Sample Midterm Questions
EC 703
Spring 2016
1. (a) Find all pure strategy Nash and subgame perfect equilibria of the game shown in
Figure 1 below.
1
b
a
2
c
4
1
2
d
c
0
0
5
0
d
1
1
Figure 1: Question 1(a)
(b) Now suppose that there is a small probability, p ∈ (0, 1), that 2 does not observe
1’s move correctly. Suppose
we model this by a move by Nature as shown in Figure
Figure 1
2 below. In particular, if 1 chooses a, Nature “tells” player 2 the correct action (i.e.,
N chooses A) with probability 1 − p and “lies” (chooses B) otherwise. Analogously, 2
sees B with probability 1 − p if 1 chooses b and sees A otherwise. Payoffs depend on
1’s actual action and 2’s action, not the observation by 2. Find all pure strategy weak
perfect Bayesian equilibria.
1
4,1
c
2
2
A
N
B
1-p
d
4,1
c
p
d
a
0,0
0,0
1
5,0
b
c
d
c
A
B
p
1-p
N
5,0
d
1,1
1,1
Figure 2: Question 1(b)
Figure 2
2. This problem is a dynamic extension of a problem from Problem Set 2. Part (a)
repeats that problem.
(a) We have N agents playing a game. Each agent simultaneously chooses whether
to contribute or not to a public good. If no agent contributes, all agents get a payoff of
0. If at least one agent contributes, then the public good is provided. In this case, every
contributor gets a payoff of 1 − c, while every agent who doesn’t contribute gets a payoff
of 1. Assume 0 < c < 1. Find the symmetric mixed strategy equilibrium — that is, the
equilibrium where all players use the same randomization.
(b) Now consider a two period version of the same model. More specifically, the
game now works as follows. In period 1, every agent simultaneously chooses whether to
contribute or not. If at least one player contributes in the first period, the public good is
provided and the game ends with payoffs of 1 − c for every player who contributes and 1
for every player who doesn’t contribute. Unlike the previous case, if no one contributes,
we move to period 2. In period 2, again, all agents simultaneously decide whether to
contribute or not with payoffs as in part (a). Assume players discount the future, so a
payoff of, say u, in the second period is worth δu where 0 < δ < 1. Find the symmetric
mixed subgame perfect equilibrium. In which period does the mixed strategy put more
probability on contributing?
(c) Extend (b) to the case of 3 periods. How would your answer change with more
than 3 periods?
3. Consider the following signaling game. Player 1 has two possible types, t1 and t2 .
The prior probability of t1 is 3/4. Player 1 has two possible messages m1 and m2 , while
player 2 has three possible actions a1 , a2 , and a3 . The payoffs are as follows. If player 1
2
is of type t1 , the payoffs are
m1
m2
a1
a2
a3
2, 3 −2, 2 1, 0
0, 4 −2, 1 1, 0
while if player 1 is type t2 , the payoffs are
a1
a2
a3
m1 −1, 0 −2, 2 −3, 3
m2 0, 0 −2, 1 1, 4
The payoff to player 1 is the first one. As usual, this is not the normal form but a summary
of the payoffs in the tree. Find all pure strategy weak perfect Bayesian equilibria.
3