Deep Thought
If I ever went to war, instead of
throwing a grenade, I’d throw one of
those small pumpkins. Then maybe
my enemy would pick up the
pumpkin and think about the
futility of war. And that would give
me the time I need to hit him with a
real grenade. ~ Jack Handey.
(Translation: Today’s lesson demonstrates why people
might not cooperate or collude even if it is in their best
interests to do so.)
BA 445 Lesson B.6 Prisoner Dilemmas
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Readings
Readings
Baye “Oligopoly” (see the index)
Dixit Chapter 4
BA 445 Lesson B.6 Prisoner Dilemmas
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Overview
Overview
BA 445 Lesson B.6 Prisoner Dilemmas
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Overview
The Prisoners’ Dilemma shows why people might not cooperate even if it is
profitable. Cournot duopoly is a prisoners’ dilemma. Lower output by all is
profitable but lower output by you is unprofitable.
Duopoly with Substitutes is a prisoners’ dilemma with firms simultaneously
choosing prices and producing gross substitutes. Profitable cooperation raises
prices.
Duopoly with Complements is a prisoners’ dilemma with firms simultaneously
choosing prices and producing gross substitutes. Profitable cooperation lowers
prices.
Advertising is a prisoners’ dilemma when advertising mostly transfers customers
between firms rather than generating new customers. Profitable cooperation
reduces advertising.
Cleaning and Other Public Goods are prisoner dilemmas when public good
purchases by all is profitable but purchases by you are unprofitable. Profitable
cooperation increases good purchases.
Noise and Other Externalities can be prisoner dilemmas. Profitable cooperation
decreases the negative externalities (like noise) and increases the positive
externalities (like entertainment).
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
Example 1: The Prisoners’
Dilemma
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
Overview
The Prisoners’ Dilemma shows why people might not
cooperate even if it is profitable. Cournot duopoly is a
prisoners’ dilemma. Lower output by all is profitable but
lower output by you is unprofitable.
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Example 1: The Prisoners’ Dilemma
Comment: A Prisoners’ Dilemma demonstrates why people
might not cooperate or collude even if it is in their best
interests to do so. The strongest form of the prisoners’
dilemma is when non-cooperation is a dominate strategy
for each person.
The first game called a “Prisoners’ Dilemma” described
prisoners, and has been used in law enforcement. The
solution to that game also solves a variety of business
applications.
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Example 1: The Prisoners’ Dilemma
In the original dilemma, two suspects are arrested by the police. The
police have insufficient evidence for a conviction, and, having
separated both prisoners, visit each of them to offer the same deal. If
one confesses for the prosecution against the other and the other
remains silent, the confessor goes free and the silent accomplice
receives the full 10-year sentence. If both remain silent, both are
sentenced to only six months in jail for a minor charge. If each
confesses against the other, each receives a five-year sentence. Each
prisoner must choose to confess or to remain silent. Each one is
assured that the other would not know about the betrayal before the
end of the investigation.
How should each prisoner act? Are there mutual gains from
cooperation? If so, can each trust the other to cooperate?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
One way to write the normal
form of the game is for
payoffs to be the negative of
the number of years of
imprisonment.
Prisoner B
Prisoner A
Confess
Silent
Confess
-5,-5
0,-10
Silent
-10,-10
-.5,-.5
Each prisoner should confess since it is the dominate
strategy.
Prisoners would both increase their payoff, from -5 to -.5
(gaining 4.5), if they each cooperated and remained silent.
Neither prisoner can trust the other to cooperate and
remain silent since Confess is the best response to the
other prisoner remaining Silent.
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
Question: Verizon and AT&T control a large share of the
U.S. telecommunications market. They simultaneously
decide on the size of manufacturing plants for the next
generation of smart phones. Suppose the firms’ goods are
perfect substitutes, and market demand defines a linear
inverse demand curve P = 20 – (QV + QA), where output
quantities QV and QA are the thousands of phones
produced weekly by Verizon and AT&T. Suppose unit costs
of production are cV = 2 and cA = 2 for both Verizon and
AT&T. Suppose Verizon and AT&T consider any quantities
QV = 4.5 or 6, and QA = 4.5 or 6. What quantity should
Verizon choose in this Cournot Duopoly?
Are there mutual gains from cooperation? Can Verizon
trust AT&T to cooperate? Can AT&T trust Verizon to
cooperate?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
Answer: To begin, fill out the normal form for this game of
simultaneous moves. For example, at Verizon quantity 4.5
and AT&T quantity 6.0, price = 20-10.5 = 9.5, so Verizon
profits = (9.5-2)4.5 = 33.75 and AT&T profits = (9.5-2)6 =
45.
AT&T
Verizon
4.5
6.0
4.5
6.0
40.5,40.5 33.75,45
45,33.75 36,36
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
AT&T
Each player should choose 6
4.5
6.0
since it is the dominate
4.5
40.5,40.5 33.75,45
strategy for each player:
Verizon
6.0
45,33.75 36,36
6 gives better payoffs for that
player compared with 4.5, no matter whether the other
player chooses 4.5 or 6.
There are mutual gains if both Verizon and AT&T cooperate
and produce 4.5. But Verizon cannot trust AT&T to
cooperate because AT&T cooperating and choosing 4.5 is
not a best response to Verizon cooperating and choosing
4.5. Likewise, AT&T cannot trust Verizon to cooperate
because Verizon cooperating and choosing 4.5 is not a
best response to AT&T cooperating and choosing 4.5.
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 2: Duopoly with Substitutes
Example 2: Duopoly with
Substitutes
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 2: Duopoly with Substitutes
Overview
Duopoly with Substitutes is a prisoners’ dilemma with firms
simultaneously choosing prices and producing gross
substitutes. Profitable cooperation raises prices.
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Example 2: Duopoly with Substitutes
Question: MillerCoors and Anheuser-Busch control a large share of the
U.S. Domestic beer market. The unit cost of a keg to both retailers is
$75. The retailers compete on price but consumers do not find the
goods to be perfect substitutes. Suppose MillerCoors and AnheuserBusch consider prices $85 and $95. If both choose price $85, each
has demand 50; if both $95, each has 40; and if one chooses $85 and
the other $95, the lower price has demand 85 and the higher price 5.
Are the two goods gross substitutes or gross complements? What price
should MillerCoors choose in this Price Competition Game?
Are there mutual gains from cooperation? Can MillerCoors trust
Anheuser-Busch to cooperate? Can Anheuser-Busch trust MillerCoors
to cooperate?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 2: Duopoly with Substitutes
Answer: To begin, fill out the normal form for this game of simultaneous
moves. For example, at Miller price $95 and Busch price $85, Miller’s
demand is 5 and Busch’s is 85, so Miller profits $(95-75)x5 = $100 and
Busch profits $(85-75)x85 = $850.
Goods are gross substitutes because a higher price for one means
higher demand for the other.
Busch
Miller
$85
$95
$85
500,500
100,850
BA 445 Lesson B.6 Prisoner Dilemmas
$95
850,100
800,800
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Example 2: Duopoly with Substitutes
Each player should choose $85
since it is the dominate strategy
for each player: $85 it gives
Miller
better payoffs for that player
compared with $95, no matter
whether the other player chooses $85 or $95.
Busch
$85
$95
$85
500,500
100,850
$95
850,100
800,800
There are mutual gains if both MillerCoors and Anheuser-Busch
cooperate and charge $95. But MillerCoors cannot trust AnheuserBusch to cooperate because Anheuser-Busch cooperating and
choosing $95 is not a best response to MillerCoors cooperating and
choosing $95.
Likewise, Anheuser-Busch cannot trust MillerCoors to cooperate
because MillerCoors cooperating and choosing $95 is not a best
response to Anheuser-Busch cooperating and choosing $95.
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 3: Duopoly with Complements
Example 3: Duopoly with
Complements
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 3: Duopoly with Complements
Overview
Duopoly with Complements is a prisoners’ dilemma with
firms simultaneously choosing prices and producing gross
substitutes. Profitable cooperation lowers prices.
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Example 3: Duopoly with Complements
Question: Wii video game consoles are made by Nintendo, and some
games are produced by third parties, including Sega. The unit cost of a
console to Nintendo is $50, and of a game to Sega is $10. Suppose
Nintendo considers prices $250 and $350 for consoles, and Sega
considers $40 and $50 for games. If they choose prices $250 and $40
for consoles and games, then demands are 1 and 2 (in millions); if
$250 and $50, then .8 and 1.6 (in millions); if $350 and $40, then .7 and
1.4 (in millions); and if $350 and $50, then .6 and 1.2 (in millions).
Are the two goods gross substitutes or gross complements? What price
should Nintendo choose if both companies choose simultaneously?
Are there mutual gains from cooperation? Can Nintendo trust Sega to
cooperate? Can Sega trust Nintendo to cooperate?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 3: Duopoly with Complements
Answer: To begin, fill out the normal form for this game of simultaneous
moves. For example, at Nintendo price $350 and Sega price $40,
Nintendo’s demand is .7 and Sega’s is 1.4, so Nintendo profits $(35050)x.7 = $210 and Sega profits $(40-10)x1.4 = $42.
Goods gross complements because a higher price for one means lower
demand for the other.
Sega
Nintendo
$250
$350
$40
200,60
210,42
BA 445 Lesson B.6 Prisoner Dilemmas
$50
160,64
180,48
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Example 3: Duopoly with Complements
Nintendo should choose $350
since it is the dominate
strategy, and Sega should
Nintendo
choose $50 since it is the
dominate strategy.
Sega
$250
$350
$40
200,60
210,42
$50
160,64
180,48
There are mutual gains if both Nintendo and Sega
cooperate and charge their lower price. But Nintendo
cannot trust Sega to cooperate because Sega cooperating
and choosing $40 is not a best response to Nintendo
cooperating and choosing $250. Likewise, Sega cannot
trust Nintendo to cooperate because Nintendo cooperating
and choosing $250 is not a best response to Sega
cooperating and choosing $40.
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 3: Duopoly with Complements
Comment: The dilemma with
Sam’s and Costco producing
gross substitutes is the
dominate strategy for each
prices goods too low.
The dilemma with Nintendo
and Sega producing gross
complements is the
dominate strategy for each
prices goods too high.
Costco
Sam's
$85
$95
$85
500,500
100,850
$95
850,100
800,800
Sega
Nintendo
$250
$350
BA 445 Lesson B.6 Prisoner Dilemmas
$40
200,60
210,42
$50
160,64
180,48
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Example 4: Advertising
Example 4: Advertising
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Example 4: Advertising
Overview
Advertising is a prisoners’ dilemma when advertising mostly
transfers customers between firms rather than generating
new customers. Profitable cooperation reduces
advertising.
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Example 4: Advertising
Comment: Advertising is a real life example of the prisoner’s dilemma. When
cigarette advertising on television was legal in the United States, competing
cigarette manufacturers had to decide how much money to spend on
advertising. The effectiveness of Firm A’s advertising was partially determined
by the advertising conducted by Firm B. Likewise, the profit derived from
advertising for Firm B is affected by the advertising conducted by Firm A. If both
Firm A and Firm B chose to advertise during a given period the advertising
cancels out, receipts remain constant, and expenses increase due to the cost
of advertising. Both firms would benefit from a reduction in advertising.
However, should Firm B choose not to advertise, Firm A would benefit by
advertising and Firm B would lose.
As in any prisoner’s dilemma, each player cannot trust the other to cooperate.
In the case of cigarette advertising, that lack of trust made cigarette
manufacturers endorse the creation in the U.S. of the Public Health Cigarette
Smoking Act banning cigarette advertising on television, understanding that this
would reduce costs and increase profits across the industry.
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 4: Advertising
Question: R.J. Reynolds Tobacco Corp. and Philip Morris
Corp. must decide how much money to spend on
advertising. They consider spending either $10,000 or
zero. If one advertises and the other does not, the
advertiser pays $10,000, then takes $100,000 profit from
the other. If each advertises, each pays $10,000 but the
advertisements cancel out and neither player takes profit
from the other.
Should R.J. Reynolds spend $10,000 or zero on
advertising if both companies choose simultaneously?
Are there mutual gains from cooperation? Can R.J.
Reynolds trust Philip Morris to cooperate? Can Philip
Morris trust R.J. Reynolds to cooperate?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 4: Advertising
Answer: To begin, fill out the normal form for this game of
simultaneous moves, with payoffs in thousands of dollars.
For example, if Reynolds advertises and Philip does not,
Reynolds pays $10,000, then takes $100,000 profit from
Philip. Hence, Reynolds makes $90,000 and Philip looses
$100,000. Write payoffs in thousands of dollars.
Philip
Reynolds
Ad
No Ad
Ad
-10,-10
-100,90
BA 445 Lesson B.6 Prisoner Dilemmas
No Ad
90,-100
0,0
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Example 4: Advertising
Philip
Each player should choose to
Ad
No Ad
advertise since it is the
Ad
-10,-10 90,-100
dominate strategy for each
Reynolds
No Ad
-100,90
0,0
player: Ad gives better
payoffs for that player compared with No Ad, no matter
whether the other player chooses Ad or No Ad.
There are mutual gains if both Reynolds and Philip
cooperate and choose No Ad. But Reynolds cannot trust
Philip to cooperate because Philip cooperating and
choosing No Ad is not a best response to Reynolds
cooperating and choosing No Ad. Likewise, Philip cannot
trust Reynolds to cooperate because Reynolds cooperating
and choosing No Ad is not a best response to Philip
cooperating and choosing No Ad.
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 5: Cleaning and Other Public Goods
Example 5: Cleaning and Other
Public Goods
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Example 5: Cleaning and Other Public Goods
Overview
Cleaning and Other Public Goods are prisoner dilemmas
when public good purchases by all is profitable but
purchases by you are unprofitable. Profitable cooperation
increases good purchases.
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 5: Cleaning and Other Public Goods
Question: Consider a New York City street on which 25 small
businesses are run, and which suffers from a serious litter problem that
detracts customers. It costs $100 annually for each business to keep
the front of their store clean. If a store owner decides to keep the front
of their store clean, all businesses on the street will have improved
sales and profits. Suppose every business on the street will have a $10
increase in annual profit for each business that decides to keep the
front of their store clean. If more than ten businesses clean their
storefronts, then all of the businesses will make more money, including
the businesses that clean. If some businesses clean but fewer than ten
do so, then the businesses that clean will lose money, while the
businesses that do not clean will gain money. Should anyone clean if
all businesses choose simultaneously? Are there mutual gains from
cooperation? Can any business trust the others to cooperate?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 5: Cleaning and Other Public Goods
Answer: No one should clean since Not Cleaning is a dominate
strategy. For any strategies by each of the other 24 stores, the extra
payoff to Store X from cleaning is a $10 increase minus a $100 cost,
which makes the payoff $90 less than for Not Cleaning.
When each of the 25 stores follows its dominate strategy, no one
cleans, and the payoff to each store is 0. But if each of the 25 stores
cleans, each receives a $10x25 increase minus a $100 cost, which
makes the payoff $150 more than in the dominance solution.
But no business can trust the others to cooperate because Store X
cooperating and choosing Cleaning is not a best response to all the
other stores cooperating and choosing Cleaning.
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Example 6: Noise and Other Externalities
Example 6: Noise and Other
Externalities
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Example 6: Noise and Other Externalities
Overview
Noise and Other Externalities can be prisoner dilemmas.
Profitable cooperation decreases the negative externalities
(like noise) and increases the positive externalities (like
entertainment).
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 6: Noise and Other Externalities
Comment: October 23, 2008 marked the date that many
downtown Fullerton regulars cried a tear. The Rockin' Taco
Cantina was shut down after being served an eviction
notice.
No more dueling pianos, no more drunk dance floors, no
more disorderly conduct outside in the alley.
Was government action needed to control disorderly
conduct?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 6: Noise and Other Externalities
Question: Consider a downtown Fullerton street on which 12 bars are
run, and which suffers from a serious drunkenness problem that
detracts customers because of the violence and smell. It costs $200
daily in foregone profit for each bar to enforce moderation stop serving
customers before they become drunk. If a bar owner decides to enforce
moderation, all bars on the street will have improved sales and profits.
Suppose every bar on the street will have a $20 increase in daily profit
for each bar that decides to enforce moderation. If more than ten bars
enforce moderation, then all of the bars will make more money,
including the bars that enforce moderation. If some businesses enforce
moderation but fewer than ten do so, then the bars that enforce
moderation will lose money, while the bars that do not will gain money.
Should anyone enforce moderation if all bars choose simultaneously?
Are there mutual gains from cooperation? Can any bar trust other bars
to cooperate?
BA 445 Lesson B.6 Prisoner Dilemmas
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Example 6: Noise and Other Externalities
Answer: No one should enforce moderation since Not Enforce
Moderation is a dominate strategy. For any strategies by each of the
other 11 bars, the extra payoff to Bar X from enforcing moderation is a
$20 increase minus a $200 cost, which makes the payoff $180 less
than for Not Enforce Moderation.
When each of the 12 bars follows its dominate strategy, no one
enforces moderation, and the payoff to each store is 0. But if each of
the 12 bars enforce moderation, each receives a $20x12 increase
minus a $200 cost, which makes the payoff $40 more than in the
dominance solution. So mutual gains are possible.
But no bar can trust the others to cooperate because Bar X cooperating
and choosing Enforce Moderation is not a best response to the other
bars choosing Enforce Moderation.
BA 445 Lesson B.6 Prisoner Dilemmas
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Review Questions
Review Questions
 You should try to answer some of the review
questions (see the online syllabus) before the next
class.
 You will not turn in your answers, but students may
request to discuss their answers to begin the next class.
 Your upcoming Exam 2 and cumulative Final Exam
will contain some similar questions, so you should
eventually consider every review question before taking
your exams.
BA 445 Lesson B.6 Prisoner Dilemmas
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BA 445
Managerial Economics
End of Lesson B.6
BA 445 Lesson B.6 Prisoner Dilemmas
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