718 research outputs found
Large Fluctuations and Fixation in Evolutionary Games
We study large fluctuations in evolutionary games belonging to the
coordination and anti-coordination classes. The dynamics of these games,
modeling cooperation dilemmas, is characterized by a coexistence fixed point
separating two absorbing states. We are particularly interested in the problem
of fixation that refers to the possibility that a few mutants take over the
entire population. Here, the fixation phenomenon is induced by large
fluctuations and is investigated by a semi-classical WKB
(Wentzel-Kramers-Brillouin) theory generalized to treat stochastic systems
possessing multiple absorbing states. Importantly, this method allows us to
analyze the combined influence of selection and random fluctuations on the
evolutionary dynamics \textit{beyond} the weak selection limit often considered
in previous works. We accurately compute, including pre-exponential factors,
the probability distribution function in the long-lived coexistence state and
the mean fixation time necessary for a few mutants to take over the entire
population in anti-coordination games, and also the fixation probability in the
coordination class. Our analytical results compare excellently with extensive
numerical simulations. Furthermore, we demonstrate that our treatment is
superior to the Fokker-Planck approximation when the selection intensity is
finite.Comment: 17 pages, 10 figures, to appear in JSTA
Evolutionary games on multilayer networks: coordination and equilibrium selection
We study mechanisms of synchronisation, coordination, and equilibrium
selection in two-player coordination games on multilayer networks. We apply the
approach from evolutionary game theory with three possible update rules: the
replicator dynamics (RD), the best response (BR), and the unconditional
imitation (UI). Players interact on a two-layer random regular network. The
population on each layer plays a different game, with layer I preferring the
opposite strategy to layer II. We measure the difference between the two games
played on the layers by a difference in payoffs while the
inter-connectedness is measured by a node overlap parameter . We discover a
critical value below which layers do not synchronise. For
in general both layers coordinate on the same strategy. Surprisingly,
there is a symmetry breaking in the selection of equilibrium -- for RD and UI
there is a phase where only the payoff-dominant equilibrium is selected. Our
work is an example of previously observed differences between the update rules
on a single network. However, we took a novel approach with the game being
played on two inter-connected layers. As we show, the multilayer structure
enhances the abundance of the Pareto-optimal equilibrium in coordination games
with imitative update rules
Stochastic learning dynamics and speed of convergence in population games
We study how long it takes for large populations of interacting agents to come close to Nash equilibrium when they adapt their behavior using a stochastic better reply dynamic. Prior work considers this question mainly for 2 × 2 games and potential games; here we characterize convergence times for general weakly acyclic games, including coordination games, dominance solvable games, games with strategic complementarities, potential games, and many others with applications in economics, biology, and distributed control. If players' better replies are governed by idiosyncratic shocks, the convergence time can grow exponentially in the population size; moreover, this is true even in games with very simple payoff structures. However, if their responses are sufficiently correlated due to aggregate shocks, the convergence time is greatly accelerated; in fact, it is bounded for all sufficiently large populations. We provide explicit bounds on the speed of convergence as a function of key structural parameters including the number of strategies, the length of the better reply paths, the extent to which players can influence the payoffs of others, and the desired degree of approximation to Nash equilibrium
Deterministic Equations for Stochastic Spatial Evolutionary Games
Spatial evolutionary games model individuals who are distributed in a spatial
domain and update their strategies upon playing a normal form game with their
neighbors. We derive integro-differential equations as deterministic
approximations of the microscopic updating stochastic processes. This
generalizes the known mean-field ordinary differential equations and provide a
powerful tool to investigate the spatial effects in populations evolution. The
deterministic equations allow to identify many interesting features of the
evolution of strategy profiles in a population, such as standing and traveling
waves, and pattern formation, especially in replicator-type evolutions
Mutation, Sexual Reproduction and Survival in Dynamic Environments
A new approach to understanding evolution [Valiant, JACM 2009], namely viewing it through the lens of computation,
has already started yielding new insights, e.g., natural selection under sexual reproduction can be interpreted
as the Multiplicative Weight Update (MWU) Algorithm in coordination games played among genes [Chastain, Livnat, Papadimitriou, Vazirani, PNAS 2014]. Using this machinery, we study the role of mutation in changing environments in the presence of sexual reproduction. Following [Wolf, Vazirani, Arkin, J. Theor. Biology], we model changing environments via a Markov chain, with the states representing environments, each with its own fitness matrix. In this setting, we show that in the absence of mutation, the population goes extinct, but in the presence of mutation, the population survives with positive probability.
On the way to proving the above theorem, we need to establish some facts about dynamics in games. We provide the first, to our knowledge, polynomial convergence bound for noisy MWU in a coordination game.
Finally, we also show that in static environments, sexual evolution with mutation converges, for any level of mutation
Stochastic evolutionary game dynamics
In this review, we summarize recent developments in stochastic evolutionary
game dynamics of finite populations.Comment: To appear in "Reviews of Nonlinear Dynamics and Complexity" Vol. II,
Wiley-VCH, 2009, edited by H.-G. Schuste
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