A measure approach to a kinetic equation modelling the collective dynamics of the rock-paper-scissors binary game (Hugo Martin, INSERM)

The rock-paper-scissors game has been studied for many years from various points of view. Game theory established a long time ago that two rational players have a unique optimal strategy, which is playing one of the three moves at random, with probability 1/3. This very simple case was extended in various ways. In a recent paper, Pouradier Duteil and Salvarani considered a large amount of rational agents, that play a r-p-s game after encountering, and exchange a certain amount of money based on the outcome. This results in a PDE modeling a population structured in wealth, that can be interpreted as a discrete heat equation on the half-line with rate of diffusion depending on the amount of players that are rich enough (i.e. that would be able to pay their dept if they loose they next game), thus introducing a nonlinearity.

The goal of my work was to adapt a methodology in the measure framework that was successfully used on equations arising from biology. On the well-known renewal and growth-fragmentation equations, as well as equations steming for instance from neuroscience, convergences in total variation norm were obtained, sometimes supplemented by explicit exponential rates of convergence. The first step was to reformulate the equation in the space of signed measures, by mean of a duality approach. The dual problem in turn provided a explicit weak limit to the initial equation, which is new compared to the previous work. Stronger results on the asymptotic behavior —convergence in total variation norm and rate of convergence— are still works in progress. The talk will be illustrated with numerical illustrations, that suggest that the convergence rate does not have a simple expression.

The PDE depends on a parameter h > 0 (that is the amount of money exchanged if the game does not result on a toss). Up to a diffusive scaling, the "exchange part" in the PDE tends to a Laplace operator as h vanishes, so this model seems an interesting candidate to a discretization using asymptotic preserving scheme. Such work would be a logical follow up of my talk.