Game Theory - End of Chapter Problem 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. You One Finger Two Fingers One Finger Friend Two Fingers b. Identify the pure-strategy Nash equilibria, if any, in this game, ordered as (Friend's Choice, Your Choice). A Nash equilibrium occurs at O (One Finger, One Finger). O (One Finger, One Finger) and (One Finger, Two Fingers). (One Finger, One Finger) and (Two Fingers, Two Fingers). none of the combinations. c. If you always play One Finger, your friend will . On average, playing this strategy, you can expect to win . On average, d. If you always play Two Fingers, your friend will playing this strategy, you can expect to win e. If you randomly mix One Finger and Two Fingers 50:50, and your friend does the same, you can expect to emerge victorious of the time. On average, playing this strategy, you can expect to win f. Suppose your day-by-day mixture is as follows: 1, 2, 1, 2, 1, 2, 1, 2. If your friend knows the pattern of your moves, this 50:50 mixture will give you playing a pure strategy. g. Economist Avinash Dixit claims that there is no better way to surprise your opponent than to surprise yourself. Which of the following are ways of randomizing your play that would give you a 50:50 mix overall but in an unpredictable fashion? You could base your moves on a random number generator, and an odd number means you play One Finger, whereas an even number means you play Two Fingers. flip a coin, and if you get heads, you play One Finger, and if you get tails, you play Two Fingers. write down a sequence of moves in advance of the game, and follow the sequence exactly as you play