Decision Making using Game Theory: An introduction for managers

100Two-person mixed-motive games of strategyPlayer 2Strategy c 1 c 2Player r 1 u 1 (r 1 ,c 1 ) , u 2 (r 1 ,c 1 ) u 1 (r 1 ,c 2 ) , u 2 (r 1 ,c 2 )1r 2 u 1 (r 2 ,c 1 ) , u 2 (r 2 ,c 1 ) u 1 (r 2 ,c 2 ) , u 2 (r 2 ,c 2 )Figure 6.1A mixed-motive game with two players.Player 2Strategy c 1 c 2Player r 1 2, 2 4, 31r 2 3, 4 1, 1Figure 6.2A mixed-motive game with no dominant strategies.u 1 (r i , c) u 1 (r j , c), c S 2Strategy r j of player 1 is now said to be inadmissible in that player 1cannot choose it and at the same time claim to act rationally.The dominance of r i over r j is said to be strict if:u 1 (r i ,c) u 1 (r j ,c), c S 2and weak if:u 1 (r i , c) u 1 (r j , c), c S 2In the previous chapter, the method of iterated elimination ofdominated strategies was described as an alternative way of solvinggame matrices (see Figures 5.11–5.13). Unfortunately, it cannot beused to solve many mixed-motive games. The matrix represented onFigure 6.2, for example, has no dominant or dominated strategies andtherefore cannot be solved using the elimination method.Instead, such games must be solved using the concept of the Nashequilibrium (Plon, 1974). A pair of strategies (r N , c N ) S 1 S 2 is said tobe a Nash equilibrium if: