Generic Properties of First-Order Mean Field Games

• Published in: Dynamic games and applications Vol. 13; no. 3; pp. 750 - 782
• Main Authors: Bressan, Alberto, Nguyen, Khai T.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2023; Springer Nature B.V
• Subjects: Communications Engineering; Computer Systems Organization and Communication Networks; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Game Theory; Games; Management Science; Mathematics; Mathematics and Statistics; Networks; Operations Research; Social and Behav. Sciences; Terminal constraints; Topology; Uniqueness
• ISSN: 2153-0785; 2153-0793
• DOI: 10.1007/s13235-022-00487-3

We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for “nearly all” mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.