DOCSLIB.ORG

Reciprocity and Overoptimism in Public Goods Games

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Reciprocity and Overoptimism in Public Goods Games Biased Beliefs and Reciprocal Behavior in Social Dilemmas Von der Fakultät für Wirtschaftswissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Wirtschafts- und Sozialwissenschaften genehmigte Dissertation vorgelegt von Dipl.-Ing. Anselm Hüwe Berichter: Univ.-Prof. Dr. rer. pol. Wolfgang Breuer Univ.-Prof. Dr. rer. pol. Christine Harbring Tag der mündlichen Prüfung: 21.04.2015 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Table of Contents 1 Introduction ............................................................................................................................. 1 2 Economic Experiments ........................................................................................................... 3 3 Game Theory and (Common) Belief in Rationality ................................................................ 8 3.1 Boundedly Rational Behavior ........................................................................................ 11 3.2 Biased Beliefs ................................................................................................................. 14 4 Social Preferences ................................................................................................................. 15 4.1 When Social Preferences Should Be Assumed, and When Not ..................................... 16 4.2 Evolutionary Analysis .................................................................................................... 18 4.3 Reciprocity ...................................................................................................................... 22 5 Remarks on the experiments ................................................................................................. 24 5.1 Reciprocity in the Public Goods Game .......................................................................... 25 5.2 Reciprocity in the Trust Game ........................................................................................ 26 5.3 Reciprocity in the Ultimatum Game ............................................................................... 28 5.4 An Integrated View of All Three Experiments .............................................................. 29 References ................................................................................................................................ 36 Explaining Individual Contributions in Public Goods Games Using (only) Reciprocity and Overoptimism ........................................................................................................................... 41 Trust, Reciprocity, and Betrayal Aversion: Theoretical and Experimental Insights ............. 100 Using the Carrot and the Stick? Theoretical and Experimental Insights Into Positive vs. Negative Reciprocity .............................................................................................................. 161 1 Introduction We cannot understand the […] economy without having a theory of how humans make decisions. Herbert Gintis (2009), p. 2. Adam Smith, who is usually considered to be the founding father of microeconomics, is most frequently cited for his image of the “invisible hand“, which expresses that everybody should intend only his own gain: By doing so, an “invisible hand” lets people simultaneously promote the public interest (Smith, 1937 [1776]). According to this concept, the existence of social dilemmas, where a conflict between self-interest and public interest exists, is denied. This “selfishness axiom” (Henrich et al., 2004) has for a long time prevailed in economic research. Typically, the selfishness axiom was (or still is) combined with the assumption of rational behavior, leading to a simple and often useful theory of human behavior (known as the homo economicus model, the neoclassical model, or the standard model). However, representations of social dilemmas, such as the public goods game (PGG), prove that selfishness is not always in the public interest. The selfishness axiom predicts that people cannot solve such social dilemmas. In contrast, experimental research in the last decades has proven that people often act in the public interest even when it causes harm to themselves. This finding, which – according to his less frequently cited book “Theory of Moral Sentiments” – even Adam Smith was aware of, has brought the understanding of social preferences for economic interactions back into the spotlight. In addition, experimental research has discovered various deviations from the rationality assumption, which is an essential part of the homo economicus model. While giving up this assumption makes models more complicated, upholding it hampers attempts to gain important new insights. For example, an important finding of this dissertation is that social preferences and seemingly irrational biases in human thinking interact with each other. Therefore, they must 1 be considered in an integrated way. More generally, the goal of this dissertation is to contribute to a better understanding of human decision making in social dilemmas. To achieve this objective, we reject both the selfishness axiom and the rationality axiom. Instead, we modify the reciprocity model of Dufwenberg and Kirchsteiger (2004) (henceforth DK), and extend it by introducing biased and therefore irrational beliefs. We extract these two essential aspects of human decision making by using experimental methods: Indeed, people have reciprocal preferences, and they believe in more reciprocal behavior of others than is actually the case. We will call such people “reciprocal believers” in the following, and our results show that this belief bias only makes sense in combination with reciprocal preferences. Briefly, our findings can be summarized as follows. People 1. cooperate because they (to some degree: wrongly) believe that others cooperate as well, 2. trust because they (to some degree: wrongly) believe that others are trustworthy, and the fear of being betrayed does not diminish trusting behavior, 3. behave in a fair way because they (to some degree: wrongly) believe that they will be punished if they do not. Furthermore, fair – meaning equal – payoffs are achieved because people want to be kind to others. Reciprocating kind behavior in one situation does not necessarily mean that unkind behavior in another situation will be reciprocated as well. The incorporation of these results into thinking about human decision making leads to a different way of designing corporations, institutions, and markets. Is it possible to auction goods anonymously and with little legal control via the internet? How much supervision is necessary to make people pay their taxes or for their bus tickets? Behavioral economists will come to very different conclusions compared to neoclassical economists: Reciprocal believers are much more successful in solving social dilemmas than homines economici, which is good news. 2 However, accepting that people are reciprocal believers is, at the same time, bad news because it implies that our economy is much more vulnerable than it would be if unbiased and selfish subjects were making economic transactions: Correct beliefs and selfishness are precisely defined. In contrast, biases are prone to framing. Akerlof and Shiller (2010) argue that people think with the help of “stories” (such as “house prices always rise”), and such stories can be significantly biased. Perceptions are influenceable or even manipulable positively as well as negatively (see Posten et al., 2014, to name only one example). With respect to social preferences, evolutionary analyses find oscillating or chaotic phases of cooperation and defection in social dilemmas (see Section 4.2). Thinking of people as reciprocal believers explains why economic developments can make sudden disruptions unjustified by “hard facts”. Three of the five irrational “animal spirits” which drive human behavior according to Akerlof and Shiller (2010) (trust, fairness, biased beliefs) are addressed in this dissertation (we do not address corruption and money illusion), and their link between economic decision making of humans and macroeconomic phenomena, such as financial crises, illustrates how important it is to understand human decision making in social dilemmas much better than has been the case up to now. 2 Economic Experiments Economists (unfortunately)… cannot perform the controlled experiments of chemists or biologists because they cannot easily control other important factors. Like astronomers or meteorologists, they generally must be content largely to observe. Paul A. Samuelson and William D. Nordhaus, Principles of Economics 12th ed. McGraw-Hill, New York 1985, p. 8 Experimental economics is an ‘exciting new development’. Paul A. Samuelson and William D. Nordhaus, Principles of Economics 14th ed. McGraw-Hill, New York 1992, p. 5 3 Science can be theoretical, observational, or experimental. In economics, research has for a long time not been experimental, and it was believed that it is not possible to conduct economic experiments. While other (natural) sciences began to conduct experiments much earlier (starting with physics in the time of Galileo, followed by chemistry, biology, and no more than about one hundred years ago, psychology, compare Friedman and Sunder, 1994), the first economic experiments were not made until the second half of the last century. Since the mid-1980s, the number of experimental papers
Load more

Recommended publications
  • Uniqueness and Symmetry in Bargaining Theories of Justice
    Philos Stud DOI 10.1007/s11098-013-0121-y Uniqueness and symmetry in bargaining theories of justice John Thrasher Ó Springer Science+Business Media Dordrecht 2013 Abstract For contractarians, justice is the result of a rational bargain. The goal is to show that the rules of justice are consistent with rationality. The two most important bargaining theories of justice are David Gauthier’s and those that use the Nash’s bargaining solution. I argue that both of these approaches are fatally undermined by their reliance on a symmetry condition. Symmetry is a substantive constraint, not an implication of rationality. I argue that using symmetry to generate uniqueness undermines the goal of bargaining theories of justice. Keywords David Gauthier Á John Nash Á John Harsanyi Á Thomas Schelling Á Bargaining Á Symmetry Throughout the last century and into this one, many philosophers modeled justice as a bargaining problem between rational agents. Even those who did not explicitly use a bargaining problem as their model, most notably Rawls, incorporated many of the concepts and techniques from bargaining theories into their understanding of what a theory of justice should look like. This allowed them to use the powerful tools of game theory to justify their various theories of distributive justice. The debates between partisans of different theories of distributive justice has tended to be over the respective benefits of each particular bargaining solution and whether or not the solution to the bargaining problem matches our pre-theoretical intuitions about justice. There is, however, a more serious problem that has effectively been ignored since economists originally J.
    [Show full text]
  • Lecture Notes
    GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies ..................................
    [Show full text]
  • Gender Differences in Responsiveness to a Homo Economicus Prime in the Gift-Exchange Game
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Mertins, Vanessa; Warning, Susanne Working Paper Gender differences in responsiveness to a homo economicus prime in the gift-exchange game IAAEU Discussion Paper Series in Economics, No. 09/2013 Provided in Cooperation with: Institute for Labor Law and Relations in the European Union (IAAEU), University of Trier Suggested Citation: Mertins, Vanessa; Warning, Susanne (2013) : Gender differences in responsiveness to a homo economicus prime in the gift-exchange game, IAAEU Discussion Paper Series in Economics, No. 09/2013, University of Trier, Institute for Labour Law and Industrial Relations in the European Union (IAAEU), Trier This Version is available at: http://hdl.handle.net/10419/80863 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence.
    [Show full text]
  • Strong Reciprocity and Human Sociality∗
    Strong Reciprocity and Human Sociality∗ Herbert Gintis Department of Economics University of Massachusetts, Amherst Phone: 413-586-7756 Fax: 413-586-6014 Email: [email protected] Web: http://www-unix.oit.umass.edu/˜gintis Running Head: Strong Reciprocity and Human Sociality March 11, 2000 Abstract Human groups maintain a high level of sociality despite a low level of relatedness among group members. The behavioral basis of this sociality remains in doubt. This paper reviews the evidence for an empirically identifi- able form of prosocial behavior in humans, which we call ‘strong reciprocity,’ that may in part explain human sociality. A strong reciprocator is predisposed to cooperate with others and punish non-cooperators, even when this behavior cannot be justified in terms of extended kinship or reciprocal altruism. We present a simple model, stylized but plausible, of the evolutionary emergence of strong reciprocity. 1 Introduction Human groups maintain a high level of sociality despite a low level of relatedness among group members. Three types of explanation have been offered for this phe- nomenon: reciprocal altruism (Trivers 1971, Axelrod and Hamilton 1981), cultural group selection (Cavalli-Sforza and Feldman 1981, Boyd and Richerson 1985) and genetically-based altruism (Lumsden and Wilson 1981, Simon 1993, Wilson and Dugatkin 1997). These approaches are of course not incompatible. Reciprocal ∗ I would like to thank Lee Alan Dugatkin, Ernst Fehr, David Sloan Wilson, and the referees of this Journal for helpful comments, Samuel Bowles and Robert Boyd for many extended discussions of these issues, and the MacArthur Foundation for financial support. This paper is dedicated to the memory of W.
    [Show full text]
  • The Logic of Costly Punishment Reversed: Expropriation of Free-Riders and Outsiders∗,†
    The logic of costly punishment reversed: expropriation of free-riders and outsiders∗ ,† David Hugh-Jones Carlo Perroni University of East Anglia University of Warwick and CAGE January 5, 2017 Abstract Current literature views the punishment of free-riders as an under-supplied pub- lic good, carried out by individuals at a cost to themselves. It need not be so: often, free-riders’ property can be forcibly appropriated by a coordinated group. This power makes punishment profitable, but it can also be abused. It is easier to contain abuses, and focus group punishment on free-riders, in societies where coordinated expropriation is harder. Our theory explains why public goods are un- dersupplied in heterogenous communities: because groups target minorities instead of free-riders. In our laboratory experiment, outcomes were more efficient when coordination was more difficult, while outgroup members were targeted more than ingroup members, and reacted differently to punishment. KEY WORDS: Cooperation, costly punishment, group coercion, heterogeneity JEL CLASSIFICATION: H1, H4, N4, D02 ∗We are grateful to CAGE and NIBS grant ES/K002201/1 for financial support. We would like to thank Mark Harrison, Francesco Guala, Diego Gambetta, David Skarbek, participants at conferences and presentations including EPCS, IMEBESS, SAET and ESA, and two anonymous reviewers for their comments. †Comments and correspondence should be addressed to David Hugh-Jones, School of Economics, University of East Anglia, [email protected] 1 Introduction Deterring free-riding is a central element of social order. Most students of collective action believe that punishing free-riders is costly to the punisher, but benefits the community as a whole.
    [Show full text]
  • Kranton Duke University
    The Devil is in the Details – Implications of Samuel Bowles’ The Moral Economy for economics and policy research October 13 2017 Rachel Kranton Duke University The Moral Economy by Samuel Bowles should be required reading by all graduate students in economics. Indeed, all economists should buy a copy and read it. The book is a stunning, critical discussion of the interplay between economic incentives and preferences. It challenges basic premises of economic theory and questions policy recommendations based on these theories. The book proposes the path forward: designing policy that combines incentives and moral appeals. And, therefore, like such as book should, The Moral Economy leaves us with much work to do. The Moral Economy concerns individual choices and economic policy, particularly microeconomic policies with goals to enhance the collective good. The book takes aim at laws, policies, and business practices that are based on the classic Homo economicus model of individual choice. The book first argues in great detail that policies that follow from the Homo economicus paradigm can backfire. While most economists would now recognize that people are not purely selfish and self-interested, The Moral Economy goes one step further. Incentives can amplify the selfishness of individuals. People might act in more self-interested ways in a system based on incentives and rewards than they would in the absence of such inducements. The Moral Economy warns economists to be especially wary of incentives because social norms, like norms of trust and honesty, are critical to economic activity. The danger is not only of incentives backfiring in a single instance; monetary incentives can generally erode ethical and moral codes and social motivations people can have towards each other.
    [Show full text]
  • Recency, Records and Recaps: Learning and Non-Equilibrium Behavior in a Simple Decision Problem*
    Recency, Records and Recaps: Learning and Non-equilibrium Behavior in a Simple Decision Problem* DREW FUDENBERG, Harvard University ALEXANDER PEYSAKHOVICH, Facebook Nash equilibrium takes optimization as a primitive, but suboptimal behavior can persist in simple stochastic decision problems. This has motivated the development of other equilibrium concepts such as cursed equilibrium and behavioral equilibrium. We experimentally study a simple adverse selection (or “lemons”) problem and find that learning models that heavily discount past information (i.e. display recency bias) explain patterns of behavior better than Nash, cursed or behavioral equilibrium. Providing counterfactual information or a record of past outcomes does little to aid convergence to optimal strategies, but providing sample averages (“recaps”) gets individuals most of the way to optimality. Thus recency effects are not solely due to limited memory but stem from some other form of cognitive constraints. Our results show the importance of going beyond static optimization and incorporating features of human learning into economic models. Categories & Subject Descriptors: J.4 [Social and Behavioral Sciences] Economics Author Keywords & Phrases: Learning, behavioral economics, recency, equilibrium concepts 1. INTRODUCTION Understanding when repeat experience can lead individuals to optimal behavior is crucial for the success of game theory and behavioral economics. Equilibrium analysis assumes all individuals choose optimal strategies while much research in behavioral economics
    [Show full text]
  • Public Goods Agreements with Other-Regarding Preferences
    NBER WORKING PAPER SERIES PUBLIC GOODS AGREEMENTS WITH OTHER-REGARDING PREFERENCES Charles D. Kolstad Working Paper 17017 http://www.nber.org/papers/w17017 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 May 2011 Department of Economics and Bren School, University of California, Santa Barbara; Resources for the Future; and NBER. Comments from Werner Güth, Kaj Thomsson and Philipp Wichardt and discussions with Gary Charness and Michael Finus have been appreciated. Outstanding research assistance from Trevor O’Grady and Adam Wright is gratefully acknowledged. Funding from the University of California Center for Energy and Environmental Economics (UCE3) is also acknowledged and appreciated. The views expressed herein are those of the author and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer- reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2011 by Charles D. Kolstad. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Public Goods Agreements with Other-Regarding Preferences Charles D. Kolstad NBER Working Paper No. 17017 May 2011, Revised June 2012 JEL No. D03,H4,H41,Q5 ABSTRACT Why cooperation occurs when noncooperation appears to be individually rational has been an issue in economics for at least a half century. In the 1960’s and 1970’s the context was cooperation in the prisoner’s dilemma game; in the 1980’s concern shifted to voluntary provision of public goods; in the 1990’s, the literature on coalition formation for public goods provision emerged, in the context of coalitions to provide transboundary pollution abatement.
    [Show full text]
  • The Impact of Beliefs on Effort in Telecommuting Teams
    The impact of beliefs on effort in telecommuting teams E. Glenn Dutcher, Krista Jabs Saral Working Papers in Economics and Statistics 2012-22 University of Innsbruck http://eeecon.uibk.ac.at/ University of Innsbruck Working Papers in Economics and Statistics The series is jointly edited and published by - Department of Economics - Department of Public Finance - Department of Statistics Contact Address: University of Innsbruck Department of Public Finance Universitaetsstrasse 15 A-6020 Innsbruck Austria Tel: + 43 512 507 7171 Fax: + 43 512 507 2970 E-mail: [email protected] The most recent version of all working papers can be downloaded at http://eeecon.uibk.ac.at/wopec/ For a list of recent papers see the backpages of this paper. The Impact of Beliefs on E¤ort in Telecommuting Teams E. Glenn Dutchery Krista Jabs Saralz February 2014 Abstract The use of telecommuting policies remains controversial for many employers because of the perceived opportunity for shirking outside of the traditional workplace; a problem that is potentially exacerbated if employees work in teams. Using a controlled experiment, where individuals work in teams with varying numbers of telecommuters, we test how telecommut- ing a¤ects the e¤ort choice of workers. We …nd that di¤erences in productivity within the team do not result from shirking by telecommuters; rather, changes in e¤ort result from an individual’s belief about the productivity of their teammates. In line with stereotypes, a high proportion of both telecommuting and non-telecommuting participants believed their telecommuting partners were less productive. Consequently, lower expectations of partner productivity resulted in lower e¤ort when individuals were partnered with telecommuters.
    [Show full text]
  • The Social Side of Homo Economicus
    TREE-1308; No. of Pages 3 Update Forum The social side of Homo economicus Daniel J. Rankin1,2 1 Department of Biochemistry, University of Zu¨ rich, Building Y27, Winterthurerstrasse 190, CH-8057 Zu¨ rich, Switzerland 2 Swiss Institute of Bioinformatics, Quartier Sorge Baˆ timent Ge´ nopode, CH-1015 Lausanne, Switzerland Many recent experiments in the field of behavioural cooperatively than traditional economic theory would pre- economics appear to demonstrate a willingness of dict [1,11]. humans to behave altruistically, even when it is not in their interest to do so. This has led to the assertion that Are humans especially cooperative? humans have evolved a special predisposition towards Cooperation in public goods games is unstable and most altruism. Recent studies have questioned this, and dem- players quickly realise that not contributing maximises onstrated that selfless cooperation does not hold up in their profit (see Figure 1 for an explanation of public controlled experiments. As I discuss here, this calls for goods games). Despite this, the public goods game fre- more economic ‘field experiments’ and highlights the quently results in higher levels of contribution than one need for greater integration of the evolutionary and would expect if individuals were selfishly rational and economic sciences. 10% of participants contribute in the long run [1,12]. This has led to the conclusion that humans have a special Kto nekradne, okra´da svoju rodinu degree of cooperation, or ‘other regarding preferences’, (He who does not steal, steals from his family) that cannot be explained from a purely selfish perspec- Slovak Saying tive and that has been used as evidence that humans behave more cooperatively than theory would predict The apparent puzzle of human cooperation [1,12].
    [Show full text]
  • The Role of Self-Interest in Elite Bargaining
    The role of self-interest in elite bargaining Brad L. LeVecka,b, D. Alex Hughesa,c, James H. Fowlera,c,d, Emilie Hafner-Burtona,c, and David G. Victora,1 aLaboratory on International Law and Regulation, School of International Relations and Pacific Studies, cDepartment of Political Science, and dDivision of Medical Genetics, University of California, San Diego, La Jolla, CA 92093; and bDepartment of Political Science, University of California, Merced, CA 95343 Edited by David G. Rand, Yale University, New Haven, CT, and accepted by the Editorial Board November 11, 2014 (received for review May 28, 2014) One of the best-known and most replicated laboratory results in evident in less experienced populations (16, 17). Some research behavioral economics is that bargainers frequently reject low has examined whether leaders are psychologically or physiolog- offers, even when it harms their material self-interest. This finding ically different from nonelite populations—for example, research could have important implications for international negotiations on anxiety levels in leaders (18) and work on how leaders may be on many problems facing humanity today, because models of selected for skills in coordination, leadership, and followership international bargaining assume exactly the opposite: that policy (19). The few existing studies that have empirically examined makers are rational and self-interested. However, it is unknown actual decision making by elites in highly particular roles—such whether elites who engage in diplomatic bargaining will similarly as professional tennis players, soccer players, and traders—sug- reject low offers because past research has been based almost gest that, through a combination of learning and attrition, elites exclusively on convenience samples of undergraduates, members are more likely to exhibit self-interested, profit-maximizing be- of the general public, or small-scale societies rather than highly havior than their less experienced counterparts (20–23).
    [Show full text]
  • Nash Equilibrium
    Lecture 3: Nash equilibrium Nash equilibrium: The mathematician John Nash introduced the concept of an equi- librium for a game, and equilibrium is often called a Nash equilibrium. They provide a way to identify reasonable outcomes when an easy argument based on domination (like in the prisoner's dilemma, see lecture 2) is not available. We formulate the concept of an equilibrium for a two player game with respective 0 payoff matrices PR and PC . We write PR(s; s ) for the payoff for player R when R plays 0 s and C plays s, this is simply the (s; s ) entry the matrix PR. Definition 1. A pair of strategies (^sR; s^C ) is an Nash equilbrium for a two player game if no player can improve his payoff by changing his strategy from his equilibrium strategy to another strategy provided his opponent keeps his equilibrium strategy. In terms of the payoffs matrices this means that PR(sR; s^C ) ≤ P (^sR; s^C ) for all sR ; and PC (^sR; sC ) ≤ P (^sR; s^C ) for all sc : The idea at work in the definition of Nash equilibrium deserves a name: Definition 2. A strategy s^R is a best-response to a strategy sc if PR(sR; sC ) ≤ P (^sR; sC ) for all sR ; i.e. s^R is such that max PR(sR; sC ) = P (^sR; sC ) sR We can now reformulate the idea of a Nash equilibrium as The pair (^sR; s^C ) is a Nash equilibrium if and only ifs ^R is a best-response tos ^C and s^C is a best-response tos ^R.
    [Show full text]