Essays in Market Design

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Joseph E. Duggan Jr., B.A., M.A., M.S.

Graduate Program in Economics

The Ohio State University

2018

Dissertation Committee:

Lixin Ye, Advisor Ramteen Sioshansi, Co-Advisor Paul J. Healy c Copyright by

Joseph E. Duggan Jr.

2018 Abstract

Economic theory has increasingly been brought to bear in designing and improving markets in the real world. Using tools drawn from economics, , and op- erations research, market design economists attempt to better understand real-world allocation problems in light of the unique constraints faced in particular markets, broadly defined.

This dissertation studies several different aspects of market design. The first chapter introduces the content of the dissertation. The second chapter consists of a comparison of two uniform price auction formats used in wholesale electricity mar- kets: centrally and self-committed markets. Both types of markets are operated by an independent third-party market operator, which solicits supply offers from gen- erators and determines how much energy they produce to serve customer demand.

In centrally committed markets, generators submit complex offers that convey all of their nonconvex operating costs and constraints. Conversely, generators submit sim- ple offers in self-committed markets that only specify the amount of energy that they are willing to produce at a given price. Thus, generators must internalize their non- convex costs and other operating constraints in submitting offers in a self-committed market. Centrally committed markets also include a provision that each generator is made whole on the basis of its submitted offers. No such guarantees exist in self- committed markets. The energy-cost ranking and incentive properties of the two

ii market designs in a multi-firm setting are studied and it is found that cost equivalence between the two market designs break down when there are three or more

firms.

The third chapter of the dissertation studies a one sided many-to-many matching model called the Stable Fixtures problem. First studied by Irving and Scott (2007), the Stable Fixtures Problem is a generalization of the .

A pair-wise stable matching is not guaranteed to exist for a given instance of the

fixtures problem. In this chapter, a psychologically motivated class of preferences that guarantees the existence of a pair-wise stable matching are explored and potential applications of this model for electricity markets are discussed.

In the fourth chapter, we model a proposed Power Purchase Agreement (PPA) in a stylized Nash-Cournot setting, demonstrating that the proposed PPA has incentive properties that can lead to socially undesirable equilibria. The proposed PPA would entail guaranteeing a predetermined profit level to a generating firm, which is paid by confiscating revenues from the non-subsidized generating firms. The PPA is ostensibly designed to ensure that generating firms with large fixed costs are able to supply power. However, we demonstrate that the proposed PPA may in fact have the exact opposite effect: there exist equilibria where the subsidized firm produces no output and the other generating firms obtain higher profits, a greater wholesale price, and there is a lower aggregate quantity produced in the market. The proposed PPA can result in (i) an inefficient firm that would otherwise exit the market in the absence of the subsidy is subsidized to produce nothing, or (ii) an economically viable firm would rather idle its generating capacity and collect its subsidy than produce.

The fifth chapter concludes with a discussion of directions for future research.

iii This work is dedicated to the the love and light of my life, Megan, and the

soon-to-be newest addition to our family:

“Before I formed you in the womb I knew you, before you were born I dedicated you”

Jeremiah 1:5

iv Acknowledgments

I would like to thank Lixin Ye for his guidance as my advisor. I would also like to extend thanks to Ramteen Sioshansi, my co-advisor and coauthor of my work, for his mentorship and for introducing me to electricity market design. I would like to thank Paul Healy for his feedback and the research meetings he ran. I would like to thank the many friends and classmates without whom I would not have completed this work.

I would like to extend thanks to my family. I would not have come this far without you. Mom and Dad, thank you for always supporting me in my dreams. To my siblings, my brothers Connor, Mike, and Jack, and my sisters Katie and Kiera, thanks for always being there for your big brother. I love you all.

I would like to thank my mother-in-law and father-in-law for their support. Fi- nally, I must extend the most heartfelt thanks to my wife, Megan. You have always been there for me and I would not have ever gotten anywhere this far in life without you. Thank you for your unceasing patience, love, support, and friendship. I love you.

v Vita

December 1, 1987 ...... Born - Boston, Massachusetts

2010 ...... B.A. in Economics and Mathematics, Boston College 2013 ...... M.A. in Economics, The Ohio State University 2016 ...... M.S. in Operations Research, The Ohio State University 2013-2017 ...... Graduate Teaching Associate, The Ohio State University. 2017-Present ...... Visiting Assistant Professor, Department of Economics and Finance, University of Dayton.

Fields of Study

Major Field: Economics

vi Table of Contents

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vi

List of Tables ...... ix

List of Figures ...... x

1. Introduction ...... 1

1.1 Background ...... 1 1.2 Scope and Contributions ...... 3 1.2.1 Incentives and Bidding Behavior in Unit-Commitment Auctions 4 1.2.2 Preferences and Stability in the Stable Fixtures Problem . . 6 1.2.3 Power Purchase Agreements ...... 8 1.3 Dissertation Structure ...... 9

2. Another Step Towards Equilibrium Offers in Unit Commitment Auctions with Nonconvex Costs: Multi-Firm ...... 11

2.1 Introduction ...... 11 2.2 System and Market Models ...... 14 2.2.1 Power System and Firm Structure ...... 14 2.2.2 Centrally Committed Market ...... 15 2.2.3 Self-Committed Market ...... 19 2.3 Market Equilibria ...... 22 2.3.1 Equilibria Under a Centrally Committed Market Design . . 23

vii 2.3.2 Equilibria Under a Self-Committed Market Design . . . . . 45 2.4 Expected-Cost Comparison of Market Designs ...... 61 2.4.1 Analysis of Market Equilibria ...... 61 2.4.2 Numerical Example ...... 65 2.5 Discussion and Concluding Remarks ...... 68

3. Subjective Distance: Random Paths to Pair-Wise Stability and the Fix- tures Problem ...... 71

3.1 Introduction ...... 71 3.2 The Stable Fixtures Problem ...... 75 3.3 Preferences and the Subjective Distance Metric ...... 77 3.4 Results ...... 79 3.4.1 Existence of Pairwise-Stable Fixture Matchings ...... 79 3.4.2 Random Paths to Pairwise-Stable Fixture Matchings . . . . 82 3.5 Other Applications ...... 86 3.5.1 Failures of Approximate Symmetry ...... 88 3.6 Discussion and Concluding Remarks ...... 89

4. A Nash-Cournot Analysis of a Proposed Power Purchase Agreement . . . 93

4.1 Introduction ...... 93 4.2 Related Literature ...... 94 4.3 Model ...... 95 4.3.1 Preliminaries ...... 95 4.3.2 Market Subsidy ...... 96 4.3.3 Effect of Subsidy on Production Levels ...... 99 4.3.4 Effect of Subsidy on Profit Levels ...... 101 4.4 A Simple 3-Firm Example ...... 103 4.5 Discussion and Concluding Remarks ...... 109

5. Contributions and Future Work ...... 111

5.1 Bidder Cost Revelation in Electric Power Auctions with Capacitated Suppliers ...... 111 5.2 Incentive Effects of Environmental Adders in Multi-Dimensional Procurement Auctions for Power Reserves ...... 113

Bibliography ...... 115

viii List of Tables

Table Page

2.1 Data for numerical example ...... 66

2.2 Total expected remuneration cost to MO under centrally and self- committed market designs for different values of N if firms follow mixed- Nash equilibria ...... 68

4.1 Welfare Comparison, F3=$10 ...... 107

4.2 Welfare Comparison, F3=$6 ...... 109

ix List of Figures

Figure Page

2.1 Expected Energy Price as a Function of Residual Load for Different Values of N if Firms Follow Mixed-Strategy Nash Equilibria . . . . . 67

x Chapter 1: Introduction

1.1 Background

Economic theory has increasingly been brought to bear in designing and improv- ing markets in the real world. Market Design can be thought of as the engineering branch of economics, (Roth (2002)). Using tools drawn from economics, game the- ory, and operations research, market design economists attempt to better understand real-world allocation problems in light of the unique constraints faced in particular markets, broadly defined. Market design has had many diverse applications, includ- ing allocating radio spectrum rights (Milgrom (2000)), matching cadets to service branches in the military (S¨onmez and Switzer (2013)), and assigning students to public schools (Abdulkadiroglu and S¨onmez (2003)). Two of the most well-known applications of market design economics are found in fields related to medicine, nei- ther of which involves markets in the traditional sense of the word. The first is the

National Resident Matching Program (NRMP), with the second being the design of exchanges for kidney donation. Both the NRMP and kidney exchange are examples of matching markets (Roth (1984), Roth et al. (2004)). The NRMP is used to match graduating medical students with residency positions, while kidney exchanges are

1 used to facilitate matches between donors and recipients of kidneys for renal trans- plantation. Market design is an especially rich area of research because economic theory can be applied to real world problems. The particulars inherent in studying a given institution generate further insights that can be used to improve the theory for future problems.

The two main areas of focus of market design economics are auctions and match- ing markets. The applications of economic auction theory are surveyed extensively by Milgrom (2004). Many of the foundational results and applications of matching theory are detailed in Roth and Sotomayor (1992). In this dissertation, we analyze competing market designs in the context of wholesale electricity markets and a par- ticular type of matching market. Using techniques of applied game theory, we study equilibrium bidding behavior and expected prices in two different types of uniform price auctions used in wholesale electricity markets. Not only does this add to our the- oretical understanding of auctions and industrial economics, it provides a benchmark for further analysis of these markets as they are restructured, offering potential guid- ance and information to policy makers and regulators. We also study the existence of stable matchings in a certain type of one-sided many-to-many matching market, showing that a psychologically motivated class of preferences ensures the existence of stable equilibria. Finally, we study the effects on prices, output, and social welfare in a modified Cournot model to explore the potential implications of a proposed market intervention.

2 1.2 Scope and Contributions

The goal of this dissertation is to explore the functioning of a variety of different markets, in order to add to our theoretical understanding of market design and ex- plore potential avenues for their improvement in the real world. We consider three important facets of market design in this dissertation, covering auctions, matching markets, and a proposed subsidization market intervention. We begin with a study of unit commitment auctions in the context of multi-firm oligopolies for wholesale electricity. We derive incentive and bidding behavior for both market designs and conduct a cost comparison across the two formats. Following the model of unit commitment auctions, we study a type of one-sided many-to-many matching market called the Stables Fixtures Problem. In this matching market, a set of n agents must be matched. Each agent has a capacity representing his maximum possible number of matches with other agents. Like the Stable Roommates Problem, there do not always exist stable matches for a given instance of the Stable Fixtures Problem. We consider a simple, psychologically motivated class of preferences that ensures the existence of pair-wise stable matchings. This class makes the matching market akin to some types of coarse matching markets, which have been studied as a possible improvement over rolling blackouts as a means of rationing electric power (McAfee

(2002)). The many-to-many nature of this one-sided matching market may also be of use in studying contracting arrangements around the integration of renewables into the electric grid. Finally, we examine a proposed Power Purchase Agreement for wholesale electricity markets. Under the proposed scheme, an inefficient firm would be guaranteed a predetermined profit level via a subsidy that is paid by confiscat- ing revenues from the non-subsidized firms. By developing a variant of the standard

3 Cournot model, we derive equilibria for the case of linear demand and demonstrate that the proposed market intervention will result in the existence of socially unde- sirable equiliibria that are detrimental to overall social welfare and may in fact have the opposite of their intended effect; namely, the subsidy, by guaranteeing a set profit level, can create an incentive for the subsidized firm to shut down, producing nothing and collecting the subsidy.

The remainder of this chapter introduces the topics and analysis of this disserta- tion.

1.2.1 Incentives and Bidding Behavior in Unit-Commitment Auctions

The organization of wholesale electricity markets is a question of immense im- portance to regulators, firms, and consumers. There are two uniform price auction formats that represent two commonly used market designs in the context of wholesale electricity markets: centrally-committed markets and self-committed markets. Under both market designs, both systems are operated by a third-party market operator.

The primary distinction between the two market designs is the manner in which bids are solicited and the system is operated/coordinated. In centrally-committed mar- kets, generating firms compete by offering complex multidimensional bids that reflect not only their marginal costs of production, but also their non-convex operating costs.

Under a self-committed market design, generating firms compete by submitting sim- ple offers that represent a minimum price at which the firm is willing to supply its

4 capacity. Importantly, this implies that generating firms must internalize the non- convexities of their production costs in their simple offers. Yet another important distinction between the two market designs is that in a centrally-committed market, generating firms are guaranteed to be “made whole” on the basis of their submitted bids. That is, these firms are guaranteed to at least break-even on the basis of their submitted offers. There are no such guarantees under a self-committed market de- sign. The first chapter of this dissertation examines incentives, bidding behavior, and expected costs between the two designs in the context of multi-firm oligopolies.

Methodology

In this chapter, we build upon the analysis of Sioshansi and Nicholson (2011) to study the energy-cost rankings and incentive properties of these two commonly used market designs. We develop a game theoretic model and derive Nash equilibria under both of the market designs. While equilibrium bidding behavior remains qualitatively consistent with the results of the setting of Sioshansi and Nicholson (2011) when demand is high, we find important differences between the duopoly setting and the multi-firm oligopoly setting when demand is low. In particular, when demand is low, we find that cost equivalence between the two market designs breaks down, a result which is driven by the fact that in certain states of low demand, inframarginal generators are able to earn positive profits under the self-committed market design, whereas all generating firms are constrained to earn zero profits under the centrally committed market design. In addition to deriving Nash equlibria analytically, we compare expected energy prices across the two market designs with a numerical ex- ample.

5 1.2.2 Preferences and Stability in the Stable Fixtures Prob- lem

The majority of the literature on matching markets has focused primarily on two- sided markets. That is, markets where there is a clear delineation between agents on one side of the market and those on the other side. For example, the canonical two-sided matching market studied is the , where agents are either men or women and each agent can only be successfully matched with an agent on the other side of the market. In this case, each agent can be matched to at most one other agent (his or her spouse - a one-to-one matching). When the restriction on the number of possible matches is relaxed for one side, the Stable

Marriage Problem becomes a College Admissions problem: the side of the market that can accept multiple matches is akin to a college which can accept many students, but any one student can only matriculate at one college (many-to-one matching).

In contrast, one-sided matching markets are those markets where any agent can, in theory, be matched with any other agent. The canonical one-sided matching model is called the Stable Roommates Problem and explores whether there is a way of assigning the agents into pairs such that no two agents would rather pair with one another rather than their assigned partner, or abandon their assigned partner and remain alone. Unlike the stable marriage problem, there do not always exist stable matches for a given instance of the Stable Roomates Problem. The Stable Fixtures

Problem is a generalization of the Stable Roommates problem where each agent has a capacity representing his maximum possible number of matches with other agents in the group. Since all agents can, in theory, be matched with more than one other agent, this is called a many-to-many matching. What makes the Stable Fixtures

6 Problem particularly interesting from a theoretical standpoint is the fact that it nests the Stable Marriage, College Admissions, and Stable Roommates Problems as special cases. As in Stable Roomates, there do not always exist stable matches for a given instance of the Stable Fixtures Problem. Examining the existence of pair-wise stable matches under certain preference domains is the goal of this chapter. The preference restriction explored here is related to a variety of different matching models. In assortative matching markets, the two sides of the market can be ranked in such a way that the highest ranked members on one side are matched with the highest ranked members of the other. Coarse matching is in some sense a particular case of assortative matching, where, for example, members of each side of the market can be designated ”high” or ”low” priority, and all agents in the same class receive the same allocation. This type of matching arrangement was studied in the context of rationing electricity by McAfee (2002)). The many-to-many nature of this matching market also affords the opportunity of expanding the analysis by incorporating contracts for integrating renewables.

Methodology

In this chapter, we examine a class of psychologically motivated preferences where agents prefer to match with agents that are “closer” to themselves in some subjective sense. As long as any subjective disagreements between the agents are not too “large,” there will exist pairwise stable matchings. If agents are allowed to form and break matches at random, the end result of the process will be a pair-wise stable matching.

7 1.2.3 Power Purchase Agreements

In this chapter, we model a proposed Power Purchase Agreement (PPA) that was put forth by several states and most recently the Department of Energy.1 The proposal is purportedly designed to benefit customers by providing them (i) with lower energy prices in the long-run and (ii) ensuring the reliability and resilience of generating capacity by ensuring that less efficient firms that nonetheless are necessary to supply power are able to do so even in the presence of high or uneconomic fixed costs. The proposal entails subsidizing certain firms by confiscating revenues from the non-subsidized firms on the basis of per-unit output. However, the proposal fails to consider the potentially adverse incentive effects of a policy intervention such as this, and in this chapter we explore the existence of socially undesirable equilibria.

Methodology

To model the PPA, we develop a variant of the standard Cournot oligopoly model.

A unique feature of the model studied here is that first, the subsidized firm is guaran- teed a certain profit level. In effect, what this means is that since the subsidized firm is guaranteed the same profit level independent of whatever output choice it makes, this firm has no strategic decision to make. Secondly, unlike Cournot models that consider an exogenously imposed tax or subsidy that is constant on a per-unit basis, under the proposed PPA, the per-unit cost of the subsidy that must be confiscated from the non-subsidized firms varies with the aggregate output in the market. Using a simple linear demand function, we derive Nash equilibria and demonstrate that the

1See https://www.ferc.gov/media/news-releases2018/2018-1/01-08-18.asp.Wrl hiMwgy4

8 proposed PPA can result in the existence of undesirable equilibria that result in losses of social welfare.

1.3 Dissertation Structure

This dissertation is organized as follows:

Chapter 1 provides an introduction to this dissertation. Background information on the economics of market design and its applicability to wholesale electricity and matching markets is discussed. We then introduce the research studied in this disser- tation.

Chapter 2 introduces the analysis of centrally versus self-committed market designs in the context of unit commitment auctions. We derive equilibrium bidding behavior and compare expected costs against the two market designs.

Chapter 3 studies the Stable Fixtures Problem, a one-sided many-to-many match- ing problem is introduced. We analyze a particular class of psychologically motivated preferences and the existence of pair-wise stable matchings under this preference structure.

Chapter 4 introduces a proposed Power Purchase Agreement (PPA) that was pro- posed in several states to allegedly generate lower energy prices for consumers in the long-run and recently championed by the Department of Energy to ensure reliability and resilience of generating capacity. Using a modified Nash-Cournot analysis, we

9 demonstrate that the proposed subsidy has deleterious incentive properties that can result in socially undesirable equilibria.

Chapter 5 concludes the dissertation with a discussion of directions for future re- search.

10 Chapter 2: Another Step Towards Equilibrium Offers in Unit Commitment Auctions with Nonconvex Costs: Multi-Firm Oligopolies

2.1 Introduction

The organization of wholesale electricity markets is a question of immense impor- tance to government regulators, firms, and consumers. However, market designs of this nature remain an open area of inquiry. Two commonly utilized market designs are centrally-committed markets versus self-committed markets. In both market designs, generators bid in a uniform price auction where bids are submitted to a third party market operator. What distinguishes these market designs is the format of the bid- ding and the way in which commitment and dispatch decisions are made. In centrally committed markets, generators submit complex offers, representing a fixed start-up cost as well as a of producing energy, thus incorporating non-convexities in their costs into the submitted offer. The market operator, with information on all relevant cost information about generators in the market, is then positioned to dis- patch generators to serve demand at lowest cost. In centrally-committed markets, generators are guaranteed to recover their as-bid start-up cost via make-whole pay- ments to ensure that no generator that is dispatched will be forced to operate at a

11 loss. In self-committed markets, generators submit a single-part bid which represents the minimum price at which the generator is willing to supply all of its capacity.

In self-committed markets, there are no make-whole payments, and generators are liable to produce all of their capacity at the uniform price determined by the highest accepted bid in the uniform price auction.

A common debate in this literature centers around the merits of these two mar- ket designs. Centrally-committed markets, it is argued, provide the market operator with the greatest amount of information. This then allows the market operator to make commitment and dispatch decisions to most efficiently meet the demand of the market. However, given that generators know that they are guaranteed to recover their reported start-up costs, it is possible that generators will overstate their true costs in their bids. Self-committed markets, in contrast, may not be as efficient as centrally-committed markets, but may in fact reduce the incentive of generators to overstate their costs. Ruff (1994) and Hogan (1994) argue that centrally-committed markets are most efficient, given that they provide the market operator, which makes dispatch and commitment decisions, with a greater amount of information. Elma- grabhy and Oren (1999) argue that commitment decisions are more efficient when left to individual firms, and therefore that self-committed markets are superior. Despite this contentious debate, there had been no systematic comparison of incentive and bidding properties of these two market designs.

Sioshansi and Nicholson (2011), were the first to study incentive properties and bidding behavior in centrally-committed and self-committed wholesale electricity mar- kets. We contribute to the literature on the design and regulation of wholesale elec- tricity markets by generalizing their analysis to that of an N-generator oligopoly. This

12 analysis sheds important light on the functioning of these two market designs. Im- portantly, we find that while many of the results derived by Sioshansi and Nicholson in the symmetric duopoly case carry over to oligopolistic markets, there are impor- tant differences that arise as the number of generators participating in the market increases. In particular, we find that expected cost-equivalence between the two mar- ket designs breaks down when there are more than two generators competing in the market. This result is driven by the fact that in a duopolistic setting, generators earn expected profits of zero regardless of whether the market is organized as a centrally- committed market or a self-committed market. However, when there are more than two generators, it is possible for generators in self-committed markets to earn posi- tive profits in certain low demand states. This occurs because of the fact that when there are more than two generators, the marginal generator is constrained to earn zero profits in equilibrium, but because bids in self-committed markets consist of a single-dimensional price offer, this implies that the uniform energy price will be large enough for generators producing at full capacity to earn positive profits in light of their fixed start-up costs.

In Section 2.2, we present the model. In Section 2.3, we derive equilibria across the two market designs. In Section 2.4, we perform an expected cost comparison of the two market designs and demonstrate our findings via simple numerical example.

Section 2.5 presents concluding remarks and further discussion regarding regulatory and policy implications of the analysis.

13 2.2 System and Market Models

Our model largely follows that used by Sioshansi and Nicholson (2011). The important distinction between our model and theirs is that we examine general N-

firm oligopolies, as opposed to restricting our attention to the duopoly case as they do.

2.2.1 Power System and Firm Structure

The power system consists of N identical generating firms that are competing in a single-shot, single-period, uniform-price auction to serve a deterministic load of l MW.

The load, l, is commonly known and must be served exactly. The generators each have capacity constraints, K > 0, on their generation, and this capacity is commonly known. The total capacity among the N generators is assumed to always be sufficient to serve the load, or that l ≤ NK. For a given load, l, we also define Nl, such that:

(Nl − 1)K < l ≤ NlK.

Nl represents the minimum number of generators that must be committed to be able to serve l MW of load. For ease of exposition, we also define:

lr = l − (Nl − 1)K,

as the residual load that remains if (Nl − 1) generators produce at their capacity of

K MW.

Each generator incurs a fixed start-up cost, S > 0, if it is committed. A generator must be committed to be able to produce energy. Otherwise, its output must be zero.

Each generator also incurs a constant marginal generating cost, c > 0, for each MW

14 produced. Thus, the actual cost to each firm of producing q MW is:   0, if q = 0; C(q) = S + cq, if 0 < q ≤ K;  +∞, if q > K.

The cost parameters, S and c, are commonly known to the generators but not to the

MO. Instead, the generators submit offers to the MO in the two market designs that are meant to convey this cost information. These offers are the sole basis on which the MO clears the market.

2.2.2 Centrally Committed Market

In a centrally committed market, each generator submits a complex offer that consists of two parameters. σi and i are the start-up and marginal generating costs, respectively, that are specified in generator i’s offer. Henceforth, we refer to these parameters as generator i’s start-up and energy offers, respectively. We let ωi =

(σi, i) denote generator i’s complex offer in the centrally committed market. We assume that there are offer caps, which we denote asσ ¯ and ¯, on the two parts of the offer, respectively. These offer caps exist in all competitive wholesale electricity markets as a means of mitigating the exercise of . Thus, we assume that 0 < σi ≤ σ¯ and 0 ≤ i ≤ ¯ for all generators. The assumption that the fixed-cost portion, σi, of generator i’s offer is strictly positive is to ensure that the equilibria studied under the centrally committed market design differ from those under a self- committed market.

Because the MO does not know the values of S and c, it commits and dispatches generators based on the costs specified in their offers. Thus, from the perspective of

15 the MO, generator i’s cost of producing q MW is:   0, if q = 0; ˜c C (q; ωi) = σi + iq, if 0 < q ≤ K;  +∞, if q > K.

Let ω = (ω1, . . . , ωn) denote the entire vector of complex offers. We use the subscript

‘−i’(e.g., −i) to denote offers submitted by generator i’s rivals.

Based on ω, the MO uses a unit commitment model to determine the generator commitments and dispatches. To formulate this unit commitment model, we define the following set of binary variables:

 1, if generator i is committed; u = i 0, otherwise. for all i = 1,...,N. We also define qi as a continuous variable indicating how many

MW are produced by generator i, for all i = 1,...,N. The unit commitment problem is then formulated as:

N X min (uiσi + iqi) (2.1) u,q i=1 N X s.t. qi = l, (2.2) i=1

0 ≤ qi ≤ uiK, ∀i = 1, . . . , n, (2.3)

ui ∈ {0, 1}, ∀i = 1, . . . , n. (2.4)

Objective function (2.1) minimizes total commitment and dispatch cost on the basis of the offers (i.e., not on the basis of the true costs, which the MO does not know).

Constraint (2.2) imposes load balance that total supply exactly equals demand. Con- straint set (2.3) imposes generator-capacity limits and requires a generator to be committed to produce a non-zero amount. Constraint set (2.4) forces the commit- ment variables to be binary.

16 The solution to the MO’s unit commitment problem depends upon the demand

level. If Nl = N, all N generators must be committed to serve the load. We refer to any load level such that Nl = N as a high-demand state. Conversely, if Nl < N then not all N generators must be committed to serve the load and at least one generator may be left inactive. We refer to such load levels as low-demand states.

In high-demand states, the start-up offers are irrelevant for determining the op- timal solution to the MO’s problem. This is because all of the generators must be committed and their start-up offers are sunk costs from the perspective of determining production levels. Thus, we can easily determine generator i’s expected production level in a high-demand state, by first defining:

c Mi = ||{j|j = i}||, (2.5)

as the number of generators that submit the same energy offer that generator i submits

c (including generator i itself). Thus, by definition, we always have that Mi ≥ 1. We can then define generator i’s expected production level in a high-demand state as:  c K, if ∃j such that j > i; qi (ω; l) = 1 c (2.6) c (l − (N − Mi )K), if j ≤ i, ∀j. Mi

Generator i has two possible production levels in a high-demand state. First, if an- other generator submits an energy offer that is strictly greater than that of generator i, then generator i produces at its capacity. The other case arises if generator i submits the highest energy offer. In this case, all of the generators that submit energy offers

c that are strictly less than that of generator i, of which there are (N − Mi ), produce

c at their maximum capacity. All but one of the remaining Mi generators that submit the highest energy offer are randomly assigned to produce K MW while one serves the residual load.

17 In low-demand states, the load can be served with strictly less than N generators being committed. Indeed, because we assume that σ1, . . . , σN > 0, it is optimal in the

MO’s problem for only Nl generators to be committed and the remaining generators to be kept offline. Once Nl generators are selected to be committed, (Nl − 1) of those

generators will be dispatched (on the basis of the marginal generating cost specified in

their offers only) to produce at their capacity of K MW. The remaining one generator

with the highest marginal generating cost in its offer serves the residual load, lr.

The energy price under a centrally committed market design is the highest of the

accepted energy offers among generators producing a positive amount of energy:

c c p = max {i|qi > 0}, i=1,...,N

c c where qi denotes generator i’s production level. In a high-demand state qi is explicitly

c given by (2.6). We characterize qi in low-demand states when deriving equilibria in Section 2.3.

Generators are paid for their energy production in a centrally committed market,

giving generator i energy revenues of:

c c p qi .

These energy payments alone may be insufficient for each generator to recover all of

its costs. This is due to the non-convex start-up cost. As such, the MO uses make-

whole payments to ensure that no generator operates at a net loss, on the basis of

the costs specified in its offer. If we let Wi denote the make-whole payment given to

generator i and Ti the total payment given to generator i we have:

c c Ti = p qi + Wi

c c c c = p qi + max{0, σi + (i − p )qi }, (2.7)

18 meaning that:

c c Wi = max{0, σi + (i − p )qi }. (2.8)

To understand (2.7) and (2.8) note that generator i receives a strictly positive make- whole payment if and only if its net operating profit from receiving energy payments alone (on the basis of its offer), which is:

c c (p − i)qi − σi, is strictly negative. In such a case, we have from (2.8) that generator i receives a make-whole payment of:

c c Wi = σi + (i − p )qi , meaning that its net operating profit from receiving the energy and make-whole pay- ments is:

c c (p − i)qi − σi + Wi = 0.

Otherwise, if:

c c (p − i)qi − σi, is non-negative, generator i does not need a supplemental make-whole payment to recover the costs specified in its offer. In such a case, we have from (2.8) that Wi = 0, as desired.

2.2.3 Self-Committed Market

In a self-committed market each generator submits a simple offer consisting of a single parameter—δi specifies the minimum price at which generator i is willing to produce. We assume that this minimum energy price must be non-negative and that it is subject to an offer cap, which we denote as δ¯. We assume that all generators,

19 including those that do not wish to commit their units, submit a minimum energy

price. If generator i does not wish to commit its unit it could choose to offer δi =

δ¯. Doing so would act to economically withhold generator i’s capacity from the

market. Indeed, with such an offer generator i is only dispatched (which would

effectively require the unit to be committed) if all other capacity in the system is

exhausted. Many restructured wholesale electricity markets have such a requirement

that generators offer their capacity at some price below the offer cap, to minimize

the likelihood of having unserved demand. We let δ = (δ1, . . . , δN ) denote the entire

vector of simple offers.

As in the centrally committed market, because the MO does not know the values

of S and c, in a self-committed market it dispatches the generators based on the

simple offers (i.e., the value of δ) only. This dispatch problem can be formulated as

the following linear optimization problem:

N X min δiqi q i=1 N X s.t. qi = l, i=1

0 ≤ qi ≤ K, ∀i = 1, . . . , n,

where the variables, q1, . . . , qN , retain the same definitions as in the unit commitment

model of the centrally committed market. The objective function and constraints of

this dispatch problem are analogous to those in the unit commitment model, except

that start-up costs are not included in the objective function nor are there binary

variables representing commitment decisions.

The solution to the dispatch problem depends upon the demand level and the rank

ordering of the offers. The solution can be expressed in closed-form by relabeling the

20 offers in increasing order. Because the generators are symmetric (except for their

offers) we can, without loss of generality, relabel them such that δ1 ≤ · · · ≤ δN . We next define:

s Mi = ||{j|δj = δi}||, as the number of generators that submit the same offer that generator i submits. We can then define generator i’s expected production level as:  K, if δi < δNl ; s  1 s qi (δ; l) = M s (l − (Nl − Mi )K), if δi = δNl ; (2.9)  i 0, if δi > δNl . Generator i has three possible production levels in a self-committed market. If its offer

is strictly lower than the Nlth offer (in ascending order), then generator i is dispatched

to produce up to its capacity of K MW. If, on the other hand, generator i’s offer is

strictly greater than the Nlth offer, then it is not dispatched and produces 0 MW.

This situation corresponds to the third case in (2.9). If generator i’s offer is equal to

the Nlth offer, then it is randomly assigned to either product K MW or to serve the

residual load. This case is analogous to the treatment of ties in the highest energy

offer given in (2.6).

The energy price in this market is the highest of the accepted offers among gen-

erators producing a positive amount:

s s p = max {δi|qi > 0}, i=1,...,N

and generators are paid for their production, earning energy revenues:

s s p qi .

Unlike in a centrally committed market, there are no make-whole payments. This is

because the MO does not have any information regarding generator start-up costs.

21 Instead, generators that choose to commit themselves and sell energy in the market

must ensure that they earn sufficient revenues from those energy sales to recover the

start-up costs that they incur. As such, we assume that the offer cap, δ¯, is sufficiently

large to ensure that any generator producing a strictly positive amount of energy is ¯ able to recover its start-up cost. That is, we assume that (δ − c)lr ≥ S.

2.3 Market Equilibria

Both market designs have different types of equilibria, depending on the demand

level. We have three types of low-demand states, in which strictly less than N gen-

erators are needed to serve the load. We refer to the first type, in which only one

generator is needed to serve the load (i.e., when l ≤ K or equivalently Nl = 1), as

weak low-demand states. We also refer to the one generator that serves the load in a weak demand state as the unique generator. The unique generator’s offer, production,

profit, and payment are all denoted by the superscript, ‘U.’

The other two types of low-demand states require at least two generators to serve

the load (i.e., Nl ≥ 2). As discussed in Section 2.2, the MO’s problem in low-

demand states (under both centrally and self-committed market designs) always has

an optimal solution in which (Nl −1) generators operate at their K MW capacity and

one generator serves the residual load, lr. We refer to the (Nl − 1) generators that

operate at their K MW capacity as inframarginal generators and the generator that

serves the residual load as the marginal generator. We also refer to any generators that produce 0 MW as inactive generators. As needed, the offers, production, and profits of and payments to inframarginal, marginal, and inactive generators are denoted by the superscripts, ‘I,’ ‘M,’ and ‘V ,’ respectively.

22 The remaining two types of low-demand states differ in terms of whether the

marginal generator operates at its K MW capacity or not. We refer to the first type of low-demand state as an integral low-demand state. In an integral low-demand state,

the load can be served by having Nl generators produce at their capacity, meaning

that the marginal generator operates at its K MW capacity. Thus, an integral low-

demand state is characterized by lr = K or equivalently by l = NlK. We refer

to the other type of low-demand state as a regular low-demand state. In a regular

low-demand we have l < NlK or equivalently that lr < K. The load in a regular

low-demand state can be satisfied by (Nl − 1) inframarginal generators that produce

at their K MW capacity and one marginal generator that produces lr < K.

The final type of demand state that we have is a high-demand state, in which

Nl = N and (N − 1)K < l. We retain the same notion of inframarginal and marginal

generators in high-demand states as we have in low-demand states. However, there

are no inactive generators in a high-demand state, because all N generators are needed

to serve the load.

We now proceed by characterizing equilibria under the two market designs in

different demand states.

2.3.1 Equilibria Under a Centrally Committed Market De- sign

Our analysis of the centrally committed market design begins with the following

lemma, which characterizes payments to generators in different demand states.

Lemma 1. Under a centrally committed market design the total payment to the unique

generator is:

T U = σU + U l,

23 while all other generators receive zero payment in a weak low-demand state.

In all other demand states the total payment to the marginal generator is:

M M M T = σ +  lr, the payment to inframarginal generator i is:

I M Ti = max{ K, σi + iK}, and the payments to any inactive generators are zero under a centrally committed market design.

Proof. In a weak low-demand state only one generator is required to serve demand.

As such, this generator sets the energy price, pc = U . Moreover, because σU > 0, this generator must receive make-whole payments to recover its offer-based start-up cost. Thus, from (2.7) we have that total payments to the unique generator are:

T U = U l + max{0, σU + (U − U )l}

= σU + U l.

Because inactive generators produce nothing and incur zero cost, they receive neither energy nor make-whole payments.

In all other demand states we have Nl ≥ 2. The marginal generator is dispatched to serve the residual load, lr, and it sets the energy price. This is because it is optimal for the MO to have the generator with the highest value of i in its offer (among the

c M Nl generators that it commits) serve the residual load. Thus, we have p =  .

Following the same logic as in the weak low-demand state, the marginal generator must receive make-whole payments because σM > 0. Thus, the total payments to the

24 marginal generator are:

M M M M M T =  lr + max{0, σ + ( −  )lr}

M M = σ +  lr.

The MO ensures that inframarginal generators earn revenues that fully cover their incurred costs (as calculated on the basis of their offers). We further know that inframarginal generators are dispatched to produce K MW. Thus, from (2.7) we have that total payments to inframarginal generator i are:

I M M Ti =  K + max{0, σi + (i −  )K}.

If:

M M max{0, σi + (i −  )K} = σi + (i −  )K,

M this means that σi + iK ≥  K and that energy payments alone are insufficient to recover the offer-based cost incurred by generator i. Thus, generator i is given a make-whole payment equal to σi + iK. Otherwise, if:

M max{0, σi + (i −  )K} = 0,

M we have that σi + iK ≤  K meaning that energy payments are sufficient to recover the offer-based cost incurred by generator i. In such a case, generator i is given an energy payment equal to M K. Thus, in the end, an inframarginal generator receives

M the larger of σi + iK and  K in total payments, meaning that:

I M Ti = max{ K, σi + iK}.

We finally have (as with weak low-demand states) that any inactive generators receive neither energy nor make-whole payments.

25 We now proceed by first characterizing equilibrium offering strategies in the three

different types of low-demand states and then examining high-demand states.

Equilibria in Low-Demand States

We begin with the following lemma, showing that in all three types of low-demand

states the marginal generator earns zero profits.

Lemma 2. In all three types of low-demand states the marginal generator earns zero

profits in a under a centrally committed market design.

Proof. Assume for contradiction that the marginal generator earns nonzero profits.

If its profits are negative, the marginal generator would prefer to be inactive and

earn zero profits. If the marginal generator is earning positive profits, its offer can

be profitably undercut by an inactive generator. Thus, the profits of the marginal

generator must be zero.

Lemma 2 implies the following corollary, which dictates the behavior of inactive

generators in a Nash equilibrium.

Corollary 1. In all three types of low-demand states all inactive generators submit offers, ωV = (σV , V ), such that:

V V σ +  lr ≥ S + clr, (2.10)

and this inequality must be binding for at least one inactive generator in a Nash

equilibrium under a centrally committed market design.

Proof. Based on Lemma 1 we know that the marginal generator receives:

M M M T = σ +  lr,

26 in total payments. We further know from Lemma 2 that the marginal generator earns zero profits. Thus we have that:

M M σ +  lr = S + clr.

Now, assume for contradiction that no inactive generator submits an offer that satisfies (2.10). This means that at least one inactive generator, generator j, submits an offer ωj = (σj, j), such that:

σj + jlr < S + clr.

However, in this case it would be optimal for the MO to commit and dispatch gener- ator j in place of the marginal generator, giving a contradiction.

To prove the second part of the corollary, assume for contradiction that no inactive generator submits an offer that makes inequality (2.10) binding. Let generator i be the inactive generator with the offer that gives the lowest value of σi + ilr. The marginal generator then has a profitable deviation in which it increases its offer to undercut σi+

ilr, which increases the make-whole payments that the marginal generator receives.

We can now demonstrate that in weak low-demand states all generators earn zero profits.

Proposition 1. In weak low-demand states the unique set of pure-strategy Nash equi- libria under a centrally committed market design is characterized by at least two of the generators submitting offers of the form ω ∈ ΩW where:

ΩW = {(σ, )|σ + l = S + cl, σ ∈ (0, σ¯],  ∈ [0, ¯]},

27 and the remaining generators submitting offers of the form:

ω ∈ {(σ, )|σ + l ≥ S + cl, σ ∈ (0, σ¯],  ∈ [0, ¯]}.

Moreover, all generators earn zero profits.

Proof. In a weak low-demand state the load is served by a single unique generator, which is also, by definition, the marginal generator. From Lemma 2, we know that the unique generator must earn zero profit. Combining this with Lemma 1 tells us

U that ω ∈ ΩW . Corollary 1 shows that at least one inactive generator must submit an offer from the set ΩW and the property regarding offers of the remaining inactive generators. Finally, from Lemma 1 we know that all generators earn zero profits.

We next characterize equilibria in integral low-demand states.

Proposition 2. In integral low-demand states the unique set of pure-strategy Nash equilibria under a centrally committed market design is characterized by at least (Nl +

1) generators submitting offers of the form ω ∈ ΩL where:

ΩL = {(σ, )|σ + K = S + cK, σ ∈ (0, σ¯],  ∈ [0, ¯]}, (2.11) and the remaining generators submitting offers of the form:

ω ∈ {(σ, )|σ + K ≥ S + cK, σ ∈ (0, σ¯],  ∈ [0, ¯]}.

Moreover, all generators earn zero profits.

Proof. In an integral low-demand state we know that (Nl−1) inframarginal generators are dispatched at capacity and one additional marginal generator serves the residual demand, which is equal to its K MW capacity. From Lemma 2 we know that this

28 marginal generator must earn zero profit. Combining this with Lemma 1 we further

know that the marginal generator’s offer must be in the set ΩL.

Next, consider the offers of the (Nl − 1) inframarginal generators and assume

for contradiction that the offer of one of them, generator i, is not in the set ΩL. If

generator i’s offer is of the form:

ωi ∈ {(σi, i)|σi + iK > S + cK, σi ∈ (0, σ¯], i ∈ [0, ¯]}.

Then, based on Lemma 1, generator i’s total payments are:

I M Ti = max{ K, σi + iK} = σi + iK.

This is because we know that the marginal generator must receive make-whole pay- ments. Thus, because generator i’s cost is greater than that of the marginal generator

(when computed on the basis of their offers), generator i must also receive make-whole payments. Thus, in this case generator i earns strictly positive profits. However, this would mean that any of the inactive generators could profitably undercut generator i.

If, instead, generator i’s offer is of the form:

ωi ∈ {(σi, i)|σi + iK < S + cK, σi ∈ (0, σ¯], i ∈ [0, ¯]}.

Based on Lemma 1, we know that generator i’s total payments are:

I M Ti = max{ K, σi + iK}.

I M If Ti =  K, then generator i is earning negative net profit (because the marginal

I generator must receive make-whole payments to recover its start-up costs). If Ti =

σi+iK, then generator i is also earning negative net profit (because of the assumption on ωi). Thus, we have contradictions showing that all of the inframarginal generators must submit offers in ΩL.

29 Corollary 1 shows that at least one inactive generator must submit an offer from

the set ΩL and the property regarding the offers of the remaining inactive generators.

Finally, Lemma 1 and the properties of the inframarginal generators’ offers that are

shown in this proof allow us to conclude that all generators earn zero profits.

Propositions 1 and 2 characterize Nash equilibria in the weak and integral low-

demand cases. Characterizing equilibria in regular low-demand cases is more chal-

lenging. We begin with the following lemma, which shows that in a Nash equilibrium

all inframarginal generators receive the same type of payment (i.e., they all either

receive make-whole or energy payments).

Lemma 3. If at least one inframarginal generator receives energy payments in a regular low-demand state under a centrally committed market design, then all infra- marginal generators must receive energy payments in a Nash equilibrium.

Proof. Assume for contradiction that there exists at least one inframarginal generator

that receives energy payments and another inframarginal generator, generator j, that

receives make-whole payments. This means that generator j receives total payments

M of σj + jK which is greater than  K, the amount earned by the inframarginal

generator receiving energy payments. The inframarginal generator receiving energy

payments then has a profitable deviation, in which it matches generator j’s offer.

The preceding lemma leads to the following corollary, which shows that in a Nash

equilibrium all inframarginal generators earn the same profits.

Corollary 2. All inframarginal generators earn the same profits in a Nash equilibrium

under a centrally committed market design in a regular low-demand state.

30 Proof. From Lemma 3 we know that all inframarginal generators receive the same

type of payments. If they all receive energy payments, then this result is trivially

true.

If, on the other hand, they receive make-whole payments assume for contradiction

that inframarginal generator i receives a greater make-whole payment than infra- marginal generator j. This implies that:

σi + iK > σj + jK.

In such a case, however, generator j can profitably deviate and match generator i’s

offer.

We now proceed by examining three possible cases involving the offer of the

marginal generator—those in which the marginal-cost portion of its offer, M , is

strictly less than, equal to, or strictly greater than its true marginal cost, c. We

show in the following lemmata and corollaries that equilibria are not possible in the

first case, while they are in the second. We further show that in the third case all

generators earn zero profits.

Lemma 4. There are no Nash equilibria in which the marginal generator submits an

offer with M < c in a regular low-demand state under a centrally committed market

design.

Proof. Assume for contradiction that the marginal generator submits an offer with

M < c. By Lemma 2 we must have σM > S and the marginal generator receives

make-whole payments to fully recover its actual incurred costs (and earn zero profits).

M M This means that σ +  lr = S + clr. Given that lr < K and the assumption that

31 M < c, it must be true that σM + M K < S + cK, which in turn implies that

M K < S + cK (because σM > 0). Thus, inframarginal generators must receive make-whole payments, otherwise they earn negative profits.

For inframarginal generators to earn non-negative profits from make-whole pay- ments, they must submit offers such that σI + I K ≥ S + cK. This, however, implies that:

σM + M K < S + cK ≤ σI + I K, meaning that it would be less costly (from the perspective of the MO) for the marginal generator to be dispatched as an inframarginal generator. This gives a contradiction showing that there are no equilibria in which M < c.

We now show that Nash equilibria, in which the marginal generator submits its

true marginal generating cost, do exist.

Lemma 5. There are pure-strategy Nash equilibria in a regular low-demand state under a centrally committed market design in which the marginal generator submits an offer with M = c. Such equilibria are characterized by the marginal generator

submitting an offer with σM = S, all inframarginal generators submitting offers of

the form ω ∈ ΩL, where ΩL is defined by (2.11), and the inactive generators submitting

offers of the form:

ω ∈ {(σ, )|σ + lr ≥ S + clr, σ ∈ (0, σ¯],  ∈ [0, ¯]},

with this inequality being binding for at least one inactive generator. Moreover, all

generators earn zero profits.

Proof. We prove that a set of offers satisfying these conditions constitute a Nash equi-

librium by examining the offers of the three different types of generators—marginal,

32 inactive, and inframarginal generators—in turn. First, we know from Lemma 2 that

the marginal generator must earn zero profits in a Nash equilibrium. Combining this

with Lemma 1 tells us that if M = c then σM = S. Corollary 1 gives us the properties of the offers of the inactive generators.

Finally, consider the offers of inframarginal generators. If an inframarginal gener- ator submits an offer from the set, ΩL, it must receive make-whole payments (because the marginal generator sets the energy price equal to c, which is insufficient to re- cover the inframarginal generator’s start-up cost) and earns zero profits. Clearly this generator has no incentive to submit an offer,ω ˜ = (˜σ, ˜), such that:

σ˜ +K ˜ < S + cK, because doing so would result in negative profits. Moreover, consider a deviation in which it submits an offer,ω ˜, such that:

σ˜ +K ˜ > S + cK.

The marginal generator is submitting an offer with true costs. Thus, we have:

σ˜ +K ˜ > S + cK = σM + M K, meaning that the inframarginal generator becomes a marginal or inactive generator

(resulting in it earning zero profits) if it deviates and offersω ˜. Thus, we conclude the the inframarginal generator has no profitable deviation from submitting an offer in the set ΩL.

We can also conclude that there are no Nash equilibria in which any inframarginal generator submits an offer that is not in the set ΩL. To see this, consider inframarginal generator i submitting an offer, ωi = (σi, i), such that:

σi + iK < S + cK.

33 In this case, generator i earns negative profits and has a profitable deviation in which it

submits a sufficiently high offer that it becomes inactive. Conversely, if inframarginal

generator i submits an offer such that:

σi + iK > S + cK,

then because the marginal generator is submitting an offer with its actual costs,

we know that it is not possible for generator i to be inframarginal (i.e., there is a

lower-cost solution to the MO’s problem in which generator i is not an inframarginal

generator).

We finally turn to the most complicated case, in which the marginal generator

submits offers with M > c. We do not fully characterize Nash equilibria in this

case. Rather, we conclude in the following lemma and corollary that there are no

pure-strategy Nash equilibria in such a setting in which any generator earns strictly

positive profits.

Lemma 6. If the marginal generator submits an offer with M > c in a regular low- demand state under a centrally committed market, then no inframarginal generator can earn strictly positive profits in a Nash equilibrium.

Proof. Assume for contradiction that there is a Nash equilibrium in which M > c and in which the inframarginal generators earn strictly positive profits. We examine two cases, those in which the inframarginal generators receive energy payments and those in which they receive make-whole payments (we know from Lemma 3 that all inframarginal generators receive the same type of payments in equilibrium).

First consider the case in which inframarginal generators receive energy payments.

M For this to be true, it must be the case that  K > σi + iK, where generator i is an

34 inframarginal generator. For this generator’s profits to be strictly positive, it must

also be the case that M K > S + cK. Moreover, because σM > 0 we must have that

σM + M K > M K > S + cK. Thus, generator i has a profitable deviation in which it submits the offer,ω ˜i = (˜σi, ˜i), such that:

M M M σ +  K > σ˜i + ˜iK >  K.

Doing so results in generator i receiving make-whole payments (as opposed to energy payments), which are greater than the energy payments it otherwise receives. Thus, we have a contradiction showing that if M > c there are no pure-strategy Nash equilibria in which inframarginal generators receive energy payments and earn strictly positive profits.

We next consider the case in which inframarginal generator i receives make-whole

M M payments. For generator i to be inframarginal, we need σ + K ≥ σi+iK. We now

M M consider cases in which this inequality is binding and not. If σ +  K > σi + iK, then generator i has a profitable deviation in which it submits the offer,ω ˜i = (˜σi, ˜i), such that:

M M σ +  K > σ˜i + ˜iK > σi + iK, as doing so increases the make-whole payments that it receives. In the other case that:

M M σ +  K = σi + iK, (2.12)

we note that for generator i to earn strictly positive profits we need that σi + iK >

S + cK. Combining this inequality with (2.12) gives:

M M σ +  K = σi + iK > S + cK.

35 This inequality gives the marginal generator a profitable deviation, in which it slightly

undercuts the offer of generator i, earning it strictly positive profits. Thus, we again

have a contradiction showing that if M > c there are no pure-strategy Nash equilibria

in which inframarginal generators receive make-whole payments and earn strictly

positive profits.

Corollary 3. If the marginal generator submits an offer with M > c in a regular

low-demand state under a centrally committed market design, then all inframarginal

generators earn zero profits in a pure-strategy Nash equilibrium.

Proof. By Lemma 6 we know that if M > c, inframarginal generators cannot earn strictly positive profits in a pure-strategy Nash equilibrium. We can further conclude that they cannot earn strictly negative profits in an equilibrium, as they would have a profitable deviation in which they increase their offers to become inactive generators.

Thus, the inframarginal generators must earn exactly zero profits in equilibrium.

Equilibria in High-Demand States

In high-demand states all generators must be active. As a result, the start-up costs

specified in the generators’ offers, σ1, σ2, . . . , σN , must all be incurred by the MO.

Thus, dispatch decisions are made solely on the basis of the marginal costs specified

in the generators’ offers, 1, 2, . . . , N . We first show, in the following proposition,

that as a result of this, there are no pure-strategy Nash equilibria in high-demand

states under a centrally committed market design.

Proposition 3. There are no pure-strategy Nash equilibria in a high-demand state

under a centrally committed market design.

36 Proof. Assume for contradiction that (σ1, 1), (σ2, 2),..., (σN , N ) constitutes a pure- strategy Nash equilibrium. We begin by noting that if any of the N generators is operating at a net profit loss, such a generator can deviate and submit the offer (¯σ, ¯).

By assumption, such offers are high enough to guarantee that a generator does not operate at a net loss. Thus, we assume henceforth that no generator is operating at a net profit loss under the assumed equilibrium.

Without loss of generality, suppose that the generators are labeled so that 1 ≤

2 ≤ · · · ≤ N . Because all N generators must be committed, we know that they are dispatched based solely on the merit order of the marginal costs that are specified in their offers. As such, generators 1 through (N − 1) are inframarginal generators, generator N is the marginal generator, and the energy price is N . We now consider two cases, depending on whether there is a tie between the marginal costs specified in the offers of generators (N − 1) and N.

Consider, first, the case in which N−1 < N . If generator (N − 1) receives make- whole payments, then generator (N −1) has a profitable deviation in which it submits an offer,ω ˜N−1 = (σN−1, N − η), with η > 0 sufficiently small. Offeringω ˜N−1 results in generator (N − 1) remaining an inframarginal generator but earning higher make- whole payments. Conversely, if generator (N − 1) receives energy payments, this means that N K > σN−1 + N−1K. Generator (N − 1) also has a profitable deviation in this case, wherein it submits the offer,ω ˜N−1 = (σN , N − η), with η > 0 sufficiently small so that σN + (N − η)K > N K. Offeringω ˜N−1 results in generator (N − 1) remaining an inframarginal generator but receiving make-whole payments, which are greater than the energy payments, N K. The profitable deviations for generator (N −

37 1) in both of these cases show that there are no pure-strategy Nash equilibria in which

there is a unique marginal generator with the highest marginal cost in its offer.

We finally consider the second case, in which N−1 = N . We know from (2.6)

that the expected profit of generator (N − 1) is:

1 c σN−1 − S + (N − c) c (l − (N − MN )K), MN

c where MN is defined by (2.5). Consider a deviation, in which generator (N − 1)

submits the offer,ω ˜N−1 = (σN−1, N − η), with η > 0 sufficiently small. Offering

ω˜N−1 causes generator (N − 1) to become an inframarginal generator that receives make-whole payments. As such, its profit changes to:

σN−1 − S + (N − η − c)K.

Note, however, that:

1 c σN−1 − S + (N − η − c)K > σN−1 − S + (N − c) c (l − (N − MN )K), MN because:

1 c K > c (l − (N − MN )K), MN whereas η can be made arbitrarily small. Thus, we have a contradiction showing that

there are no pure-strategy Nash equilibria in which there are multiple generators tied

with the highest energy offer, proving the desired result.

The logic of the proof is as follows. If generator N submits an offer with a marginal

cost that is strictly greater than the marginal cost in generator (N − 1)’s offer, then

generator (N − 1) has an incentive to submit an offer that is just below that of

generator N to receive higher make-whole payments. Indeed, in this case all of the

38 inframarginal generators have an incentive to deviate in this way. Conversely, if there is a tie for the highest marginal cost in the offers, then one of the generators that is tied can profitably undercut the energy offer in generator N’s offer. Although doing so results in a slight decrease in make-whole payments, the deviating generator produces more energy. Moreover, the reduction in make-whole payments can be controlled by keeping the value of η sufficiently small.

Because there are no pure-strategy Nash equilibria in a high-demand state under a centrally committed market design, we now proceed to characterizing and then deriving an analytical expression for a mixed-strategy Nash equilibrium. We let

Fi(σi, i) denote the cumulative distribution function of generator i’s mixed strategy.

− + We further let Φi denote the support of Fi and more specifically define i and i as

N the infimum and supremum energy offers, respectively, in Φi. We also let Φ = ∩i=1Φi denote the common support of the N cumulative distribution functions.

Lemma 7. The infimum energy offers of the mixed strategies used by the generators in a high-demand state under a centrally committed market design are equal in a Nash equilibrium.

− − Proof. Assume for contradiction that i < j . In such a case, generator i has a profitable deviation wherein it moves the density that is assigned to the interval,

− − − [i , j ), to j − η, with η > 0 sufficiently small. By doing so, generator i increases its expected profit without decreasing the probability that it is an inframarginal generator. Thus, we have a contradiction showing that the infimum energy offers in a mixed-strategy Nash equilibrium must be equal.

39 Lemma 8. None of F1(σ1, 1),F2(σ2, 2),...,FN (σN , N ) have a mass point on Φ in a

mixed-strategy Nash equilibrium in a high-demand state under a centrally committed

market design.

Proof. Assume for contradiction that there exists at least one mass point of Fi(σi, i)

for some generator, i. Let ˆi denote one of these mass points. There exist η > 0 and ρ > 0 such that some generator j 6= i can profitably deviate by moving the density that is assigned to the interval, [ˆi, ˆi + η), to the offer ˆi − ρ, contradicting the assumption of an equilibrium mass point.

To see that this is a profitable deviation, we consider the following three possi- ble cases of whether generator j is marginal or inframarginal before and after the deviation. First, if generator j is inframarginal before and after the deviation, there is no profit loss if it receives energy payments. Otherwise, if it receives make-whole payments its profit loss is at most (η + ρ)K. Second, if generator j is marginal before and after the deviation, there is a profit loss of at most (η + ρ)K. Finally, if genera- tor j is marginal before the deviation and inframarginal after the deviation, its profit increases by at least (ˆi − c)(K − lr) − ρK − ηlr. For η and ρ sufficiently small, the profit increase in the third case outweighs the profit losses in the first two.

Lemma 9. Φi is a connected set for all generators, i = 1,...,N, in a mixed-strategy

Nash equilibrium in a high-demand state under a centrally committed market design.

Proof. Assume for contradiction that there exists an interval, [ˆi, ˆi + η] with η >

0, on which generator i places zero density in a mixed-strategy Nash equilibrium.

Generator j 6= i has a profitable deviation in which the density that is assigned to the interval, (ˆi − ρ, ˆi), is assigned to the offer, ˆi + η − ξ, where ρ > 0 and ξ ∈ (0, η),

40 contradicting the assumption that the support of generator i’s mixed-strategy is not

connected.

To see that this is a profitable deviation, we consider the following three possible

cases of whether generator j is marginal or inframarginal before and after the devi-

ation. First, if generator j is inframarginal before and after the deviation, there is

no profit change if it receives energy payments. Otherwise, if it receives make-whole

payments its profits increase by at least (η − ξ)K. Second, if generator j is marginal before and after the deviation, its profits increase by at least (η − ξ)lr. Finally, if

generator j is inframarginal before the deviation and marginal after the deviation, its

profits change by at most (ˆi − c)(lr − K) + (η − ξ)lr + ρK. Thus, generator j only

stands to have a profit loss in the third case. However, ρ can be chosen to make the

probability of this event arbitrarily close to zero.

Lemma 10. The supremum energy offers of the mixed strategies that are used by

the generators in a high-demand state under a centrally committed market design are

equal in a Nash equilibrium.

+ + Proof. Assume for contradiction that i < j . Generator i could profitably deviate by

+ + moving the density that is assigned to the interval, (i −η, i ], with η > 0 sufficiently

+ + + small, to j − ρ, with j − ρ > i . By doing so, generator i increases its expected profits without decreasing the probability that it is an inframarginal generator. This contradiction shows that the supremum energy offers must be equal.

Lemma 11. The generators submit σ1 = σ2 = ··· = σN =σ ¯ in a mixed-strategy

Nash equilibrium in a high-demand state under a centrally committed market design.

41 Proof. There are only mixed-strategy Nash equilibria in a high-demand state under

a centrally committed market design. Moreover, we have that Φ1 = Φ2 = ··· =

ΦN . Thus, each generator has a strictly positive probability of being the marginal

generator. From Lemma 1 we know that the marginal generator’s profit is strictly

increasing in σM . Moreover, the profits of the inframarginal generators are non-

decreasing in σI . Finally, we know that the assignment of the N generators to being

marginal or inframarginal does not depend on the values of σ1 = σ2 = ··· = σN in

a high-demand state. Thus, it is expected-profit-maximizing for each generator to

submit the highest possible value for the start-up cost in its offer,σ ¯.

As a result of Lemma 11, we can write the cumulative distribution function of

generator i’s mixed strategy as Fi(i)(i.e., we no longer write it as depending on

σi). This is because we know that generator i submits σi =σ ¯ in a Nash equilib- rium. We can now derive an analytical expression that characterizes each generator’s equilibrium mixed strategy in the following proposition.

Proposition 4. There is a symmetric mixed-strategy Nash equilibrium in a high- demand state under a centrally committed market design. The cumulative distribution functions of the mixed strategies that are used by the generators satisfy the delay differential equation:

1  F () F ( +σ/K ¯ )N−1K  f() = + , N − 1 c −  (l − NK)(c − )F ()N−2 where f() and F () are the probability density and cumulative distribution functions, respectively, of the mixed strategies.

42 Proof. We begin by assuming that all of generator i’s rivals follow the given by the probability density function, f(), and the associated cumu- lative distribution function, F (). We can then write generator i’s expected profit, as a function of its own offer, i, as:

c N−1 E [πi (i)] = F (i) [¯σ − S + (i − c)lr]

 N−1 N−1 + F (i +σ/K ¯ ) − F (i) [¯σ − S + (i − c)K] Z ¯ max max N−1 + [−S + (−i − c)K]d F (−i ) . i+¯σ/K N−1 = F (i) [¯σ − S + (i − c)lr] (2.13)

 N−1 N−1 + F (i +σ/K ¯ ) − F (i) [¯σ − S + (i − c)K] Z ¯ max max N−2 max max + [−S + (−i − c)K](N − 1)F (−i ) f(−i )d−i , i+¯σ/K max where −i represents the maximum energy offer of generator i’s rivals. This profit expression consists of three terms. The first:

N−1 F (i) [¯σ − S + (i − c)lr], is the expected profit earned, conditional on generator i being marginal. The second:

 N−1 N−1 F (i +σ/K ¯ ) − F (i) [¯σ − S + (i − c)K], is the expected profits earned, conditional on generator i being inframarginal and receiving make-whole payments. The final is the expected profit, conditional on generator i being inframarginal and receiving energy payments.

The first-order necessary condition (FONC) for maximizing (2.13) with respect to

i is:

N−2 N−1 (N − 1)F (i) f(i)[(i − c)(l − NK)] + F (i) (l − NK)

N−1 + F (i +σ/K ¯ ) K = 0.

43 This can be rewritten as:

 N−1  1 F (i) F (i +σ/K ¯ ) K f(i) = + N−2 . N − 1 c − i (l − NK)(c − i)F (i) Finally, because we are assuming a symmetric equilibrium, we drop the subscript, i, which gives the desired delay differential equation:

1  F () F ( +σ/K ¯ )N−1K  f() = + . N − 1 c −  (l − NK)(c − )F ()N−2

Lemma 12. The supremum energy offers of the mixed strategies used by the gener-

ators in a high-demand state under a centrally committed market design are equal to

¯ in a Nash equilibrium.

+ + + + Proof. Assume for contradiction that 1 = 2 = ··· = N =  < ¯ in a Nash equilibrium. Then generator i has a profitable deviation in which it moves the density that is assigned to the interval, (+ − η, +], where η > 0, to the energy offer, ¯, contradicting the assumption of a Nash equilibrium in which + < ¯.

To see that this is a profitable deviation, we consider the following three possi- ble cases of whether generator i is marginal or inframarginal before and after the deviation. First, if generator i is inframarginal before and after the deviation, there is no change in its profits if it receives energy payments. Otherwise, if it receives make-whole payments its profits increase by at least (¯−+)K. Second, if generator i

+ is marginal before and after the deviation, its profits increase by at least (¯ −  )lr.

Finally, if generator i is inframarginal before the deviation and marginal after the

+ deviation, its profits change by at most (¯ − c)lr − ( − c)K. Thus, generator i only stands to have a profit loss in the third case. However, η can be chosen to make the probability of this event arbitrarily close to zero.

44 We conclude our analysis of Nash equilibria in a high-demand state under a cen- trally committed market design by computing the expected profits of the N genera- tors.

Proposition 5. In a symmetric mixed-strategy Nash equilibrium in a high-demand state under a centrally committed market design each generator earns expected profits equal to:

c E [π ] =σ ¯ − S + (¯ − c)lr.

Proof. From (2.13) we have that generator i earns an expected profit of:

c N−1 E [πi (¯)] = F (¯) [¯σ − S + (¯ − c)lr]

+ F (¯ +σ/K ¯ )N−1 − F (¯)N−1 [¯σ − S + (¯ − c)K] Z ¯ max max N−2 max max + [−S + (−i − c)K](N − 1)F (−i ) f(−i )d−i ¯+¯σ/K

=σ ¯ − S + (¯ − c)lr, if it submits an energy offer equal to ¯. Because the mixed strategies constitute a

Nash equilibrium, all energy offers in Φ must yield the same expected profit for each generator. Thus, we have that:

c E [πi ()] =σ ¯ − S + (¯ − c)lr,

∀ ∈ Φ, i = 1,...,N, showing the desired result.

2.3.2 Equilibria Under a Self-Committed Market Design

We proceed with the analysis of Nash equilibria under a self-committed market design by examining different demand states, beginning with low-demand states.

45 Equilibria in Low-Demand States

First, we show in the following proposition that in weak-demand states all gener- ators earn zero profits in equilibrium.

Proposition 6. In weak low-demand states the unique set of pure-strategy Nash equi- libria under a self-committed market design is characterized by all of the generators submitting offers of the form: S δ ≥ c + , (2.14) l with this inequality binding for at least two of them. All generators earn zero profits.

Proof. Without loss of generality, assume that the generators are labeled so that:

δ1 ≤ δ2 ≤ · · · ≤ δN .

Because we are in a weak low-demand state, we know that generator 1 is the unique generator, which is dispatched to produce l MW, and its profits are −S + (δ1 − c)l.

First, assume for contradiction that δ1 < c + S/l. In this case, generator 1’s profits are negative and it has a profitable deviation in which it submits an offer that is strictly greater than δ2. Doing so makes generator 1 inactive and its profits zero.

Thus, we have a contradiction showing that in an equilibrium, all generators submit offers satisfying (2.14).

Next, assume for contradiction that δ1 > c+S/l. In this case, generator 1’s profits are strictly positive and all of generators 2 through N have profitable deviations in ˜ which generator i > 1 submits the offer, δi, such that:

S c + < δ˜ < δ . l i 1

46 Such an offer results in generator i becoming the unique generator (in place of gen- erator 1) and earning strictly positive profits. This contradiction shows that in an equilibrium, inequality (2.14) must be binding for at least one generator.

To show that (2.14) must be binding for at least two generators, assume for contradiction that δ1 = c + S/l while δ2 > c + S/l. In this case generator 1 earns zero profits. However, generator 1 has a profitable deviation in which it submits the offer, ˜ δ1, such that: S c + < δ˜ < δ . l 1 2 Doing so allows generator 1 to remain the unique generator but earn strictly positive profits. This contradiction shows that in an equilibrium at least two generators must submit offers that make (2.14) binding.

Finally, we have that the unique generator exactly breaks even and earns zero profits. Moreover, because the remaining generators are inactive, they earn zero profits as well.

The logic of the previous proof also extends to the integral low-demand state. We show in the following proposition that all generators earn zero profits in such states as well.

Proposition 7. In integral low-demand states the unique set of pure-strategy Nash equilibria under a self-committed market design is characterized by (Nl −1) generators submitting offers of the form: S δ ≤ c + , (2.15) K and the remaining generators submitting offers of the form:

S δ ≥ c + . (2.16) K 47 Of these remaining generators, at least two submit offers that make (2.16) binding.

All generators earn zero profits.

Proof. Without loss of generality, assume that the generators are labeled so that:

δ1 ≤ δ2 ≤ · · · ≤ δN .

Because we are in an integral low-demand state, we know that generators 1 through Nl

are dispatched to produce exactly K MW and their profits are −S + (δNl − c)K.

First, assume for contradiction that δNl < c + S/K. In this case, the profits of

generators 1 through Nl are all negative. Each of these generators has a profitable

deviation in which it submits an offer that is strictly greater than δNl+1. Doing so makes the deviating generator inactive and its profits zero. Thus, we have a

contradiction showing that in an equilibrium δNl , δNl+1, . . . , δN must satisfy (2.16).

Next, assume for contradiction that δNl > c + S/K. In this case, generator Nl’s

profits are strictly positive and all of generators (Nl + 1) through N have profitable ˜ deviations in which generator i > Nl submits the offer, δi, such that:

S c + < δ˜ < δ . K i Nl

Such an offer results in generator i becoming marginal (in place of generator Nl)

and earning strictly positive profits. This contradiction shows that in an equilibrium,

inequality (2.16) must be binding for at least one generator. We can further conclude

that the offers of generators 1 through (Nl − 1) must satisfy (2.15).

To show that (2.16) must be binding for at least two generators, assume for

contradiction that δNl = c + S/K while δNl+1 > c + S/K. In this case, generator Nl earns zero profits. However, this generator has a profitable deviation in which it

48 ˜ submits the offer, δNl , such that:

S c + < δ˜ < δ . K Nl Nl+1

Doing so allows generator Nl to remain marginal but earn strictly positive profits.

This contradiction shows that in an equilibrium at least two generators must submit offers that make (2.16) binding.

Finally, we have that the inframarginal and marginal generators earn zero profits.

Moreover, because the remaining generators are inactive, they earn zero profits as well.

Unlike weak and integral low-demand states, regular low-demand states entail at least one inframarginal generator operating at full capacity while the marginal gener- ator operates below its capacity. This distinction between regular and the two other types of low-demand states results in very different equilibria. In regular low-demand states, the marginal generator is still constrained to earn zero profits. However, in- framarginal generators earn strictly positive profits. We demonstrate this property of equilibria in a regular low-demand state in the following proposition.

Proposition 8. In regular low-demand states the unique set of pure-strategy Nash equilibria under a self-committed market design is characterized by (Nl −1) generators submitting offers of the form: S δ ≤ c + , (2.17) K and the remaining generators submitting offers of the form:

S δ ≥ c + . (2.18) lr

49 Of these remaining generators, at least two submit offers that make (2.18) binding.

The marginal and inactive generators earn zero profits while the inframarginal gen- erators earn strictly positive profits.

Proof. Without loss of generality, assume that the generators are labeled so that:

δ1 ≤ δ2 ≤ · · · ≤ δN .

Because we are in a regular low-demand state, we know that generators 1 through (Nl−

1) are inframarginal and dispatched to produce K MW, while generator Nl is marginal and produces lr < K.

First, assume for contradiction that δNl < c + S/lr. In this case, generator Nl’s profits are negative and it has a profitable deviation in which it submits an offer that

is strictly greater than δNl+1. Doing so makes generator Nl inactive and its profits

zero. Thus, we have a contradiction showing that in an equilibrium δNl , δNl+1, . . . , δN must satisfy (2.18).

Next, assume for contradiction that δNl > c + S/lr. In this case, generator Nl’s profits are strictly positive and all of generators (Nl + 1) through N have profitable ˜ deviations in which generator i > Nl submits the offer, δi, such that:

S ˜ c + < δi < δNl . lr

Such an offer results in generator i becoming marginal (in place of generator Nl) and earning strictly positive profits. This contradiction shows that in an equilibrium, inequality (2.18) must be binding for at least one generator.

To show that (2.18) must be binding for at least two generators, assume for

contradiction that δNl = c + S/lr while δNl+1 > c + S/lr. In this case, generator Nl

50 earns zero profits. However, this generator has a profitable deviation in which it ˜ submits the offer, δNl , such that:

S ˜ c + < δNl < δNl+1. lr

Doing so allows generator Nl to remain marginal but earn strictly positive profits.

This contradiction shows that in an equilibrium at least two generators must submit

offers that make (2.18) binding.

To derive (2.17), assume for contradiction that:

S c + < δ . K Nl−1

˜ In this case, generator Nl has a profitable deviation in which it submits the offer, δNl , such that: ˜ δNl < δNl−1.

Doing so makes generator Nl become inframarginal and generator (Nl − 1) marginal.

Moreover, the energy price under this deviation is:

S ps = δ > c + . Nl−1 K

This means that generator Nl’s profits are:

 S  −S + (δ − c)K > −S + c + − c K > 0, Nl−1 K

after the deviation. Thus, we have a contradiction showing that in an equilibrium,

δ1, δ2, . . . , δNl−1 must satisfy (2.17). Finally, we have that the inactive generators earn zero profits, because they pro-

duce nothing. The marginal generator also earns zero profit, because the energy price

51 is exactly high enough for it to recover its costs. The profits of each inframarginal generator are equal to:

 S  K  −S + (δNl − c)K = −S + c + − c K = S · − 1 > 0, lr lr because lr < K.

The equilibria that are characterized in Proposition 8 are made possible by the fact that there is at least one inframarginal generator that is producing at capacity while the inframarginal generator is producing less than capacity. This scenario is not possible in weak and integral low-demand states. The marginal generator must produce below its full capacity but must recover its fixed and variable costs. Because its costs must be recovered through energy payments only (due to the lack of a make- whole-payment mechanism under a self-committed market design), the energy price must be sufficiently high to recover both cost components. This higher energy price allows the inframarginal generators to earn strictly positive profits. The inframarginal generators ‘enforce’ the equilibrium by submitting offers sufficiently low that neither the marginal nor the inactive generators have profitable deviations in which they undercut an inframarginal generator.

Before concluding our analysis of low-demand states in a self-committed market, we remark on one of the important distinctions between the multi-firm oligopoly setting that is studied here and the two-firm setting that Sioshansi and Nicholson

(2011) examine. In the two-firm setting, low-demand states that do not require both of the generators to serve the load result in all generators earning zero profits under both the centrally and self-committed market designs. This equivalence no

52 longer holds in the multi-firm setting. Indeed, we find that ceteris paribus, the self- committed market design is more expensive (in terms of remuneration to generators) in low-demand states than a centrally committed design is.

Equilibria in High-Demand States

We finally turn to the case of equilibria in high-demand states under a self- committed market design. Unlike the case of a centrally committed design, both pure- and mixed-strategy Nash equilibria exist in high-demand states under a self- committed design. We begin by first characterizing the set of pure-strategy Nash equilibria in the following proposition.

Proposition 9. In high-demand states the unique set of pure-strategy Nash equilibria under a self-committed market design is characterized by (N−1) generators submitting offers of the form: l δ ≤ (δ¯ − c) r + c, K and the remaining generator submitting an offer of the form:

δ = δ.¯

All generators earn non-negative profits.

Proof. Without loss of generality, assume that the generators are labeled so that:

δ1 ≤ δ2 ≤ · · · ≤ δN .

Because we are examining a high-demand state, we know that generators 1 through (N−

1) are inframarginal and dispatched to produce K MW, while generator N is marginal and produces lr < K, and that the energy price is δN .

53 We first show that generator N does not have a profitable deviation. Obviously if generator N submits an offer less than δN but greater than δN−1, it remains the

marginal generator and its profits strictly decrease because the energy price decreases.

Next, consider a deviation in which generator N undercuts one of the other generators.

In such an instance, it becomes an inframarginal generator and the energy price is at

most: l (δ¯ − c) r + c. K

Thus, its profits after this deviation are at most:

 l  −S + (δ¯ − c) r + c − c K = −S + (δ¯ − c)l . K r

However, its profits before the deviation are:

¯ −S + (δ − c)lr,

which is non-negative by assumption. This means that the marginal generator does

not have a profitable deviation in which it undercuts the offer of an inframarginal

generator.

This, thus, shows that the marginal generator must submit an offer equal to

δ¯ in a pure-strategy Nash equilibrium. Moreover, we can also conclude that the ¯ inframarginal generators must submit offers no greater than (δ − c)lr/K + c for the

marginal generator not to have a profitable deviation. We finally have that the profits

of inframarginal generators are −S + (δ¯ − c)K > 0.

The intuition behind this set of pure-strategy Nash equilibria is fairly straightfor-

ward. Because the generator submitting the highest offer is marginal and sets the

energy price, this generator has an incentive to submit the highest offer possible. At

54 the same time, the inframarginal generators must submit offers that are sufficiently

small to ensure that the marginal generator has no incentive to undercut one of them. ¯ The threshold offer, (δ − c)lr/K + c, is sufficiently small to ensure this. Because the

market has a uniform price that is set by the marginal generator, the offers submit-

ted by the inframarginal generators can be any value below this threshold (including

offers that are below cost). We also note that because lr < K we have that the

threshold offer: l (δ¯ − c) r + c ∈ (c, δ¯), K is valid per the market rules.

We finally turn to characterizing and deriving a closed-form expression for a mixed-

strategy Nash equilibrium in high-demand states under a self-committed market de-

sign. We let Gi(δi) denote the cumulative distribution function of generator i’s mixed

− + strategy. We further let Ψi denote the support of Gi and define δi and δi as the

N infimum and supremum offers, respectively, in Ψi. We also let Ψ = ∪i=1Ψi denote the common support of the N cumulative distribution functions.

Lemma 13. The infimum offers of the mixed strategies that are used by the generators

in a high-demand state under a self-committed market design are equal in a Nash

equilibrium.

− − Proof. Assume for contradiction that δi < δj . In such a case, generator i has a profitable deviation wherein it moves the density that is assigned to the interval,

− − − [δi , δj ), to δj − η with η > 0 sufficiently small. By doing so, generator i increases its expected profit without decreasing the probability that it is an inframarginal

generator. Thus, we have a contradiction showing that the infimum energy offers in

a mixed-strategy Nash equilibrium must be equal.

55 Lemma 14. None of G1(δ1),G2(δ2),...,GN (δN ) have a mass point on Ψ in a mixed-

strategy Nash equilibrium in a high-demand state under a self-committed market de-

sign.

Proof. Assume for contradiction that there exists at least one mass point of Gi(δi) ˆ for some generator, i. Let δi denote one of these mass points. There exist η > 0 and

ρ > 0 such that some generator j 6= i can profitably deviate by moving the density ˆ ˆ ˆ that is assigned to the interval, [δi, δi + η), to the offer, δi − ρ, contradicting the

assumption of an equilibrium mass point.

To see that this is a profitable deviation, we consider the following three possi-

ble cases of whether generator j is marginal or inframarginal before and after the

deviation. First, if generator j is inframarginal before and after the deviation, there

is no profit loss as it continues to earn payments determined by the marginal gen-

erator’s offer. Second, if generator j is marginal before and after the deviation,

there is a profit loss of at most (η + ρ)K. Finally, if generator j is marginal before

the deviation and inframarginal after the deviation, its profit increases by at least ˆ (δi − c)(K − lr) − ρK − ηlr. For η and ρ sufficiently small, the profit increase in the

third case outweighs the profit losses in the second.

Lemma 15. Ψi is a connected set for all generators, i = 1,...,N, in a mixed-strategy

Nash equilibrium in a high-demand state under a self-committed market design.

ˆ ˆ Proof. Assume for contradiction that there exists an interval, [δi, δi + η], with η >

0, on which generator i places zero density in a mixed-strategy Nash equilibrium.

Generator j 6= i has a profitable deviation in which the density that it assigns to the ˆ ˆ ˆ interval, (δi − ρ, δi), is assigned to the offer, δi + η − ξ, where ρ > 0 and ξ ∈ (0, η),

56 contradicting the assumption that the support of generator i’s mixed strategy is not

connected.

To see that this is a profitable deviation, we consider the following three possible

cases of whether generator j is marginal or inframarginal before and after the devia-

tion. First, if generator j is inframarginal before and after the deviation, there is no

profit change, as it continues receiving payments that are determined by the offer of

the marginal generator. Second, if generator j is marginal before and after the devi-

ation, its profits increase by at least (η − ξ)lr. Finally, if generator j is inframarginal before the deviation and marginal after the deviation, its profits change by at most ˆ (δi − c)(lr − K) + (η − ξ)lr + ρK. Thus, generator j only stands to have a profit loss

in the third case. However, ρ can be chosen to make the probability of this event

arbitrarily close to zero.

Lemma 16. The supremum offers of the mixed strategies that are used by the gen-

erators in a high-demand state under a self-committed market design are equal in a

Nash equilibrium.

+ + Proof. Assume for contradiction that δi < δj . Generator i could profitably deviate

+ + + by moving density in the interval, (δi −η, δi ], with η > 0 sufficiently small, to δj −ρ,

+ + with δj − ρ > δi . By doing so, generator i increases its expected profits without decreasing the probability that it is an inframarginal generator. This contradiction

demonstrates that the supremum energy offers must be equal.

Lemma 17. The supremum offers of the mixed strategies used by the generators in

a high-demand state under a self-committed market design are equal to δ¯ in a Nash

equilibrium.

57 + + + + ¯ Proof. Assume for contradiction that δ1 = δ2 = ··· = δN = δ < δ in a Nash equilibrium. Then generator i has a profitable deviation in which it moves the density

that is assigned to the interval, (δ+ − η, δ+], where η > 0, to the energy offer, δ¯,

contradicting the assumption of a Nash equilibrium in which δ+ < δ¯.

To see that this is a profitable deviation, we consider the following three possible cases of whether generator i is marginal or inframarginal before and after the devia- tion. First, if generator i is inframarginal before and after the deviation, there is no change in its profits as it continues to receive payments determined by the offer of the marginal generator. Second, if generator i is marginal before and after the devia-

¯ + tion, its profits increase by at least (δ − δ )lr. Finally, if generator i is inframarginal before the deviation and marginal after the deviation, its profits change by at most

¯ + (δ − c)lr − (δ − c)K. Thus, generator i only stands to have a profit loss in the third case. However, η can be chosen to make the probability of this event arbitrarily close to zero.

We now derive a closed-form expression of each generator’s equilibrium mixed

strategy in high-demand states under a self-committed market design. This deriva-

tion follows the approach that is taken in Proposition 5. Namely, we first express gen-

erator i’s expected profit as a function of its offer, assuming that all other generators

follow the equilibrium. We next derive the FONC characterizing generator i’s optimal

offer. We finally impose the condition that the generators are following a symmetric

equilibrium to obtain a differential equation characterizing the mixed strategies. We

can further explicitly solve the differential equation to obtain a closed-form expression

for the equilibrium cumulative distribution function.

58 Proposition 10. There is a symmetric mixed-strategy Nash equilibrium in a high-

demand state under a self-committed market design. The cumulative distribution

functions of the mixed strategies used by the generators are given by:

δ − cλ G(δ) = , δ¯ − c

where: l λ = r . (2.19) (K − lr)(N − 1)

Proof. We assume that all of generator i’s rivals follow the symmetric equilibrium

that is given by the probability density function, g(δ), and the cumulative distribution

function, G(δ). Generator i’s expected profit, as a function of its offer, δi, is then

given by:

s N−1 E [πi (δi)] = G(δi) [−S + (δi − c)lr] Z δ¯ max max N−1 + [−S + (δ−i − c)K]d G(δ−i ) , (2.20) δi

max where δ−i represents the maximum offer of generator i’s rivals. Differentiating (2.20)

with respect to δi gives:

N−2 N−1 (N − 1)G(δi) g(δi)(δi − c)(lr − K) + G(δi) lr = 0,

which is the FONC for generator i’s profit-maximizing offer. This FONC can be

rewritten as:

G(δi) g(δi) − λ = 0, δi − c where λ is given by (2.19). Because we assume a symmetric equilibrium, we can eliminate the subscripts, giving:

G(δ) g(δ) − λ = 0. (2.21) δ − c 59 Differential equation (2.21) can be explicitly solved by first defining the integrating

factor:  Z δ λ  δ − c−λ µ(δ) = exp − dx = , a x − c a − c where a is an arbitrary constant. Multiplying both sides of (2.21) by µ(δ) gives:

δ − c−λ δ − c−λ G(δ) g(δ) − λ = 0. a − c a − c δ − c

Integrating the left-hand side of this equation with respect to δ gives:

δ − c−λ G(δ) − b = 0, a − c where b is the constant of integration. Thus, we have that:

δ − cλ G(δ) = b · . a − c

Because we know that G(δ¯) = 1, we have that:

a − cλ b = , δ¯ − c

which then gives: δ − cλ G(δ) = , δ¯ − c which is the desired result.

We conclude our analysis of the mixed-strategy Nash equilibria in a high-demand

state under a self-committed market design by determining the expected profits of

the generators.

Proposition 11. In a symmetric mixed-strategy Nash equilibrium in a high-demand

state under a self-committed market design each generator earns expected profits equal

to:

s ¯ E [π ] = −S + (δ − c)lr.

60 Proof. From (2.20) we have that generator i earns an expected profit of:

 s ¯  ¯ N−1 ¯ E πi (δ) = G(δ) [−S + (δ − c)lr] Z δ¯ max max N−1 ¯ + [−S + (δ−i − c)K]d G(δ−i ) = −S + (δ − c)lr, (2.22) δ¯ if it submits an offer equal to δ¯. Because the mixed strategies constitute a Nash

equilibrium, all offers in Ψ must yield the same expected profit for all generators.

Thus, we have that:

s ¯ E [πi (δ)] = −S + (δ − c)lr,

∀δ ∈ Ψ, i = 1,...,N, which is the desired result.

2.4 Expected-Cost Comparison of Market Designs

We conduct a cost comparison of the two market designs in this section. We begin

by first showing analytical results, which rely on the properties of the equilibria that

are derived in Section 2.3. We then demonstrate the cost comparisons between the

two market designs using a numerical example.

2.4.1 Analysis of Market Equilibria

We begin in this section by showing conditions under which the two market de-

signs are expected-cost-equivalent and cases in which cost equivalence fails to hold.

Importantly, we find cases in which cost equivalence holds in the duopoly case (cf., the work of Sioshansi and Nicholson (2011) but fails to hold in the multi-firm case.

We begin with cases in which the two market designs are cost-equivalent.

Proposition 12. Centrally and self-committed market designs are always cost-equivalent in weak and integral low-demand states.

61 Proof. This result follows immediately from Propositions 1, 2, 6, and 7, which show that all generators earn zero profits under both market designs in weak and integral low-demand states.

This cost-equivalence finding has an analogue to the duopoly case that Sioshansi and Nicholson (2011) examine. They show that the two market designs are cost- equivalent for all low-demand states. A point of departure between the duopoly and multi-firm cases involves regular low-demand states. As the following proposition shows, cost equivalence between the two market fails to hold in the multi-firm case that is examined here.

Proposition 13. Centrally and self-committed market designs are not cost-equivalent in regular low-demand states.

Proof. From Lemma 5 we know that in regular low-demand states there are pure- strategy Nash equilibria under a centrally committed market design in which all generators earn zero profits. On the other hand, we know from Proposition 8 that in a regular low-demand state the unique set of pure-strategy Nash equilibria under a self-committed market design results in inframarginal generators earning strictly positive profits. Thus, cost equivalence fails to hold.

Strictly positive profits of inframarginal generators under a self-committed market design arises because the marginal generator must recover its fixed start-up cost.

Because this cost must be recovered through energy payments only, the energy price is above the level needed for inframarginal generators to recover their costs. This situation does not arise under a centrally committed market design, because start-up costs are recovered through make-whole payments.

62 We conduct our cost comparisons of the two market designs in high-demand states

by adding the following assumption.

Assumption 1. The offer caps of the two market designs are set to satisfy:

¯ σ¯ − S + (¯ − c)E [lr|l > (N − 1)K] = −S + (δ − c)E [lr|l > (N − 1)K] .

Assumption 1 is premised on the idea that whichever of a centrally committed or self-committed market is adopted, the market clears repeatedly (e.g., hourly) at different load levels. The assumption states that the offer caps of the two markets should be set in such a way that generators are not disadvantaged under one market design relative to another (i.e., by one market design having an unduly low cap compared to the other design). The way that Assumption 1 compares the offer caps under the two market designs is based on the expected profits of a generator submitting offers equal to the caps in high-demand states. A generator submitting an offer at that cap will be marginal with probability 1. Thus, the two terms in the equality defining Assumption 1 are the expected profits of a generator offering at the cap.

It should be noted that the ‘equivalent’ offer cap under one market design can be determined from the offer cap of the other market design without knowing the true generator costs. This is because the equality in Assumption 1 can be simplified to:

¯ σ¯ +E ¯ [lr|l > (N − 1)K] = δE [lr|l > (N − 1)K] .

This is desirable, as policymakers may not know true generator costs when designing markets and setting offer caps (indeed, if a policymaker knows true generator costs, one would not need to solicit offers in the market as is examined in this chapter).

63 We now show the cost-comparison properties of the two market designs in high-

demand states when Assumption 1 holds.

Proposition 14. Centrally and self-committed market designs are expected-cost-equivalent

in high-demand states if Assumption 1 holds and the generators follow mixed-strategy

Nash equilibria.

Proof. From Propositions 5 and 11, respectively, we have that expected generator

profits in high-demand states are:

c E [π ] =σ ¯ − S + (¯ − c)lr,

under a centrally committed market design and:

s ¯ E [π ] = −S + (δ − c)lr, if generators follow mixed-strategy Nash equilibria under a self-committed market design. Expected-cost-equivalence would require the expectation (with respect to lr) of these two terms to be equal, or:

c s El [E [π ] |l > (N − 1)K] = El [E [π ] |l > (N − 1)K]

¯ σ¯ − S + (¯ − c)E [lr|l > (N − 1)K] = −S + (δ − c)E [lr|l > (N − 1)K] , which is the equality in Assumption 1.

Corollary 4. Centrally and self-committed market designs are not expected-cost- equivalent in high-demand states if Assumption 1 holds and the generators follow a pure-strategy Nash equilibrium under a self-committed market design.

Proof. From Proposition 9 we know that if generators follow a pure-strategy Nash

equilibrium under a self-committed market design the energy price is equal to δ¯. We

64 further know from Proposition 10 that if the generators follow a mixed-strategy Nash equilibrium under a self-committed market design there is a non-zero probability that they submit energy offers that are strictly less than δ¯ (meaning that the energy price is less than δ¯). Because the mixed-strategy Nash equilibrium under a self-committed market design is expected-cost-equivalent to the mixed-strategy Nash equilibrium under a centrally committed market design, it must be the case that the pure-strategy

Nash equilibrium under a self-committed market design is more costly.

These two results regarding cost comparisons in high-demand states have ana- logues in the duopoly case. Sioshansi and Nicholson (2011) show that the two market designs are expected-cost-equivalent in the duopoly if Assumption 1 holds and gen- erators follow mixed-strategy Nash equilibria. Otherwise, if duopolists follow a pure- strategy Nash equilibrium under a self-committed market design, the self-committed market is more costly.

2.4.2 Numerical Example

We conclude this section with a simple numerical example that illustrates equi- librium prices and remuneration costs under the two market designs. Table 2.1 lists the parameter values that are assumed in the example. The price caps that given in the table satisfy Assumption 1 if lr is uniformly distributed in high-demand states.

We examine cases with different values for N. Increasing the number of firms has two impacts on market equilibria. First, higher values of N mean that the load must be higher for the market to be in a high-demand state. Second, having more firms in the market tends to result in less aggressive offers (and lower expected energy prices) in high-demand states.

65 Table 2.1: Data for numerical example Parameter Value c $30/MWh S $10000 K 500 MW E [lr|l > (N − 1)K] 250 MW ¯ $1000/MWh σ¯ $25000 δ¯ $1100

Figure 2.1 shows expected energy prices under the two market designs as a func-

tion of the residual load with different values of N, if the firms follow mixed-strategy

Nash equilibria. Expected energy prices under a self-committed market design are computed analytically, using the closed-form expression for the equilibrium cumu- lative distribution function. Expected energy prices under a centrally committed market design are numerically approximated by first solving the delay differential equation using finite differences to obtain the equilibrium cumulative distribution function. The cumulative distribution function is then integrated using numerical quadrature to approximate the expected energy price.

Figure 2.1 shows that with a greater number of firms, the expected energy price decreases under both market designs. The figure also shows that for low residual-load levels the expected energy price under a self-committed market design is lower than that under a centrally committed design. This is because lower residual-load levels imply greater profit losses from being the marginal as opposed to an inframarginal generator (because less energy is produced and sold). As such, generators submit more aggressive offers, resulting in a lower expected price, for lower residual-load levels. At

66 Figure 2.1: Expected Energy Price as a Function of Residual Load for Different Values of N if Firms Follow Mixed-Strategy Nash Equilibria

higher residual-load levels the expected energy prices under the two market designs reverse.

Table 2.2 summarizes the total expected remuneration cost in the high-demand cases that are shown in Figure 2.1, assuming that the residual loads in high-demand cases are uniformly distributed between 50 MW and 450 MW and that the firms follow mixed-strategy equilibria. As expected from Proposition 14, the expected settlement costs are the same under the two market designs. The table shows that the make- whole payments represent a non-trivial portion of the total cost, constituting about

5% of costs with N = 2 and decreasing to 2% of costs with N = 10.

67 Table 2.2: Total expected remuneration cost to MO under centrally and self- committed market designs for different values of N if firms follow mixed-strategy Nash equilibria Centrally Committed Self-Committed N = 2 N = 3 N = 10 N = 2 N = 3 N = 10 Expected Energy Price 645 598 548 678 620 557 [$/MWh] Expected Make-Whole 28 30 46 n/a n/a n/a Payments [$ thousand] Expected Settlement 535 803 2675 535 803 2675 Cost [$ thousand]

2.5 Discussion and Concluding Remarks

We have studied centrally-committed and self-committed wholesale electricity markets in the context of an N-generator oligopoly. We find that the equilibrium re- sults concerning high states of demand are qualitatively unchanged from the duopoly to the oligopoly. That is, when demand is high, there exist only mixed-strategy

Nash equilibria in centrally-committed markets, whereas there exist both pure and mixed-strategy Nash equilbria in self-committed markets. Indeed, it is possible to set offer caps across the two market designs in such as way as to ensure expected cost-equivalence in both market designs when demand is high. However, there is a stark point of departure between the two market designs when demand is low. As is the case in the duopoly model, when demand is low, all generators are constrained to earn zero profits in the centrally-committed market. However, in the self-committed market, inframarginal generators will earn positive profits in regular states of low demand. This difference is driven by the fact that when there are more than two

68 generators, a low demand state may require that at least one generator produce at capacity while the marginal generator produces a positive amount of energy that is strictly less than its capacity. Given that demand is low, the marginal generator is unable to bid above cost. Doing so would result in it being profitably undercut by one of the inactive generators. Given the single-dimensional bids in a self-committed market, this means that the the bid that will result in the marginal generator pro- ducing at cost, which determines the uniform energy price in the market, will result in inframarginal generators earning positive profits. This scenario is ruled out in the duopoly case because the only low-demand states will require just one generator to produce, a result replicated when the oligopolistic market encounters weak demand states. The addition of more than two generators results in the possibility of richer demand dynamics. This result holds potential policy relevance, as it suggests, coun- terintuitively, that under certain conditions, self-committed markets may actually become less competitive as the number of generators increases.

We emphasize that this model, while generating unique insights above and be- yond the results obtained in the symmetric duopoly analysis, is still highly styl- ized. However the results obtained do provide unique insights into the dynamics of centrally-committed and self-committed markets that will be of interest to reg- ulators and market designers. We have assumed that firms are symmetric in costs and capacities and that demand is known with certainty. Promising work remains to be done by relaxing these assumption and exploring more general models of these market designs. In addition, further explorations of these models will generate impor- tant insights from a practical regulatory standpoint. LaCasse (1995) studies how the

69 possibility of prosecution alters incentives for firms to collude in government procure- ments. Fabra (2003) considers the susceptibility of discriminatory auctions versus uniform auctions to in a . Given the important role of reg- ulators in designing and improving markets for wholesale electricity, understanding how centrally-committed and self-committed markets either hinder or facilitate col- lusion is an important question. In addition, in analyzing a stylized model of how these markets function in practice, we have not incorporated constraints on . Chao and Wilson (2002) consider an optimal auction mechanism for the procurement of electricity reserves. A promising direction in which to take future work would be examining these markets from a theoretical perspec- tive, explicitly taking into account incentive compatibility constraints. Finally, the study of electricity auctions, while a growing aspect of the theoretical literature, re- mains a promising arena for future experimental work. Denton et al. (2001) conduct an experimental analysis for designing spot markets for electricity, while Rassenti et al. (2003) study discriminatory price auctions for electricity in an experimental set- ting. An experimental analysis of centrally-committed markets versus self-committed markets would generate useful information regarding the behavior of participants in these markets as well as potential mechanisms for their improvement.

70 Chapter 3: Subjective Distance: Random Paths to Pair-Wise Stability and the Fixtures Problem

3.1 Introduction

The Stable Fixtures problem (Irving and Scott 2007) is a generalization of the

Stable Roommates problem. The classical Stable Roommates problem entails match-

ing each of 2n individuals so that no two people prefer each other over their assigned partners. The Stable Fixtures problem generalizes the Stable Roommates problem from a one-sided one-to-one matching model to a one-sided many-to-many match- ing model by allowing each individual to have a unique capacity representing his maximum possible number of matches. As in the Stable Roommates problem, there are instances of the Stable Fixtures problem for which there are no stable solutions

(Irving and Scott 2007). Irving and Scott develop an that determines, for any given instance of the Stable Fixtures problem, if a stable solution exists. The

Stable Fixtures problem is of theoretical interest because it nests several different matching models: the Stable Roommates problem, the Stable Marriage problem, and the College Admissions problem. The relationship between the Stable Roommates

71 and Stable Marriage problems has been extensively studied, 2 as has the relation- ship between the Stable Marriage and College Admissions problems. 3 As noted by

Chung (2000), the Stable Marriage problem is the only point of contact between the

Stable Roommates problem and the College Admissions problem. Understanding sta- bility in the more general Stable Fixtures problem has the potential to yield insights into our understanding of how preferences, stability, and interact in analyzing different matching economies.

Gale and Shapley (1962) proved that the two-sided marriage and college admis- sions markets always admit stable matches. However, they also demonstrated that the one-sided generalization of the Stable Marriage problem, the Stable Roommates problem, does not always admit stable matches. Tan (1991) developed necessary and sufficient conditions for the existence of stable matches in the Stable Roommates problem. These results were generalized to the weak preferences case by Chung (2000) to obtain a sufficient condition for the existence of stable matches in the roommates problem.

Okumura (2014) considers a one-sided many-to-many matching model that is similar, but not identical, to the Stable Fixtures problem. In Okumura’s framework, agents are teams that are looking to schedule games with one another. Unlike the

Stable Fixtures problem, each team may play multiple games against the same op- ponent. Okumura’s model is a generalization of matching models under dichotomous preferences; that is, teams have an ideal number of games that they are willing to play against each acceptable opponent, but are indifferent between acceptable teams.

2See Gusfield and Irivng (1989) for an in depth review of the Stable Roommates Problem and its relationship with the Stable Marriage problem 3See Roth and Sotomayor (1992) for an extensive overview of two-sided matching.

72 Okumura examines stability and efficiency of matchings and the strategy-proofness

of a direct mechanism in this context.

Another strand of the matching literature concerns itself with decentralized match-

ing markets. In the absence of a centralized algorithmic mechanism, it is common

for many markets to allow agents to freely form matches among themselves at ran-

dom. Roth and Vande Vate (1990) proved a random paths result for the Stable

Marriage problem, generalized by Chung (2000) to the Stable Roommates problem.

Kojima and Unver¨ (2008) demonstrated a random paths to pairwise-stability result for two-sided many-to-many matching markets.

Coarse matching (McAfee (2002)), is a type of matching where the agents are broken into two broad classes, and matched on the basis of the class. McAfee (2002), using a model of coarse matching, illustrated that efficiently gains could be had by using a coarse matching scheme to ration electricity, rather than relying on rolling blackouts. Hospitals, for example, have a high priority under this system, and, as such, are guaranteed access to power in a way that lower priority agents are not. This type of rank based matching system has similarities with what are called assortative matching markets (surveyed in Hoppe at al. (2009). The preference class examined herein has much in common with this notion of assortative matching, as the agents are all able to be ranked in a manner conducive to deriving pair-wise stable matchings.

It is possible that this type of matching scheme may prove useful in certain energy market contracting applications, such as household adoption of renewables and their integration into the grid.

73 Bartholdi and Trick (1986) demonstrated that when preferences are derived from a simple psychological model, there always exists a stable matching for the Room- mates Problem. The idea behind their preference restriction is intuitive: agents have preferences that are derived from a common framework that allow the agents to be ordered sequentially. For example, they consider roommates who wish to live with people who have similar preferences for setting the thermostat. Another example is an individual who prefers to live with someone who comes from a town closer to his own hometown over someone who comes from farther away. This class of preferences has an intuitive psychological appeal because agents prefer other agents who are closer or “more like them” in the sense of the metric. Implicit in this framework is a type of symmetry; agents agree on their differences. We show how this can be relaxed by introducing the notion of “subjective distance” between and among agents. This reflects the possibility that agents may not agree completely on their differences, but so long as those disagreements are not too large the existence of a stable matching is guaranteed. We demonstrate that this class of preferences confers a nice structure to the Stable Fixtures problem: when agents can be sequentially ordered in this way, preferring those who are closer to those who are farther away, a stable solution will always exist.

We present two main results in this chapter. Generalizing the work of Bartholdi and Trick (1986), we extend their results pertaining to the existence of one-sided one-to-one stable roommates matches to the one-sided many-to-many stable fixtures framework by introducing the notion of the subjective distance metric. We first demonstrate via an algorithm that the subjective distance preference restriction is sufficient to ensure the existence of a pairwise-stable matching in the Stable Fixtures

74 problem. We then demonstrate that starting from an arbitrary matching, a pairwise-

stable matching can be achieved via a decentralized process of randomly satisfying a

finite sequence of blocking pairs. We conclude with a discussion of how these results

can be extended and directions for future research.

3.2 The Stable Fixtures Problem

To define the Stable Fixtures problem, let X = {x1, x2, . . . xn} denote the set

of agents. For all xi ∈ X , there exists an integer ci which we call xi’s capacity,

representing the maximum number of possible matches for xi. Every agent xi ∈ X

has a preference ordering over X ∪ ∅ and his preference relation is denoted by i. For each xi ∈ X , let i denote the strict preference relation derived from i. We assume that the preference ordering is a linear order, that is, antisymmetric, complete, and

4 transitive; thus preferences are assumed to be strict. If xi prefers xk to xl, then we write xk i xl. An instance of the Stable Fixtures problem is completely defined by the collection of agents, X , their capacities, c = (c1, c2, . . . , cn), and the preference profiles of the agents, = ( 1, 2,..., n), that is, (X , c, ).

When ci = 1 for every xi ∈ X , this is the Stable Roommates problem. If ci = 1 for every xi ∈ X , and the agents can be partitioned into two sets, M ⊂ X and W ⊂ X such that M ∩ W = ∅, M ∪ W = X , and agents in M only have preferences over agents in W and vice versa, this is the Stable Marriage problem. Allowing agents on one side of the aforementioned partition to have capacities greater than one is the

College Admissions problem.

Definition 1 (Acceptable). If xj i ∅, xj is acceptable to xi.

4The assumption of strict preferences is used for simplicity; this can be relaxed and the main results still hold.

75 In words, xj is acceptable to xi if and only if xi prefers being matched with xj over remaining unmatched or leaving excess capacity open. In the event that xi would rather remain unmatched or leave excess capacity open to matching with xj, we state that xj is unacceptable to xi, which we denote by ∅ i xj.

Definition 2 (Acceptable Pair). A pair {xi, xj} is an acceptable pair if xi is accept- able to xj and xj is acceptable to xi.

Definition 3 (Matching). A matching, µ, is a set of pairs of agents {xi, xj} ⊂ X such that, for all xi ∈ X ,

|{xj : {xi, xj} ∈ µ}| ≤ ci.

The size of µ is the number of pairs in µ. The members of the set µ(xi) = {xj :

{xi, xj} ∈ µ} are referred to as the matches of xi in µ. We denote the set of all possible matchings by M.

Definition 4 (Individually Rational). A matching µ is said to be individually rational if no agent is matched to an agent he considers unacceptable.

Definition 5 (Blocking Pair). An acceptable pair {xi, xj} ∈/ µ is a blocking pair for matching µ, or blocks µ if

(i) Either xi has fewer than ci matches or prefers xj to at least one of his matches

in µ; and

(ii) Either xj has fewer than cj matches or prefers xi to at least one of his matches

in µ.

76 In words, this says that for {xi, xj} to be a blocking pair, either xi must have

excess capacity or xi must prefer xj to one of his current matches, and either xj has excess capacity or prefers xi to one of his current matches.

Definition 6 (Pairwise-Stable). 5 A matching for which there is no blocking pair is said to be pairwise-stable. Otherwise, the matching is said to be pairwise-unstable.

3.3 Preferences and the Subjective Distance Metric

In this section, we build a framework for examining a certain class of preferences.

Suppose that a given agent, xi ∈ X , is looking to form partnerships with other agents in X . In choosing his matches, xi desires to be matched with agents he deems to be “closer” to him according to some distance measure. This can be thought of as representing some notion of similarity, compatibility, or any other trait that xi desires.

For example, this could encompass where other agents hail from geographically or their political views. To capture this idea, we introduce the notion of the subjective distance metric.

Definition 7 (Subjective Distance Metric). A subjective distance metric is a map- ping, si : X → R+ such that si(j) > 0 for all xi, xj ∈ X such that i 6= j.

The subjective distance metric represents the “distance” an agent perceives be- tween himself and a fellow agent. The distance metric is called “subjective” because we do not require that si(j) = sj(i), that is, we do not require that xi’s view of the distance between himself and xj be the same as xj’s view of the distance between himself and xi. This relaxes the symmetry assumption of Bartholdi and Trick (1986).

5There are other notions of stability in many-to-many matching markets. See Echenique and Oviedo (2006).

77 We are interested in the case where agents would like to match with those who are perceived to be closest in terms of subjective distance.

Definition 8 (Narcissistic). A subjective distance metric is narcissistic if, for all xi ∈ X , si(i) = 0 and agents prefer those who are perceived to be closer to those who are farther away.

When the subjective distance metric is narcissistic, we write that each agent in some sense represents his own ideal point. The smaller the perceived distance between agents, the more preferred they are.

When agent preferences are determined by narcissistic subjective distance metrics, we have that:

xj i xk ⇔ si(j) < si(k) for all xi, xj, xk ∈ X .

Remark 1. The above formulation implicitly defines every agent as acceptable to every other agent. We can accommodate unacceptability into the framework in the following way: if agent xi deems xj unacceptable, we write si(j) = ∞.

We now define a particular restriction on the subjective distance metrics that will prove useful in demonstrating our main results.

Definition 9 (Approximate Symmetry). A subjective distance metric satisfies ap- proximate symmetry if, for all xi, xj, xm, xn ∈ X

si(j) ≤ sm(n) ⇒ max{si(j), sj(i)} ≤ min{sn(m), sm(n)}.

This condition imposes some degree of structure on preferences. We move away from the requirement of complete symmetry of Bartholdi and Trick (1986), but rather

78 only require that preferences be close enough to capture the effects of pure symmetry.

The idea is that even if agents are not in perfect agreement regarding their differences,

there exists some common framework that allows them to evaluate one another in

such a way that their perceptions of one another are not radically different.

Definition 10 (Subjective Distance Vector). Let s = (si(j))xi,xj ∈X be the vector of

subjective distances between all agents. We call s the Subjective Distance Vector.

Definition 11 (Ordered Subjective Distance Vector). Consider only entries si(j) in s such that i 6= j, ordered from smallest to largest. The resulting vector is the Ordered

Subjective Distance Vector, denoted by s.

Remark 2. When the subjective distance metrics satisfy approximate symmetry, then

s can be written so that, for all xi, xj ∈ X , either si(j) immediately follows or imme-

diately precedes sj(i).

3.4 Results

3.4.1 Existence of Pairwise-Stable Fixture Matchings

The following lemma will be useful in constructing our algorithm to prove the

existence of Stable Fixtures matchings when agent preferences are consistent with

subjective distance metrics satisfying approximate symmetry.

Lemma 18 (Bartholdi and Trick). If among all available choices, agent xi most prefers agent xj, and agent xj most prefers agent xi, then in any pairwise-stable matching xi and xj must be matched.

Proof. If xi and xj are not matched, they form a blocking pair.

79 We now demonstrate that the existence of a pairwise-stable fixtures matching is guar-

anteed when preferences are consistent with subjective distance metrics satisfying

approximate symmetry.

Theorem 1. If agent preferences are consistent with subjective distance metrics sat-

isfying approximate symmetry then there exist pairwise-stable fixture matchings.

Proof. We demonstrate a constructive algorithm for obtaining a pairwise-stable fix-

ture matching:

Step 1: Begin with the first entry in ¯s. Let si(j) be this entry. Since neither xi nor

xj currently has any matches, we match them. Remove si(j) from ¯s. By approximate

symmetry, the next entry in ¯s is sj(i). Because xi and xj are matched, this entry

can be removed as well. Reduce both ci and cj by 1. If either xi or xj has filled

his capacity, remove any remaining subjective distances corresponding to that agent

from ¯s. If neither xi nor xj has filled his capacity, no entries are removed from ¯s.

Define ¯s1 as the vector of subjective distances remaining following Step 1.

Step k: If ¯sk−1 = ∅, the algorithm terminates. If ¯sk−1 6= ∅, we match the agents corresponding to the first entry in ¯sk−1. Remove the subjective distances correspond- ing to a match between these two agents from ¯sk−1, and reduce each agent’s capacity by 1. If either agent has filled his capacity, we remove all subjective distances corre- sponding to that agent from ¯sk−1 and rename the resulting vector ¯sk.

80 This algorithm will terminate after a finite number of steps (there are a finite

number of agents, and therefore a finite number of subjective distances to consider),

when either all subjective distances have been removed or there is a single agent re-

maining with excess capacity. We now demonstrate that the resulting matching is

pairwise-stable. Assume for contradiction that the matching resulting at the termi-

nation of the above algorithm is not pairwise-stable. Then there exists a blocking

pair, {xi, xj}, such that

(i) either xj i xk for some xk that xi is matched with, or xi has not filled his quota

and has excess capacity remaining, and

(ii) either xi j xl for some xl that xj is matched with, or xj has not filled his quota

and has excess capacity remaining.

Assume that xj i xk for some xk that xi is matched with. This implies that si(j) < si(k). But since xi and xj are not matched, it must be the case that when the algorithm reached si(k), xj’s entries must have already been removed, meaning that his quota was filled. Therefore, there is no agent, xl, matched with xj such that xi j xl. We now assume that xi has not filled his quota and therefore has excess capacity available. This implies that there is no one remaining to whom he can be matched as all other agents must have filled their quotas with agents closer to them than xi. Thus, {xi, xj} can not be a blocking pair, and we have obtained the contra- diction.

The matching is pairwise stable.

81 3.4.2 Random Paths to Pairwise-Stable Fixture Matchings

We have demonstrated that when preferences are consistent with subjective dis-

tance metrics satisfying approximate symmetry, there exists a pairwise-stable solution

to the Stable Fixtures problem. A natural question is whether stable matchings can

be obtained through a decentralized matching process as opposed to a centralized

algorithmic mechanism.

Our main result shows that when preferences are consistent with subjective dis-

tance metrics satisfying approximate symmetry, a pairwise-stable matching can al-

ways be attained from a pairwise-unstable matching by satisfying a finite sequence

of blocking pairs. Starting from an arbitrary matching µ, if {xi, xj} form a blocking

pair and it is true that both |µ(xi)| < ci and |µ(xj)| < cj, then we can simply match

both agents to generate a new matching, µ0. In this case, no other agents are affected

by the match. However, it is possible that an agent xi may have his entire capacity

filled under µ, that is, |µ(xi)| = ci. If {xi, xj} form a blocking pair for matching µ

in this case, this means that xi must “dump” one of his current matches in order to

match with xj. In this case, xi will dump his least preferred match among his current

matches, that is, xi will dump xk ∈ µ(xi) such that si(k) ≥ si(l) for all xl ∈ µ(xi). In words, xi will dump the current match of his who is farthest away from him, accord- ing to the perceived subjective distance, of all his current matches under µ in favor of matching with xj. The dumped agent will then gain one unit of excess capacity .

Definition 12 (Satisfying the Blocking Pair). Let X be a set of agents with pref- erences consistent with subjective distance metrics satisfying approximate symmetry.

Let µ ∈ M be a matching. Let {xi, xj} ⊂ X be a blocking pair for µ. A new matching,

0 µ , is obtained from µ by satisfying the blocking pair {xi, xj} if:

82 0 0 (i) xj ∈ µ (xi) and xi ∈ µ (xj)

(ii) If |µ(xi)| = ci, then ∃xk ∈ µ(xi) s.t. si(k) ≥ si(l) ∀xl ∈ µ(xi) and xi dumps xk

in favor of matching with xj

(iii) If |µ(xj)| = cj, then ∃xm ∈ µ(xj) s.t. sj(m) ≥ sj(h) ∀xh ∈ µ(xj) and xj dumps

xm in favor of matching with xi

0 0 (iv) If xk = xm = xd, then µ (xd) = µ(xd)\{xi, xj} and if xk 6= xm, then µ (xk) =

0 µ(xk)\{xi} and µ (xm) = µ(xm)\{xj}.

0 (v) ∀xr ∈ X \{xi, xj, xk, xm}, µ (xr) = µ(xr).

Condition (i) states that after satisfying the blocking pair, xi and xj must now be matched with each other. Conditions (ii) and (iii) state that if xi or xj is currently matched at full capacity under the original matching µ, they must dump their least preferred current match to satisfy the blocking pair. Condition (iv) states that under the new matching µ0, dumped agents remain matched to the same set of agents that they were matched with under µ minus the members of the satisfied blocking pair.

Condition (v) states that all other agents not affected by the blocking pair have the same matches under the new matching µ0 as under the old matching µ.

Remark 3. Any individually irrational matching can be transformed into an individ- ually rational matching by having agents dump any unacceptable matches.

83 We now demonstrate that starting from an arbitrary matching we can achieve a

stable matching by sequentially satisfying a finite number of blocking pairs.

Lemma 19. When agent preferences are consistent with subjective distance metrics

satisfying approximate symmetry, for any matching µ, there exists a finite sequence

of matchings (µ1, µ2, . . . , µT ), such that µ1 = µ, µT is pairwise-stable, and for each

t = 1, 2,...,T − 1, there is a blocking pair for µt such that µt+1 is obtained from µt

by satisfying that blocking pair.

Proof. We provide a constructive algorithm that will transform the current matching

µ into a stable matching in a finite number of steps:

Step 1: Let µ1 = µ. Consider the first entry of s that corresponds to a pair of agents {xi, xj} that are not matched under µ1. This represents the first potential blocking pair. If these agents do not form a blocking pair, then either xi or xj must be matched to capacity and does not wish to dump any of his current matches. As subjective distances are increasing, no future blocking pair will arise involving these two agents together. Remove any subjective distances corresponding to this agent from s. Define µ2 = µ1. If, however, {xi, xj} does constitute a blocking pair, we have

two possible cases:

Case 1 : If |µ1(xi)| < ci and |µ1(xj)| < cj, we match xi and xj. Since nei-

ther xi nor xj is currently matched at full capacity, no other agents are affected. Call the resulting matching µ2.

84 Case 2 : If |µ1(xi)| = ci or |µ1(xj)| = cj, then either xi or xj are at full capacity under the current matching, and must dump their least preferred agent to satisfy the blocking pair. We satisfy the blocking pair and any dumped agents gain one unit of excess capacity. Call the resulting matching µ2.

Define ¯s1 as the vector of subjective distances remaining following Step 1.

Step k: Consider the first entry of sk−1 corresponding to a pair of agents {xi, xj} that are not matched according to the matching µk−1. This represents a potential blocking pair. If these agents do not form a blocking pair, then either xi or xj must be matched to capacity and does not wish to dump any of his current matches. As subjective distances are increasing, no future blocking pairs involving this agent can form. Remove any subjective distances corresponding to this agent from sk−1. Define

µk = µk−1 If, however, {xi, xj} does constitute a blocking pair, we have the same two

possible cases as above and proceed accordingly.

This algorithm terminates in a finite number of iterations resulting in the match-

ing, µT . The proof of stability follows the same argument as given in the proof of

Theorem 1.

The critical step in the above proof is that blocking pairs can be satisfied sequen-

tially based on subjective distance. Any time an agent is dumped when a blocking

pair is satisfied, the dumped agent has a greater subjective distance than the newly

85 matched agent. This means that a dumped agent will not create any new instability among the matches that have been generated in previous steps of the algorithm.

Having proved the above lemma, the random paths to pairwise-stability result is an immediate consequence of the standard Markov-chain argument. Starting from an arbitrary matching µ, a random process can generate a sequence of matchings by satisfying a single randomly chosen blocking pair. The probability of any one block- ing pair being chosen is positive for all such blocking pairs for a given matching. The following theorem results from the fact that for any matching, every blocking pair has a positive probability of being chosen.

Theorem 2. If agent preferences are consistent with subjective distance metrics sat- isfying approximate symmetry, then a decentralized process of allowing randomly cho- sen blocking pairs to match will converge to a pairwise-stable fixtures matching with probability one.

3.5 Other Applications

An immediate application of Theorems 1 and 2 is that by demonstrating that a given class of preferences is consistent with the subjective distance metric formulation, we can guarantee the existence of pairwise-stable fixture matchings and the attendant random paths to pairwise-stability result, extending results from the one-sided one- to-one Stable Roommates problem to the one-sided many-to-many Stable Fixtures

Problem. We now provide two examples of preference domains that are consistent with the subjective distance metric derivation.

86 The first application of our theorems is extending the results of Bartholdi and

Trick (1986). Bartholdi and Trick considered a preference restriction for the Stable

Roommates problem where agent attributes can be represented by points in a metric

space, every agent strictly prefers agents who are more similar according to the metric,

and every agent prefers having a roommate to not having one. They prove that this

preference restriction guarantees the existence of Stable Roommate matchings.

Corollary 5. If agents can be represented as points in a metric space, every agent strictly prefers agents closer to him to those farther away, and strictly prefers having a match to not, then there exists pairwise-stable fixture matchings.

Proof. Let d(i, j) represent the distance between agents xi, xj ∈ X . For all xi, xj ∈ X , define si(j) = d(i, j). By definition of a metric space, d(i, j) = d(j, i) for all xi, xj ∈ X .

Thus approximate symmetry is satisfied and a pairwise-stable matching exists.

Another domain of interest is Dichotomous preferences. Under Dichotomous pref-

erences, each agent partitions the set of all agents into two groups. He prefers all

agents in one group to the other, and is indifferent among agents within the same

group.

Definition 13 (Dichotomous Preferences). A preference profile is Dichotomous if every agent classifies all agents into two groups in such a way that within each group he is indifferent among members.

Corollary 6. If the preference profile is Dichotomous, there exists pairwise-stable

fixture matchings.

Proof. Assume that each agent xi ∈ X partitions the set of agents into two sets, agents who are acceptable as matches and agents who are not acceptable. For all

87 xi, xj ∈ X , define  1 if {xi, xj} is an acceptable pair, si(j) = ∞ if{xi, xj} is not an acceptable pair.

Thus Dichotomous preferences are consistent with subjective distance metrics sat-

isfying approximate symmetry, and therefore pairwise-stable fixture matchings ex-

ist.

3.5.1 Failures of Approximate Symmetry

We now examine an instance of the Stable Fixtures problem discussed by Irving

and Scott (2007) which has no stable solution. We show that approximate symmetry

fails to hold for this example. Assume that six agents, each with two units of capacity,

have the following preferences:

x1 : x2 1 x4 1 x3

x2 : x3 2 x5 2 x1

x3 : x1 3 x6 3 x2

x4 : x5 4 x1 4 x6

x5 : x6 5 x2 5 x4

x6 : x4 6 x3 6 x5.

Irving and Scott (2007) prove that this instance of the Stable Fixtures problem has no stable solution.6 We now demonstrate that the above preferences are not consistent with subjective distance metrics satisfying approximate symmetry. Given the preferences of the agents, we must have:

6For the complete proof of instability, see Irving and Scott (2007).

88 s1(2) < s1(3)

s2(3) < s2(1)

s3(1) < s3(2).

Since s1(2) < s1(3), this implies, by approximate symmetry, that

max{s1(2), s2(1)} ≤ min{s1(3), s3(1)}.

Thus, s1(2) < s3(1). We also know that since s3(1) < s3(2), by approximate symmetry,

max{s3(1), s1(3)} ≤ min{s3(2), s2(3)}.

Therefore, s3(1) < s3(2). This then implies that s1(2) < s3(2), so by approximate symmetry:

max{s1(2), s2(1)} ≤ min{s3(2), s2(3)}.

However, this contradicts our initial assumption that s2(3) < s2(1). Thus, in this example where there exists no pairwise-stable matching, approximate symmetry fails to hold.

3.6 Discussion and Concluding Remarks

We have demonstrated that for a psychologically appealing class of preferences, a pairwise-stable matching always exists for the Stable Fixtures problem. We have also demonstrated that a decentralized matching process will converge to a pairwise-stable matching with probability one by satisfying random blocking pairs from any unstable matching. These results represent an attempt to extend previous work on one-sided

89 matching models from the one-to-one to the many-to-many case, and constitute a step towards better understanding one-sided many-to-many matching models which remain to be studied in further detail. There is much more work that remains to be done to enhance our understanding of the theory.

One-sided many-to-many matching is of theoretical interest because it provides a general framework that can encapsulate many of the most commonly studied match- ing markets. Understanding the interaction between preferences and stability in this type of unified framework that nests Stable Roommates, Stable Marriage, and College

Admissions as special cases is of theoretical and also practical interest. For example, it is conceivable to consider markets where firms act as both employers (hiring inde- pendent consultants or signing contracts with suppliers) while simultaneously being employed to provide particular services. When the separation between workers and

firms is not clearly delineated, this type of one-sided many-to-many matching model may be of interest.

Sufficient conditions that guarantee stability in the Roommates Problem have been found, and a natural direction for future research is looking into general suf-

ficient conditions that guarantee the existence of Stable Fixture matchings. The subjective distance metric satisfies Chung’s “No Odd Rings” sufficient condition for the existence of Stable Roommates matchings, and an open question is whether and how his results may be further generalized to the Stable Fixtures problem. This has the potential to deepen our understanding of the relationships between one-sided and two-sided matching markets and one-to-one, many-to-one, and many-to-many matchings. This chapter provides a sufficient condition for stable matchings under a restricted, psychologically appealing class of preferences. Examining other preference

90 profiles and whether they can guarantee stability is a promising potential direction for future research that may help us understand general sufficient conditions for the existence of stable matchings in the Stable Fixtures problem.

Further work remains to be done to study more general conditions for the existence of random paths to pairwise-stable matchings in the Stable Fixtures problem. Kojima and Unver¨ proved that for a two-sided many-to-many matching model, as long as one side has responsive preferences while the other side has substitutable preferences, a random path to a pairwise-stable matching always exists. The subjective distance metric preferences are responsive, and responsive preferences satisfy substitutability.

Understanding the interplay of these types of preference domains in one-sided many- to-many matching models will be of interest in developing new results on random paths to pairwise-stable matchings.

Another promising direction for future research is to consider modeling the Sta- ble Fixtures Problem using linear programming techniques. Abeledo and Rothblum

(1994) demonstrated how the existence of stable matchings in the Stable Marriage and Stable Roomates problems can be determined using linear inequalities and inte- ger programming techniques. Chung (2000) notes that his no odd rings condition can be derived using the techniques of Abeledo and Rothblum. This may prove fruitful in further examining the Stable Fixtures Problem.

Irving developed a general algorithm for finding solutions to the Stable Fixtures problem, but the strategy-proofness of this algorithm, whether agents can misreport their preferences to attain a more preferred matching, has not yet been examined.

Notions of Stability in one-sided many-to-many matching problems may also hold promise for future research. Echenique and Oviedo (2004) discuss various notions

91 of stability in two-sided many-to-many matching markets. In this paper, we focus on examining pairwise-stability, but there are other notions of stability that are not equivalent to pairwise stability in many-to-many markets, such as group stability, set- wise stability, and stability. Exploring these stability concepts in the one-sided many-to-many matching framework may be of interest. This type of model may also be of interest in the context of contracting for either retail or wholesale electricity, particularly with repect to the integration of renewables. For instance, households that install solar panels become are, in effect, both producers and consumers of elec- tricity. Their relationships with one another and the overall grid may potentially be modeled as a kind of one-sided many-to-many matching market explored here.

92 Chapter 4: A Nash-Cournot Analysis of a Proposed Power Purchase Agreement

4.1 Introduction

The changing energy landscape has resulted in the need for a variety of new approaches to problems of market design and regulation. One such issue that must be addressed is how to best ensure reliability and resilience. In January 2018, the

Federal Energy Regulatory Commission rejected a Department of Energy proposal ostensibly designed to ensure access to power in areas served by older, less efficient generating firms7. The idea behind the proposal is fairly straight-forward; generating

firms would be guaranteed a predetermined profit level with the expectation that without the profit guarantee, these generating firms would exit the market, leaving consumers without access to power. However, as will be demonstrated in this work, the proposed PPA has a number of problematic incentive properties created by this particular market intervention. Namely, we demonstrate that the subsidy, rather than ensuring the generation of a firm that would otherwise exit the market, can in fact create incentives for the subsidized firm to produce nothing and simply collect the subsidy. This can then lead to a number of undesirable effects in the broader

7See https://www.ferc.gov/media/news-releases2018/2018-1/01-08-18.asp.Wrl hiMwgy4

93 wholesale electricity market, namely, a higher wholesale price for electricity, lower aggregate quantity, higher profits for the generating firms, lower consumer surplus, and lower aggregate social welfare.

4.2 Related Literature

This analysis of the PPA draws on and contributes to two main branches of the literature. The first has to do with using Cournot models to analyze electricity markets. The second has to do with the existence of equilibria in certain variants of the standard Cournot model.

Cournot models have been a staple in the study of eco- nomics. For a survey of the many applications of the Cournot model, see Tirole

(1988). Cournot models have been particularly relevant to the study of electricity markets in both the economics and operations research literatures. Green and New- berry (1992) compare aspects of the Cournot solution to supply function equilibria in their analysis of the British electricity spot market. Vasin and Kartunova (2016) survey and generalize results on in the context of electricity markets.

The existence or non-existence of equilibria in Cournot markets has been stud- ied extensively in the game theory and theoretical industrial organization literature.

Roberts and Sonnenschein (1976) explored the existence of Cournot equilibria in the absence of concave profit functions. Novshek (1985) demonstrated that if marginal revenues are declining as the aggregate output of other firms increases, there exists a

Cournot equilibrium. Various applications of the Cournot model are studied in Vives

(2001) and Daughety (2005). Important to our Nash-Cournot analysis of the PPA is

94 the role of taxes and subsidies in Cournot markets. Levin (1985) studied taxation in

a Cournot oligopoly. However, in Levin’s model, the tax paid by each firm is leveled

on a per-unit basis and is independent of the aggregate quantity produced. As will

be seen in our ensuing analysis, the nature of how the subsidy is collected and paid to

the subsidized generating firms means that the per-unit cost to each generating firm

will vary with the aggregate quantity of output produced in the market. Corchon

(2008) studies the magnitude of welfare loss under differently sized Cournot markets.

4.3 Model

4.3.1 Preliminaries

Consider an electricity market with n competing generating firms. Assume that each firm i has fixed cost of production, Fi, and a constant marginal cost of production ci, so that the total cost of each firm i is represented by the cost function:

Ci(qi) = ciqi + Fi.

Electricity prices are determined by the linear inverse demand function P (Q) =

A − BQ, where A, B ∈ R+ and :

n X Q = qi, i=1 is aggregate supply.

Firms simultaneously make production decisions and the market clears by whole- sale price equating demand and aggregate supply.

95 4.3.2 Market Subsidy

Assume that one firm, denoted firm s, is offered a subsidy. The profit of the

subsidized firm can be written as function of its own production, qs, and the aggregate

production of its rivals, Q−s:

n X Q−s = qi, i=1,i6=s as:

πs(qs,Q−s) = L, where L represents the profit level guaranteed to firm s by the subsidy. The profit level

of the subsidized firm is guaranteed by confiscating revenue from the non-subsidized

firms on the basis of their per-unit output. Therefore, the profit of the non-subsidized

firms can be written as:

(L − ((P (Q) − c)q − F )) π (q ,Q ) = P (Q) · q − s s q − cq − F , i i −i i Q i i i (4.1) (L − (((A − BQ) − c)q − F )) = (A − BQ) · q − s s q − cq − F . i Q i i i The second term in the above equation represents the revenues that are confiscated

from the non-subsidized firms. The numerator represents the effective size of the

subsidy, that is, the difference between the guaranteed profit level, L, and the profits

earned by the firm based on its own production and the market price alone.

The idea behind the proposal guaranteeing a profit level, L, for the subsidized firm

is that in the absence of the subsidy, this firm will not be able to produce. That is,

this firm will shut down and exit the market, leaving consumers without access to the

96 power this firm would have generated. However, we next demonstrate the existence

of an equilibrium where the subsidized firm produces nothing and simply accepts the

guaranteed profit level of L. This comes at the expense of the consumer in that there

is a smaller amount of electricity produced and a higher wholesale price.

To determine the existence of an equilibrium where the subsidized firm refuses to

∗ produce, we begin by setting qs = 0. Note that since the subsidized firm is guaranteed the same profit level regardless of its production, this implies that the subsidized firm

has effectively no strategic decision to make: its payoff is entirely independent of

its choice of output. This implies the existence of a continuum of equilibria. By

exploring the case where qs = 0, we examine one particularly “bad” equilibrium that

exists under the proposed subsidy scheme. This means that the profit maximization

problem faced by non-subsidized firm i is:

L˜ max {(A − BQ)q − q − cq − F }, (4.2) qi i Q i i i ˜ ˜ where L represents the effective size of the subsidy, that is, L = L + Fs. Since the subsidized firm is producing nothing, this means that the subsidized firm must be compensated to cover both its fixed cost as well as the guaranteed profit level.

Therefore, the first order necessary condition for an optimum is:

˜ ˜ ∂ L Lqi πi = (A − BQ) − Bqi − + 2 − ci = 0. (4.3) ∂qi Q Q Simplifying and rearranging terms yields:

2 2 ˜ ˜ 2 (A − BQ)Q − BqiQ − LQ + Lqi − ciQ = 0, (4.4) 2 ˜ (A − BQ − Bqi − ci)Q = L(Q − qi).

97 We now check that the second order condition is satisfied, that is,

∂2 2 πi ≤ 0. ∂qi First, note that:

∂2 L˜ L˜ 2L˜q π = −B − B + + − i − 2c Q, ∂q2 i Q2 Q2 Q3 i i (4.5) 2L˜ 2L˜q = −2B + − i − 2c Q. Q2 Q3 i Therefore, the second order condition is satisfied provided that:

2L˜ 2L˜q −2B + − i − 2c Q ≤ 0 Q2 Q3 i

Rearranging and consolidating terms yields:

2L˜ 2L˜q −2B + − i − 2c Q ≤ 0, Q2 Q3 i 3 ˜ ˜ 4 −2BQ + 2LQ − 2Lqi − 2ciQ ≤ 0,

3 ˜ 4 −2BQ + 2L(Q − qi) − 2ciQ ≤ 0, (4.6) ˜ 3 4 2L(Q − qi) ≤ 2BQ + 2ciQ ,

˜ 3 2L(Q − qi) ≤ 2Q (B + ciQ),

˜ 3 L(Q − qi) ≤ Q (B + ciQ).

Combining the FONC in (4.4) with (4.6) implies that at a Nash equilibrium, it must be the case that:

98 2 3 (A − BQ − Bqi − ci)Q ≤ Q (B + cQ),

A − BQ − Bqi − ci ≤ Q(B + ciQ),

2 A − BQ − Bqi − ci ≤ QB + ciQ , (4.7)

2 A − ci ≤ 2BQ + Bqi + ciQ ,

2 A − ci ≤ B(2Q + qi) + ciQ .

Given our assumption of an inverse demand function of the form P (Q) = A−BQ,

this implies that since B is a free variable, we ensure that the second order sufficient

condition is satisfied.

4.3.3 Effect of Subsidy on Production Levels

We now wish to demonstrate that the equilibrium production level of each non-

subsidized firm is non-increasing in the aggregate output of its rivals. First, notice

that an equilibrium production level of each non-subsidized firm i ∈ {1, . . . , n}, i 6= s, is implicitly defined by the FONC in (3). We can now rewrite the FONC as:

∂ ∗ πi(qi (Q−i),Q−i) = 0. (4.8) ∂qi

∗ We now write qi (Q−i) to denote that firm i’s optimal production level depends on the aggregate output of its rivals. This means that aggregate production can now

∗ be written as Q = qi (Q−i) + Q−i. With this new notation, we can now rewrite the FONC in (3) as:

˜ ˜ ∗ ∗ ∗ L Lqi (Q−i) (A − B(qi (Q−i) + Q−i)) − Bqi (Q−i) − ∗ + ∗ 2 − ci = 0. qi (Q−i) + Q−i (qi (Q−i) + Q−i) (4.9)

99 Taking the total derivative of the above expression with respect to Q−i yields:

˜ d ∗ ˜ d ∗ L( q (Q−i) + 1) L q (Q−i) d ∗ d ∗ dQ−i i dQ−i i −B( qi (Q−i) + 1) − B qi (Q−i) + ∗ 2 + ∗ 2 dQ−i dQ−i (qi (Q−i) + Q−i) (qi (Q−i) + Q−i) ˜ ∗ 2Lqi (Q−i) d ∗ − ∗ 3 ( qi (Q−i) + 1) = 0. (qi (Q−i) + Q−i) dQ−i (4.10)

The goal is to determine how the optimal production level of a non-subsidized

firm is changing in the aggregate output of its rivals. Therefore, we must isolate the

d ∗ ∗ 3 term q (Q−i). We multiply by sides of the above expression by (q (Q−i) + Q−i) , dQ−i i i yielding:

d ∗ ∗ 3 ∗ 3 −2B qi (Q−i)(qi (Q−i) + Q−i) − B(qi (Q−i) + Q−i) dQ−i ˜ d ∗ ∗ +L(2 qi (Q−i) + 1)(qi (Q−i) + Q−i) (4.11) dQ−i ˜ ∗ d ∗ −2Lqi (Q−i)( qi (Q−i) + 1) = 0. dQ−i By multiplying out terms we can next obtain:

d ∗ ∗ 3 ∗ 3 −2B qi (Q−i)(qi (Q−i) + Q−i) − B(qi (Q−i) + Q−i) dQ−i ˜ d ∗ ∗ ˜ ∗ +2L qi (Q−i)(qi (Q−i) + Q−i) + L(qi (Q−i) + Q−i) (4.12) dQ−i ˜ ∗ d ∗ ˜ ∗ −2Lqi (Q−i) qi (Q−i) − 2Lqi (Q−i) = 0. dQ−i By further rearranging the terms, we can obtain the following expression:

! d ∗ ∗ 3 ˜ ∗ ˜ ∗ qi (Q−i) − 2B(qi (Q−i) + Q−i) + 2L(qi (Q−i) + Q−i) − 2Lqi (Q−i) dQ−i (4.13) ∗ 3 ˜ ∗ ˜ ∗ = B(qi (Q−i) + Q−i) − L(qi (Q−i) + Q−i) + 2Lqi (Q−i)

100 .

d ∗ This allows us to write q (Q−i) as: dQ−i i

∗ 3 ˜ ∗ ˜ ∗ d ∗ B(qi (Q−i) + Q−i) − L(qi (Q−i) + Q−i) + 2Lqi (Q−i q (Q−i) = . (4.14) i ∗ 3 ˜ ∗ ˜ ∗ dQ−i −2B(qi (Q−i) + Q−i) + L(qi (Q−i) + Q−i) − 2Lqi (Q−i By examining the terms in both the numerator and denominator of the above expression, we can determine how the optimal production level of a non-subsidized

firm is changing in the aggregate output of its rivals. The optimal production level of a non-subsidized firm will be decreasing in the the aggregate output of its rivals

∗ 3 ˜ ∗ provided that (B(qi (Q−i) + Qi) − L(qi (Q−i) + Q−i) > 0.

4.3.4 Effect of Subsidy on Profit Levels

We now examine how the profit level of a non-subsidized firm changes with the ag- gregate output of its rivals. We first note that the equilibrium profit of non-subsidized

firm i can be written as:

∗ ∗ ∗ πi(Q−i) = πi(qi (Q−i),Q−i) = qi (Q−i)(A − B(qi (Q−i) + Q−i)) ˜ L ∗ − ∗ − ciqi (Q−i) − Fi. qi (Q−i) + Q−i

Taking the total derivative of the above expression with respect to Q−i yields the

following:

d d ∗ d ∗ ∗ d ∗ πi(Q−i) = qi (Q−i)A − 2B qi (Q−i) − Bqi (Q−i) − BQi qi (Q−i) dQ−i dQ−i dQ−i dQ−i ˜ L d ∗ d ∗ + ∗ 2 ( qi (Q−i + 1) − ci qi (Q−i). (qi (Q−i) + Q−i) dQ−i dQ−i For profit to be non-increasing in the aggregate production of its rivals, it must

be the case that the above expression is non-positive. This is satisfied if:

101 ˜ d ∗ L d ∗ ∗ qi (Q−i)A + ∗ 2 ( qi (Q−i) + 1) − Bqi (Q−i) dQ−i (qi (Q−i) + Q−i) dQ−i d d ∗ d ∗ ≤ 2B qi(Q−i) + BQ−i qi (Q−i) + c qi (Q−i), dQ−i dQ−i dQ−i which is satisfied if:

˜ ˜ d ∗ L L ∗ qi (Q−i)(A + ∗ 2 ) + ∗ 2 − Bqi (Q−i) dQ−i (qi (Q−i) + Q−i) (qi (Q−i) + Q−) d ∗ ≤ qi (Q−i)(2B + BQ−i + c), dQ−i which is satisfied again given the choice of the inverse demand function P (Q) =

A − BQ.

This leads to the following proposition:

Proposition 1. Given a linear demand function, the subsidy scheme under the pro- posed power purchase agreement can lead to equilibria where (i) the subsidized firm produces nothing; (ii) aggregate market quantity is lower than it would be in the ab- sence of the subsidy; (iii) wholesale electricity prices are higher; (iv) aggregate firm profits are higher; (v) consumer welfare is lower; and (vi) aggregate social welfare is lower.

We now illustrate the above result by providing a simple constructive example demonstrating the existence of a Nash equilibrium where the subsidized firm produces no output using a generic linear inverse demand function. As will be shown, the subsidy results in a number of effects that run absolutely counter to the desired purpose of the PPA.

102 4.4 A Simple 3-Firm Example

Let N = 3 and let demand be P (Q) = 14−Q. For simplicity, let ci = 2 for all i. We

further assume that each firm i has a fixed cost of production, Fi. Let F1 = F2 = $3.

For firm 3, we will consider two different cases, where the fixed cost exceeds total revenues in the competitive non-subsidized equilibrium, and where the fixed cost does not exceed total revenues in the competitive non-subsidized equilibrium. We will now examine firm behavior under two scenarios: (i) in the absence of the subsidy; (ii) when one firm is subsidized at the expense of the other two. In the absence of the subsidy, this implies that the profit maximization problem for firm i is:

maxqi {(14 − (qi + Q−i))qi − 2qi − Fi}.

For all three firms, the fixed cost does not affect the choice of profit-maximizing output. Therefore, without loss of generality, we solve the program for firm 1, yielding

FONC:

14 − 2q1 − q2 − q3 − 2 = 0,

12 − 2q1 − q2 − q3 = 0.

∗ ∗ Given the symmetry among firms, this implies that at a Nash equilibrium qi = q for each firm i = 1, 2, 3. Therefore, we have that:

12 − 4q∗ = 0

q∗ = 3. Since q∗ = 3, this implies that aggregate production at the Nash equilibrium is

Q∗ = 9. Therefore, the market price is P (Q∗) = 14 − Q∗ = $5. Thus, for both firm 1

103 and firm 2, profit is P (Q∗) · q∗ − 2q∗ − 3 = $6.

˜ ˆ We will consider two possible fixed costs for firm 3: F3 = $10 and F3 = $6. In the first case, given firm 3’s fixed cost of 10, it earns negative economic profits of

−$1. In the second case it earns positive economic profits of $3. It is important to note the distinction between these two cases. In the first, the competitive leaves the firm with revenue insufficient to cover its fixed costs and, as such, this

firm will presumably exit the market in the long-run. In the second case, the firm is economically viable, but earns lesser profits than its rivals, Firms 1 and 2. We now discuss how the subsidy affects the outcome of the market under both fixed-cost scenarios for Firm 3.

Assume a policy is enacted that guarantees the third firm a positive profit level .

That is, the firm is guaranteed a profit level of L = $3.5. This profit level can be de- composed into two parts: the fixed cost of $10 and the economic profit of $3.5. Thus, the effective size of the subsidy if firm 3 produces nothing is L˜ = $10+$3.5 = $13.5 if ˜ ˜ ˆ it has fixed cost F3 = $10 and L = $6 + 3.5 = $9.5 if F3 = $6. This is the size of the subsidy that must be split among firms 1 and 2 after the imposition of this policy. We now demonstrate the existence of a Nash equilibrium where firm 3 produces nothing under both fixed cost scenarios.

∗ Let q3 = 0. How will the other firms react? Without loss of generality, we provide ˜ the profit maximization problem for firm 1 when F3 = $10 :

$13.5 maxq1 {(14 − (q1 + q2))q1 − · q1 − 2q1 − 3}. q1 + q2

104 .

Given that the subsidized firm produces nothing, the above profit maximization

problem yields a first order condition of the form:

13.5 13.5q1 14 − 2q1 − q2 − + 2 − 2 = 0. (4.15) q1 + q2 (q1 + q2) Given the symmetry of firms 1 and 2, this implies that at a Nash equilibrium,

∗ ∗ ∗ q1 = q2 = q . Substituting this into the FONC above, we obtain:

13.5 13.5q∗ 14 − 3q∗ − + − 2 = 0 2q∗ (2q∗)2 13.5 13.5q∗ 14 − 3q∗ − + − 2 = 0 2q∗ 4q∗2 13.5 13.5 14 − 3q∗ − + − 2 = 0 2q∗ 4q∗ 27 13.5 14 − 3q∗ − + − 2 = 0 4q∗ 4q∗ 13.5 14 − 3q∗ − − 2 = 0 4q∗ We can use the above FONC to derive the Nash equilibrium output levels for firms

1 and 2:

13.5 14 − 3q∗ − − 2 = 0 4q∗ 52q∗ − 12q∗2 − 13.5 − 8q∗ = 0

44q∗ − 12q∗2 − 13.5 = 0. √ √ ∗ 22− 322 ∗ 22+ 322 The solutions of the above quadratic formula are: q = 12 and q = 12 . √ ∗ 22+ 322 The latter value is the true optimum, where q = 12 ≈ 3.33. To verify that this is indeed an optimum, we examine the Second Order conditions (SOC) by taking the

derivative of the FONC in (17):

105 13.5 13.5 27q1 −2 + 2 + 2 − 3 ≤ 0. (4.16) (q1 + q2) (q1 + q2) (q1 + q2)

√ ∗ 22+ 322 Given that q1 = q2 = q = 12 at the optimum, the above SOC is satisfied (with value -1.39). Thus q∗ is an optimal output quantity in this Nash equilibrium.

This implies that the aggregate production in the market is Q∗ = 2q∗ ≈ 6.66, meaning that the overall price in the market is P (Q∗) = $7.34. The profit earned by firms 1

∗ and 2 at this Nash equilibrium, denoted by π1 = π2 = π can be written as:

$13.5 π∗ = P (Q∗)q∗ − q∗ − 2 ∗ q∗ − 3 2q∗ = $7.34(3.33) − $6.75 − $2 ∗ (3.33) − 3

≈ $8.03

Therefore, after the introduction of the subsidy, the subsidized firm can cease production altogether and produce nothing, and the for firms 1 and 2 will result in: (i) lower aggregate output; (2) a higher market price; (3) higher profits for the non-subsidized firms; (4) lower consumer surplus; and (5) overall lower social welfare. These results are summarized in Table 4.1:

While the above example is certainly highly stylized, it demonstrates how the proposed subsidy may have perverse incentive effects. Given that the subsidized firm

106 Table 4.1: Welfare Comparison, F3=$10 Table 4.1 Welfare Comparison, F3=$10 Firm 3 Fixed Cost of $10 No Subsidy Subsidy Overall Output 9 6.66 Market Price $5 $7.34 Aggregate Firm Profits $11 $19.56 Consumer Surplus $40.50 $22.16 Social Welfare $51.50 $41.73

is guaranteed a profit level of L, there are a continuum of possible Nash equilibria.

However, what the above example shows is that this policy can generate what most policy makers, regulators, and consumers would agree is a “bad” outcome: namely, the non-subsidized firm could produce nothing and all firms end up with profits at least weakly higher (in the case of the subsidized firm) or definitely higher, in the case of the non-subsidized firms, with lower overall aggregate production.

The above analysis continues to hold in the case that the fixed cost of firm 3 is ˆ F3 = $6. In this case, since a smaller amount of revenue needs to be confiscated from the non-subsidized firms, the overall affect on welfare moves in the same direction. In this case, the profits of Firms 1 and 2 are unchanged from the unsubsidized market equilibrium scenario. However, instead of incurring a loss of −$1, now firm three earns positive economic profit of $3, because now its total revenues are enough to offset its fixed cost of production. However, we will demonstrate that the proposed subsidy essentially pays Firm 3 to idle economically viable generating capacity.

107 ˆ Assuming that Firm 3 now has a Fixed cost of F3 = $6, its Nash equilibrium quantity of output in the non-subsidized case remains 3 units. So again, overall market quantity in the non-subsidized case is 9 units, market price is $5, and firms 1 and 2 each earn profits of $6 while firm 3 earns profits of $3. Now, assume that the same subsidy policy is enacted which guarantees Firm 3 a profit level of $3.5. This means that a total of $9.5 must be confiscated from the non subsidized firms. We can now replicate the above analysis, starting with the profit maximization problem of Firm 1:

$9.5 maxq1 {(14 − (q1 + q2))q1 − · q1 − 2q1 − 3}. q1 + q2 .

By the aforementioned procedure, in this case the optimal production quantity,

q∗, will satisfy the following equation:

44q∗ − 12q∗2 − 9.5 = 0.

√ ∗ 22+ 370 In this case, the optimal production quantity for firms 1 and 2 is q = 12 ≈ 3.44. Therefore, aggregate market output is approximately 6.88, and market price is $7.13. Therefore, under the proposal when Firm 3 has fixed cost of $6, the profit earned by firms 1 and 2 is:

$9.5 π∗ = P (Q∗)q∗ − q∗ − 2 ∗ q∗ − 3 2q∗ = $7.13(3.44) − $4.75 − $2 ∗ (3.44) − 3

≈ $9.87

108 .

The effects of the proposed subsidy are certainly undesirable: an economically viable generating firm is paid to idle its capacity, and all firms end up with higher profits, a greater wholesale electricity price, fewer overall units produced, and lower consumer surplus. These results are summarized in Table 4.2:

Table 4.2: Welfare Comparison, F3=$6 Welfare Comparison, F3=$6 Firm 3 Fixed Cost of $6 No Subsidy Subsidy Overall Output 9 6.88 Market Price $5 $7.13 Aggregate Firm Profits $15 $19.56 Consumer Surplus $40.50 $23.61 Social Welfare $55.50 $46.85

4.5 Discussion and Concluding Remarks

In this analysis, we have demonstrated the incentive shortcomings of a proposed power purchase agreement. The proposed subsidy, purportedly designed to ensure the reliability and resilience of generating capacity, actually creates incentives that cause the existence of “undesirable” equilibria. The rationale underlying the Depart- ment of Energy’s proposed power purchase agreement is that in the absence of this subsidy, firms will exit the market and consumers will lose out on needed generating capacity. In fact, what the above analysis demonstrates is that this proposal may

109 counterintuitively induce equilibria with the exact opposite of the desired effect. We derived equilibrium behavior for the case of linear demand, and using a simple nu- merical example, we demonstrate how the proposed subsidy can result in inefficient

firms preferring to produce nothing and collect their guaranteed profit level via the subsidy, and economically viable firms choosing to idle their capacity and produce nothing when they would have been operated profitably in the absence of the subsidy.

The above model is highly stylized, but we believe it provides a useful foundation from which to examine these types of policy interventions in electricity markets.

Indeed, as the Federal Energy Regulatory Commission prepares to further analyze power purchase agreements in this vein, this model will provide a useful foundation for more complex analysis. There is much promising research to be done in this area, by exploring more general demand functions, introducing multiple periods, and explicitly considering reliability in the analysis.

110 Chapter 5: Contributions and Future Work

In this dissertation, we have explored several areas of relevance to the economics of market design and regulation, with a particular emphasis on wholesale electricity market design. The research in this dissertation lays the foundation for a variety of future research endeavors. Two possible directions for future research that build upon the work herein by making use of developments in the literature on scoring auctions are explored in this chapter.

5.1 Bidder Cost Revelation in Electric Power Auctions with Capacitated Suppliers

One of the ways in which the model of Chapter 1 is highly stylized is the assump- tion that all firms have identical capacities. Here, I am proposing to build upon the analysis of Chapter 2 by explicitly considering more general auction environments to study electricity market design. Scoring auctions have been a growing area of study in the economics literature. Che (1993) first studied an auction model inspired by the manner in which the Department of Defense procures weapons systems. Che studies a single buyer who wishes to evaluate proposals from a set of competing firms on the basis of the two dimensions of price and quality. Each firm privately holds information about its cost of completing the project. Che examines the strengths and

111 weaknesses of a variety of scoring rules to rank the two-dimensional bids and deter- mine the ultimate winner of the contract. Branco (1997) generalizes Che’s model to incorporate the possibility that there may be correlation in the firms costs of com- pleting the project. In both Che (1993) and Branco (1997), the private information held by firms is simply the cost of completing the project. When private information is single dimensional in this way, the analysis of the problem is relatively straight- forward, as standard auction theory techniques can be applied. However, when firms have private information across several different dimensions, the analysis becomes much more difficult. Bushnell and Oren (1994) consider a model based on electricity markets where the private information of firms consists of two types of costs: the start-up cost of activating a generator and the marginal cost associated with the pro- duction of each MWh of electricity. They derive a scoring rule that results in truthful revelation of the marginal costs. Chao and Wilson (2002) perform a similar exer- cise in the context of procurement auctions for power reserves. Asker and Cantillon

(2008) study a particular class of scoring auctions where the private information of the competing firms can include multiple dimensions beyond just cost. They derive a technique for using a scoring rule to reduce a complex multidimensional auction into a single-dimensional setting, allowing for commonly known results from standard auc- tion theory to be applied to this more general setting. However, they rely on several fairly restrictive assumptions that prevent generalizing their results much beyond this setting. One weakness of Che (1993), Branco (1998), and Asker and Cantillon (2008) is that all three deal with auctions where only a single winner is chosen. However, in many procurement environments, particularly wholesale electricity, capacity con- straints require that more than one firm be chosen to service demand. This represents

112 an important gap in the literature on scoring auctions that this research aims to help

fill. Iyengar and Kumar (2008) study a setting where capacitated suppliers compete to provide a homogenous good and derive the optimal (revenue-maximizing) procure- ment mechanism. A crucial assumption in their analysis is that firms are unable to bid above capacity. However, the firms studied in their model have only marginal costs of production. The techniques that are utilized by Iyengar and Kumar (2008) may prove useful in extending the analysis of Bushnell and Oren (1994) to generalize the study of cost revelation in electric power auctions to include capacitated suppliers.

5.2 Incentive Effects of Environmental Adders in Multi-Dimensional Procurement Auctions for Power Reserves

In this work, I am proposing to explore the incentive effects of environmental adders in multi-dimensional procurement auctions for power reserves. This work will represent a step towards understanding the role of environmental adders in procure- ment auctions for reserve capacity, by building upon the analyses of Bushnell and

Oren (1994) and Chao and Wilson (2002). In particular, I am proposing to use the principle of incentive compatibility to examine settlement and scoring rules in the context of procuring reserve capacity when carbon emissions must be taken into ac- count. I believe that results pertaining to environmental adders in the context of auctions for non-utility generation may be extended to the study of reserve capac- ity markets which may then have policy relevance when deciding how to introduce new capacity into electricity markets. There has been a growing literature on the study of multidimensional scoring auctions in economics. In these auctions, suppliers

113 compete via the submission of complex multidimensional bids which may contain in- formation regarding capacities, start-up costs, time-to-completion, quality, etc. The complex bids are then ranked on the basis of an aggregate score to determine win- ning bidders. Che (1993) studies several competing scoring auction formats inspired by weapons system procurements conducted by the Department of Defense. Branco

(1997) considers the design of multidimensional auctions with correlation among the costs of the competing firms. Asker and Cantillon (2008) study scoring auctions when the competing firms have multidimensional private information regarding the types of contracts they can provide. In the context of electricity markets, Bushnell and

Oren (1994) analyze efficient selection and operation of generation sources when bid- ders compete over both fixed and variable costs. Bushnell and Oren (1994) consider the incorporation of environmental adders into auctions for non-utility generation.

Chao and Wilson (2002) derive simple scoring and settlement rules that incentize bidders to truthfully reveal their private information. This study will provide two main contributions. By studying the role of environmental adders in auctions for reserve capacity, it will contribute to the literature on scoring auctions, power system economics, and energy economics. This study will also have practical policy rele- vance in that it will aid our understanding of how the introduction of environmental adders, taking into account carbon emissions, may impact bidder cost revelation in auctions for power reserves. This research may potentially provide insights useful for new capacity investment decisions.

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