1 answer one example of such an equilibrium note that

1

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