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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 210 Lesson II.6 Prisoner Dilemmas
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Overview
Overview
BA 210 Lesson II.6 Prisoner Dilemmas
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Lesson Overview
Lesson II.6 Prisoner Dilemmas
Example 1: The Prisoners’ Dilemma
Example 2: Bertrand Duopoly
Example 3: Duopoly with Substitutes
Example 4: Duopoly with Complements
Example 5: Cournot Duopoly
Example 6: Advertising
Summary
Review Questions
Lesson II.7 Repeated Dilemmas
BA 210 Lesson II.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
Example 1: The Prisoners’ Dilemma
BA 210 Lesson II.6 Prisoner Dilemmas
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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
Question: 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 210 Lesson II.6 Prisoner Dilemmas
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Example 1: The Prisoners’ Dilemma
Answer: 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.
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Example 2: Bertrand Duopoly
Example 2: Bertrand Duopoly
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Example 2: Bertrand Duopoly
Question: Sam’s Club and Costco both sell emergency food
supplies in a weather-proof bucket that provides 275 delicious
easy-to-prepare meals, including potato soup and corn chowder.
The unit cost to both retailers is $75. The retailers compete on
price: the low-price retailer gets all the market and they split the
market if they have equal prices. Suppose they consider prices
$85 and $95, and suppose market demands at those prices are 100
and 80.
What price should Costco choose in this Price Competition
Game, also called a Bertrand Duopoly (named after French
mathematician Joseph Louis François Bertrand (1822-1900)).
Are there mutual gains from cooperation? Can Costco trust
Sam’s to cooperate? Can Sam’s trust Costco to cooperate?
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Example 2: Bertrand Duopoly
Answer: To begin, fill out the normal form for this game of
simultaneous moves. For example, at Sam's Club price $95 and
Costco price $85, Costco gets the entire market demand of 100.
Hence, Sam's makes $0 and Costco makes $(85-75)x100 =
$1,000.
Costco
Sam's
$85
$95
$85
500,500
0,1000
BA 210 Lesson II.6 Prisoner Dilemmas
$95
1000,0
800,800
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Example 2: Bertrand Duopoly
Costco
Each player should choose $85
$85
$95
since it is the dominate strategy
$85
500,500 1000,0
for each player: $85 it gives
Sam's
$95
0,1000 800,800
better payoffs for that player
compared with $95, no matter whether the other player chooses
$85 or $95.
There are mutual gains if both Sam’s and Costco cooperate and
charge $95. But Costco cannot trust Sam’s to cooperate because
Sam’s cooperating and choosing $95 is not a best response to
Costco cooperating and choosing $95. Likewise, Sam’s cannot
trust Costco to cooperate because Costco cooperating and
choosing $95 is not a best response to Sam’s cooperating and
choosing $95.
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Example 3: Duopoly with Substitutes
Example 3: Duopoly with Substitutes
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Example 3: Duopoly with Substitutes
Question: Sam’s Club and Costco both sell emergency food
supplies in a weather-proof bucket that provides 275 delicious
easy-to-prepare meals, including potato soup and corn chowder.
The unit cost to both retailers is $75. The retailers compete on
price but consumers do not find the goods to be perfect
substitutes. Suppose Sam’s Costco 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 Costco choose in this Price Competition Game?
Are there mutual gains from cooperation? Can Costco trust
Sam’s to cooperate? Can Sam’s trust Costco to cooperate?
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Example 2: Bertrand Duopoly
Answer: To begin, fill out the normal form for this game of
simultaneous moves. For example, at Sam’s Club price $95 and
Costco price $85, Sam’s demand is 5 and Costco’s is 85, so
Sam’s profits $(95-75)x5 = $100 and Costco profits $(85-75)x85
= $850.
Goods gross substitutes because a higher price for one means
higher demand for the other.
Costco
Sam's
$85
$95
$85
500,500
100,850
BA 210 Lesson II.6 Prisoner Dilemmas
$95
850,100
800,800
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Example 2: Bertrand Duopoly
Costco
Each player should choose $85
$85
$95
since it is the dominate strategy
$85
500,500 850,100
for each player: $85 it gives
Sam's
$95
100,850 800,800
better payoffs for that player
compared with $95, no matter whether the other player chooses
$85 or $95.
There are mutual gains if both Sam’s and Costco cooperate and
charge $95. But Costco cannot trust Sam’s to cooperate because
Sam’s cooperating and choosing $95 is not a best response to
Costco cooperating and choosing $95. Likewise, Sam’s cannot
trust Costco to cooperate because Costco cooperating and
choosing $95 is not a best response to Sam’s cooperating and
choosing $95.
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Example 4: Duopoly with Complements
Example 4: Duopoly with
Complements
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Example 4: 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?
Are there mutual gains from cooperation? Can Nintendo trust
Sega to cooperate? Can Sega trust Nintendo to cooperate?
BA 210 Lesson II.6 Prisoner Dilemmas
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Example 4: 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 $(350-50)x.7 = $210 and Sega profits $(4010)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 210 Lesson II.6 Prisoner Dilemmas
$50
160,64
180,48
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Example 4: Duopoly with Complements
Nintendo should choose $350
since it is the dominate strategy,
And Sega should choose $50
since it is the dominate strategy.
Sega
Nintendo
$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.
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Example 4: 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 210 Lesson II.6 Prisoner Dilemmas
$40
200,60
210,42
$50
160,64
180,48
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Example 5: Cournot Duopoly
Example 5: Cournot Duopoly
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Example 5: Cournot Duopoly
Question: Intel and AMD simultaneously decide on the size of
manufacturing plants for the next generation of microprocessors
for consumer desktop computers. Suppose the firms’ goods are
perfect substitutes, and market demand defines a linear inverse
demand curve P = 20 – (QI + QA), where output quantities QI and
QA are the thousands of processors produced weekly by Intel and
AMD. Suppose unit costs of production are cI = 2 and cA = 2 for
both Intel and AMD. Suppose Intel and AMD consider any
quantities QI = 4.5 or 6, and QA = 4.5 or 6.
What quantity should Intel choose in this Cournot Duopoly?
Are there mutual gains from cooperation? Can Intel trust AMD
to cooperate? Can AMD trust Intel to cooperate?
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Example 5: Cournot Duopoly
Answer: To begin, fill out the normal form for this game of
simultaneous moves. For example, at Intel quantity 4.5 and
AMD quantity 6.0, price = 20-10.5 = 9.5, so Intel profits = (9.52)4.5 = 33.75 and AMD profits = (9.5-2)6 = 45.
AMD
Intel
4.5
6.0
4.5
6.0
40.5,40.5 33.75,45
45,33.75 36,36
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Example 5: Cournot Duopoly
AMD
Each player should choose 6
4.5
6.0
since it is the dominate strategy
4.5
40.5,40.5 33.75,45
for each player: 6 it gives
Intel
6.0
45,33.75 36,36
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 Intel and AMD cooperate and
produce 4.5. But Intel cannot trust AMD to cooperate because
AMD cooperating and choosing 4.5 is not a best response to Intel
cooperating and choosing 4.5. Likewise, AMD cannot
trust Intel to cooperate because Intel cooperating and choosing
4.5 is not a best response to AMD cooperating and choosing 4.5.
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Example 6: Advertising
Example 6: Advertising
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Example 6: 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.
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Example 6: 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?
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 210 Lesson II.6 Prisoner Dilemmas
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Example 6: 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 210 Lesson II.6 Prisoner Dilemmas
No Ad
90,-100
0,0
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Example 6: Advertising
Philip
Each player should choose to
Ad
No Ad
advertise since it is the dominate
Ad
-10,-10 90,-100
strategy for each player: Ad
Reynolds
No Ad
-100,90
0,0
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 210 Lesson II.6 Prisoner Dilemmas
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Review Questions
Review Questions
 You should try to answer some of the questions above
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 210 Lesson II.6 Prisoner Dilemmas
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BA 210
Introduction to Microeconomics
End of Lesson II.6
BA 210 Lesson II.6 Prisoner Dilemmas
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