Reducing Inequalities Among UnequalsJournal articleMathieu Faure and Nicolas Gravel, International Economic Review, Volume 62, Issue 1, pp. 357-404, 2021

This article establishes an equivalence between four incomplete rankings of distributions of income among agents who are vertically differentiated with respect to some nonincome characteristic (health, household size, etc.). The first ranking is the possibility of going from one distribution to the other by a finite sequence of income transfers from richer and more highly ranked agents to poorer and less highly ranked ones. The second ranking is the unanimity among utilitarian planners who assume that agents' marginal utility of income is decreasing with respect to both income and the source of vertical differentiation. The third ranking is the Bourguignon (Journal of Econometrics, 42 (1989), 67-80) Ordered Poverty Gap dominance criterion. The fourth ranking is a new dominance criterion based on cumulative lowest incomes.

Learning with minimal information in continuous gamesJournal articleSebastian Bervoets, Mario Bravo and Mathieu Faure, Theoretical Economics, Volume 15, Issue 4, pp. 1471-1508, 2020

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.

Convergence in games with continua of equilibriaJournal articleSebastian Bervoets and Mathieu Faure, Journal of Mathematical Economics, Volume 90, pp. 25-30, 2020

In game theory, the question of convergence of dynamical systems to the set of Nash equilibria has often been tackled. When the game admits a continuum of Nash equilibria, however, a natural and challenging question is whether convergence to the set of Nash equilibria implies convergence to a Nash equilibrium. In this paper we introduce a technique developed in Bhat and Bernstein (2003) as a useful way to answer this question. We illustrate it with the best-response dynamics in the local public good game played on a network, where continua of Nash equilibria often appear.

Stability in games with continua of equilibriaJournal articleSebastian Bervoets and Mathieu Faure, Journal of Economic Theory, Volume 179, Issue C, pp. 131-162, 2019

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.

Online Learning and Game Theory. A Quick Overview with recent results and applicationsJournal articleMathieu Faure, Pierre Gaillard, Bruno Gaujal and Vianney Perchet, ESAIM: Proceedings and Surveys, Aurélien Garivier (Eds.), Volume 51, pp. 246-271, 2015

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

Convergence of generalized urn models to non-equilibrium attractorsJournal articleMathieu Faure and Sebastian Schreiber, Stochastic Processes and their Applications, Volume 125, Issue 8, pp. 3053-3074, 2015

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.

Reinforcement Learning with Restrictions on the Action SetJournal articleMario Bravo and Mathieu Faure, SIAM Journal on Control and Optimization, Volume 53, Issue 1, pp. 287-312, 2015

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.

Quasi-stationary distributions for randomly perturbed dynamical systemsJournal articleMathieu Faure and Sebastian Schreiber, Annals of Applied Probability, Volume 24, Issue 2, pp. 553-598, 2014

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.

Ergodic properties of weak asymptotic pseudo-trajectories for set valued dynamical systemsJournal articleMathieu Faure and Gregory Roth, Stochastics and Dynamics, Volume 13, Issue 1, pp. 1250011, 2013
Consistency of Vanishingly Smooth Fictitious PlayJournal articleMichel Benaïm and Mathieu Faure, Mathematics of Operations Research, Volume 38, Issue 3, pp. 437-450, 2013

We discuss consistency of vanishingly smooth fictitious play , a strategy in the context of game theory, which can be regarded as a smooth fictitious play procedure , where the smoothing parameter is time dependent and asymptotically vanishes. This answers a question initially raised by Drew Fudenberg and Satoru Takahashi.