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

Richard H. Brown
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Richard H. Brown: Operations Evaluation Group, Massachusetts Institute of Technology, P.O. Box 2176, Potomac Station, Alexandria, Virginia

Operations Research, 1957, vol. 5, issue 1, 63-67

Abstract: A two-person zero-sum game is specified as follows: Let B be any positive real number, and let (theta) be a real number between zero and one, exclusive. Player I chooses a number, t 1 , between 0 and B + 1, inclusive, and player II chooses a number, t 2 , between 0 and B , inclusive II then pays I the amount (phi)( t 1 , t 2 ) where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\begin{array}{r@{\quad}l@{\qquad}l} \mbox{(i)} & \mbox{if } t_{1}\leqq t_{2}, & \phi(t_{1}, t_{2})=1 - \theta,\\ \mbox{(ii)} & \mbox{if } t_{2} Best pure strategies do not exist. Let the mixed strategies of I and II be specified by the distribution functions F and G , respectively, and let E ( F , G ) be the expected value of (phi). Although (phi) is discontinuous, it is shown that the game has a well-determined value \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mbox{lub}_{F}\, \mbox{glb}_{G}\, E(F, G)= \mbox{glb}_{F}\, \mbox{lub}_{G}\, E(F, G)=(1-\theta) /(1-\theta^{n+1}),$$\end{document} where n is the largest integer not exceeding B . Optimal strategies for each player are given explicitly.

Date: 1957
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