# Non-Cooperative Game Theory

• Tapan Biswas
Chapter

## Abstract

In Chapter 9 we saw that the pure strategy solution of a non-cooperative game may not exist. In such a case a player benefits from knowing his opponent’s strategy. It is suggested that players should play the game like a lottery, with probabilities attached to the strategies so that no player is certain about his rival’s strategy. Such strategies are called mixed strategies. Note that pure strategies are special cases of mixed strategies. A pure strategy implies the selection of one strategy with probability one and others with probability zero. In this section, we shall discuss the optimal probability mixture for the mixed strategy, or optimal mixed strategy, as it is called. Consider a two-person zero-sum game (TZNC) with two players A and B. Denote the mixed strategies of the two players as S A and S B :
$${S_A} = \left[ {\begin{array}{*{20}{c}} {{A_1}{A_2}{A_3}} \\ {{p_1}{p_2}{p_3}} \end{array}} \right]\;\quad {S_B} = \left[ {\begin{array}{*{20}{c}} {{B_1}{B_2}{B_3}} \\ {{q_1}{q_2}{q_3}} \end{array}} \right]\;$$
Given a pair of strategies (S A , S B ), the expected pay-offs of A and B are pAq and −pAq respectively.