课程名称︰赛局与讯息
课程性质︰选修
课程教师︰蔡崇圣
开课学院：社会科学院
开课系所︰经济学系
考试日期（年月日）︰2024/4/10
考试时限（分钟）：130
试题 :
1. [20%] Consider the game with the following extensive form:
3,0
X／
2 ／
●
／︰＼
C／ ︰ Y＼
1 ／ ︰ 7,5
● ︰
／ ＼ ︰ 4,6
A／ D＼ ︰ X／
1 ／ ＼︰／
● ●
＼ 2 ＼
B＼ Y＼
＼ 2,1
6,4
a. [3%] Draw the normal form of this game
b. [3%] Find the set of rationalizable strategies.
c. [6%] Find all the Nash equilibria.
d. [8%] Find all the subgame perfect Nash equilibria (SPNE). Is there any
Nash equilibrium that is not a SPNE? If so, explain the reason for that.
2. [20%] Consider the following normal-form game (x > 0):
2
L M N
┌───┬───┬───┐
U │ 5,3 │ 4,2 │ 2,1 │
├───┼───┼───┤
1 C │ x,3 │ 3,1 │ 0,6 │
├───┼───┼───┤
D │ 7,x │ 0,5 │ 1,4 │
└───┴───┴───┘
The value of x is such that "C" is a dominated strategy.
a. [3%] Find the range of x.
b. [3%] After imposing the condition you found in a., find the set of
rationalizable strategies.
c. [14%] Find all the Nash equilibria, including the pure- and mixed-strategy
ones. Denote the probability that player 1 plays U by p, and the probability
that player 2 plays L by q. In each Nash equilibrium, describe the best
response functions and draw them on the (p,q) coordinates. Indicate the Nash
equilibrium on the figure.
3. [20%] Consider a game between a government (player 1) and a criminal (player
2). The government makes an effort to stop the crime, x ∈ [0,1], which is
also the probability that the criminal is caught. The criminal chooses the
level of crime, y≧0. If the criminal is caught (with probability x), there
is a penalty with a rate f > 0 imposed on the crime level, resulting in a
payment of fy paid by the criminal to the government. If the criminal is at
large (with probability 1-x), the he/she gains y, which causes a social cost
-y to the government. Therefore, the government's payoff is:
u_1 = xfy - (1 - x)y - k_1‧x^2
where k_1‧x^2 is the effort cost incurred by the government,
and the criminal's payoff is:
u_2 = -xfy + (1 - x)y - k_2‧y^2
where k_2‧y^2 is the cost of committing y level of crime.
They make the decisions simultaneously.
a. [7%] Find the best response function for each player. Draw a figure to
show both of them.
b. [5%] What is the Nash equilibrium (x*,y*)?
c. [8%] Determine whether each of the following statements is true or false
by showing the changes of the best response functions and the
equilibrium. Explain your finding.
(i) Raising the penalty rate f can effectively reduce the crime level.
(ii) The more difficult for the government to stop a crime (a higher k_1),
the higher the crime level is.
4. [20%] Consider the following alternating-offer bargaining game with three
periods. Two players, 1 and 2 want to divide $100. The game proceeds as
follows:
Period 1: Player 1 first makes a proposal, (x,100 - x), which means that
player 1 gets x and player 2 gets 100 - x. If player 2 accepts the
proposal, then each player obtains what is proposed. If player 2 rejects
it, they continue to the next period.
Period 2: Player 2 makes another proposal, (y,100 - y). If player 1 accepts
it, then each player obtains what is proposed. If player 1 rejects it,
they continue to next period.
Period 3: Player 1 makes the final proposal, (z,100 - z). If player 2
accepts it, then each player obtains what is proposed. If player 2 rejects
it, the game ends and they split $100 equally, which means that they each
obtain $50.
There is a discount factor δ_i ∈ (0,1) for player i between any two
periods.
a. [3%] Draw the extensive form of this game.
b. [13%] Fully characterize the SPNE of this game. In which period will the
proposal pass? How much does each player obtain in the equilibrium?
c. [4%] Discuss the effect of the level of "patience" on the players'
equilibrium payoffs. More specifically, when δ_1 increases (given δ_2),
who will get more and who will get less? When δ_2 increases (given δ_1)
, who will get more and who will get less? Explain your finding.
5. [20%] Consider the lawsuit game taught in class with some modification. A
plaintiff (player 1, she) first decides whether or not to bring a lawsuit
against a defendant (player 2, he). If no lawsuit is filed, then they both
obtain 0. If a lawsuit is filed, then it costs c to the plaintiff. She then
makes a settlement offer of s > 0, and the defendant decides whether to
accept it or to reject it. If the defendant accepts the offer, then he
needs to pay s to the plaintiff. If the defendant rejects the offer, the
plaintiff then decides either to give up or to go to trial. If the plaintiff
gives up, then no money transfer is made (although the plaintiff still needs
to pay c). If the case goes to trial, the plaintiff receives a compensation
from the defendant with an amount x if she wins, while she receives nothing
if she loses. It is believed that the plaintiff will win the case with
probability γ and lose with probability 1 - γ. Moreover, during the trial,
the plaintiff incurs an attorney fee k_p, and the defendant incurs an
attorney fee k_d. However, after the outcome of the trial is determined, the
loser needs to pay the attorney fees for both players. That is, if the
plaintiff wins the lawsuit, the defendant needs to pay the fees k_p + k_d and
the plaintiff does not pay any fee; while if the defendant wins, the
plaintiff needs to pay the fees k_p + k_d while the defendant does not pay
any fee.
a. [3%] Draw the extensive form of this game.
b. [11%] Fully characterize the SPNE of the game. In particular, describe
the conditions for the equilibrium where a positive amount of settlement is
made and accepted, and those for the equilibrium where no lawsuit is filed.
c. [6%] Determine each of the following statements is true or false.
Explain your finding.
(i) The higher probability that the plaintiff wins the case (γ is higher
), the more likely that the lawsuit is filed and a settlement is made in
the equilibrium.
(ii) The attorney fee for the plaintiff k_p does not affect the amount of
settlement (if it takes place in the equilibrium).