The Solution of a Certain Two-Person Zero-Sum Game

A two-person zero-sum game is specified as follows: Let B be any positive real number, and let θ be a real number between zero and one, exclusive. Player I chooses a number, t₁, between 0 and B + 1, inclusive, and player II chooses a number, t₂, between 0 and B, inclusive II then pays I the amount ϕ(t₁, t₂) where

Best pure strategies do not exist. Let the mixed strategies of I and II be specified by the distribution functions F and G, respectively, and let E(F, G) be the expected value of ϕ. Although ϕ is discontinuous, it is shown that the game has a well-determined value where n is the largest integer not exceeding B. Optimal strategies for each player are given explicitly.