Microeconomics class assignment on Game Theory. For part 1 of the assignment I have attached the reading paper.
game theory class exam 2021.dvi 1 2 1 2 1 2 1 2 (1 ) 1 ( ) 1 2 1 0 = 2 = 1000 998 = 1000 Learning, Mutation, and Long Run Equilibria in Games Learning, Mutation, and Long Run Equilibria in Games Author(s): Michihiro Kandori, George J. Mailath and Rafael Rob Source: Econometrica, Vol. 61, No. 1 (Jan., 1993), pp. 29-56 Published by: The Econometric Society Stable URL: https://www.jstor.org/stable/2951777 Accessed: 16-10-2021 09:01 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
[email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica This content downloaded from 192.41.114.224 on Sat, 16 Oct 2021 09:01:34 UTC All use subject to https://about.jstor.org/terms Econometrica, Vol. 61, No. 1 (January, 1993), 29-56 LEARNING, MUTATION, AND LONG RUN EQUILIBRIA IN GAMES BY MICHIHIRO KANDORI, GEORGE J. MAILATH, AND RAFAEL ROB We analyze an evolutionary model with a finite number of players and with noise or mutations. The expansion and contraction of strategies is linked-as usual-to their current relative success, but mutations-which perturb the system away from its deter- ministic evolution-are present as well. Mutations can occur in every period, so the focus is on the implications of ongoing mutations, not a one-shot mutation. The effect of these mutations is to drastically reduce the set of equilibria to what we term "long-run equilibria." For 2 x 2 symmetric games with two symmetric strict Nash equilibria the equilibrium selected satisfies (for large populations) Harsanyi and Selten's (1988) criterion of risk-dominance. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilib- rium. KEYWORDS: Evolutionary game theory, evolution, bounded rationality, learning, Markov chains, strict equilibria, risk dominance, equilibrium selection. 1. INTRODUCTION WHILE THE NASH EQUILIBRIUM CONCEPT has been used extensively in many diverse contexts, game theory has been unsuccessful in explaining how players know that a Nash equilibrium will be played. Moreover, the traditional theory is silent on how players know which Nash equilibrium is to be played if a game has multiple equally plausible Nash equilibria. Introspective (eductive) theories that attempt to explain equilibrium play "directly" at the individual decision- making level impose very strong informational assumptions and so are widely recognized as having serious deficiencies (see, for example, Binmore (1987, 1988)). As a consequence, attention has shifted to "evolutionary" explanations of equilibrium, motivated by the work of biologists in evolutionary game theory (in particular, the seminal work of Maynard Smith and Price (1973), see also Maynard Smith (1982)).2 Two features of this approach distinguish it from the introspective approach. First, players are not assumed to be so "rational" or "knowledgeable" as to correctly guess (anticipate) the other players' choices. Second (and instead), an explicit dynamic process is specified describing how players adjust their choices over time as they learn (from experience) about the 'This paper has benefited from the comments of In-Koo Cho, Dean Foster, Drew Fudenberg, Jacob Glazer, Christopher Harris, Michael Woodford, Peyton Young, and many seminar audiences, as well as from an editor and three anonymous referees. Financial support from the NSF is gratefully acknowledged by the first and second authors (SES-9108351 and SES-8908451). 2Maynard Smith and Price (1973) introduced the notion of an evolutionary stable strategy (ESS), a notion of stability against mutations. Subsequent work (such as Taylor and Jonker (1978)) has studied when an ESS is an attractor of the replicator dynamic (in which the rate of growth in the fraction of the population playing a particular strategy is equal to the deviation of that strategy's fitness or payoff from the average fitness or payoff). The biological literature is surveyed in Hines (1987). 29 This content downloaded from 192.41.114.224 on Sat, 16 Oct 2021 09:01:34 UTC All use subject to https://about.jstor.org/terms 30 M. KANDORI, G. J. MAILATH, AND R. ROB other players' choices. Therefore, this approach tries to explain how an equilib- rium emerges, based on trial-and-error learning (instead of introspective-type arguments). The evolutionary approach is surveyed by van Damme (1987) and Mailath (1992). Many papers have explored this approach, including Friedman (1991), Fudenberg and Maskin (1990), Nachbar (1990), and Binmore and Samuelson (1992). The work of Nelson and Winter (which culminated in their book (1982)) should also be mentioned for its emphasis on the importance of evolutionary ideas in explaining economic change. The purpose of the present paper is to extend these ideas and-more importantly-to add a new perspective on the problem of equilibrium selection. Specifically, we follow the pioneering work of Foster and Young (1990), who were the first to argue that in games with multiple strict Nash equilibria,3 some equilibria are more likely to emerge than others in the presence of continual small stochastic shocks. In the present paper, we introduce a discrete frame- work to address this issue,4 and develop a general technique to determine the most likely, or long run, equilibrium (this concept is essentially the stochastically stable equilibrium of Foster and Young (1990)). We then apply this technique to the class of symmetric 2 x 2 games, and show that, for coordination games, the long run equilibrium coincides with the risk dominant equilibrium (see Harsanyi and Selten (1988)).5 We show this to be the case independent of all but the most crude features of the dynamics. Thus, our framework provides a link between the evolutionary approach and the risk-dominance criterion.6 Our specification of dynamics draws heavily on the biological literature. In that literature, animals are viewed as being genetically coded with a strategy and selection pressure favors animals which are fitter (i.e., whose strategy yields a higher reproductive fitness-or payoff-against the population). The focus there has been on the concept of evolutionary stable strategy (ESS) due to Maynard Smith and Price (1973) and its relationship to various dynamics. Selten (1991) provides a nontechnical discussion of the degree to which this literature is suited to the study of social phenomena. While our model can be interpreted in like manner, we intend it more as a contribution to the growing literature on bounded rationality and learning. Accordingly, the hypotheses we employ here reflect limited ability (on the players' part) to receive, decode, and act upon information they get in the course of playing games. In particular, we consider the situation where a group of players is repeatedly matched to play a game. The following three hypotheses form the basis of our analysis. (i) Not all agents need react instantaneously to their environment (the 3And so multiple evolutionary stable strategies. 4Moreover, the stochastic shocks are based on the micro-structure of the game. 5See page 46 for a definition. If the game is a coordination game with zeroes off-the-diagonal (or -more generally, if the two strategies have identical security levels, i.e., payoffs from miscoordination), then the risk dominant equilibrium coincides with the Pareto dominant equilibrium. 6This is only shown for the symmetric 2 x 2 case. It is not clear that a similar link exists for the more general case. This content downloaded from 192.41.114.224 on Sat, 16 Oct 2021 09:01:34 UTC All use subject to https://about.jstor.org/terms LEARNING, MUTATIONS, AND L.R. EQUILIBRIA 31 inertia hypothesis); (ii) when agents react, they react myopically (the myopia hypothesis); and (iii) there is a small probability that agents change their strategies at random (the mutation, or experimentation hypothesis). We regard these as descriptions of boundedly rational behavior; we argue below that they can also be justified as rational behavior under some circumstances. We first lay out the general motivations of the three hypotheses, and then discuss some specific interpretations. The logic underlying these hypotheses is as follows. A strategy in a game is, in general, a complicated object, specifying what actions one should take given various contingencies one is able to observe. However, players' observations are imperfect, their knowledge of how payoffs depend on strategy choices may be tenuous,7 and changing one's strategy may be costly. The presence of inertia is then suggested by the existence of such uncertainties and adjustment costs.8 In turn, to the extent that there is substantial inertia present, only a small fraction of agents are changing their strategies simultaneously. In this case, those who do move are justified in acting myopically: they know that only a small segment of the population changes its behavior at any given point in time and, hence, strategies that proved to be effective today are likely to remain effective for some time in the future. Thus, taking myopic best responses is justified as being fully rational when players have a high discount rate compared to the speed of adjustment. The myopia hypothesis also captures a second aspect of learning which we feel is important; namely, imitation or emulation. Here the idea is that the world is a complicated place and agents cannot calculate best responses to their stage environment. People learn what are good strategies by observing what has worked well for other people. The myopia assumption amounts to saying that at the same time that players are learning, they are not taking into account the long run implications of their strategy choices. Thus, agents act as if each stage game is the last.9 Mutation plays a central role in our analysis. With some small probability, each agent plays an arbitrary strategy.10 One economic interpretation is that a player exits with some probability and is replaced with a new player who knows nothing about the game and so chooses a strategy at random (as in Canning (1989)). The model admits a variety of interpretations, with differing types of bounded