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

¹ School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, P.R. China.
juanli@sdu.edu.cn
² School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China.
wenqianglis2009@gmail.com

Received: 28 January 2016
Revised: 23 May 2016
Accepted: 2 June 2016

Abstract

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 ≤ t ≤ T) guarantees that a randomized strategy along a partition π of [ 0,T ] remains a randomized NAD strategy with respect to any finer partition π′ (π ⊂ π′). 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.