Find all Nash equilibria in pure strategies for the following games. First check for dominant strategies. If there are none, solve using iterated elimination of dominated strategies. Explain your reasoning.
For each of the four games in Exercise S1, identify whether the game is zero-sum or non-zero-sum. Explain your reasoning.
Another method for solving zero-sum games, important because it was developed long before Nash developed his concept of equilibrium for non-zero-sum games, is the minimax method. To use this method, assume that no matter which strategy a player chooses, her rival will choose to give her the worst possible payoff from that strategy. For each zero-sum game identified in Exercise S2, use the minimax method to find the game’s equilibrium strategies by doing the following:
(a) For each row strategy, write down the minimum possible payoff to Rowena (the worst that Colin can do to her in each case). For each column strategy, write down the minimum possible payoff to Colin (the worst that Rowena can do to him in each case).
(b) For each player, determine the strategy (or strategies) that gives each player the best of these worst payoffs. This is called a “minimax” strategy for each player. (Because this is a zero-sum game, players’ best responses do indeed involve minimizing each other’s payoff, so these minimax strategies are the same as the Nash equilibrium strategies. John von Neumann proved the existence of a minimax equilibrium in zero-sum games in 1928, more than 20 years before Nash generalized the theory to include zero-sum games.)