Political
Science 272: Problem Set 1
Spring 2005.
Due in my mailbox at 12:00 noon, Monday, February 28.
Write your answers neatly on a separate sheet of paper (attach as many sheets as necessary). Show all calculations, game diagrams, etc. Partial credit will be given for getting most of the way to an answer. Please write your name on your work and staple the pages.
1.
Strategic-form games.
Find and circle all Nash equilibria in pure strategies
and cross out all dominated strategies.
|
|
|
Column |
|
|
|
Column |
|
|
|
Column |
|
|
|
|
C |
D |
|
|
C |
D |
|
|
C |
D |
|
|
C |
5,5 |
3,3 |
|
C |
2,-2 |
3,-3 |
|
C |
2,2 |
0,4 |
|
Row |
D |
0,0 |
6,7 |
Row |
D |
1,-1 |
5,-5 |
Row |
D |
4,0 |
1,1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Column |
|
|
|
Column |
|
|
|
Column |
|
|
|
|
C |
D |
|
|
C |
D |
|
|
C |
D |
|
|
C |
3,3 |
1,5 |
|
C |
1,5 |
3,3 |
|
C |
0,1 |
1,0 |
|
Row |
D |
4,2 |
0,0 |
Row |
D |
0,0 |
2,7 |
Row |
D |
1,0 |
0,1 |
2.
Dominated strategies
Solve the
following game. The utilities in
the cells are for (Row, Column).
Circle all Nash equilibria.
Cross out all strictly dominated strategies. Hint: Eliminate as many strategies as possible
first and then iterate the elimination of dominated strategies.
|
|
1 |
2 |
3 |
4 |
5 |
6 |
|
A |
2, 4 |
4, 5 |
2, 2 |
0, 3 |
-2, -6 |
1, 1 |
|
B |
1, 3 |
0, 0 |
0, 0 |
0, 4 |
0, 0 |
0, 0 |
|
C |
-2, -6 |
1, 1 |
1, 1 |
2, 2 |
-2, 5 |
-2, 5 |
|
D |
0, 4 |
4, 1 |
0, 0 |
0, 0 |
3, 3 |
0, 0 |
|
E |
3, 3 |
0, 4 |
3, 3 |
3, 2 |
3, 2 |
-5, 4 |
|
F |
3, 3 |
3, 3 |
2, 4 |
3, 2 |
3, 3 |
2, 4 |
|
G |
1, -5 |
0, 0 |
1, 1 |
-6, 6 |
0, 4 |
1, 1 |
|
H |
3, 3 |
2, 4 |
5, 3 |
0, 0 |
1, 1 |
3, 4 |
|
I |
2, 4 |
-2, 0 |
-3, 4 |
-2, 5 |
-2, 3 |
1, 3 |
|
J |
4, 4 |
5, 3 |
3, 2 |
-3, 2 |
5, 3 |
5, 4 |
3.
Lotteries.
George W.
Bush wants to fight a war with Iraq, and prefers to do so with multilateral
support. Should he ask
the UN Security Council for a resolution authorizing the use of force? The political fallout from starting the
war will be greatly reduced, particularly for Bush’s ally Tony Blair, if there
is such a resolution. However,
asking for such a resolution and facing a rejection by the Security Council
will increase the political cost of war.
Bush estimates that the Security Council will approve a resolution with
probability p, and that the war will have a satisfactory outcome
(victory) with probability q.
Bush’s cardinal utilities for each outcome are as follows: U(Victory|Resolution)=8; U(Victory|No
request)=5; U(Victory|Rejection)=3; U(Defeat|Resolution)=-5; U(Defeat|No
request)=-7; U(Defeat|Rejection)=-10; U(Peace)=0
(a) Express Bush’s expected utility of going to war
(b) Express his expected utility of asking for a resolution
(c) How high does the probability of passing a resolution, p, have to be for Bush to take the issue to the Security Council?
(d) How high does the probability of victory, q, have to be
for Bush to initiate a war if he requests a resolution and the Security Council
rejects his request?
4. Mixed Strategy Nash Equilibrium
The numbers
in the cells are cardinal utilities for (Row, Column). Solve for the mixed strategy Nash
equilibrium.
|
|
C |
D |
|
C |
3, -3 |
-2, 2 |
|
D |
-2, 2 |
3,-3 |
5. Repeated games.
Assume that two players find themselves in a repeated
game of Prisoner’s Dilemma (i.e., DC>CC>DD>CD for each player), and
each attaches the following cardinal utilities to the respective outcomes:
U(CC)=5, U(DC)=9, U(CD)=0, U(DD)=3. The game is infinitely repeated, and each
actor discounts future payoffs, using a common discount factor, 
.
(a) Find the Nash equilibrium of the stage game (a single-play of PD).
(b) Find the discount factor necessary to prevent a player from defecting if both players play grim trigger strategies: play C in the first period; always continue to play C if the other player does likewise, and always defect if the other player has ever defected.
(c) Does this pair of strategies form a Nash equilibrium? Does it form a subgame perfect equilibrium (SPE)? Why (or why not)?
(d) Now find the discount factor necessary to prevent a player from defecting if both players play tit-for-tat strategies: play C in the first period; always play whatever the other player played in the previous period thereafter.
(e) Does
this pair of strategies form a Nash equilibrium? Does it form a subgame perfect
equilibrium? Why (or why not)?