University of California Irvine Game Theory Discussion Questions
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Problem Set 5, Econ 171
1. Recall the story of Alice and Bob, who like to be together but have di?erent
taste in movies. Each can choose to go to either Movie A or Movie B. For Alice,
the payo? from going to Movie A is 3 if Bob is there and 1 if he is not. Alices
payo? from going to Movie B is 2 if Bob is there and 0 if he is not. Bobs payo?
from going to A is 2 if Alice is there and 0 if she is not. Bobs payo? from going
to Movie B is 3 if Alice is there and 1 if she is not.
Each is about to go to a movie. They forgot to settle on which one the last
time they met. Bob has lost his cell phone, so they cannot contact each other.
A) Find a mixed strategy Nash equilibrium in which each of them chooses
to go his or her own favorite movie with some probability p between 0 and 1.
B) In the mixed strategy Nash equilibrium, what is the probability that they
wind up at the same movie?
2.
Consider the game shown below.
Player 2
Player 1
Top
Bottom
Left
6,4
4,8
Right
2,6
4,1
A) For this game, find the best reply function for each player and plot these
functions as is done in Figures 7.4 and 7.5 of your text.
1
B) What are the equilibrium mixed strategies of the two players.
C) In the mixed strategy Nash equilibrium, what is the expected payo? of each
player.
3. Bart and Lisa are playing a game of Rock-Paper- Scissors. They hold their
hands behind their backs and on the count of 3 show whether they are playing
rock, paper, or scissors. When she makes her choice, Lisa doesnt know what
Bart is going to do. Before he decides what to do, Bart peeks behind Lisas
back. If Lisa is going to play scissors, Bart can tell by looking that she will play
scissors. If she is going to play rock or paper, Bart cant tell which of these
she will do. The payo?s in this game are the same as in ordinary Rock-PaperScissors. The winner gets a payo? of 1, the loser gets -1. Rock beats scissors,
scissors beats paper, paper beats rock. If both players show the same thing,
both get 0. Lisa knows that Bart peeks and she knows that he can tell if she
is going to do scissors, but when she isnt going to do scissors, he cant tell
whether she will do paper or rock.
A) Show this game in extensive form.
2
B) How many strategies are possible for Lisa? How many strategies are possible
for Bart?
C) Show this game in strategic form.
D) Which if any of the possible strategies for Bart are weakly dominated? If
Lisa believes that Bart will not use weakly dominated strategies, are there any
strategies that she will not use? Explain
3
E) If Bart knows that Lisa knows that Bart is rational, what strategies will he
not use.
F) Find a mixed strategy Nash equilibrium for this game. In equilibrium, what
are the expected payo?s for Bart? for Lisa?
4. Two countries are disputing a piece of territory. Each country has two
possible strategies, Compromise or Invade. If both countries compromise, they
each get a payo? of zero. If one country compromises and the other invades, the
country that invades gets a payo? of 1 and the country that compromises gets
-1. If both countries invade, there is a war and the payo? for both countries is
x, where x > 1.
A) Does this game have any pure strategy Nash equilibria? If so, what are
the Nash equililbrium profiles?
B) Find a mixed strategy Nash equilibrium where each fights with a probability p where 0 < p < 1. (Your answer will be a function of x.)
C) In the mixed strategy Nash equilibrium that you found in Part B, what
is the probability that there is a war? What is the probability that there is
compromise? (Your answers will depend on x.
D) As x gets larger, the consequences of war become more terrible, but war
becomes less likely. What happens to the expected cost of war in the mixed
strategy Nash equilibrium? That is, the probability that a war happens times
4
the cost of a war.
E) Suppose that payo?s are as before, except that a war is more costly to
Player 2 than to Player 1. Find the probability that each player will fight in
the mixed strategy Nash equilibrium. Which player is more likely to fight, the
one for which war is less costly or the one for which war is more costly?
5. (This problem is inspired by Edgar Allen Poes short story, The Purloined
Letter. Poe very clearly understands the puzzling nature of game theory and
explains them in a charming way I have put copies (in English and in Mandarin
translation) on Gauchospace in the Readings folder.)
A blackmailer has stolen a letter, whose contents, if revealed would destroy
the reputation of Lady S. She has contracted with Inspector Dupin to find that
letter. There are two places where the blackmailer could have hidden the letter.
One of these places is easy to access, both for the blackmailer and for Dupin.
The other place is difficult to access for both of them. Dupin has permission to
look in only one of the two places. For the blackmailer, there are two possible
strategies. Hide the letter in the easy place. Hide the letter in the hard place.
For Dupin there are two possible strategies. Look in the easy place. Look in the
hard place. The payo? to Dupin is 3 if he finds the letter in the easy place and 2
if he finds the letter in the hard place. If he looks in the east place and doesnt
find the letter, his payo? is 0 and if he looks in the hard place and doesnt find
the letter, his payo? is -1. The payo? to the blackmailer if he hides it in the
easy place and Dupin doesnt find it is 3 and if he hides it in the hard place and
Dupin doesn t find it, his payo? is 2. The payo? to the blackmailer if he hides
it in the easy place and Dupin finds it is 0. If he hides it in the hard place and
Dupin finds it, his payo? is -1.
5
A) Show this game in strategic form.
B) Does this game have any pure strategy Nash equilibria? Explain
C) If Dupin believes that the blackmailer is equally likely to hide the letter
in the easy place as in the hard place, what is Dupins best response?
D) Is there a Nash equilibrium in which the blackmailer is equally likely to
hide the letter in the easy as in the hard place? Explain your answer.
6
E) Find a mixed strategy Nash equilibrium for the game between Dupin and
the blackmailer.
F) In the mixed strategy Nash equilibrium, what is the probability that
Dupin finds the letter?
6. This problem is inspired by an incident in the movie, A Beautiful Mind,
about John Nash.
Two boys meet two girls in a bar. Both boys like both girls, but both prefer
Girl A to Girl B. Both boys assign a cardinal utility of 5 to talking with Girl
A and 4 to talking with Girl B, and a utility of 0 to talking with neither. The
boys attitudes towards gambles about outcomes are given by expected utilities.
For example the expected utility of talking to Girl A with probability p and to
nobody with probability 1-p is 5p+0(1 p) = 5p. The expected utility of talking
to Girl B with probability p and nobody with probability 1-p is 4p+0(1 p) = 4p.
The girls like both boys equally. If both boys approach the same girl, she
will choose one of them at random. The girl who wasnt approached is o?ended
and wont talk to the rejected boy. If one boy approaches A and the other
approaches B, then each of the girls will talk to the boy who approached her.
A) The decision about who to approach can be modeled as a game between
the two boys, where each boy has two possible strategies Approach A and Approach B. Show this game in strategic form,
7
B) Find a symmetric Nash equilibrium for this game.
C) In symmetric Nash equilibrium, what is the probability that at least one
boy approaches Girl A? Girl B? What is the probability that both girls are
approached by one of the boys?
8
Problem Set 5, Econ 171
1. Recall the story of Alice and Bob, who like to be together but have di?erent
taste in movies. Each can choose to go to either Movie A or Movie B. For Alice,
the payo? from going to Movie A is 3 if Bob is there and 1 if he is not. Alices
payo? from going to Movie B is 2 if Bob is there and 0 if he is not. Bobs payo?
from going to A is 2 if Alice is there and 0 if she is not. Bobs payo? from going
to Movie B is 3 if Alice is there and 1 if she is not.
Each is about to go to a movie. They forgot to settle on which one the last
time they met. Bob has lost his cell phone, so they cannot contact each other.
A) Find a mixed strategy Nash equilibrium in which each of them chooses
to go his or her own favorite movie with some probability p between 0 and 1.
B) In the mixed strategy Nash equilibrium, what is the probability that they
wind up at the same movie?
2.
Consider the game shown below.
Player 2
Player 1
Top
Bottom
Left
6,4
4,8
Right
2,6
4,1
A) For this game, find the best reply function for each player and plot these
functions as is done in Figures 7.4 and 7.5 of your text.
1
B) What are the equilibrium mixed strategies of the two players.
C) In the mixed strategy Nash equilibrium, what is the expected payo? of each
player.
3. Bart and Lisa are playing a game of Rock-Paper- Scissors. They hold their
hands behind their backs and on the count of 3 show whether they are playing
rock, paper, or scissors. When she makes her choice, Lisa doesnt know what
Bart is going to do. Before he decides what to do, Bart peeks behind Lisas
back. If Lisa is going to play scissors, Bart can tell by looking that she will play
scissors. If she is going to play rock or paper, Bart cant tell which of these
she will do. The payo?s in this game are the same as in ordinary Rock-PaperScissors. The winner gets a payo? of 1, the loser gets -1. Rock beats scissors,
scissors beats paper, paper beats rock. If both players show the same thing,
both get 0. Lisa knows that Bart peeks and she knows that he can tell if she
is going to do scissors, but when she isnt going to do scissors, he cant tell
whether she will do paper or rock.
A) Show this game in extensive form.
2
B) How many strategies are possible for Lisa? How many strategies are possible
for Bart?
C) Show this game in strategic form.
D) Which if any of the possible strategies for Bart are weakly dominated? If
Lisa believes that Bart will not use weakly dominated strategies, are there any
strategies that she will not use? Explain
3
E) If Bart knows that Lisa knows that Bart is rational, what strategies will he
not use.
F) Find a mixed strategy Nash equilibrium for this game. In equilibrium, what
are the expected payo?s for Bart? for Lisa?
4. Two countries are disputing a piece of territory. Each country has two
possible strategies, Compromise or Invade. If both countries compromise, they
each get a payo? of zero. If one country compromises and the other invades, the
country that invades gets a payo? of 1 and the country that compromises gets
-1. If both countries invade, there is a war and the payo? for both countries is
x, where x > 1.
A) Does this game have any pure strategy Nash equilibria? If so, what are
the Nash equililbrium profiles?
B) Find a mixed strategy Nash equilibrium where each fights with a probability p where 0 < p < 1. (Your answer will be a function of x.)
C) In the mixed strategy Nash equilibrium that you found in Part B, what
is the probability that there is a war? What is the probability that there is
compromise? (Your answers will depend on x.
D) As x gets larger, the consequences of war become more terrible, but war
becomes less likely. What happens to the expected cost of war in the mixed
strategy Nash equilibrium? That is, the probability that a war happens times
4
the cost of a war.
E) Suppose that payo?s are as before, except that a war is more costly to
Player 2 than to Player 1. Find the probability that each player will fight in
the mixed strategy Nash equilibrium. Which player is more likely to fight, the
one for which war is less costly or the one for which war is more costly?
5. (This problem is inspired by Edgar Allen Poes short story, The Purloined
Letter. Poe very clearly understands the puzzling nature of game theory and
explains them in a charming way I have put copies (in English and in Mandarin
translation) on Gauchospace in the Readings folder.)
A blackmailer has stolen a letter, whose contents, if revealed would destroy
the reputation of Lady S. She has contracted with Inspector Dupin to find that
letter. There are two places where the blackmailer could have hidden the letter.
One of these places is easy to access, both for the blackmailer and for Dupin.
The other place is difficult to access for both of them. Dupin has permission to
look in only one of the two places. For the blackmailer, there are two possible
strategies. Hide the letter in the easy place. Hide the letter in the hard place.
For Dupin there are two possible strategies. Look in the easy place. Look in the
hard place. The payo? to Dupin is 3 if he finds the letter in the easy place and 2
if he finds the letter in the hard place. If he looks in the east place and doesnt
find the letter, his payo? is 0 and if he looks in the hard place and doesnt find
the letter, his payo? is -1. The payo? to the blackmailer if he hides it in the
easy place and Dupin doesnt find it is 3 and if he hides it in the hard place and
Dupin doesn t find it, his payo? is 2. The payo? to the blackmailer if he hides
it in the easy place and Dupin finds it is 0. If he hides it in the hard place and
Dupin finds it, his payo? is -1.
5
A) Show this game in strategic form.
B) Does this game have any pure strategy Nash equilibria? Explain
C) If Dupin believes that the blackmailer is equally likely to hide the letter
in the easy place as in the hard place, what is Dupins best response?
D) Is there a Nash equilibrium in which the blackmailer is equally likely to
hide the letter in the easy as in the hard place? Explain your answer.
6
E) Find a mixed strategy Nash equilibrium for the game between Dupin and
the blackmailer.
F) In the mixed strategy Nash equilibrium, what is the probability that
Dupin finds the letter?
6. This problem is inspired by an incident in the movie, A Beautiful Mind,
about John Nash.
Two boys meet two girls in a bar. Both boys like both girls, but both prefer
Girl A to Girl B. Both boys assign a cardinal utility of 5 to talking with Girl
A and 4 to talking with Girl B, and a utility of 0 to talking with neither. The
boys attitudes towards gambles about outcomes are given by expected utilities.
For example the expected utility of talking to Girl A with probability p and to
nobody with probability 1-p is 5p+0(1 p) = 5p. The expected utility of talking
to Girl B with probability p and nobody with probability 1-p is 4p+0(1 p) = 4p.
The girls like both boys equally. If both boys approach the same girl, she
will choose one of them at random. The girl who wasnt approached is o?ended
and wont talk to the rejected boy. If one boy approaches A and the other
approaches B, then each of the girls will talk to the boy who approached her.
A) The decision about who to approach can be modeled as a game between
the two boys, where each boy has two possible strategies Approach A and Approach B. Show this game in strategic form,
7
B) Find a symmetric Nash equilibrium for this game.
C) In symmetric Nash equilibrium, what is the probability that at least one
boy approaches Girl A? Girl B? What is the probability that both girls are
approached by one of the boys?
8
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