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So far, we have considered only what are known as pure
strategies, where players simply choose one of the actions
available to them. However, employing pure strategies also makes
the player’s actions predictable and easily countered. Indeed,
there is no guarantee that a game must have a Nash equilibrium in
pure strategies, as we can imagine many that will inevitably result
in a endless sequence of counter moves. In this problem, we will
investigate one such game, and find a Nash equilibrium by expanding
the scope of the player’s choices beyond only pure
strategies.
Suppose that two players, Abe and Liz, are playing a game
known as matching pennies, in which both simultaneously reveal a
penny as either heads H or tails T . Abe wins $1 from Liz if both
players choose the same side, while Liz wins $1 from Abe if they
choose different sides.
1. (10) Write the payoff matrix for this game and show that
none of the four pure strategy outcomes is a Nash
equilibrium.
Matching pennies is an example of a game that has no Nash
equilibria in pure strategies, as if both players commit to a
single action, one of them will always want to switch. However, a
Nash equilibrium does in fact exist in this game, but to find it we
consider the players employing mixed strategies, where they choose
their actions randomly rather than deciding on one or the other
with certainty.
Suppose that Abe, frustrated with his inability to win
consistently picking either heads or tails, decides that
rather than picking one or the other, he will just flip the
coin and play whatever it lands on. That is, he
randomly chooses his action, choosing either heads or tails,
each with probability
2. (10) Liz observes Abe’s novel new strategy and thinks about
how best to respond to it. What is her average payoff if she plays
heads? What if she plays tails?
3. (10) Liz resolves that the best way to deal with Abe’s
mixed strategy is to employ a mixed strategy of her own, playing
heads with probability 2 and tails with probability 1 . What is
Liz’s expected payoff?
Does it make sense for her to employ the mixed strategy?
4. (10) Abe now observes Liz’s mixed strategy. What is his
average payoff if he plays heads? What if he plays tails? Does it
makes sense for him to continue randomizing between heads and
tails?
5. (10) Find the mixed strategy Nash equilibrium in the
matching pennies game.