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39
Basins of attraction, longrun stochastic stability, and the speed of stepbystep evolution
 Review of Economic Studies
"... The paper examines the behaviour of ‘‘evolutionary’ ’ models with εnoise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the ‘‘long run stochastic stability’ ’ of a convention is related to the speed with ..."
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Cited by 62 (0 self)
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The paper examines the behaviour of ‘‘evolutionary’ ’ models with εnoise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the ‘‘long run stochastic stability’ ’ of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2B2 games is generalized to the selection of 1 2dominant equilibria in arbitrary games. Other applications involve twodimensional local interaction and cycles as long run stochastically stable sets. 1.
EVOLUTIONARY DRIFT AND EQUILIBRIUM SELECTION
, 1996
"... This paper develops an approach to equilibrium selection in game theory based on studying the equilibriating process through which equilibrium is achieved. The differential equations derived from models of interactive learning typically have stationary states that are not isolated. Instead, Nash equ ..."
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Cited by 52 (2 self)
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This paper develops an approach to equilibrium selection in game theory based on studying the equilibriating process through which equilibrium is achieved. The differential equations derived from models of interactive learning typically have stationary states that are not isolated. Instead, Nash equilibria that specify the same behavior on the equilibrium path, but different outofequilibrium behavior, appear in connected components of stationary states. The stability properties of these components often depend critically on the perturbations to which the system is subjected. We argue that it is then important to incorporate such drift into the model. A su±cient condition is provided for drift to create stationary states with strong stability properties near a component of equilibria. This result is used to derive comparative static predictions concerning common questions raised in the literature on refinements of Nash equilibrium
Evolutionary Game Dynamics in Finite Populations
, 2004
"... We introduce a model of stochastic evolutionary game dynamics in finite populations which is similar to the familiar replicator dynamics for infinite populations. Our focus is on the conditions for selection favoring the invasion and/or fixation of new phenotypes. For infinite populations, there are ..."
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Cited by 48 (12 self)
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We introduce a model of stochastic evolutionary game dynamics in finite populations which is similar to the familiar replicator dynamics for infinite populations. Our focus is on the conditions for selection favoring the invasion and/or fixation of new phenotypes. For infinite populations, there are three generic selection scenarios describing evolutionary game dynamics among two strategies. For finite populations, there are eight selection scenarios. For a fixed payoff matrix a number of these scenarios can occur for different population sizes. We discuss several examples with unexpected behavior.
Evolving Aspirations and Cooperation
 Journal of Economic Theory
, 1998
"... This paper therefore builds on [3], in which a model of consistent aspirationsbased learning was introduced ..."
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Cited by 42 (2 self)
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This paper therefore builds on [3], in which a model of consistent aspirationsbased learning was introduced
Deterministic approximation of stochastic evolution in games
, 2002
"... This paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The processes are Markov chains, and the approximation is defined in continuous time as a system of ordinary differential equations of the type studied ..."
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Cited by 37 (3 self)
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This paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The processes are Markov chains, and the approximation is defined in continuous time as a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the longrun behavior of the discrete stochastic process, for large populations, and its deterministic flow approximation. In particular, we provide probabilistic bounds on exit times from and visitation rates to neighborhoods of attractors to the deterministic flow. We sharpen these results in the special case of ergodic processes.
Fast Equilibrium Selection by Rational Players Living in a Changing World
 Econometrica
, 1996
"... We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 ..."
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Cited by 29 (7 self)
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We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 \Theta 2 game. The payoff matrix changes over time according to some Brownian motion. Players observe these payoffs and the population distribution of actions as they evolve. The game has frictions: opportunities to change strategies arrive from independent random processes, so that the players are locked into their actions for some time. We solve the game using only backwards induction. As the frictions disappear, each player ignores what the others are doing and switches at her first opportunity to the risk dominant equilibrium. History dependence emerges in some cases when frictions remain positive. As an application we show how frictions and aggregate cost shocks can lead to the selecti...
Stochastic Game Theory: Adjustment to Equilibrium Under Noisy Directional Learning
, 1999
"... This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the FokkerPlanck equation. The dynamic process is ..."
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Cited by 26 (13 self)
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This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the FokkerPlanck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This "logit equilibrium" forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.
Learning to be Imperfect: . . .
, 1995
"... This paper studies interactive learning processes that are subject to constant perturbations or "noise." We argue that payoffs in the Ultimatum Game are such that responders are more apt to be "noisy" than are proposers and show that as a result the learning process readily leads to outcomes that ar ..."
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Cited by 13 (1 self)
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This paper studies interactive learning processes that are subject to constant perturbations or "noise." We argue that payoffs in the Ultimatum Game are such that responders are more apt to be "noisy" than are proposers and show that as a result the learning process readily leads to outcomes that are Nash equilibria but not subgameperfect. We conclude that game theorists should not restrict attention to the subgameperfect equilibrium when predicting laboratory behavior in the
Nash Equilibrium and the Evolution of Preferences
 Journal of Economic Theory
, 1997
"... A population of players of players is randomly matched to play a normal form game G. The payoffs in this game represent the fitness associated with the various outcomes. Each individual has preferences over the outcomes in the game and chooses an optimal action with respect to those preferences. How ..."
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Cited by 13 (0 self)
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A population of players of players is randomly matched to play a normal form game G. The payoffs in this game represent the fitness associated with the various outcomes. Each individual has preferences over the outcomes in the game and chooses an optimal action with respect to those preferences. However, these preferences needn't coincide with the fitness payoffs. When evolution selects individuals on the basis of the fitness of the actions they choose, the distribution of aggregate play must be a Nash equilibrium of G. Weak additional assumptions on the evolutionary process imply perfect equilibrium.