Mean-field-type games

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