ANR CONVERGENCE

I am the PI of the ANR JCJC project ``Convergence of Zero-Sum Games Dynamics'', for the period 2022-2025. The other members are:

• Luc Attia (Paris Dauphine University)

• Charles Bertucci (Ecole Polytechnique, CNRS)

• Krishnendu Chatterjee (IST Austria)

• Joon Kwon (Agro Paris Tech, INRAE)

• Miquel Oliu-Barton (Paris Dauphine University)

• Vianney Perchet (ENSAE)

• Raimundo Saona (IST Austria)

Description of the project

In a zero-sum game, two players choose simultaneously a strategy, and receive opposite payoffs. Under standard assumptions, a minmax theorem assures that the game has a value and optimal strategies, which represent the outcome of the game played by rational players. Zero-sum game dynamics have motivated a vast literature in Mathematics and Computer Science, notably through the seminal model of stochastic game (Shapley, 1953). This consists in the repetition of a game depending on a variable called state. This state evolves along time, following a stochastic process influenced by players actions. Investigating long-term properties of this model and its extensions has highlighted deep mathematical problems in various topics, including Probability Theory, Convex Analysis, Topology, or Algebraic Geometry. Recently, this model has enabled to solve important problems in PDE theory.

Moreover, the rise of Machine Learning has inspired new research directions and applications of zero-sum games in the last decade (e.g. GANs, Poker, Go), with regards notably to the computation of value and optimal strategies, for which dynamic procedures have proved to be very efficient.

This project pursues the three following objectives:

• Describe the limit behavior of long stochastic games, through the Mertens conjectures, and study its applications to the PDE problem of stochastic homogenization of Hamilton-Jacobi equations

• Study computability and online learning procedures in long stochastic games

• Build procedures that quickly converge to value and optimal strategies of a fixed zero-sum game