Transcribed Text
Remarks: The problem set contains 6 exercises. Always motivate your solutions! It is
often helpful to look up the way we solved similar problems in our lecture notes.
The maximum number of points you can achieve in this problem set is 120 points. To get
the full score of 30% for your final course grade, however, you need only 100 points. All
points above the total of 200 points needed from both problem sets that you achieve will
count already for your final exam (if you need some or all of these extra points, because
you may get full points in the final anyway). For example, if you score 115 points in the
first problem set and 105 points in the second problem set, thus 220 points in total, you
have already achieved 20 out of 100 points of the final exam.
1) A governor needs to hire a bureaucrat to evaluate an issue. With probability p the
bureaucrat has the same ideology as the governor and, if hired, prefers making an
effort to evaluate the issue over not making an effort. And, with probability 1-p both
players have different ideologies and, if hired, the bureaucrat prefers making no
effort to shed light on the issue. The governor's preferences are known to both
players.
a)
What are the pure Nash equilibria and the subgame-perfect equilibria of the two
extensive games if the governor knows with certainty that the bureaucrat has the
same ideology as herself (i.e., p = 1; the extensive game on the left hand side) and
if the governor knows with certainty that the bureaucrat has a different ideology
(i.e., p = 0; the extensive game on the right hand side)?
[15 points]
Governor and bureaucrat have the same ideology.
Governor and bureaucrat have different ideologies.
Governor
Governor
Hire
Not hire
Hire
Not hire
Bureaucrat
Bureaucrat
0,0
0,0
Effort
No effort
Effort
No effort
4,4
-2,-2
4,2
-2,4
b) Draw an extensive game that includes "nature" as the first player, where nature
determines with probability p that the bureaucrat has the same ideology as the
governor and with probability 1-p that the bureaucrat has a different ideology
than the governor. Whereas the bureaucrat knows her own ideology, the governor
does not know the ideology of the bureaucrat. Find all sequential equilibria (i.e.,
beliefs and optimal strategies) of this extensive game!
[15 points]
1
2) Consider an ambitious party member who decides on whether or not to challenge the
party leader's predictions about an important issue with uncertain outcomes
("nature" randomly determines whether the leader's prediction is correct, LC, or
incorrect, LI, where each outcome occurs with probability 1/2). If the challenger's
prediction turns out to be true, her reputation within the party will increase by 1. But
if her prediction turns out to be wrong, her reputation will decrease by 1. Finally, if
she does not challenge the leader her reputation remains unchanged.
"Nature"
p(LC)= = 1/2 p(LI)=1/2
Challenge
-1
1
Party
member
Don't
challenge
a) What are the expected payoffs of the ambitious party member for each of her
actions "challenge' and don't challenge"? Will she challenge the leader?
[5 points]
b) Assume that the ambitious party member has the opportunity to search for more
information about the issue's outcome. The search costs are equal to 0.1.
Specifically, if she searches she will either receive a signal that "the leader's
prediction is correct" or that "the leader's prediction is incorrect". Her signal is
true with probability .75 and it is not true with probability .25. Given the search
costs, will she search for more information? [15 points]
3) Consider the following 3-player step-level public goods game. Each player i can
make up to two units of costly effort, or ei € {0,1,2}. If at least k = 2 units of effort
are made, irrespective of who of the players makes these efforts, the public good is
provided and each player gets a revenue of 3. If less than k = 2 units of effort are
made the public good is not provided and each player gets a revenue of 0. The payoff
function for each i is given by
3
if
>
k
=
2
IT; = + 0 if > j=1,2,3 ej ej <k =="" 2.<br="">j=1,2,3
a) Describe all pure strategy Nash equilibria of the game!
[10 points]
b) What are the individual payoffs of each player in each of the pure strategy Nash
equilibria? And what are the respective aggregate (or group) payoffs? [10 points]
c) What is the socially optimal outcome (i.e., the highest possible aggregate payoff)
when only pure strategies are allowed? Is the social optimum achieved by any of
the pure strategy Nash equilibria? Is the social optimum 'fair' in the sense that all
players have the same individual payoff?
[5 points]
2
</k>5) In the extensive game with imperfect information of the Prisoner's Dilemma (PD),
"nature" first chooses a "pro-social' type of player 1 with probability p and a "selfish'
type of player 1 with probability 1- p. Player 1 can observe her own type and knows
that player 2 is 'pro-social'. On the other hand, player 2 knows her own type 'pro-
social' but is not certain about player l's type. Find all sequential equilibria
(consisting of beliefs and actions) of this game!
[15 points]
Nature
p
1-p
Player 1:
Player 1:
pro-social type
selfish type
C
D
C
D
Pl. 2
Pl. 2
C
D
C
D
C
D
C
D
3,3
0,1
1,0
2,2
2,3
0,1
3,0
1,2
6) Firm A wants to acquire Firm T. Firm A is not informed about Firm T's exact value,
X. However, Firm A knows that X is uniformly distributed on the real line [0,100],
that is, the minimum value is 0, the maximum value is 100, and all values from 0 to
100 are equally likely to occur. Under the management of Firm A the value of Firm T
is Mx, where M 1. Firm A makes a bid, b E [0,100], which Firm T either accepts
or rejects. If the bid is accepted, Firm T earns b and Firm A earns Mx - b. If the bid
is rejected, Firm T earns X and Firm A earns 0. Determine the threshold M such that
Firm A's expected payoff for any b > 0 is strictly positive for M > M and strictly
negative for M < M.
[20 points]
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