Columbia University Department of Economics
S4415D Game Theory, Problem Set 2
Due Monday June 5th in class. Q1.
Consider the following extensive form game between 3 players.
Z

4, 4, 0 Y

2, 5, 1 F

1, 7, 0 E

3, 0, 1 P

4, 4, 0 O

2, 5, 1 D

1, 7, 0 C

3, 0, 1 3
N
2
M B 1 1 2 A X
3
W
1 (a) List all pure strategies for each player.
(b) Use the game tree to find a backward induction solution for the game. Is it unique? What
sre its payoffs?
(c) Find one Nash equilibrium for the game which is not a backward induction solution. (You
are not required to write out the associated normal form, but of course can if you wish).
(d) How many subgames does the game have? Is the backwards induction solution you found in
(b) subgame perfect? Is the Nash equilibrium you found in (c) subgame perfect?
1 Q2.
Consider the following ‘Brooklyn Bridge’ game, except that this time, Player 1 chooses her
location (Brooklyn or Manhattan) before Player 2 does, so Player 2 observes Player 1’s choice
before choosing his own. The payoffs for the four eventual outcomes are listed in the table below.
1\2
B
M B
5, 3
0, 0 M
1, 2
4, 5 (a) Draw a game tree for the game, being careful to label all the players, actions, and payoffs.
(b) Use the game tree to find a backward induction solution for the game. Is it unique? Is it
subgame perfect?
(c) Write out the associated normal form, and find a Nash equilibrium for the game which is
not a backwards induction solution. Explain briefly why it is still a Nash equilibrium.
(d) Draw a game tree for the version of this game in which the players choose simultaneously.
Find the mixed strategy Nash equilibrium. What are the expected payoffs for each player in
this equilibrium? Q3.
Consider a classic ‘centipede game’ which works as follows:
Two players, Alan and Bella, start with a pot of value 1. Alan must choose whether to take
the pot (strategy T), in which case he gets the payoff of 1 while Bella gets nothing, or to pass
the pot to Bella (strategy P). If he passes the pot, its value increases to 2 and it becomes Bella’s
turn to choose - she can either take the pot (T), in which case she gets the payoff of 2 while Alan
gets nothing, or she can pass the pot to Alan (P). If she passes the pot to Alan, the value of the
pot increases to 3, and it becomes Alan’s turn again. Each time the pot is passed, it increases
in value by 1 and the game continues. The game continues until either a player takes the pot at which point the player taking it receives its value and the other player receives nothing, and
the game ends - or the value of the pot exceeds 100, in which case both players receive a payoff
of 0 and the game ends.
(a) Draw a partial game tree for this problem, indicating the first few nodes and the last few
nodes. Feel free to omit nodes in the middle, indicating where you’ve skipped them with an
ellipsis (...).
(b) Describe one of Player 1’s pure strategies. (You don’t have to write it out in full if it’s very
long).
(c) What is the subgame-perfect Nash equilibrium for this game? What are the equilibrium
payoffs?
(d) Are there any Nash equilibria that are not subgame perfect? (Note: it’s completely impractical to draw the associated normal form here, so don’t try). 2 Q4.
Consider the following sequential game between two players: In the first stage, Player 1 moves
and chooses between action A and action B. This move is observed by both the players. In the
second stage, the two players play a simultaneous-move game. If player 1 chose A in the first
stage, then the two players play Game A in the second. If player 1 chose B in the first stage,
then the two players play Game B in the second. The payoffs of Game A and B are summarized
below:
Game B: Game A: Player 1 UA
DA Player 2
LA
RA
10, 5 0, 0
0, 0 5, 10 Player 1 UB
DB Player 2
LB
RB
k, k
0, 0
0, 0 −1, −1 where k > 0.
(a) Find all Nash equilibria in Game A.1
(b) Find all Nash equilibria in Game B.
(c) Draw the extensive-form game tree. Make sure you include all multi-node information sets,
players, actions, and payoffs.
(d) How many pure strategies does player 2 have in the complete sequential game?
(e) How many pure strategies does player 1 have in the sequential game?
(f) For all values of k > 0, find all the Subgame Perfect Nash Equilibrium.2 Q5.
A landlord owns a√farm and hires a worker. The output of the farm as a function of the worker’s
effort level, e, is e. The landlord can not directly observe the worker’s effort, but she does
get to write a contract ahead of time, specifying the share α of output that will be kept by the
worker. After observing the contract, the worker gets to choose his effort level,
√ e. Effort is costly
to the worker. Given α and e, the landlord’s utility is vL (α,
e)
=
(1
−
α)
e (the output less
√
the worker’s share), and the worker’s utility is vW (α, e) = α e − e (his share of output minus a
linear effort cost). Assume α ∈ [0, 1] and e ∈ [0, 1].
(a) Is this extensive form game one of perfect information? Is it a finite game?
(b) Use backward induction to find the level of α the landlord will set, and the effort level e that
will occur on the equilibrium path.
(c) Suppose that a social planner can choose e. Suppose the planner aims to maximize total
utility vL (α, e) + vW (α, e). What level of e will the social planner choose?
(d) Suppose now the social planner still wants to maximize total utility, but she cannot specify
e (perhaps because she too cannot observe effort). Instead, the social planner only gets to
set α. What level of α will the social planner set? (Hint: use backward induction again.)
1 Don’t forget about mixed strategies!
that this involves finding different Nash equilibria for different values of k, not one NE that holds for all
values of k > 0.
2 Note 3

 

SuperTutor

(15)

Status NEW Posted 05 Jun 2017 06:06 AM My Price 20.00

-----------

Attachments

file 1496644840-Solutions file.docx preview (51 words )

S-----------olu-----------tio-----------ns -----------fil-----------e -----------Hel-----------lo -----------Sir-----------/Ma-----------dam----------- T-----------han-----------k y-----------ou -----------for----------- yo-----------ur -----------int-----------ere-----------st -----------and----------- bu-----------yin-----------g m-----------y p-----------ost-----------ed -----------sol-----------uti-----------on.----------- Pl-----------eas-----------e p-----------ing----------- me----------- on----------- ch-----------at -----------I a-----------m o-----------nli-----------ne -----------or -----------inb-----------ox -----------me -----------a m-----------ess-----------age----------- I -----------wil-----------l b-----------e q-----------uic-----------kly----------- on-----------lin-----------e a-----------nd -----------giv-----------e y-----------ou -----------exa-----------ct -----------fil-----------e a-----------nd -----------the----------- sa-----------me -----------fil-----------e i-----------s a-----------lso----------- se-----------nt -----------to -----------you-----------r e-----------mai-----------l t-----------hat----------- is----------- re-----------gis-----------ter-----------ed -----------onÂ----------- th-----------is -----------web-----------sit-----------e -----------Tha-----------nk -----------you----------- -----------

Not Rated(0)

• Buy Answer