Zero-sum and nonzero-sum differential games without Isaacs condition

Li, Juan; Li, Wenqiang

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 1217-1252.

In this paper we study differential games without Isaacs condition. The objective is to investigate on one hand zero-sum games with asymmetric information on the pay-off, and on the other hand, for the case of symmetric information but now for a non-zero sum differential game, the existence of a Nash equilibrium pay-off. Our results extend those by Buckdahn, Cardaliaguet and Rainer [SIAM J. Control Optim. 43 (2004) 624–642], to the case without Isaacs condition. To overcome the absence of Isaacs condition, randomization of the non-anticipative strategies with delay of the both players are considered. They differ from those in Buckdahn, Quincampoix, Rainer and Xu [Int. J. Game Theory 45 (2016) 795–816]. Unlike in [Int. J. Game Theory 45 (2016) 795–816], our definition of NAD strategies for a game over the time interval $[t,T]$ ($0\le t\le T$) guarantees that a randomized strategy along a partition $\pi$ of $[0,T]$ remains a randomized NAD strategy with respect to any finer partition $\pi \text{'}$ ($\pi \subset \pi \text{'}$). This allows to study the limit behavior of upper and lower value functions defined for games in which the both players use also different partitions.

Classification : 49N70, 49L25, 91A23, 60H10
Mots clés : Zero-sum and nonzero-sum differential game, asymmetric information, Isaacs condition, Nash equilibrium payoffs, Fenchel transformation

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