Two risk neutral players, A and B, simultaneously choose bets on the outcome of a fair coin toss. Each player can place a $4 bet on either outcome (heads or tails), pay $8 to bet on both outcomes, or pay nothing to bet on neither outcome. If exactly one player bets on the correct outcome, that player wins $12 and the other player wins nothing. If both players bet on the correct outcome, they each win $6. If neither player bets on the correct outcome, they both win nothing. Thus, for example, if A bets on both heads and tails and B bets only on heads, then if heads comes up, A ends up with a net loss of $2 while B ends up with a net gain of $2, and if tails comes up, A ends up with a net gain of $4 while B ends up with a net loss of $4. (a) Find all pure strategy Nash equilibria of this game. Solution: Regardless of whether or not they are betting on tails, the marginal expected benefit to betting on heads is 12-4 = 2 if the other player is not betting on heads, and-6-4-1 if the other player is betting on heads. Similarly for tails. Therefore, both players are choosing a best response if and only if they are betting on exactly the outcome(s) that the other player is not betting on. There are four such strategy profiles, which are the pure strategy NE: (both, neither), (H,T), (T, H), and (neither, both). (b) Find a mixed strategy Nash equilibrium of this game in both players choose to bet on both outcomes with positive probability, or show that no such equilibrium exists. Solution: Consider a symmetric strategy profile in which each player bets on neither with probability p and both with probability 1-p (and places 0 probability on the other two possible bets). Given that B is using this strategy, A is indifferent between neither and both if 0 = 4p-2(1-p), which implies that p = 1/3. For this value of p, the expected payoff to betting on heads is 0, which is no larger than the payoff to betting on neither or both; similarly for betting on tails. Therefore, this strategy profile with p-1/3 forms a NE.

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