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Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zerosum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value [Formula: see text]). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closedloop controls (instead of openloop controls); and (ii) we must allow for path dependent controls, even if the problem is in a statedependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or pathdependent PDE), which is quite efficient for numerically computing the set value of the game.more » « less

The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the oneperiod (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of risk minimization problems. The results are applied to several important CAIs, such as the gaintoloss ratio, the riskadjusted return on capital and the tailvalueatrisk based CAI. In the second part of the paper we investigate the acceptability maximization in a discrete time dynamic setup. Using robust representations of CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we establish an intriguing dichotomy: if the corresponding family of DCRMs is recursive (i.e. strongly time consistent) and assuming some recursive structure of the market model, then the acceptability maximization problem reduces to just a one period problem and the maximal acceptability is constant across all states and times. On the other hand, if the family of DCRMs is not recursive, which is often the case, then the acceptability maximization problem ordinarily is a timeinconsistent stochastic control problem, similar to the classical meanvariance criteria. To overcome this form of timeinconsistency, we adapt to our setup the setvalued Bellman's principle recently proposed in [
23 ] applied to two particular dynamic CAIs  the dynamic riskadjusted return on capital and the dynamic gaintoloss ratio. The obtained theoretical results are illustrated via numerical examples that include, in particular, the computation of the intermediate meanrisk efficient frontiers.