Michel Benaïm

Deterministic Approximation of Stochastic Evolution in Games

This paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The processes are Markov chains, and the approximation is defined in continuous time as a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the long-run behavior of the discrete stochastic process, for large populations, and its deterministic flow approximation. In particular, we provide probabilistic bounds on exit times from and visitation rates to neighborhoods of attractors to the deterministic flow. We sharpen these results in the special case of ergodic processes.

Stochastic Approximations and Differential Inclusions

The dynamical systems approach to stochastic approximation is generalized to the case where the mean differential equation is replaced by a differential inclusion. The limit set theorem of Benaim and Hirsch is extended to this situation. Internally chain transitive sets and attractors are studied in detail for set-valued dynamical systems. Applications to game theory are given, in particular to Blackwells approachability theorem and the convergence of fictitious play.

A Dynamical System Approach to Stochastic Approximations

It is known that some problems of almost sure convergence for stochastic approximation processes can be analyzed via an ordinary differential equation (ODE) obtained by suitable averaging. The goal of this paper is to show that the asymptotic behavior of such a process can be related to the asymptotic behavior of the ODE without any particular assumption concerning the dynamics of this ODE. The main results are as follows: a) The limit sets of trajectory solutions to the stochastic approximation recursion are, under classical assumptions, almost surely nonempty compact connected sets invariant under the flow of the ODE and contained in its set of chain-recurrence. b) If the gain parameter goes to zero at a suitable rate depending on the expansion rate of the ODE, any trajectory solution to the recursion is almost surely asymptotic to a forward trajectory solution to the ODE.

Stochastic Approximations and Differential Inclusions, Part II: Applications

We apply the theoretical results on “stochastic approximations and differential inclusions” developed in Benaim et al. [M. Benaim, J. Hofbauer, S. Sorin. 2005. Stochastic approximations and differential inclusions. SIAM J. Control Optim.44 328--348] to several adaptive processes used in game theory, including classical and generalized approachability, no-regret potential procedures (Hart and Mas-Colell [S. Hart, A. Mas-Colell. 2003. Regret-based continuous time dynamics. Games Econom. Behav.45 375--394]), and smooth fictitious play [D. Fudenberg, D. K. Levine. 1995. Consistency and cautious fictitious play. J. Econom. Dynam. Control19 1065--1089].

Asymptotic pseudotrajectories and chain recurrent flows, with applications

We present a general framework to study compact limit sets of trajectories for a class of nonautonomous systems, including asymptotically autonomous differential equations, certain stochastic differential equations, stochastic approximation processes with decreasing gain, and fictitious plays in game theory. Such limit sets are shown to be internally chain recurrent, and conversely.

Persistence in fluctuating environments

AbstractUnderstanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions. One day is fine, the next is black.—The Clash

Qualitative properties of certain piecewise deterministic Markov processes

We study a class of Piecewise Deterministic Markov Processes with state space Rd x E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hormander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.

Recursive algorithms, urn processes and chaining number of chain recurrent sets

This paper investigates the dynamical properties of a class of urn processes and recursive stochastic algorithms with constant gain which arise frequently in control, pattern recognition, learning theory, and elsewhere. It is shown that, under suitable conditions, invariant measures of the process tend to concentrate on the Birkhoff center of irreducible (i.e. chain transitive) attractors of some vector field

Exponential concentration for first passage percolation through modified Poincaré inequalities

F: {\Bbb R}^d \rightarrow {\Bbb R}^d

Quantitative ergodicity for some switched dynamical systems

obtained by averaging. Applications are given to simple situations including the cases where