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27
The Projection Dynamic and the Geometry of Population Games
 GAMES ECON. BEHAV.
, 2008
"... The projection dynamic is an evolutionary dynamic for population games. It is derived from a model of individual choice in which agents abandon their current strategies at rates inversely proportional to the strategies ’ current levels of use. The dynamic admits a simple geometric definition, its re ..."
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Cited by 20 (4 self)
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The projection dynamic is an evolutionary dynamic for population games. It is derived from a model of individual choice in which agents abandon their current strategies at rates inversely proportional to the strategies ’ current levels of use. The dynamic admits a simple geometric definition, its rest points coincide with the Nash equilibria of the underlying game, and it converges globally to Nash equilibrium in potential games and in stable games.
Survival of dominated strategies under evolutionary dynamics,”
 Theoretical Economics,
, 2011
"... Abstract We prove that any deterministic evolutionary dynamic satisfying four mild requirements does not eliminate strictly dominated strategies in some games. We also show that existing elimination results for evolutionary dynamics are not robust to small changes in the specifications of the dynam ..."
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Cited by 18 (10 self)
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Abstract We prove that any deterministic evolutionary dynamic satisfying four mild requirements does not eliminate strictly dominated strategies in some games. We also show that existing elimination results for evolutionary dynamics are not robust to small changes in the specifications of the dynamics. Numerical analysis reveals that dominated strategies can persist at nontrivial frequencies even when the level of domination is not small.
Chaos in the Cobweb Model with a New Learning Dynamic
, 2008
"... The new learning dynamic of Brown, von Neuman and Nash (1950) is introduced to macroeconomic dynamics via the cobweb model with rational and naive forecasting strategies. This dynamic has appealing properties such as positive correlation and inventiveness. There is persistent heterogeneity in the fo ..."
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The new learning dynamic of Brown, von Neuman and Nash (1950) is introduced to macroeconomic dynamics via the cobweb model with rational and naive forecasting strategies. This dynamic has appealing properties such as positive correlation and inventiveness. There is persistent heterogeneity in the forecasts and chaotic behavior with bifurcations between periodic orbits and strange attractors for the same range of parameter values as in previous studies. Unlike Brock and Hommes (1997), however, there exist intuitively appealing steady states where one strategy dominates, and there are qualitative differences in the resulting dynamics of the two approaches. There are similar bifurcations in a parameter that represents how aggressively agents switch to better performing strategies.
Basins of Attraction and Equilibrium Selection Under Different Learning Rules
, 2009
"... A deterministic learning model applied to a game with multiple equilibria produces distinct basins of attraction for those equilibria. In symmetric twobytwo games, basins of attraction are invariant to a wide range of learning rules including best response dynamics, replicator dynamics, and fict ..."
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Cited by 4 (3 self)
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A deterministic learning model applied to a game with multiple equilibria produces distinct basins of attraction for those equilibria. In symmetric twobytwo 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 threebythree 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 pure, uniformly evolutionarily stable strategies are almost never initial best responses. The existence of parasitic or misleading actions allows subtle differences in the learning rules to accumulate.
Deterministic Equations for Stochastic Spatial Evolutionary Games 1
"... Spatial evolutionary games model individuals playing a game with their neighbors in a spatial domain and describe the time evolution of strategy profile of individuals over space. We derive integrodifferential equations as deterministic approximations of strategy revision stochastic processes. Thes ..."
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Cited by 3 (1 self)
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Spatial evolutionary games model individuals playing a game with their neighbors in a spatial domain and describe the time evolution of strategy profile of individuals over space. We derive integrodifferential equations as deterministic approximations of strategy revision stochastic processes. These equations generalize the existing ordinary differential equations such as replicator dynamics and provide powerful tools for investigating the problem of equilibrium selection. Deterministic equations allow the identification of many interesting features of the evolution of a population’s strategy profiles, including traveling front solutions and pattern formation.
Stochastic Evolutionary Game Dynamics: Foundations, Deterministic Approximation, and Equilibrium Selection
"... Abstract. We present a general model of stochastic evolution in games played by large populations of anonymous agents. Agents receive opportunities to revise their strategies by way of independent Poisson processes. A revision protocol describes how the probabilities with which an agent chooses each ..."
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Abstract. We present a general model of stochastic evolution in games played by large populations of anonymous agents. Agents receive opportunities to revise their strategies by way of independent Poisson processes. A revision protocol describes how the probabilities with which an agent chooses each of his strategies depend on his current payoff opportunities and the current behavior of the population. Over finite time horizons, the population’s behavior is wellapproximated by a mean dynamic, an ordinary differential equation defined by the expected motion of the stochastic evolutionary process. Over the infinite time horizon, the population’s behavior is described by the stationary distribution of the stochastic evolutionary process. If limits are taken in the population size, the level of noise in agents ’ revision protocols, or both, the stationary distribution may become concentrated on a small set of population states, which are then said to be stochastically stable. Stochastic stability analysis allows one to obtain unique predictions of very long run behavior even when the mean dynamic admits multiple locally stable states. We present a full analysis of the asymptotics of the stationary distribution in twostrategy games under noisy best protocols, and discuss extensions of this analysis to other settings. 1.
Population games and deterministic evolutionary dynamics
, 2014
"... Population games describe strategic interactions among large numbers of small, anonymous agents. Behavior in these games is typically modeled dynamically, with agents occasionally receiving opportunities to switch strategies, basing their choices on simple myopic rules called revision protocols. Ove ..."
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Population games describe strategic interactions among large numbers of small, anonymous agents. Behavior in these games is typically modeled dynamically, with agents occasionally receiving opportunities to switch strategies, basing their choices on simple myopic rules called revision protocols. Over finite time spans the evolution of aggregate behavior is well approximated by the solution of a differential equation. From a different point of view, every revision protocol defines a map—a deterministic evolutionary dynamic—that assigns each population game a differential equation describing the evolution of aggregate behavior in that game. In this chapter, we provide an overview of the theory of population games and deterministic evolutionary dynamics. We introduce population games through a series of examples and illustrate their basic geometric properties. We formally derive deterministic evolutionary dynamics from revision protocols, introduce the main families of dynamics—imitative/biological, best response, comparison to average payoffs, and pairwise comparison—and discuss their basic properties. Combining these streams, we consider classes of population games in which members of these families of dynamics converge to equilibrium; these classes include potential games, contractive games, games solvable by iterative solution concepts, and supermodular games. We relate these classes to the classical notion of an evolutionarily stable state (ESS) and to recent work on deterministic equilibrium selection. We present a variety of examples of cycling and chaos under evolutionary dynamics, as well as a general result on survival of strictly dominated strategies. Finally, we provide connections to other approaches to game dynamics, and indicate applications of evolutionary game dynamics to economics and social science.
The projection dynamic, the replicator dynamic, and the geometry of population games
, 2006
"... Every population game defines a vector field on the set of strategy distributions X. The projection dynamic maps each population game to a new vector field: namely, the one closest to the payoff vector field among those that never point outward from X. We investigate the geometric underpinnings of t ..."
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Cited by 2 (0 self)
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Every population game defines a vector field on the set of strategy distributions X. The projection dynamic maps each population game to a new vector field: namely, the one closest to the payoff vector field among those that never point outward from X. We investigate the geometric underpinnings of the projection dynamic, describe its basic gametheoretic properties, and establish a number of close connections between the projection dynamic and the replicator dynamic.