Problem Set 5
Economics 703
Spring 2016
Due: Friday February 26
1. In each of the two cases below, we have a game with two players, each of whom has
two types. There is no “state of nature” in this problem, so beliefs are entirely about the
type of the other player. The types of player 1 are T1 = {a, b}. The types of player 2 are
T2 = {c, d}.
a) Suppose the beliefs of the types are p1 (c | a) = 1/4, p1 (c | b) = 3/4, p2 (a | c) = 1/4,
and p2 (a | d) = 3/4. Show that a common prior exists.
b) Now suppose the beliefs are p1 (c | a) = 1/2, p1 (c | b) = 1/2, p2 (a | c) = 1/3, and
p2 (a | d) = 2/3. Show that there cannot be a common prior.
2. Consider the following game of incomplete information. The two players are either
playing the game G1 which is
b1
b2
a1
3,0
0,1
a2
0,0
3,1
a3
2,0
2,1
b1
b2
a1
3,0
0,1
a2
0,1
3,0
a3
2,2
2,0
or the game G2 which is
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a) Assume that player 2 knows which game they are playing, but player 1 does not.
Instead, player 1 views each game as equally likely. Write this as a game of incomplete
information. Find all pure strategy Bayes–Nash equilibria.
b) Now suppose that player 2 knows which game they are playing but doesn’t know
whether or not player 1 knows. More specifically, the common prior is that 2 always
knows the game, while player 1 knows the game being played with probability 1/2. If
player 1 does not know the game being played, he views the two games as equally likely.
Write this as a game of incomplete information. Find all pure strategy Bayes–Nash
equilibria.
3. Consider the following signaling game. There are two types, t1 and t2 , where the prior
probability of t1 is p < 1/2. The message set is {m1 , m2 }, the action set {a1 , a2 }. The
payoffs when player 1 is type t1 are
a1
a2
m1 5, 0 −1, 1
m2 0, 0 −1, 1
while the payoffs when player 1 is type t2 are
a1
a2
m1 5, 1 −1, 0
m2 0, 1 −1, 0
(Note that these tables are intended to be a simple way to summarize the payoffs that
go at the terminal nodes of the game tree — this is not a normal form of a game!)
Find all pure strategy weak perfect Bayesian equilibria. Are some of the equilibria more
“intuitively reasonable” than others?
4. Another signaling game with two types and the same notation as in question 1. Again,
the probability that player 1 is type 1 is p < 1/2. The payoffs when player 1 is type t1
are
a1
a2
m1 0, 1
1, 0
m2 −4, 1 −4, 1
while the payoffs when player 1 is type t2 are
m1
m2
a1
a2
3, 0 1, 1
3, 1 −4, 0
Find all pure strategy weak perfect Bayesian equilibria. Are some of the equilibria more
“intuitively reasonable” than others?
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