Contribution to the Linear Complementarity Problem and Completely Mixed Games

Abstract

This dissertation focuses on the linear complementarity problem (LCP ), two-person zero-sum matrix games, and Q-tensors. A matrix game is considered completely mixed if all the optimal pairs of strategies in the game are completely mixed. In this thesis, we provide new characterizations of Kaplansky’s results (1945 and 1995) on completely mixed games. Pang proved that within the class of semimonotone matrices, R0-matrices are Q- matrices and conjectured that the converse is also true. Gowda proved that the conjecture is true for symmetric matrices. We prove that semimonotone Q-matrices are R0-matrices up to order 3 and provide a counterexample to show that this statement does not hold for matrices of order 4 and higher. We also provide an application of this result using completely mixed games. Stone proposed that fully semimonotone Q0-matrices are P0-matrices. In this thesis, we establish that this conjecture holds true for matrices with certain sign patterns. Since fully semimonotone matrices are semimonotone and Z-matrices are Q0, we demonstrate that semimonotone Z-matrices are P0. Gowda proved that a Z-matrix with value zero is completely mixed if and only if it is irreducible. We provide new equivalent conditions for this statement. Additionally, we present results on completely mixed games, exploring their connection to various classes of matrices. We also extend some results of Q-matrices to Q-tensors.