Zero-Sum State-Blind Stochastic Games with Vanishing Stage Duration

TL;DR

This paper investigates the limiting behavior of zero-sum state-blind stochastic games as stage duration approaches zero, showing convergence of the game values to a viscosity solution of a PDE.

Contribution

It extends the analysis of game value limits to state-blind stochastic games, establishing convergence to a PDE solution as stage duration vanishes.

Findings

Game values converge to a viscosity solution of a PDE as h approaches 0.

The result generalizes previous perfect observation cases to state-blind scenarios.

Provides a mathematical framework for continuous-time approximation of state-blind stochastic games.

Abstract

In stochastic games with stage duration h, players act at times 0, h, 2h, and so on. The payoff and leaving probabilities are proportional to h. As h approaches 0, such discrete-time games approximate games played in continuous time. The behavior of the values when h tends to 0 was already studied in the case of stochastic games with perfect observation of the state. We examine the same question for the case of state-blind stochastic games. Our main finding is that, as h approaches 0, the value of any state-blind stochastic game with stage duration h converges to the unique viscosity solution of a partial differential equation.