### Chapter 11

```Chapter 10
Strategic Choice in Oligopoly,
Monopolistic Competition, and
Everyday Life
5 additional Questions
Even-numbered Qs.
Chapter 10, Problem 2
Consider the following “dating game”, which has two players, A and
B, and two strategies, to buy a movie ticket or a baseball ticket.
The payoffs, given in points, are as shown in the matrix below.
Note that the highest payoffs occur when both A and B attend the
same event.
Assume that players A and B buy their tickets separately and
simultaneously. Each must decide what to do knowing the available
choices and payoffs but not what the other has actually chosen.
Each player believes the other to be rational and self-interested.
Buy movie ticket
Buy movie ticket
A
Buy baseball ticket
B
Buy baseball ticket
2 for A
0 for A
3 for B
0 for B
1 for A
3 for A
1 for B
2 for B
a) Does either player have a dominant strategy?
“When one player has a strategy that yields a higher payoff no
matter which choice the other player makes.”
Dominated strategy – “The other strategy available to the player
that yield a payoff strictly smaller than that of the dominant
strategy”
a) Does either player have a dominant strategy?
“ A player yields a higher payoff no matter what the other players
in a game choose.”
Assumption:
If A assumes B buys movie ticket, A will buy movie ticket.
If A assumes B buys baseball ticket, A will buy baseball ticket.
If B assumes A buys movie ticket, B will buy movie ticket.
If B assumes A buys baseball ticket, B will buy baseball ticket.
Therefore, there is no dominant strategy. Each player buys the
ticket according to other player’s choice.
b) How many potential equilibria are there? (Hint: To see whether a
given combination of strategies is an equilibrium, ask whether
either player could get a higher payoff by changing his or her
strategy.)
•
•
Nash Equilibrium “any combination of strategies in which each
player’s strategy is his or her best choice, given the other players’
choices”
There can be an equilibrium when players do not have a dominant
strategy
There are 2 potential equilibria, the upper-left cell and the lowerright cell.
According to the assumption, in each cell, neither player has any
incentive to change strategies.
c) Is this game a prisoner’s dilemma? Explain.
•
Prisoner’s dilemma “a game in which each player has a
dominant strategy, and when each plays it, the resulting payoffs
are smaller than if each had played a dominated strategy.”
Therefore, this game is not a prisoner’s dilemma because neither
player has a dominant strategy.
d) Suppose player A gets to buy his or her ticket first. Player B does
not observe A’s choice, but knows that A chose first. Player A
knows that player B knows he or she chose first. What is the
equilibrium outcome?
Buy movie ticket
2 for A
Buy movie ticket
A
B
Buy baseball ticket
0 for A
3 for B
0 for B
1 for A
3 for A
1 for B
2 for B
Buy baseball ticket
Each player believes the other to be rational and self-interested and each
knowing the payoffs.
If A chooses first and buys a movie ticket, given A’s choice, B will also buy
movie ticket. A will get a payoff of 2.
If A chooses first and buys baseball ticket, given A’s choice, B will also buy
baseball ticket. A will get a payoff of 3.
The highest payoff for A is to buy baseball ticket.
The equilibrium outcome is where both A and B will buy baseball ticket.
e) Suppose the situation is similar to part d, except that player B
chooses first. What is the equilibrium outcome?
Buy movie ticket
2 for A
Buy movie ticket
A
B
Buy baseball ticket
0 for A
3 for B
0 for B
1 for A
3 for A
1 for B
2 for B
Buy baseball ticket
Each player believes the other to be rational and self-interested and each
knowing the payoffs.
If B chooses first and buys a movie ticket, given B’s choice, A will also buy
movie ticket. B will get a payoff of 3.
If B chooses first and buys baseball ticket, given B’s choice, A will also buy
baseball ticket. B will get a payoff of 2.
The highest payoff for B is to buy movie ticket.
The equilibrium outcome is where both A and B will buy movie ticket.
Chapter 10, Problem 4
The owner of a thriving business wants to open a new office in a
distant city. If he can hire someone who will manage the new
office honestly, he can afford to pay that person a weekly salary of
\$2,000 (\$1,000 more than the manager would be able to earn
elsewhere) and still earn an economic profit of \$800. The owner’s
concern is that he will not be able to monitor the manager’s
behavior and that the manager would therefore be in a position to
embezzle money from the business. The owner knows that if the
remote office is managed dishonestly, the manager can earn
\$3,100, while causing the owner an economic loss of \$600 per
week.
a) If the owner believes that all managers are narrowly self-interested
income maximizers, will he open the new office?
Construct the game tree:
Manager manages honestly
-Owner gets \$800
-Manager gets \$2,000
Owner opens remote
office
A
Owner does not open
remote office
Manager manages dishonestly
-Owner gets -\$600
-Manager gets \$3,100
-Owner gets \$0
-Manager gets \$1,000
elsewhere
Working backward:
• If owner opens the remote office, the potential manager’s best
strategy is to manage dishonestly which gives him \$1,100 more and
owner gets -\$600.
• If owner does not open the remote office, owner gets \$0.
• Since \$0 is better than -\$600, the owner will not open the new
office.
b)
Suppose the owner knows that a managerial candidate is a devoutly
religious person who condemns dishonest behavior, and who would be
willing to pay up to \$15,000 to avoid the guilt she would feel if she were
dishonest. Will the owner open the remote office?
Construct the game tree:
Owner opens
remote office
A
Owner does not
open remote
office
Manager manages honestly
-Owner gets \$800
-Manager gets \$2,000
Manager manages dishonestly
-Owner gets -\$600
-Manager gets \$3,100 \$15,000 = -\$11,900
-Owner gets \$0
-Manager gets \$1,000
elsewhere
Working backward:
• If owner opens the remote office, the potential manager’s best
strategy is to manage honestly which gives him \$2,000 and owner
gets \$800.
• If owner does not open the remote office, owner gets \$0.
• Therefore, the owner will open the new office.
Chapter 10, Problem 6
Newfoundland’s fishing industry has recently declined sharply due
to overfishing, even though fishing companies were supposedly
bound by a quota agreement. If all fishermen had abided by the
agreement, yields could have been maintained at high levels.
a) Model this situation as a prisoner’s dilemma in which the
players are Company A and Company B and the strategies are to
keep the quota and break the quota. Include appropriate payoffs
in the matrix. Explain why overfishing is inevitable in the absence
of effective enforcement of the quota agreement.
a)
Company B
Keep Quota
Keep Quota
Break Quota
Second Best for both
Worst for A
Best for B
Best for A
Worst for B
Third Best for both
Company A
Break Quota
If A breaks its quota while B keeps it, then A will get the largest possible profit and
B will get the smallest.
If B Breaks its quota while A keeps it, then B will get the largest possible profit and
A will get the smallest.
Both will get a higher profit if both keep the quota than if both break it.
The payoffs are perfectly symmetric.
Each dominant strategy is to break the quota, which means that both will do so
unless some way can be found to enforce the quota.
b)
Provide another environmental example of a prisoner’s dilemma.
Air pollution.
If I pollute from my factory and no one else does, then I gain
from not having to install pollution-control equipment, as well as
from clean air; since my own pollution has only a negligible effect
on air quality.
However, if all other industrialists think this way, the air will
become polluted, and all will be worse off than if none had
polluted.
c)
In many potential prisoner’s dilemma, a way out of the dilemma
for a would-be cooperator is to make reliable character judgments
about the trustworthiness of potential partners. Explain why this
solution is no available in many situations involving degradation of
the environment.
In situation involving environmental degradation, the players
usually do not know each other.
When interactions are anonymous, there is no opportunity to
make character judgments.
In such cases, legal enforcement is often necessary.
Chapter 10, Problem 8
Consider the following game. Harry has four quarters. He can
offer Sally from one to four of them. If she accepts his offer, she
keeps the quarters Harry offered her and Harry keeps the others.
If Sally declines Harry’s offer, they both get nothing (\$0). They
play the game only once, and each cares only about the amount of
money he or she ends up with.
Let X be the number of quarters Harry proposes to Sally, where X = 1, 2, 3, 4.
If Harry proposes X quarters to Sally and she accepts,
- Sally keeps X quarters or (\$0.25)(X)
- Harry keeps 4-X quarters or (\$0.25)(4-X)
If Sally declines Harry’s offer,
- Sally gets \$0
- Harry gets \$0
a) Who are the players? What are each player’s strategies?
Construct a decision tree for this ultimatum-bargaining game.
Harry and Sally are the players.
Harry’s strategies involve the number of quarters he offers Sally,
his choice of X
Sally’s strategies are to accept or to refuse Harry’s offer.
Sally: keep (\$0.25)X
Harry: keep (\$0.25)(4-X)
Sally accepts
A
B
Harry proposes X quarters
for Sally
Harry keeps 4-X
Sally refuses
Sally: \$0
Harry:\$0
b) Given their goal, what is the optimal choice for each player?
b) Given their goal, what is the optimal choice for each player?
•
•
•
At B on the decision tree, if Sally accepts the offer, she gets
(\$0.25)X.
If she refuses, she gets \$0.
Therefore, Sally’s best choice is to accept the offer, no matter what
X is.
•
Knowing that Sally will accept the offer no matter what X is, Harry
will offer as little quarter as he can to Sally so as to enjoy the
highest payoff.
•
Harry offers 1 quarter to Sally and keeps 3
•
•
Sally accepts his offer and receive (\$0.25)(1) = \$0.25.
Harry keeps 3 quarters and receive (\$0.25)(4-1)=\$0.75.
Chapter 10, Problem 10
Jill and Jack both have two pails that can be used to carry water
down from a hill. Each makes only one trip down the hill, and each
pail of water can be sold for \$5. Carrying the pails of water down
requires considerable effort. Both Jill and Jack would be willing to
pay \$2 each to avoid carrying one bucket down the hill, and an
additional \$3 to avoid carrying a second bucket down the hill.
a) Given market prices, how many pails of water will each child fetch
from the top of the hill?
In this part of the question, each player’s payoffs are independent
of the action taken by the other.
Each pail of water sells for \$5.
To avoid carrying one bucket costs \$2.
To avoid carrying a second bucket costs \$3.
Since the cost of carrying each bucket is less than \$5, Jill and Jack
will each carry 2 buckets.
b) Jill and Jack’s parents are worried that the two children don’t
cooperate enough with one another. Suppose they make Jill and
Jack share equally their revenues from selling the water. Given
that both are self-interested, construct the payoff matrix for the
decisions Jill and Jack face regarding the number of pails of water
each should carry. What is the equilibrium outcome?
b) When the two children are forced to share revenues, their payoff
matrix is as follows:
Carry 1 Pail
Carry 1 Pail
Jack
Carry 2 Pails
Jill
Carry 2 Pails
\$3 for Jill
\$3 for Jack
\$2.5 for Jill
\$5.5 for Jack
\$5.5 for Jill
\$2.5 for Jack
\$5 for Jill
\$5 for Jack
To calculate their payoff:
Jack and Jill both carry 1 pail, that is, 2 pails in total:
Jack: \$5(2)/2 - \$2(1) = \$3
Jill: \$5(2)/2 - \$2(1) = \$3
Jack carries 2 pails, Jill carries 1 pail, that is, 3 pails in total:
Jack: \$5(3)/2 - \$2(1) - \$3(1) = \$2.5
Jill: \$5(3)/2 - \$2(1) = \$5.5
Jack carries 1 pail, Jill carries 2 pails, that is, 3 pails in total:
Jill: \$5(3)/2 - \$2(1) - \$3(1) = \$2.5
Jack: \$5(3)/2 - \$2(1) = \$5.5
Jack and Jill both carry 2 pails, that is, 4 pails in total:
Jack: \$5(4)/2 - \$2(1) - \$3(1) = \$5
Jill: \$5(4)/2 - \$2(1) - \$3(1) = \$5
The payoffs are perfectly symmetric.
If Jack assumes Jill carries 1 pail, Jack will carry 1 pail.
If Jack assumes Jill carries 2 pails, Jack will carry 1 pail.
If Jill assumes Jack carries 1 pail, Jill will carry 1 pail.
If Jill assumes Jack carries 2 pails, Jill will carry 1 pail.
The dominant strategy for both Jill and Jack is to carry only one
bucket down the hill.
This game is a prisoner’s dilemma.
“ a game in which each player has a dominant strategy, and when
each plays it, the resulting payoffs are smaller than if each had
played a dominated strategy.”
If each follows his dominant strategy, carry 1 pail, both will earn
less profit than if both carry 2 pails.
Additional Question #1
A dominant strategy occurs when
A)
B)
C)
D)
E)
One player has a strategy that yields the highest payoff
independent of the other player’s choice.
Both players have a strategy that yields the highest payoff
independent of the other’s choice.
Both players make the same choice.
The payoff to a strategy depends on the choice made by the
other player.
Each player has a single strategy.
Ans: A
• Let’s illustrate this by an example:
• Player 1’s dominant strategy is {Top}, because it gives
him a higher payoff than {Bottom}, no matter what
Player 2 chooses.
• Player 2’s dominant strategy is {Right}.
2
Left
Right
Top
(100, 30)
(80, 90)
Bottom
(60, 60)
(70, 100)
1
• Therefore, a dominant strategy is a strategy that
yields the highest payoff compared to other available
strategies, no matter what the other player’s choice
is.
• A rational player will always choose to play his
dominant strategy (if there is any in the game),
because this maximises his payoff.
• The other strategy available to the player that yield a
payoff strictly smaller than that of the dominant
strategy is called a ‘dominated strategy’ (e.g. Player
2’s [Left})
• Dominant strategies may not exist in all games. It all
depends on the payoff matrix.
Additional Question #2
The prisoner’s dilemma refers to games where
A)
B)
C)
D)
E)
Neither player has a dominant strategy.
One player has a dominant strategy and the other does not.
Both players have a dominant strategy.
Both players have a dominant strategy which results in the
largest possible payoff.
Both players have a dominant strategy which results in a
lower payoff than their dominated strategies.
Ans: E
• The prisoner’s dilemma is a coordination game.
• Both players have a dominant strategy, but the result
of which is a lower payoff than the dominated
strategies.
2
1
Confess
Deny
Confess
Deny
(-3, -3)*
(0, -6)
(-6,
(-1, -1)
0)
Additional Question #3a
Jordan
Comedy
Comedy
Lee
Documentary
Lee: 3
Jordan: 5
Lee: 2
Jordan: 2
Documentary
Lee: 1
Jordan: 1
Lee: 5
Jordan: 3
3a) The payoff matrix shows the utilities from seeing Comedy or
Documentary.
The game has ? Nash Equilibrium.
A)
B)
C)
D)
E)
0
1
2
3
4
Ans: C
• Let’s look at the payoff matrix to find out the N.E.
• {C, C} and {D, C} are the Nash Equilibria.
• Hence, there are 2 N.E. in this game.
• The N.E. is also known as pure strategy N.E., the adjective “pure strategy” is
to distinguish it from the alternative of “mixed strategy” N.E. A mixed
strategy N.E. is a N.E. in which players will randomly choose between two or
more strategies with some probability.
Jordan
Comedy
Documentary
Comedy
(3, 5)
(1, 1)
Documentary
(2, 2)
(5, 3)
Lee
Additional Question #3b
3b) By allowing for a timing element in this game, i.e., letting
either Jordan or Lee buy a ticket first and then letting the
other choose second, assuming rational players, the
equilibrium is ? , based on ? .
A)
B)
C)
D)
E)
Still uncertain; who buys the 2nd ticket.
Now determinant; who buys the 1st ticket.
Now determinant; who buys the 2nd ticket.
Still uncertain; who buys the 1st ticket.
Now determinant; who is more cooperative.
Ans: B
• By allowing a timing element, the game is now a sequential
game.
• That means, one player moves first, and buys the first ticket.
• The other player observes any action taken (i.e. knows what
ticket has been bought), and then makes his / her decision.
• Actions are not taken simultaneously anymore.
• Whoever chooses an action can now predict how the other
player is going to react.
• E.g. If Lee chooses {Comedy}, he can be sure that Jordan will
choose {Comedy} as well, because this gives Jordan a higher
payoff than picking {Documentary}.
• Therefore, the first mover has the advantage (called First
Mover Advantage) to take actions first, hence securing his or
her own payoff by predicting the response from the other
player.
• A rational (self-interested) player will always pick the action
that maximises his or her own payoff (irregardless of others’)
• Therefore, the result is determinant, as soon as we know who
is buying the 1st ticket.
Additional Question #4
A commitment problem exists when
A)
B)
C)
D)
E)
Players cannot make credible threats or promises.
Players cannot make threats.
There is a Prisoner’s Dilemma.
Players cannot make promises.
Players are playing games in which timing does not matter.
Ans: A
• In games like the prisoner’s dilemma, players have trouble
arriving at the better outcomes for both players…. Because
– Both players are unable to make credible commitments
that they will choose a strategy that will ensue a better
outcomes for both players (either in the form of credible
threats or credible promises)
• This is known as the commitment problem.
Additional Question #5
Suppose Dean promises Matthew that he will always select
the upper branch of either Y or Z. If Matthew believes Dean
and Dean does in fact keep his promise, the outcome of the
game is
A)
B)
C)
D)
E)
Unpredictable.
Matthew and Dean both get \$1,000.
Matthew gets \$500; Dean gets \$1,500.
Matthew gets \$1.5m; Dean gets \$1m.
Matthew gets \$400; Dean gets \$1.5m.
Ans: D
• If Dean will indeed goes for the upper branch, then Matthew
can either earn \$1,000 by choosing the upper branch (i.e.,
arriving the node Y), or \$1.5m by picking the lower branch
(i.e., arriving the node Z).
• As Matthew is a rational individual, he will choose a lower
branch (i.e., arriving the node Z).
(1000, 1000)
Dean
Y
(500, 1500)
X
Matthew
*
Z
Dean
(1.5m, 1m)
(400, 1.5m)
End of Chapter 10
```