Final Exam of the Game Theory
Dec. 19 2022
Note 1: The notations without extra guidance in this exam are consistent with those in class.
Note 2: All answers must be explained, or no points will be given.
1. Signaling games
There are a frog and a princess. The frog could either say he was a ’prince’ or a ’frog.’
The princess could either kiss the frog, in which case he might turn into a prince. Or, she
could eat him. It was well known that 10% of the frogs in the kingdom were actually princes
who had a spell cast upon them. Only frogs that were actually princes would turn into
princes when kissed by a princess. Frogs like to kiss princesses and don’t like to be eaten.
Frogs who are really princes especially like to kiss princesses (because then frogs turn into
princes). Frogs who are not really princes cannot say ’Prince’ without taking lessons in
elocution. Princesses like to eat frogs, but they prefer to kiss frogs who are really princes.
Princesses don’t like to kiss frogs who are not princes. Payoffs that are consistent with these
facts are shown below.
(10,100)
Frog
kiss
kiss
(10,100)
eat
(-10,5)
Say ’Frog’ Say ’Prince’
(-10,5)
eat
0.1 True Prince
Princess
Chance
Princess
0.9 True Frog
(10,-10)
kiss
kiss
(5,-10)
eat
(-10,5)
Say ’Frog’ Say ’Prince’
(0,5)
eat
Frog
a. Is this a signaling game? If so, who are the sender, the receiver, the signals? Can you tell
whether the signals are costly to some types of senders? (10 pts)
b. Please whether there are any separating equilibria. (10 pts)
c. Please whether there are any pooling equilibria. (10 pts)
2. Common valuations
• players: {1, · · · , n}
1
• states: (t1 , · · · , tn ) of the signals that the players may receive
• actions: each player’s set of actions in the set of the possible bids
• signals: τi (t1 , · · · , tn ) = ti
• beliefs: everyone’s signal is independent
• pay-offs:
ui (b, (t1 , · · · , tn )) =
{
gi (t1 ,··· ,tn )−P (b)
m
if
0
bj ≤ bi ∀j ̸= i, bj = bi for m players
bj > bi for some j ̸= i
a. Show that when α = γ = 1, for any value of λ > 0 the second-price sealed-bid auction
has an (asymmetric) Nash equilibrium in which each Type t1 of Player 1 bids (1 + λ)t1 and
each Type t2 of Player 2 bids (1 + 1/λ)t2 . (10 pts)
b. Show that the first-price sealed-bid auction has a symmetric Nash equilibrium in which
each Type ti of Player i bids (α + γ)ti /2. (10 pts)
3. Reporting a crime
A crime is observed by a group of n people. Each person would like the police to be
informed, but prefers that someone else make the phone call. Specifically, suppose that each
person attaches the value v to the police being informed and bears the cost c if she makes
the phone call, where v > c > 0.
a. Define the game. (5 pts)
b. Find the symmetric mixed Nash equilibrium, and show that the probability that each
person reports a crime is decreasing in the size of the group. (8 pts)
c. When the size of group become larger, does the probability that a crime is reported
become larger or smaller? (6 pts)
4. A centipede game
Below is a six-period centipede game.
2
a. Find the subgame perfect equilibrium. (5 pts)
b. Describe the experiment result in the textbook. Does the experiment result fit the
prediction of the game theory? (6 pts)
5.
a. Solve the equilibrium of the above Bayesian game. (10 pts)
b. Solve the equilibrium if Player 2 is informed. (10 pts)
c. Does more information hurt in this case? (5 pts)
3