Zero-sum stochastic differential games without the Isaacs condition: random rules of priority and intermediate Hamiltonians

TL;DR

This paper introduces a novel representation for the value function in zero-sum stochastic differential games without the Isaacs condition, using random rules of priority and intermediate Hamiltonians, with asymptotic discretization methods.

Contribution

It provides a new way to characterize the value function via a convex combination of Hamiltonians, involving random or deterministic rules depending on the state dependence.

Findings

Value function represented as convex combination of Hamiltonians.

Random rules of priority depend on coin tosses with state-dependent probabilities.

Asymptotic discretization methods for space and time are developed.

Abstract

For a zero-sum stochastic game which does not satisfy the Isaacs condition, we provide a value function representation for an Isaacs-type equation whose Hamiltonian lies in between the lower and upper Hamiltonians, as a convex combination of the two. For the general case (i.e. the convex combination is time and state dependent) our representation amounts to a random change of the rules of the game, to allow each player at any moment to see the other player's action or not, according to a coin toss with probabilities of heads and tails given by the convex combination appearing in the PDE. If the combination is state independent, then the rules can be set all in advance, in a deterministic way. This means that tossing the coin along the game, or tossing it repeatedly right at the beginning leads to the same value. The representations are asymptotic, over time discretizations. Space discretization is possible as well, leading to similar results.