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Spatially Inhomogeneous Evolutionary Games
Author(s) -
Ambrosio Luigi,
Fornasier Massimo,
Morandotti Marco,
Savaré Giuseppe
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21995
Subject(s) - mathematics , uniqueness , replicator equation , banach space , stability (learning theory) , position (finance) , convergence (economics) , field (mathematics) , equivalence (formal languages) , mathematical economics , mathematical analysis , pure mathematics , computer science , sociology , economics , population , demography , finance , machine learning , economic growth
We introduce and study a mean‐field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represent‐dimensional state space (pairs ( x , σ ) of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean‐field formulation, when the number N of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces that could be of independent interest. © 2021 Wiley Periodicals LLC.