We study how risk-sensitive players act in situations where the outcome is influenced not only by the state-action profile but also by the distribution of it. In such interactive decision-making problems, the classical mean-field game framework does not apply. We depart from most of the mean-field games literature by presuming that a decision-maker may include its own-state distribution in its decision. This leads to the class of mean-field-type games. In mean-field-type situations, a single decision-maker may have a big impact on the mean-field terms for which new type of optimality equations are derived. We establish a finite dimensional stochastic maximum principle for mean-field-type games where the drift functions have a p-norm structure which weaken the classical Lipschitz and differentiability assumptions. Sufficient optimality equations are established via Dynamic Programming Principle but in infinite dimension. Using de Finetti-Hewitt-Savage theorem, we show that a propagation of chaos property with 'virtual' particles holds for the non-linear McKean-Vlasov dynamics.
@cite_17: Abstract In the standard formulation of a game, a player's payoff function depends on the states and actions of all the players. Yet, real world applications suggest to consider also a functional of the probability measure of states and actions of all the players. In this paper, we consider cooperative mean-field type games in which the state dynamics and the payoffs depend not only on the state and actions but also on their probability measure. We establish stochastic maximum principle and provide a time-dependent payoff allocation procedure for coalitions. The allocated payoff considers not only fairness property but also the cost of making the coalition. Finally, time consistency and subgame perfectness solution concept equations are established.
Substantial progress have been done in the last decade in mean-field games in the non-cooperative setup. However, very little is known about cooperative mean-field games. In @cite_17 we have introduced cooperative mean-field type games in which the state dynamics and the payoffs depend not only on the state and actions but also on their probability measure. We establish a time-dependent payoff allocation procedure for coalitions of mean-field type. The allocated payoff considers not only fairness property but also the cost of making the coalition. Both time consistency and subgame perfectness solution concept equations are established.