Interval Invariance in the Evolutionary Game of Coordination

Abstract

The object of this article is the study of invariant sets in evolutionary games. Proving the invariance of a dynamic system to an invariant set consists in proving that if the system is initialized in the invariant set, then the state of the system remains forever in the invariant set. A (dynamically) invariant system must verify system of equations. Interval analysis makes it possible to prove (exactly) that the contraposition of this system of equations has no solutions. Solving the contraposition of this system consists in proving that there is no state of the invariant set leaving the invariant set (in positive time) according to the dynamics of the system. The studied dynamical systems are evolutionary game theory learning algorithms such as Replicator Dynamic and Best Response. Evolutionary algorithms (differential equations) like Replicator Dynamic allow to calculate Nash evolutionary equilibria (stable Nash under certain conditions) of evolutionary games such as the evolutionary game of coordination that is used for illustration purpose. Evolutionary games can converge to different equilibria depending on the initial condition. It is therefore important to study the invariance of such systems to predict the fixed points of learning algorithms.