Decision Making using Game Theory: An introduction for managers

131Solving games without Nash equilibrium points using mixed strategiesPlayer 2Strategy c 1 c 2 . . . . . . . . c nr 1 U 11 , V 11 U 12 , V 12 . . . . . . . . U 1n , V 1nPlayer r 2 U 21 , V 21 U 22 , V 22 . . . . . . . . U 2n , V 2n1 :::::::::::::::r m U m1 , V m1 U m1 , V m1 . . . . . . . . U m n , V m nFigure 6.17The abbreviated general matrix for mixed strategies in mixed-motive games.matrix. It is a pair of strategies (r i , c j ) such that:∑ U ij is maximum in its column c j , and∑ V ij is maximum in its row r i .A mixed strategy for player 1 is a selection of probabilities p i such that0 p i 1 andΣp i 1, for i 1tomA mixed strategy for player 2 is a selection of probabilities q j such that0 q j 1 andΣq j 1, for j 1tonA mixed strategy becomes pure if p i (or q j ) is 1 and the probabilitiesassigned to every other strategy are zero.If player 1 and player 2 choose their strategies according to theirmixed strategies (p and q, respectively), each has no way of knowing theother player’s strategy, though the question remains, of course, as tohow these mixed strategies and resulting pay-oVs are calculated.If player 2 chooses strategy c j , then player 1 has a pay-oV:Σp i u ij , for i 1tomHowever, player 2 is selecting strategy c j with a probability q j , so thecumulative pay-oV for player 1 is:Ψ 1 ΣΣp i q j u ij , for i 1tom and for j 1tonand for player 2: