Calculus of Variations and Geometric Measure Theory

S. Almi - M. Morandotti - F. Solombrino

A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

created by morandott on 15 May 2020
modified by solombrin on 25 Apr 2021


Published Paper

Inserted: 15 may 2020
Last Updated: 25 apr 2021

Journal: Journal of Evolution Equations
Pages: 34
Year: 2021
Doi: 10.1007/s00028-021-00702-5

ArXiv: 2005.07479 PDF


An multi-step 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 of multi-step type, 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.

Keywords: inhomogeneous evolutionary games, replicator equation, reversible Markov chains, minimizing movements scheme, multi-step Lagrangian scheme