EC 513 Problem Set 3
Spring 2015
1. For each of the following games, nd all Nash equilibria in (pure and) mixed
strategies.
a)
c
d
a 1, 1 0, 0
b 0, 0 0, 0
b)
c
d
a 1, 1 0, 0
b 0, 0 3, 3
c)
c
d
a 2, 2 1, 2
b 10, 1 0, 0
d)
c
d
a 3, 3 0, 5
b 5, 0 1, 1
e) In
CAS EC513(A1) Game Theory
Occasional Lecture Notes
Boston University
Hsueh-Ling Huynh
Rubinstein Bargaining Model
The Game
Two people (I and II) must agree on how to divide a sum of money, and they bargain according to
the game form depicted below. It is
EC 513 Problem Set 8
1. [Second price auctions with private values]
(a) (Weak domination in a second-price sealed-bid auction) Show that for
each type
vi
of each player
i in a second-price sealed-bid auction with imperfect
information about valuations the
EC513 Answers to Problem Set 2
1
a) Player 1 does not have a dominant or dominated strategy, strict or weak.
Player 2 has a strictly dominant strategy of
dominated
b2 ,
a1
then
strictly dominates
a2
b1 .
If we eliminate the strictly
in the remaining game.
EC513 Answers to Problem Set 8
1
(a) Weak domination in second-price sealed-bid action
pix plyer iD nd hoose id for every type of every other plyerF lyer iD
who does not know the other plyers9 typesD is unertin of the highest id of
the other plyersF henot
EC 513 Answers to Problem Set 3
1
a) This game has two pure strategy equilibria: (a, c) and (b, d). There are
no (other) mixed Nash equilibria. To see this, suppose we could nd aq for
player 2 that would make player 1 indierent. Then it would have to sati
EC 513 Problem Set 4
1. Suppose the following game is played exactly twice.
A
B
A 10, 3 2, 4
B 9, 1 1, 0
What are the subgame perfect equilibria?
2. Suppose the following game is played exactly twice.
A
B
A 10, 10 0, 9
B
9, 0
1, 1
For each of the four pos
Economics 513: Problem Set 4
B. Lipman
EC 513 Problem Set 6
Spring 2011
For the games below and on the following pages, nd the normal form and
the set of Nash equilibria. Also, nd the set of subgame perfect equilibria and
For the games below and on the fo
EC513 Answers for Problem Set 1
1
a) The backward induction strategies are (b, ce, h), the path is
beh, and the outcome is (5, 3, 6).
b) The backward induction strategies are (d1 d2 , D1 D2 ) with path
d1 and outcome (1, 0).
c) The backward induction stra
EC513 Answers to Problem Set 7
Spring 2015
1
(a) Clearly, the equilibrium price can't exceed $210, since no one would pay
that much. Suppose it exceeds $200. Then all sellers want to sell, so the expected value of a car to a buyer is (1/3)(210) + (1/3)(12
Economics 513: Problem Set 1
EC 513 Problem Set 1
B. Lipman
Spring 2011
Spring 2015
1. For each of the extensive form games with perfect information shown below, nd the
1. For each of the extensive form games with perfect information shown below,
backward
EC 513 Problem Set 2
Spring 2015
1. For each of the following games, nd the strictly dominant and dominated
strategies (if any) and weakly dominant and dominated strategies if any. See if the
game can be solved by successive elimination of strictly domina
EC 513 Problem Set 4
Spring 2015
1. Suppose the following game is played exactly twice.
A
B
A 10, 3 2, 4
B 9, 1 1, 0
What are the subgame perfect equilibria?
2. Suppose the following game is played exactly twice.
A
B
A 10, 10 0, 9
B
9, 0
1, 1
For each of
EC 513 Problem Set 5
Spring 2015
1. (When can a pure/mixed strategy be evolutionarily stable?)
(Evolutionary stability and weak domination) Let
stable action.
Does
a
a
be an evolutionarily
necessarily weakly dominate every other action?
possible that some
EC 513 Problem Set 7
Spring 2015
1. Suppose there are three kinds of cars: terric ones, acceptable ones, and
lemons. The terric ones are worth $210 to a buyer and the lemons are worth
$30 to a buyer. The value to a seller of any kind of car is $10 less th
Economics 513: Problem Set 4
EC 513 Problem Set 6
B. Lipman
Spring 2015
Spring
2011
For the games below and on the following pages, nd the normal form and
the set of Nash equilibria. Also, nd the set of subgame perfect equilibria and
For the games below a
EC513 Answers for Problem Set 1
Spring 2015
1
a) The backward induction strategies are (b, ce, h), the path is
beh, and the outcome is (5, 3, 6).
b) The backward induction strategies are (d1 d2 , D1 D2 ) with path
d1 and outcome (1, 0).
c) The backward in
EC 513 Answers to Problem Set 5
Spring 2015
1
Evolutionary stability and weak domination
The ESS a does not necessarily weakly dominate every other action in the
game. For example, in the game in Figure 395.1 of the text, X is an ESS but
does not weakly d
EC 513 Answers to Problem Set 4
Spring 2015
1
Note that A is a strictly dominant strategy for player 1 and B is 2's best
strategy when 1 plays A. So (A,B) is the only Nash equilibrium of the game.
For the repeated version, this means that the only subgame
EC 513 Answers to Problem Set 5
1
Evolutionary stability and weak domination
The ESS a does not necessarily weakly dominate every other action in the
game. For example, in the game in Figure 395.1 of the text, X is an ESS but
does not weakly dominate Y.
N