We now start to study repeated games with incomplete information. In the present chapter we consider the simplest class of those games, namely, two-person zero-sum games in which one player, say, player I, is fully informed about the state of nature, while the other player, player II, knows only the prior distribution according to which the state is chosen.

GENERAL PROPERTIES

In this section we prove some general properties of a one-shot game with incomplete information, which later will be applied to various versions of the game: finitely or infinitely repeated games or discounted games. The game considered here is a two-person zero-sum game of the following form: chance chooses a state k from a finite set K of states (games) according to some probability p ∈ Π = Δ(K). Player I (the maximizer) is informed which k was chosen but player II is not. Players I and II then choose simultaneously σk ∈ Σ and τ ∈ J, respectively, and finally Gk(σk, τ) is paid to player I by player II. The sets Σ and J are some convex sets of strategies, and the payoff functions Gk(σk, τ) are bi-linear and uniformly bounded on Σ × J.

In normal form this is a game in which the strategies are σ ∈ ΣK and τ ∈ J, respectively, and the payoff function is Gp(σ, τ) = ∑kpkGk (σk, τ). Denote this game by Г(p).

Theorem V.1.1.w(p) = infτsupσGp(σ, τ) and w(p) = supσinfτGp(σ, τ) are concave.

Proof. The proof is the same for both functions. We write it for w(p). Let (pe)e∈E be finitely many points in Δ(K), and let α = (αe)e∈E be a point in Δ(E) such that ∑e∈E αepe = p; we claim that w(p) ≥ ∑e∈E αew(pe).