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

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- Spring '11
- JohnPaddy
- Game Theory, Nash, b-c b-c b-c