Mean-field-type games

This article examines games in which the payoffs and the state dynamics depend not only on the state-action profile of the decision-makers but also on a measure of the state-action pair. These game situations, also referred to as mean-field-type games, involve novel equilibrium systems to be solved. Three solution approaches are presented: (i) dynamic programming principle, (ii) stochastic maximum principle, (iii) Wiener chaos expansion. Relationship between dynamic programming and stochastic maximum principle are established using sub/super weak differentials. In the non-convex control action spaces, connections between the second order weaker differentials of the dual function and second order adjoint processes are provided. Multi-index Wiener chaos expansions are used to transform the non-standard game problems into standard ones with ordinary differential equations. Aggregative and moment-based mean-field-type games are discussed.