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Games with continuous, convex pay-off. (English) Zbl 0041.25703

Contrib. Theory of Games, Ann. Math. Stud. 24, 181-192 (1950).
The authors study a game where the strategies of the first player form an arbitrary set \(A\) and those of the second player a compact and convex region \(B\), whereas the pay-off is for every \(x\) in \(A\) a continuous convex function of \(y\). They prove that, under these conditions, the first player has an optimal mixed strategy composed of at most \(n+1\) pure strategies, when \(n\) is the dimension of \(B\), and that the second player has an optimal pure strategy. If the second player has a \(p\)-dimensional set of optimal pure strategies, then the first has an optimal mixed strategy consisting of at most \(n-p+1\) strategies with positive probability. A way of computing the solution is indicated.
Reviewer: Stefan Vajda

MSC:

91A99 Game theory

Citations:

Zbl 0041.25302