Decision Making using Game Theory: An introduction for managers

101Representing mixed-motive games and the Nash equilibriumPlayer 2Strategy c 1 c 2 c 3Player r 11,0 0, 3 3, 11r 20, 2 1, 1 4, 0r 30, 2 3, 4 6, 2Figure 6.3A Nash equilibrium.∑ u 1 (r N , c N ) u 1 (r, c N ), r S 1∑ u 2 (r N , c N ) u 2 (r N , c), c S 2In other words, r N is bigger than any other r in the same column and c Nis bigger than any other c in the same row. In the example representedon Figure 6.3, three player 1 pay-oVs are maximum in their columns −(1, 0), (6, 2) and (3, 4) – but only the last one is simultaneouslymaximum in its row for player 2.So a Nash equilibrium is a unique pair of strategies from whichneither player has an incentive to deviate since, given what the otherplayer has chosen, the Nash equilibrium is optimal. In many ways, theconcept of the Nash equilibrium is a self-fulWlling prophecy.If both players know that both know about the Nash equilibrium,then they will both want to choose their Nash equilibrium strategy.Conversely, any outcome that is not the result of a Nash equilibriumstrategy will not be self-promoting and one player will always want todeviate. For example, the matrix on Figure 6.2 has two Nash equilibria– (4, 3) and (3, 4) – and it can be seen that, if either of the other pairs ofstrategies were chosen, there would be regret from at least one of theplayers.It is worth noting that, if a game can be solved using the method ofiterated elimination, then the game must have a single unique Nashequilibrium that would, of course, give the same solution. For example,the game represented on Figure 6.4 has a Nash equilibrium at (4, 4) andcould easily have been solved by the principles of dominance andinadmissibility, since strategy r 1 dominates r 2 for player 1, and strategyc 1 dominates c 2 for player 2. Neither player has any incentive to deviatefrom these strategies r 1 and c 1 .