Geometric aspects of game theory – GAGA

Game theory is the mathematical language for studying strategic interactions between agents, be they humans, firms, bacteria, or even computers. This is a rapidly developing subject, with growing applications in economics, social sciences, computer sciences, evolutionary biology and engineering.

Many game theoretical topics and tools have a strong geometrical or topological flavor: the structure of a game's equilibrium set, the design of algorithms computing equilibria, Blackwell approachability theory, the metric character of the replicator game dynamics, the use of semi-algebraicity in stochastic games, and many others. The objective of this project is to perform a systematic study of these Geometric Aspects of GAmes, and by so doing, to establish new links between research directions that so far appeared unrelated (such as Hessian-Riemannian metrics and discrete choice theory).

More precisely, we intend to focus on the following issues:
- The systematic and rigorous treatment of static geometric structures in games: equilibrium sets and sets of equilibrium payoffs, o-minimality in a game's definition data, spatial evolutionary games and the Hodge - de Rham decomposition of population games.

- The development of geometric game dynamics and algorithms, namely the derivation of game dynamics of first and second order based on Hessian-Riemannian metrics, and the development of value/equilibrium computation algorithms.

- The geometric features of online learning: approachability in stochastic games, the use of geometry in the design of optimal learning algorithms, and the examination of metric-driven no-regret learning schemes.

Importantly, apart from its purely mathematical value, the proposed research also touches on a wide array of applications, ranging from the study of local interactions in bacterial populations, to resource allocation problems in massively parallel computing grids, and to transmission signal optimization in MIMO-enabled devices (smartphones, tablets, etc.).

Of course, a prerequisite to the success of this project is the use of tools and techniques from a broad pool of mathematical fields, ranging from convex analysis, optimization, probabilty and statistics, to geometry and topology (not to mention the project's envisioned application areas). That said, the project team is uniquely suited to tackle this challenge: many of the project partners have an interdisciplinary background, so the team's collective spectrum of skills and expertise more than compensates for the above requirements. On that account, and adding to its scientific content, this project will also serve to cement the links between its partners and, by potentially being the first ANR JCJC on games, to launch to the forefront a new generation of the French school of game theory.