ON POSITIVE GAME MATRICES AND THEIR EXTENSIONS T. E. S. RAGHAVAN Summary In this paper we study some relations between the optimal strategies, the characteristic roots and the characteristic vectors of a positive square matrix whose rows and columns correspond to the pure strategy spaces of players in a zero-sum two-person game. Further we study a property of the game values of positive matrices that commute. The results are extended to infinite games on the unit square, with positive kernels as pay-off functions. Introduction It is well known that any two zero-sum two-person games have the same sets of optimal strategies for the two players when their pay-off kernels differ just by a constant. The optimal strategy sets could there- fore be analysed by considering a new game with a positive pay-off that differs from the original only by a constant. In the particular case where the pure strategy spaces are finite we can further reduce the positive pay-off matrix to a positive square matrix. This reduction is just the restatement of Theorem 3 of Kaplansky [3]. By such a reduction, we get the optimal strategies of the new game to be a subset of the optimal strategies of the old game. As Karlin puts it, in his book [10]: "What is far less understood is the relationship of the form of the pay-off kernel to the form of the solutions." So it seems possible to analyse the interrelations between the optimal strategy spaces and the kernel of a zero-sum two-person game, when we first study the strategy spaces of positive game kernels. For convenience the paper is divided into two sections. In the first section we consider finite games with positive pay-off and in the second section we generalise them to the infinite games with positive kernels on the unit square. I Preliminaries We shall only consider pay-off matrices of the type %><>, * , j = l , 2, ...,», Received 13 April, 1964; revised 6 September, 1964. [JOURNAL LONDON MATH. SOC, 40 (1965), 467-477]