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.

Accepted:

DOI: 10.1051/cocv/2016044

Keywords: Zero-sum and nonzero-sum differential game, asymmetric information, Isaacs condition, Nash equilibrium payoffs, Fenchel transformation

^{1}; Li, Wenqiang

^{1, 2}

@article{COCV_2017__23_3_1217_0, author = {Li, Juan and Li, Wenqiang}, title = {Zero-sum and nonzero-sum differential games without {Isaacs} condition}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1217--1252}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016044}, zbl = {1369.49053}, mrnumber = {3660466}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016044/} }

TY - JOUR AU - Li, Juan AU - Li, Wenqiang TI - Zero-sum and nonzero-sum differential games without Isaacs condition JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1217 EP - 1252 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016044/ DO - 10.1051/cocv/2016044 LA - en ID - COCV_2017__23_3_1217_0 ER -

%0 Journal Article %A Li, Juan %A Li, Wenqiang %T Zero-sum and nonzero-sum differential games without Isaacs condition %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1217-1252 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016044/ %R 10.1051/cocv/2016044 %G en %F COCV_2017__23_3_1217_0

Li, Juan; Li, Wenqiang. Zero-sum and nonzero-sum differential games without Isaacs condition. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1217-1252. doi : 10.1051/cocv/2016044. http://archive.numdam.org/articles/10.1051/cocv/2016044/

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