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