Implementing Lindahl Allocation - Incorporating Experimental Observations into Mechanism Design Theory
Item Type text; Electronic Dissertation
Authors Van Essen, Matthew J.
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/195026 Implementing Lindahl Allocations —
Incorporating Experimental Observations into
Mechanism Design Theory
by
Matthew J. Van Essen
Copyright c Matthew J. Van Essen 2010 ° A Dissertation Submitted to the Faculty of the
Department of Economics
In Partial Fulfillment of the Requirements
For the Degree of
Doctor of Philosophy
In the Graduate College
The University of Arizona
2010 2
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation
prepared by Matthew J. Van Essen entitled “Implementing Lindahl Allocations –
Incorporating Experimental Evidence into Mechanism Design Theory”
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of Doctor of Philosophy
______Date: 04/30/2010 John Wooders
______Date: 04/30/2010 Mark Walker
______Date: 04/30/2010 Martin Dufwenberg
______Date: 04/30/2010 Rabah Amir
Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.
______Date: 04/30/2010 Co-Dissertation Supervisor: John Wooders
______Date: 04/30/2010 Co-Dissertation Supervisor: Mark Walker 3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the copyright holder.
Matthew J. Van Essen
SIGNED:
4
ACKNOWLEDGMENTS
It is difficult to find someone to read a paper about Lindahl mechanisms without putting up a fight (let alone five papers). I am therefore very grateful to my primary advisors John Wooders and Mark Walker for not only reading every word of every poorly written paper, but for their constructive comments, their career advice, and their constant pushing and the general discussions which has made me a better researcher. I benefited immensely from their collective expertise in game theory and experimental economics. These papers also benefited from comments made by Rabah Amir and Martin Dufwenberg. Many of the insights made in this dissertation were learned while taking Rabah’s game theory course and Martin’s behavioral game theory course. Their doors were always open to me. For their attention, discussion, and comments throughout the program I owe a tremendous amount of gratitude to my friends Natalia Lazzati, Jenny Hawkins, Alec Smith, and members of “The Brain Trust.” For teaching me where my academic interests in economics lie, I thank John Fox (economic history) and Dina Shatanawi (labor). Finally, and most importantly, I thank my parents and my fiancé Tiffiny Guidry for their steadfast support throughout graduate school. 5
DEDICATION
This dissertation is dedicated to my Dad who taught me the importance of strategic thinking in grueling games of Risk®; and to my Mom who taught me the more important strategic lesson...never takeover your spouse’s continent.
6
Ta b l e o f Co n t e n t s
List of Figures ...... 10
List of Tables ...... 13
Abstract ...... 15
Chapter 1. Lindahl Equilibrium: A Brief Overview ...... 17
1.1. Introduction ...... 17
1.2.APublicGoodEconomy...... 19
1.2.1.ParetoOptimalConditions...... 20
1.3.LindahlEquilibrium...... 25
1.3.1.Lindahl’sCostSharingScheme(1919):AnExample...... 25
1.3.2.AnAlternativeFormulation...... 28
1.3.3.Samuelson’sCritique...... 31
1.4.IncentiveCompatibleLindahlMechanisms...... 38
1.4.1.TheWalkerMechanism...... 39
1.5.Discussion...... 43 7
Table of Contents–Continued
Chapter 2. Out-of-Equilibrium Performance of Three Lindahl
Mechanisms1 ...... 45
2.1. Introduction ...... 45
2.2.TheoreticalPreliminaries...... 48
2.2.1.PublicGoodsandtheLindahlOutcome...... 48
2.2.2.TheMechanisms...... 50
2.3. The Experiment ...... 52
2.4. Convergence ...... 57
2.4.1.ConvergenceofParticipants’Requests...... 57
2.4.2.PublicGoodProvisionLevel...... 59
2.4.3. Lindahl Taxes ...... 61
2.5.WelfareandBudgetComparisons...... 62
2.5.1.WelfarefromProvisionofthePublicGood...... 62
2.5.2.ViolationsofIndividualRationality...... 64
2.5.3.UnbalancedBudgets...... 65
2.5.4.TheRoleofParameterValues...... 67
2.6.Conclusion...... 68
Chapter 3. A Simple Supermodular Mechanism that Implements
Lindahl Allocations ...... 80 1 This is joint work with Natalia Lazzati and Mark Walker. 8
Table of Contents–Continued
3.1. Introduction ...... 80
3.2. Preliminaries ...... 85
3.3.ThePublicGoodEconomy...... 87
3.3.1.TheMechanism...... 88
3.3.2.PreferenceandWealthAssumptions...... 91
3.4. Implementation ...... 93
3.4.1.ImplementationinQuasi-LinearEnvironments...... 98
3.5.Out-of-EquilibriumTaxPenaltiesoftheNewMechanism...... 104
3.6.Conclusion...... 107
Chapter 4. A Note on Chen’s Lindahl Mechanism ...... 109
4.1. Introduction ...... 109
4.2.StabilityandtheChenMechanism...... 110
4.3.Conclusion...... 117
Chapter 5. Information Complexity, Punishment, and Stability
in Two Nash Efficient Lindahl Mechanisms ...... 118
5.1. Introduction ...... 118
5.2.TheoreticalPreliminaries...... 124
5.2.1.TheBasicPublicGoodsProblem...... 124
5.2.2.IncentiveCompatibleLindahlMechanisms...... 125 9
Table of Contents–Continued
5.2.3. Stability of the Myopic Best Reply Learning Algorithm in the
CHandPSmechanisms...... 128
5.3. Research Hypotheses...... 130
5.3.1.HypothesesbasedonStabilityParameters...... 131
5.3.2.HypothesesbasedonMechanismComparison...... 136
5.4.ExperimentalDesignandEnvironment...... 141
5.5.ExperimentalProcedures...... 142
5.6.ExperimentalResults...... 146
5.6.1.HypothesesbasedonStabilityParameters...... 146
5.6.2.HypothesesBasedonMechanismComparison...... 156
5.7.Conclusion...... 164
Appendix A. Stability of Chen, Kim, and Walker Mechanisms .. 198
A.1.TheWalkermechanism...... 198
A.2.TheKimmechanism...... 199
A.3. The Chen mechanism...... 200
Appendix B. Proofs: A Simple Supermodular Mechanism that Im-
plements Lindahl Allocations ...... 202
References ...... 219 10
List of Figures
Figure 1.1.VerticalSummationofDemand...... 24
Figure 1.2.Lindahl’sCrossingDemandsFigure...... 27
Figure 2.1.VLWExperimentParameters...... 54
Figure 2.2. Deviation of Requests from Equilibrium ...... 70
Figure 2.3.AveragePublicGoodProvision(ChenandKim)...... 71
Figure 2.4.AveragePublicGoodProvision(Walker)...... 72
Figure 2.5. Public Good Deviation from Equilibrium (Chen and Kim) . . . 73
Figure 2.6.PublicGoodDeviationfromEquilibrium(Walker)...... 74
Figure 2.7.AverageSurplus(ChenandKim)...... 75
Figure 2.8. Deviation from Lindahl Tax Profile...... 76
Figure 2.9.AverageSurplus(Walker)...... 77
Figure 2.10. Percentage of Participants with Violations of Individual Ratio-
nality...... 78
Figure 2.11.AverageBudgetImbalance...... 79
Figure 3.1.Consumer1’sBestResponseProblem...... 96
Figure 3.2.Consumer2’sBestResponseProblem...... 97
Figure 5.1.DivergenceofInformationRequirement...... 139 11
List of Figures–Continued
Figure 5.2.AveragePublicGoodProvision(PS2,SPMvs.NSPM)..... 178
Figure 5.3. Average Public Good Provision (CH2, SPM vs. NSPM) . . . . 179
Figure 5.4.AveragePublicGoodProvision(PS6,SPMvs.NSPM)..... 180
Figure 5.5. Average Public Good Provision (CH6, SPM vs. NSPM) . . . . 181
Figure 5.6. Average Absolute Deviation from Equilibrium (PS2, SPM vs.
NSPM)...... 182
Figure 5.7. Average Absolute Deviation from Equilibrium (CH2, SPM vs.
NSPM)...... 183
Figure 5.8. Average Absolute Deviation from Equilibrium (PS6, SPM vs.
NSPM)...... 184
Figure 5.9. Average Absolute Deviation from Equilibrium (CH6, SPM vs.
NSPM)...... 185
Figure 5.10. Efficiency(PS2,SPMvs.NSPM)...... 186
Figure 5.11. Efficiency(CH2,SPMvs.NSPM)...... 187
Figure 5.12. Efficiency(PS6,SPMvs.NSPM)...... 188
Figure 5.13. Efficiency(CH6,SPMvs.NSPM)...... 189
Figure 5.14.Average#ofIRViolations(PS2,SPMvs.NSPM)...... 190
Figure 5.15.Average#ofIRViolations(CH2,SPMvs.NSPM)...... 191
Figure 5.16.Average#ofIRViolations(PS6,SPMvs.NSPM)...... 192
Figure 5.17.Average#ofIRViolations(CH6,SPMvs.NSPM)...... 193 12
List of Figures–Continued
Figure 5.18. Efficiency(SPM,PS2vs.CH2)...... 194
Figure 5.19. Efficiency(NSPM,PS2vs.CH2)...... 195
Figure 5.20. Efficiency(SPM,PS6vs.CH6)...... 196
Figure 5.21. Efficiency(NSPM,PS6vs.CH6)...... 197 13
List of Tables
Table 2.1. VLW Tables 1-7 ...... 69
Table 5.1.ExperimentParametersandTreatmentSummary...... 143
Table 5.2. Session Summary ...... 144
Table 5.3.ClosenesstoEquilibriumMessages(SPMvs.NSPM)...... 150
Table 5.4. Efficiency(SPMvs.NSPM)...... 153
Table 5.5.ViolationsofIR(SPMvs.NSPM)...... 155
Table 5.6.ClosenesstoEquilibriumMessages(PSvs.CH)...... 158
Table 5.7. Efficiency(PSvs.CH)...... 159
Table 5.8.ViolationsofIR(PSvs.CH)...... 161
Table 5.9.StandardDeviation(PSvs.CH)...... 163
Table 5.10. Average (10 Round) Public Good Provision (2 Player Groups) . 166
Table 5.11. Average (10 Round) Public Good Provision (6 Player Groups) . 167
Table 5.12. Average (10 Round) Standard Deviation of Public Good Provision
(2 Player Groups) ...... 168
Table 5.13. Average (10 Round) Standard Deviation of Public Good Provision
(6 Player Groups) ...... 169
Table 5.14. Average (10 Round) Total Tax Burden (2 Player Groups) . . . . 170 14
List of Tables–Continued
Table 5.15. Average (10 Round) Total Tax Burden (6 Player Groups) . . . . 171
Table 5.16. Average (10 Round) Efficiency(2PlayerGroups)...... 172
Table 5.17. Average (10 Round) Efficiency(6PlayerGroups)...... 173
Table 5.18. Average (10 Round) Absolute Deviation from Equilibrium Strat-
egy (2 Player Groups) ...... 174
Table 5.19. Average (10 Round) Absolute Deviation from Equilibrium Strat-
egy (6 Player Groups) ...... 175
Table 5.20. Average (10 Round) # of Violations of Individual Rationality (2
PlayerGroups)...... 176
Table 5.21. Average (10 Round) # of Violations of Individual Rationality . . 177 15
Abstract
Mechanism design theory has given economists a set of tools for designing institutions to achieve socially desirable outcomes. Unfortunately, the behavioral assumptions that these theories often rest are somewhat unrealistic. Testing these institutions in a laboratory setting gives us insight into what assumptions or properties of institu- tions make them behaviorally successful. Moreover these insights allow us to create new theories that offer, in principle, better actual performance. Thus, the interplay between experimental economics and economic theory seems vital in mechanism de- sign to insure successful institutions. It is in this spirit that this dissertation precedes focusing entirely with mechanisms that were designed to achieve the Lindahl alloca- tion in a public goods environment. The first chapter experimentally examines three such mechanisms in a laboratory setting. It finds that the mechanism that gets the closest to the Lindahl allocation is the one that induces a game with very strong stability of equilibrium properties. Unfortunately this mechanism also has some clear disadvantages: first, it is very complicated; second, payoffs to consumers while learn- ing to play equilibrium are very low; and last, the mechanism gets more complicated when more people participate. The second chapter uses the insights from the first experiment to create a new institution which avoids some of the concerns outlined above while maintaining the strong stability of equilibrium property. The third chap- 16 ter contributes a missing stability result into the literature. The final chapter of the dissertation experimentally compares the new mechanism introduced in chapter 2 with the most successful mechanism from the first experiment. The treatments in this experiment are designed to stress the above observed trouble areas. 17
Chapter 1
Lindahl Equilibrium: A Brief Overview
1.1 Introduction
This dissertation deals with mechanisms that were designed to implement the Lindahl allocations in a public goods environment. A simple example of a public good is the amount of insecticide sprayed in a neighborhood. All people in the neighborhood enjoy the benefits of the spray, and it is not possible to exclude one neighbor from enjoying the benefitsofthespraybecausehedidnothelppayforit.Thus,this and similar public good institutions are usually, but not always, plagued by under provision of the public good due the failure of consumers to appreciate the benefits that spillover to others when they purchase units of the good. When viewed as an externality, the classical Pigouvian solution becomes clear. Charge different individ- uals different prices depending on their marginal valuation schedule. If agent’s take into account all the benefits and costs of their purchasing decisions, they tend to make efficient choices. In this situation the optimal solution is to charge a price, or a
Lindahl tax, according to each individual’s marginal benefit(evaluatedattheoptimal level of the public good). 18
While elegant in principle, Lindahl taxation is often criticized for its unrealistic requirements about information. Perhaps most glaring of these requirements is that efficient taxes require knowledge of individual preferences. This seems problematic when, knowing that revealing their preferences will influence how much tax they will pay, individuals have strong incentives to keep the government from learning these preferences. Incentive schemes can, however, be designed to combat this problem and allocate Pareto optimal allocations. Unfortunately, the set of Pareto efficient allocations in a public good economy can be quite large and contains allocations that are not always better than consumer’s initial holdings. The Lindahl allocation is a specific Pareto optimal allocation that has the additional property that no consumer is ever worse-off after the receiving the allocation. Thus, a Lindahl allocation is a very desirable allocation for economists to try and implement with mechanisms. We will only be concerned with mechanisms designed to achieve this specificParetooptimal allocation.
Since this dissertation is intimately connected with the notion of a Lindahl equi- librium and its various properties the first chapter is devoted to developing some key concepts. we first characterize the set of Pareto optimal allocations in a simple gen- eral equilibrium environment. Second, through an example, we introduce Lindahl’s original 1919 formulation of the concept and, subsequently, expand our example to a more general and familiar Walrasian context. Third, after a discussion of the wel- fare properties of equilibrium, we look at another example to illustrate the primary 19 incentive critique of Lindahl equilibria due to Samuelson. Forth, we discuss potential answers to Samuelson’s critique by first examining the consequences of repeated in- teraction and by introducing the incentive compatible Lindahl mechanism literature.
Finally, we preview the remainder of the dissertation.
1.2 A Public Good Economy
The basic public good problem can be described as follows. There are 3 agents in an economy (2 consumers indexed = and 1 producer). Each of the consumers are endowed with some amount of the private good 0, and have preferences over two goods and ,where is a public good common to all players and is a private good. Each consumer has preferences that are represented by a twice continuously differentiable payoff function that is strictly increasing in the private good as well as strictly quasi-concave. The producer can use the private good to make units of a public good according to the constant returns to scale production
function = ()= ,where is the real marginal cost of production — i.e., the amount of the private good given up to produce 1 unit .1
1 Since the production technology exhibits CRS, in a competitive equilibrium the firm’s profitis always zero so there is no need to account for additional income in the feasibility constraints in the
Walrasian problem introduced later. 20
1.2.1 Pareto Optimal Conditions
In a public good economy the consumption of any non-rival/non-excludable good by one consumer does not prevent another consumer from enjoying that same good. This fact implies that the characterization of Pareto efficiency might appear somewhat different than in the standard private good setting. It is therefore useful to start this survey by deriving the marginal conditions for efficiency in order to establish a baseline knowledge.
The Pareto optimal allocations are the solutions to the following maximization problem
max ( ) ()
Subject to the following constraints:
0 (1.1) ≥
1 + + − () + (1.2) ≤
( ). (1.3) ≤
Using these constraints we set up the constrained maximization problem for this simplified economy.2 The Lagrangian for this problem is
1 ( yλ)=( ) [ + + − () ] [ ( )], − − − − − where and are the Lagrangian multipliers. Applying Kuhn-Tucker the first order
2 Note that is just some positive scalar used for convenience. 21 marginal conditions for Pareto efficiency are:
1 1 + 0(with = if 0) − 0 ( − ()) ≤ 0(with = if 0). − ≤
The interior conditions are first that there is no waste in the private good market
— i.e.,
+ = + ; − and second, that the public good be produced until the marginal social benefitequals the marginal cost of production — i.e.,
1 + = 0() ≡
+ = .
Equation (4) is often called the “Samuelson Marginal Condition” since Samuelson was the first to rigorously define the requirements for Pareto efficiency in his seminal
1954 paper on public goods.
Determining the Pareto efficient conditions in an economy is a good starting place to conduct welfare analysis. However, it has a number of potential shortcomings.
First, the set of Pareto allocations is too big to have much predictive power. Second, some Pareto allocations may be good for one person, but may leave the others worse off than the initial endowment. The first example demonstrates both criticisms in a simple two consumer economy with a constant returns to scale production technology motivating the appeal of Lindahl equilibria. 22
Example 1. Consider the following economy composed of Ann and Bob. Each has preferences for two goods—a private good () and a public good ().Furthermore assume these preferences are represented, respectively, by utility functions
1 2 ( )= (6 ) − 2 −
1 2 ( )= (8 ) − 2 −
Initially, both consumers have no units of the public good and 20 units of the private good. The government has the capability of producing the public good using the private
1 good as an input according to production technology ()= 4 ,where4 is the real marginal cost of production.
1. What are the Pareto efficient allocations for this economy?
2. Is there a Pareto allocation that makes one of the consumers worse off (relative
to the initial endowment)?
The first question is simply computational problem solved using the Samuelson
Marginal condition. Ann’s marginal rate of substitution is =6 ,andBob’s − marginal rate of substitution is =8 Taking these equations and − applying the Samuelson Marginal Conditions we have that
+ =4
14 2 =4 − =5. 23
Sinceittakes4 units of the private good to produce each unit of the public good, the total cost of producing the Pareto optimal amount of the public good is 20.Thus,
Pareto allocations are the ones satisfying
=5
+ =20.
From the above conditions we see that there are a continuum of Pareto efficient allocations for this simple two consumer economy. As a comparison let us calculate each individual’s utility from the initial endowment ( )=(0 20).ForAnn,we have that (0 20) = 2 andforBobwehave(0 20) = 4. Now consider the − allocation ( )=(5 1 19). Clearly this allocation is Pareto optimal since it satisfies the above restrictions. However, Ann is bearing almost the entire cost of production (she commits 19 units of the private good to production while Bob only
1 commits 1 unit to production). Her utility at this new allocation is (5 1) = 2
(0 20) = 2. So, while the allocation is Pareto optimal it leaves Ann worse off than she was at her initial endowment — i.e., this allocation is not in the “core” of this economy.
In some sense, Pareto allocations that leave individuals pining for their initial endowments are unsatisfactory. In looking for a normative benchmark allocation we would like one that is both individually rational (i.e., consumers are at least as well of when compared to their initial endowment) and efficient. In the private good case, 24
Figure 1.1. Vertical Summation of Demand the Walrasian equilibrium allocation satisfies both of these requirements. Ideally, we would like to find a similar “price taking” allocation for a public good economy. It turns out that Lindahl equilibria are precisely the Walrasian equilibria of a public good economy. This analogy is probably best seen from Bowen’s 1943 diagram illustrating the well known “vertical summation” of demand curves and the efficient provision of a public good.3 The social optimum is found where social value equals marginal cost
(or = ).
Note,P that if Consumer 1 is charged a price 1 and Consumer 2 is charged a price of 2 they will each demand the same Pareto optimal quantity of the public good.
Moreover, each unit produced collects 1+ 2 as revenue, which is precisely the
3 Bowen’s 1943 paper is a remarkable contribution to the theory of public goods and voting.
Apparently, unaware of Lindahl’s paper (which wasn’t translated for a few more years), he intro- duces the well known vertical summation diagram, a procedure for distributing the cost of a public good (i.e., the Lindahl procedure), and pre-sages the information and incentive problems of Lindahl taxation! 25 revenue needed to cover the cost . Thus, the allocation is Pareto optimal. ∗ Furthermore, since, at the prices these consumers face, the public good demanded is on their demand curve, the allocation must be individually rational. It turns out that this natural pricing mechanism yields an allocation which we will dub the Lindahl allocation. The next section, presents the Lindahl scheme under a slightly different guise.
1.3 Lindahl Equilibrium
1.3.1 Lindahl’s Cost Sharing Scheme (1919): An Example
As an introduction to Lindahl’s scheme, let us return to the earlier example of Ann and Bob. Suppose we inform both consumers that they will pay a fraction of the marginal cost. Let and denote the percentage of the production cost that Ann and Bob must pay, where , 0 and + =1. Thiscostshareactslikea ≥ price and it is straightforward to determine each consumer’s demand as a function of their own or the other’s cost share. Note that by referring to individual demand behavior we are implicitly requiring individually rational behavior by each consumer.
In the example, Ann’s demand for the public good as a function of her per-unit 26 cost share is
=6 −
=6 (4). −
Similarly, Bob’s demand for the public good as a function of his per-unit cost share is
=8 4. −
Since + =1we can rewrite Bob’s demand as a function of Ann’s cost share variable,
=8 4(1 ). − −
Since a public good is by definition non-exludable and non-rival, an optimal allocation will require both consumers to demand the same quantity of the public good. We now solve for a cost sharing arrangement such that both Ann and Bob individually demand the same amount of the public good.
=
6 4 =8 4(1 ) − − − 1 3 = and = 4 4 =5. ⇒
The figure below plots Ann and Bob’s demand for the public good as a function of
Ann’s cost share. 27
x 10
9 Bob's Demand
8
7
6
5
4 Ann's Demand 3
2
1
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Cost Share (Ann)
Figure 1.2. Lindahl’s Crossing Demands Figure
There are a couple of important things to notice about this example. First, the
Lindahl allocation is Pareto optimal. Recall that the allocation computed for this example is
=5 1 =20 ( )(4)(5) = 15 − 4 3 =20 ( )(4)(5) = 5. − 4
It is simple to see that this allocation satisfies the equations derived from Example 1 and is thus Pareto optimal. Second, at this allocation both Ann and Bob are better off than the initial endowment since
29 (0 20) = 2 = (5 15) 2 1 (0 20) = 12 = (5 15). − 2
Thus, the Lindahl allocation is in the core for this example. It should be pointed out that Lindahl did not really motivate his scheme as a solution to the free riding 28 problem, but rather sought to provide an algorithm for finding a Pareto optimal allocation in a public good setting when preferences were known to the participants.
This is an important distinction as we will find out when we discuss Samuelson’s critique of the Lindahl concept. However, the Lindahl concept has considerable merit on its own. In particular, this allocation has become the standard for efficiency in public good economies. In the next section, we provide a modern presentation of the
Lindahl equilibrium which more closely resembles the standard Walrasian set-up.
1.3.2 An Alternative Formulation
Lindahl’s proposal can be re-defined to look like a standard Walrasian setting, i.e., a setting where consumers and producers take prices as given and choose demand and supply quantities respectively. A public economy equilibrium, or Lindahl equilibrium,
is a then a set of prices ( ) andanallocation((), ()) such that: both consumers and producers, taking the prices as given, are each maximizing their respective objective functions subject to appropriate constraints; and their choices clear the market. In the public economy, for the market to clear we need: first, that consumers each demand the same amount of the public good; second, this amount is the profit maximizing quantity for the producer; finally, there is no waste of the 29 private good. These conditions are represented by the following equations:
= = ,
1 + = + − (). −
Given prices, the agents in the economy solve their respective maximization prob- lems. Consumers choose bundles of goods to maximize their utility subject their budget constraint, i.e.,
max () s.t. + · ≤
The producer of the public good chooses in order to maximize profits, i.e.,
1 max ()=( + ) − (). · −
Taking these maximization problems we can solve for the standard first order condi- tions. The Kuhn-Tucker marginal condition for Consumer are
(with = if 0) ≤
(with = if 0). ≤
Therefore, if both goods are positive the first order condition requires = .
Consumer solves the symmetrical problem resulting in the condition = .
The first order condition for the producer is + = (i.e., = ). Putting 30
the conditions together imply that in any equilibrium we must have + =
(i.e., the Samuelson marginal condition must hold). The last requirement is for the market to clear. Thus, from the two conditions, any Lindahl equilibrium is Pareto optimal.4
Other welfare results follow from the Lindahl equilibrium. First, the Lindahl equi- librium is always in the core for public good economies (see Foley (1970)). Second,
Lindahl equilibria are associated with two fundamental welfare theorems for public good economies akin to the ones from a private good economy. These two theorems show that: (i) Lindahl equilibrium are Pareto optimal; and (ii) any Pareto opti- mal allocation can be achieved as a Lindahl equilibrium after some redistribution of income. 4 Although this formulation may seem quite different from Lindahl’s original cost sharing scheme, it is always possible to rewrite Lindahl prices as we have definedtheminthissectionintermsof cost shares. This is since the first order condition requires
+ = .
If we divide both sides by the marginal cost, then we have
+ =1,
where represents Player ’s cost share and represents Player ’s cost share . 31
1.3.3 Samuelson’s Critique
Despite the appeal of the Lindahl equilibrium from a welfare point of view, the out- come as the result of a decentralized market process may not be realistic. In fact, if we price according to individual benefit, then there is, essentially, a market for each individual. Thus, due to individual’s “market” power, in this situation, it may not be reasonable to assume that consumers act as price takers. This was the point of
Samuelson in his famous 1954 article, one such point about the Lindahl solution is given below.
...now it is in the selfish interest of each person to give false signals, to
pretend to have less interest in a given collective consumption activity
than he really has, etc. I must emphasize this: taxing according to a
benefit theory of taxation can not solve the computational problem in a
decentralized manner possible for the first category of “private” goods to
which the ordinary market pricing applies and which do not have “external
effects” basic to the notion of collective consumption goods.
Samuelson’s critique appears to damn the possibility that a decentralized mecha- nism could be used to make Pareto efficient allocation decisions. If asked to reveal this information, consumers typically have an incentive to misrepresent their preferences, hoping to free ride on other participants’ payments. 32
A Strategic Analysis of Benefit Taxation To illustrate Samuelson’s critique of benefit taxation (i.e., the Lindahl scheme) let us consider the following simple example using a direct mechanism implicit in the Lindahl pricing scheme.
Example 2. Consider the following economy composed of Ann and Bob. Each has preferences for two goods—a private good () and a public good ().Furthermore assume these preferences are represented, respectively, by utility functions
1 2 ( )= ( ) − 2 −
1 2 ( )= ( ) , − 2 − where , R++ are Ann and Bob’s preference parameters and satisfy the follow- ∈ ing inequality 1 3 3
Initially, both consumers have no units of the public good and 20 units of the private good. The government has the capability of producing the public good at a marginal cost of zero.
Assume the two agent have knowledge of each others preferences while the gov- ernmentdoesnotknowthepreferenceparameters(althoughknowstheyaregreater than zero) and must commit to an allocation mechanism. Specifically, the government chooses the direct mechanism where each agent reports their preference parameter and he determines the reported Lindahl allocation. 33
A Direct Lindahl Mechanism
˜ In this mechanism, each agent will announce a strictly positive report R++ to ∈ the government who will produce the reported socially efficient level of public good calculated as follows: ˜ + ˜ ˜(˜ ˜ )= , 2 and charge each agent their associated “reported” Lindahl price
˜ ˜ ˜ ˜ ˜ ˜ ˜ + ( )= = − − − Ã 2 ! 2 per unit.
˜ ˜ ˜ ˜ Note that ( )+( )=0for all reports and that the “truth telling” outcome of this mechanism by construction is the Lindahl allocation — i.e.,
+ ( )= 2 andeachunitpays
= − − 2 per unit.
Analysis of the “Lindahl Game” (Single Shot)
We now look at the Nash equilibrium of “Lindahl Game” induced by this mecha-
˜ nism starting with Ann’s best response problem. Suppose Bob reports , then Ann’s 34
˜ best response is to choose a report to the government to maximize:
˜ ˜ ˜ ˜ ˜ ˜ max 20 ( ) ˜ ( )+ (˜ ( )). ˜ − ·
The first order condition for this problem reduces to the following economic state- ment:
˜ ˜ ˜ ˜ = ( )+˜ ( ).
˜ ˜ ˜ ˜ We can make similar calculations for Bob. Clearly, since ( )+( )=0
˜ ˜ and ˜ ( ) 0,wehave 0. Thus, the NE allocation always under provides the public good relativeP to the social optimum. For illustration, we now compute the Nash equilibrium of the game.
˜ ˜ 2 1 ˜ Ann’s best response function is ()= and, by symmetry, Bob’s best 3 − 3 ˜ ˜ 2 1 ˜ response is ()= . The Nash equilibrium is the intersection of these 3 − 3 ˜ ˜ ˜ ˜ two functions. Plugging () into () we have
˜ 2 1 2 1˜ = ( ), 3 − 3 3 − 3
˜ 3 1 ˜ 3 1 5 which reduces to = .Inasimilarmanner, = . The unique 4 − 4 4 − 4 Nash equilibrium of this game is
˜ ˜ 3 1 3 1 ( )=( ). 4 − 4 4 − 4 5 The best reply mapping of the best response functions is a contraction mapping. So, there is a unique Nash equilibrium which is stable. 35
Inputting these strategies into the allocation rules of the mechanism gives us
+ + = = , 4 2 which is sub-optimal, while the Lindahl prices in the Nash equilibrium are actually the correct ones— since
˜ ˜ 3 1 3 1 ( 4 4 ) ( 4 4 )) ˜ = − = − − − 2 2
= − = . 2
So in the NE allocation, the consumer faces the same price as in the Lindahl equilibrium, but receives half of the output implying that the profit received by a consumer in the equilibrium is smaller than that individual would receive in the
Lindahl scheme.
Our result of the “one-shot Lindahl Game” illustrates Samuelson’s critique, but it should be pointed out that it does not mean that there is no such institution where agents acting in their own incentives arrive at a Lindahl equilibrium. In fact, the same direct institution analyzed in this section can be used to support a Lindahl
Allocation as a Nash equilibrium of a repeated game so long as agents care enough about the future. The next section demonstrates this observation.
Analysis of the “Infinitely Repeated Lindahl Game” For the parameters used in our example, the game induced by the direct Lindahl mechanism shares many similarities 36 with the standard Cournot duopoly problem. In particular, we can support the social optimum
Specifically, let:
denote the profitto if both players follow a truth telling strategy; •
denote the profitto if both players follow their Nash equilibrium strate- • gies;
denote the payoff to when he reported truthfully, but the other agent • chooses his best response to truth telling;
denote the payoff to when he best responds to truth telling, but the other • agent continues to report truthfully.
1 It is straight forward to show that since 3 3,wehave
. Now, if we allow this “Lindahl game” to be repeated an infinite number of times, it is possible to support the Lindahl allocation as a Nash equilibrium allocation
(solongasagent’scaresufficiently about the future). Our game analysis starts by defining strategies for the infinitely repeated game.
A strategy for player ,inaninfinitely repeated game, is a rule that assigns an action in each period as a function of the past history. A grim trigger strategy in this setting is a very simple rule that has player choose the truth telling action every period so long as her opponent has never an played an action different from his truth 37 telling action. If she observes something different than truth telling, the strategy calls her to play her Nash equilibrium strategy thereafter. Formally, in each period ,the grim trigger specifies the following action
ˆ ˆ ,if = for all − − = ⎧ . ⎪ ⎨⎪ , otherwise We now show that both⎪ players following a grim trigger strategy is a Nash equi- ⎩⎪ librium of the infinitely repeated Lindahl game.
Suppose the other player follows the grim trigger strategy. The payoff stream for choosing to play a grim trigger strategy as well yields
2 + + + = , ··· 1 − where 0 1 istherateinwhichplayer discounts future payoffs. In a similar manner, deviating in any period from the grim trigger strategy, at best, yields a payoff stream of 2 + + + = + . ··· 1 − The grim trigger is a therefore a best response to a grim trigger strategy if ≥ (1 ) + . − Since , we need sufficiently close to 1 for this to occur. Specifically, we need that 1 − 0, − 38
− a condition which is possible since 1 0 is always true. Thus, even in − a mechanism where agents’ reports directly influence their own payments it is not hopeless to expect Lindahl allocations be obtained.
In the next section, we introduce a branch of research that shows that incentive compatible Lindahl mechanisms exist even in a one-shot setting. Thus, leaving us with the impression that Samuelson’s critique may only apply to a limited set of public good mechanisms.
1.4 Incentive Compatible Lindahl Mechanisms
The previous section proposed a simple direct mechanism with the property that truth telling corresponded to a Lindahl allocation. Our analysis of the one-shot version of this mechanism, as postulated by Samuelson, showed that agents have a strict incentive to mis-report their preferences resulting in a sub-optimal outcome.
This is not the case for all mechanisms. In a pioneering paper, Groves & Ledyard
(1977) proposed a mechanism whose Nash equilibria are Pareto optimal, but in which participants’ shares of the cost are not the Lindahl shares. Hurwicz (1979) and Walker
(1981) are the first to adapt the Groves-Ledyard idea to create “Lindahl mechanisms”
— mechanisms whose Nash equilibria are Lindahl allocations. Since the Hurwicz and
Walker articles there have been several alternative Lindahl mechanisms proposed, the Walker mechanism is perhaps the simplest and clearly illustrates how this class 39 of mechanisms work.
In this section, we examine the Walker mechanism as a simple introduction into incentive compatible Lindahl mechanisms. We then take insights from Walker to introduce some of the issues related to mechanism design theory that will be explored in the remainder of this dissertation.
1.4.1 The Walker Mechanism
A mechanism is a set rules that specify: (1) what choices, or messages, economic agents get to send to a planner; and (2) how those choices get mapped into outcomes.
A mechanism is not a game, but rather a game form. When we apply individual preferences to the outcomes of the mechanism, we have a game. A mechanism, therefore, induces a game. A public good mechanism needs to map messages into an allocation— i.e., a level of the public good and the amount each players private good consumption is reduced by a tax .
In the Walker mechanism each player announces a request .Themechanism sums the players’ messages to determine the level at which the public good will be provided:
(m)= . =1 X Denote by m the profile of all players’ requests: m =(1 ). Each individual’s 40 tax is given by
(m)= + +2 +1 (m) − µ ¶ where +2 = 2 and +1 = 1. It is useful to define ’s personalized price as
(m )= + +2 +1.Notethat’s personalized price in this mechanism is − − independent of his own message. Furthermore, the budget is balanced regardless of
the requests, i.e. =1 (m)=(m) for all m. P To illustrate how this mechanism works consider the following example:
Example 3. Consider the following economy composed of Ann, Bob, and Carol. Each has preferences for two goods—a private good () and a public good ().Furthermore assume these preferences are represented, respectively, by utility functions
1 2 ( )= (6 ) − 2 −
1 2 ( )= (8 ) − 2 −
1 2 ( )= (5 ) . − 2 −
Initially, the consumers have no units of the public good and 20 units of the private good. The government has the capability of producing the public good using the private
1 good as an input according to production technology ()= 4 ,where4 is the real marginal cost of production. 41
First, notice that the Lindahl prices for each consumer in this example are
()=(1 3 0).
These prices result in the allocation of ( )=(5 15 5 20) which is Pareto efficient.
Now suppose we implemented the Walker scheme. Each consumer submits a real number to the government who:
1. Produces the public good equal to the sum of the three messages, i.e.,
(m)= + + .
2. Charges a tax to each consumer according to the function
4 (m )= + +1 +2. − 3 −
Given rules of this institution we might expect that consumers act in a strategic manner. Consumer , for instance, takes the rules of the institution to to solve the following maximization problem
max ((m) 20 (m ) (m)) − − ·
The first order condition for this problem is
4 6 (∗ + + )= + , − 3 − or
(∗ ; m )= (m ). − − 42
It is straightforward to see that since each consumer’s personalized price is in- dependent of his own actions it acts like a price. Furthermore since can be any real number, taking the other players decisions as fixed, consumer ’s problem is sim- ply to choose a public good level that maximizes their utility given the personalized
price . In a Nash equilibrium, (∗ ∗ ∗ ), therefore each consumer set their
(∗; m∗ )=(m∗ ) and have no incentive to change ∗ given the choices of − − the other consumers. Looking at the three consumers’ equilibrium conditions together we have
4 6 (∗ + ∗ + ∗ )= + ∗ ∗ − 3 − 4 8 (∗ + ∗ + ∗ )= + ∗ ∗ − 3 − 4 5 (∗ + ∗ + ∗ )= + ∗ ∗ − 3 −
Adding the three conditions together we get
19 ∗ =4, −
∗ =15.
Thus, the public good produced in equilibrium is 15. However, this is the same as the Pareto optimal amount that we derived above using the the Samuelson Marginal
Condition. This is not a coincidence. Notice that utility maximization from each
consumer results in setting their = (m∗ ) and since (m )=4= − − P for all ,wehave =4= in equilibrium which is simply the Samuelson P 43
Marginal condition. Furthermore, in equilibrium, the consumers’ three personalized price functions are
(m∗ )=1 −
(m∗ )=3 −
(m∗ )=0. −
However, these personalized prices are equal to the Lindahl prices. The Nash allo- cation of the Walker mechanism in this example is Lindahl. This is in fact true for remarkably general circumstances.
We have demonstrated in this example that every Nash equilibrium allocation must correspond to a Lindahl allocation. Walker’s mechanism also has the property that for every Lindahl allocation, we can find a Nash equilibrium allocation that yields exactly the same allocation. In the jargon of the mechanism design literature, we would say that the Walker equilibrium fully implements the Lindahl correspondence in Nash equilibrium.6
1.5 Discussion
This chapter has motivated the notion of a Lindahl equilibrium as a normative ideal in a public good economy, and has shown that Lindahl institutions, like the Walker
6 Or rather, due to issues along the boundary, the Walker mechanism fully implements the "con- strained" Lindahl correspondence in Nash equilibrium. 44 mechanism, are not necessarily incompatible with self interested agents. In the next several chapters, we ask the following research questions:
1. Do any of these so-called incentive compatible Lindahl mechanisms work in
practice? If they do work, how long does it take participants to learn equilib-
rium? If they don’t work, what might explain their failure?
2. What are the allocative properties of these mechanisms when not in equilibrium?
Since it presumably takes time for participants to learn how to play equilibrium,
and while they are learning they may be forfeiting surplus. Out-of-equilibrium
performance allows us gauge true economic cost of implementing the mechanism
by comparing the out-of-equilibrium allocations with those they would receive
in equilibrium .
3. What are the properties that make some mechanisms more/less successful than
other mechanisms? Do issues such as complexity, dynamic stability, and out-
of-equilibrium punishment matter in actual performance?
4. Can we do a better job at designing mechanisms to make public good decisions? 45
Chapter 2
Out-of-Equilibrium Performance of Three
Lindahl Mechanisms1
2.1 Introduction
In his classic 1919 paper, Erik Lindahl proposed a cost sharing procedure for financing public goods and he maintained that use of his procedure would (in modern termi- nology) produce Pareto efficient outcomes. Incentive compatible mechanisms that implement Lindahl’s outcome as a Nash equilibrium were first proposed by Hurwicz
(1979) and Walker (1981). A drawback of these early Lindahl mechanisms was the instability of their equilibria, as shown by Kim (1987).
Several authors have provided solutions to the instability problem by incorporating some form of dynamic stability into the design of the mechanism. Examples include the mechanisms introduced by Vega-Redondo (1989), de Trenqualye (1989), Kim
(1993), and Chen (2002). All four mechanisms attain Lindahl outcomes as Nash equilibria, as in the Hurwicz and Walker mechanisms. The first two mechanisms are stable under myopic best reply, and Kim’s mechanism is globally stable under a
1 This is joint work with Natalia Lazzati and Mark Walker. 46 gradient adjustment process. Chen’s mechanism is supermodular for some parameter values,2 and is therefore stable under a wide variety of out-of-equilibrium behavior by participants.
It is not enough, however, to evaluate an allocation mechanism only in terms of the efficiency and stability of its equilibria. Whenever there is a change in the underlying economic conditions (preferences, costs, etc.) and a corresponding change in a mechanism’s equilibrium, the mechanism is likely to be out of equilibrium for a period of time – perhaps a considerable period – before it attains an equilibrium or even comes close to an equilibrium. It is therefore important to know how long alternative mechanisms require to reach or approximate an equilibrium, and how far short of efficiency these out-of-equilibrium outcomes will be.
Previous experimental examinations of the stability of public goods allocation mechanisms have found that the Groves-Ledyard mechanism, in which equilibria are
Pareto optimal but not Lindahl, converges to near its equilibrium under some assign- ments of parameter values (Chen & Plott (1996), Chen & Tang (1998)). Many ex- perimental studies have found convergence to Nash equilibrium in mechanisms whose equilibria are not Pareto optimal, such as the mechanism of voluntary contributions and the Vickrey-Clarke-Groves mechanism. We are aware of only two prior stud-
2 More precisely, if the mechanism’s strategy (message) spaces are compact, then there are some utility functions and some values of the mechanism’s parameters for which the resulting game is supermodular. 47 ies that examine stability of a Lindahl mechanism: Chen & Tang (1998) and Healy
(2004) found that the Walker mechanism did not converge – as expected, given its theoretical instability under any plausible out-of-equilibrium behavior.
The present paper reports on laboratory experiments designed to compare the out-of-equilibrium performance of the Walker (1981), Kim (1993), and Chen (2002) mechanisms. While each mechanism achieves Lindahl allocations at its Nash equi- libria, the mechanisms differ in several important respects. Our primary aim is to compare the effects of these differences on the mechanisms’ convergence to equilibrium and on their out-of-equilibrium welfare properties.
We find that the Chen and Kim mechanisms converge toward their respective equilibrium levels of public good provision, but ultimately remain at some distance from equilibrium. As in previous research, the Walker mechanism does not converge.
Both the Chen and Kim mechanisms attain levels of the public good that yield a substantial portion of the maximum possible consumer surplus, generally between
70% and 95%. The Walker mechanism produces considerably less than 50% of the possible surplus. Surprisingly, none of the mechanisms converges to the Lindahl taxes.
The most pronounced difference between the mechanisms is in their violations of individual rationality and their failure to balance the budget. On both counts the Kim mechanism performs well, with minimal violations of individual rationality and with budget imbalances that are small relative to the consumer surplus the mechanism produces. In contrast, the parameter values that make the Chen mechanism super- 48 modular produce extreme tax obligations, many violations of individual rationality, and enormous budget deficits and surpluses. On balance, we find the performance of the Kim mechanism to be superior to both the Chen and Walker mechanisms.
The remainder of the paper will proceed as follows. Section 2 summarizes the theoretical concepts that form the basis of our experiment; Section 3 describes the experiment; and Sections 4 and 5 present the experimental results. Section 6 sum- marizes the results and provides some concluding remarks.
2.2 Theoretical Preliminaries
2.2.1 Public Goods and the Lindahl Outcome
We analyze the three mechanisms in the same economic setting: there is a single public good, produced from a second (private) good via a constant-returns-to-scale technology, and there are individuals, or participants, for whom the public good is to be provided. An outcome is denoted by ( 1 ): denotes the level at which the public good is provided, and denotes the amount of the private good participant
contributes to finance the provision of the public good. It’s convenient to think of the private good as money and as a tax or transfer paid by . Each participant evaluates outcomes according to a utility function ( )=() ,where( ) − · is strictly concave and differentiable; is referred to as ’s valuation function.We assume that no participant’s initial holding of the private good is exhausted by his 49
tax .
Let denote the per-unit cost of producing the public good – i.e., each unit of the public good requires units of the private good as input. The Pareto optimal
outcomes are the ones that satisfy the Samuelson condition =1 0()= and also
P balance the budget – i.e.,onesforwhich =1 = . Notethatstrictconcavity P of each ensures that there is at most one value of that is consistent with Pareto optimality.WerefertothisasthePareto value of ,denoted∗.
Lindahl proposed charging each participant a share of the per-unit cost for each
unit of the public good that is provided – i.e., = for each ,and =1 = . P If each participant takes his share as given, profit-maximization will lead him to request a public-good provision level at which his marginal value 0() is equal to
. Lindahl suggested that his procedure would be in equilibrium when the shares
1 are set so as to induce every participant to request the same amount of
the public good. Clearly, such an equilibrium will be Pareto optimal: =1 0()=
P =1 = . P But how do we determine the shares 1 ? At any provision level ,par- ticipant ’s share should be his marginal value 0(). But if asked to reveal this information, participants typically have an incentive to misrepresent their preferences, hoping to free ride on other participants’ payments. In a pioneering paper, Groves
& Ledyard (1977) proposed a mechanism whose Nash equilibria are Pareto optimal, but in which participants’ shares of the cost are not the Lindahl shares. Hurwicz, 50
Walker, and the subsequent authors mentioned above adapted the Groves-Ledyard idea to create “Lindahl mechanisms” – mechanisms whose Nash equilibria are Lin- dahl allocations.
2.2.2 The Mechanisms
A mechanism uses messages or actions by the participants to calculate an outcome
( 1 ), the level at which the public good will be provided and the tax/transfer paid by each of the participants. (Note that this transfer may be negative – a rebate
– for some participants.)
The Walker Mechanism Each participant announces a request .Themecha- nism sums the participants’ requests to determine the level at which the public good will be provided:
= . (1) =1 X Denote by r the profile of all participants’ requests: r =(1 ).Eachparticipant’s tax is given by
(r)= + +2 +1 (2) − ³ ´ where +2=2 and +1=1. Note that in this mechanism the budget is balanced
regardless of the requests, i.e., (r)= for all r. =1 P 51
The Chen Mechanism Each participant announces a request ;theprofile r determines the public good level according to (1), just as in the Walker mechanism.
The Chen Mechanism requires that each participant also announce a second number,
, which the mechanism interprets as a prediction about the level at which the public good will be provided. The profiles of requests and predictions are r =(1 ) and p =(1 ). Each participant’s tax in the Chen Mechanism is
1 2 2 (r p; )= + + ( ) + ( ) (3) − 2 − 2 − Ã = = ! = X6 X6 X6 where and are parameters specified by the mechanism’s designer. Note that each participant’s tax depends on the accuracy of all participants’ predictions, through the
2 terms ( ) . The mechanism is a generalization of Kim’s earlier mechanism. −
The Kim Mechanism The Kim Mechanism is the special case of the Chen Mech-
anism in which =1and =0– i.e.,theKimtaxis (r p)= (r p;1 0).
Properties of the Mechanisms The Walker Mechanism is the simplest of the three, an important consideration when actually implementing a mechanism. More- over, the mechanism’s budget is balanced identically for any profile of requests by the participants, whether the profile is an equilibrium or not. However, as Kim (1987) has shown, the mechanism is unstable under any plausible dynamic behavior by the participants.
The Kim Mechanism is not as simple as the Walker Mechanism: each participant 52 must augment his public-good request with a prediction about the result of the other participants’ requests. Further, the mechanism’s budget is generally unbalanced when out of equilibrium. However, the mechanism is globally stable under the continuous- time gradient adjustment process, and in some circumstances is stable under discrete- time Cournot best reply, as shown in the Appendix, below.
The Chen Mechanism is clearly the most complicated of the three, and like the
Kim Mechanism its budget is typically not balanced except in equilibrium. But because it is supermodular for some combinations of parameter values and individuals’ preferences, it is possible, for a range of preferences, to choose corresponding values for and that will make the mechanism stable under a wide variety of out-of- equilibirium behavior.
2.3 The Experiment
The experiment consists of applying each of the three Lindahl mechanisms to the same simple public-goods allocation problem, or environment.Wefirst describe this common environment, and then the mechanisms.
The environment consists of three participants – i.e., =3. The participants’
2 valuation functions all have the form ()= ; the parameter values − and are as shown in Table 0. The cost function for providing the public good is 53
()=12 – i.e., =12. The unique Pareto public good level ∗ is therefore 3 12 =1 − 66 12 ∗ = = − =9. P 3 (2)(3) 2 =1 In order to define the three mechanismsP for the experiment, we must specify (a) the message spaces that will be made available to the participants and (b) the values of the
Chen Mechanism’s parameters and .Therequest space willbethesameforallthree mechanisms, consisting of the numbers 5 499 4981499 15 .The ∈ {− − − } set of possible provision levels for the public good (the sum of the three participants’ requests) is therefore the set 15 14994499 45 , and we therefore allow the {− − } participants to select their predictions from this set. The parameter values for the
Chen mechanism are set at =21and =8.
Figure 2.1 displays the parameter values and the three mechanisms’ equilibrium messages and resulting surplus.
We show in an appendix that with these parameter values the Walker Mechanism is unstable under the discrete-time best reply dynamic, the Kim Mechanism is stable under best reply, and the Chen Mechanism induces a supermodular game and is thus stable under a wide range of dynamic behavior.
Six laboratory sessions were conducted, two sessions with each mechanism. All sessions were conducted in the Economic Science Laboratory at the University of
Arizona. All subjects were undergraduate students at the University, recruited via e-mail from the ESL’s online subject database. 54
Table 0: Experimental Parameters and Equilibria
Parameters Equilibrium Strategies Lindahl Surplus at
Player ∗ ∗ ∗ ∗ Share Tax Equilibrium
1221 13 3 9 4 36 81 − 216153 19 9 2 18 81 − 7 − − 23 3281597 9109081
Total 66 3 9 9 9 12 108 243
Figure 2.1. VLW Experiment Parameters
In each session the subjects were first randomly assigned into groups of three; there were four to six such groups in each session. The subjects’ parameter values were revealed to them privately; subjects were not provided with any information about the parameter values of the other two subjects in their group. Each three- person group remained together throughout the session, participating in 40 rounds, or time periods, of that session’s mechanism.
In each period the subjects communicated their messages – their requests and, in the Kim and Chen mechanisms, their predictions –fromcomputerterminals to a central server, and they received information in return from the server. Written instructions were provided at each computer terminal. The subjects were given time 55 to read the instructions, after which the experimenter read the instructions aloud and entertained questions. All sessions were conducted by the first author.
The software for the experiment includes two tools to aid subjects in their deci- sion making. Each subject was provided with a “What-if-Scenario” profitcalculator3, which allowed the subject to input hypothetical messages for the other two group members and explore how, against those hypothetical messages, his own decisions would affect his profit. This is a substitute for providing subjects with payoff ta- bles: the complexity of the mechanisms’ outcome functions would require multiple extremely complex tables. This calculator, which allows a subject to answer any
“what if” question that could have been answered with payoff tables, but to do so more transparently, appears to be a better decision-making aid than payoff tables.
Subjects were also able to access a screen that showed, for all prior rounds, all three subjects’ messages as well as the resulting public good level and the subject’s own profit. Subjects were not required to use these decision aids, but most subjects made use of the profit calculator on almost every round.
As in Healy (2006), we did not use practice rounds, but instead allowed subjects
five minutes to practice with the “What-if-Scenario” profit calculator. This provided each subject with some experience using the software, without allowing subjects to learn anything about other subjects’ parameters or behavior. After this five-minute practice time with the calculator, each group played 40 periods with one of the three
3 See Healy (2006) for a discussion of the "What-if-Scenario" calculator. 56 public goods mechanisms.
Each of the 40 decision periods proceeded in the same fashion. Subjects were first asked to submit their requests and, in the Kim and Chen mechanisms, their predic- tions as well. When all three participants had submitted their messages, the outcome
(the public good level and the participants’ taxes) was calculated and the following information about the just-completed period was communicated to each participant: all three group members’ decisions; the resulting amount of the public good that was provided; and the subject’s own revenue, tax, and profit. The subject’s cumulative profit was reported only at the end of the experiment, although, as described above, a subject could access a screen displaying all information he’d been provided at prior periods. Subjects were also required to record their information by hand on a record sheet. This task was included in order to ensure that at least some of a subject’s attention would be directed to how much he was earning.
At the end of each session one of the 40 periods was selected at random and each subject was paid six cents for every experimental dollar earned in that period.
Subjects remained in the same group for the entire session and were paid privately at the end of the session. No subject participated in more than one session. Sessions typically lasted about 90 minutes. 57
2.4 Convergence
We first consider how the subjects’ behavior in each mechanism compares with the mechanism’s equilibrium: do the subjects’ actions tend to become close to their equilibrium actions as they interact repeatedly in the mechanism, and does this con- vergence occur at different rates in the three mechanisms? Then we will assess how well the mechanisms accomplish the task they were designed for: how well do they converge to the Pareto public good level and to the Lindahl taxes?
2.4.1 Convergence of Participants’ Requests
In order to compare the mechanisms’ convergence to their respective equilibrium strategy profiles, we need a measure of the distance between an observed profile of requests, =(123), and a mechanism’s equilibrium profile ∗ =(1∗2∗3∗).Here, and throughout our analysis, we use the so-called (average) “city block” metric (or
1 metric) as our measure of distance. Thus, we define the request deviation of group
in period as 1 := ( 1 ∗ + 2 ∗ + 3 ∗ ) 3 | − 1| | − 2| | − 3| i.e., as the average absolute deviation of the participants’ requests from their equi- librium values.
Figure 1 shows the time series, for each mechanism, of the period-by-period request deviations averaged across all nine groups who participated in the mechanism, i.e., 58 the average request deviations,
1 9 := 9 =1 X Table 2.1 presents the information in Figure 2.2 in an alternative concise format: the table divides the forty periods into four segments of ten periods each, and displays the average deviations further averaged across each ten-period block.
There is little evidence that the Walker mechanism is converging. Indeed, in the early periods the requests in the Walker mechanism are moving farther from their equilibrium values. In the last 13 periods the deviations lie within or very close to the interval [4,5]. This failure to converge is qualitatively consistent with the theoretical instability of the mechanism, shown in the appendix.
The deviations in the Kim and Chen mechanisms exhibit a decreasing trend, eventually remaining in the intervals [2,3] and [1,2], respectively, over the final 15 periods. This is qualitatively consistent with these mechanisms’ theoretical stability.
The Chen mechanism’s average deviations from equilibrium are substantially smaller than those of the Kim mechanism in every period with the exception of two early periods.
Result 1: The deviations from equilibrium requests are consistently smallest in the Chen mechanism and largest in the Walker mechanism. There is no evidence of convergence in the Walker mechanism. The requests in the Chen and Kim mechanisms grow closer to equilibrium through most of the experiment, but it is unclear how close 59 to equilibrium they would eventually converge over a longer time horizon.
2.4.2 Public Good Provision Level
The primary raison d’etre of all three mechanisms is to achieve Pareto outcomes. The
Pareto outcomes in our experiment are those in which the public good is provided at
the level = ∗ =9.
Figures 2.3 and 2.4 and Table 2.1 present a simple description of the mechanisms’ comparative success at meeting this objective. Averaging across the nine three-person groups, each mechanism tends to yield public good provision levels that exceed the
equilibrium (and Pareto) level = ∗ =9. Over time the average public good level tends to decrease, moving closer to the Pareto level.
However, the time series of average public good levels depicted in Figures 2.3 and
2.4 obscure a substantial amount of variation across the respective nine groups in each mechanism: some groups chose public good levels that were much smaller than and some chose levels much larger than . Using the average therefore obscures the extent to which the public good levels are converging over time, or failing to converge.
A clearer picture of the mechanisms’ convergence to the equilibrium public good level is provided in Figures 3a and 3b, which display the average absolute deviations from equilibrium, 1 9 := 9 =1 | − | X 60
where denotes the public good level chosen by group in period .
In both the Chen and Kim mechanisms the average deviation decreased for approximately the first twenty periods, but thereafter showed no further decrease, remaining for the most part within the interval [2 4]. The average deviations in the
Walker mechanism increased over about the first twenty periods, displayed extreme period-by-period fluctuations throughout, and never declined to the levels attained by the other two mechanisms. Table 3 presents the average deviations from the Pareto public good level in the 10-period-block format introduced in Table 2.1, above.
We summarize Figures 2.5 and 2.6 and Table 3 in the following statement:
Result 2: The public good levels attained by the Chen and Kim mechanisms become closer to the equilibrium level during the early periods but do not converge more closely to the equilibrium level in subsequent periods. The Walker mechanism’s public good level is consistently farther from the equilibrium level than the levels in the other two mechanisms.
The deviations from equilibrium — in effect, the “errors” — produced by the Chen and Kim mechanisms may seem large: even in the later periods, these “errors” are generally between 25% and 40% of the target level ∗ =9. However, when we measure a mechanism’s performance by the welfare it produces, as in Section 5 below, the Chen and Kim mechanisms appear to be relatively more successful at the task of choosing a public good provision level. 61
2.4.3 Lindahl Taxes
We now ask how closely the taxes converge to the Lindahl taxes the mechanisms are designed to achieve. The Lindahl tax profile is =( ∗∗∗)=(36 18 90).At 1 2 3 − each period we define group ’s deviation from as follows:
1 = ( 1 ∗ + 2 ∗ + 3 ∗ ) 3 | − 1| | − 2| | − 3| i.e.,theaverageamountthateachparticipant’s tax deviates from his Lindahl tax.
Figure 4 depicts the time series of average deviations
1 9 = 9 =1 X for the nine three-person groups in each mechanism. Table 4 presents these average deviations in the 10-period-block format introduced in Table 2.1, above.
Figure 4 and Table 4 reveal striking differences among the three mechanisms: the
Kim mechanism’s tax profiles are consistently closer to the Lindahl taxes than the tax profiles in the Walker mechanism, which in turn are much closer to the Lindahl taxes than are the tax profiles in the Chen mechanism. Over the last 20 periods the average deviation from the Lindahl tax profile was 30 in the Kim mechanism,
97 in the Walker mechanism, and 408 in the Chen mechanism. In both the Walker and Chen mechanisms these average deviations significantly exceed the target taxes themselves: ( )=(36 18 90) (see Table 0). Thus, neither mechanism is 1 2 3 − even approximately achieving the Lindahl taxes. 62
Result 3: All three mechanisms consistently deviate from the Lindahl taxes. The deviations follow a clear ranking: the Kim mechanism’s taxes are closest to Lindahl; the Walker mechanism the second closest; and the Chen mechanism’s taxes are con- sistently much farther from Lindahl than are the taxes in the other two mechanisms.
2.5 Welfare and Budget Comparisons
The attraction of Lindahl mechanisms is that they attain good allocations — at their equilibria. The allocations are Pareto efficient at the mechanisms’ equilibria; the allocations are individually rational at the mechanisms’ equilibria; and the Lindahl taxes the mechanisms impose are proportional to participants’ marginal benefits — at the mechanisms’ equilibria.
Butbecausewecanexpectthemechanismstobesignificantly out of equilibrium for a significant proportion of time, it is important to evaluate the welfare properties of their allocations when out of equilibrium as well as in equilibrium. Indeed, as
Figure 2.2 suggests, the mechanisms may remain significantly out of equilibrium for averylongtime.
2.5.1 Welfare from Provision of the Public Good
We focus our attention first on the welfare associated with the public good levels that were chosen by the nine three-person subject groups in each mechanism, ignoring for 63 the time being any budget imbalances. The induced utility functions in the experi- ment were quasilinear, i.e., they have the form ( )=() . Therefore the − welfare a mechanism achieves for its participants directly from providing the public good is simply the consumer surplus,
():=1()+2()+3() −
Assuming a balanced budget (i.e., that the taxes collected, 1 + 2 + 3,equalthe cost of the public good), () is the surplus created by providing units of the public good instead of not providing any of the public good.
The maximum possible surplus is attained when = ∗ =9. This maximum surplus is ∗ =243(see Table 0). Figures 2.7 and 2.9 display the time series of average surplus values, 1 9 := 9 =1 X attained by subjects in each mechanism, where denotes the surplus attained by group in period .
As one would expect, the time series in Figures 2.7 and 2.9 closely reflect those in
Figures 2.5 and 2.6, the average deviations from the Pareto public good level. The sur- pluses in the Chen and Kim mechanisms show improvement during the early periods and then show little if any improvement over the final twenty or so periods, remaining mostly between 175 and 225 and never exceeding 230. Similarly, the surpluses in the
Walker mechanism reflect the Figure 2.6 deviations from equilibrium provision levels, 64 worsening over the early periods, continuing to fluctuate significantly, and remaining well below the surplus levels attained by the other two mechanisms.
Note that a surplus of 175 is more than 70% of the potential surplus ∗ =243,and a surplus of 225 is more than 90% of ∗. Thus, as suggested in the preceding section, while the public good provision levels in the Chen and Kim mechanisms generally deviate by 25% to 40% from the equilibrium level, the provision levels nevertheless yield substantial gains in welfare when compared to non-provision of the public good.
Result 4: The surpluses obtained in the Chen and Kim mechanisms from pro- viding the public good increase during the early periods. In the later periods the surpluses remain approximately between 70% and 95% of the potential surplus, av- eraging 78% and 77%, respectively, over the final 20 periods. The surpluses in the
Walker mechanism display much larger fluctuations and remain well below those at- tained by the other two mechanisms, averaging only 26% of the potential surplus over the final 20 periods.
2.5.2 Violations of Individual Rationality
Lindahl allocations, in addition to being Pareto optimal, are also individually rational: eachparticipantisatleastaswelloff at the Lindahl allocation as he would have been had the public good not been provided. Formally, Lindahl allocations ( 1) 65
satisfy the inequality ( ) (0 0) for each participant . Lindahl mechanisms ≥ therefore, by their definition, yield individually rational outcomes at their equilibria.
However, their disequilibrium outcomes are not Lindahl allocations and therefore need not be individually rational – some participants may be made worse off than if the mechanism were not used and the public good not provided.
Figure 2.10 and Table 2.1 describe the violations of individual rationality produced by each of the three mechanisms. In the Kim mechanism IR violations decreased to about five percent of all participant outcomes in the final 20 periods. But in the Chen and Walker mechanisms the number of IR violations did not decrease, remaining at about 30% in the Walker mechanism and at about 40% in the Chen mechanism.
Result 5: The Kim mechanism produces far fewer violations of individual ratio- nality than either the Chen or Walker mechanism.
2.5.3 Unbalanced Budgets
Until now, we have ignored the fact that both the Chen and Kim mechanisms typically fail to balance the budget when out of equilibrium. An unbalanced budget can reduce or eliminate the welfare gains (the consumer surplus) achieved by providing the public good. For example, if more taxes are collected from the mechanism’s participants than required in order to produce the public good, then the participants in the mechanism are sacrificing some, or perhaps all, of their surplus. Conversely, if the taxes fail to cover the cost of the public good, then the amount by which the cost exceeds the taxes 66 must come from outside the mechanism — an additional cost that must be borne by someone.
In either case – tax collections that are too large, or tax collections that are too small – we regard the amount of the budget imbalance as a cost of using the mechanism. For the Walker mechanism this cost is always zero: the mechanism’s budget is always balanced, whether in or out of equilibrium. Figure 7 and Table 7 show the average magnitude of the budget imbalance for the Chen and Kim mechanisms.
The average budget imbalance produced by the Kim mechanism over the last 20 periods was 48. The Chen mechanism produced imbalances that averaged just under
1000 over the last 20 periods.
Recall from Table 5 that the Kim mechanism produced, on average, about 187 units of consumer surplus over the last 20 periods. Thus, the average budget imbal- ance of 48 is a cost worth bearing: the average surplus the Kim mechanism produced was nearly four times as large.
On the other hand, the Chen mechanism over the last 20 periods produced average budget imbalances of 995 and average consumer surplus of only 190. The cost of using the mechanism, on average, was more than five times the consumer surplus it produced. Indeed, the maximum possible consumer surplus is only ∗ =243;the average imbalance was more than four times as great as ∗.
Result 6: The Kim mechanism produces budget imbalances that are significantly smaller than the consumer surplus it produces. The Chen mechanism produces budget 67 imbalances that are significantly larger than the consumer surplus it produces.
2.5.4 The Role of Parameter Values
The large budget imbalances and the numerous violations of individual rationality in the Chen mechanism are related to the mechanism’s parameters and ,which were set at =8and =21in our experiment. Smaller values would likely re- duce the mechanism’s budget imbalances and violations of individual rationality, but supermodularity imposes a lower bound on and . As shown in the Appendix, supermodularity in the Chen mechanism requires (for quasilinear-quadratic utilities) that ( 1) +1+2max 1 . Fortheeconomicenvironment − { } ≤ ≤ in our experiment ( =3and 1 = 2 = 3 =1), one easily verifies that the smallest values of and that satisfy this condition are =3and =9. Indeed, related experimental research on efficient mechanisms (e.g., Chen and Gazzale, Chen and Tang, Healy) suggests that large values of “punishment parameters” such as are important for inducing convergence. The values =8and =21were chosen to give the Chen mechanism a good chance to converge, and these values also yield simple, integer-valued outcome functions that were likely to be more easily understood by subjects. 68
2.6 Conclusion
The Chen and Kim mechanisms perform similarly in some respects: after several early periods of adjustment, each mechanism generally provides the public good at a level that produces 70% to 95% of the possible consumer surplus. However, neither mechanism converges to the Lindahl taxes. The Walker mechanism achieves a much smaller fraction of the possible surplus and fails to converge to either its equilibrium public good level or the Lindahl taxes.
In other important respects, the Kim mechanism significantly outperforms the other two mechanisms. After a few initial periods, the Kim mechanism produces very few violations of individual rationality. These violations occur in the Chen and Walker mechanisms more than 30% of the time, on average. The magnitude of the budget imbalances in the Kim mechanism average about 25% of the consumer surplus the mechanism produces. In the Chen mechanism the budget imbalances average about
five times the amount of consumer surplus produced. The Walker mechanism always produces a balanced budget.
Thus, when we take account of the fact that these mechanisms will often be out of equilibrium, the Kim mechanism significantly outperforms the Chen and Walker mechanisms. 69 Public Good Level BB Consumer Surplus * Number of IR Violations Periods Chen Kim Walker Chen Kim Walker Chen Kim Walker 1 ‐ 10 13.1 10.5 10.1 69 134 137 118 55 63 10 ‐ 20 10.9 11.4 10.7 204 150 56 83 41 99 20 ‐ 30 10.1 10.0 11.7 190 188 25 99 12 77 30 ‐ 40 8.9 9.6 10.2 189 186 100 117 15 75 * Budget‐balanced consumer surplus = Value from public good minus total cost of public good
"City Block" Distance from "City Block" Distance from CB Distance from Equil'm Request Profile Equil'm Prediction Profile Lindahl Tax Profile Periods Chen Kim Walker Chen Kim Walker Chen Kim Walker 1 ‐ 10 2.6 3.3 3.9 4.4 3.5 na 811 40 71 10 ‐ 20 1.5 3.1 5.4 2.6 3.1 na 394 39 110 20 ‐ 30 1.6 2.8 4.8 2.4 2.7 na 437 29 108 30 ‐ 40 1.5 2.4 4.4 2.7 2.3 na 378 30 86
Maximum Absolute Mean Absolute Minimum Absolute Budget Imbalance Budget Imbalance Budget Imbalance Periods Chen Kim Walker Chen Kim Walker Chen Kim Walker 1 ‐ 10 6692 241 0 2038 92 0 142 14 0 10 ‐ 20 2629 185 0 875 72 0 102 7 0 20 ‐ 30 4208 153 0 994 54 0 71 4 0 30 ‐ 40 3973 163 0 947 43 0 80 3 0 70
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Chapter 3
A Simple Supermodular Mechanism that
Implements Lindahl Allocations
3.1 Introduction
The reliance on unregulated markets for the provision of public goods presents well known challenges to efficiency. For economists, this problem continues to motivate the search for alternative institutions which yield Pareto optimal outcomes. One problem with this approach is that some Pareto outcomes may not be desirable for everyone involved. Some participants could end up being worse off than they were with their original endowment, a common critique, for example, of the Groves-Ledyard
(G-L) mechanism. G-L is an institution that overcomes the free riding problem — the incentive to enjoy the public good’s benefits while not sharing in its cost — but the mechanism may leave some participants worse off than before they participated. In contrast, Lindahl allocations, while also Pareto optimal, make no one worse off than he was to begin with (i.e., are individually rational). Lindahl allocations are therefore attractive outcomes when there are public goods. 81
In addition to being Pareto optimal and individually rational, Lindahl allocations share another important property of Walrasian (“competitive”) allocations of private goods: every individual’s payment for each unit of the public good is equal to the marginal value he places on the good. In the Walrasian setting consumers all face the same price and they demand potentially different quantities of a good; in the Lindahl setting they face distinct “personalized” prices and, in equilibrium, each consumer demands the same quantity of the public good. Actually implementing a Lindahl scheme, however is problematic, since it is not exactly clear how the personalized prices are to be determined. Perhaps one could use surveys, but there may be incen- tives for participants in the surveys to misrepresent their preferences in order to pay a lower price. This has led to the development of incentive compatible public goods mechanisms.
The purpose of this paper is to introduce a new incentive compatible mechanism which attains Lindahl allocations as Nash equilibria. This is true for economies with an arbitrary number of consumers and general preference environments. In addition to this Nash “implementation” result, the mechanism has several other attractive properties, which have been motivated by experimental research: the mechanism retains its structural simplicity as the number of consumers increases—there are no new penalty terms added to the mechanism when the economy increases in size; the minimum dimension of data needed to compute payoffs is smaller than other mechanisms with comparable properties; the components of the mechanism have a 82 clear economic interpretation; and for quasi-linear preference environments the unique equilibrium is stable under a wide variety of learning behavior.
The mechanism introduced here is not the first to implement Lindahl allocations.
Hurwicz (1979) and Walker (1981) were the first to present such mechanisms, but
Kim (1987) has shown that both mechanisms are quite unstable. While some sort of dynamic stability is desired, there is no agreement about how people’s behavior ad- justs when out of equilibrium. In mechanism-design experiments, however, a common empirical finding is that in mechanisms with theoretically robust dynamic stability properties, subjects’ behavior tends to converge. Supermodular mechanisms have been particularly successful.1 This empirical regularity was presaged by the theoret- ical stability results established by Milgrom and Roberts (1990a) for supermodular games. In this light, Chen’s (2002) theoretical contribution is of particular interest.
She presented the first Lindahl mechanism that is supermodular in quasi-linear en- vironments for some values of the mechanism parameters. Thus, motivated by the observation that supermodular mechanisms tend to perform better in the laboratory, she creates a Lindahl mechanism that induces a supermodular game. Two fundamen- tal issues in her paper are worth noting.
First, in Chen’s paper, the environment used to establish that the game is super- modular and the uniqueness of Nash equilibrium does not satisfy the conditions stated in the Milgrom and Roberts stability result. Thus, no stability of equilibrium results
1 For example see Chen and Tang (1998), Chen and Gazzale (2004), and Healy (2004). 83 can be inferred from the fact that her mechanism induces a supermodular game with a unique equilibrium. As a cautionary example in this paper, we show that there is a simple incentive compatible Lindahl mechanism that, in the same environment as Chen’s, always induces a supermodular game with a unique, dynamically unstable equilibrium. This observation suggests that we need to appeal to an alternative tech- nique to guarantee that the game induced by our new mechanism yields a unique, dynamically stable Nash equilibrium. We do this by introducing a mechanism that, under some choices of the mechanism parameters, induces a game whose best reply mapping is a contraction mapping. Interestingly, we also show that if the game in- duced by our mechanism is supermodular, the sufficient conditions for the best reply mappingtobeacontractionaresatisfied giving us a relatively simple condition to check for stability.2
The second issue with Chen’s mechanism is a practical one — the severe method in which it taxes out-of-equilibrium behavior. Van Essen, Lazzati, and Walker (2007), hereafter VLW, experimentally tested three Lindahl mechanisms, including the Chen mechanism, in a laboratory environment. The experimental evidence suggests that while the Chen mechanism produces public good levels near the Pareto optimal level, when not in equilibrium the mechanism generates large amounts of tax waste and participants frequently did worse than their initial endowment. Furthermore, the
2 Our mechanism induces a supermodular game that does not fit the Milgrom and Roberts result, however, for this mechanism, this is sufficient to guarantee dynamic stability. 84 subjects in the experiment did not show any signsofgettingclosetotheirequilibrium messages, so these poor out-of-equilibrium properties do not diminish much over time. VLW attribute this divergence to the complexity of the mechanism and the number of penalty terms the Chen mechanism needs to induce the supermodular game. Additionally, since the Chen mechanism adds more of these penalty terms
(one term per consumer) as more consumers are added to the economy, one would conjecture that these large tax payments would only get worse for larger group sizes.
The Lindahl mechanism presented in this paper addresses these concerns by arranging the mechanism so the number of penalty terms is fixed at two for any number of players.
The remainder of the paper will proceed as follows: Section 2 provides a simple definition of a supermodular game and summarizes some of the important properties these games exhibit, including the Milgrom and Roberts result; Section 3 outlines the basic public goods problem and mechanism environments; Section 4 contains the bulk of the paper’s theoretical results concerning implementation and stability;
finally, Section 5 discusses the actual implementation of the new mechanism and how the out-of-equilibrium tax penalties of the new mechanism compare to the penalties from the Chen mechanism. 85
3.2 Preliminaries
Supermodularity and contraction mappings play a significant role in several of the results we will develop. In this section we review some definitions, framed in terms of the strategy spaces and the payoff functions we will use. The strategy spaces are subsets of Euclidean spaces and the payoff functions are twice continuously differ- entiable (or 2). More general definitions of a supermodular game can be found in
Topkis (1998) or Milgrom and Roberts (1990a). A general treatment of contraction mappings can be found in Ortega and Rheinboldt (1970).
A normal form game is defined by a set of players, a strategy set for each player, and a payoff function for each player. Denote the set of players ,where = 1 . { } 2 Let R be player ’s strategy space with an arbitrary element m =(12), ⊆ where = is the collection of all players’ strategy spaces. For each player ×=1 let : R be a payoff function which maps strategy profiles into a numerical → payoff.
A supermodular game is characterized by payoff functions that satisfy both the supermodularity property and the increasing differences property.
Definition 1: A 2 payoff function is supermodular if a player’s own actions are strategic complements—i.e. for each
(m) 0 12 ≥ 86
Definition 2: A 2 payoff function has increasing differences if a player’s own actions are strategic complements with the actions of all other players—i.e. for each
(m) 0 ≥ for =1 2 and =1 2.
A game is supermodular if the payoffs for all players satisfy both properties:
Definition 3: A game is supermodular if for each player , is a non-empty subset of R2 and has the supermodularity and increasing difference properties.
Supermodular games have properties that make them attractive for mechanism design. If the strategy space is compact and the payoff function is 2, then Milgrom and Roberts (1990a) show that:
(1) The set of serially undominated strategy profiles has a maximum and a mini-
mum element, and these elements are Nash equilibria;
(2) Under a wide class of dynamic adjustment processes the predicted behavior
converges to the set of profiles bounded by the two extreme Nash equilib-
ria. These dynamic processes include best-response dynamics, fictitious play,
Bayesian learning, and others.
When the Nash equilibrium is unique, the predictive power of these results is increased: property (1) implies that the game is dominance solvable and property (2) 87 says that the unique Nash equilibrium is “stable” under a wide range of adaptive behavior. While these properties are attractive, as mentioned in the introduction, the environment in which we will be working does not satisfy the above Milgrom and
Roberts stability conditions — i.e., the players’ message space for the new mechanism
(and the Chen mechanism) is R2, which is not compact. We will therefore instead appeal to the Contraction Mapping Theorem for our stability results.
Definition 4: Let ( ) be a metric space. A self map on is said to be a contraction if there exists a real number 0 1 such that
(()()) ( ) ≤ for all , . ∈
Contraction Mapping Theorem: Let ( ) be a non-empty complete metric space and let : be a contraction. Then there exists a unique point ∗ → ∈ such that (∗)=∗. Furthermore if 0 is any point of and 1 = (0), 2 = (1),
3 = (2), etc., then
lim = ∗. →∞
3.3 The Public Good Economy
Our setting applies to 2 consumers. For simplicity of exposition, we restrict ≥ attention to economies with one private good, one public good, and a constant re- 88 turns to scale production technology. However, it it straightforward to generalize the results to include economies with many private and public goods. The quantity of the public good will be denoted by , and the private good for consumer by ,where consumers are indexed by subscript . Each consumer is characterized by the convex
2 consumption set = R+, an initial endowment of the private good 0, and no initial endowment of the public good. The public good is produced, using the private good as an input (quantity denoted ), with a constant returns to scale production
technology ()= – i.e., each unit of the public good requires units (0) of the private good. Thus is the constant (real) marginal cost of production. An
+1 allocation in this simple economy is an ( +1)-tuple( 1 ) R+ . ∈
3.3.1 The Mechanism
A mechanism maps consumers’ strategies (or messages) into an outcome (an alloca- tion). We consider a mechanism in which consumers report messages to a “planner” who uses this information to determine an amount of the public good to produce and
2 a tax for each consumer. The message space of consumer is = R with generic element m =() Let m =(m1 m ) denote the profile of all players’ mes- sages. Consumer ’s action should be interpreted as a request from the consumer to the planner for units of the public good. Notice that negative requests are allowed.
Consumer ’s other action, , is interpreted as his statement about the amount of the public good that will be produced. Rather than write (1122) for a strategy 89
profile, we write (12 1 )=(r s). These messages are collected by the planner and used to determine an amount of the public good and a tax for each player according to outcome functions (r s) and (r s) respectively. For positive
real numbers , ,and,let (r s)= (r s) ( (r s)) be a mechanism − =1 ³ ´ with outcome functions defined as follows:
1 (r s)= =1 X 2 2 (r s)= (r s) (r s)+ ( (r s)) + (+1 (r s)) · 2 − 2 −
where
1 (r s)= +1 , − 1 − Ã = ! − X6 3 where +1 = 1. (r s) can be thought of as ’s personalized price for the public good and the remaining two terms as statement penalties must pay.
The mechanism works as follows: the planner collects each consumer’s request and produces an amount of the public good equal to the average request. In addition, the requests and statements are used to determine each consumer’s tax, which is the sum of the two penalty terms and the term involving the personalized price. The
first statement penalty for consumer is increasing in the amount by which his own statement differs from the actual amount of the public good produced, and the other
3 For some non-equilibrium messages the payoffs are not completely well defined. That is they will take consumers outside of their consumption set . This same weakness is shared by the
Groves-Ledyard, Hurwicz, Walker, Kim, and Chen mechanisms. However it should be noted that at each interior equilibrium there is a neighborhood on which feasibility is assured. 90 statement penalty is increasing in the amount by which his neighbor’s (consumer
+1)statement,+1,differs from the actual public good production. Since (r s) is independent of s and since preferences are increasing in , it is clear that in a Nash equilibrium every consumer’s statement will be correct: = (r s).Consequently, in equilibrium both penalty terms will be zero for every consumer, and the consumer will therefore simply pay the price (r s) for each unit of the public good. Note that
(r s)is independent of both and .
The personalized price function, (r s), has an intuitive economic interpretation.
The price is higher for a consumer who is perceived by his neighbor (consumer +1)to demand more of the good than others and the price is lower if he himself is perceived to
request less than others. The term = 1 corresponds to the amount of the public 6 − P good if consumer didnotparticipateinthemechanism.Theterm+1 represents consumer +1’s statement about the level or quantity of the public good. Thus if
= 1 +1, it means that consumer +1believes that consumer ’s request 6 − willP lower the level of the public good produced. As a consequence, ’s personalized
price is less than an equal share of the marginal cost. If = 1 +1, then the 6 − reverse is true and consumer ’s personalized prize is greaterP than an equal share of
the marginal cost. If = 1 = +1,thepersonalizedpriceisanevenshareofthe 6 − P marginal cost. The first term in (r s) is , the per-capita cost of the public good. 91
3.3.2 Preference and Wealth Assumptions
The coupling of the mechanism (r s) and a preference environment defines a game. Our results require only that all of the Lindahl equilibria allocations be in the interior of the consumption set and that preference are continuos, convex, and strictly increasing in the private good. We introduce two types of preference environments:
first, a “regular” environment where preferences satisfy the usual set of consistency conditions; second, a “quasi-linear” environment that satisfies some additional properties. In both cases, the assumptions on and are sufficient for Lindahl allocations to be in the interior of each consumer’s consumption space. However, there are many other environments for which the Lindahl allocations will be interior and to which our implementation results will also apply.
Definition 5: A regular preference environment isoneinwhichforeachplayer has a complete and transitive preference relation that satisfies the following prop- º erties:
1. (Continuity): For every (¯ ¯) , the sets ( ) ( ) (¯ ¯) and ∈ { | º }
( ) (¯ ¯) ( ) are closed in . { | º }
2. (Convexity): If ( ) (¯ ¯),then( +(1 )¯ +(1 )¯) (¯ ¯) º − − º for any [0 1]. ∈
3. (Strictly Increasing in ): If ¯ ,thenforany0, ( ¯) ( ). Â 92
4. (Strict Preference of Interior Allocations to Boundary Allocations): If (¯ ¯) ∈ ++ ++ and ( ) ,then(¯ ¯) ( ),where and are the inte- ∈ Â
rior and the boundary of the consumption set respectively (this assumption
(together with 1) implies that all bundles on the boundary are indifferent to
one another — i.e., the boundary comprises a single indifference set.)
These (Cobb-Douglas type) preferences (along with strictly positive income) en- sure that each consumer’s allocation in a Lindahl equilibrium in the interior of the consumption set.
Definition 6: denotes the set of standard 2 quasi-linear environments — i.e., those in which,
1. For each , there is a real-valued function such that ( )= + ().
2 2. is , where its second derivative is bounded from above and below by ¯
2() ¯ and K respectively —i.e., K 2 0. ¯ −∞ ¯ ≤ ≤
Ω (0) ( ) 3. and — i.e., that there is unique, interior Pareto
optimalP level the publicP good that does not exhaust the economy’s private good
supply, where Ω = . P () 4. For each , 0 — i.e., each consumer has enough wealth to − ≥ cover his or her Lindahl taxes. 93
Items (1)-(4) in the definition of are sufficient to guarantee that for each ∈ , there is a unique Lindahl equilibrium which is in the interior of each consumer’s consumption set.
3.4 Implementation
The first result of the paper shows that the game induced by the mechanism (r s) implements Lindahl allocations as Nash equilibrium outcomes. Implementation is an exact correspondence between Lindahl and Nash outcomes. In other words, any
Lindahl allocation can be achieved as the allocation of a Nash equilibrium; and at any Nash equilibrium the equilibrium allocation is Lindahl.
Theorem 4. The mechanism implements the Lindahl allocations for any ∈ and any . ∈
Proof. See Appendix.
The conditions needed for the existence of Lindahl equilibria can be found in
Milleron (1972) or Foley (1970). Theorem 1 does not impose any restrictions on the positive parameters , ,and. These are free parameters which will be manipulated later in the paper to create a family of stable Lindahl mechanisms. Furthermore, 94 unlike many Lindahl mechanisms in the literature, the new mechanism applies to economies with two consumers.4
In order to illustrate the dual nature of the theorem the next example may be useful.
Consider a two-consumer economy, where each consumer is endowed with =20 units of the private good. Suppose it takes 4 units of the private good to produce each unit of the public good (i.e., =4) and that Consumer 1’s and Consumer
1 2 2’s preferences can be represented by the utility functions 1( 1)=1 (6 ) − 2 − 1 2 111 and 2( 2)=2 (8 ) respectively. The mechanism (m) implements − 2 − the Lindahl allocations of this economy.
Implementation of Lindahl allocations requires firstthatanyLindahlallocation can be achieved as a Nash equilibrium of the mechanism, and at any Nash equilib- rium, the equilibrium allocation is Lindahl. For this example, we start with the first requirement.
From the utility functions we solve for both Player 1’s and Player 2’s demand for the public good (or their marginal rates of substitution) which are 1 =6 and −
2 =8 respectively. Using the Samuelson marginal condition (i.e., that at a −
Pareto optimal quantity of the public good 1 + 2 =4), the Pareto optimal level of the public good for these two consumers is =5. Inserting this quantity into each consumer’s demand for the public good, we find that the corresponding
4 3 is the usual restriction. ≥ 95
Lindahl prices for Consumer 1 and 2 are ¯1 =1and ¯2 =3respectively. Therefore this example has a unique Lindahl allocation and it is in the interior of each consumer’s consumption space.
Suppose (¯1 ¯2 ¯1 ¯2) is a Nash equilibrium of the game induced by mechanism
111(m). If the Lindahl allocation is to be achieved as a Nash equilibrium, then two equations must hold: first, the average request must equal the Pareto optimal amount, i.e., ¯ +¯ (r s)= 1 2 =5; 2 second, Player 1’s personalized price function must equal his Lindahl price ¯1 =1, i.e.,
1 (r s)= ¯2 +¯2 =1. 2 −
Any equilibrium that achieves this allocation requires Consumer 2’s statement to be correct (i.e., ¯2 =5), it follows from the second equation that ¯2 =6.Thus, the strategy profile [(¯1 ¯1) (¯2 ¯2)] = [(4 5) (6 5)] is the only profile which could achieve the Lindahl outcome as an equilibrium. We now show that this profile is a
Nash equilibrium by checking that Consumer 1 is best responding to Consumer 2’s strategy and vice versa.
Consumer 1’s best response problem is to maximize his utility subject to a feasible set defined by Consumer 2’s strategy and the mechanism. Since in a best response
1 1 Consumer 1’s strategy satisfies 1 = 2 1 + 2 ¯2 (i.e., 1 = (1 ¯21 ¯2)), Consumer 96
y 24 Consumer 1's Utility 22 20 18 16 14 12 Slope = - Lindahl Price = -1 10 8 6 Feasible Set 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2 s
Figure 3.1. Consumer 1’s Best Response Problem
1’s best response problem simplifies to
1 2 max 1(1 20 1 (5 1) ). 1 − − 2 −
The first order condition yields ¯1 =5, which implies ¯1 =4and verifies that Con- sumer 1’s strategy (4 5) is his best response to Consumer 2’s strategy. A graphical depiction of Consumer 1’s best response problem is illustrated below.
A similar argument can be used to show that Consumer 2’s best response to
(¯1 ¯1)=(4 5) is (¯2 ¯2)=(6 5). NoticethatConsumer1’sactionsdefine a per- sonalized price equal to the Lindahl price ¯2 =3for Consumer 2. The graphical depiction of Consumer 2’s best response problem is given below.
Sincebothplayersarebestrespondingtoeachothersactions,theuniqueLindahl allocation of this example is achieved as a Nash equilibrium. 97
y 24 Consumer 2's Utility 22 20 18 16 14 12 10 8 6 4 2 Feasible Set 0 2 4 6 8 10 12 14 16 18 20 22 24 -2 s -4 Slope = - Lindahl Price = -3
Figure 3.2. Consumer 2’s Best Response Problem
The second implication of Theorem 1 says that it is also possible to go in the other direction. Namely, if (¯1 ¯2 ¯1 ¯2) is a Nash equilibrium of the mechanism, the equilibrium allocation is Lindahl. To demonstrate this in our example suppose
(¯1 ¯2 ¯1 ¯2) is a Nash equilibrium. Then the first order condition (with respect
¯1+¯2 to statement ) yields ¯ = 2 for each . Inserting this expression into each consumer’s first order condition (with respect to their request) we have
¯1 +¯2 6 =2 ¯2 +¯2 − 2 −
and
¯1 +¯2 8 =2 ¯1 +¯1. − 2 − for Consumer 1 and Consumer 2 respectively. The unique solution of this pair of
¯1+¯2 equations is ¯1 =4, ¯2 =6, which coupled with the optimal statements ¯ = 2 for 98 each yields the Lindahl equilibrium ¯1 =1, ¯2 =3,and =5. Thus, the Nash allocation is Lindahl, completing the example.
3.4.1 Implementation in Quasi-Linear Environments
In this section, we show that the new Lindahl mechanism induces a supermodular game in quasi-linear environments for some values of the mechanism’s parame- ters. Furthermore, we identify sufficient conditions for uniqueness and the stability of equilibrium in this environment. This aligns the desirable welfare properties of
Lindahl equilibrium with a set of desirable behavioral properties one would like in practice. We begin however with the following useful implication of Theorem 1 which follows directly from the assumptions on primitives that give us of a unique Lindahl equilibrium in the environment.
Remark: For any ,themechanism has a unique Nash equilibrium. ∈
For players in the environment and with an appropriate choice of mech- anism parameters, the new mechanism induces a supermodular game. Recall from
2 Definition 6 that in the environment 2 is bounded from below by for all
0. Theorem 2 therefore gives a sufficient condition for the game to be globally ≥ supermodular. 99
Theorem 5. For any , the mechanism induces a supermodular game if ∈ +min ≤ 1 − ∈ ( 1) − + min ∈ − ∙ ³ ´ ¸ Proof. See Appendix.
Theorem 2 provides conditions under which the mechanism induces a supermod- ular game. If the strategy set for each player is a compact rectangle in R2,then the game induced by the mechanism satisfies the Milgrom and Roberts conditions referred to above. However, simply compactifying the strategy set has a number of troubling consequences. Perhaps the most obvious of these is that the uniqueness result in the remark no longer applies. There may now exist boundary equilibria of the mechanism which are not Lindahl equilibria.5 In the next section, we discuss a solution to this problem.
Stability One of our goals is to find preference environments for which the new mech- anism induces a game with a unique and stable equilibrium. Thus far it has been shown that, in quasi-linear environments, the mechanism has a unique Nash equilib- rium and has the increasing difference and supermodular properties. These properties
5 This is an issue since Milgrom and Roberts only show that adaptive behavior converges to the bounds of the outermost Nash equilibria. If there are equilibria on the boundary then there is no predictive power. 100 were also shown for the Chen mechanism in her 2002 paper. These two properties are typically not enough to guarantee stability of equilibrium. In fact, it is relatively straightforward to devise mechanisms (with an unbounded strategy space) that are supermodular with a unique, unstable equilibrium. For pedagogical reasons, we pro- vide the following two-player, variation of the Walker mechanism as an example.
Specifically, each consumer chooses a 2 dimensional message in R2,wherefor each agent =() is an arbitrary element. These messages are collected by the planner and used to determine an amount of the public good and a tax for each player
according to outcome functions (r s) and (r s) respectively. For any positive real
numbers , ,and,let (r s)= (r s) ( (r s)) be a mechanism with − =1 ³ ´ outcome functions defined as follows:
(r s)=1 2 −
2 1 2 1(r s)=( 2 2) ( )+ (1 2) + (2 1) − − · 2 − 2 −
2 1 2 2(r s)=( + 1 + 1) ( )+ (1 2) + (2 1) . · 2 − 2 −
To distinguish this mechanism from the new mechanism we refer to it as (NS for “not stable”). It is relatively straightforward to show that, for any choice of the
parameters, will Nash implement the Lindahl allocations of a general environ- ment and in quasi-linear environments the mechanism induces a supermodular game with a unique equilibrium, but the unique equilibrium is unstable.6 In other words,
6 It has eigenvalues that are positive (which rules out stability in continuous time) and outside 101 we can replicate Theorem 1, Corollary 1, and Theorem 2 (minus the conditions on
the mechanism parameters) for .
The existence of such a mechanism would seem to contradict the Milgrom and
Roberts stability theorem, but recall that an unbounded strategy space does not meet the criteria of their theorem. Specifically, the strategy space needs to be a complete lattice. If the strategy space is compactified for these problematic mechanisms, there would be boundary equilibria, and the Milgrom and Roberts stability result (which now applies) only predicts behavior will coincide between the extrema equilibria — which is not very useful. One way we can rule this sort of thing out is by creating conditions that ensure compacting the strategy space would not create new equilibria.
The most natural method to do this is to look for conditions that make the best reply mapping a contraction, for the following reasons. First, if the best reply map- ping is a contraction, the equilibrium will be unique whether the strategy set is R2 or a compact rectangle in R2 . This observation makes the theorem immediately relevant to the problems observed in the previous section. Second, the Contraction
Mapping Theorem provides an algorithm for finding the unique fixed point of the game. We will elaborate on the application of this part of the theorem in Corollary
1. Clearly, a contraction mapping is a powerful tool. However, the sufficient condi- tions for such a mapping are sometimes difficult to use. In this section, we provide the somewhat surprising result that if the new mechanism induces a supermodular the unit circle (which rules out stability in discrete time). 102 game, then the best reply map is always a contraction. Thus, we can replace the less tractable sufficient conditions found by brute force calculating the slopes of the reac- tion curves with the relatively simple parameter conditions from Theorem 2. Then, taking advantage of the Contraction Mapping Theorem, we show that the sufficient conditions for uniqueness and stability of the Nash equilibrium are satisfied even if the strategy space is compactified.7 This can also be viewed as an alternative proof to the remark. While we later argue there is no need to compactify the strategy space, the discussion is useful since it highlights several issues in this literature.
The following theorem reports the contraction result.
Theorem 6. If , ,and satisfy the supermodularity restrictions of Theorem 2, then the best reply mapping is a contraction.
Proof. See Appendix.
Theorem 2 guarantees uniqueness of equilibrium so long as the strategy space is a complete metric space. Throughout the paper the complete metric space R2 (with the usual metric) has been used. Consequently, Theorem 2 provides an alternative proof to the uniqueness result in Corollary 1. Since the best reply mapping is a contraction and since the compact rectangle in R2 (with the usual metric) is still 7 I am grateful to PJ Healy for many comments that have greatly improved this section of the paper. 103 a complete metric space, we can compactify its strategy space and remain confident that our Nash equilibrium is unique (and finite). Therefore, if one were inclined to compactify the strategy space the mechanism will induce a game with a unique equilibrium that also satisfies the Milgrom and Roberts’ dynamic stability properties.
Thus, at the cost of shrinking the set of applicable preference environments to quasi- linear environments, we gain the property that only rationalizable strategies coincide with the Nash strategies and that the unique equilibrium is stable under “adaptive” learning dynamics such as fictitious play, -period average best response, and Bayesian learning.
Unfortunately, compactifying the strategy set in this manner is an unacceptable way of guaranteeing stability for this class of mechanisms. Despite the fact that we alwayshaveauniqueequilibrium,wecannotbesurethattheequilibriumcorresponds to the Lindahl outcome unless the strategy sets are compactified in such a way to keep the original equilibrium strategies in the strategy space. A planner would, in general, not have enough information to guarantee that equilibrium messages would be in the interior of the compactified message space. Thus, by arbitrarily compactifying the message space, we could actually eliminate the nice equilibrium outcome and prevent rational players from learning to achieve the Lindahl allocation. Fortunately, using the result from Theorem 3, stability of equilibrium can be ensured under some learning dynamics without resorting to compacting the strategy space. We formalize this statement in the following corollary. 104
Corollary 7. If , ,and satisfy the supermodularity restrictions of Theorem 2, then the unique equilibrium of the induced game is stable under the myopic best reply learning algorithm.
The corollary follows immediately from the “Successive Approximations” result of the Contraction Mapping Theorem. Thus, starting at any initial strategy profile and iterating the best reply mapping we are guaranteed by the Contraction Mapping
Theorem to converge to the unique equilibrium — i.e., the equilibrium is stable un- der the myopic best reply learning algorithm. Thus, in quasi-linear environments, without resorting to compactifying procedures, supermodularity of the game induced by the new mechanism actually ensures existence, uniqueness, and global stability of equilibrium.8
3.5 Out-of-Equilibrium Tax Penalties of the New Mechanism
In equilibrium, all Lindahl mechanisms yield the same, nice welfare properties. How- ever, in practice, we do not expect people to immediately find equilibrium. We have
8 Some non-equilibrium messages might take consumers outside of their consumption set .This is a little troubling when considering out-of-equilibrium dynamics. Kim (1993) gets around this issue by taking the consumption set to be the whole of R2. For our environment, we can always find a neighborhood around equilibrium where all messages are feasible and equilibrium is locally feasible and stable. 105 argued throughout this paper that ensuring the dynamic stability of equilibrium is essential for participants in a Lindahl mechanism to find their way to equilibrium.
The loss in welfare to consumers and the surplus/ deficitintaxrevenuethatthe government incurs while consumers are learning to play equilibrium can be consid- ered the cost of implementing the mechanism. These issues are highlighted in the
VLW experiment where the Chen mechanism’s out-of-equilibrium tax penalties are often quite severe. In this section, we briefly highlight a structural property of the new mechanism that was chosen specifically to provide a better alternative to the
Chen mechanism — i.e., the manner in which the new mechanism penalizes incorrect statements.
In the new mechanism, for each consumer , his personalized price function de- pends only on the statement of consumer +1. The personalized price of Chen’s mechanism depends on the statements off all the other players. This seemingly in- nocuous choice of personalized price function actually yields a potential welfare issue related to the statement penalties. In order to get the right complementarity between actions in quasi-linear environments, this choice of personalized price requires Chen to include a separate squared difference penalty for each consumer in the economy.
2 In other words, a term ( (r s)) is added to the Chen tax function for each 2 − consumer = in the economy. While in equilibrium each of these terms will be 6 equal to zero and drop out of the tax function, when out of equilibrium, even small incorrect statements by each player can quickly increase the taxes each consumer has 106 to pay (the magnitude of the penalties depends on the specific parameterization of the mechanism). This welfare issue was documented by Van Essen, Lazzati, and Walker
(2008), where, in an experiment, subjects’ incorrect statements often created large losses for all consumers, as well as generated large revenue swings to the government, and overall losses in welfare.
Since consumers in the new mechanism have only one penalty term connected to the statement of their neighbor, statement penalties for each consumer in a similar
(parametric) situation to the situation mentioned above will also be significantly smaller. Additionally, from a welfare perspective, individuals are shielded from large incorrect guesses by everyone other than their partner. In actual implementation of the mechanism, it is easy to imagine that one consumer in a group may be a little slow to correct his statement. In the Chen mechanism, every participant pays for this slowness, while in the new mechanism only one other person is affected. Lastly, since for any economy size , the new mechanism’s personalized price for consumer
depends only on the statement of his neighbor +1, it maintains this bilateral structure as increases.9 The degree to which this structural difference in penalties matters in actual implementation is an empirical question and is currently the subject of ongoing research.
9 This circular ordering structure has been used by Hurwicz (1979) and Walker (1981) in their Lin- dahl mechanisms. Saijo (1988) also uses the same circular ordering structure to obtain a significant strategy space reduction of Maskin’s Canonical mechanism. 107
3.6 Conclusion
We have introduced a new incentive compatible mechanism capable of implementing
Lindahl allocations as Nash equilibria. While a simplified economy with two goods was used for the exposition, it is straightforward to generalize the mechanism to ac- commodate economies with an arbitrary number of private and public goods. We have seen in an example that a incentive compatible Lindahl mechanism inducing a supermodular game is not enough to get a dynamically stable Nash equilibrium.
This observation led us to use the contraction mapping as our tool to produce sta- bility. Ironically, for this mechanism, we have shown that inducing a supermodular game is sufficient to guarantee that the best reply mapping is a contraction. Thus, supermodularity gives us a relatively simple condition to produce stability. Finally, we remark that the new mechanism has several desirable behavioral properties that suggest its out-of-equilibrium performance will improve on the difficulties with the
Chen mechanism that were observed in the VLW experiment.
There are several interesting areas for future research. For example, it is known that two-dimensional stable Lindahl mechanisms can be found in quasi-linear pref- erence environments (i.e., the mechanism introduced in this paper). And while the stability results in quasi-linear environments are important, it is unknown what is the maximum preference domain for stable environments. A natural extension of the quasi-linear environments could be those defined by generalized Bergstrom-Cornes 108 preferences. It would also be nice to know if it is possible to find a Lindahl mecha- nism that is stable for some environments and always in budget balance; or a stable, one-choice-variable, Lindahl mechanism. Additional research on Lindahl and Wal- rasian contractive mechanisms is currently being explored by Healy and Mathevet
(2009), who show, in a manner akin to the Milgrom and Roberts’ results, contrac- tive mechanisms induce games for which a wide variety of learning rules (other than myopic best reply) converge to the equilibrium bounds in this framework. Finally, we need more experiments on implementation theory. Experiments give us a better handle on what mechanism characteristics work or do not work in a more applied environment. 109
Chapter 4
A Note on Chen’s Lindahl Mechanism
4.1 Introduction
Mechanisms that induce supermodular games with a unique equilibrium have tended to be very successful in the lab.1 Milgrom and Roberts (1990), hereafter MR, have shown if a supermodular game has: first, strategy spaces which are complete lattices; and second, a unique equilibrium; then its equilibrium is stable under a variety of learning behavior such as myopic best reply and fictitious play. Motivated by the aforementioned experimental studies as well as the MR result, Chen (2002) proposed a parametric family of Lindahl mechanisms (i.e., mechanisms that Nash implement
Lindahl allocations) which, under some conditions of the mechanism parameters, induced a supermodular game with a unique equilibrium. Citing MR, she concluded that the equilibrium of the induced game would be robustly stable. This conclusion is slightly premature.
TheeconomicenvironmentinwhichChenworks does not satisfy the conditions of MR — in fact, requires that MR not be satisfied. While the Chen mechanism does
1 See Chen and Tang(1998), Healy (2004), or Chen and Gazzale (2004). 110 induce a supermodular game, the strategy space is not compact (a requirement of the MR result) thus, no inferences on stability of equilibrium can be made. As a cautionary note both Van Essen (2009) and Healy and Mathevet (2009), in the same environment as Chen’s, have provided examples of other Lindahl mechanisms that induce supermodular games with a unique, unstable equilibrium.2 However, neither paper shows that the Chen mechanism is unstable. It therefore remains to be shown whether the Chen mechanism does induce a game with a unique, stable equilibrium.
In this paper, we show sufficient conditions for the Chen mechanism to induce a game whose best reply map is a contraction. These conditions are difficult to work with in practice. However, if the game induced by the Chen mechanism is supermodular, then the sufficient conditions for a contraction mapping are satisfied. This gives us a relatively easy to check sufficient condition for stability.
4.2 Stability and the Chen Mechanism
Our setting applies to 2 consumers. We restrict attention to economies with one ≥ private good, one public good, and a constant returns to scale production technology.
The quantity of the public good will be denoted by , and the private good for consumer by , where consumers are indexed by subscript . Each consumer is
2 Van Essen’s example is with 2 players and induces a supermodular game no matter what value of the mechanism parameters are chosen. Healy and Mathevet’s example holds for 4 “even ≥ economies.” 111
2 characterized by the convex consumption set = R+, an initial endowment of the private good 0, and no initial endowment of the public good. The public good is produced, using the private good as an input (quantity denoted ), with a constant
returns to scale production technology ()= – i.e., each unit of the public good requires units (0)oftheprivategood.Thus istheconstant(real) marginal cost of production. An allocation in this simple economy is an ( +1)-
+1 tuple ( 1 ) R+ . ∈ The Chen mechanism is an institution in which consumers report messages to a
“planner” who uses this information to determine an amount of the public good to produce and a tax for each consumer. The message space of consumer is
2 = R with generic element =() Consumer ’s action should be in- terpreted as a request from the consumer to the planner for units of the public good.
Negative requests are allowed. Consumer ’s other action, , is interpreted as his statement about the amount of the public good that will be produced. We write
(12 1 )=( ) for a strategy profile. These messages are collected by the planner authority and used to determine an amount of the public good and a tax for each player according to outcome functions ( ) and ( ) respec- tively. The mechanism has exogenous parameters , 0 and 0 which can be ≥ manipulated to affect stability.
Let ( )= ( ) ( ( )) be the Chen mechanism with outcome − =1 ³ ´ functions defined as follows: 112
( )= =1 X 1 2 2 ( )= ( ) ( )+ ( ( )) + ( ( )) · 2 − 2 − = X6 where
1 ( )= − − Ã = = ! X6 X6 can be thought of as ’s personalized price for the public good.
Finally, in order to get a unique interior Nash and Lindahl equilibrium, we restrict attention to the following class of economic, quasi-linear environments.
Quasi-linear Economic Environment: denotes the set of standard 2 quasi-linear environments — i.e., those in which,
1. For each , there is a real-valued function such that ( )= + ().
2. is 2, where its second derivative is bounded from above and below— i.e., for
¯ ¯ each ,thereexistsK R such that K 11 0. ¯ ∈ − −∞ ¯ ≤ ≤
Ω 3. 1(0) and 1( ) — i.e., that there is unique, interior Pareto
optimalP level the publicP good that does not exhaust the economy’s private good
supply, where Ω = . P 4. For each , ( ) 0 — i.e., each consumer has enough wealth to − 1 ≥ cover his or her Lindahl taxes. 113
In the environment, a mechanism induces a supermodular game if for each
the following inequalities hold: first, for all = ,wehave 0, 0, 6 ≥ ≥ 0,and 0;second, 0.3 Chen proves that her mechanism Nash ≥ ≥ ≥ implements the Lindahl allocations for general environments (including )andgives conditions for which her mechanism induces a supermodular game.
Theorem (Chen (2002)): For each ,themechanism( ) Nash ∈ implements the unique Lindahl allocation of the economy. Furthermore, if , ,and
are set such that
[1 ) and [1 +( 1) ] ∈ − ∞ ∈ − −
then the game induced by ( ) will be supermodular, where =minK. ¯
Since the message space of the Chen mechanism is R2,wecannotappealtotheMR stability result for supermodular games. However, MR is not needed, as shown in the following corollary to her theorem. If the Chen mechanism induces a supermodular game, then the best reply mapping is a contraction. Furthermore, by applying the successive approximations result from the Contraction Mapping Theorem, we know the equilibrium is stable (at least under myopic best reply).
Corollary: If the Chen mechanism induces a supermodular game, then the best reply mapping will be a contraction and the unique equilibrium of the game induced by ( ) will be stable under myopic best reply learning behavior.
3 See Amir (2005) or Vives (2001) for a good introduction on the theory of supermodular games. 114
As a proof, we directly calculate the slopes of the best response system and show that when the mechanism induces a supermodular game, the best response system satisfies the following sufficient condition for the best response mapping to be a contraction— i.e., for each ,wehavethat
∗ + ∗ 1 = ¯ ¯ = ¯ ¯ X6 ¯ ¯ X6 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∗ ¯ + ¯ ∗ ¯ 1, = ¯ ¯ = ¯ ¯ X6 ¯ ¯ X6 ¯ ¯ ¯ ¯ ¯ ¯ 4 where ∗ and ∗ are ’s optimal actions¯ ¯ given¯ the¯ actions of his rivals. In words, this condition reads that if all players other than change each of their actions and by 1,player’s total change, for each his actions, is less than 1.
When preferences are added to a mechanism they induce a game. The Chen mechanism induces a game where each player has following preferences
2 2 ( ) ( ) ( ) ( ) . − − 2 − − 2 − = X X X6 X The first order conditions that define ’s best response system are:
[]: ( ) ( )+( )+ ( )=0 1 · − − − = X X6 X []: ( )=0. − − X Let ∗ = ∗( ) and ∗ = ∗( ) be the solution to the two first order − − − − conditions. Plug these solutions into each other to get the following augmented FOC:
4 That this is a sufficient condition follows from the mean value theorem and that [k ¯] — 11 ∈ ¯ by the Weirstrass theorem, for any fixed set of parameters, the total derivative of the best response system is bounded. 115
(∗ + ) ( )+ ( ∗ )=0 (4.1) 1 − − − = = X6 X6 X ∗ ∗ =0 (4.2) − − = X6 Differentiating equation (1) with respect to we get
∗ ∗ 11( +1)+ ( 1) ( 1) = 0. − − − −
We can then solve for ∗ 11 + ( 1) = − − ( 1) − − 11
Differentiating equation (1) with respect to we get
∗ ∗ 11 + ( 1) =0. − − −
We can then solve directly for
∗ + = − ( 1) − − 11
∗ ∗ Now checking the first sufficient condition = + = 1,weneed 6 6 ¯ ¯ ¯ ¯ + ( 1) P ¯ ¯ +P ¯ ¯ 11 − − + ¯ −¯ ¯ ¯ 1. ( 1) ( 1) = 11 = ¯ 11 ¯ 6 ¯ − − ¯ 6 ¯ − − ¯ X ¯ ¯ X ¯ ¯ ¯ ¯ ¯ ¯ Since the mechanism¯ parameters are such¯ that¯ the induced game¯ is supermodular,
then ∗ and ∗ are positive for each . We dispense with the absolute values in our
first sufficient condition yielding
+ ( 1) + 11 − − + − 1. ( 1) ( 1) = 11 = 11 X6 − − X6 − − 116
Simplifying this expression reads
( 1) . − − −( 1) 11 µ ¶ − Since 0 (by assumption of being in ) and from supermodularity 11 ≥ therefore this condition is satisfied.
∗ ∗ Next, we check the second condition = + = 1.Differentiating 6 6 ¯ ¯ ¯ ¯ equation (2) we find ∗ = ∗ +1and P∗ = ¯ ∗ ,thereforeweneedthat¯ P ¯ ¯ ¯ ¯ ¯ ¯
+ ( 1) + 11 − − +1 + − 1. ( 1) ( 1) = 11 = ¯ 11 ¯ 6 ¯ − − ¯ 6 ¯ − − ¯ X ¯ ¯ X ¯ ¯ ¯ ¯ ¯ ¯ Ourgameissupermodular,therefore¯ ∗ and¯ ∗ ¯for all and are¯ positive — i.e., we have
( 1) + − ( 1) 11 − +1 1. − ( 1) Ã − − 11 ! Simplifying, this expression reads
1 ( 1) . − − − ( 1) 11 µ ¶ − Since 0 (by assumption of being in )and (from the supermodularity 11 ≥ conditions) this condition is satisfied.
Thus, if the parameters of the Chen mechanism induces a supermodular game, the best response mapping will be a contraction and the equilibrium will be unique and stable under the myopic best reply learning behavior. 117
4.3 Conclusion
We have shown that having Chen’s Lindahl mechanism induce a supermodular game, in the quasi-linear economic environment, is sufficient to guarantee stability of the unique Nash equilibrium of the game. This property is not a universal phenomenon among incentive compatible Lindahl mechanisms as demonstrated by both Healy and
Mathevet (2009) and Van Essen (2009), but it is not unique either. Van Essen (2009) introduces another Lindahl mechanism for which this is the case. Why this condition holds for some mechanisms and not others is an open question. 118
Chapter 5
Information Complexity, Punishment, and
Stability in Two Nash Efficient Lindahl
Mechanisms
5.1 Introduction
The dynamic stability of an equilibrium has long served as a robust and effective indi- cator of whether an equilibrium will be attained in the lab.1 These results are critical as they give insights into real economic situations and highlight the importance of dynamic stability analysis. Recent experiments on Nash efficient public good mech- anisms have continued this line of inquiry by looking at the properties of games that are induced by various mechanisms. Robustly, the distinguishing characteristic of successful mechanisms has been whether the game induced has a unique, stable equi- librium.2 While these observations are important, stability is certainly not the only
1 See, for example, Cox and Walker (1998), Chen and Tang (1998), Chen and Gazzale (2004),
Healy (2004), or Van Essen, Lazzati, and Walker (2010). 2 Nash efficient mechanisms are mechanisms that induce games whose Nash allocations are Pareto efficient. 119 mechanism property of interest. Rather, mechanisms should be evaluated based on a menu of properties including, but not limited to, dynamic stability, efficiency, budget balancedness, information complexity, severity of punishment, strength of rewards, etc. This evaluation process was envisioned by Hurwicz over 50 years ago.
The members of such a domain (of mechanisms) can then be appraised
in terms their various “performance characteristics” and, in particular,
of their (static and dynamic) optimality properties, their informational
efficiency, and the compatibility of their postulated behavior with self-
interest (or other motivational variables). —Hurwicz (1960)
One case in point is the family of so-called incentive compatible Lindahl mecha- nisms. A mechanism, or institution, in this class induces a game whose Nash equilibria allocations exactly coincide with the Lindahl allocations of that economic environ- ment. While this literature has focused on dynamic stability, other performance characteristics have been largely ignored. In this paper, we look to evaluate the performance characteristics of two, dynamically stable, incentive compatible Lindahl mechanisms in a laboratory environment.
Incentive compatible mechanisms that implement Lindahl outcomes as a Nash equilibria were first proposed by Hurwicz (1979) and Walker (1981). Unfortunately, in environments with a unique Nash equilibrium, these mechanisms have the property 120 that the equilibrium is dynamically unstable.3 Since Hurwicz and Walker, several authors who have provided solutions to this instability problem by incorporating some form of dynamic stability into the design of the mechanism. Examples include the mechanisms introduced by Vega-Redondo (1989), de Trenqualye (1989), Kim
(1993), Chen (2002), Van Essen (2009), and Healy and Mathevet (2009). All these mechanisms attain Lindahl outcomes as Nash equilibria, as in the Hurwicz and Walker mechanisms. Moreover, the first two mechanisms are stable under myopic best reply,
Kim’s mechanism is globally stable under a gradient adjustment process, and the last three mechanisms are stable under a wide variety of learning dynamics (discrete time and continuous time) for some parameter values.4
Despite the theoretical interest in the Lindahl allocation and its role in the mech- anism design literature, there have been few experiments testing incentive compatible
Lindahl mechanisms. In fact, until recently, the only Lindahl mechanism that had been tested was the one due to Walker (1981). Both Chen and Tang (1998) and Healy
(2004) tested the Walker mechanism finding that behavior was highly unstable and never observed convergence to equilibrium. Van Essen, Lazzati, and Walker (2008), hereafter VLW, was the first experiment to compare several incentive compatible Lin- dahl mechanisms in the lab. Specifically, they tested the mechanisms due to Chen
(2002), Kim (1992), and Walker (1981). These mechanisms were chosen to compare
3 In words, if people follow a best response learning style and do not start at equilibrium, then their behavior will diverge away from equilibrium. See Kim (1987). 4 —i.e., they induce games whose best reply systems are contraction mappings. 121 the behavioral properties of mechanisms that induced games with stable equilibrium to mechanisms that induce games with unstable equilibrium. The Walker mechanism, for instance, is highly unstable under any adaptive learning dynamic. In contrast, for quasi-linear environments, both the Kim and Chen mechanism induce games with a unique, stable equilibrium. The Kim mechanism, for the experimental parameters, was stable under the myopic best reply algorithm (not monotonic), and the Chen mechanism was stable under a wide variety of learning behavior (monotonic). While both Chen and Kim do converge in some respect toward their equilibrium, VLW find that the out-of-equilibrium outcomes yielded by the Chen mechanism were very neg- ative in terms of efficiency and consumer welfare. The Kim mechanism, on the other hand, yielded much better outcomes. These results are somewhat surprising given the previous work on stability and Nash efficient mechanisms. Motivated by this study,
Van Essen (2009) develops an alternative Lindahl mechanism designed to mitigate two of the most glaring issues with the Chen mechanism: its complexity as the number of consumers increases; and the degree to which it taxes out-of-equilibrium behavior.
In this paper, we experimentally examine the mechanisms due to Chen (2002) and the paired-statement mechanism (hereafter PS mechanism) due to Van Essen (2009).
In particular, we examine how changes in stability parameters and group size affect the overall performance of the mechanisms. Both of these mechanisms, in standard quasi-linear environments, induce games with a unique Nash equilibrium and have identical stability properties. We develop a simple learning model to derive two sets 122 of stability parameters (for each group size) and our experimental predictions. The
first set of parameters allows both mechanisms to induce supermodular games which satisfy the conditions of the Milgrom and Roberts’ stability theorem.5 Due to the empirical success that this class of games regularly exhibits, we use this treatment as a benchmark. However, supermodularity, in the experimental environment, is only a sufficient condition for stability. Thus, there is nothing to say that this supermodular set of parameters is optimal for performance. For our second set of parameters, we use a simple learning model to suggest an alternative set of parameters (which do not induce supermodular games) which, for both mechanisms yields very fast convergence to equilibrium under myopic best reply. We predict this set of parameters will out- perform the supermodular parameters. The experimental data largely confirms this prediction. Treatments using the non-supermodular parameters yield outocmes much closer to their equilibrium allocations (with several groups actually achieving Lindahl allocations). Furthermore, these groups achieve much higher levels of overall efficiency on average and have fewer violations of individual rationality.
The second purpose of this experiment is to provide an in depth comparison of two, seemingly similar, incentive compatible Lindahl mechanisms. While not a true
“apples to apples” comparison, we perform a “between mechanism” comparison of
5 Milgrom and Roberts (1990) have shown that supermodular games with compact strategy sets and a unique equilibrium are stable under a wide variety of adaptive learning dynamics, including myopic best reply, Bayesian learning, and fictitious play. 123 the convergence to equilibrium, efficiency, and welfare performance of the two mecha- nisms for each group size — holding constant stability conditions. These comparisons highlight several, seemingly innocuous, structural differences between the mechanisms which seem to matter a great deal in actual implementation: the severity of out-of- equilibrium punishment; and the amount of information consumers need in order to compute their payoff. The experiment hypotheses are based on conjectures made in
Van Essen (2009). Along these lines, we provide a number of arguments that suggest that the performance of the Chen mechanism will diminish relative to the PS mecha- nism when group size increases. Again, the experimental results seem to support this hypothesis. Treatments using the PS mechanism achieve higher levels of overall effi- ciency, on average, and have fewer violations of individual rationality when compared to similar treatments using the Chen mechanism.
The plan for the rest of this paper is as follows: in section 2, we describe the public good environment, define the two mechanisms and the mechanisms’ stability properties for a simple myopic learning algorithm; in section 3, we derive our research hypothesis; sections 4 and 5 presents the experimental design and procedures; in sections 6 we present our experimental results; section 7 concludes. 124
5.2 Theoretical Preliminaries
5.2.1 The Basic Public Goods Problem
The basic public good problem can be described as follows. There are +1agents in an economy ( consumers indexed =1 and 1 producer). Each consumer is endowed with some amount of the private good , and have preferences over two goods and ,where is a public good common to all players and is a private good. Assume that preferences can be represented by a payoff function of the form
2 ( )= + . The producer can use the private good to make units − of the public good according to the constant returns to scale production function
= ()= ,where is the marginal cost of production.
Socially efficient outcomes, or Pareto optimal allocations are the ones that max-
− imize social surplus (i.e., = 2 )andefficiently use the remainder amounts of the private good (i.e., = ). − One particularly niceP Pareto optimalP allocation is the one defined by Erik Lindahl in 1919. The Lindahl allocation is a Pareto optimal allocation where individuals pay a tax proportional to their marginal benefit of the public good (evaluated at the Pareto optimal level of the public good). In other words, each individual pays a price equal
to = 2 per unit. When faced with these prices, each consumer demands − the Pareto optimal level of the public good and pays a tax of =( 2 ) − which, by construction, is a Pareto optimal allocation. Despite the nice properties 125 of the Lindahl allocation, it is uncertain how the planner could determine enough information about individual preferences to derive the appropriate Lindahl prices.
This has created interest in the public good mechanism design literature. In the next section, we describe two incentive compatible Lindahl mechanisms due to Chen
(2002) and Van Essen (2009). These mechanisms are of interest since, when agents are acting strategically, the equilibrium allocation of the game induced by these two mechanisms corresponds to the Lindahl allocation. Thus, acting in their own interest and strategically, consumers arrive at the Lindahl allocation without the assistance of the planner.
5.2.2 Incentive Compatible Lindahl Mechanisms
In this section, we describe two different incentive compatible Lindahl mechanisms due to Chen (2002) and Van Essen (2009). For simplicity, we refer to the mechanism from Chen’s 2002 paper as the CH mechanism and index it by the capital letter
. Furthermore, we refer to the mechanism introduced in Van Essen (2008) as the
“Paired Statement” mechanism and index it by .
A mechanism takes consumers’ actions (or messages), and maps them into an out- come (or allocation). Here we consider mechanisms in which consumers report mes- sages to a “planner,” or the government, who uses this information to determine an amount of the public good to produce and a tax for each consumer. The message space
2 of consumer is = R with generic element =() Let =(1 ) 126
denote the profile of all players’ messages. Consumer ’s action should be inter- preted as a request from the consumer to the planner for units of the public good.
Negative requests are allowed. Consumer ’s other action, , is interpreted as his statement about the amount of the public good that will be produced. Rather than write (1122) for a strategy profile, we write (12 1 )=( ).
These messages are collected by the planner and used to determine an amount of the public good and a tax for each player according to outcome functions ( ) and
( ) respectively. We write ( ) and ( ) to refer to outcome functions defined by Chen. The outcome functions for the Paired Statement mechanism are similarly defined with subscript .
Let ( )= ( ) ( ( )) be a mechanism with outcome func- − =1 ³ ´ tions defined as follows:
The Chen (CH) Mechanism The Chen mechanism takes messages from con- sumers and produces an outcome according to functions
( )= =1 X 2 2 ( )= ( ) ( )+ ( ( )) + ( ( )) · 2 − 2 − = X6 where
1 ( )= − − Ã = = ! X6 X6 127 can be thought of as ’s personalized price for the public good and , 0 are positive parameters.6
The Paired Statement (PS) Mechanism The Chen mechanism takes messages from consumers and produces an outcome according to functions
1 ( )= =1 X 2 2 ( )= ( ) ( )+ ( ( )) + (+1 ( )) · 2 − 2 −
where
( )= +1 − 1 − Ã = ! X6 − can be thought of as ’s personalized price for the public good and , , 0 are positive parameters. Furthermore, we write +1 = 1.
Notice that both of the mechanisms have exogenous parameters , , which can be manipulated to affect stability.
Both of these mechanisms fully Nash implement the Lindahl allocations of gen- eral preference environments.7 Moreover, for quasi-linear preference environments and a judicious choice of mechanism parameters, the mechanisms induce games with a unique, dynamically stable Nash equilibrium. This is the environment we are in- terested in for this paper. In the next section, we discuss the stability properties of
6 Chen’s mechanism was orginally presented with =1. 7 See Maskin (1999). 128 each of these mechanisms for the quadratic preferences described in the beginning of this section.
5.2.3 Stability of the Myopic Best Reply Learning Algorithm in the CH
andPSmechanisms
In this section, we derive a simple, discrete time, myopic best reply learning algorithm for both the CH and the PS mechanisms. While simple, this model clearly indicates how each of the mechanisms’ “free” parameters can be manipulated to stabilize equi- librium and speed convergence.
The CH and PS mechanisms induce games. Nash equilibrium behavior by each agentrequiresthatgiventheactionsofothers( ), chooses () to maximize − −
her utility (( ) ( )). The best reply system for is derived from the first − order conditions (FOC) of this maximization problem. Next, we adopt the FOC into a simple myopic (or Cournot) learning algorithm. In this learning algorithm, consumers take the actions of their rivals in the previous period as fixed, and choose the request and statement to maximize their utility. This assumption yields the following sets of learning algorithms. In anticipation of a discussion on stability, we also include sufficient conditions for the stability of these two learning algorithms.
Myopic Best Reply Learning Algorithm (CH Mechanism) By using the
FOC, we can solve for the best response system. The myopic best reply learning 129 algorithm is found by putting superscripts on all of consumer ’s actions and putting
1 time superscripts on all actions taken by consumers other than . This results − in the following system.
( 1) 2 1 1 = − + − − − − − − 2 + ( 1) 2 + ( 1) − 2 + ( 1) = = − − X6 − X6 1 = ( )+ − − − = X6 We want to ensure that if agents are following this sort of learning rule, that their behavior converges to equilibrium behavior in a reasonable amount of time. Alterna- tively, we want the unique equilibrium of the game induced by the Chen mechanism to be stable under the myopic best reply learning algorithm. A sufficient condition for the Chen mechanism to be stable under myopic best reply is to require that, for each , if every other player increases their actions by 1 unit, then wants to change his request and his statement by strictly less than one.8 In other words, to require that for each ,both
( 1) 2 ( 1) − − − +( 1) − 1 − 2 + ( 1) − ¯2 + ( 1)¯ ¯ − ¯ ¯ − ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ and ¯ ¯ ¯ ¯ ( 1) 2 ( 1) − − − +1 +( 1) − 1 − 2 + ( 1) − ¯2 + ( 1)¯ ¯ − ¯ ¯ − ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ are satisfied. ¯ ¯ ¯ ¯
8 See, for example, Namate and Tse (1981). 130
Discrete Time Best Reply System (PS) Similarly to the Chen mechanism, we use the FOC to solve for the best reply system and add time superscripts to get the following learning algorithm for the PS mechanism.
1 2 1 1 =( − )+( − − − ) − ( − )+1− 2 + 2 + − 2 + = X6 1 1 1 = ( )+ − − − = X6 Akin to the CH mechanism, the corresponding sufficient condition for stability of the
PS mechanism is to require that for each ,both
1 2 ( 1) − − − + − 1 − ¯ 2 + ¯ 2 + ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ and ¯ ¯ ¯ ¯ 1 1 2 1 ( 1) − − − + + − 1 − ¯ Ã 2 + ! ¯ 2 + ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ are satisfied. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ This concludes the theoretical preliminaries of the paper. In the next section, we investigate how changes to the mechanism environment might affect the behavior of consumers using these mechanisms and formulate several testable hypothesis.
5.3 Research Hypotheses
In this paper, we are interested in how changes of the mechanisms’ stability parameters and different group sizes affect the performance of the PS and CH mechanisms. The 131 cleanest test of these items will come from a within “mechanism and group size” com- parison, however, we will also make some predictions about the relative performance of these two mechanisms based on the structural differences of the two mechanisms.
In the next three sections we formulate the testable predictions for the experiment.
5.3.1 Hypotheses based on Stability Parameters
The best response system for each mechanism and the postulation that people are behaving akin to a myopic best reply maximizer gives us several testable predictions about the effect of a change in each mechanisms’ stability parameters. We start with a parameter condition that has been seen frequently in mechanism design experiments and discuss its implications toward stability.
Supermodularity and Stability The first set of parameters we would like to investigate are ones that let the mechanisms induce supermodular games. Some experimental studies have lauded the virtues of mechanisms that induce supermodular games.9
Interestingly, for these two mechanisms, if the game induced by these mechanism is supermodular, then the players’ best response system will be a contraction mapping and the unique equilibrium of the game will be stable under the myopic best reply
9 See, for example, Chen and Plott (1996), Chen and Tang (1998) which studt the Groves-Ledyard
Mechanism, and Chen and Gazzale (2004) which look at Varian’s compensation mechanism. 132 learning algorithm.10
Definition 1: The CH or PS mechanism induces a supermodular game if, for each
, the payoff functions of the induced game satisfy both the increasing differences and supermodularity conditions.11 In other words, we require that, for each player ;
1. (Increasing Differences) For all = ,wehave 0, 0, 0, 6 ≥ ≥ ≥ and 0; ≥
2. (Supermodularity) 0. ≥
It is straightforward to verify that the CH mechanism induces a supermodular game if and only if for each , the condition
( 1) +2max + ≥ ≥ − is satisfied. Similarly, the PS mechanism induces a supermodular game if and only if for each , the conditions
2max ≤ 1 − − ∈ ( 1) − + +max2 ∈ ∙ ³ ´ ¸ 10Van Essen (2009) proves this statement for the PS mechanism and also provides a counter example to the claim that supermodularity is a sufficient condition for stability for all Lindahl mechanisms. Van Essen (2010) proves this statement for the Chen mechanism. 11See Amir (2005) or Vives (2001) for a good introduction on the theory of supermodular games. 133 are satisfied. In the experiment, we will have a set of parameters that induce su- permodular games for a given mechanism and group size. We index conditions that satisfy the supermodularity conditions by the name SPM Treatment (for super- modular).
Other Parameter Conditions For both the PS and the CH mechanisms inducing a supermodular game is only a sufficient condition for dynamic stability. In other words,wemaybeabletofind parameters that lead to better allocations using a set of parameters that do not satisfy the supermodularity conditions we listed earlier. The following argument offers one suggestion for a “optimal set” of stability parameters.
First, it is clear that in both mechanisms the optimal statement should equal the expected level of the public good. This behavior minimizes the expected incorrect guess tax a player would have to pay in the mechanism. Changing either of the mechanisms’ parameters only influences the optimal statement through its influence on the optimal request. Intuitively, if a change in the mechanisms’ parameters is one that stabilizes the optimal request for all agents — i.e., decreases the impact of small change in other players’ strategies on the own optimal requests — the variance of the optimal statement will decrease as well. Reducing the variance of the optimal request is akin to saying we want, for each player , to reduce the impact of other players’ strategies on the optimal request. However, we can manipulate the impact of other players’ strategies the optimal request through a judicious choice of each mechanism’s 134 parameters. The following two conditions illustrate.
Invariant Optimal Request Condition (Chen): If = and ( 1) − − −
2 =0, then the best reply system for the Chen mechanism looks as follows:
= − 2 + ( 1) − 1 = ( )+ − − − = X6 where player ’s optimal request is invariant in the decisions of the other group mem- bers.
Player ’s optimal statement will still depend on the requests of his rivals. We can make the same observation for the PS mechanism.
Invariant Optimal Request Condition (PS): If = and 1 2 =0, − − − then the best reply system for the PS mechanism looks as follows:
= − 2 +
1 1 1 = ( )+ − − − = X6 where player ’s optimal request is invariant in the decisions of the other group mem- bers.
The invariant request conditions for each mechanism do not satisfy the super- modularity conditions we listed earlier.12 Therefore, for the rest of the paper, we
12 This is since for the CH mechanism, supermodularity requires ( 1) 2 0;and − − − ≥ ( 1) the PS mechanism requires − 2 0. − − ≥ 135 refer to these two invariant conditions as the NSPM treatment (NSPM for non- supermodular) for their respective mechanisms. Since for each player the optimal request in the NSPM is invariant to the decisions of others, it is reasonable to expect that this treatment will lead to faster convergence to equilibrium— i.e., the optimal request is always a best response. If everyone is near their optimal request, the op- timal statements will also be close to their equilibrium levels. This observation is
Hypothesis 1.
Hypothesis 1: (NSPM and Equilibrium) The deviation of messages from equi- librium, for both mechanisms, will be smaller in the NSPM treatment compared to the SPM treatment — holding constant mechanism and group size.
A consequence of equilibrium (and near equilibrium) behavior is that, in the
NSPM treatments, both the public good level and the mechanisms’ taxes should be closer to the Lindahl prescribed level. It follows that efficiency should also be higher in these treatments.
Hypothesis 2: (NSPM and Efficiency) Efficiency will be higher in the NSPM treatment compared to the SPM treatment — holding constant mechanism and group size. 136
5.3.2 Hypotheses based on Mechanism Comparison
The hypothesis based on stability parameters apply to a within mechanism compari- son. We would, however, like to be able to say something about relative performance of the two mechanisms when we hold these stability parameters constant. Compar- ing the Chen and PS mechanisms, we have two mechanisms designed to achieve the
Lindahl allocation. In fact, based on previous experimental evidence, there is strong reason to expect that these mechanisms might actually be successful. This is since both mechanisms induce games with a unique, stable equilibrium. Moreover, our learning model predicts that, for appropriate choices of the mechanism parameters, the two mechanisms will have very similar dynamics converging toward equilibrium.
Thus, nothing we have argued thus far would lead us to expect different performance from one of the mechanisms. Despite these similarities, however, there are several reasons to expect that the performance of the PS mechanism will yield better out- of-equilibrium outcomes than the Chen mechanism (at least for larger group sizes) and perhaps faster convergence toward equilibrium. For the remainder of the section we highlight several of these key differences which lead to several additional testable hypotheses.
Differences in Penalty Taxes The first key difference between these two mechanisms is the structure of the statement penalty tax. For every consumer in the economy, 137 the Chen mechanism requires a separate statement penalty
2 ( ( )) 2 − be added to consumer tax function. Hence there are penalty terms. The PS mechanism always has two squared penalty terms for each consumer regardless of the number of consumers. It now becomes easy to conjecture about why this difference should matter. When the mechanism is in equilibrium these penalty terms are zero. However, when not in equilibrium it is possible that consumers’ statements are neither coordinated nor correct resulting in higher tax penalties. Simple day to day experience tells us that it is harder to coordinate with group members when more people are involved. It stands to reason that since minimizing the statement penalty tax in Chen mechanism requires coordination from more people, there will be more, relative to the PS mechanism, penalties from the lack of coordination. This lack of coordination will be observed in both higher efficiency losses and more violations of individual rationality.
Differences in Information Complexity The second key difference between the two mechanisms is the minimum dimension of data subjects need to compute their payoffs.
This information can be used as a rudimentary index of information complexity and, using this index, we can form hypotheses about the behavioral properties of different mechanisms. 138
"Paired Statement" Mechanism Data
1. The total request of all other players
= X6 2. The statement of their “neighbor” player +1
+1.
Chen Mechanism Data:
1. The total request of all other players
= X6 2. The individual statements of all of the other players. Player 1, for example,
would need 23 .
As the number of players participating in a mechanism increases, the number of individual pieces of data required by the Chen mechanism also increases. This is in contrast with the PS mechanism, where the information requirement stays constant.
So for any consumers participating in the mechanism, the Chen mechanism requires
pieces of data while the PS mechanism only requires 2. The graph below illustrates the divergence in the information requirement of these two mechanisms as increases.
The information complexity of a mechanism can be thought of as a “cost” for participants to evaluating their options. In order to find a best response, for instance, 139
6 Dim. of Data
Chen
4
PS 2
0 2 3 4 5 6 N
Figure 5.1. Divergence of Information Requirement a player would have to first input his beliefs about these sufficient statistics and then compare payoffs for all combinations of his actions. If this cost his higher, it takes longer to evaluate payoffs and, presumably, find a best response. In this experiment, we look at groups of 2 and groups of 6. In groups of 2, both of the mechanisms have the same information complexity. In groups of 6, the subjects in the PS mechanism still only need a minimal amount of information to compute their payoff (in fact 2 pieces of information), while for the CH mechanism, it depends on the parameter conditions. In the NSPM CH treatment, the invariant optimal request condition is satisfied, thus we do not expect the increase in information complexity to matter much. However, in the SPM CH treatment, beliefs about what others are doing matters a great deal to payoffs and consequently how players determine their best 140 reply and the cost of these beliefs is significant. Thus, coupled with our argument from before about out-of-equilibrium tax penalties in the six player group, there is strong reason to expect the performance of the CH mechanism to diminish relative to the PS mechanism when group size is 6.
Hypothesis 3: (Efficiency) Efficiency will be lower in the CH Mechanism for groups of 6 compared to the PS mechanism —holding constant stability conditions.
Hypothesis 4: (Violations of IR between Mechanisms) There will be a higher frequency of violations of individual rationality in the CH mechanism for groups of 6 relative to the PS mechanism— holding constant stability conditions.
Outcome Variance Finally, we notice that the PS mechanism produces the public good according to a average request rule rather than a summation rule. This seem- ingly innocuous difference in outcome functions has the potential for creating a signif- icant deviation in mechanism performance. Specifically, for large , the level of the public good level changes little when each subject is making a small change in their
1 own requests. In other words, a one unit change in request by a subject leads to a unit change in the public good (rather than a 1 unit change in the public good that would have resulted from using the summation rule). For any given changes to sub- jects’ requests, the level of the public good will fluctuate more in the CH mechanism than the PS mechanism and this difference increases with . 141
Hypothesis 5: (Variance of Public Good Allocations) The standard deviation of the public good will be smaller for groups participating in the PS Mechanism relative to the CH mechanism.
5.4 Experimental Design and Environment
We run 8 treatments. We first varied the number of players in groups: =2and
=6.Forthe =2treatment, there are two types of players, where each player will have separate payoff functions (i.e., 1, 2).Thetype1playerhasapayoff function
2 equal to 1( 1)=1 +10 and the type 2 player has a profitfunctionequalto − 2 2( 2)=2+15 . These profit functions can be indexed by the parameter types −
and ,wherethepossibletypesintheexperimentare() (10 1) (15 1) . ∈ { } The Pareto efficient level of the public good is =8and the corresponding Lindahl taxes for Player 1 and Player 2 are 16 and 56 respectively. Thus, the payoff in equilibrium is 32 for both players. In the 6 player environment, there are three players of each type. Specifically, (11)=(33)=(55)=(10 1) and
(22)=(44)=(66)=(15 1).TheParetoefficient level of the public good is =11and the corresponding Lindahl taxes are 11 foralloftheType1 − players and 44 for the Type 2 players with a profitequalto605 for all players.
The second variation came from the choice of mechanism. Thus, for each group size, we used two mechanisms: the CH Mechanism and the PS Mechanism. The 142 request space and the statement space for the PS mechanism, for groups of two and groups of six, were both [5 15], in discrete steps of 00001. The request space for the
Chen mechanism, for both groups of two and groups of six, was [0 10], in discrete steps of 00001. Finally, the statement space for the Chen mechanism was [0 20] for groups of 2 and [0 60] for groups of six, in discrete steps of 00001.Thedifferences in the statement spaces were dictated by the differences in the public good outcome function. The request spaces for both mechanisms were the same size.
Finally, for each mechanism and group size, we vary the mechanisms’ stability parameters (SPM and NSPM). The chart below indicates the mechanism parameters used in the experiment for each treatment and the equilibrium messages corresponding to the preference parameters listed earlier. The environment category is coded by:
“Mechanism,” “Stability Treatment,” and “Group Size.”
The following table summarizes the treatments and the dates in which they were run.
5.5 Experimental Procedures
Laboratory sessions were conducted in the Economic Science Laboratory at the Uni- versity of Arizona. All subjects were undergraduate students recruited via E-mail from the ESL’s online subject database. For each treatment we have ten groups.
In each session, only one treatment was conducted. Sessions were composed of five 143
Environment 1 2 3 4 5 6
PS,SPM,2 2052 675 925 8 −−−− PS,SPM,6 10 0510975 1225 975 1225 975 1225 11
PS, NSPM, 2 1051 55105 8 −−−− PS, NSPM, 6 5055 85135851358513511
CH,SPM,2 505253545 8 −−−− CH,SPM,6 12 0521625 2042 1625 2042 1625 2042 11
CH, NSPM, 2 2051 275 525 8 −−−− CH, NSPM, 6 6051 142 225 142 225 142 225 11
Table 5.1. Experiment Parameters and Treatment Summary groups who interacted through their decisions over computer terminals. Subjects were randomly assigned to a computer (and group) where written instructions were provided. They were given time to read the instructions, after which the experimenter read the instructions aloud and entertained questions.
The software for the experiment includes two tools to aid subjects in their decision making. Each subject was provided with a "what-if-scenario" profitcalculator,which allowed the subject to input trial summary statistics corresponding to the other group members and explore how, against those trial sufficient statistics, his own decisions would affect his profit. This is equivalent to giving subjects a payoff table, but the complexity of the mechanisms’ outcome functions would require multiple extremely 144
2 Player Groups 6 Player Groups
1) PS SPM 5 Groups (3/25/2009) 5 Groups (4/08/2009)
5 Groups (3/26/2009) 5 Groups (4/10/2009)
2) PS NSPM 5 Groups (3/27/2009) 5 Groups (4/30/2009)
5 Groups (3/31/2009) 5 Groups (5/19/2009)
3) CH SPM 5 Groups (4/01/2009) 5 Groups (5/20/2009)
5 Groups (4/07/2009) 5 Groups (5/20/2009)
4) CH NSPM 5 Groups (4/07/2009) 5 Groups (5/21/2009)
5 Groups (4/08/2009) 5 Groups (5/27/2009)
Total # of Participants 80 240
Table 5.2. Session Summary 145 complex tables, suggesting that this calculator would be a better decision-making aid.
In order to eliminate biases of subjects inputting only integers decisions were entered into the computer through the use of sliders. Subjects were also able to access a screen that shows, for all prior rounds, summary statistics, the resulting public good level, and the subject’s own profit. Subjects were not required to use these decision aids.
We did not use practice rounds. Each group played 50 periods in one of the sixteen treatments. No subject participated in more than one treatment. Each period of the experiment had the same structure. Subjects were given 45 seconds to submit their requests and their statements. When all participants in the group submitted their messages, the outcome (public good level and taxes) was calculated and the following information about the just-completed period was communicated to each participant: sufficient statistics of the other players’ decisions as we have defined above; the resulting amount of the public good that was provided; and the subject’s own profit. Only per-round profit was reported, although, as described above, a subject could access a screen displaying all information he’d been provided at prior periods. Subjects were also required to record their information by hand on a record sheet. This task was included in order to ensure that at least some of a subject’s attention would be directed to how much he was earning. At the end of each session one of the 50 periods was selected at random and each subject was paid 15 cents for every experimental dollar earned in that period. Subjects remained in the same 146 group for the entire session and were paid individually at the end of the session to maintain their anonymity. No subject participated in more than one session. Sessions all lasted less than 90 minutes.
5.6 Experimental Results
5.6.1 Hypotheses based on Stability Parameters
Our first treatment is designed to look at how changing the mechanisms’ stability parameters impacts the performance of the mechanisms for both groups of 2 and groups of 6. We start by looking at the public good provision per round.
Public Good Provision In order to get a preliminary idea about the influence of the stability properties on convergence, we look at the public good each treatment pro- duces on average each round. This gives an indication about which mechanism envi- ronments seem to be successful. Figures (5.2)-(5.5) illustrate the level of the public good produced in each round, average across the five groups in the treatment, and the predicted Pareto efficient level of the public good (8 for groups of 2 and 11 for groups of 6). Furthermore, a cursory look at these figures provides some support that these mechanisms are approaching their equilibrium level of public good provision.
Moreover, the production in the NSPM treatment appears to get closer to the PO level of the public good more rapidly than its SPM counterpart. Tables 5.10-5.13 147 present this data at the individual level, where the average public good provision and the standard deviation of the public good provision, for each treatment, are averaged within the group over the 10 rounds blocks [1,10], [11, 20], [21-30], [31-40], and [41-50].
Recall, however, that our primary research question is to determine whether these mechanisms (under any treatment) converge to the Lindahl allocation (i.e., whether the mechanisms converge toward equilibrium). Convergence of the public good is certainly a necessary condition for this to be true, however, analysis of the how far the messages submitted deviated from the equilibrium messages is needed.
Equilibrium Messages Since the message space is rather large and number of rounds is small, we do not expect that subjects will find their exact equilibrium strategies. As a consequence we focus on how “close” these mechanisms get to their equilibrium. To do this, we use an augmented version of the metric proposed by VLW. Specifically, we propose to use the average “city block” distance from equilibrium. This is a simple, non-subjective measure that allows for comparisons between the mechanisms. Specif- ically, at each round , we observe a 2 tuplet of actions (1 1 ) and − measure the city block (i.e., absolute value) distance between the observed requests and the equilibrium ones and in order to collect this information in on statistic we average all of these distances. The next definition formalizes this idea.
Definition 2: Distance from equilibrium messages at time , is the average
“city block” distance between the observed messages, at time , and the equilibrium 148 messagesforthegroup.Thisisgivenbytheequation
1 1 = + . 2 2 =1 | − | =1 | − | X X
Bymeasuringdistanceinthismannerweareabletogetacommonstatisticfor each of the different mechanisms at each round. Figures (5.6)-(5.9), plot the average
1 distance from equilibrium 5 for each treatment as a function of time plotting the
SPM and NSPM treatments on the same chart for comparison. Again, for statistical analysis we look at this information by group to maintain independence of observa- tions. Tables 5.18 and 5.19 summarize the average distance from equilibrium for each treatment and each independent observation for group sizes of 2 and 6 respectively.
Data is averaged within groups over 10 round blocks. A quick look at these numbers indicate that a fair number of groups appear to get very close to equilibrium and sus- tain that closeness once achieved. In particular, Group 7 in PS2_NSPM and Group
9 in CH2_NSPM seem to sustain their closeness to equilibrium for over 30 rounds of the experiment. Only groups in NSPM treatments, however, ever dip below 1. This observation is not replicated in the six player treatments where several groups in the
PS6_SPM treatment appear to get quite close to equilibrium. Our first research hy- pothesis is that messages submitted in the NSPM treatments are statistically closer to equilibrium than their SPM counterparts. In order to test this conjecture, we use nonparametric Mann-Whitney tests, hereafter MW Test, on the data in Charts 5 and 149
6 where we run a separate testforeach10roundblock.13
In the MW test, data is labeled as either coming from group or group ,then ranked from smallest to largest (1 is assigned to the smallest observation, 2 to the second smallest, etc.). These ranks are added up according to group, where group
’s rank sum is denoted . Under the null (i.e., that the data from these groups are beingdrawnfromthesamedistribution),theseranksumsshouldnotbetoodifferent; otherwise, we may wish to reject the null. For all of the MW tests done in this paper, we have 10 independent observations for each group. Thus, for all groups and ,we
20(20+1) have that the total rank sum adds up to + = 2 =210. Using a standard
MW table, if 88 or 122, then we reject the null at the 10% level in favor of the corresponding one sided alternative. Similarly, 83 or 127 then we reject at the 5% level in favor of a one sided alternative and at the 1% level if 83 or 127.
Table 5.3 reports the rank sum of the NSPM treatments, , and the proba- bility of observing a rank sum as extreme as (i.e., if is small then we report the Pr() and if is large, we report Pr()). The rank sum for the SPM treatment can be found by taking =210 . −
Result 1: Groups in the NSPM PS2 and NSPM CH6 treatments submit strate- gies that are significantly closer to equilibrium than groups in either the SPM PS2
13For a textbook treatment of the Mann-Whitney test see, for example, Siegel and Castellan’s
1988 nonparametric statistics book. 150
Closeness to Equilibrium Messages (NSPM vs. SPM)
Rank Sum NSPM and Pr()
Environment [1-10] [11-20] [21-30] [31-40] [41-50]
1PS2 94 88 71 80 76
(0.2179) (0.0526) (0.0045) (0.0315) (0.0144)
2PS6 92 89 91 92 93
(0.1763) (0.1237) (0.1575) (0.1763) (0.1965)
3CH2 112 99 87 88 90
(0.6847) (0.3421) (0.0952) (0.1088) (0.1399)
4CH6 81 90 80 83 75
(0.0376) (0.1399) (0.0315) (0.0526) (0.0116)
Table 5.3. Closeness to Equilibrium Messages (SPM vs. NSPM) 151 treatment or the SPM CH6 treatment. We can reject the null in block [1-10] for the NSPM CH6 treatment, [11-20] for the NSPM PS2 treatment, and blocks [21-30],
[31-40], [41-50] for both treatments. There is weak evidence that groups in the NSPM
CH2 submit strategies that are significantly closer to equilibrium than groups in the
SPM CH2 treatment. We can reject the null in block [21-30]. Blocks [31-40] and
[41-50] are on the edge of statistical significance.
Efficiency We now turn to efficiency. It is possible that even if the mechanisms do not attain their equilibrium, the allocations they produce out-of-equilibrium be nearly efficient (although probably not Lindahl). We first define a notion of efficiency that takes into account that these mechanisms are not balanced out-of-equilibrium. In other words, the tax revenue in these mechanisms need not always add up to the cost of providing the public good. This imbalance needs to be taken into account when looking at efficiency. The following definition captures this idea.
Definition 3: Efficiency at time , for group , ,isthe’s total payoff minus any waste (surplus or deficit) in tax collection divided by what the group could have earned if the allocation was PO—i.e.,
= − | − |. P P P The previous definition is a generalization of the measure used by Chen and Tang 152 taking into account that the mechanisms may not be in budget balance.14 Chen and Tang only look at the balanced mechanisms due to Groves and Ledyard (1977) and Walker (1981). Figures (5.10)-(5.13) plot the average group efficiency per round, whereitappearsthattheefficiency for the NSPM treatment is unambiguously higher than the efficiency for the SPM treatment. Tables 5.16 and 5.17 summarize the efficiency information for the individual groups and treatments averaging over 10 round blocks. Table 5.4 reports the rank sum of the NSPM treatments, ,and the probability of observing a rank sum as extreme as .
Result 2: Efficiency levels are significantly higher for groups in the NSPM treat- ments. We reject the null for all treatments and all blocks except for the CH2 groups in block [1-10].
Violations of Individual Rationality (IR) The efficiency numbers can be deceptive. If one consumer gets a high payoff,thiscanmaskalowpayoff of another. One reason
Lindahl allocations are attractive because consumers are never worse off than their original endowment. However, these mechanisms only achieve Lindahl allocations in equilibrium. It is therefore useful to determine how many times subjects violated individual rationality, hereafter IR. In the experiment, subjects’ initial endowments were equal to zero. Violation of IR is just an occurrence where the subject’s payoff fell
14 Note, also, that implicity assumes that if tax revenue is needed to finance deficits, the taxes used to raise the revenue are non-distortionary in nature — for instance, lump sum taxes. 153
Efficiency (NSPM vs. SPM)
Rank Sum NSPM and Pr()
Environment [1-10] [11-20] [21-30] [31-40] [41-50]
1PS2 133 125 130 143 136
(0.0177) (0.0716) (0.0315) (0.0014) (0.0093)
2PS6 125 132 129 132 126
(0.0716) (0.0216) (0.0376) (0.0216) (0.0615)
3CH2 108 123 130 144 140
(0.4267) (0.0952) (0.0315) (0.0010) (0.0034)
4CH6 124 134 124 121 133
(0.0827) (0.0144) (0.0827) (0.1237) (0.0177)
Table 5.4.Efficiency (SPM vs. NSPM) 154 below zero. Figures (5.14)-(5.17), plot the average number of violations of IR in each round, where SPM and NSPM treatments on the same graph for comparison. Again, for statistical analysis we look at this information by group to maintain independence of observations. Table 5.20 and 5.21 summarize the average number of IR violations in each 10 round block, for each treatment and each independent observation, for group sizes of 2 and 6 respectively. Since the average number of violations of IR is not a continuous variable, we cannot use a Mann-Whitney test without manipulation.
We use a nonparametric test of central tendency, the median test.
The null hypothesis in a median test is that the two groups of interest, group and group are drawn from distributions with the same median. In this test, data is labeled as coming from either group or group andthenpooled.Themedianof the pooled data is calculated. Data is then sorted according to: first, which group it came from; and, second, whether the data is above or below the median. This sorting has a known distribution and one can calculate the probability that the observed data occurs as well as occurrence that are more extreme.15 Our alternative hypothesis is that, for each group size, the median of the NSPM treatment is smaller than the median of the SPM treatment. In Chart 13 below, we report the probability (under the null hypothesis) of the having the observed number of NSPM data points above the median or less. If this number is less than 0.10, we reject the null of equal medians in favor of a lower median for the NSPM treatment.
15See, for example, p.124 Siegel and Castellan. 155
Violations of IR (NSPM vs. SPM)
Median Test Pr()
Environment [1-10] [11-20] [21-30] [31-40] [41-50]
1PS2 0.089 0.328 0.089 0.089 0.001
2PS6 0.012 0.001 0.012 0.311 0.001
3CH2 0.550 0.550 0.012 0.550 0.550
4CH6 0.672 0.672 0.586 0.586 0.586
Table 5.5. Violations of IR (SPM vs. NSPM) 156
The chart indicates that there is stronger support for our research hypothesis in the PS treatments. There appears little difference in the median number of violations of IR for the CH mechanism treatments.
Result 3: The median number of violations of individual rationality is signifi- cantly smaller for the NSPM PS6 treatment in all blocks except [31,40] and for the
NSPM PS2 treatment for all blocks except [11-20]. There is one block where we can reject the null in the CH2 treatments and none in the CH6 treatments.
Inthenextsection,wecomparethePSandCHmechanismsbytheoverallwelfare they provide to their participants and by the efficiency of the allocations they produce.
5.6.2 Hypotheses Based on Mechanism Comparison
Despite some similarities, the PS and CH mechanisms are fundamentally different mechanisms with different properties that could, in principle, lead to vastly different economic outcomes. Part of the purpose of this paper is to evaluate the “performance characteristics” of these two mechanisms. This comparison allows us to evaluate structural differences in the mechanisms, highlighting certain properties which could lead to more successful mechanisms. Moreover, this process allows us to direct future mechanism design theory. Iterating this comparative process, one would hope that incorporating these observations will yield to a mechanism design literature that converges to more successful mechanisms. In this section, we compare the CH and 157
PS mechanism according to (1) how close they get to their equilibrium strategies;
(2) the efficiency of the allocations they produce out-of-equilibrium (and very near equilibrium in some cases); and finally, (3) how frequently consumers’ participation constraint is violated.
Closeness to Equilibrium Our first comparison between the two mechanisms is on their convergence to equilibrium. We restrict ourselves to comparing the mechanisms in similar group sizes and stability conditions. Tables 5.18 and 5.19 report, for each
10 round block and each group , the average absolute deviation from equilibrium
. Table 5.6 reports the results of our MW test.
Result 4: Reported messages of the PS mechanism, relative to the CH mecha- nism, are significantly closer to equilibrium in the SPM_6 and the NSPM_6 treat- ments. Messages are initially closer in the NSPM_2 treatment, but in the last two blocks there are not significant differences between the two mechanisms.
Efficiency Using the same efficiency measure as earlier, we compare the CH and PS mechanisms.Inaneffort to provide a better comparison, we hold constant whether the group size is 2 or 6 and whether the parameter treatment is SPM or NSPM.
Figures (5.18)-(5.21) graph the average group efficiency for each mechanism. Now, for each stability condition and group size, the two mechanisms are plotted together for comparison. The graphs seem to indicate that efficiency levels in the PS6 treatments 158
Closeness to Equilibrium Messages (CH vs. PS)
Rank Sum PS and Pr()
Environment [1-10] [11-20] [21-30] [31-40] [41-50]
1SPM_2 98 91 87 100 93
(0.3153) (0.1575) (0.0952) (0.3697) (0.1965)
2NSPM_2 82 79 76 90 92
(0.0446) (0.0262) (0.0144) (0.1399) (0.1763)
3SPM_6 55 63 71 66 73
(0.000) (0.0004) (0.0045) (0.001) (0.0073)
4NSPM_6 58 63 78 64 84
(0.000) (0.0004) (0.0262) (0.0005) (0.0615)
Table 5.6. Closeness to Equilibrium Messages (PS vs. CH) 159
Efficiency (CH vs. PS)
Rank Sum PS and Pr()
Environment [1-10] [11-20] [21-30] [31-40] [41-50]
1SPM_2 106 119 124 116 124
(0.4853) (0.1575) (0.0827) (0.2179) (0.0827)
2NSPM_2 134 131 144 121 122
(0.0144) (0.0262) (0.001) (0.1237) (0.1088)
3SPM_6 155 147 149 143 145
(0.000) (0.0004) (0.0002) (0.0014) (0.0008)
4NSPM_6 155 155 151 155 151
(0.000) (0.000) (0.0001) (0.000) (0.0001)
Table 5.7.Efficiency (PS vs. CH) are significantly higher than the CH6 treatments as hypothesized. The distinction is not as clear in the 2 player treatments. Again, the information about efficiency is reported in Tables 5.16 and 5.17. In this section, we run MW test comparing efficiency numbers across mechanisms holding constant stability and group size. Table
5.7 contains the rank sum values for the PS mechanism and the associated one-sided p-values.
Result 5: Forthesixplayergroups,efficiency is significantly higher in the PS mechanism treatments relative to the CH mechanism treatments. For the two player 160 groups, efficiency is initially higher in the PS_NSPM mechanism when compared to the CH_NSPM mechanism, however, for the last two 10 period blocks we are unable to reject the null hypothesis that the two are drawn from the same distribution. The
PS_SPM mechanism does attain significantly higher efficiency than the CH_SPM mechanism for the majority of the experiment. Although in the last 10 round block we are able to reject the null and say that it attains higher efficiency.
Violations of Individual Rationality Recall that our first observation, when comparing the two mechanisms, is that consumers in the CH mechanism get penalized if their statement is incorrect and if any of members of their cohorts’ statements are incorrect.
When is large (and the mechanism is not in equilibrium), these penalties have the potential to be quite large. These large penalties can lead to a large tax burden for each consumer and, potentially, to violations of individual rationality.16 We now investigate this observation. Recall, Tables 5.20 and 5.21 report the 10 round average number of violations of individual rationality for each treatment and independent observation for the 2 and 6 player groups respectively. Tables 5.8 reports the results from the nonparametric median tests.
Result 6: Except the SPM_2 treatment, the PS mechanism unambiguously produces fewer violations of the participation constraint than the CH mechanism.
16A consumer violates individual rationality if their profit drops below the value of their initial endowment (in this case =0). 161
Violations of IR (CH vs. PS)
Median Test Pr()
Environment [1-10] [11-20] [21-30] [31-40] [41-50]
1SPM_2 0.089 0.328 0.089 0.672 0.328
2NSPM_2 0.089 0.089 0.001 0.089 0.012
3SPM_6 0.001 0.001 0.012 0.012 0.012
4NSPM_6 0.000 0.001 0.000 0.000 0.001
Table 5.8. Violations of IR (PS vs. CH) 162
Outcome Variance Finally, our last conjecture about the two mechanisms was derived from the difference in the outcome function used to produce the public good. We speculated that the standard deviation of the public good would be smaller in the
PS mechanism because small changes in subjects’ requests would not affect the level of the public good produced as much as equal sizes changes in the Chen mechanism.
Looking at Figures 5.2-5.5, there appears to be some truth in this conjecture. Charts
2 and 3 report the standard deviation of the public good produced by independent observations. Table 5.9 reports the results of MW tests on the standard deviation of the public good production.
Result 7: The standard deviation of the public good for the PS mechanism is significantly smaller in all treatments relative to the Chen mechanism.
Note this result seems to have some explanatory power about why the efficiency of the PS6 treatments were surprisingly high. Recall, that an important part of the best reply system was the ability to predict what the level of the public good is at each round. For instance, correct predictions minimized incorrect statement penalties a subject might receive. One can imagine that if the data available to subjects did not
fluctuate much, subject could better predict on the public good production. With better predictions, subjects would receive lower penalties, higher payoffs, and might perhaps faster convergence toward equilibrium. 163
Standard Deviation (CH vs. PS)
Rank Sum PS and Pr()
Environment [1-10] [11-20] [21-30] [31-40] [41-50]
1SPM_2 84 82 90 63 70
(0.0615) (0.0446) (0.1399) (0.0004) (0.0034)
2NSPM_2 61 82 70 81 92
(0.0002) (0.0446) (0.0034) (0.0376) (0.1763)
3SPM_6 55 58 59 58 56
(0.000) (0.000) (0.0001) (0.000) (0.000)
4NSPM_6 55 55 55 55 64
(0.000) (0.000) (0.000) (0.000) (0.0005)
Table 5.9. Standard Deviation (PS vs. CH) 164
5.7 Conclusion
This experiment looked at how changes in stability parameters and group size af- fected the performance of two Nash efficientLindahlmechanisms.Wehaveshown that, contrary some work in this area, there is strong evidence to reject claims that a mechanism inducing a supermodular game is some how a necessary condition for convergence. In fact for both mechanisms, using a simple best reply learning model, we were able to derive a set of non-supermodular parameters that yielded, on aver- age, higher efficiency, fewer violations of individual rationality, and pushed groups closer to equilibrium than a set of supermodular parameters. These differences are likely to disappear as the supermodular parameters approach our request invariant parameters. Moreover, rough comparisons of the performance of the two mechanisms suggests that the structural differences in the PS mechanism and the CH mechanism are substantially important to actual implementation. The PS mechanism was, on average, much close to its equilibrium messages, yielded higher efficiency, and fewer violations of individual rationality.
Future work in this area seems to lie in correcting sources of inefficiencies in these mechanisms. As in VLW, we find that the lack of out-of-equilibrium budget balance is a significant source of inefficiency for this class of mechanisms. An important next step in Lindahl mechanisms will be to determine whether one can devise a scheme that is both stable and always in budget balance. An example that leaves hope for 165 such a mechanism is Varian’s compensation mechanism which can be adopted to have both of properties in economies with public goods and yields Lindahl allocations.17
However, Varian’s mechanism uses subgame perfection to attain this result and it is difficult to imagine implementing the compensation mechanism for groups larger than two.
17A version of this mechanism was tested in Chen and Gazzale (2002). Average (10 Round) Public Good Provision (2 Player Groups) 166 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS2_SPM [1,10] 8.92 11.23 9.42 10.09 9.86 9.48 6.69 8.95 7.86 12.64 9.51 [11,20] 8.84 11.45 8.76 8.61 10.35 7.77 6.14 9.15 8.14 11.13 9.03 [21,30] 10.05 11.76 8.94 10.26 10.61 8.44 6.74 9.17 9.18 10.76 9.59 [31,40] 10.30 11.65 8.90 10.29 9.93 6.84 6.71 8.60 7.99 10.17 9.14 [41,50] 9.00 10.30 8.47 8.32 10.19 9.39 8.30 8.81 8.44 9.65 9.09
2) PS2_NSPM [1,10] 10.36 9.09 9.14 7.39 6.91 8.26 8.87 7.76 12.34 10.39 9.05 [11,20] 8.76 9.25 8.63 8.34 7.05 7.97 8.48 8.43 8.36 10.31 8.56 [21,30] 8.79 9.07 8.63 8.43 7.07 7.92 8.21 7.65 8.60 9.54 8.39 [31,40] 7.84 9.60 8.69 7.38 7.23 7.40 7.89 8.86 8.41 9.63 8.29 [41,50] 7.63 9.42 8.78 8.35 7.12 7.34 8.01 8.26 8.29 9.39 8.26
3) CH2_SPM [1,10] 7.43 9.34 9.10 12.36 12.70 9.56 12.44 8.69 9.81 13.81 10.52 [11,20] 11.70 9.76 8.80 12.10 12.43 9.49 10.29 9.67 10.23 13.67 10.81 [21,30] 8.04 10.61 8.20 12.16 11.66 10.73 7.60 9.23 10.09 14.83 10.31 [31,40] 10.83 9.03 7.26 9.96 11.59 9.09 7.13 9.20 9.11 8.71 9.19 [41,50] 10.57 9.60 7.44 11.41 11.19 8.54 5.01 8.39 8.06 5.59 8.58
4) CH2_NSPM [1,10] 12.04 9.48 9.70 9.67 10.30 10.47 13.14 12.84 9.03 10.01 10.67 [11,20] 8.04 8.91 10.13 9.05 8.01 11.19 13.56 8.03 7.50 12.00 9.64 [21,30] 10.94 8.67 9.32 8.91 4.43 9.30 9.00 8.87 7.77 7.49 8.47 [31,40] 8.37 8.03 10.84 8.27 5.83 7.89 9.01 8.06 8.69 6.70 8.17 [41,50] 7.63 8.13 8.99 9.66 4.07 9.67 7.71 9.14 8.80 5.50 7.93
*Pareto Optimal Level (x=8) Average (10 Round) Public Good Provision (6 Player Groups) 167 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS6_SPM [1,10] 9.74 10.06 11.08 10.38 11.54 9.52 10.10 9.60 12.00 10.63 10.46 [11,20] 8.88 10.85 10.68 10.55 11.66 9.85 9.60 9.68 12.60 11.37 10.57 [21,30] 9.99 10.43 10.28 10.58 10.69 11.96 9.59 10.14 12.49 11.56 10.77 [31,40] 10.67 10.76 10.06 10.80 10.33 12.04 9.52 9.81 11.40 11.68 10.71 [41,50] 10.07 11.01 10.19 10.81 11.68 10.76 9.50 10.06 11.98 11.77 10.78
2) PS6_NSPM [1,10] 11.68 11.38 11.78 10.61 10.65 11.06 10.52 11.35 11.18 10.60 11.08 [11,20] 12.35 12.13 11.78 10.49 11.15 11.48 10.97 11.81 11.86 11.56 11.56 [21,30] 10.78 12.51 12.06 10.86 10.85 10.72 11.34 11.21 11.44 11.08 11.29 [31,40] 10.40 11.77 11.50 10.66 10.86 10.72 11.34 11.21 11.44 11.08 11.10 [41,50] 10.57 11.63 11.31 10.91 10.61 10.58 11.43 11.54 11.36 11.08 11.10
3) CH6_SPM [1,10] 21.61 26.75 16.13 16.94 17.20 24.62 35.16 22.93 10.63 21.82 21.38 [11,20] 16.69 10.54 16.21 15.13 12.57 21.44 26.11 16.71 7.51 21.37 16.43 [21,30] 14.26 10.16 18.40 12.40 12.81 18.14 17.56 19.84 8.26 19.12 15.10 [31,40] 13.84 9.73 11.81 14.63 17.46 11.35 16.10 15.16 7.29 14.82 13.22 [41,50] 11.41 19.77 9.14 12.49 13.56 10.80 16.91 15.27 7.08 17.39 13.38
4) CH6_NSPM [1,10] 15.76 20.44 20.76 17.73 21.27 13.54 15.90 17.87 15.20 11.86 17.03 [11,20] 12.53 18.05 15.88 13.74 16.07 14.18 14.51 17.41 13.28 8.91 14.46 [21,30] 11.11 14.81 16.13 9.76 15.27 13.27 13.23 14.06 11.48 11.51 13.06 [31,40] 13.04 12.52 13.48 11.15 14.39 10.41 13.52 12.03 14.43 10.87 12.58 [41,50] 10.84 12.09 14.49 10.59 12.44 12.41 12.16 10.36 10.62 8.64 11.46
*Pareto Optimal Level (x=11) Average (10 Round) Standard Deviation of Public Good Provision (2 Player Groups) 168 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS2_SPM [1,10] 2.07 1.00 0.55 2.04 0.18 1.46 0.62 0.82 1.23 1.81 1.18 [11,20] 1.27 1.07 0.83 1.50 0.47 1.93 1.11 0.71 1.71 2.00 1.26 [21,30] 1.04 0.81 0.63 2.48 0.28 1.19 0.51 0.65 1.63 2.06 1.13 [31,40] 1.96 0.63 0.57 1.17 0.26 1.16 0.85 0.82 1.60 1.95 1.10 [41,50] 1.35 1.49 1.64 0.57 0.41 1.10 1.04 1.00 2.04 1.47 1.21
2) PS2_NSPM [1,10] 0.78 0.99 1.84 0.67 1.19 1.53 0.92 1.32 1.93 0.78 1.20 [11,20] 1.20 1.21 1.05 1.19 1.28 1.09 0.30 2.01 0.49 1.05 1.09 [21,30] 0.79 0.87 0.69 1.09 0.41 1.34 0.22 1.75 1.22 0.29 0.87 [31,40] 0.72 1.13 0.43 0.68 0.31 0.70 0.42 2.32 1.78 0.80 0.93 [41,50] 0.27 1.13 0.49 0.68 0.83 0.76 0.02 1.73 1.44 0.20 0.75
3) CH2_SPM [1,10] 3.50 1.49 2.34 1.08 1.40 0.72 2.02 1.55 1.16 2.22 1.75 [11,20] 3.60 1.11 2.21 1.29 1.85 1.49 4.60 0.89 2.73 1.22 2.10 [21,30] 2.98 1.04 2.03 1.40 1.56 1.17 3.46 0.51 1.70 0.65 1.65 [31,40] 3.59 2.42 2.30 3.42 2.47 1.41 1.83 1.42 2.92 2.90 2.47 [41,50] 2.83 1.31 2.09 4.01 1.78 0.70 2.00 2.26 3.51 2.70 2.32
4) CH2_NSPM [1,10] 2.33 3.04 2.05 1.81 3.02 1.87 3.60 4.47 1.37 2.98 2.65 [11,20] 0.55 2.16 2.01 1.67 1.53 1.10 2.76 1.83 0.29 2.79 1.67 [21,30] 4.19 1.46 2.34 2.04 1.73 2.24 0.76 2.29 0.55 3.31 2.09 [31,40] 0.85 1.58 1.60 1.72 1.94 1.40 2.02 1.02 0.24 2.87 1.52 [41,50] 0.45 0.90 0.84 1.27 1.41 2.38 0.00 1.57 0.61 0.97 1.04 Average (10 Round) Standard Deviation of Public Good Provision (6 Player Groups) 169 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS6_SPM [1,10] 0.38 0.89 0.67 0.51 0.87 0.52 0.61 0.51 0.82 1.21 0.70 [11,20] 0.58 0.34 0.54 0.24 0.90 0.84 0.29 0.32 0.47 0.71 0.52 [21,30] 0.73 0.52 0.55 0.29 1.17 0.61 0.22 0.62 0.93 0.40 0.60 [31,40] 0.54 0.26 0.33 0.21 0.83 0.72 0.24 0.47 0.76 0.25 0.46 [41,50] 0.67 0.36 0.22 0.13 1.20 0.63 0.28 0.54 0.59 0.15 0.48
2) PS6_NSPM [1,10] 0.72 0.73 0.49 0.47 0.55 0.91 0.36 0.65 0.57 0.64 0.61 [11,20] 0.66 0.31 0.61 0.32 0.55 0.33 0.43 0.33 0.30 0.21 0.41 [21,30] 0.66 0.36 0.83 0.31 0.22 0.27 0.22 0.42 0.36 0.14 0.38 [31,40] 0.67 0.22 0.61 0.32 0.43 0.25 0.19 0.30 0.36 0.07 0.34 [41,50] 0.57 0.53 0.36 0.18 1.15 0.12 0.15 0.31 0.48 0.07 0.39
3) CH6_SPM [1,10] 5.82 7.27 5.11 4.75 5.71 4.00 5.41 5.72 1.85 4.03 4.97 [11,20] 4.52 2.20 5.68 0.83 2.63 4.96 3.85 5.31 0.87 3.84 3.47 [21,30] 2.72 0.84 4.03 2.38 2.57 3.78 5.32 4.30 0.74 2.71 2.94 [31,40] 2.72 1.26 3.96 1.73 1.21 1.87 4.76 5.46 0.70 1.82 2.55 [41,50] 2.11 7.04 2.48 1.61 4.42 1.44 3.59 5.45 0.80 2.62 3.16
4) CH6_NSPM [1,10] 4.26 6.43 3.90 2.77 6.92 3.70 1.94 6.09 3.27 3.18 4.25 [11,20] 1.60 3.32 4.01 4.33 2.45 3.39 2.09 5.47 1.93 1.41 3.00 [21,30] 2.76 0.92 3.28 1.99 2.51 3.89 2.14 3.86 1.35 2.65 2.53 [31,40] 2.45 3.81 3.26 4.63 2.83 0.97 2.95 2.20 1.47 2.92 2.75 [41,50] 1.59 0.48 3.29 2.18 1.11 0.58 2.51 2.98 1.77 0.43 1.69 Average (10 Round) Total Tax Burden (2 Player Groups) 170 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS2_SPM [1,10] 131.81 121.74 42.80 126.90 84.78 109.47 85.67 68.20 141.29 108.94 102.16 [11,20] 79.39 125.79 123.58 78.73 91.53 65.25 67.84 58.58 156.07 56.76 90.35 [21,30] 63.59 151.66 85.97 64.46 52.74 65.36 71.34 59.66 111.42 73.71 79.99 [31,40] 99.99 131.76 128.48 105.89 65.64 73.31 84.44 61.93 157.67 58.91 96.80 [41,50] 113.48 101.64 134.35 71.66 78.45 77.57 70.62 63.97 128.93 66.37 90.70
2) PS2_NSPM [1,10] 76.49 126.90 107.95 76.53 71.32 87.38 83.01 92.53 102.19 87.32 91.16 [11,20] 70.01 105.50 111.50 72.62 83.45 90.43 71.30 85.23 84.28 100.17 87.45 [21,30] 79.04 84.96 88.49 83.47 70.15 83.59 73.29 82.37 74.50 94.98 81.48 [31,40] 76.36 82.46 91.49 70.25 66.33 75.10 73.07 78.05 107.09 95.51 81.57 [41,50] 71.96 86.13 88.48 73.47 74.32 74.41 72.31 81.12 81.25 85.28 78.87
3) CH2_SPM [1,10] 106.83 70.53 69.88 195.30 107.41 156.39 91.64 80.33 89.93 114.00 108.23 [11,20] ‐22.39 82.67 163.84 119.38 146.02 109.32 85.04 70.91 73.24 136.94 96.50 [21,30] 90.01 83.20 137.86 110.80 112.66 122.53 275.57 83.66 179.80 139.47 133.56 [31,40] 130.80 93.46 163.96 98.66 91.55 84.59 81.64 78.64 142.20 114.06 107.95 [41,50] 172.75 72.48 139.12 116.78 112.98 86.09 75.09 68.10 136.91 77.35 105.76
4) CH2_NSPM [1,10] 126.07 154.39 122.50 94.11 93.93 105.77 128.71 103.75 56.22 93.40 107.89 [11,20] 85.10 128.69 75.84 97.84 98.47 99.92 120.66 86.04 54.52 116.16 96.32 [21,30] 97.43 97.89 48.36 126.64 47.31 59.26 89.29 76.96 71.05 97.48 81.17 [31,40] 98.49 88.76 47.20 125.61 51.66 60.19 94.82 59.28 90.71 64.77 78.15 [41,50] 73.47 108.39 14.26 101.21 45.91 57.31 86.26 81.94 83.42 130.59 78.28
*Lindah Equilibrium Level (T=72) Average 10 Round Total Tax Burden (6 Player Groups) 171 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS6_SPM [1,10] ‐155.71 ‐89.66 394.13 107.66 ‐637.28 ‐143.08 ‐489.97 227.60 ‐505.76 221.10 ‐107.10 [11,20] 121.15 ‐120.27 47.98 232.09 ‐570.68 ‐43.53 ‐385.13 376.29 ‐168.39 ‐350.50 ‐86.10 [21,30] 8.14 ‐66.23 318.85 87.28 ‐87.28 ‐119.74 ‐126.23 560.10 ‐92.76 ‐75.04 40.71 [31,40] ‐22.01 ‐98.29 305.32 ‐9.67 ‐114.64 294.06 ‐129.09 601.28 111.80 28.90 96.77 [41,50] ‐109.86 ‐116.92 366.24 39.23 ‐770.36 101.53 14.97 470.88 ‐107.14 ‐42.87 ‐15.43
2) PS6_NSPM [1,10] ‐44.51 ‐451.76 71.50 ‐24.95 ‐77.06 21.38 ‐108.31 ‐103.34 ‐158.62 46.11 ‐82.96 [11,20] ‐189.81 ‐356.97 74.16 ‐43.56 ‐14.63 160.29 ‐58.31 10.91 71.86 95.66 ‐25.04 [21,30] 55.70 ‐142.54 46.18 ‐31.97 51.19 120.42 ‐34.26 145.65 109.60 175.64 49.56 [31,40] 103.34 ‐54.90 89.12 35.89 103.35 198.16 41.83 66.27 172.70 93.18 84.89 [41,50] ‐98.34 ‐24.25 145.98 ‐68.45 172.70 134.77 81.51 ‐2.67 92.04 16.56 44.98
3) CH6_SPM [1,10] 1135.68 554.77 236.94 2222.77 1065.82 3732.99 ‐2394.07 ‐1228.08 425.92 ‐1923.32 382.94 [11,20] 901.24 2558.57 237.20 ‐2103.53 1535.00 1595.02 10712.06 78.26 368.08 ‐948.40 1493.35 [21,30] 413.72 1521.45 ‐594.79 ‐325.21 327.54 96.99 7236.73 ‐2040.83 ‐190.74 1640.78 808.56 [31,40] 430.54 136.46 3758.87 ‐1015.84 3653.15 156.60 ‐95.87 1674.43 ‐57.55 865.34 950.61 [41,50] 746.26 ‐3438.86 916.15 ‐393.60 390.14 138.11 46.60 ‐608.59 ‐176.00 ‐683.85 ‐306.36
4) CH6_NSPM [1,10] 973.67 445.92 1037.32 1600.47 3155.22 1563.77 ‐87.74 755.66 849.94 338.29 1063.25 [11,20] 429.53 659.19 1659.67 1204.71 2166.51 1269.02 140.08 232.63 10.65 221.10 799.31 [21,30] 70.38 527.13 1402.28 471.56 1701.65 ‐105.38 152.57 ‐12.57 ‐20.54 33.37 422.05 [31,40] ‐177.82 207.81 2264.03 546.30 674.51 190.11 163.58 440.29 ‐67.98 ‐54.52 418.63 [41,50] 255.53 232.14 155.79 259.92 375.75 62.29 510.64 42.03 257.80 48.63 220.05
*Lindah Equilibrium Level (T=99) Average (10 Round) Efficiency (2 Player Groups) 172 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS2_SPM [1,10] ‐1.65 ‐0.45 0.20 ‐1.67 0.00 0.05 0.17 0.72 ‐1.34 ‐0.06 ‐0.40 [11,20] 0.55 ‐0.27 ‐1.21 ‐0.35 ‐0.07 0.56 0.40 0.94 ‐1.63 0.48 ‐0.06 [21,30] 0.82 ‐0.80 ‐0.35 ‐0.41 0.06 0.72 0.53 0.86 ‐0.98 0.37 0.08 [31,40] 0.04 ‐0.19 ‐1.24 ‐0.51 0.16 0.38 0.17 0.76 ‐2.30 0.28 ‐0.24 [41,50] ‐0.18 0.20 ‐1.87 0.18 0.02 0.81 0.84 0.86 ‐1.76 0.45 ‐0.05
2) PS2_NSPM [1,10] 0.76 ‐0.44 ‐0.28 0.62 0.46 0.44 0.76 0.13 0.06 0.68 0.32 [11,20] 0.85 0.15 ‐0.11 0.86 0.34 0.39 0.95 0.35 0.62 0.61 0.50 [21,30] 0.87 0.69 0.59 0.63 0.74 0.53 0.95 0.26 0.74 0.67 0.67 [31,40] 0.79 0.76 0.50 0.82 0.90 0.63 0.92 0.33 ‐0.17 0.65 0.61 [41,50] 0.86 0.65 0.65 0.92 0.62 0.70 0.99 0.53 0.71 0.92 0.75
3) CH2_SPM [1,10] ‐0.81 0.18 ‐0.03 ‐2.03 0.16 ‐1.50 ‐0.55 0.33 0.42 ‐1.07 ‐0.49 [11,20] 0.36 0.31 ‐2.22 ‐0.09 ‐1.09 ‐0.09 ‐1.06 0.67 ‐0.28 ‐0.35 ‐0.38 [21,30] ‐0.48 0.53 ‐1.60 ‐0.58 ‐0.11 ‐0.25 ‐5.85 0.75 ‐1.91 ‐0.18 ‐0.97 [31,40] ‐1.23 ‐0.34 ‐2.26 ‐0.67 ‐0.26 0.43 0.21 0.36 ‐2.27 ‐0.43 ‐0.65 [41,50] ‐1.75 0.29 ‐1.53 ‐1.31 ‐0.65 0.56 ‐0.30 0.11 ‐1.19 ‐0.50 ‐0.63
4) CH2_NSPM [1,10] ‐0.47 ‐1.39 ‐0.56 0.36 ‐0.34 ‐0.04 ‐1.42 ‐0.35 0.91 0.11 ‐0.32 [11,20] 0.52 ‐0.59 0.29 0.30 0.02 0.32 ‐0.42 0.17 0.95 ‐0.02 0.15 [21,30] 0.08 ‐0.12 0.53 ‐0.52 0.41 0.08 0.39 0.32 0.69 ‐0.19 0.17 [31,40] 0.26 0.33 0.84 ‐0.64 0.57 0.74 0.36 0.85 0.53 0.43 0.43 [41,50] 0.83 ‐0.12 0.97 0.38 0.39 0.74 0.45 0.75 0.75 ‐1.64 0.35 Average (10 Round) Efficiency (6 Player Groups) 173 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS6_SPM [1,10] 0.72 ‐0.10 ‐1.76 0.14 0.87 0.93 0.99 ‐0.34 0.94 ‐0.41 0.20 [11,20] ‐0.23 0.68 0.07 0.14 0.80 0.61 0.98 ‐0.74 0.74 0.90 0.39 [21,30] 0.38 0.80 ‐0.38 0.76 0.19 0.72 0.93 ‐1.84 0.11 0.85 0.25 [31,40] 0.49 0.84 ‐0.23 0.93 0.42 ‐0.54 0.91 ‐1.91 ‐0.19 0.79 0.15 [41,50] 0.33 0.90 ‐0.60 0.85 0.91 0.09 0.65 ‐1.12 0.70 0.96 0.37
2) PS6_NSPM [1,10] 0.77 1.00 0.63 0.91 0.98 0.85 0.92 0.86 0.96 0.50 0.84 [11,20] 0.96 0.99 0.56 0.99 0.93 0.59 0.76 0.99 0.70 0.82 0.83 [21,30] 0.51 0.98 0.53 0.99 0.96 0.72 1.00 0.53 0.85 0.55 0.76 [31,40] 0.70 0.99 0.64 0.94 0.78 0.35 0.95 0.92 0.60 0.95 0.78 [41,50] 0.90 0.80 0.64 0.99 0.23 0.72 0.94 0.98 0.77 0.97 0.79
3) CH6_SPM [1,10] ‐17.06 ‐15.56 ‐9.37 ‐14.62 ‐5.85 ‐27.97 ‐21.59 ‐11.23 ‐4.41 ‐5.83 ‐13.35 [11,20] ‐7.51 ‐13.64 ‐11.04 0.85 ‐10.85 ‐17.73 ‐60.92 ‐9.69 ‐1.34 ‐10.05 ‐14.19 [21,30] ‐5.78 ‐7.17 ‐4.24 ‐0.59 ‐3.85 ‐5.18 ‐38.57 ‐4.29 0.72 ‐11.21 ‐8.02 [31,40] ‐5.60 ‐1.30 ‐19.60 0.51 ‐23.56 ‐1.52 ‐6.25 ‐8.78 0.40 ‐5.22 ‐7.09 [41,50] ‐4.11 ‐3.36 ‐6.78 0.32 ‐3.35 ‐1.02 ‐8.04 ‐6.41 0.71 ‐2.69 ‐3.47
4) CH6_NSPM [1,10] ‐6.74 ‐5.63 ‐9.62 ‐7.55 ‐19.59 ‐9.38 ‐2.12 ‐11.23 ‐6.00 ‐2.59 ‐8.04 [11,20] ‐1.22 ‐3.85 ‐7.74 ‐6.14 ‐10.83 ‐7.67 ‐1.25 ‐5.56 ‐0.41 ‐0.42 ‐4.51 [21,30] ‐1.79 ‐1.58 ‐7.33 ‐1.61 ‐8.02 ‐1.66 ‐1.13 ‐2.45 0.62 ‐1.17 ‐2.61 [31,40] ‐0.72 ‐1.21 ‐11.38 ‐4.33 ‐3.25 ‐0.08 ‐2.16 ‐2.49 0.06 ‐0.04 ‐2.56 [41,50] ‐0.77 0.01 ‐1.38 ‐1.13 ‐0.95 ‐0.06 ‐1.95 ‐1.12 ‐0.32 0.80 ‐0.69 Average (10 Round) Absolute Deviation from Equilibrium Strategy (2 Player Groups) 174 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS2_SPM [1,10] 2.80 3.42 1.32 3.10 1.85 1.62 1.35 1.36 1.64 4.50 2.30 [11,20] 1.22 3.70 1.77 1.40 2.33 2.00 1.70 1.11 2.25 2.67 2.01 [21,30] 1.79 4.18 1.12 2.31 2.06 1.33 1.12 1.26 1.94 2.50 1.96 [31,40] 2.61 3.91 1.47 2.44 1.70 1.15 1.21 0.86 1.90 2.09 1.93 [41,50] 1.92 2.79 1.94 0.63 2.11 1.39 0.90 1.05 2.06 1.73 1.65
2) PS2_NSPM [1,10] 2.12 2.34 2.41 1.31 1.58 1.49 1.03 1.31 4.33 2.23 2.01 [11,20] 1.59 2.14 1.74 1.05 1.35 1.04 0.34 1.84 0.89 2.52 1.45 [21,30] 1.19 1.47 1.03 1.04 1.11 1.03 0.20 1.57 1.17 1.77 1.16 [31,40] 0.66 1.72 1.30 1.10 0.83 0.91 0.16 1.80 1.83 1.84 1.22 [41,50] 0.44 1.69 1.05 0.82 1.13 0.56 0.05 1.70 1.09 1.41 0.99
3) CH2_SPM [1,10] 2.75 1.55 1.79 3.82 3.49 1.84 3.15 1.25 1.44 4.25 2.53 [11,20] 2.48 1.41 2.20 3.14 3.73 1.41 2.69 1.13 1.84 4.34 2.44 [21,30] 1.81 1.84 2.20 3.20 2.89 2.23 3.47 0.93 2.28 5.15 2.60 [31,40] 3.13 1.51 1.86 3.34 2.76 0.98 1.57 1.08 1.97 2.31 2.05 [41,50] 2.72 1.17 1.81 3.85 2.60 0.60 2.26 1.12 1.83 2.42 2.04
4) CH2_NSPM [1,10] 3.14 2.59 1.72 3.26 3.41 2.05 3.67 3.83 1.53 2.03 2.72 [11,20] 0.53 1.88 2.16 2.54 2.91 2.28 3.92 1.38 1.14 2.94 2.17 [21,30] 1.63 1.26 2.17 2.63 3.12 2.33 1.29 1.15 0.87 2.54 1.90 [31,40] 0.90 1.07 1.88 2.67 2.92 1.37 1.42 1.14 0.52 2.26 1.62 [41,50] 0.51 0.90 1.65 3.05 2.98 1.96 0.21 1.39 0.42 2.09 1.52 Average (10 Round) Absolute Deviation from Equilibrium Strategy (6 Player Groups) 175 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS6_SPM [1,10] 2.26 1.66 1.52 1.31 1.66 2.05 2.13 2.00 2.02 1.76 1.84 [11,20] 2.66 0.98 1.29 0.85 1.49 1.66 2.33 1.49 1.98 1.06 1.58 [21,30] 2.30 1.04 1.26 0.83 1.59 1.19 1.99 1.57 1.99 0.81 1.46 [31,40] 1.81 0.70 1.28 0.72 1.78 1.42 2.05 1.53 1.24 0.69 1.32 [41,50] 1.58 0.66 1.38 0.73 1.78 0.92 1.86 1.61 1.42 0.75 1.27
2) PS6_NSPM [1,10] 1.72 2.24 1.47 1.62 1.44 1.92 1.89 1.55 1.68 1.69 1.72 [11,20] 1.83 1.63 1.25 1.28 1.04 1.48 1.35 1.25 0.98 0.59 1.27 [21,30] 2.27 1.72 1.45 1.05 0.80 1.15 1.09 1.39 0.82 0.48 1.22 [31,40] 2.06 1.43 1.06 0.95 0.91 1.17 1.06 1.05 0.89 0.18 1.07 [41,50] 1.78 1.36 0.91 0.74 1.67 1.19 1.11 1.14 0.69 0.20 1.08
3) CH6_SPM [1,10] 6.61 9.84 4.32 4.38 3.89 8.55 13.06 6.96 4.55 5.35 6.75 [11,20] 4.01 1.63 4.15 2.26 2.32 6.39 11.12 3.58 3.59 5.12 4.42 [21,30] 1.93 1.13 4.53 1.65 1.80 4.59 6.49 4.50 4.08 5.01 3.57 [31,40] 2.25 1.28 3.28 1.92 4.72 1.33 3.72 3.45 3.99 2.73 2.87 [41,50] 1.28 4.13 2.47 1.38 3.09 0.99 3.75 3.06 4.13 3.14 2.74
4) CH6_NSPM [1,10] 3.43 5.60 6.04 4.76 7.57 4.09 3.05 4.66 3.10 1.74 4.40 [11,20] 1.35 4.70 4.13 3.46 4.34 2.96 2.49 3.87 1.65 1.43 3.04 [21,30] 0.97 2.82 3.99 1.53 3.98 2.08 1.93 2.02 1.11 1.09 2.15 [31,40] 1.52 1.91 3.25 2.30 2.76 1.16 2.02 1.54 2.17 1.43 2.00 [41,50] 0.86 1.01 2.07 1.73 1.35 1.27 1.81 1.67 1.03 1.46 1.43 Average (10 Round) # of Violations of Individual Rationality (2 Player Groups) 176 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS2_SPM [1,10] 0.70 0.60 0.40 1.20 0.40 0.50 0.30 0.10 0.90 0.60 0.57 [11,20] 0.10 0.50 1.10 0.40 0.80 0.20 0.10 0.10 1.10 0.20 0.46 [21,30] 0.10 0.90 0.80 0.30 0.00 0.10 0.00 0.00 0.80 0.50 0.35 [31,40] 0.70 0.50 1.00 1.00 0.00 0.30 0.30 0.00 1.40 0.40 0.56 [41,50] 0.80 0.60 1.10 0.70 0.50 0.20 0.00 0.00 1.00 0.40 0.53
2) PS2_NSPM [1,10] 0.00 0.80 0.50 0.00 0.10 0.00 0.10 0.30 0.70 0.10 0.26 [11,20] 0.00 0.60 0.60 0.00 0.10 0.40 0.00 0.20 0.00 0.00 0.19 [21,30] 0.00 0.00 0.10 0.00 0.00 0.20 0.00 0.30 0.00 0.00 0.06 [31,40] 0.00 0.00 0.30 0.00 0.00 0.00 0.00 0.30 0.70 0.00 0.13 [41,50] 0.00 0.00 0.10 0.00 0.00 0.10 0.00 0.20 0.10 0.00 0.05
3) CH2_SPM [1,10] 0.80 0.60 0.90 0.60 0.50 1.00 0.60 0.20 0.30 0.90 0.64 [11,20] 0.30 0.30 1.10 0.50 0.70 0.40 0.60 0.00 0.60 0.90 0.54 [21,30] 0.70 0.30 0.70 0.90 0.80 0.70 1.60 0.00 0.70 1.00 0.74 [31,40] 0.80 0.40 1.10 1.10 0.60 0.00 0.00 0.30 0.70 0.00 0.50 [41,50] 1.00 0.30 1.10 1.30 0.40 0.10 1.00 0.40 0.80 0.60 0.70
4) CH2_NSPM [1,10] 1.00 1.20 0.60 0.30 0.60 0.70 1.20 0.40 0.10 0.40 0.65 [11,20] 0.20 0.90 0.30 0.40 0.70 0.40 0.80 0.50 0.10 0.60 0.49 [21,30] 0.60 0.50 0.20 0.80 0.30 0.50 0.30 0.30 0.40 0.50 0.44 [31,40] 0.30 0.30 0.00 0.80 0.50 0.10 0.30 0.10 0.40 0.40 0.32 [41,50] 0.00 0.20 0.00 0.20 0.50 0.10 0.10 0.40 0.10 0.60 0.22
*Maximum =2 Average (10 Round) # of Violations of Individual Rationality (6 Player Groups) 177 Treatment Rounds Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Avg 1) PS6_SPM [1,10] 1.80 1.60 2.60 1.90 1.00 1.50 1.50 1.90 0.90 1.90 1.66 [11,20] 2.00 0.80 1.60 1.40 0.90 0.70 0.60 2.30 1.10 0.60 1.20 [21,30] 1.40 0.90 2.00 1.00 2.00 1.00 0.20 2.10 1.50 0.70 1.28 [31,40] 1.40 0.40 1.40 0.90 1.40 2.00 0.10 1.80 1.90 0.50 1.18 [41,50] 1.10 0.90 1.40 0.70 0.70 1.80 0.70 1.90 0.80 0.10 1.01
2) PS6_NSPM [1,10] 0.80 0.20 0.90 0.70 0.60 1.30 0.50 0.50 0.20 1.30 0.70 [11,20] 0.30 0.10 0.20 0.10 0.50 0.80 0.40 0.20 0.10 0.20 0.29 [21,30] 1.00 0.20 0.60 0.00 0.20 0.40 0.00 0.70 0.20 0.50 0.38 [31,40] 0.90 0.10 0.60 0.10 0.70 0.90 0.00 0.00 0.00 0.00 0.33 [41,50] 0.40 0.60 0.40 0.00 0.90 0.20 0.00 0.10 0.30 0.00 0.29
3) CH6_SPM [1,10] 3.60 3.80 3.00 4.30 3.70 3.90 2.30 2.00 3.00 1.80 3.14 [11,20] 3.00 3.60 3.50 1.50 2.70 3.70 5.30 2.90 2.70 2.50 3.14 [21,30] 2.60 2.30 2.80 1.90 3.20 2.20 5.60 2.20 1.20 3.70 2.77 [31,40] 2.60 2.40 5.10 1.10 2.90 2.50 2.50 3.60 1.30 3.30 2.73 [41,50] 3.20 2.30 3.60 1.20 2.40 2.30 3.00 2.40 0.30 2.60 2.33
4) CH6_NSPM [1,10] 4.20 3.30 3.70 4.40 4.80 2.70 2.00 3.80 3.70 3.30 3.59 [11,20] 2.30 3.00 4.50 4.00 4.10 3.70 2.60 3.20 1.90 1.70 3.10 [21,30] 2.60 2.10 3.70 3.10 4.80 2.70 2.60 3.60 1.20 2.00 2.84 [31,40] 2.30 2.10 3.80 3.70 3.50 1.70 2.40 2.60 1.90 1.20 2.52 [41,50] 2.50 1.70 2.30 3.10 2.40 1.70 2.70 2.30 2.20 0.10 2.10
*Maximum =6 178
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Appendix A
Stability of Chen, Kim, and Walker Mechanisms
With the parameters used in the experiment, we show that the Walker mechanism is unstable under the best reply dynamic, the Kim Mechanism is stable under best reply, and the Chen mechanism (with the compact message space used in our experiment) induces a supermodular game and is therefore robustly stable under a wide range of adjustment behavior. Chen (2002) showed that neither the Walker nor the Kim mechanism is supermodular.
A.1 The Walker mechanism
If there are three players and each one has preferences of the form ( )=+ − 2 , then players’ best reply functions in the Walker mechanism can be represented by the following system of linear difference equations,
+1 21 1 21+1 1 3 0 − − 1 21 21 1 21 ⎡ ⎤ ⎡ − − ⎤ ⎡ ⎤ ⎡ ⎤ +1 22+1 22 1 2 3 = 0 − + − ⎢ 2 ⎥ ⎢ 22 22 ⎥ ⎢ 2 ⎥ ⎢ 22 ⎥ ⎢ ⎥ ⎢ − − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ +1 23 1 23+1 3 3 ⎢ ⎥ ⎢ − 0 ⎥ ⎢ ⎥ ⎢ − ⎥ ⎢ 3 ⎥ ⎢ 23 23 ⎥ ⎢ 3 ⎥ ⎢ 23 ⎥ ⎢ ⎥ ⎢ − − ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 199
In our experiment 1 = 2 = 3 =1,sothecoefficient matrix reduces to
1 3 0 2 2 ⎡ − − ⎤ 3 0 1 ⎢ 2 2 ⎥ ⎢ − − ⎥ ⎢ ⎥ ⎢ 1 3 0 ⎥ ⎢ 2 2 ⎥ ⎢ − − ⎥ ⎣ ⎦ 1 1 The eigenvalues of the matrix are 1 =1+ √3 , 2 =1 √3 ,and3 = 2,all 2 − 2 − of which lie outside the unit circle. The system is therefore unstable.
A.2 The Kim mechanism
Kim established that his mechanism is globally stable under the continuous-time gradient adjustment process, but time is discrete in our experiment. We show here
2 that for three players with utility functions of the form ( )= + ,the − Kim mechanism is stable under myopic best reply. Express player ’s payoff function in the Kim mechanism as follows:
2 1 1 2 ( )= ( + ) ( ) − − − − 2 − = = X6 X6 The first-order conditions of player ’s maximization problem are
1 = 2 + + =0 − − − − = = X6 X6 = =0 − 200
If player is best responding he will choose and in period +1according to
1 1 1 ( 2) 1 +1 1 − = + 1 2 − − 2 − − − − − " µ ¶ = # X6 +1 +1 = + = X6 Adding up these conditions for the three individuals, we obtain
+1 8 4 1 2 = − 3 3 − − − X +1 1 4 2 − The homogeneous part of this system is + 3 + 3 =0,forwhichthe characteristic equation is 4 2 2 + + =0 3 3
2 1 2 1 The characteristic roots are 1 = + √2 and 2 = √2 ,eachofwhichis − 3 3 − 3 − 3 inside the unit circle: 2 = = 1 2 3 | | | | r and the system is therefore globally stable for any initial conditions.
A.3 The Chen mechanism
The conditions for Chen’s mechanism to be supermodular – i.e.,forthegamedefined by the mechanism to be supermodular – depend on the values of the environment parameters (utility and cost functions) and the parameters of the mechanism ( and
). For the utility parameters used in our experiment, we derive restrictions on and
that will make Chen’s mechanism supermodular. 201
We first write player ’s payoff function as follows:
2 1 2 2 ( )= ( + ) ( ) ( ) − − − − 2 − − 2 − = = = X6 X6 X6
The function is supermodular in (),because
2 =1 0 ≥
and has increasing differences in () () () and (), = if ∀ 6 and only if the following inequalities hold:
2 = + 0 − ≥ 2 = 2 + 1 ( 1) 0 − − − − ≥ 2 =0 0 ≥ 2 =1 0 ≥
Therefore, the game is supermodular if and only if ( 1) +1+2max 1 − { } ≤
.Inourexperiment, =3, =8, =21,andmax 1 =1; hence ≤ { } the game is supermodular. Note that the Kim mechanism is not supermodular: the inequality condition above fails to hold when =0and =1. 202
Appendix B
Proofs: A Simple Supermodular Mechanism that
Implements Lindahl Allocations
The strategy for the proof of Theorem 1 will be as follows: first, we demonstrate that a Nash allocation is Pareto optimal via an argument similar to the one used by
Groves and Ledyard (1979); second, using the fact that the Nash allocation is Pareto optimal, we use an "unbiasedness" proof similar to Foley (1970) p. 68-69 and Chen
(2002) to establish that the outcome is Lindahl; finally, we show that any Lindahl allocation is achieved as a Nash allocation of the mechanism using a technique we believe was first used by Walker (1981).
Lemma 8. Suppose the strategy profile (¯r¯s) is a Nash equilibrium of for , ∈ where (¯ ¯) is consumer ’s Nash allocation, then the following statements are true:
1. For any bundle ( ) ,thereisapair() such that = (¯r ¯s ). ∈ − −
2. The private good consumed by consumer in equilibrium is ¯ (¯r¯s) ≡ −
1 3. Consumer ’s statement and tax are ¯ = ¯ and (¯r¯s)= (¯r¯s) =1 · (¯r¯s) for all P 203
4. If a feasible allocation ( ) is weakly preferred to the Nash allocation ∈
(¯ ¯), then the preferred bundle is at least as expensive as consumer ’s initial
wealth (i.e, + (¯r ¯s ) ). − − ≥
5. If a feasible allocation ( ) is strictly preferred to the Nash allocation ∈
(¯ ¯), then the preferred bundle is more expensive than consumer ’s initial
wealth (i.e., + (¯r ¯s ) ). − −
6. Consumer ’s equilibrium allocation is in the interior of his consumption set
++ (¯ ¯) ,andthereexists() such that (¯r ¯s ) 0 and ∈ − − (¯r ¯s ) . − −
1 Proof. L1.1 Since only depends on the requests of individuals, set = (+ ¯) = 6 P or = ¯ − = 6 P L1.2 Consider the following two bundles ( (¯r¯s) ¯) and ( (¯r¯s) ˆ) where ∈ 0 ˆ (¯r¯s) Since preferences are complete, transitive, and strictly ≤ −
increasing in ,wehave( (¯r¯s) ¯) ( (¯r¯s) ˆ) for all ˆ. Â
L1.3 Since ( ) is a Nash equilibrium, then for each consumer
((¯r¯s) (¯r¯s)) ((¯r s ) (¯r s )) for all . − º − − −
From the functional form of the tax function and since preferences are complete,
1 transitive, and strictly increasing in ,foreach, ¯ = =1 ¯. It follows
directly that (¯r¯s)= (¯r¯s) (¯r¯s). P · 204
L1.4 Suppose not. Then + (¯r ¯s ) ¯ + (¯r¯s)=.Since is − − continuos and using the fact that preferences are continuos, convex, and strictly
increasing in ,thereexists(´ ´ ´) such that ((´¯r ´¯s ) ´) , − − ∈ ´ + (´¯r ´¯s ) ,and((´¯r ´¯s ) ´) (¯ ¯). However, this − − ≤ − − Â means that there is an individually feasible bundle which is strictly preferred
to the Nash allocation. This contradicts the assumption that (¯r¯s) is a Nash
equilibrium.
L1.5 Suppose not. Then (¯ ¯) is not a best response which contradicts the assump-
tion that (¯r¯s) is a Nash equilibrium.
L1.6 This lemma follows directly from continuity, strictly increasing preferences in the
private good, and strict preference for interior allocation in the environment.
Lemma 9. Suppose consumer could purchase units of the public good at a price of ,
where is defined as consumer ’s equilibrium marginal tax rate (i.e., (¯r¯s)), ≡ then the following statements are true:
1. If a feasible allocation ( ) is weakly preferred to the Nash allocation ∈
(¯ ¯) was then the preferred bundle is at least as expensive as the Nash alloca-
tion (i.e., + ¯ + ¯). · ≥ ·
2. If an allocation achieved in the mechanism is less expensive than the Nash al- 205
location (i.e., + (¯r ¯s ) ¯ + (¯r¯s)) , then the same allocation − −
is less expensive if the public good could be purchased at a price of (i.e.,
+ ¯ + ¯). · ·
3. If a feasible allocation ( ) is strictly preferred to the Nash allocation ∈
(¯ ¯), then the preferred bundle is more expensive than the Nash allocation
(i.e., + ¯ + ¯). · ·
Proof. L2.1 By definition, = (¯r¯s). By assumption, preference are convex so
certainly the set of bundles that are weakly preferred to (¯ ¯) is convex and
(¯ ¯) is on the boundary of the set. Let the set of affordable bundles be denoted
= ( ) +ˆ (;¯r ¯s ) ,whereˆ (;¯ s)= (r s) + ∈ | − − ≤ − · © 2 2 ª ( ) + (¯s+1 ) . is convex since ˆ is a convex function of .Bypart 2 − 2 −
(2) of Lemma 1, (¯ ¯) is on the boundary of set . From part (5) of Lemma
1, we have the intersection of the set of weakly preferred bundles to (¯ ¯)
(denote )andthebudgetset is empty. From the Separating Hyperplane
Theorem, there exists a hyperplane through (¯ ¯) that separates and .
The vector ( 1) defines this hyperplane. Also from the Separating Hyperplane
Theorem, we have that + and ¯ + ¯ = ,where =0. It follows · ≥ · 6
that + ¯ + ¯. · ≥ · 206
L2.2 If + (¯r ¯s ) ¯ + (¯r¯s)=, we can expand each of these ex- − −
2 2 pressions to + (¯r ¯s ) (¯r ¯s )+ ( ) + (¯+1 ) − − · − − 2 − 2 − ¯ + (¯r¯s) (¯r¯s).Let = (¯r ¯s ) and ¯ = (¯r¯s). By construc- · − − tion, the personalized price function (¯r ¯s )= (¯r¯s)=.wecan − −
subtract the two squared terms on the LHS to get + ¯ + ¯. · ·
L2.3 Suppose not. By part (1) of Lemma 2, we have + =¯ + ¯.Sincepref- · ·
erences are continuous there exists a neighborhood of ( ) denoted ( )
such that for all (ˆ ˆ) ( ) , (ˆ ˆ) (¯ ¯). Parts1and6ofLemma ∈ ∩ Â
1andpart2ofLemma2implythatthereexistsabundle(´ ´) such that ∈
´ + ´ ¯ + ¯ = + = .Let · · ·
(ˆ ˆ) (ˆ ˆ)=(´ +(1 ) ´ +(1 ) ) for all (0 1) . ≡ { ∈ | − − ∈ }
All points in this line between (´ ´) and ( ) have a value smaller than
(¯ ¯). However since the consumption set is convex it follows that there exists
a which is small enough such that ( ) —i.e. there exists a bundle ∩
(ˆ ˆ) such that (ˆ ˆ) (¯ ¯) and ˆ + ˆ ¯ + ¯ which leads to a  · · contradiction of part (1) of Lemma 2.
The next lemma and its proof are almost identical to those in the First Funda- mental Welfare Theorem for private good economies (see Debreu 1959). 207
Lemma 10. Suppose (¯r¯s) isaNashequilibriumof for , then the Nash ∈ allocation (¯r¯s) ( (¯r¯s)) is Pareto optimal. − =1 £ ¤ Proof. Suppose ¯ (¯)=1 is not a Pareto optimal allocation and that ()=1 is a feasible, Pareto£ superior¤ allocation. From part 3 of Lemma 2, we have£ that ¤
+ ¯ + ¯ for all . · ·
Summing across all consumers, we have
+ ¯ + ¯. =1 =1 · =1 =1 · X X X X By construction, =1 = =1 (¯r¯s)=. Re-writing the above strict in- equality, we have P P
+ ¯ + ¯ = . =1 · =1 · =1 X X X Thus, the Pareto superior bundle is not feasible.
Lemma 11. The affordable feasible set, denoted ,isaconvexset,where
(1 1 ) () = ⎧ | ∈ ⎫ . ( ) ⎪ =1 − ⎪ ⎨ where = = for all = and ⎬ 6 ≤ ⎪ ⎪ Furthermore, the⎩ point (¯1 ¯ ¯1 ¯ ), associated with the Nash equilibrium,⎭ is on the boundary of . 208
Proof. To show that is convex choose two arbitrary profiles
(1 1 ) (´1 ´ ´1 ´ ) ∈
For [0 1], the convex combination of these two vectors is ∈
(1 +(1 )´1 +(1 )´ 1 +(1 )´1 +(1 )´ ) . − − − −
First, since is convex, ( +(1 )´ +(1 )´) for all . Second, − − ∈ because both = = and ´ =´ =´ for all = ,then+(1 )´ = +(1 6 − − )´ = +(1 )´. Finally, ( ) implies ( ). − ≤ =1 − ≤ =1 − P P Similarly, ´ ( ´) implies (1 ) ´ (1 ) ( ´).Adding ≤ =1 − − ≤ − =1 − these two conditionsP together, we have the following inequality,P
( ( +(1 )´)) +(1 )´ =1 − − . − ≤ P verifying that the set is convex.
To see that (¯1 ¯ ¯1 ¯ ) is in the boundary of the set. Recall from the
Lemma 3 that the Nash allocation is Pareto optimal—i.e. ¯+ ¯ = .Re- =1 · =1 ( ¯) =1 − P P arranging this expression, we have ¯ = which is clearly on the boundary of .
Proof. [Proof of Theorem 1] The proof for Theorem 1 is done in three parts. In the
first part of the proof we show that for any if (¯r¯s) is a Nash equilibrium of ∈ , the corresponding allocation (¯r¯s) ( (¯r¯s)) is a Lindahl equilibrium − =1 h i 209 and for each , (¯r¯s) is the corresponding Lindahl price. It is first shown that the personalized price associated with the Nash equilibrium per unit tax (¯r¯s) defines a separating hyperplane between the feasible allocation set andthepreferredset
; second, we show that the Nash allocation is the allocation that maximizes a consumer’s preferences subject to a budget constraint when facing the personalized price (¯r¯s); finally, we show that the tax revenue equals the cost of producing the public good.
In the second part of the proof, we show that for if ¯1 ¯ is the profile ∈ ¡ ¢ of Lindahl prices and ¯ ¯ ¯ is the corresponding Lindahl allocation, − · =1 ³ ´ then it must correspond to¡ a Nash equilibrium¢ of the mechanism. we do this by first showing that the messages that could achieve this allocation in the mechanism are unique. Subsequently that this profile of strategies is a Nash equilibrium of the game induced by the mechanism.
Part 3 of the proof estabilishes Nash implementation for the environment.
(Part 1): Consider the point (¯1 ¯ ¯1 ¯ ) associated with the Nash al- location for each consumer. From Lemma 4, we have that the feasible set is convex and that the point (¯1 ¯ ¯1 ¯ ) is on its boundary. Similarly from
Lemma5,wehavethattheset is convex and point (¯1 ¯ ¯1 ¯ ) is on the boundary. Notice that the intersection of the interiors of and have no points in common. To see this suppose that these sets do have points in the interior that are common. Then there is a strictly cheaper feasible point that is weakly preferred by 210
all consumers. However, this contradicts the fact that (¯1 ¯ ¯1 ¯ ) is Pareto optimal (Lemma 3). Therefore by the Separating Hyperplane Theorem, there exists
a vector (1 1 ) =0and R such that for all points in the weakly ··· ··· 6 ∈ preferred set ,
( ) + =1 · =1 · ≥ X X In addition, since the vector (¯1 ¯ ¯1 ¯ ) is in the boundary of both and
,
( ) ¯ + ¯ = =1 · =1 · X X Since (¯r¯s) is a Nash equilibrium, the hyperplane that crosses through (¯ ¯) is defined
by the vector of ( )=( 1) for each where = = (¯r¯s) (Lemmas 1 and
2). This should be thought of as consumer ’s personalized price.
Next, we show that the bundle (¯ ¯) maximizes the preferences of consumer subject to ’s budget constraint when facing (¯r¯s) as his personalized price.
Suppose () (¯ ¯) while =¯ and =¯ for all = .Thispointisin  6 set . From the separating hyperplane defined above we have,
(¯ ¯) + (¯ ¯) ¯ + ¯. Ã =1 ! · =1 ≥ Ã =1 ! · =1 X X X X All terms in this expression are the same except those belonging to consumer .
Thus the expression can be simplified to + (¯ ¯) ¯ + (¯ ¯) ¯.From · ≥ · part 3 of Lemma 2, since the bundle () (¯ ¯) equality cannot hold so Â
(¯ ¯) + (¯ ¯) ¯ +¯ · · 211
The personalized price for consumer is independent ´’s actions—i.e., (¯ ¯)=
( ¯ ¯ ) for all and . Using this fact we am going to rewrite the above − − expression to be
( ¯ ¯ ) + (¯ ¯) ¯ +¯. − − · ·
Now adding two appropriately chosen positive terms on the LHS, we have
2 2 ( ¯ ¯ ) + + ( ) + (¯+1 ) (¯ ¯) ¯ +¯. − − · 2 − 2 − ·
However, this is equivalent to + ( ¯ ¯ ) ¯ + (¯ ¯)=,where = − −
1 1 ( + = ¯) and ¯ = (¯ + = ¯). Thus, any bundle that is strictly preferred to 6 6 theNashbundleisnotaP ffordableP by the consumer—i.e., the Nash allocation maximizes consumer ’s preferences subject to a budget constraint.
The last part of the argument requires tax revenue to equal the total cost of production.
If we add up the tax revenue, we have that
(¯ ¯)= (¯ ¯) (¯ ¯) =1 =1 · X X ¯ = + ¯+1 (¯ ¯) − 1 · =1 Ã = ! X X6 − ¯ = + ¯+1 (¯ ¯) − 1 · Ã =1 = =1 ! X X6 − X =( (¯ ¯)+(¯ ¯)) (¯ ¯) − · = (¯ ¯). · 212
Thus the allocation is feasible and this is a Lindahl allocation, where ¯1 ¯ will be the profile of Lindahl prices. ¡ ¢
(Part 2): For all ,let¯ =¯. Consider the following system of linear equations and variables (1 )
1 + 2 + + = ¯ ··· ·
1 = ¯ ¯+1 ( 1) for =1 1 − − − − − = ∙ µ ¶ ¸ X6 It is straightforward to verify that the coefficient matrix of this system of × equations is non-singular with a rank of . Thus, the system has a unique solution which we will call (¯ ¯). It remains to show that (¯r¯s) is a Nash equilibrium.
Since the allocation (¯ ( ¯ ¯) ) is Lindahl, (¯ ¯ ¯) ( ¯ ) − · =1 − · º − · 1 for all .Let = ( + = ¯)=(¯r ¯s),then 6 − P 1 1 (¯ ¯ ¯) ( + ¯) ¯ ( + ¯) − · º − · Ã = = ! X6 X6 for all .
Similarly, since preferences are strictly increasing in ,itisalsotruethat
(¯ ¯ ¯) − · 1 1 1 2 ( + ¯) ¯ ( + ¯) ( ¯) º − · − 2 − Ã = = =1 X6 X6 X 1 1 2 (+1 ¯) −2 − − = ! X6 213
for all , .
By construction, the public good
¯1 + +¯ ¯ = ··· = (¯r¯s) consumer i’s Lindahl price was
¯ = (¯r¯s)
1 and ¯ = ¯ for all . =1 PluggingP in these expressions into the above inequality, we have
((¯r¯s) (¯r¯s)) −
((¯r ¯ ) (¯r ¯s )) º − − − − −
for all , .Therefore (¯r¯s) is a Nash equilibrium of the mechanism
(Part 3) For any ,if(¯r¯s) is a Nash equilibrium, then the stictly increas- ∈ ing utility in the private good implies each ’s statement is correct and first order
() conditions for the Nash equilibrium must be = = (¯r¯s). Since, by
construction = (¯r¯s)=, the equilibrium production of the pub- lic good is Pareto.P Moreover,P since in equilibrium ’s private good allocation is
() 0 the allocation is Lindahl. By assumption, the environment − has a unique Lindahl allocation. In a manner similar to part 2 of above we can show that this unique Lindahl allocation is implemented by a unique Nash equilibrium. 214
2 2 Proof. [Proof of Theorem 2] Since = R , it is a sublattice of R .Bydefinition of being in the environment is 2 and therefore trivially satisfies the continuity requirement. To see that has the supermodularity property, we appeal to the fact that the utility function is 2. we therefore need to check the following cross-partial derivative 2 0 ≥ Checking this, we see that
2 = 0 ≥
for each consumer . The increasing difference property requires checking the following five conditions:
2 (1) 0 for all = ≥ 6 2 (2) 0 for all = and = +1 ≥ 6 6 2 (3) 0 +1 ≥ 2 (4) 0 for all = ≥ 6 2 (5) 0 +1 ≥
Checking each of these in turn we have
2 1 1 2 = 2 2 + 2 2 ( 1) − − − In order for the above expression to be positive we need
1 2 − + for all = ≥ − 2 6 µ ¶ 215
A more compact way of writing this is
1 2 − + min ≥ − 2 µ ∈ ¶ Condition 2 is trivially satisfied since
2 =0 for all = and = +1 6 6
Checking Condition 3 we have
2 = + +1 −
This expression is positive for all if and only if
≤
Condition 4 and 5 are always satisfied since
2 2 = 0 and =0 +1
Therefore, for the mechanism to be supermodular the following is sufficient.
1 − + min ∈ − ∙ µ ∈ ¶ ¸ Finally, this interval is non-empty if and only if 1 +min is true. ≤ − ∈
Proof. [Proof of Theorem 3] First, we characterize the best replies. Applying the mechanism to each consumer’s utility we arrive at the augmented utility function
1 1 ( ) + +1 − − 1 =1 Ã = ! =1 X − X6 X 1 2 1 2 ( ) (+1 ) . −2 − − 2 − =1 X X 216
Best responding requires consumers choices to satisfy first order conditions — i.e.„
1 1 1 1 : + +1 + ( )+ (+1 )=0 − − 1 − − Ã = ! =1 − X6 X X 1 : ( )=0 − − =1 X
If we let ∗( ) and ∗( ) be the solutions to these first order conditions. − − − −
1 1 Clearly, ∗( )= ∗( )+ =1 ,ifweplug∗ into the condition. − − − − P The new condition is
1( ) 1 1 1 · + +1 + (+1 ∗( )) )=0 − − 1 − − − − Ã = ! − X6 X
We can think of each decision being chosen by a separate agent: 1 agent for ∗( ) − − and one agent for the ∗( ). Since we have already accounted for the interaction − − between own decisions in the determination of the best replies, we can think of the game as one with 2 independent players choosing according to the specified reac- tion functions. The problem of showing a contraction reduces to the one of Vives.1
Therefore a sufficient condition for the best reply map to yield a contraction is that, for each , the absolute total change in ∗( ) and ∗( ) (evaluated at any − − − − point ( )) is bounded by 1. In other words, we require − −
∗( ) ∗( ) − − + − − 1 = ¯ ¯ = ¯ ¯ X6 ¯ ¯ X6 ¯ ¯ ¯ ∗( ¯ ) ¯∗( )¯ ¯ − ¯− + ¯ − − ¯ 1 +1 = ¯ ¯ ¯ ¯ X6 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 See, for example, p. 47. ¯ ¯ ¯ ¯ 217
We compute these slopes directly. Differentiating the new first order condition with respect to .
11 ∗( ) ∗( ) − − − − 2 (1 + )+ 2 2 =0 ( 1) − − − ∗( ) ∗( ) 11(1 + − − )+ − − =0 ( 1) − − − ∗( ) 11 + =( 11) − − ( 1) − − − + ∗( ) 11 ( 1) − − − − = − 11
Differentiating with respect to +1
11 ∗( ) ∗( ) − − − − 2 + 2 =0 +1 − − +1
∗( ) ∗( ) + = − − 11 − − − +1 − +1
∗( ) + − − = − +1 − 11
Asufficient condition for ∗( ) to be a contraction is that − −
11 + ( 1) + − − + − 1. = ¯ 11 ¯ 11 6 ¯ − ¯ ¯ − ¯ X ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Suppose , ,and satisfy¯ the supermodularity¯ ¯ conditions¯ from Theorem 1, then the slopes are all positive leaving
( 1) + ( 1) + − 11 − − − − 11 0 . − 11
This condition is always satisfied since 11 0. 218
1 1 Now consider ∗( )= ∗( )+ =1 . − − − − P ∗( ) 1 ∗ 1 − − = + 1 11 + ( 1) 1 = − − + Ã − 11 ! 1 ( 11)+ ( 1) = (− − − +1) − 11 =( ) ( 1)( ) − − 11
∗( ) 1 ∗ − − = +1 +1 1 + = − µ − 11 ¶ + = − − 11 Adding up across each player and checking the sufficient condition.
∗( ) ∗( ) − − + − − 1 +1 = ¯ ¯ ¯ ¯ X6 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Supermodularity ensures the¯ slopes are¯ all positive.¯ Therefore,¯
+ + − 1 − 11 − 11 − 11
11 0
and since we have 11 0 by assumption, the second condition is satisfied. Since this is true for each , the best reply map is a contraction. 219
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