The dynamic instability of dispersed price equilibria

We adopt an evolutionary framework to explain price dispersion as a time varying phenomenon. By developing a finite strategy analogue of the Burdett and Judd (1983) price dispersion model, we show that all dispersed price equilibria are unstable under the class of perturbed best response dynamics. Instead, numerical simulations using the logit dynamic show that price dispersion manifests itself as a limit cycle. We verify that limit cycles persist even when the finite strategy model approaches the original continuous strategy model. For a particularly simple case of the model, we prove the existence of a limit cycle.     

Evolutionary Analyses of Tacit Communication in Van Huyck, Battalio, and Beil's Game Experiments

This paper provides evolutionary interpretations of Van Huyck, Battalio, and Beil's experimental results on coordination game with auction. A set-valued solution concept is defined for a finite population model under a best response evolutionary dynamic and applied to their games. It is shown that our solution concept captures the role of auction as a tacit communication device and predicts the equilibria chosen by players in the real experiments. We also discuss the possibility of constructing a more realistic evolutionary learning model to explain the experimental results.Journal of Economic LiteratureClassification Numbers: C72, C92.     

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.     

Laboratory Federalism: The Open Method of Coordination (OMC) as an Evolutionary Learning Process

In view of the concept of laboratory federalism, the Open Method of Coordination (OMC), adopted by the EU as a mode of governance, can be interpreted as an imitative learning dynamics of the type considered in evolutionary game theory. Its iterative design and focus on good practice are captured by the behavioral rule “imitate the best.” In a redistribution game with utilitarian governments and mobile welfare recipients, we compare the outcomes of imitative behavior (long‐run evolutionary equilibria) and decentralized best‐response behavior (Nash equilibria). The learning dynamics leads to coordination on a strict subset of Nash equilibria, favoring policy choices that can be sustained by a simple majority of Member States.     

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     

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.     

Equilibrium selection in coordination games with simultaneous play

Summary. We apply the dynamic stochastic framework proposed in recent evolutionary literature to a class of coordination games played simultaneously by the entire population. In these games payoffs, and hence best replies, are determined by a summary statistic of the population strategy profile. We demonstrate that with simultaneous play, the equilibrium selection depends crucially on how best responses to the summary statistic remain piece-wise constant. In fact, all the strict Nash equilibria in the underlying stage game can be made stochastically stable depending on how the best response mapping generates piece-wise constant best responses. Received: February 12, 2001; revised version: October 29, 2001     

Forward and backward dynamics in implicitly defined overlapping generations models

In dynamic economic models derived from optimization principles, the forward equilibrium dynamics may not be uniquely defined, while the backward dynamics is well defined. We derive properties of the global forward equilibrium paths based on properties of the backward dynamics. We propose the framework of iterated function systems (IFSs) to describe the set of forward equilibria and apply the IFS framework to a one- and a two-dimensional version of the overlapping generations (OLG)-model. We show that if the backward dynamics is chaotic and has a homoclinic orbit (a “snap-back repeller”), the set of forward equilibrium paths converges to a fractal attractor. Forward equilibria may be interpreted as sunspot equilibria, where a random sunspot sequence determines equilibrium selection at each date.     

Splitting the baby in two: solving Solomon's dilemma with boundedly rational agents

We study the dynamic implementation of the first-best forKing Solomon's Dilemma, on the assumption that boundedly rational players find their way to equilibrium using monotonic evolutionary dynamics. We find that all the mechanisms proposed by the literature fail to implement the first best dynamically and provide an alternative solution.     

Evolution with Delay

This paper analyses evolutionary models with delays in players’ observations. I apply best response dynamics to symmetric 2 × 2 games, in particular the coordination game and the game with no symmetric pure strategy equilibrium. When the dynamics have no delay, their solutions converge to symmetric Nash equilibria monotonically. However when I introduce delays to the dynamics, although these dynamics represented by differential difference equations are seemingly simple, it shows that they have periodic solutions. JEL Classification Number: C79.     

Evolutionary Stability of Pure-Strategy Equilibria in Finite Games

Sufficient conditions for pure-strategy Nash equilibria of finite games to be (Lyapunov) stable under a large class of evolutionary dynamics, the regular monotonic selection dynamics, are discussed. In particular, it is shown that in almost all finite extensive-form games, all the pure-strategy equilibria are stable. In such games, all mixed-strategy equilibria close to pure-strategy equilibria are also stable.Journal of Economic LiteratureClassification Numbers: C70, C72.     

Basins of attraction and equilibrium selection under different learning rules

A deterministic learning model applied to a game with multiple equilibria produces distinct basins of attraction for those equilibria. In symmetric two-by-two games, basins of attraction are invariant to a wide range of learning rules including best response dynamics, replicator dynamics, and fictitious play. In this paper, we construct a class of three-by-three symmetric games for which the overlap in the basins of attraction under best response learning and replicator dynamics is arbitrarily small. We then derive necessary and sufficient conditions on payoffs for these two learning rules to create basins of attraction with vanishing overlap. The necessary condition requires that with probability one the initial best response is not an equilibrium to the game. The existence of parasitic or misleading actions allows subtle differences in the learning rules to accumulate.     

Implementation,Elimination of Weakly Dominated Strategies and Evolutionary Dynamics

This paper studies convergence and stability properties of T. Sjöström's (1994, Games Econom. Behav.6, 502–511) mechanism, under the assumption that boundedly rational players find their way to equilibrium using monotonic evolutionary dynamics and best-reply dynamics. This mechanism implements most social choice functions in economic environments using as a solution concept one round of deletion of weakly dominated strategies and one round of deletion of strictly dominated strategies. However, there are other sets of Nash equilibria, whose payoffs may be very different from those desired by the social choice function. With monotonic dynamics, all these sets of equilibria contain limit points of the evolutionary dynamics. Furthermore, even if the dynamics converge to the “right” set of equilibria (i.e., the one which contains the solution of the mechanism), it may converge to an equilibrium which is worse in welfare terms. In contrast with this result, any interior solution of the best-reply dynamics converges to the equilibrium whose outcome the planner desires. Journal of Economic Literature Classification Numbers: C72, D70, D78.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

Dynamics in near-potential games

We consider discrete-time learning dynamics in finite strategic form games, and show that games that are close to a potential game inherit many of the dynamical properties of potential games. We first study the evolution of the sequence of pure strategy profiles under better/best response dynamics. We show that this sequence converges to a (pure) approximate equilibrium set whose size is a function of the “distance” to a given nearby potential game. We then focus on logit response dynamics, and provide a characterization of the limiting outcome in terms of the distance of the game to a given potential game and the corresponding potential function. Finally, we turn attention to fictitious play, and establish that in near-potential games the sequence of empirical frequencies of player actions converges to a neighborhood of (mixed) equilibria, where the size of the neighborhood increases according to the distance to the set of potential games.     

Evolutionary Stability in the Investment-Opportunism-Retaliation Game

Can economically efficient outcomes be obtained and sustained in the absence of externally enforced property rights? We study the evolutionary properties of a game that exhibits two well-defined Nash equilibria: one generates an inefficient outcome while the other set generates an efficient outcome supported by the potential for retaliation. Although standard forward-looking refinements eliminate the efficient equilibrium, neither equilibrium type satisfies strict evolutionary stability criteria. However, both types of equilibrium define strategies that are neutrally stable, which makes them vulnerable to drift in dynamic environments. We conduct computer simulation experiments in which players learn adaptively via a tournament selection mechanism called sophisticated experimentation. Our simulations demonstrate that while the system spends a disproportionately high proportion of time in the inefficient equilibrium set, the efficient equilibrium is pervasive as the system drifts back and forth between the equilibrium sets, never settling on one or the other.     

Cycles and multiple equilibria in the market for durable lemons

Summary. The paper investigates the nature of market failure in a dynamic version of Akerlof (1970) where identical cohorts of a durable good enter the market over time. In the dynamic model, equilibria with qualitatively different properties emerge. Typically, in equilibria of the dynamic model, sellers with higher quality wait in order to sell and wait more than sellers of lower quality. The main result is that for any distribution of quality there exist an infinite number of cyclical equilibria where all goods are traded within a certain number of periods after entering the market. Received: December 21, 2000; revised version: September 5, 2001     

Learning,Matching, and Aggregation

Fictitious play and stimulus–response/reinforcement learning are examined in the context of a large population where agents are repeatedly randomly matched. We show that the aggregation of this learning behavior can be qualitatively different from learning at the level of the individual. This aggregate dynamic belongs to the same class of simply defined dynamic as do several formulations of evolutionary dynamics. We obtain sufficient conditions for convergence and divergence which are valid for the whole class of dynamics. These results are therefore robust to most specifications of adaptive behavior.Journal of Economic LiteratureClassification Numbers: C72, D83.     

A Simple Supermodular Mechanism that Implements Lindahl Allocations

This paper introduces a new incentive‐compatible mechanism which, for general preference environments, implements Lindahl allocations as Nash equilibria. We provide an example that having a mechanism induce a supermodular game is not typically sufficient to guarantee dynamic stability of equilibrium. However, for the new mechanism, inducing a supermodular game guarantees that the best reply mapping is a contraction. Thus, this new mechanism provides a connection between the desirable welfare properties of Lindahl allocations and the theoretical/convergence properties of games whose best reply mappings are a contraction.