Economics 316
Fall 2013
Martin J. Osborne
Problem Set 8
1. Consider an independent private values sealed-bid second-price auction. We saw in class that such an auction has a Nash equilibrium in
which every players bid is equal to her valuation. Does it al
Econ 1051: Section 2 Notes
TA: Zhenyu Lai ([email protected]), 2/11/2011
Voting with Tie-Breaking
Nash Equilibrium
Volunteering Time
In Class: Discussion of PS2
1. Problem Set 1, Question 3
Q: The winner is decided by majority voting. In case of a tie,
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 5
1. Consider the mixed strategy Nash equilibrium of the model of expert
diagnosis studied in class. How does the equilibrium change if both
E and E decrease by the same amount (a major repair becomes
VSE - Introduction to Game Theory
Problem set #1 - Due Tuesday, March 16, 2010 - Suggested Solution
Problem 1: Two people enter a bus. Two adjacent cramped seats are free. Each person
must decide whether to sit or stand. Sitting alone is more comfortable
Boazii University
g c
Department of Economics
Spring 2014
EC 206 MICROECONOMICS II
Problem Set 3 - Answer Key
1. There are four grilled chestnut sellers who are choosing where to locate on Istiklal Avenue. Each has only
three options to locate their vendo
UNIVERSITY OF TORONTO
Faculty of Arts and Science
ECO 316 (Applied Game Theory)
Instructor: Martin J. Osborne
Midterm Exam
28 February 2013
Duration: 1 hour 50 minutes
No aids allowed
The number in brackets at the start of each question is the number of p
Economics 316
Fall 2013
Martin J. Osborne
Selected solutions to Problem Set 11
1. The condition for a player not to want to deviate is
b
1
a + 1
or
a b
a1 .
As a increases, this lower bound increases. That is, if the payoff to D
increases then the play
Economics 316
Fall 2013
Martin J. Osborne
Selected solutions to Problem Set 9
1. (a) The game may be represented as follows.
1
E
C
D
2
G
F
H
2, 0, 0
3
I
0, 2, 3
2, 3, 1
1, 2, 2
3, 1, 0
(b) Players: cfw_1, 2. Terminal histories: cfw_(C, E, G ), (C, E, H ),
Economics 316
Fall 2013
Martin J. Osborne
Selected solutions to Problem Set 10
1. (a) From the answer to question 6 on Problem Set 2, the best response
function of rm 2 is the function b2 dened by
b2 (q1 ) =
1
4 ( q1 )
0
if q1
if q1 > .
Firm 1s subgame p
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 10
1. (a) Find the subgame perfect equilibrium of Stackelbergs duopoly
game under the assumptions of question 6 on Problem Set 2, and
compare the equilibrium outcome with the Nash equilibrium of
Courno
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 11
1. Find the range of values of the discount factor (in terms of a and b)
for which the strategy pair in which both players use the strategy Unrelenting punishment is a Nash equilibrium of the follow
Economics 316
Fall 2013
Martin J. Osborne
Selected solutions to Problem Set 8
2. In the rst-price auction, a bidder with valuation v bids 1 v. The prob2
ability she wins is the probability that the other bidders valuation is
less than v, which is v (given
Economics 316
Fall 2013
Martin J. Osborne
Solutions to selected problems on Problem Set 5
1. If both E and E decrease by the same amount, E E remains constant, so the equilibrium value of p decreases: experts become less
honest. The equilibrium value of q
Economics 316
Winter 2013
Martin J. Osborne
Solutions to Problem Set 6
1. For any output q2 of rm 2, rm 1s prot if its output is q1 is q1 (
q1 q2 c). Thus rm 1s optimal output is 1 ( c q2 ). This output
2
is equal to q1 when q1 = 1 ( c q2 ), or q2 = c 2q
Economics 316
Fall 2013
Martin J. Osborne
Solutions to selected problems on Problem Set 3
1. The unique Nash equilibrium remains (m, m); the argument is exactly
the same as before. (The dividing line between the supporters of two
candidates with different
Economics 316
Fall 2013
Martin J. Osborne
Solutions to selected problems on Problem Set 4
2. Let a1 be the payoff you assigned for the worst outcome for player 1,
b1 the payoff for the next best outcome for player 1, c1 for the next best
outcome, and d1 f
Economics 316
Fall 2013
Martin J. Osborne
Solutions to selected problems on Problem Set 2
1. The pair (c, c) of prices remains a Nash equilibrium. The argument is
the same as before. The prot of each rm is zero for p = (c, c); if a
rm increases its price,
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 9
1. (a) Represent the following extensive game in a diagram like the one
I used in class for the entry game.
Players cfw_1, 2, 3
Terminal histories cfw_(C, F), (C, G ), ( D, H ), ( D, I ), E
Player fu
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 6
1. Consider the example of Cournots duopoly game considered in class.
Let qi be an output of rm 1 with q1 < qm . Show that the value of
rm 2s output for which q1 is the optimal (prot-maximizing) outp
PAGE 2
1. (a) [5] Two players simultaneously announce bids. A bid is either $1, $2, or $3.
Each player receives the amount she bids unless her bid is exactly $1 more than
the other players bid, in which case she receives her bid plus $4. Each player
care
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 4
1. Find all the mixed strategy Nash equilibria of the games in Figure 1.
T
B
L
6, 0
3, 2
R
0, 6
6, 0
T
B
L
2, 1
2, 0
R
0, 1
1, 2
T
B
L
2, 2
2, 0
R
0, 0
1, 1
Figure 1. The games for problem 1.
2. Find
Economics 316
Fall 2013
Martin J. Osborne
Solutions to selected problems on Problem Set 3
1. The unique Nash equilibrium remains (m, m); the argument is exactly
the same as before. (The dividing line between the supporters of two
candidates with different
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 4
1. Find all the mixed strategy Nash equilibria of the games in Figure 1.
T
B
L
6, 0
3, 2
R
0, 6
6, 0
T
B
L
2, 1
2, 0
R
0, 1
1, 2
T
B
L
2, 2
2, 0
R
0, 0
1, 1
Figure 1. The games for problem 1.
2. Find
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 3
1. Consider a variant of Hotellings model with two parties in which the
voters preferences are asymmetric. Specically, suppose that each
voter cares twice as much about policy differences to the left
Economics 316
Fall 2013
Martin J. Osborne
Solutions to selected problems on Problem Set 1
2. The rst game is the same as Bach or Stravinsky? The reason is that
the players preferences are the same and their preferences in Bach
or Stravinsky?, even though
Economics 316
Winter 2013
Martin J. Osborne
Problem Set 2
1. Consider a generalization of the example of Bertrands model considered in class in which the demand function D is not necessarily linear.
Suppose that D is any continuous function for which D (
Student number:
UNIVERSITY OF TORONTO
Faculty of Arts and Science
ECO 316 (Applied Game Theory)
Instructor: Martin J. Osborne
Midterm Exam
28 February 2013
Duration: 1 hour 50 minutes
No aids allowed
This examination paper consists of 16 pages and 5 quest
Economics 316
Fall 2013
Martin J. Osborne
Problem Set 1
1. Two animals are ghting over some prey. Each can be passive or aggressive. Each prefers to be aggressive if its opponent is passive, and
passive if its opponent is aggressive; given its own stance,