Each day, you and a friend play odds/evens to see who gets the last doughnut. On command, you each extend either one or two fingers. If the sum of the fingers is odd, you get the doughnut. If the sum of the fingers is even, your friend gets the doughnut.
a. Let the payoffs from winning be 1, and from losing, 0. Fill in the payoff matrix below:

b. Find the pure-strategy Nash equilibria in this game, if any.
c. If you always play one finger, how will your friend respond? How much can you expect to win, on average?
d. If you always play two fingers, how will your friend respond? How much can you expect to win, on average?
e. If you mix one finger and two fingers 50:50, and your friend does the same, what fraction of the time will you emerge victorious? How much can you expect to win, on average? Does mixing give you a higher average payout than playing a pure strategy?
f. Suppose your day-by-day mixture is as follows: 1, 2, 1, 2, 1, 2, 1, 2. Will your 50:50 mixture give you a higher payout than playing a pure strategy? Why or why not?
g. Economist Avinash Dixit claims that there is no better way to surprise your opponent than to surprise yourself. Suggest an easy way to randomize your play that gives you a 50:50 mix overall, but that does so in an unpredictable fashion.

Transcribed Image Text:

You One Two FingerFingers One Finger Friend Two Fingers