An alternate Lagrangian scheme for spatially inhomogeneous evolutionary games

TL;DR

This paper introduces an innovative Lagrangian scheme for approximating nonlinear continuity equations in spatially inhomogeneous evolutionary games, with proven convergence and adaptations for specific systems like replicator dynamics.

Contribution

It presents a novel alternate Lagrangian scheme that combines position and strategy updates, with convergence analysis and variants for special cases.

Findings

Scheme converges in probability measure space.

Variants for replicator systems and Markov chains are effective.

Explicit and implicit label evolution steps are analyzed.

Abstract

An alternate Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics and is alternate, as it consists of the following two steps: first the distribution of strategies or labels is updated according to a best performance criterion and then this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.