Question: 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. 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 1 . 2

2. 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. 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 /3and tails with probability 1/3What is Liz's expected payoff?
Does it make sense for her to employ the mixed strategy?

4. 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. Find the mixed strategy Nash equilibrium in the matching pennies game.