Maxminimizers vs. Nash equilibrium actions:-
The game in Figure 1 has a unique Nash equilibrium, in which player 1's strategy is ( ¼, 3/4 ) and player 2's strategy is ( 2/3 , 1/3 ). In this equilibrium player 1's payoff is 4.
Now consider the maxminimizer for player 1. Player 1's payoff as a function of the probability that she assigns to T is shown in Figure 2. We see that the max minimizer for player 1 is ( 1 3 , 2 3 ), and this strategy guarantees player 1 a payoff of 4. Thus in this game player 1's max minimizer guarantees that she obtain her payoff in the unique equilibrium, while her equilibrium strategy does not. If player 1 is certain that player 2 will adhere to the equilibrium then her equilibrium strategy yields her equilibrium payoff of 4, but if player 2 chooses a different strategy then player 1's payoff may be less than 4 (it also may be greater than 4). Player 1's maxminimizer, on the other hand, guarantees a payoff of 4 regardless of player 2's behavior.
