  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.
