94 research outputs found
Local stability under evolutionary game dynamics
We prove that any regular ESS is asymptotically stable under any impartial pairwise comparison dynamic, including the Smith dynamic; under any separable excess payoff dynamic, including the BNN dynamic; and under the best response dynamic. Combined with existing results for imitative dynamics, our analysis validates the use of ESS as a blanket sufficient condition for local stability under evolutionary game dynamics.Evolutionary game dynamics, ESS
Orders of limits for stationary distributions, stochastic dominance, and stochastic stability
A population of agents recurrently plays a two-strategy population game. When an agent receives a revision opportunity, he chooses a new strategy using a noisy best response rule that satisfies mild regularity conditions; best response with mutations, logit choice, and probit choice are all permitted. We study the long run behavior of the resulting Markov process when the noise level is small and the population size is large. We obtain a precise characterization of the asymptotics of the stationary distributions as approaches zero and approaches infinity, and we establish that these asymptotics are the same for either order of limits and for all simultaneous limits. In general, different noisy best response rules can generate different stochastically stable states. To obtain a robust selection result, we introduce a refinement of risk dominance called \emph{stochastic dominance}, and we prove that coordination on a given strategy is stochastically stable under every noisy best response rule if and only if that strategy is stochastically dominant.Evolutionary game theory, stochastic stability, equilibrium selection
Riemannian game dynamics
We study a class of evolutionary game dynamics defined by balancing a gain
determined by the game's payoffs against a cost of motion that captures the
difficulty with which the population moves between states. Costs of motion are
represented by a Riemannian metric, i.e., a state-dependent inner product on
the set of population states. The replicator dynamics and the (Euclidean)
projection dynamics are the archetypal examples of the class we study. Like
these representative dynamics, all Riemannian game dynamics satisfy certain
basic desiderata, including positive correlation and global convergence in
potential games. Moreover, when the underlying Riemannian metric satisfies a
Hessian integrability condition, the resulting dynamics preserve many further
properties of the replicator and projection dynamics. We examine the close
connections between Hessian game dynamics and reinforcement learning in normal
form games, extending and elucidating a well-known link between the replicator
dynamics and exponential reinforcement learning.Comment: 47 pages, 12 figures; added figures and further simplified the
derivation of the dynamic
Survival of dominated strategies under evolutionary dynamics
We prove that any deterministic evolutionary dynamic satisfying four mild requirements fails to 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.Evolutionary game theory, evolutionary game dynamics, nonconvergnece, dominated strategies
Local stability under evolutionary game dynamics
We prove that any regular ESS is asymptotically stable under any impartial pairwise comparison dynamic, including the Smith dynamic; under any separable excess payoff dynamic, including the BNN dynamic; and under the best response dynamic. Combined with existing results for imitative dynamics, our analysis validates the use of ESS as a blanket sufficient condition for local stability under evolutionary game dynamics
Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics
We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games
An Evolutionary Approach to Congestion
Using techniques from evolutionary game theory, we analyze potential games with continuous player sets, a class of games which includes a general model of network congestion as a special case. We concisely characterize both the complete set of Nash equilibria and the set of equilibria which are robust against small disturbances of aggregate behavior. We provide a strong evolutionary justification of why equilibria must arise. We characterize situations in which stable equilibria are socially efficient, and show that in such cases, evolution always increases aggregate efficiency. Applying these results, we construct a parameterized class of congestion tolls under which evolution yields socially optimal play. Finally, we characterize potential games with continuous player sets by establishing that a generalization of these games is precisely the limiting version of finite player potential games (Monderer and Shapley (1996)) which satisfy an anonymity condition.
Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory
Sandholm WH, Staudigl M. Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory. Center for Mathematical Economics Working Papers. Vol 505. Bielefeld: Center for Mathematical Economics; 2014.We consider a model of stochastic evolution under general noisy best response
protocols, allowing the probabilities of suboptimal choices to depend on their payoff
consequences. Our analysis focuses on behavior in the small noise double limit: we
first take the noise level in agents’ decisions to zero, and then take the population
size to infinity. We show that in this double limit, escape from and transitions between
equilibria can be described in terms of solutions to continuous optimal control
problems. These are used in turn to characterize the asymptotics of the the stationary
distribution, and so to determine the stochastically stable states. The control problems
are tractable in certain interesting cases, allowing analytical descriptions of the escape
dynamics and long run behavior of the stochastic evolutionary process
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