Abstract:

We introduce asymptotic analysis of stochastic games with short-stage duration. The play of stage k, $k\geq 0$, of a stochastic game $\Gamma_\delta$ with stage duration $\delta$ is interpreted as the play in time $k\delta\leq t<(k+1)\delta$, and therefore the average payoff of the $n$-stage play per unit of time is the sum of the payoffs in the first $n$ stages divided by $n\delta$, and the $\lambda$-discounted present value of a payoff $g$ in stage $k$ is $\lambda^{k\delta} g$. We define convergence, strong convergence, and exact convergence of the data of a family $(\Gamma_\delta)_{\delta>0}$ as the stage duration $\delta$ goes to $0$, and study the asymptotic behavior of the value, optimal strategies, and equilibrium. The asymptotic analogs of the discounted, limiting-average, and uniform equilibrium payoffs are defined. Convergence implies the existence of an asymptotic discounted equilibrium payoff, strong convergence implies the existence of an asymptotic limiting-average equilibrium payoff, and exact convergence implies the existence of an asymptotic uniform equilibrium payoff.