An approximation theorem for infinite games

Abstract:

We consider infinite, two person zero sum games played as follows: On the nth move, players A, B select privately from fixed finite sets, ${A_n},{B_n}$, the result of their selections being made known before the next selection is made. A point in the associated sequence space $\Omega = \prod \nolimits _{n = 1}^\infty {({A_n} \times {B_n})}$ is thus produced upon which B pays A an amount determined by a payoff function defined on $\Omega$. We show that if the payoff functions of games $\{ {G_n}\}$ are upper semicontinuous and decrease pointwise to a function which is the payoff for a game, G, then ${\text {Val}}({G_n}) \downarrow {\text {Val}}(G)$. This shows that a certain class of games can be approximated by finite games. We then give a counterexample to possibly more general approximation theorems by displaying a sequence of games $\{ {G_n}\}$ with upper semicontinuous payoff functions which increase to the payoff of a game G, and where ${\text {Val}}({G_n}) = 0$ for all n but ${\text {Val}}(G) = 1$.