Chapter 10: Thinking Strategically
A. Introduction to game
theory:
Prisoner’s dilemma
Prisoner’s dilemma:
• Two prisoners (Horace and Jasper) have
committed a big crime
• Police have evidence to convict for a small
crime, need confession to convict for a big
crime
Police put Horace and Jasper in separate
rooms, offer Horace the following deal:
• If you confess and Jasper doesn’t, we’ll let
you go free and put him in jail for 20 years
• If Jasper confesses and you don’t, he
goes free, you get 20 years
MB MC
The Payoff Matrix
for a Prisoner’s Dilemma
Jasper’s Choice
• If neither of you confess, you both get 1
year
• If you both confess, you both get 5 years
• Jasper gets offered exactly the same deal
Confess
Remain
silent
Jasper gets
5 years
Jasper gets
20 years
Confess
Horace gets
5 years
Horace gets
0 years
Horace’s Choice
Jasper gets
0 years
Remain
silent
Horace gets
20 years
Jasper gets
1 years
Horace gets
1 years
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
1
MB MC
Horace reasons: if Jasper confesses,
my best option is to confess
MB MC
Horace reasons: if Jasper remains
silent, my best option is to confess
Jasper’s Choice
Confess
Jasper’s Choice
Remain
silent
Jasper gets
5 years
Remain
silent
Confess
Jasper gets
20 years
Jasper gets
5 years
Confess
Jasper gets
20 years
Confess
Horace gets
5 years
Horace gets
0 years
Horace gets
5 years
Horace’s Choice
Horace gets
0 years
Horace’s Choice
Jasper gets
0 years
Remain
silent
Horace gets
20 years
Jasper gets
1 year
Horace gets
1 year
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
Jasper gets
0 years
Remain
silent
Horace gets
20 years
Jasper gets
1 year
Horace gets
1 year
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
Horace concludes:
No matter what Jasper does, my best
option would be to confess
Definition:
If a given strategy yields a higher payoff than
any other strategy, no matter what the other
players in the game choose, then it is called a
dominant strategy
Confession is a dominant strategy for Horace
Confession is also a dominant strategy for Jasper
MB MC
Both confess is the Nash equilibrium
Definition: If each
player’s strategy is
the best he or she
can choose given the
other player’s chosen
strategy, the
strategies are
characterized as the
Nash equilibrium of
the game
Jasper’s Choice
Confess
Remain
silent
Jasper gets
5 years
Jasper gets
20 years
Confess
Horace gets
5 years
Horace gets
0 years
Horace’s Choice
Jasper gets
0 years
Remain
silent
Horace gets
20 years
John Nash
Jasper gets
1 year
Horace gets
1 year
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
2
MB MC
The Prisoner’s Dilemma
Any game has 3 basic
elements:
• the players
• list of possible actions
(strategies)
• payoffs
For prisoner’s dilemma:
Prisoner’s Dilemma
z
• Horace and Jasper
• confess or remain
silent
• 0, 1, 5 or 20 years
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
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
MB MC
Chapter 10
The Payoff Matrix for
an Advertising Game
American’s Choice
A. Prisoner’s dilemma
B. Application: strategic interaction in
advertising
Raise ad
spending
Leave ad
spending
the same
American
gets $5,500
Raise ad
spending
United gets
$5,500
American
gets $2,000
United gets
$8,000
United’s Choice
Leave ad
spending
the same
American
gets $8,000
United gets
$2,000
American
gets $6,000
United gets
$6,000
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
United’s dominant strategy is to raise spending
American’s dominant strategy is to raise spending
MB MC
This advertising game is a
prisoner’s dilemma
Raising spending is the Nash equilibrium
Raise ad
spending
American
gets $5,500
Raise ad
spending
United’s
Choice
Leave ad
spending
the same
American’s Choice
Leave ad
American’s
spending
Choice
the same
United gets
$5,500
Raise ad
spending
American
gets $2,000
American
gets $5,500
Raise ad
spending
United gets
$8,000
Leave ad
spending
the same
United gets
$5,500
American
gets $2,000
United gets
$8,000
United’s Choice
American
gets $8,000
United gets
$2,000
American
gets $6,000
United gets
$6,000
Leave ad
spending
the same
American
gets $8,000
United gets
$2,000
American
gets $6,000
United gets
$6,000
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
3
MB MC
MB MC
Equilibrium When One
Player Lacks a Dominant Strategy
American’s Choice
What if the payoffs were different?
Raise ad
spending
Leave ad
spending
the same
American
gets $4,000
Raise ad
spending
United gets
$3,000
American
gets $3,000
United gets
$8,000
United’s Choice
Leave ad
spending
the same
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
American
gets $5,000
United gets
$4,000
MB MC
American’s dominant strategy is to raise spending
Nash equilibrium: American
raises spending, United does not
United therefore assumes American spends more
Raise ad
spending
American’s Choice
Leave ad
American’s
spending
Choice
the same
American
gets $4,000
United’s
Choice
Leave ad
spending
the same
United gets
$5,000
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
United has no dominant strategy
Raise ad
spending
American
gets $2,000
United gets
$3,000
Raise ad
spending
American
gets $3,000
American
gets $4,000
Raise ad
spending
United gets
$8,000
Leave ad
spending
the same
United gets
$3,000
American
gets $3,000
United gets
$8,000
United’s Choice
American
gets $5,000
United gets
$4,000
American
gets $2,000
United gets
$5,000
Leave ad
spending
the same
American
gets $5,000
United gets
$4,000
American
gets $2,000
United gets
$5,000
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
Chapter 10
A. Prisoner’s dilemma
B. Application: strategic interaction in
advertising
C. Application: stability of a two-player
cartel
We earlier looked at instability of a cartel
when one member can “cheat” by
increasing production without the other
participants knowing.
If instead the cheater knows that the others
will find out and respond, how might that
change the incentives?
4
MB MC
2.00
Price $/bottle
Example:
A duopoly of two firms (Aquapure and
Mountain Spring) have exclusive rights to
bottle water from a spring.
The Market Demand
for Mineral Water
1.00
Marginal cost = zero
MR
D
2,000
1,000
Bottles/day
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
MB MC
One possibilitiy: each
firm chooses P = $1 and
sells 500 bottles to earn
$500/day
2.00
Price $/bottle)
Players: two firms
Strategies: a price each charges (any
number between 0 and $2)
Payoffs: price firm charges times number of
bottles it would sell at that price
The Market Demand
for Mineral Water
This would correspond to
the firms acting together
as an optimizing
monopoly
1.00
MR
D
2,000
1,000
Bottles/day
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
MB MC
Aquapure lowers P
• P = $.90/bottle
• Q = 1,100 bottles/day
2.00
Price $/bottle)
Is this a Nash equilibrium?
Suppose that Mountain Spring charges $1
but Aquapure charges $0.90.
Then Aquapure would sell 1,100 bottles at
profit of $990 > $500.
The Temptation to
Violate a Cartel Agreement
Mountain Spring retaliates
• P = $.90/bottle
• Both firms split 1,100
bottles/day @ $.90
• Profit = $495/day
1.00
0.90
MR
D
1,000 1,100
2,000
Bottles/day
Copyright c 2004 by The McGraw-Hill
Companies, Inc. All rights reserved.
5
Chapter 10: Thinking Strategically
Nash equilibrium: price falls to marginal cost
(in this case, zero)
Have perfect competition even with only two
firms
A. Prisoner’s dilemma
B. Application: strategic interaction in
advertising
C. Application: stability of a two-player
cartel
D. Game theory in more complicated
settings
Possible extensions:
• What if I don’t know for sure the other
player’s payoffs?
• What if I’m not sure that the other player is
rational?
• What if we’re going to play the game many
times over into the future (called a
repeated game)?
Definition:
an algorithm is a formal algebraic or
computational procedure for choosing
strategies
Open competitions between algorithms in 50 repeats
of prisoner’s dilemma with same opponent/partner
And the winner is …
tit for tat
Round 1: cooperate
Round t: do what other
player did in t - 1
6
Chapter 10
Ultimatum bargaining game
A. Prisoner’s dilemma
B. Application: strategic interaction in
advertising
C. Application: stability of a two-player
cartel
D. Game theory in more complicated
settings
E. Games in which timing matters
Example: Ultimatum bargaining game
– Experimenter gives $100 to Tom
– Tom proposes how to divide $100 with
Michael
– Tom must give Michael at least $1
(X = Tom and $100 - X = Michael)
– If Michael accepts the proposal, Tom and
Michael get the money
– If Michael does not accept the proposal, the
money goes to the experimenter
Tom’s Best Strategy in
an Ultimatum Bargaining Game
Decision Tree for Tom
Possible Moves and Payoffs
$X for Tom
$(100 – X) for Michael
$99 for Tom
$1 for Michael
Michael
accepts
A
Michael
accepts
B
Tom proposes
$X for himself,
$(100 – X) for
Michael
A
Michael
refuses
$0 for Tom
$0 for Michael
B
Michael
refuses
Tom proposes
$99 for himself,
$1 for Michael
$0 for Tom
$0 for Michael
• Tom can give Michael a take-it-or-leave-it offer
• Tom will propose $1
• Michael will accept
• The outcome is a Nash Equilibrium
Game
with an Acceptance
New Rule: Michael
can commit in advance
Threshold
to the minimum offer he will accept
Credible Threat
• A threat to take an action that is in the threatener’s
interest to carry out
• Can Michael threaten Tom that he would refuse
x = 1$?
Tom proposes
$X < $(100 - Y) for himself
$(100 - X) > Y for Michael
A
Michael announces
that he will reject any
offer less than $Y
$X for Tom
$(100 – X) for Michael
B
Tom proposes
$X > $(100 - Y) for himself
$(100 - X) < Y for Michael
$0 for Tom
$0 for Michael
7
Nash equilibrium for this modification of the
game: Y = $99
That is, Tom gets $1, Michael gets $99
Tom reasons, if I specify X > (100 – Y),
then I get nothing
Results from experiments with real people
playing original ultimatum bargaining
game:
Few people actually choose the Nash
equilibrium X = $99
If someone does play X = $99, usually the
other person refuses
Most common selection: X = $50
One possible conclusion:
People don’t always behave rationally just
to get the most money.
On the other hand …
would you refuse Tom’s offer if he gets
$99 million and you get $1 million?
Perhaps notions of fairness and moral
principles help provide a commitment
mechanism that would otherwise be
lacking
If not, perhaps rational game theory is the
right way to understand multimillion dollar
business decisions after all
8
