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