Strong Nash Equilibria and Mixed Strategies
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Implementation Theory*
Chapter 5 IMPLEMENTATION THEORY* ERIC MASKIN Institute for Advanced Study, Princeton, NJ, USA TOMAS SJOSTROM Department of Economics, Pennsylvania State University, University Park, PA, USA Contents Abstract 238 Keywords 238 1. Introduction 239 2. Definitions 245 3. Nash implementation 247 3.1. Definitions 248 3.2. Monotonicity and no veto power 248 3.3. Necessary and sufficient conditions 250 3.4. Weak implementation 254 3.5. Strategy-proofness and rich domains of preferences 254 3.6. Unrestricted domain of strict preferences 256 3.7. Economic environments 257 3.8. Two agent implementation 259 4. Implementation with complete information: further topics 260 4.1. Refinements of Nash equilibrium 260 4.2. Virtual implementation 264 4.3. Mixed strategies 265 4.4. Extensive form mechanisms 267 4.5. Renegotiation 269 4.6. The planner as a player 275 5. Bayesian implementation 276 5.1. Definitions 276 5.2. Closure 277 5.3. Incentive compatibility 278 5.4. Bayesian monotonicity 279 * We are grateful to Sandeep Baliga, Luis Corch6n, Matt Jackson, Byungchae Rhee, Ariel Rubinstein, Ilya Segal, Hannu Vartiainen, Masahiro Watabe, and two referees, for helpful comments. Handbook of Social Choice and Welfare, Volume 1, Edited by K.J Arrow, A.K. Sen and K. Suzumura ( 2002 Elsevier Science B. V All rights reserved 238 E. Maskin and T: Sj'str6m 5.5. Non-parametric, robust and fault tolerant implementation 281 6. Concluding remarks 281 References 282 Abstract The implementation problem is the problem of designing a mechanism (game form) such that the equilibrium outcomes satisfy a criterion of social optimality embodied in a social choice rule. -
Journal of Mathematical Economics Implementation of Pareto Efficient Allocations
Journal of Mathematical Economics 45 (2009) 113–123 Contents lists available at ScienceDirect Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco Implementation of Pareto efficient allocations Guoqiang Tian a,b,∗ a Department of Economics, Texas A&M University, College Station, TX 77843, USA b School of Economics and Institute for Advanced Research, Shanghai University of Finance and Economics, Shanghai 200433, China. article info abstract Article history: This paper considers Nash implementation and double implementation of Pareto effi- Received 10 October 2005 cient allocations for production economies. We allow production sets and preferences Received in revised form 17 July 2008 are unknown to the planner. We present a well-behaved mechanism that fully imple- Accepted 22 July 2008 ments Pareto efficient allocations in Nash equilibrium. The mechanism then is modified Available online 5 August 2008 to fully doubly implement Pareto efficient allocations in Nash and strong Nash equilibria. The mechanisms constructed in the paper have many nice properties such as feasibility JEL classification: C72 and continuity. In addition, they use finite-dimensional message spaces. Furthermore, the D61 mechanism works not only for three or more agents, but also for two-agent economies. D71 © 2008 Elsevier B.V. All rights reserved. D82 Keywords: Incentive mechanism design Implementation Pareto efficiency Price equilibrium with transfer 1. Introduction 1.1. Motivation This paper considers implementation of Pareto efficient allocations for production economies by presenting well-behaved and simple mechanisms that are continuous, feasible, and use finite-dimensional spaces. Pareto optimality is a highly desir- able property in designing incentive compatible mechanisms. The importance of this property is attributed to what may be regarded as minimal welfare property. -
Collusion Constrained Equilibrium
Theoretical Economics 13 (2018), 307–340 1555-7561/20180307 Collusion constrained equilibrium Rohan Dutta Department of Economics, McGill University David K. Levine Department of Economics, European University Institute and Department of Economics, Washington University in Saint Louis Salvatore Modica Department of Economics, Università di Palermo We study collusion within groups in noncooperative games. The primitives are the preferences of the players, their assignment to nonoverlapping groups, and the goals of the groups. Our notion of collusion is that a group coordinates the play of its members among different incentive compatible plans to best achieve its goals. Unfortunately, equilibria that meet this requirement need not exist. We instead introduce the weaker notion of collusion constrained equilibrium. This al- lows groups to put positive probability on alternatives that are suboptimal for the group in certain razor’s edge cases where the set of incentive compatible plans changes discontinuously. These collusion constrained equilibria exist and are a subset of the correlated equilibria of the underlying game. We examine four per- turbations of the underlying game. In each case,we show that equilibria in which groups choose the best alternative exist and that limits of these equilibria lead to collusion constrained equilibria. We also show that for a sufficiently broad class of perturbations, every collusion constrained equilibrium arises as such a limit. We give an application to a voter participation game that shows how collusion constraints may be socially costly. Keywords. Collusion, organization, group. JEL classification. C72, D70. 1. Introduction As the literature on collective action (for example, Olson 1965) emphasizes, groups often behave collusively while the preferences of individual group members limit the possi- Rohan Dutta: [email protected] David K. -
Arxiv:0803.2996V1 [Q-Fin.GN] 20 Mar 2008 JEL Classification: A10, A12, B0, B40, B50, C69, C9, D5, D1, G1, G10-G14
The virtues and vices of equilibrium and the future of financial economics J. Doyne Farmer∗ and John Geanakoplosy December 2, 2008 Abstract The use of equilibrium models in economics springs from the desire for parsimonious models of economic phenomena that take human rea- soning into account. This approach has been the cornerstone of modern economic theory. We explain why this is so, extolling the virtues of equilibrium theory; then we present a critique and describe why this approach is inherently limited, and why economics needs to move in new directions if it is to continue to make progress. We stress that this shouldn't be a question of dogma, but should be resolved empir- ically. There are situations where equilibrium models provide useful predictions and there are situations where they can never provide use- ful predictions. There are also many situations where the jury is still out, i.e., where so far they fail to provide a good description of the world, but where proper extensions might change this. Our goal is to convince the skeptics that equilibrium models can be useful, but also to make traditional economists more aware of the limitations of equilib- rium models. We sketch some alternative approaches and discuss why they should play an important role in future research in economics. Key words: equilibrium, rational expectations, efficiency, arbitrage, bounded rationality, power laws, disequilibrium, zero intelligence, mar- ket ecology, agent based modeling arXiv:0803.2996v1 [q-fin.GN] 20 Mar 2008 JEL Classification: A10, A12, B0, B40, B50, C69, C9, D5, D1, G1, G10-G14. ∗Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe NM 87501 and LUISS Guido Carli, Viale Pola 12, 00198, Roma, Italy yJames Tobin Professor of Economics, Yale University, New Haven CT, and Santa Fe Institute 1 Contents 1 Introduction 4 2 What is an equilibrium theory? 5 2.1 Existence of equilibrium and fixed points . -
An Equilibrium-Conserving Taxation Scheme for Income from Capital
Eur. Phys. J. B (2018) 91: 38 https://doi.org/10.1140/epjb/e2018-80497-x THE EUROPEAN PHYSICAL JOURNAL B Regular Article An equilibrium-conserving taxation scheme for income from capital Jacques Temperea Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universiteitsplein 1, 2610 Antwerpen, Belgium Received 28 August 2017 / Received in final form 23 November 2017 Published online 14 February 2018 c The Author(s) 2018. This article is published with open access at Springerlink.com Abstract. Under conditions of market equilibrium, the distribution of capital income follows a Pareto power law, with an exponent that characterizes the given equilibrium. Here, a simple taxation scheme is proposed such that the post-tax capital income distribution remains an equilibrium distribution, albeit with a different exponent. This taxation scheme is shown to be progressive, and its parameters can be simply derived from (i) the total amount of tax that will be levied, (ii) the threshold selected above which capital income will be taxed and (iii) the total amount of capital income. The latter can be obtained either by using Piketty's estimates of the capital/labor income ratio or by fitting the initial Pareto exponent. Both ways moreover provide a check on the amount of declared income from capital. 1 Introduction distribution of money over the agents involved in additive transactions follows a Boltzmann{Gibbs exponential dis- The distribution of income has been studied for a long tribution. Note that this is a strongly simplified model of time in the economic literature, and has more recently economic activity: it is clear that in reality global money become a topic of investigation for statistical physicists conservation is violated. -
Implementation and Strong Nash Equilibrium
LIBRARY OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/implementationstOOmask : } ^mm working paper department of economics IMPLEMENTATION AND STRONG NASH EQUILIBRIUM Eric Maskin Number 216 January 1978 massachusetts institute of technology 50 memorial drive Cambridge, mass. 021 39 IMPLEMENTATION AND STRONG NASH EQUILIBRIUM Eric Maskin Number 216 January 1978 I am grateful for the financial support of the National Science Foundation. A social choice correspondence (SCC) is a mapping which assoc- iates each possible profile of individuals' preferences with a set of feasible alternatives (the set of f-optima) . To implement *n SCC, f, is to construct a game form g such that, for all preference profiles the equilibrium set of g (with respect to some solution concept) coincides with the f-optimal set. In a recent study [1], I examined the general question of implementing social choice correspondences when Nash equilibrium is the solution concept. Nash equilibrium, of course, is a strictly noncooperative notion, and so it is natural to consider the extent to which the results carry over when coalitions can form. The cooperative counterpart of Nash is the strong equilibrium due to Aumann. Whereas Nash defines equilibrium in terms of deviations only by single individuals, Aumann 's equilibrium incorporates deviations by every conceivable coalition. This paper considers implementation for strong equilibrium. The results of my previous paper were positive. If an SCC sat- isfies a monotonicity property and a much weaker requirement called no veto power, it can be implemented by Nash equilibrium. -
On the Verification and Computation of Strong Nash Equilibrium
On the Verification and Computation of Strong Nash Equilibrium Nicola Gatti Marco Rocco Tuomas Sandholm Politecnico di Milano Politecnico di Milano Carnegie Mellon Piazza Leonardo da Vinci 32 Piazza Leonardo da Vinci 32 Computer Science Dep. Milano, Italy Milano, Italy 5000 Forbes Avenue [email protected] [email protected] Pittsburgh, PA 15213, USA [email protected] ABSTRACT The NE concept is inadequate when agents can a priori Computing equilibria of games is a central task in computer communicate, being in a position to form coalitions and de- science. A large number of results are known for Nash equi- viate multilaterally in a coordinated way. The strong Nash librium (NE). However, these can be adopted only when equilibrium (SNE) concept strengthens the NE concept by coalitions are not an issue. When instead agents can form requiring the strategy profile to be resilient also to multilat- coalitions, NE is inadequate and an appropriate solution eral deviations, including the grand coalition [2]. However, concept is strong Nash equilibrium (SNE). Few computa- differently from NE, an SNE may not exist. It is known tional results are known about SNE. In this paper, we first that searching for an SNE is NP–hard, but, except for the study the problem of verifying whether a strategy profile is case of two–agent symmetric games, it is not known whether an SNE, showing that the problem is in P. We then design the problem is in NP [8]. Furthermore, to the best of our a spatial branch–and–bound algorithm to find an SNE, and knowledge, there is no algorithm to find an SNE in general we experimentally evaluate the algorithm. -
Egalitarianism in Mechanism Design∗
Egalitarianism in Mechanism Design∗ Geoffroy de Clippely This Version: January 2012 Abstract This paper examines ways to extend egalitarian notions of fairness to mechanism design. It is shown that the classic properties of con- strained efficiency and monotonicity with respect to feasible options become incompatible, even in quasi-linear settings. An interim egali- tarian criterion is defined and axiomatically characterized. It is applied to find \fair outcomes" in classical examples of mechanism design, such as cost sharing and bilateral trade. Combined with ex-ante utilitari- anism, the criterion characterizes Harsanyi and Selten's (1972) interim Nash product. Two alternative egalitarian criteria are proposed to il- lustrate how incomplete information creates room for debate as to what is socially desirable. ∗The paper beneficiated from insightful comments from Kfir Eliaz, Klaus Nehring, David Perez-Castrillo, Kareen Rozen, and David Wettstein. Financial support from the National Science Foundation (grant SES-0851210) is gratefully acknowledged. yDepartment of Economics, Brown University. Email: [email protected] 1. INTRODUCTION Developments in the theory of mechanism design, since its origin in the sev- enties, have greatly improved our understanding of what is feasible in envi- ronments involving agents that hold private information. Yet little effort has been devoted to the discussion of socially desirable selection criteria, and the computation of incentive compatible mechanisms meeting those criteria in ap- plications. In other words, extending the theory of social choice so as to make it applicable in mechanism design remains a challenging topic to be studied. The present paper makes some progress in that direction, with a focus on the egalitarian principle. -
Maximin Equilibrium∗
Maximin equilibrium∗ Mehmet ISMAILy March, 2014. This version: June, 2014 Abstract We introduce a new theory of games which extends von Neumann's theory of zero-sum games to nonzero-sum games by incorporating common knowledge of individual and collective rationality of the play- ers. Maximin equilibrium, extending Nash's value approach, is based on the evaluation of the strategic uncertainty of the whole game. We show that maximin equilibrium is invariant under strictly increasing transformations of the payoffs. Notably, every finite game possesses a maximin equilibrium in pure strategies. Considering the games in von Neumann-Morgenstern mixed extension, we demonstrate that the maximin equilibrium value is precisely the maximin (minimax) value and it coincides with the maximin strategies in two-player zero-sum games. We also show that for every Nash equilibrium that is not a maximin equilibrium there exists a maximin equilibrium that Pareto dominates it. In addition, a maximin equilibrium is never Pareto dominated by a Nash equilibrium. Finally, we discuss maximin equi- librium predictions in several games including the traveler's dilemma. JEL-Classification: C72 ∗I thank Jean-Jacques Herings for his feedback. I am particularly indebted to Ronald Peeters for his continuous comments and suggestions about the material in this paper. I am also thankful to the participants of the MLSE seminar at Maastricht University. Of course, any mistake is mine. yMaastricht University. E-mail: [email protected]. 1 Introduction In their ground-breaking book, von Neumann and Morgenstern (1944, p. 555) describe the maximin strategy1 solution for two-player games as follows: \There exists precisely one solution. -
Arxiv:1701.06410V1 [Q-Fin.EC] 27 Oct 2016 E Od N Phrases
ECONOMICS CANNOT ISOLATE ITSELF FROM POLITICAL THEORY: A MATHEMATICAL DEMONSTRATION BRENDAN MARKEY-TOWLER ABSTRACT. The purpose of this paper is to provide a confession of sorts from an economist to political science and philosophy. A confession of the weaknesses of the political position of the economist. It is intended as a guide for political scientists and philosophers to the ostensible policy criteria of economics, and an illustration of an argument that demonstrates logico-mathematically, therefore incontrovertibly, that any policy statement by an economist contains, or is, a political statement. It develops an inescapable compulsion that the absolute primacy and priority of political theory and philosophy in the development of policy criteria must be recognised. Economic policy cannot be divorced from politics as a matter of mathematical fact, and rather, as Amartya Sen has done, it ought embrace political theory and philosophy. 1. THE PLACE AND IMPORTANCE OF PARETO OPTIMALITY IN ECONOMICS Economics, having pretensions to being a “science”, makes distinctions between “positive” statements about how the economy functions and “normative” statements about how it should function. It is a core attribute, inherited largely from its intellectual heritage in British empiricism and Viennese logical positivism (McCloskey, 1983) that normative statements are to be avoided where possible, and ought contain little by way of political presupposition as possible where they cannot. Political ideology is the realm of the politician and the demagogue. To that end, the most basic policy decision criterion of Pareto optimality is offered. This cri- terion is weaker than the extremely strong Hicks-Kaldor criterion. The Hicks-Kaldor criterion presupposes a consequentialist, specifically utilitarian philosophy and states any policy should be adopted which yields net positive utility for society, any compensation for lost utility on the part arXiv:1701.06410v1 [q-fin.EC] 27 Oct 2016 of one to be arranged by the polity, not the economist, out of the gains to the other. -
Pareto Efficiency by MEGAN MARTORANA
RF Fall Winter 07v42-sig3-INT 2/27/07 8:45 AM Page 8 JARGONALERT Pareto Efficiency BY MEGAN MARTORANA magine you and a friend are walking down the street competitive equilibrium is typically included among them. and a $100 bill magically appears. You would likely A major drawback of Pareto efficiency, some ethicists claim, I share the money evenly, each taking $50, deeming this is that it does not suggest which of the Pareto efficient the fairest division. According to Pareto efficiency, however, outcomes is best. any allocation of the $100 would be optimal — including the Furthermore, the concept does not require an equitable distribution you would likely prefer: keeping all $100 for distribution of wealth, nor does it necessarily suggest taking yourself. remedial steps to correct for existing inequality. If the Pareto efficiency says that an allocation is efficient if an incomes of the wealthy increase while the incomes of every- action makes some individual better off and no individual one else remain stable, such a change is Pareto efficient. worse off. The concept was developed by Vilfredo Pareto, an Martin Feldstein, an economist at Harvard University and Italian economist and sociologist known for his application president of the National Bureau of Economic Research, of mathematics to economic analysis, and particularly for his explains that some see this as unfair. Such critics, while con- Manual of Political Economy (1906). ceding that the outcome is Pareto Pareto used this work to develop efficient, might complain: “I don’t his theory of pure economics, have fewer material goods, but I analyze “ophelimity,” his own have the extra pain of living in a term indicating the power of more unequal world.” In short, giving satisfaction, and intro- they are concerned about not only duce indifference curves. -
Maximin Equilibrium
Munich Personal RePEc Archive Maximin equilibrium Ismail, Mehmet Maastricht University October 2014 Online at https://mpra.ub.uni-muenchen.de/97322/ MPRA Paper No. 97322, posted 04 Dec 2019 13:32 UTC Mehmet Ismail Maximin equilibrium RM/14/037 Maximin equilibrium∗ Mehmet ISMAIL† First version March, 2014. This version: October, 2014 Abstract We introduce a new concept which extends von Neumann and Mor- genstern’s maximin strategy solution by incorporating ‘individual ra- tionality’ of the players. Maximin equilibrium, extending Nash’s value approach, is based on the evaluation of the strategic uncertainty of the whole game. We show that maximin equilibrium is invariant under strictly increasing transformations of the payoffs. Notably, every finite game possesses a maximin equilibrium in pure strategies. Considering the games in von Neumann-Morgenstern mixed extension, we demon- strate that the maximin equilibrium value is precisely the maximin (minimax) value and it coincides with the maximin strategies in two- person zerosum games. We also show that for every Nash equilibrium that is not a maximin equilibrium there exists a maximin equilibrium that Pareto dominates it. Hence, a strong Nash equilibrium is always a maximin equilibrium. In addition, a maximin equilibrium is never Pareto dominated by a Nash equilibrium. Finally, we discuss max- imin equilibrium predictions in several games including the traveler’s dilemma. JEL-Classification: C72 Keywords: Non-cooperative games, maximin strategy, zerosum games. ∗I thank Jean-Jacques Herings