PSC/ECO 288: Game Theory
Spring 2011
Answer Key: Assignment #1
1. Recall the game
Player 1
U
UM
DM
D
Player
L
C
9, 1 2, 3
2, 2 6, 5
4, 5 3, 0
5, 6 4, 6
2
R
7, 2
7, 3
6, 4
8, 8
(a) D strictly dominates DM .
u1 (D, L) = 5 > 4 = u1 (DM, L)
u1 (D, C ) = 4 > 3
PSC/ECO 288: Game Theory
Spring 2011
Answer Key: Assignment #6
1. (a) Here is the game in extensive form.
Nature
strong ()
weak (1 )
1
f
y
f
y
2
f
y
1, 1
2
f
1, 0
y
0, 1
f
0, 0
y
1, 1
f
1, 0
0, 1
Note that = p(strong), f = ght, and y = yield.
The game usi
PSC/ECO 288: Game Theory
Spring 2011
Assignment #5: Due in class Wednesday, March 23
Name :
1. (6 points) There are two urns, one with m 1 red balls and the
other with n 1 white balls. Two players play a game of perfect
information. Whenever a player gets
PSC/ECO 288: Game Theory
Spring 2011
Assignment #7: Due in class Wednesday, April 13
Name :
1. (25 points) Do exercise 319.3 from the book. (Look for weakly
sequential equilibria, not Nash. Draw the extensive form. Assume k > 0.)
1
PSC/ECO 288: Game Theory
Spring 2011
Answer key to assignment #7
1. Assume k > 0. The extensive form can be drawn as follows:
Nature
H (.5)
L(.5)
1
1
see
raise
raise
pass
1, 1
see
1
2
pass
meet
meet
1, 1
1, 1 1 + k, (1 + k) 1, 1 (1 + k), 1 + k
A weakly se
PSC/ECO 288: Game Theory
Spring 2011
Answer Key for Assignment #8
For the following, an equilibrium will be stated in the following order
[(x, y ), (z, w), U , D ]. Here x denotes the action of the S type of player 1,
y denotes the action of the W type of
PSC/ECO 288: Game Theory
Spring 2011
Assignment #4: Due in class Wednesday, March 2
Name :
1. (7 points) A legislature consisting of three representatives is about
to vote on a salary increase for its members. Legislators 1, 2, and 3
vote sequentially yes
PSC/ECO 288: Game Theory
Spring 2011
Answer Key: Assignment #3
1. (a) Suppose there is a mixed strategy Nash equilibrium such that
1 (U ) = 1 (U M ) = 0 and 1 (DM ) > 0 and 1 (D) > 0. It must
be that
U1 (DM, 2 ) = U1 (D, 2 )
(1)
U1 (DM, 2 ) U1 (U, 2 )
(2)
PSC/ECO 288: Game Theory
Spring 2011
Assignment #3: Due in class Wednesday, February 16
Name :
1. (9 points) Consider the following game.
Player 1
U
UM
DM
D
Player 2
L
R
6, 1 8, 3
4, 9 8, 4
7, 2 6, 9
5, 4 9, 3
(a) Is there a mixed strategy Nash equilibriu
PSC/ECO 288: Game Theory
Spring 2011
Assignment #6: Due in class Wednesday, April 6
Name :
1. (10 points) Do exercise 282.1 from the textbook.
2. (15 points) For the Bayesian game displayed below nd a Nash equilibrium such that the low type of both player
PSC/ECO 288: Game Theory
Spring 2011
Assignment #8: Due in class Wednesday, April 20
Name :
1. (25 points) For the signaling game displayed in the next page, nd
all weakly sequential equilibria in pure strategies. In particular, are there
any pure strateg
PSC/ECO 288: Game Theory
Spring 2011
Assignment #9: Due in class Wednesday, April 27
Name :
1. (11 points) Consider the following game in normal form
Player 1
L
1, 1
0, 0
0, 0
U
M
D
Player
C
0, 0
5, 7
7, 0
2
R
0, 0
0, 9
4, 4
Assume the game is repeated T
PSC/ECO 288: Game Theory
Spring 2011
Answer key for assignment #9
1. One possible combination of strategies that ensures (M, C) prevails in
t = 1 is
s1 () = M
1
s2 (a1 ) =
1
D if a1 = (M, C )
U
otherwise
s1 () = C
2
s2 (a1 ) =
2
R if a1 = (M, C )
L
otherw
PSC/ECO 288: Game Theory
Answer Key: Assignment #2
Spring 2011
1. (a)
N
= cfw_1, 2
Ai = cfw_xi : 0 xi 1 = [0, 1] , i = 1, 2
ui (x1 , x2 ) =
xi if x1 + x2 1
, i = 1, 2
0 if x1 + x2 > 1
(b) If the opponent demands an amount x less than 1, then the best resp
PSC/ECO 288: Game Theory
Spring 2011
Answer Key: Assignment #4
1. (a) The extensive form.
1
Y
N
2
2
Y
3
Y
6,6,6
Y
N
3
N
Y
6,6,8 6,8,6
N
3
N
Y
0,2,2
8,6,6
3
N
Y
2,0,2 2,2,0
N
2,2,2
(b) The unique subgame perfect equilibrium is: 1:(N ), 2:(N, Y ), 3:(N,
PSC/ECO 288: Game Theory
Spring 2011
Answer Key: Assignment #5
1. Player 2 has a winning strategy if n + m is odd and player 1 has a
winning strategy when n + m is even. To see this, consider the two
cases separately:
(a) n + m is odd. Then, player 1 play
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #8: Due in class Wednesday, April 22
Please write clearly and make sure to justify all your answers.
1. (25 points total) Find a WSE in the following dynamic game of
incomplete informa
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #9: Due in class Wednesday, April 29
Please write clearly and make sure to justify all your answers.
1. For the signaling game displayed below, nd all weakly sequential equilibria in p
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #7: Due in class Wednesday, April 15
1. (10 points) Do exercise 282.1 from the textbook.
2. (15 points) For the Bayesian game displayed below nd a Nash equilibrium such that the low ty
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #6: Due in class Wednesday, April 8
Please write clearly and make sure to justify all your answers.
1. (14 points total) Consider the following game in normal form
L
1, 1
0, 0
0, 0
U
M
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #2: Due in class Wednesday, February 11
Please write clearly and make sure to justify all your answers.
1. (10 points total) Consider a two-player divide-the-dollar game with
the follo
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #5: Due in class Wednesday, March 18
Please write clearly and make sure to justify all your answers.
1. (25 points total) Two siblings bargain for the division of a pie. At the
end of
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #4: Due in class Wednesday, March 4
Please write clearly and make sure to justify all your answers.
1. (7 points total) A legislature consisting of three representatives is
about to vo
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #3: Due in class Wednesday, February 18
Please write clearly and make sure to justify all your answers.
1. (13 points total) Consider the following game.
Player 1
U
UM
DM
D
Player 2
L
PSC/ECO 288
Game Theory
Prof. Tasos Kalandrakis
Spring 2015
Assignment #1: Due in class Wednesday, February 4
Please write clearly and make sure to justify all your answers.
1. (13 points total) Consider a two-player game in which two players
split 7 doll
PSC/ECO 288: Game Theory
Spring 2011
Assignment #2: Due in class Wednesday, February 9
Name :
1. (8 points) Consider a two-player divide-the-dollar game with the following rules. Both players simultaneously announce demands xi
[0, 1], i = 1, 2. If demand
PSC/ECO 288: Game Theory
Spring 2011
Assignment #1: Due in class Wednesday, February 2
Name :
1. (7 points) For the following game in strategic form:
Player 1
U
UM
DM
D
Player
L
C
9, 1 2, 3
2, 2 6, 5
4, 5 3, 0
5, 6 4, 6
2
R
7, 2
7, 3
6, 4
8, 8
(a) Identif
Lecture 12, 2/28
More on Nash & SPNE
Ultimatum Bargaining
Nash & SPNE
In an extensive form game with perfect information:
A Nash equilibrium (in pure strategies) is a
profile s* such that
ui(si*, s-i*) ui(si, s-i*)
for all i and all si in Si.
A subgame
Lecture 11, 2/21
Strategy
Nash equilibrium in extensive form games
Subgame perfect Nash equilibrium
Midterm review
Strategy
A (pure) strategy, si in Si, for player i in an
extensive form game is a complete contingent
plan.
It specifies an action choice