Abstract
A 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 a multi-step scheme, as it develops on the following two stages: 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.
Highlights
The capability of changing strategy as an adaptive response to the modification of the surrounding environment in order to maximize a certain payoff is of paramount importance in decision-making processes
Replicator-type models [21] are a particular class of dynamical models that feature this adaptivity and are well suited for studying the evolution of strategies according to their success: Given a pool of strategies, the occurrence of each of them evolves according to their performance with respect to all the others; in this way, if a strategy gives a payoff which is higher compared to the average of all strategies, it is enhanced; otherwise, it is suppressed
The work of FS was supported by the project Variational methods for stationary and evolution problems with singularities and interfaces PRIN 2017 financed by the Italian Ministry of Education, University, and Research
Summary
The capability of changing strategy as an adaptive response to the modification of the surrounding environment in order to maximize a certain payoff is of paramount importance in decision-making processes. Replicator-type models [21] are a particular class of dynamical models that feature this adaptivity and are well suited for studying the evolution of strategies according to their success: Given a pool of strategies, the occurrence of each of them evolves according to their performance with respect to all the others; in this way, if a strategy gives a payoff which is higher compared to the average of all strategies, it is enhanced; otherwise, it is suppressed This criterion, in the basic replicator model, is the only one that determines the evolution of the occurrence of the strategies, which is independent from all other factors, in particular from the position of the agents that play those strategies. The work of FS was supported by the project Variational methods for stationary and evolution problems with singularities and interfaces PRIN 2017 financed by the Italian Ministry of Education, University, and Research
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