Reduction of Silverman-like games to games on bounded sets

A two person zero sum game is regarded as Silverman-like if the strategy sets are sets of real numbers bounded below, the payoff function is bounded, the maximum payoff is achieved whenever the second player's numbery exceeds the first player's numberx by “too much”, and the minimum is achieved wheneverx exceedsy by “too much”. Explicit upper bounds are obtained for pure strategies to be included in an optimal mixed strategy in such games. In particular, if the strategy sets are discrete, the games may be reduced to games on specified finite sets.