Michel Benaïm | University of Neuchâtel (original) (raw)
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Papers by Michel Benaïm
Annals of Applied Probability, Feb 1, 2019
Comptes Rendus Mathematique, Aug 1, 2016
Journal of Dynamics and Differential Equations, 1996
We present a general framework to study compact limit sets of trajectories for a class of nonauto... more 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 approximarion processes with decreasing gain, and fictitious plays in game theory. Such limit sets are shown to be internally chain recurrent, and conversely,
Probability Theory and Related Fields, May 6, 2016
Stochastics And Partial Differential Equations: Analysis And Computations, Sep 8, 2015
Annals of Probability, 1997
Electronic Communications in Probability, 2018
Journal of Economic Theory, Jul 1, 2009
We propose a new concept for the analysis of games, the TASP, which gives a precise prediction ab... more We propose a new concept for the analysis of games, the TASP, which gives a precise prediction about non-equilibrium play in games whose Nash equilibria are mixed and are unstable under fictitious play-like learning processes. We show that, when players learn using weighted stochastic fictitious play and so place greater weight on more recent experience, the time average of play often converges in these "unstable" games, even while mixed strategies and beliefs continue to cycle. This time average, the TASP, is related to the best response cycle first identified by . Though conceptually distinct from Nash equilibrium, for many games the TASP is close enough to Nash to create the appearance of convergence to equilibrium. In other games, the TASP may be quite distant from Nash.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 9, 2018
Communications in Mathematical Sciences, 2019
arXiv (Cornell University), Oct 11, 2016
arXiv (Cornell University), Oct 10, 2019
This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are e... more This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the bias in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the bias depends on the past of the trajectory and is designed to approximate the free energy. The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the bias as time goes to infinity, and that the limit is close to the ideal bias, as an auxiliary parameter of the algorithm goes to 0. The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019
Annals of Applied Probability, Feb 1, 2019
Comptes Rendus Mathematique, Aug 1, 2016
Journal of Dynamics and Differential Equations, 1996
We present a general framework to study compact limit sets of trajectories for a class of nonauto... more 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 approximarion processes with decreasing gain, and fictitious plays in game theory. Such limit sets are shown to be internally chain recurrent, and conversely,
Probability Theory and Related Fields, May 6, 2016
Stochastics And Partial Differential Equations: Analysis And Computations, Sep 8, 2015
Annals of Probability, 1997
Electronic Communications in Probability, 2018
Journal of Economic Theory, Jul 1, 2009
We propose a new concept for the analysis of games, the TASP, which gives a precise prediction ab... more We propose a new concept for the analysis of games, the TASP, which gives a precise prediction about non-equilibrium play in games whose Nash equilibria are mixed and are unstable under fictitious play-like learning processes. We show that, when players learn using weighted stochastic fictitious play and so place greater weight on more recent experience, the time average of play often converges in these "unstable" games, even while mixed strategies and beliefs continue to cycle. This time average, the TASP, is related to the best response cycle first identified by . Though conceptually distinct from Nash equilibrium, for many games the TASP is close enough to Nash to create the appearance of convergence to equilibrium. In other games, the TASP may be quite distant from Nash.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 9, 2018
Communications in Mathematical Sciences, 2019
arXiv (Cornell University), Oct 11, 2016
arXiv (Cornell University), Oct 10, 2019
This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are e... more This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the bias in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the bias depends on the past of the trajectory and is designed to approximate the free energy. The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the bias as time goes to infinity, and that the limit is close to the ideal bias, as an auxiliary parameter of the algorithm goes to 0. The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019