Game Theory and Statistical Mechanics.

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The investigator will be working on several projects at the intersection of game theory, statistical mechanics, and probability. One series of projects is centered around the deep and largely unexplored connections between game theory and statistical mechanics. On one hand, game theoretical concepts enrich greatly the modelling tools of statistical mechanics and, on the other hand, recent advances in non-equilibrium statistical mechanics shed a new light on game theory. In particular the investigator will work on the connections between certain zero-sum games and the concepts of irreversibility and entropy production. Another series of projects involve the development of multiscale modelling via stochastic processes, deterministic equations, and hybrid stochastic-deterministic models in evolutionary games. These tools are needed to model complex systems with multiple spatio-temporal scales such as flexible real-time frequency trading in telecommunication networks or monopolistic competitions between companies vying for spatially distributed customers. The investigator will work in collaboration with mathematicians, economists and computer scientists with the goals to develop adequate mathematical tools and to construct realistic models. Game theory starting with Von Neumann, Nash, and many others has always been, and still is, a crucial tool in mathematical economics. Starting in the early 1980's, in particular due to the biologist Maynard-Smith, evolutionary game theory was born and it combines the traditional static concepts of strategic equilibria with the implementation of learning mechanisms and strategy transmissions via dynamical mechanisms. Evolutionary games are now used notably, in population biology, social sciences, computer sciences, as well as physics. The investigator's work is motivated on one hand by the deep theoretical connections between game theory and physics (in particular non-equilibrium statistical mechanics). On the other hand the investigator also aims to develop the mathematical tools to model complex multiscale problems with a combination of game theory and probability. A notable challenge is to construct an agent-based model for spectrum trading in mobile telecommunication networks. Problems of this type are expected to be crucially important in the near future due to the explosive development of mobile telecommunication devices.