Abstract:

We introduce asymptotic analysis of stochastic games with short-stage duration. The play of stage k, $k\geq 0$, of a stochastic game $\Gamma_δ$ with stage duration $δ$ is interpreted as the play in time $kδłeq t<(k+1)δ$, 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 $nδ$, and the $łambda$-discounted present value of a payoff $g$ in stage $k$ is $łambda^kδ g$. We define convergence, strong convergence, and exact convergence of the data of a family $(\Gamma_δ)_δ>0$ as the stage duration $δ$ 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.