# Faure

# Mathieu Faure

- Chercheur

Consider a two-player normal-form game repeated over time. We introduce an adaptive learning procedure, where the players only observe their own realized payoff at each stage. We assume that agents do not know their own payoff function and have no information on the other player. Furthermore, we assume that they have restrictions on their own actions such that, at each stage, their choice is limited to a subset of their action set. We prove that the empirical distributions of play converge to the set of Nash equilibria for zero-sum and potential games, and games where one player has two actions.

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 study one of the main concept of online learning and sequential decision problem known as regret minimization. We investigate three different frameworks, whether data are generated accordingly to some i.i.d. process, or when no assumption whatsoever are made on their generation and, finally, when they are the consequences of some sequential interactions between players. The overall objective is to provide a comprehensive introduction to this domain. In each of these main setups, we define and analyze classical algorithms and we analyze their performances. Finally, we also show that some concepts of equilibria that emerged in game theory are learnable by players using online learning schemes while some other concepts are not learnab

We analyze quasi-stationary distributions {Î¼ Îµ } Îµ>0 of a family of Markov chains {X Îµ } Îµ>0 that are random perturbations of a bounded, continuous map F:Mâ†’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 âŠ‚M such that F(M 0 )=M 0 and F(Mâˆ–M 0 )=Mâˆ–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 Mâˆ–M 0 ), then the weak* limit points of Î¼ Îµ 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.