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

TL;DR

This paper investigates zero-sum and nonzero-sum differential games without Isaacs condition, establishing the existence of Nash equilibrium payoffs and characterizing game values under asymmetric information using advanced mathematical tools.

Contribution

It introduces a novel approach to analyze differential games without Isaacs condition, proving the existence of Nash equilibria and characterizing game values with asymmetric information.

Findings

Limits of upper and lower value functions coincide as partition mesh tends to zero

Existence of Nash equilibrium payoff in nonzero-sum differential games without Isaacs condition

New characterization of the zero-sum game value under asymmetric information

Abstract

In this paper we study the zero-sum and nonzero-sum differential games with not assuming Isaacs condition. Along with the partition $\pi$ of the time interval $[0,T]$, we choose the suitable random non-anticipative strategy with delay to study our differential games with asymmetric information. Using Fenchel transformation, we prove that the limits of the upper value function $W^\pi$ and lower value function $V^\pi$ coincide when the mesh of partition $\pi$ tends to 0. Moreover, we give a characterization for the Nash equilibrium payoff (NEP, for short) of our nonzero-sum differential games without Isaacs condition, then we prove the existence of the NEP of our games. Finally, by considering all the strategies along with all partitions, we give a new characterization for the value of our zero-sum differential game with asymmetric information under some equivalent Isaacs condition.