Set Values for Nonzero Sum Games With Multiple Equilibriums
Nonzero sum games typically have multiple Nash equilibriums (or no equilibriums), and unlike zero sum games, they may have different values at different equilibriums. While most works in the literature focus on the existence of individual equilibriums, we propose instead to study the value set over all 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 features in 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 utility so as to choose an "optimal" equilibrium among the set we have analyzed, with social welfare as a possible application. This problem is typically time inconsistent when viewed dynamically. We shall propose a so called moving scalarization, a dynamic aggregated utility, to recover the time consistency. The talk is based on an ongoing work joint with Feinstein and Rudloff.