100% 4 100% 11 100% 5 100% 12 100% 6 100% 13 100% 7 100% 14 100% 8 100% 15 100%

13 Summary: Pooling Principals

This work describes a new variant of a known problem in economics referred to as the principal-agent problem. The economics’s principal-agent conﬁguration is concerned with the contract between the principal and the agent who responds to principal’s contract proposal. In this work we examine the opposite conﬁguration of one agent with contractual relation with many principals. This work’s setting is about principals who subcontract out- side their organization for the repair services of their revenue generating units to an agent. The principal, each principal, proposes a contract to our agent. We assume that the agent is of good standing – every principal wants to contract our agent. Thus, when we discuss the agent’s pooling the principals, we essentially mean that he, the agent, is selecting the contracts. Once this point is clear we can proceed with our assumptions, analysis, and results.

The setting is that of a single risk-neutral repair agent who selects from the contract oﬀers by multiple risk-neutral principals who outsource the repair services of their revenue generating units to an agent. Our work is an extension of the contractual analysis of a single agent

194 and a single principal that has been studied extensively by Zeng and Dror (2015a). To the best of our knowledge, little attention has been directed to the principal-agent performance based contract analysis in a single agent-multiple principals setting with heterogeneous principals and interdependencies between the failures of diﬀerent revenue generating units. Our work attempts to ﬁll this void.

We claim four contributions. First, we propose a mathematical model of a system of a single agent and multiple principals that captures the interdependencies between the failures of different revenue generating units due to the commonalities (components, designs and operational procedures, etc.) shared by them. Second, by assuming that the agent is in good standing thus is able to choose any subset of principals with the desired properties, we are able to formulate the agent’s optimization problem of choosing the ‘best’ set of principals to contract with and solve it by means of numerical simulation. Our simulation results provide an key insight that the agent prefers total negative interdependencies between the units under the contract. In such case the agent achieves a convexly increasing expected profit rate with respect to the principals set he contracts with. Third, we refine our mathematical model of single agent and multiple principals by incorporating the agent’s preference of perfectly negative interdependencies. The refined model is tractable and allows us to derive an explicit mathematical solution for the agent’s optimal expected profit rate. More importantly we prove the convexity of the agent’s optimal expected profit rate as a function of the principals set he contract with, which is consistent with our simulation results. Finally, we formulate the single agent multiple principals contractual interplay as a transferable utility cooperative game and discuss two profit rate allocation schemes, namely the Shapley value and the Louderback’s value, to assess the contributions of the principals to the agent’s optimal expected profit rate.

Although our analysis demonstrates strong potential for the agent to proﬁt by pooling principals with desired interdependencies, the theoretical framework we propose is not without limitations due to the simpliﬁcation and tractability assumptions. For example, the

195 convexity, even the superadditivity, of the agent’s expected proﬁt rate may be in jeopardy with diminishing negative interdependencies and strengthened positive interdependencies. The agent does not pool all the principals!

14 A Comparison of Performance Based Service Contracts

Consider a principal-agent setting with a principal operating a revenue generating entity for which she outsources the maintenance and repair function to an expert service agent. The principal proposes a service contract to the agent that the agent either rejects or accepts. We consider only performance based service contracts of the type discussed in Kim et al. (2010). We have identified three different types of performance based contracts for the principal to choose from. This paper compares the economic ramifications of these contracts for both the agent and the principal. Naturally one can expand such comparison beyond the performance based contracts considered in detail here. For instance, such analysis can consider other types of contract terms in addition to the compensation rate and the penalty rate in Zeng and Dror (2015a) and Zeng and Dror (2015b)).

Assume that the principal’s revenue generating unit operates for a random period time before failure, and it remains inoperative until an agent repairs it. We refer to the time periods when the unit is inoperative as downtimes, and the time periods when the unit is operational as uptimes. The procedure of a principal subcontracting the repair services to an agent is straight forward: The principal proposes a contract for the agent’s repair services whenever the unit fails. If the agent accepts the contract, then he acquires the service capacity that can not be observed and contracted by the principal, maximizing his expected proﬁt. At the end of the contract period, the principal compensates the agent as prescribed by the contract. The contract period is assumed to be undetermined (inﬁnite) in Zeng and Dror (2015a) and Zeng and Dror (2015b).

A contract structure is determined by the number and the types of contract terms that

196 are controlled by the principal and the agent. It is understood that under a principal-agent contract, the principal compensates the agent for his repair service, and the principal charges the agent a penalty when the unit is inoperative in order to provide an incentive for the agent to repair the failed unit as soon as possible. For example, a performance based contract in Zeng and Dror (2015a) and Zeng and Dror (2015b) consists of two contract terms controlled by the principal: the compensation rate w > 0 and the penalty rate p > 0. w represents the amount of money the principal pays the agent for each unit of contract time (e.g., minutes, hours, days, etc.), regardless the state of principal’s unit. p represents the amount of money the principal collects from the agent for each unit of downtime.

In this work we focus on contract structures in which only principal controlled terms are diﬀerent. The agent controls the service capacity all the time. In order to describe the contract structure in a more generalized manner, we denote by wu the compensation rate paid by the principal to the agent for each unit of uptime, denote by wd the compensation rate paid to the agent for each unit of downtime, and denote by pd the penalty rate the principal collects from the agent for each unit of downtime. We are speciﬁcally interested in the following three contract structures.

Contract Structure I. Consider a contract structure like the one used in Zeng and Dror (2015a) and Zeng and Dror (2015b). That is, the agent receives a compensation for each unit of contract time regardless of the state of the principal’s unit, and the agent is charged a penalty for each unit of principal’s downtime during the contract period. This case corresponds to wu = wd ∈ (0, +∞) and pd ∈ (0, +∞).

Contract Structure II. Under this contract structure, the agent receives a compensation only for each unit of uptime instead of each unit of time, and the agent is charged a penalty for each unit of downtime. This case corresponds to wu ∈ (0, +∞), wd = 0 and pd ∈ (0, +∞).

Contract Structure III. Under this contract structure, the agent is rewarded for each unit

197 of uptime during the contract period, and no penalty is collected from the agent during the

downtime. This case corresponds to wu ∈ (0, +∞) and wd = pd = 0.

We plan to examine the principal’s and the agent’s optimal contracting strategies when the principal can choose from the three contract structures introduced above, in a setting of a single principal and a single agent with undetermined (infinite) contract period, as in Zeng and Dror (2015a). We want to address two questions: (1) Given that the principal has more than one contract structures to choose from, which contract structure benefits her the most under different values of exogenous parameters? (2) Assume an agent of good standing, what is the agent’s preference over different contract structures when he pools the service demands of multiple principals?

For completeness we repeat the proof of Observation 4.1 in Zeng and Dror (2015a) that Contract Structure I is equivalent to Contract Structure II for a risk-neutral principal and a risk-neutral agent. However this equivalency does not necessarily hold when the agent is risk-averse. Regarding Contract Structure III, we suspect that the principal’s optimal contract terms and the agent’s optimal service capacity will deviate signiﬁcantly from those under Contract Structure I and Contract Structure II, because the principal controls only one variable (the compensation rate) instead of two (the compensation rate and the penalty rate).

The outline of our analysis is as follows. First, assume a risk-neutral principal and a risk- neutral agent, we examine the principal’s and the agent’s optimal contracting strategies under different contract structures and identify the optimal contract structure for the principal under different values of exogenous parameters. Second, we extend our analysis to the setting with a risk-averse or a risk-seeking agent, using the same risk representation adopted in Zeng and Dror (2015a). Finally we focus on the agent and identifies the contract structure that benefits the agent the most when he pools the service demands of multiple principals.

198 References

Shoshana Anily and Moshe Haviv. Cooperation in service systems. Operations Research, 58 (3):660–673, 2010.

Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath. Coherent measures of risk. Mathematical ﬁnance, 9(3):203–228, 1999.

Patrick Bolton and Mathias Dewatripont. Contract theory. MIT press, 2005.

Samuel Burer and Moshe Dror. Newsvendor games: convex optimization of centralized inventory operations. Top, 20(3):707–728, 2012.

Girolamo Cardano and T Richard Witmer. Ars magna or the rules of algebra, volume 1545. Dover New York, 1993.

M Jeya Chandra. A study of multiple ﬁnite-source queueing models. Journal of the Opera- tional Research Society, pages 275–283, 1986.

Moshe Dror, Bruce C Hartman, and Wei Chang. The cost allocation issue in joint replen- ishment. International Journal of Production Economics, 135(1):242–254, 2012.

Emel Filiz-Ozbay, Jonathan Guryan, Kyle Hyndman, Melissa Schettini Kearney, and Erkut Y Ozbay. Do lottery payments induce savings behavior: Evidence from the lab. Technical report, National Bureau of Economic Research, 2013.

Drew Fudenberg and Jean Tirole. Moral hazard and renegotiation in agency contracts. Econometrica: Journal of the Econometric Society, pages 1279–1319, 1990.

Keita Fukunaga and Wallace E Huﬀman. The role of risk and transaction costs in contract design: evidence from farmland lease contracts in us agriculture. American Journal of Agricultural Economics, 91(1):237–249, 2009.

Sanford J Grossman and Oliver D Hart. An analysis of the principal-agent problem. Econo- metrica: Journal of the Econometric Society, pages 7–45, 1983.

199 Roger Guesnerie and Jean-Jacques Laﬀont. A complete solution to a class of principal-agent problems with an application to the control of a self-managed ﬁrm. Journal of Public Economics, 25(3):329–369, 1984.

Nikolaus Hansen and Andreas Ostermeier. Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation. In Evolutionary Computation, 1996., Proceedings of IEEE International Conference on, pages 312–317. IEEE, 1996.

Nikolaus Hansen and Andreas Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary computation, 9(2):159–195, 2001.

Lani Haque and Michael J Armstrong. A survey of the machine interference problem. Eu- ropean Journal of Operational Research, 179(2):469–482, 2007.

Milton Harris and Artur Raviv. Some results on incentive contracts with applications to ed- ucation and employment, health insurance, and law enforcement. The American economic review, pages 20–30, 1978.

Bruce C Hartman and Moshe Dror. Optimizing centralized inventory operations in a cooperative game theory setting. IIE Transactions, 35(3):243–257, 2003.

Bengt H¨olmstrom.Moral hazard and observability. The Bell Journal of Economics, pages 74–91, 1979.

Bengt H¨olmstromand Paul Milgrom. Aggregation and linearity in the provision of intertem- poral incentives. Econometrica: Journal of the Econometric Society, pages 303–328, 1987.

Christian Igel, Nikolaus Hansen, and Stefan Roth. Covariance matrix adaptation for multi- objective optimization. Evolutionary computation, 15(1):1–28, 2007.

Canek Jackson and Rodrigo Pascual. Optimal maintenance service contract negotiation with aging equipment. European Journal of Operational Research, 189(2):387–398, 2008.

Daniel Kahneman and Amos Tversky. Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, pages 263–291, 1979.

200 Frank Karsten, Marco Slikker, and Geert-Jan van Houtum. Analysis of resource pooling games via a new extension of the erlang loss function. Technical report, Beta Working Paper series 344, 2011.

Frank Karsten, Marco Slikker, and Geert-Jan Van Houtum. Resource pooling and cost allocation among independent service providers. Operations Research, Articles in Advance: 1–13, 2015.

Sang-Hyun Kim, Morris A Cohen, Serguei Netessine, and Senthil Veeraraghavan. Con- tracting for infrequent restoration and recovery of mission-critical systems. Management Science, 56(9):1551–1567, 2010.

Tamara G Kolda, Robert Michael Lewis, and Virginia Torczon. Optimization by direct search: New perspectives on some classical and modern methods. SIAM review, 45(3): 385–482, 2003.

Haim Levy. Stochastic dominance and expected utility: survey and analysis. Management Science, 38(4):555–593, 1992.

Gregory Lewis and Patrick Bajari. Moral hazard, incentive contracts, and risk: Evidence from procurement. The Review of Economic Studies, page rdu002, 2014.

Robert Michael Lewis, Virginia Torczon, and Michael W Trosset. Direct search methods: then and now. Journal of Computational and Applied Mathematics, 124(1):191–207, 2000.

Joseph G Louderback. Another approach to allocating joint costs: a comment. Accounting Review, pages 683–685, 1976.

David G Luenberger. Microeconomic theory, volume 486. McGraw-Hill New York, 1995.

James G March and Zur Shapira. Managerial perspectives on risk and risk taking. Manage- ment science, 33(11):1404–1418, 1987.

Eric Maskin and Jean Tirole. The principal-agent relationship with an informed principal: The case of private values. Econometrica: Journal of the Econometric Society, pages 379–409, 1990.

201 Eric Maskin and Jean Tirole. The principal-agent relationship with an informed principal, ii: Common values. Econometrica: Journal of the Econometric Society, pages 1–42, 1992.

Roger B Myerson. Mechanism design by an informed principal. Econometrica: Journal of the Econometric Society, pages 1767–1797, 1983.

John A Nelder and Roger Mead. A simplex method for function minimization. The computer journal, 7(4):308–313, 1965.

Lionel Page, David A Savage, and Benno Torgler. Variation in risk seeking behaviour following large losses: A natural experiment. European Economic Review, 2014.

Bezalel Peleg and Peter Sudh¨olter. Introduction to the theory of cooperative games [electronic resource], volume 34. Springer, 2007.

John W Pratt. Risk aversion in the small and in the large. Econometrica: Journal of the Econometric Society, pages 122–136, 1964.

Sheldon M Ross. Introduction to probability models. Access Online via Elsevier, 2006.

Stephen A Ross. The economic theory of agency: The principal’s problem. The American Economic Review, 63(2):134–139, 1973.

Michael Rothschild and Joseph E Stiglitz. Increasing risk: I. a deﬁnition. Journal of Eco- nomic theory, 2(3):225–243, 1970.

Allan R Sampson and Robert L Smith. Assessing risks through the determination of rare event probabilities. Operations Research, 30(5):839–866, 1982.

Lloyd S Shapley. Cores of convex games. International journal of game theory, 1(1):11–26, 1971.

Joseph E Stiglitz. Incentives and risk sharing in sharecropping. The Review of Economic Studies, pages 219–255, 1974.

Joseph E Stiglitz. Equilibrium in product markets with imperfect information. The American Economic Review, 69(2):339–345, 1979.

202 Ad Straub. Cost savings from performance-based maintenance contracting. International Journal of Strategic Property Management, 13(3):205–217, 2009.

Amos Tversky and Daniel Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty, 5(4):297–323, 1992.

H Peyton Young. Cost allocation. Handbook of game theory with economic applications, 2: 1193–1235, 1994.

Shuo Zeng and Moshe Dror. Formulating Principal-Agent Service Contracts for a Revenue Generating Unit. In press, SpringerBriefs in Operations Management, 2015a.

Shuo Zeng and Moshe Dror. Pooling principals. 2015b. Working Paper, University of Arizona.

203