ECON3014: Game Theory. A¢ liate Exam, December 2007.
Answers
1.
L C R
U 1,-2 -2,1 0,0
M -2,1 1,-2 0,0
D 0,0 0,0 1,1
a) (D,R) is a pure strategy Nash equilibrium (you need to show why).
(U,L) is not a NE since player 2 can do better by deviating to C. Similarly, for
every other pure strategy pro…le, some player can do better by deviating.
b) Let p be the probability that U is played, let M be played with prob
(1 p): For 2 to be indi¤erent between L and R,
2p + (1 p) = p 2(1 p) (1)
1
p= (2)
2
So the payo¤ of 2 in this candidate equilibrium is 0:5 < 0: But then playing
R is better since it gives a payo¤ of 0. So there is no such mixed equilibrium
where both players randomize across their …rst two strategies.
s
c) Solving for 2’ optimal strategy. must play C after U, L after M and R
after D. So 1 chooses R:
4. Suppose that each bidder bids a constant fraction of his valuation. Let
s
w be my opponent’ valuation, and v be my own. My expected payo¤ from
bidding b given v is
Z b=
U (b; v) = [v 0:5b 0:5 w] dw (3)
0
b=
w2
= vw 0:5bw (4)
4 0
vb 0:5b2 0:5b2
U (b; v) = (5)
2
di¤erentiate to get …rst order condition for b :
2
v b= (6)
3
2
So a symmetric equilibrium has b = 3 v for both bidders.
1
2. Two people select a policy that a¤ects them both by alternately vetoing
policies until only one remains. First person 1 vetoes a policy. If more than
one policy remains, person 2 then vetoes a policy. If more than one policy still
remains, person 1 then vetoes another policy. The process continues until only
one policy has not been vetoed.
a) Suppose there are three possible policies, X, Y, and Z. Person 1 prefers
X to Y to Z, and person 2 prefers Z to Y to X.
In stage 2, 2 will choose his preferred option. So if 1 vetoes X, the outcome
is Z, and if he vetoes Y the outcome is Z, and if he vetoes Z the outcome is Y.
So the subgame perfect equilibrium has one vetoing Z.
b) Suppose that there are three possible policies, X, Y, and Z. Person 1
prefers X to Y to Z, and person 2 prefers Y to X to Z. Find the subgame perfect
equilibrium.
Now if 1 vetoes X, the outcome is Y, and if he vetoes Y the outcome is X,
and if he vetoes Z the outcome is Y. So the subgame perfect equilibrium has
one vetoing Y.
3. Regardless of ; if player 2 chooses L; 1 must choose the strategy (B,T).
If 2 chooses R, 1 must choose (T; B) : So these are the candidate equilibria.
If 1 chooses (B,T),
supppose that 1 chooses BT.,
s
2’ payo¤ from L is
u2 (L; BT ) = 1 + 4(1 ) (7)
while his payo¤ from R isf
u2 (R; BT ) = 0 + 0(1 ) = 0:; (8)
So the payo¤ from L is greater than payo¤ from R for every between 0
and 1.
So regardless of ; there is a pure strategy equilibrium: (B,T) for player 1,
L for player 2.
Now let us consider an equilibrium where 1 plays (T,B).
u2 (L; T B) = 2 + 0(1 ); (9)
u2 (R; T B) = 3 + 2(1 ) (10)
Since u2 (R; T B) u2 (L; T B) for every 2 [0; 1]; there is a pure strategy
equilibrium where 1 plays (T,B) and 2 plays R, for every value of :
2