Discounted Stochastic Games: The Finite Case

Abstract

Recall that in the λ-discounted game Γ_λ(z) with initial state z ₁ = z the payoff given a profile of strategies σ, γ ^z_λ (σ), is equal to the expectation, with respect to the distribution induced on plays by z and σ, of the discounted sum of the sequence of stage rewards {r _{m m} }:

$$ \gamma \frac{z}{\lambda }\left( \sigma \right) = E\frac{z}{\sigma }\left( {\sum {\frac{\infty }{{m = 1}}} \lambda {{(1 - \lambda )}^{m - 1}}{r_m}} \right) $$

This chapter considers the finite case where the state space S and each action space A ⁱ, i in I, are finite.