Risk-Sensitive Mean-Field-Type Games with Lp-norm Drifts

TL;DR

This paper develops a new framework for risk-sensitive mean-field-type games with p-norm drifts, deriving optimality conditions and showing propagation of chaos for non-linear dynamics.

Contribution

It introduces a novel class of mean-field-type games with p-norm drifts, deriving finite-dimensional maximum principles and infinite-dimensional dynamic programming equations.

Findings

Derived a stochastic maximum principle for p-norm drift functions.

Established sufficient optimality conditions via dynamic programming.

Proved propagation of chaos for non-linear McKean-Vlasov dynamics.

Abstract

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.