First Exam, Econ 171, January, 2016
There are 5 questions. Answer as many as you can. Good luck!
Problem 1.
(40 points) Player 1 chooses an integer from {5, 6, . . . , 15},
player 2 chooses an integer from {6, 7, . . . 16}, and player 3 chooses an integer
from {8, 9 . . . 18}. Letting xi be the number chosen by player i, the payoff to
player i is 100 − xi if x1 + x2 + x3 > 20 and is −xi when x1 + x2 + x3 ≤ 20.
Players move simultaneously, without seeing each others’ choices.
A) Find the strategies for each player that are NOT strictly or weakly dominated.
For Player 1, we see that the sum of the numbers chosen by Players 2 and
3 must be at least 14. If Player 1 chooses a number x1 greater than 7, the total
of the three players’ numbers will be greater than 21 and the payoff to player
1 1 will be −x1 . It is then easy to check that playing x1 is strictly dominated
choosing any number smaller than x1 . Thus all choices greater than 7 are strictly
dominated by 7. Choosing 5 and 6, and 7 however are neither strictly nor weakly
dominated. Each of these could be the best response, depending on what the other
players do. So for Player 1, the set of undominated strategies is 5, 6, 7. Similar
reasoning shows that the sets of undominated strategies forrv Players 2 and 3
are {6, 7, 8} and {8, 9, 10}.
B) If no player plays a weakly or strictly dominated strategy, what value or
values can the total x1 + x2 + x3 take?
Any integer in the interval 19-25.
C) If the outcome is a Nash equiibrium, what value or values can the the total
x1 + x2 + x3 take?
21
D ) List all of the Nash equilibrium strategy profiles for this game.
{ 5,6,10}, {5,7,9}, {5,8,8}, {6,6,9}, {6,7,8}, {7,6,8}
Problem 2.
(40 points) The players in this game are Pope Urban VIII,
Galileo, and the Inquisitor. The game begins when the Pope makes a decision
of whether to refer Galileo’s case to the Inquisition or not refer the case to the
Inquisition. If the Pope decides to not refer, the game ends and the payoffs are
3 for the Pope, 5 for Galileo, and 3 for the Inquisitor. If the Pope decides to
refer the case, Galileo is called before the Inquisitor and Galileo has two possible
actions, confess to Inquisitor or not confess to Inquisitor. If Galileo confesses,
then the game is over and Galileo is punished. In this case, the Pope’s payoff
is 5, Galileo’s payoff is 3, and the Inquisitor’s payoff is 4. If Galileo does not
confess, the Inquisitor must decide whether to torture Galileo, or not torture
Galileo. If the Inquisitor decides to not torture, the game is over and the Pope
gets a payoff of 2, Galileo gets a payoff of 4, and the inquisitor gets a payoff of 2.
If the Inquistor decides to torture, then Galileo must decide whether to confess
after torture or not confess after torture. If Galileo confesses after torture, the
game ends and the Pope gets a payoff of 4, Galileo gets a payoff of 0, and the
Inqusitor gets a payoff of 5. If Galileo does not confess after torture, the game
ends and the Pope gets a payoff of 1, Galileo gets a payoff of 1, and the Inquisitor
gets a payoff of 1.
A) Draw a game tree that represents the game between the Pope, Galileo and
the Inquisitor game in extensive form.
I will leave this for you to do.
B) List all of the possible strategies for the pope, all of the possible strategies
for Galileo, and all of the possible strategies for the Inqusitor.
For the Pope: Refer, Don’t refer
For Galileo: (Confess to Inquisitor and Confess after torture), (Confess to
Inquisitor and do not confess after torture), (Do not confess to Inquisitor and
do not confess after torture), (Do not confess to Inquisitor and confess after
torture)
For the Inquistor: Torture, Don’t Torture
C) Represent the 3-player game between the Pope, Galileo, and the Inquisitor
in strategic form. Do this using two game matrices, one of which applies when
the Pope refers and one of which applies when the Pope does not refer.
D) Find all of the Nash equilibrium strategy profiles for this game.
See the answers to C and D on the next page.
ANSWER: After using the best-reply method (where a best reply is denoted by an
asterisk), we have the figure below.
Pope Urban VIII: Refer
Inquisitor
Galileo
3*,4*,5*
3,4*,5*
C/DNC
3*,4*,5*
3,4*,5*
DNC/C
1,5*,4*
DNC/DNC
4*,2*,2
Pope Urban VIII: Do Not Refer
Inquisitor
DNT
C/C
Galileo
DNT
C/C
5*,3*,3
5*,3*,3
C/DNC
5*,3*,3
5*,3*,3
DNC/C
5*,3*,3
5*,3*,3*
DNC/DNC
5*,3*,3*
5*,3*,3*
The Nash equilibria are (C/C, torture, refer), (C/DNC, torture, refer), (DNC/C, DNT,
do not refer), (DNC/DNC, torture, do not refer) and (DNC/DNC, DNT, do not refer).
Problem 3. (40 points) Bart and Lisa are playing rock-paper-scissors. The
rules are as usual. They put their hands behind their backs and make the
symbol for either rock, paper or scissors and then on the count of three, both
show their hands. If they both show the same thing they both get a payoff of
zero. If one shows rock and the other shows paper, the one who showed paper
gets a payoff of 1 and the one who showed rock gets -1. If one shows rock and
the other shows scissors, the one who showed rock gets a payoff of 1 and the one
show showed scissors gets -1. If one shows scissors and the other shows paper,
the one who showed scissors gets a payoff of 1 and the one who showed paper
gets -1. The unusual thing about this game is that Bart cheats and Lisa knows
it. By careful peeking, Bart can always tell if Lisa is going to play paper, but
he can’t tell whether she is going to play rock or scissors. Lisa knows that this
is the case.
A) Draw an extensive form representation of this game, showing the information
sets for Bart and Lisa and the payoffs that result from each possible play of the
game.
I will leave that to you. B) How many strategies are possible for Lisa? How
many strategies are possible for Bart?
There are 3 strategies for Lisa. These are Rock, Paper, Scissor. Bart has
two information sets and 9 possible strategies. A strategy for Bart can be written
as x/y, where x is what he will do if Lisa plays paper and y is what he will do
if she plays rock or scissors.
C) Construct a game matrix, showing the representation of tthe game between
Bart and Lisa in strategic form.
The columns represent Bart’s strategies. The rows represent Lisa’s strategies. The first entry in each cell represents Lisa’s payoff and the second represents Bart’s.
R
P
S
R/R
0,0
1,-1
-1,1
R/P
-1,1
1,-1
1,-1
R/S
1,-1
1,-1
0,0
P/R
0,0
0,0
-1,1
P/P
-1,1
0,0
1,-1
P/S
1,-1
0,0
0,0
S/R
0,0
-1,1
-1,1
S/P
-1,1
-1,1
1,-1
S/S
1,-1
-1,1
0,0
D) On this matrix cross out the strategies that can be eliminated by iterated
elimination of weakly and strictly dominated strategies. Does this game have a
Nash equilibrium (in pure strategies)?
The first six columns are weakly dominated for Bart. When we cross out
these columns we are left with the game represented by
R
P
S
S/R
0.0
-1,1
-1,1
S/P
-1,1
-1,1
1,-1
S/S
1,-1
-1,1
0,0
Now for the second iteration. We see that in this matrix, Paper is weakly
dominated for Lisa. This leaves the payoff matrix
R
S
S/R
0.0
-1,1
S/P
-1,1
1,-1
S/S
1,-1
0,0
But in this matrix, the strategy S/S is strictly dominated for Bart. So we
are left with the matrix
R
S
S/R
0.0
-1,1
S/P
-1,1
1,-1
Iterated elimination of weakly domnated strategies will take us no further
than this two-by-two matrix. If there is a Nash equilibrium for the game between
Bart and Lisa, then this equilibrium must be a Nash equilibrium for the reduced
game shown in the last two-by-two matrix. But this game has no pure strategy
Nash equilibrium. So the original game has no pure strategy Nash equilibrium.
Problem 4.
(40 points) Few people are aware that Shakespeare wrote an
alternative ending for Romeo and Juliet, in which the lovers fled for Florence,
where they were captured and accused of the murder of Count Paris.
Romeo and Juliet are taken to separate rooms and questioned. They are
told that if Romeo testifies against Juliet and she does not testify against him,
he will be set free and she will be sentenced to 6 years of prison. They are told
that if Juliet testifies against Romeo and he does not testify against her, she will
be set free and he will be sentenced to 6 years of prison. If each testifies against
the other, they both will be sentenced to 5 years in prison (separate prisons).
If both refuse to testify, they will each be convicted of minor crimes and each
will be sentenced to 1 year of prison. They are given no chance to communicate
before they make their choices.
A) Write out the strategic form of this game, where the payoffs for each lover
are given as minus the number of years that he or she spends in prison.
Testify
Refuse
Testify
-5,-5
-6,0
Refuse
0,-6
-1,-1
B) Romeo and Juliet are so in love that they care about more than the length of
their own sentences. For each year that Juliet spends in prison, Romeo suffers
as much as he would spending two years in prison himself–and vice versa. Thus,
for example if both refuse to testify, Romeo and Juliet each suffer the equivalent
of three years of imprisonment, one for the year he or she spends in prison and
two for the year his or her loved one spends in prison. Write out the strategic
form matrix of the resulting game.
Testify
Refuse
Testify
-15,-15
-6,-12
Refuse
-12, -6
-3,-3
C) Suppose that instead of pressing each to testify against the other, the authorities had asked each to confess to having committed the murder himself
or herself. If both confess to the murder, each will serve 3 years. If neither
confesses, each serves 1 year. If one confesses and the other does not, the one
who confesses serves 6 years and the one who does not goes free. Write out the
modified payoff matrix for the game, taking into account that Romeo suffers
twice as much for each year that Juliet spends in prison as he does for each year
that he serves himself, and that Juliet suffers twice as much for each year that
Romeo spends in prison as for each year that she she herself spends in prison.
Find the Nash equilibrium (or equilibria) for this game.
Testify
Refuse
Testify
-9,-9
-12,-6
Refuse
-6 ,-12
-3,-3
The only Nash equilibrium is where both refuse to testify.
There are two Nash equilibria. One in which both testify and one in which
both refuse.
Problem 5.
Sixty players play the following game. Each player requests a number of dollars,
1,2,3,4, or 5. A player who requests d dollars will receive d dollars if no other
player requests requests fewer than d dollars, but this player will receive 0 dollars
if at least one other player requests fewer than d dollars. Players payoffs in this
game are given by the number of dollars they receive.
A) (20 points) Find all of the Nash equilibria for this game.
There are 5 Nash equilibria. All choose 5, alll choose 4, all choose 3, all
choose 2, all choose 1
B) (Points as stated in problem) For this part of the question we will actually
play a game. Each person in the class can request a number of points 1,2,3,4, or
5 to be added to his or her score. A student who requests p points will have p
points added to his or her score if no other student requests fewer than p points,
but will have no points added if at least one other student requests fewer than
p points. How many points do you request?
People who chose 1 got 1 point. Everybody else got 0 for this part.