Fictitious play in games: The transition between periodic and chaotic behaviour

In the 1960s Shapley provided an example of a two-player fictitious game with periodic behaviour. In this game, player A aims to copy B's behaviour and player B aims to play one ahead of player A. In this paper we generalise Shapley's example by introducing an external parameter. We show that the periodic behaviour in Shapley's example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again, but now both players aim to play one ahead of the other. In this paper we adopt a geometric (dynamical systems) approach. Here we prove rigorous results on continuity of the dynamics and on the periodic behaviour, while in the sequel to this paper we shall describe the chaotic behaviour.     

p-Dominance and Equilibrium Selection under Perfect Foresight Dynamics

This paper studies equilibrium selection based on a class of perfect foresight dynamics and relates it to the notion of p-dominance. A continuum of rational players is repeatedly and randomly matched to play a symmetric n×n game. There are frictions: opportunities to revise actions follow independent Poisson processes. The dynamics has stationary states, each of which corresponds to a Nash equilibrium of the static game. A strict Nash equilibrium is linearly stable under the perfect foresight dynamics if, independent of the current action distribution, there exists a consistent belief that any player necessarily plays the Nash equilibrium action at every revision opportunity. It is shown that a strict Nash equilibrium is linearly stable under the perfect foresight dynamics with a small degree of friction if and only if it is the p-dominant equilibrium with p<1/2. It is also shown that if a strict Nash equilibrium is the p-dominant equilibrium with p<1/2, then it is uniquely absorbing (and globally accessible) for a small friction (but not vice versa). Set-valued stability concepts are introduced and their existence is shown. Journal of Economic Literature Classification Numbers: C72, C73.     

Epistemic conditions for rationalizability

In this paper I present conditions, not involving common knowledge of rationality, that lead to (correlated) rationalizability. The basic observation is that, if the actual world belongs to a set of states where the set Z of action profiles is played, everyone is rational and it is mutual knowledge that the action profiles played are in Z, then the actions played at the actual world are rationalizable actions. Alternatively, if at the actual world the support of the conjecture of player i is Dⁱ, there is mutual knowledge of: (i) the game being played, (ii) that the players are rational, and (iii) that for every i the support of the conjecture of player i is contained in Dⁱ, then every strategy in the support of the conjectures is rationalizable. The results do not require common knowledge of anything and are valid for games with any number of players.     

Communication in Repeated Games with Private Monitoring

The paper considers repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed. We address the following question: What is the minimal level of observability that is required to obtain efficient outcomes? The main result is that the limit set of sequential equilibrium payoffs, when the discount factor tends to one, contains the set of individual rational payoffs whenever each player is observed by at leasttwoother players.Journal of Economic LiteratureClassification Number: C73.     

How to talk to multiple audiences

Stochastic fictitious play (SFP) assumes that agents do not try to influence the future play of their current opponents, an assumption that is justified by appeal to a setting with a large population of players who are randomly matched to play the game. However, the dynamics of SFP have only been analyzed in models where all agents in a player role have the same beliefs. We analyze the dynamics of SFP in settings where there is a population of agents who observe only outcomes in their own matches and thus have heterogeneous beliefs. We provide conditions that ensure that the system converges to a state with homogeneous beliefs, and that its asymptotic behavior is the same as with a single representative agent in each player role.     

A note on control costs and logit rules for strategic games

We consider best response dynamics with endogenous noise based on a finite game in strategic form. A player can reduce the noise level by expending an extra effort and incurring some disutility or control costs. We specify control costs that result in logit adjustment rules. The stochastically stable states of the dynamic process are partial Nash configurations, that is, states where at least one player plays a best response against the others. If the game has a potential, then the stochastically stable states coincide with the Nash equilibria on which the potential is maximized. RID="*" ID="*" Instructive comments of a referee are gratefully acknowledged. Correspondence to:H. Haller     

Information at equilibrium

Summary. In a game with rational expectations, individuals simultaneously refine their information with the information revealed by the strategies of other individuals. At a Nash equilibrium of a game with rational expectations, the information of individuals is essentially symmetric: the same profile of strategies is also an equilibrium of a game with symmetric information; and strategies are common knowledge. If each player has a veto act, which yields a minimum payoff that no other profile of strategies attains, then the veto profile is the only Nash equilibrium, and it is is an equilibrium with rational expectations and essentially symmetric information; which accounts for the impossibility of speculation. Received: June 20, 2001; revised version: January 9, 2002 RID="*" ID="*" We wish to thank Pierpaolo Battigalli, Fran?oise Forges, Franco Donzelli, Leonidas Koutsougeras, Aldo Rustichini, Rajiv Vohra and Nicholas Yannelis for their comments. Correspondence to: H. Polemarchakis     

What to maximize if you must

The assumption that decision makers choose actions to maximize their preferences is a central tenet in economics, often justified formally or informally by appealing to evolutionary arguments. In contrast, we show that in almost every game and for almost every family of distortions of a player's actual payoffs, some degree of this distortion is beneficial to the player, and will not be driven out by any evolutionary process involving payoff-monotonic selection dynamics. Consequently, under any such selection dynamics the population will not converge to payoff-maximizing behavior. We also show that payoff-maximizing behavior need not prevail when preferences are imperfectly observed.     

Evolution of harvesting strategies: replicator and resource dynamics

Abstract. Economic theories of managing renewable resources, such as fisheries and forestry, traditionally assume that individual harvesters are perfectly rational and thus able to compute the harvesting strategy that maximizes their discounted profits. The current paper presents an alternative approach based on bounded rationality and evolutionary mechanisms. It is assumed that individual harvesters face a choice between two harvesting strategies. The evolution of the distribution of strategies in the population is modeled through a replicator dynamics equation. The latter captures the idea that strategies yielding above average profits are demanded more than strategies yielding below average profits, so that the first type ends up accounting for a larger part in the population. From a mathematical perspective, the combination of resource and evolutionary processes leads to complex dynamics. The paper presents the existence and stability conditions for each steady-state of the system and analyzes dynamic paths to the equilibrium. In addition, effects of changes in prices are analyzed. A main result of the paper is that under certain conditions both strategies can survive in the long-run. Correspondence to: J. Noailly     

Conventions,local interaction,and automata networks

This paper deals with the evolution of conventions in a finite population of boundedly rational players. We apply methods from the theory of automata networks to analyze the effects of different interaction structures. It is shown that the size of the reference group for each player has an important impact on the evolutionary stability of a particular convention, and that the interaction structure crucially determines the distribution of conventions in the population.     

Reputation with equal discounting in repeated games with strictly conflicting interests

We analyze reputation effects in two-player repeated games of strictly conflicting interests. In such games, player 1 has a commitment action such that a best reply to it gives player 1 the highest individually rational payoff and player 2 the minmax payoff. Players have equal discount factors. With positive probability player 1 is a type who chooses the commitment action after every history. We show that player 1's payoff converges to the maximally feasible payoff when the discount factor converges to one. This contrasts with failures of reputation effects for equal discount factors that have been demonstrated in the literature.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

One-Shot Public Mediated Talk

We show that any correlation device with rational coefficients can be generated by a mechanism, where each player sends a private message to a mediator who in turn makes a public deterministic announcement. It is then shown that the mechanism can be adapted also to situations with differential information, where the correlation device itself depends on the players' private messages that may vary with their realized types. All the mechanisms suggested are immunized against individual deviations. Therefore, by using them, players can implement any correlated or communication equilibrium.Journal of Economic LiteratureClassification Number: C72.     

Reputation,Social Identity and Social Conflict

We interpret the psychology literature on social identity and examine its implications. We model a population of agents from two exogenous and well defined social groups. Agents are randomly matched to play a reduced‐form bargaining game. We show that this struggle for resources drives a conflict through the rational destruction of surplus. We assume that the population contains both rational players and behavioral players. Behavioral players aggressively discriminate against members of the other social group. The existence and specification of the behavioral player is motivated by the social identity literature. For rational players, group membership has no payoff‐relevant consequences. We show that rational players can contribute to the conflict by aggressively discriminating and that this behavior is consistent with existing empirical evidence. Our paper relates to the empirical literature which finds that social heterogeneity tends to be increasing in economic variables which we interpret as indicating inefficiency. We provide an explanation that, as social groups compete for surplus, disagreement and inefficiency can result. Our work also relates to the social conflict literature, which examines the relationship between macro level factors such as unemployment and civil disturbances. This literature finds that the amount of social conflict tends to be increasing in the inequitability of the environment.     

Gradualism and Irreversibility

This paper considers a class of two–player dynamic games in which each player controls a one–dimensional action variable, interpreted as a level of cooperation. The dynamics are due to an irreversibility constraint: neither player can ever reduce his cooperation level. Payoffs are decreasing in one&'s own action, increasing in one&'s opponent&'s action. We characterize efficient symmetric equilibrium action paths; actions rise gradually over time and converge, when payoffs are smooth, to a level strictly below the one–shot efficient level, no matter how little discounting takes place. The analysis is extended to incorporate sequential moves and asymmetric equilibria.     

Coevolution of economic and ecological systems

This paper analyzes a model of economy–environment coevolution in which economic activities induce the genetic evolution of a biological species. This model is applied to the problem of pesticide resistance management. Just as in Munro (Environ Resour Econ, 9:429–449, 1997), we consider three main types of interactions: (1) a large pest population reduces economic revenues, (2) economic activities select for resistant genes and (3) the spread of resistant genes affects the size of the pest population. The model differs from Munro in that it includes evolutionary modeling of economic strategies. Economic agents are assumed to be boundedly rational, i.e they cannot compute the optimal level of pesticide use that minimizes resistance among pests. Economic evolution represents the change in the distribution of pesticide strategies in the population of economic agents and is modeled by a replicator dynamics equation. The interactions between economic evolution of pesticide strategies, pest population dynamics and genetic evolution of resistance of pests are studied in a system of three differential equations. We explore the dynamics and stability properties of the system using numerical simulations.     

Farsightedness in a Coalitional Great Fish War

We explore the implications of the farsightedness assumption on the conjectures of players in a coalitional Great Fish War model with symmetric players, derived from the seminal model of Levhari and Mirman (Bell J Econ 11:649–661, 1980). The farsightedness assumption for players in a coalitional game acknowledges the fact that a deviation from a single player will lead to the formation of a new coalition structure as the result of possibly successive moves of her rivals in order to improve their payoffs. It departs from mainstream game theory in that it relies on the so-called rational conjectures, as opposed to the traditional Nash conjectures formed by players on the behavior of their rivals. For values of the biological parameter and the discount factor more plausible than the ones used in the current literature, the farsightedness assumption predicts a wide scope for cooperation in non-trivial coalitions, sustained by credible threats of successive deviations that defeat the shortsighted payoff of any prospective deviator. Compliance or deterrence of deviations may also be addressed by acknowledging that information on the fish stock or on the catch policies actually implemented may be available only with a delay (dynamic farsightedness). In that case, the requirements are stronger and the sizes and number of possible farsighted stable coalitions are different. In the sequential move version, which could mimic some characteristics of fishery models, the results are not less appealing, even if the dominant player or dominant coalition with first move advantage assumption provides a case for cooperation with the traditional Nash conjectures.     

Global dynamics in repeated games with additively separable payoffs

This paper studies the global dynamics of a class of infinitely repeated two-player games in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that each player's action is a stationary function of the opponent's last action. We completely characterize IREs and their dynamics in terms of certain indifference curves. Our results are used to show that in a prisoners' dilemma game with mixed strategies, gradual cooperation occurs when the players are sufficiently patient, and that in a certain duopoly game, kinked demand curves emerge naturally.     

Analogies and theories: The role of simplicity and the emergence of norms

We consider the dynamics of reasoning by general rules (theories) and by specific cases (analogies). When an agent faces an exogenous process, we show that, under mild conditions, if reality happens to be simple, the agent will converge to adopt a theory and discard analogical thinking. If, however, reality is complex, analogical reasoning is unlikely to disappear. By contrast, when the agent is a player in a large population coordination game, and the process is generated by all playersʼ predictions, convergence to a theory is much more likely. This may explain how a large population of players selects an equilibrium in such a game, and how social norms emerge. Mixed cases, involving noisy endogenous processes are likely to give rise to complex dynamics of reasoning, switching between theories and analogies.     

Congestion games with malicious players

We study the equilibria of non-atomic congestion games in which there are two types of players: rational players, who seek to minimize their own delay, and malicious players, who seek to maximize the average delay experienced by the rational players. We study the existence of pure and mixed Nash equilibria for these games, and we seek to quantify the impact of the malicious players on the equilibrium. One counterintuitive phenomenon which we demonstrate is the “windfall of malice”: paradoxically, when a myopically malicious player gains control of a fraction of the flow, the new equilibrium may be more favorable for the remaining rational players than the previous equilibrium.