ECON 0200
Summer 2012
University of Pittsburgh
Michael LeGower
Homework #3
Write your answers to the following questions on separate sheets of paper. Show all of your
work. Your answers are due in class on Monday, June 4th, 2012.
1. Blaise Pascal, a Frenc
ECON 0200
Summer 2012
University of Pittsburgh
Michael LeGower
Homework #4
1. Suppose you have an opportunity to invest in a company that is working on an important
research project, e.g. a cure for cancer. If you invest, you have to put up $1 million. If
ECON 0200
University of Pittsburgh
LeGower
Summer 2012
Michael
Homework #5
1. Consider a world in which there are two rival species- A and B. Some proportion of species A is
weak; the rest are strong. There is some feature that all strong members of speci
ECON 0200
University of Pittsburgh
LeGower
Summer 2012
Michael
Homework #5
Write your answers to the following questions on separate sheets of paper. Show all of your work. Your
answers are due in class on Monday, June 18th, 2012.
1. Consider a world in w
ECON 0200
Summer 2012
University of Pittsburgh
Michael LeGower
Homework #2
1. Write the following games out in the form requested. Be sure to label strategies and
players.
a. Stag hunt. Two hunters, Pete and Joe, are out in the woods, hunting game. If
the
ECON 0200
Summer 2012
University of Pittsburgh
Michael LeGower
Homework #3
1. Blaise Pascal, a French mathematician and philosopher, argued that we have to make a
wager as to whether or not we choose to believe in God (Lets assume monotheism is
the only p
ECON 0200
Summer 2012
University of Pittsburgh
Michael LeGower
Homework #4
Write your answers to the following questions on separate sheets of paper. Show all of your
work. Your answers are due in class on Monday, June 11th, 2012.
1. Suppose you have an o
ECON 0200
Summer 2012
University of Pittsburgh
Michael LeGower
Homework #2
Write your answers to the following questions on separate sheets of paper. Show all of your
work. Your answers are due in class on Wednesday, May 30th, 2012.
1. Write the following
52
CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
3. Consider a setting in which player 1 moves first by choosing among three
actions: a, b, and c. After observing the choice of player 1, player 2 chooses
among two actions: it and y. Consider the
178
CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
20. It is 1850 during the days of the Gold Rush in California, and 100 miners
are simultaneously deciding which of three plots to go to and mine for gold.
In order to simplify matter
214
CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
from the interval [1,10] in order to maximize the monetary value of the oil
that it extracts. Find the Nash equilibrium extraction rates. (Note: You can
assume that the payoff function is h
Exercises
players. A central underlying assumption is that the game is common knowl-
edge to the players. This means not only that players agree on the game that is
being played but also that each player knows what the other players believe
about the game
140
CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO 0R THREE PLAYERS
For another change in assumption, suppose a village that chooses the
defensive strategy receives a payoff of 6 (not 30) but, in addition, realizes
the expected benet
Exercises
3. It is the morning commute in Congestington, DC. Of 100 drivers, each
driver is deciding whether to take the toll road or take the back roads. The
toll for the toll road is $10, while the back roads are free. In deciding on
a route, each drive
Exertises l 37
10. When there are multiple Nash equilibria, one approach to selecting among
them is to eliminate all those equilibria which involve one or more players
using a weakly dominated sUategy. For the voting game in Figure 4.23, Find
all of the N
Exercises 21 3
13. Players 1, 2, and 3 are playing a game in which the strategy of player 1' is
denoted x,- and can be any nonnegative real number. The payoff function for
player 1 is
1
Vrlxifzrxsl = 1115213 (5)0602,
for player 2 is
1
Vzlxlzxa) = xzxz. (E
134
CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO 0R THREE PLAYERS
1. One of the critical moments early on in the The Lord of the Rings trilogy
is the meeting in Rivendell to decide who should take the One Ring to
Mordor. Gimli
94 CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONAIJTY IS COMMON KNOWLEDGE
18. Consider the threeplayer game below. Player 1 selects a row, either a1, bly
orcl. Player 2 selects a column, either a2, 1);, or (:2. Player 3 selects a matri
210
CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
5. At a company, 20 employees are making contributions for a retirement gift.
Each person is choosing how many dollars to contribute from the interval
[0,10]. The payoff to personi is b, X
Exercises 53
up one of the two doors not selected by the contestant. In opening up a
door, a rule of the show is that Monty is prohibited from opening the door
with the good prize. After Monty opens a door, the contestant is then given
the opportunity to
172
CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
example, if three diners are present and each orders a different meal, then
the payoff to the one ordering the pasta dish is
14+21+
21_(30
3 )=2121.67= 0.67
the payoff for the person
174
CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
i's score is assumed to be determined from the formula 5, = a, + xiz, where
a,- > 0 and 2, > 0. Li,- is related to the innate ability of the student and is what
she would score if sh
90
CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONAIJTY IS COMMON KNOWLEDGE
8. Consider the three-player game shown. Player 1 selects a row, either (.11,
bl or (:1. Player 2 selects a column, either 61; or 172. Player 3 selects a
matri
Exercises
3. Return to the team project game in Chapter 3, and suppose that a frat boy
is partnered with a sorority girl. The payoff matrix is shown below. Find all
Nash equilibria.
Team Project
Sorority girl
IIMWMMI
Immmuuml
MEEHEEI
M-Enul
4. Consider
Exercises 1 39
17. Four political candidates are deciding whether or not to enter a race for an
elected office where the decision depends on who else is throwing his or her
hat into the ring. Suppose candidate A prefers not to enter if candidate B is
expe
Exercises 1 39
17. Four political candidates are deciding whether or not to enter a race for an
elected office where the decision depends on who else is throwing his or her
hat into the ring. Suppose candidate A prefers not to enter if candidate B is
expe
CHAPTER 3: ELIMINATlNG THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALlTY IS COMMON KNOWLEDGE
those two options or abstain by not submitting a ballot. For Julie to be
admitted, she must receive at least six votes in favor of admittance. Letting
m be the numb
136
CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO 0R THREE PLAYERS
7. Return to the Kidnapping game, whose strategic form is shown below. Find
all of the Nash equilibria.
Kidnapping
Vivica (kin of victim)
Do not kidnap/Kill
Guy Do
Exercises
wins the item. Ifm bidders submit the highest bid then they share the item,
so each receives a benefit of 100im. All biddersinot just the highest
bidderspay a price equal to their bid. Letting bi denote bidderis bid, bidder
is payoff then equals