Bohnenblust, H. F.; Karlin, S.; Shapley, L. S. 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 5 Documents MSC: 91A99 Game theory Keywords:games with continuous convex pay-off Citations:Zbl 0041.25302 PDFBibTeX XML