Answer. One example of such an equilibrium (note that the Player 2’s behavior off-the-equilibrium path is not essential) is displayed below. 1 2 3 n n n n n n n y y y y y y y 0 0 0 0 0 -c 0 -c 0 -c 0 0 b b-c b-c b-c b b-c b-c b-c b b-c b-c b-c Note that the following is also an equilibrium: 1 2 3 n n n n n n n y y y y y y y 0 0 0 0 0 -c 0 -c 0 -c 0 0 b b-c b-c b-c b b-c b-c b-c b b-c b-c b-c 2. Simultaneous and Strategic Turnout. Suppose that three voters are faced with the dilemma of voting for a tax cut. Each voter has a simple choice, since they all prefer that taxes be cut – the only difficulty is that turning out to vote is somewhat costly. Specifically, each voter receives a benefit of b from the tax cut being passed and incurs a cost of c < b from turning out to vote for the tax cut. All voters make their decision privately and simultaneously. (a) Express this as a normal form game. Answer. A V A V A (0, 0, 0) (b,b-c,b) V (b-c,b,b) (b-c,b-c,b) A V A (b,b,b-c) (b,b-c,b-c) V (b-c,b,b-c) (b-c,b-c,b-c) Player 1 chooses Row, Player 2 chooses Column, Player 3 chooses Table 2
(b) Derive all pure strategy Nash equilibria. Answer. Every pure strategy Nash equilibrium involves exactly one player voting and the others abstaining. Verification of this is straight-forward. (c) Derive a mixed strategy Nash equilibrium (i.e., an equilibrium in which at least one player is playing both actions with positive probability). Answer. A mixed strategy Nash equilibrium must involve at least two players mixing with non-zero probability on both actions. I will solve for the symmetric mixed strategy equilibrium. To make this a bit simpler, let the players be denoted by i, j , and k . (This makes the symmetry useful in the derivations.) Let the mixed strategy
You've reached the end of your free preview.
Want to read all 4 pages?
- Spring '11
- Game Theory, Nash, b-c b-c b-c