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 ΓΘ. 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 VΘ of a repeated two-person zero-sum game ΓΘ with a public uncertain duration process Θ. We study asymptotic properties of the normalized value vΘ = VΘ/E(θ) as the expected duration E (θ) goes to infinity. We extend and unify several asymptotic results on the existence of lim vn and lim vλ and their equality to lim vΘ. This analysis applies in particular to stochastic games and repeated games of incomplete information.