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Nonzero sum games typically have multiple Nash equilibriums (or noequilibrium), and unlike the zero sum case, they may have different values atdifferent equilibriums. Instead of focusing on the existence of individualequilibriums, we study the set of values over all equilibriums, which we callthe set value of the game. The set value is unique by nature and always exists(with possible value $\emptyset$). Similar to the standard value function incontrol literature, it enjoys many nice properties such as regularity,stability, and more importantly the dynamic programming principle. There aretwo main features in order to obtain the dynamic programming principle: (i) wemust use closed-loop controls (instead of open-loop controls); (ii) we mustallow for path dependent controls, even if the problem is in a state dependent(Markovian) setting. We shall consider both discrete and continuous time modelswith finite time horizon. For the latter we will also provide a dualityapproach through certain standard PDE (or path dependent PDE), which is quiteefficient for numerically computing the set value of the game.

Dynamic Set Values for Nonzero Sum Games with Multiple Equilibriums