Mixed equilibria are unstable in games of strategic complements

In games with strict strategic complementarities, properly mixed Nash equilibria—equilibria that are not in pure strategies—are unstable for a broad class of learning dynamics.     

Quasi-option value under strategic interactions

We consider a simple two-period model of irreversible investment under strategic interactions between two players. In this setup, we show that the quasi-option value may cause some conceptual difficulties. In case of asymmetric information, decentralized investment decisions fail to induce first-best allocations. Therefore a regulator may not be able to exercise the option to delay the decision to develop. We also show that information-induced inefficiency may arise in a game situation and that under certain assumptions inefficiency can be eliminated by sending asymmetric information to the players, even when the regulator faces informational constraints. Our model is potentially applicable to various global environmental problems.     

The folk theorem for irreducible stochastic games with imperfect public monitoring

This paper introduces stochastic games with imperfect public signals. It provides a sufficient condition for the folk theorem when the game is irreducible, thus generalizing the results of Dutta (1995) [5] and Fudenberg, Levine, and Maskin (1994) [9]. To do this, the paper extends the concept of self-generation (Abreu, Pearce, and Stacchetti, 1990 [1]) to “return generation,” which explicitly tracks actions and incentives until the next time the state returns to its current value, and asks that players not wish to deviate given the way their continuation payoffs from the time of this return depend on the public signals that have been observed.     

Evolution in Bayesian games II: Stability of purified equilibria

We study the evolutionary stability of purified equilibria of two-player normal form games, providing simple sufficient conditions for stability and for instability under the Bayesian best response dynamic.     

A multinomial probit model of stochastic evolution

A strategy revision process in symmetric normal form games is proposed. Following Kandori et al. (Econometrica 61 (1993) 29), members of a population periodically revise their strategy choice, and choose a myopic best response to currently observed play. Their payoffs are perturbed by normally distributed Harsanyian trembles, so that strategies are chosen according to multinomial probit probabilities. As the variance of payoffs is allowed to vanish, the graph theoretic methods of the earlier literature continue to apply. The distributional assumption enables a convenient closed form characterisation for the weights of the rooted trees. An illustration of the approach is offered, via a consideration of the role of dominated strategies in equilibrium selection.     

From evolutionary to strategic stability

A connected component of Nash equilibria is (dynamically) potentially stable if there exits an evolutionary selection dynamics from a broad class for which the component is asymptotically stable. A necessary condition for potential stability is that the component's index agrees with its Euler characteristic. Second, if the latter is nonzero, the component contains a strategically stable set. If the Euler characteristic would be zero, the dynamics (that justifies potential stability) could be slightly perturbed so as to remove all zeros close to the component. Hence, any robustly potentially stable component contains equilibria that satisfy the strongest rationalistic refinement criteria.     

Public randomization in the repeated prisoner’s dilemma game with local interaction

We consider a situation in which each agent observes only their neighbor’s actions and locally interacts with them by playing prisoner’s dilemma games. Introducing a public randomization, we construct an equilibrium which sustains cooperation and in which cooperation eventually resumes after any history.     

Uniform selection in global games

This paper brings together results which are required in order to extend the global games approach to settings where the game structure is endogenous. More precisely, it shows that the selection argument of Carlsson and van Damme [Global games and equilibrium selection, Econometrica 61(5) (1993) 989-1018] holds uniformly over appropriately controlled families of games. Those results also give proper justification for the inversion of limits which is often implicit in applied work taking comparative statics on the selected risk-dominant equilibrium.     

Dynamic psychological games

The motivation of decision makers who care for various emotions, intentions-based reciprocity, or the opinions of others may depend directly on beliefs (about choices, beliefs, or information). Geanakoplos, Pearce and Stacchetti [J. Geanakoplos, D. Pearce, E. Stacchetti, Psychological games and sequential rationality, Games Econ. Behav. 1 (1989) 60-79] point out that traditional game theory is ill-equipped to address such matters, and they pioneer a new framework which does. However, their toolbox - psychological game theory - incorporates several restrictions that rule out plausible forms of belief-dependent motivation. Building on recent work on dynamic interactive epistemology, we propose a more general framework. Updated higher-order beliefs, beliefs of others, and plans of action may influence motivation, and we can capture dynamic psychological effects (such as sequential reciprocity, psychological forward induction, and regret) that were previously ruled out. We develop solution concepts, provide examples, explore properties, and suggest avenues for future research.     

Minimax regret and strategic uncertainty

This paper introduces a new solution concept, a minimax regret equilibrium, which allows for the possibility that players are uncertain about the rationality and conjectures of their opponents. We provide several applications of our concept. In particular, we consider price-setting environments and show that optimal pricing policy follows a non-degenerate distribution. The induced price dispersion is consistent with experimental and empirical observations (Baye and Morgan (2004) [4]).     

Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics

This note shows, by means of two simple, three-strategy games, the existence of stable periodic orbits and of multiple, interior steady states in a smooth version of the Best-Response Dynamics, the Logit Dynamics. The main finding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, occur under the Logit Dynamics, even for three-strategy games. We also show that the Logit Dynamics displays another bifurcation which cannot occur under the Replicator Dynamics: the fold bifurcation, with non-monotonic creation and disappearance of steady states.     

Finitely repeated games with monitoring options

We study finitely repeated games where players can decide whether to monitor the other players? actions or not every period. Monitoring is assumed to be costless and private. We compare our model with the standard one where the players automatically monitor each other. Since monitoring other players never hurts, any equilibrium payoff vector of a standard finitely repeated game is an equilibrium payoff vector of the same game with monitoring options. We show that some finitely repeated games with monitoring options have sequential equilibrium outcomes which cannot be sustained under the standard model, even if the stage game has a unique Nash equilibrium. We also present sufficient conditions for a folk theorem, when the players have a long horizon.     

Large population potential games

We offer a parsimonious definition of large population potential games, provide some alternate characterizations, and demonstrate the advantages of the new definition over the existing definition, but also show the equivalence of the two definitions.     

Large deviations and equilibrium selection in large populations

This paper uses the theory of large deviations to analyse equilibrium selection in one-dimensional games with large populations where the system evolves according to a jump Markov process. The equilibria selected maximise a quasi-potential function which can be determined by solving a polynomial equation. Estimates of waiting times are also given. It shows that the equilibria about which there is more noise are less likely to be selected and clarifies the role of the limiting deterministic dynamic in selection.     

EVOLUTIONARY DRIFT IN PREFERENCE AND EQUILIBRIUM SELECTION*

This paper studies the situation where myopic players repeatedly face a society with two actions where their common fitness exhibits economies of scale. Both states in which all players choose the same actions are equilibria. In each period, players adjust their actions based on their preferences, which are in turn shaped by natural selection. The preferences of the players need not match the underlying fitness. When rare mutations are introduced into the evolutionary process, their preferences may drift without affecting equilibrium behaviour. This paper shows that these drifts influence the results of equilibrium selection.     

An evolutionary approach to learning in a changing environment

The evolutionary fitness of the ability to learn a best response (at a cost) is explored in an environment where the underlying game changes stochastically over time. We provide a sufficient condition for asymptotic dominance by the learners. It does not require that the learners are the most fit type in any single state of the environment. We identify a necessary and sufficient condition that ensures the survival of the learners when learning costs are small. Last, we describe a medium-run phenomenon where the learners “spike”—their share temporarily rises and gets arbitrarily close to one following a regime shift.     

Stable games and their dynamics

We study a class of population games called stable games. These games are characterized by self-defeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of strategies which revising agents are abandoning. We prove that the set of Nash equilibria of a stable game is globally asymptotically stable under a wide range of evolutionary dynamics. Convergence results for stable games are not as general as those for potential games: in addition to monotonicity of the dynamics, integrability of the agents' revision protocols plays a key role.     

Risk-dominance and perfect foresight dynamics in N-player games

In perfect foresight dynamics, an action is linearly stable if expectation that people will always choose the action is self-fulfilling. A symmetric game is a PIM game if an opponent's particular action maximizes the incentive of an action, independently of the rest of the players. This class includes supermodular games, games with linear incentives and so forth. We show that, in PIM games, linear stability is equivalent to u-dominance, a generalization of risk-dominance, and that there is no path escaping a u-dominant equilibrium. Existing results on N-player coordination games, games with linear incentives and two-player games are obtained as corollaries.     

Evolution in games with randomly disturbed payoffs

We consider a simple model of stochastic evolution in population games. In our model, each agent occasionally receives opportunities to update his choice of strategy. When such an opportunity arises, the agent selects a strategy that is currently optimal, but only after his payoffs have been randomly perturbed. We prove that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games. We conclude by contrasting the evolutionary process studied here with stochastic fictitious play.     

A comment on “Egalitarianism and efficiency in repeated symmetric games” by V. Bhaskar [Games Econ. Behav. 32 (2000) 247-262]

We identify an error in Bhaskar?s (2000) Proposition 4. We provide counterexamples to this result and demonstrate that it is not correctable.