1. You and a friend are in an Italian restaurant and the owner offers both of yo

Question

1. You and a friend are in an Italian restaurant and the owner offers both of you an 8- slice pizza under the following conditions: each one of you must simultaneously announce how many slices you would like to have (i.e., each player i is belong {1,2} names her desired amount of pizza slices 0 <= s(i )<= 8 ). If s1 + s2 <= 8 the players get their demands. (The owner eats any left over pizza). If  s 1 + s 2 >8 then the players get nothing. Assume that more pizza is better for both players.

Part a. Assigning utility indices for each player according to the number of slices of pizza that are both requested and received (utility being set to zero in all other situations), write the payoff matrix.

Part b. All strategies that survive iterated elimination of strictly dominated strategies (in two-player games) are said to be rationalizable strategies. Delineate the set of rationalizable strategies, explaining briefly.

Part c. How would you describe the general nature of the best responses that each player should choose?

Part d. Identify any and all pure strategy Nash equilibria for this game. Explain your reasoning.

Explanation / Answer

1.

2. A strictly dominated strategy for either player would be choosing zero slices, since there is no pay off for this. Also, knowing that your friend would choose at least one slice, you should not choose 8, since this will give no pizza and thus no pay off. Your friend should also not choose 8 slices for the same reason. These are the only strictly dominated strategies. All other choices have the possibility of providing some pay out, depending on what your friend chooses. For instance, you choosing only 1 slice is worse for you than 2 slices, except in the instance where your friend chooses 7, in which case you both receive no pizza rather than you only getting one.

The set of rationalizable strategies is then as follows

3. In general the player should chose a number of slices that is equal to 8 minus the number of slices he thinks the other person will choose. If he chooses more, then neither of them get pizza, but if less is chosen, then at least you both get pizza. Obviously the optimal choice is for you each to choose 4 peices of pizza, since neither of you can do any better without making the other worse off.

4. The Nash equilibria exist on the diagonal above the first 0,0 entry in each row/column, or where the payoffs add up to exactly 8. Here if either player unilaterally chooses fewer slices, they are worse off, and similarly, if they unilaterally choose more, they would be worse off again, so there is no incentive to change their choice.

Friend








0 1 2 3 4 5 6 7 8 You 0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
1 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 0,0
2 2,0 2,1 2,2 2,3 2,4 2,5 2,6 0,0 0,0
3 3,0 3,1 3,2 3,3 3,4 3,5 0,0 0,0 0,0
4 4,0 4,1 4,2 4,3 4,4 0,0 0,0 0,0 0,0
5 5,0 5,1 5,2 5,3 0,0 0,0 0,0 0,0 0,0
6 6,0 6,1 6,2 0,0 0,0 0,0 0,0 0,0 0,0
7 7,0 7,1 0,0 0,0 0,0 0,0 0,0 0,0 0,0
8 8,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0

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