a) This is a very simple question. Following is the illustration of finding Nash equilibria for multi-player games, only in case you don't know how to do it.
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-4 |
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2 |
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-8 |
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Player III chooses A
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-6 |
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B |
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1 |
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Player III chooses B
Finding Nash equilibria for multi-player games is no different from finding that for two-player games. The idea is still to find the best responses for each possible strategy combination of other players.
First we do that for player I. We look at the payoff for player I, and yellow the largest one on each column. They're (A,A,A), (B,B,A), (A,A,B) and (B,B,B).
For player II, we compare along the row. Looking at player II's payoff, we blue the largest one on each row. They're (A,A,A), (B,B,A), (A,A,B), (B,B,B).
For player III, we compare the two cells located at the same position on the two matrix. Again, you gray the larger one. So gray (A,A,A), (B,B,B, (A,B,A), (A,B,B), (B,A,A), (B,A,B).
Remember the idea is to identify the best responses. The cells with all three colors are Nash equilibria because they represent the best responses by each player. They are (A,A,A) and (B,B,B).
a) Consider the case when everyone votes. For any individual, knowing that everybody else are going to vote, whether he votes or not is not going to affect the election result. Thus it would not be rational for him to vote. Therefore he would not vote. By contradiction, everybody voting is not a Nash equilibrium.
b) Consider the case when nobody votes. There would be no election result. Assume this gives a payoff of 0 to everyone. Then for any individual, he will have incentives to vote because if he votes and nobody else does, the results is going to be in his favor and he would get a payoff of 9. Therefore this is not an equilibrium.
Superadditivity of a function f means that f of a union of disjoint sets must be greater than or equal to the sum of f of those sets... i.e., the whole is greater than (or equal) to the sum of its parts.
Mathematically it is:
v(A U B) >= v(A) + v(B)
a) Since this function is additive, it is superadditive.
b) This function is NOT superadditive.
c) This function is superadditive, since when you combine together groups, this
could only increase the number of pairs that know each other! (Think about the
people who know people in the other group.)
