Asymptotic behaviors of a kinetic approach to the collective dynamics of a rock-paper-scissors binary game

TL;DR

This paper analyzes the long-term behavior of a kinetic model for the rock-paper-scissors game, proving well-posedness and describing asymptotic decay using a duality approach and Harris-type theorems.

Contribution

It introduces a novel measure-based kinetic model for the rock-paper-scissors game and provides explicit asymptotic descriptions despite nonlinearity.

Findings

Proved well-posedness of the kinetic equation.

Described explicit asymptotic behavior in large time.

Established subgeometric decay in total variation norm.

Abstract

This article studies the kinetic dynamics of the rock-paper-scissors binary game in a measure setting given by a non local and non linear integrodifferential equation. After proving the wellposedness of the equation, we provide a precise description of the asymptotic behavior in large time. To do so we adopt a duality approach, which is well suited both as a first step to construct a measure solution by mean of semigroups and to obtain an explicit expression of the asymptotic measure. Even thought the equation is non linear, this measure depends linearly on the initial condition. This result is completed by a decay in total variation norm, which happens to be subgeometric due to the nonlinearity of the equation. This relies on an unusual use of a confining condition that is needed to apply a Harris-type theorem, taken from a recent paper [2] that also provides a way to compute explicitly the constants involved in the aforementioned decay in norm.