A kinetic equation modelling the collective dynamics of the rock-paper-scissors binary game
The rock-paper-scissors game has been studied for ages 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 (in the case of a 'balanced' game). 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 diffusion rate that depends 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 this 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 method to obtain an explicit weak limit to the initial equation, which is new compared to the previous work.
The asymptotic behaviour is obtained using an unusual time rescaling. Like previously mentionned works, we obtain a decay in (weighted) total variation norm and an explicit limit, but unlike these works, the limit measure strongly depends on the initial measure (yet in a linear fashion) and the decay rate is subgeometric. This work is submited, and a preprint can be found on: https://hal.archives-ouvertes.fr/hal-03798269.