Risk-Sensitive and Robust Mean Field Games

Tamer Basar
University of Illinois at Urbana-Champaign

Mean field games (MFGs) constitute a class of non-cooperative stochastic differential games, where there is a large number of players or agents who interact with each other through a mean field coupling term—also known as the mass behavior or the macroscopic behavior in statistical physics—included in the individual cost functions and/or each agent’s dynamics generated by a controlled stochastic differential equation, capturing the average behavior of all agents. One of the main research issues in MFGs with no hierarchy in decision making is to study (existence, uniqueness and characterization of) Nash equilibria with an infinite population of players under specified information structures and further to study finite-population approximations, that is to explore to what extent an infinite-population Nash equilibrium provides an approximate Nash equilibrium for the finite-population game, and what the relationship is between the level of approximation and the size of the population.

Following a general overview of such games first with a finite number of players, the talk will dwell on two classes of MFGs: those characterized by risk sensitive (that is, exponentiated) objective functions (known as risk-sensitive MFGs) and those that have risk-neutral (RN) objective functions but with an additional adversarial driving term in the dynamics (known as robust MFGs). In stochastic optimal control, it is known that risk-sensitive (RS) cost functions lead to a behavior akin to robustness, leading to establishment of a connection between RS control problems and RN minimax ones. The talk will explore to what extent a similar connection holds between RS MFGs and robust MFGs, particularly in the context of linear-quadratic problems, which will allow for closed-form solutions and explicit comparisons between the two in both infinite- and finite-population regimes and with respect to the approximation of Nash equilibria in going from the former to the latter. The talk will conclude with a brief discussion of extensions to hierarchical decision structures with a small number of players at the top of the hierarchy (leaders) and an infinite population at the bottom (followers).

(This is based mostly on joint work with Jun Moon)

Presentation (PDF File)

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