9,249 research outputs found
On the robustness of learning in games with stochastically perturbed payoff observations
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
Spatial heterogeneity promotes coexistence of rock-paper-scissor metacommunities
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
required for regional persistence. For dispersal rates below , 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 , 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
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
Genetic draft, selective interference, and population genetics of rapid adaptation
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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