While payoff-based learning models are almost exclusively devised for finite action games, where players can test every action, it is harder to design such learning processes for continuous games. We construct a stochastic learning rule, designed for games with continuous action sets, which requires no sophistication from the players and is simple to implement: players update their actions according to variations in own payoff between current and previous action. We then analyze its behavior in several classes of continuous games and show that convergence to a stable Nash equilibrium is guaranteed in all games with strategic complements as well as in concave games, while convergence to Nash equilibrium occurs in all locally ordinal potential games as soon as Nash equilibria are isolated.

The stability of Nash equilibria has often been studied by examining the asymptotic behavior of the best-response dynamics. This is generally done in games where interactions are global and equilibria are isolated. In this paper, we analyze stability in contexts where interactions are local and where there are continua of equilibria. We focus on the public good game played on a network, where the set of equilibria is known to depend on the network structure (Bramoullé and Kranton, 2007), and where, as we show, continua of equilibria often appear. We provide necessary and sufficient conditions for a component of Nash equilibria to be asymptotically stable vis-à-vis the best-response dynamics. Interestingly, we demonstrate that these conditions relate to the structure of the network in a simple way. We also provide corresponding results for several dynamical systems related to the best response.

Generalized Polya urn models have been used to model the establishment dynamics of a small founding population consisting of k di?erent genotypes or strategies. As population sizes get large, these population processes are well-approximated by a mean limit ordinary di?erential equation whose state space is the k simplex. We prove that if this mean limit ODE has an attractor at which the temporal averages of the population growth rate is positive, then there is a positive probability of the population not going extinct (i.e. growing without bound) and its distribution converging to the attractor. Conversely, when the temporal averages of the population growth rate is negative along this attractor, the population distribution does not converge to the attractor. For the stochastic analog of the replicator equations which can exhibit non-equilibrium dynamics, we show that verifying the conditions for convergence and non-convergence reduces to a simple algebraic problem. We also apply these results to selection-mutation dynamics to illustrate convergence to periodic solutions of these population genetics models with positive probability.

We analyze quasi-stationary distributions {μ ε } ε>0 of a family of Markov chains {X ε } ε>0 that are random perturbations of a bounded, continuous map F:M→M , where M is a closed subset of R k . Consistent with many models in biology, these Markov chains have a closed absorbing set M 0 ⊂M such that F(M 0 )=M 0 and F(M∖M 0 )=M∖M 0 . Under some large deviations assumptions on the random perturbations, we show that, if there exists a positive attractor for F (i.e., an attractor for F in M∖M 0 ), then the weak* limit points of μ ε are supported by the positive attractors of F . To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of metapopulations, competing species, host-parasitoid interactions and evolutionary games.