Repeated games with public uncertain duration process

We consider repeated games where the number of repetitions θ is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process Θ that defines the probability law of the signals that players receive at each stage about the duration. To each repeated game Γ and uncertain duration process Θ is associated the Θ-repeated game Γ_Θ. A public uncertain duration process is one where the uncertainty about the duration is the same for all players. We establish a recursive formula for the value V_Θ of a repeated two-person zero-sum game Γ_Θ with a public uncertain duration process Θ. We study asymptotic properties of the normalized value v_Θ = V_Θ/E(θ) as the expected duration E (θ) goes to infinity. We extend and unify several asymptotic results on the existence of lim v_n and lim v_λ and their equality to lim v_Θ. This analysis applies in particular to stochastic games and repeated games of incomplete information.