On Completely Mixed Games

Parthasarathy Thiruvankatachari (), Ravindran Gomatam () and Sunil Kumar ()
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Parthasarathy Thiruvankatachari: Chennai Mathematical Institute
Ravindran Gomatam: Indian Statistical Institute
Sunil Kumar: Indian Statistical Institute

Journal of Optimization Theory and Applications, 2024, vol. 201, issue 1, No 12, 313-322

Abstract: Abstract A matrix game is considered completely mixed if all the optimal pairs of strategies in the game are completely mixed. In this paper, we establish that a matrix game A, with a value of zero, is completely mixed if and only if the value of the game associated with $$A +D_i $$ A + D i is positive for all i, where $$D_i$$ D i represents a diagonal matrix where ith diagonal entry is 1 and else 0. Additionally, we address Kaplansky’s question from 1945 regarding whether an odd-ordered symmetric game can be completely mixed, and provide characterizations for odd-ordered skew-symmetric matrices to be completely mixed. Moreover, we demonstrate that if A is an almost skew-symmetric matrix and the game associated with A has value positive, then $$A +D_i \in Q$$ A + D i ∈ Q for all i, where $$D_i$$ D i is a diagonal matrix whose ith diagonal entry is 1 and else 0. Skew-symmetric matrices and almost skew-symmetric matrices with value positive fall under the class of $$P_0$$ P 0 and $$Q_0$$ Q 0 , making them amenable to processing through Lemke’s algorithm.

Keywords: Q matrices; Completely mixed games; Symmetric games; Skew-symmetric matrices; 90C33; and; 91A05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02395-5

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