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Set Values for Nonzero Sum Games With Multiple Equilibriums

Nonzero sum games typically have multiple NashÌýequilibriums (or no equilibriums), and unlike zeroÌýsum games, they may have different values at different equilibriums. While most works in the literature focus on the existence of individualÌýequilibriums, we propose instead to study the valueÌýset overÌýall possible equilibriums. It turns out that this value set has many nice properties such as regularity, stability, and more importantly the dynamic programming principle. There are two main featuresÌýin order to obtain the DPP: (i) we must use closed-loop controls (instead of open-loop controls), and (ii) we must allow for path dependent controls and hence path dependent values, even if the problem is in a state dependent setting. We next impose an additional aggregated utilityÌýso as to choose an "optimal" equilibrium among the set we have analyzed, withÌýsocial welfare asÌýa possible application. This problem is typically time inconsistent when viewed dynamically. We shall propose a so called moving scalarization, a dynamicÌýaggregated utility, to recoverÌýthe time consistency. The talk is based on an ongoing work joint with Feinstein and Rudloff.