9,249 research outputs found

    On the robustness of learning in games with stochastically perturbed payoff observations

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    Motivated by the scarcity of accurate payoff feedback in practical applications of game theory, we examine a class of learning dynamics where players adjust their choices based on past payoff observations that are subject to noise and random disturbances. First, in the single-player case (corresponding to an agent trying to adapt to an arbitrarily changing environment), we show that the stochastic dynamics under study lead to no regret almost surely, irrespective of the noise level in the player's observations. In the multi-player case, we find that dominated strategies become extinct and we show that strict Nash equilibria are stochastically stable and attracting; conversely, if a state is stable or attracting with positive probability, then it is a Nash equilibrium. Finally, we provide an averaging principle for 2-player games, and we show that in zero-sum games with an interior equilibrium, time averages converge to Nash equilibrium for any noise level.Comment: 36 pages, 4 figure

    Evolutionary game theory

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    Game Theory

    Spatial heterogeneity promotes coexistence of rock-paper-scissor metacommunities

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    The rock-paper-scissor game -- which is characterized by three strategies R,P,S, satisfying the non-transitive relations S excludes P, P excludes R, and R excludes S -- serves as a simple prototype for studying more complex non-transitive systems. For well-mixed systems where interactions result in fitness reductions of the losers exceeding fitness gains of the winners, classical theory predicts that two strategies go extinct. The effects of spatial heterogeneity and dispersal rates on this outcome are analyzed using a general framework for evolutionary games in patchy landscapes. The analysis reveals that coexistence is determined by the rates at which dominant strategies invade a landscape occupied by the subordinate strategy (e.g. rock invades a landscape occupied by scissors) and the rates at which subordinate strategies get excluded in a landscape occupied by the dominant strategy (e.g. scissor gets excluded in a landscape occupied by rock). These invasion and exclusion rates correspond to eigenvalues of the linearized dynamics near single strategy equilibria. Coexistence occurs when the product of the invasion rates exceeds the product of the exclusion rates. Provided there is sufficient spatial variation in payoffs, the analysis identifies a critical dispersal rate dd^* required for regional persistence. For dispersal rates below dd^*, the product of the invasion rates exceed the product of the exclusion rates and the rock-paper-scissor metacommunities persist regionally despite being extinction prone locally. For dispersal rates above dd^*, the product of the exclusion rates exceed the product of the invasion rates and the strategies are extinction prone. These results highlight the delicate interplay between spatial heterogeneity and dispersal in mediating long-term outcomes for evolutionary games.Comment: 31pages, 5 figure

    Imitation Dynamics with Payoff Shocks

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    We investigate the impact of payoff shocks on the evolution of large populations of myopic players that employ simple strategy revision protocols such as the "imitation of success". In the noiseless case, this process is governed by the standard (deterministic) replicator dynamics; in the presence of noise however, the induced stochastic dynamics are different from previous versions of the stochastic replicator dynamics (such as the aggregate-shocks model of Fudenberg and Harris, 1992). In this context, we show that strict equilibria are always stochastically asymptotically stable, irrespective of the magnitude of the shocks; on the other hand, in the high-noise regime, non-equilibrium states may also become stochastically asymptotically stable and dominated strategies may survive in perpetuity (they become extinct if the noise is low). Such behavior is eliminated if players are less myopic and revise their strategies based on their cumulative payoffs. In this case, we obtain a second order stochastic dynamical system whose attracting states coincide with the game's strict equilibria and where dominated strategies become extinct (a.s.), no matter the noise level.Comment: 25 page

    Stochastic Replicator Dynamics

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    Genetic draft, selective interference, and population genetics of rapid adaptation

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    To learn about the past from a sample of genomic sequences, one needs to understand how evolutionary processes shape genetic diversity. Most population genetic inference is based on frameworks assuming adaptive evolution is rare. But if positive selection operates on many loci simultaneously, as has recently been suggested for many species including animals such as flies, a different approach is necessary. In this review, I discuss recent progress in characterizing and understanding evolution in rapidly adapting populations where random associations of mutations with genetic backgrounds of different fitness, i.e., genetic draft, dominate over genetic drift. As a result, neutral genetic diversity depends weakly on population size, but strongly on the rate of adaptation or more generally the variance in fitness. Coalescent processes with multiple mergers, rather than Kingman's coalescent, are appropriate genealogical models for rapidly adapting populations with important implications for population genetic inference.Comment: supplementary illustrations and scripts are available at http://webdav.tuebingen.mpg.de/interference
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