Managerial Economics &
Business Strategy
Chapter 10
Game Theory: Inside Oligopoly
A Market-Share Game
•
•
•
•
Two managers want to maximize market share.
Strategies are pricing decisions.
Simultaneous moves.
One-shot game.
The Market-Share Game
in Normal Form
Manager 1
Manager 2
Strategy
P=$10
P=$5
P=$1
P=$10
.5, .5
.8, .2
.9, .1
P=$5
.2, .8
.5, .5
.8, .2
What is the Nash Equilibrium???
P = $1
.1, .9
.2, .8
.5, .5
Market-Share Game
Equilibrium
Manager 1
Manager 2
Strategy
P=$10
P=$5
P=$1
P=$10
.5, .5
.8, .2
.9, .1
P=$5
.2, .8
.5, .5
.8, .2
Nash Equilibrium
P = $1
.1, .9
.2, .8
.5, .5
What do we see?
• Game theory can be used to analyze situations
where “payoffs” are non monetary!

Here we were looking at market shares
• Typically we will focus on environments where
businesses want to maximize profits.

Typical payoffs are measured in monetary units.
Is the Nash Equilibrium always best?
Firm A
Firm B
Strategy Low Price High Price
Low Price
0,0
50,-10
High Price -10,50
10,10
• What is the Nash equilibrium?

Low Price – Low Price
• Is it the best?

Could collude and both charge high prices
• What if we “agree” to charge high prices, but then firm A
cheats?

Firm A will increase profits (can’t rat them out because collusion is illegal)
• Everyone fears cheating…we don’t agree.
Examples of Coordination
Games
• Coordination games  should we produce
things to coordinate with our rivals, or do
something completely different
• Laser disks vs. DVDs
• Industry standards


size of floppy disks.
size of CDs.
• National standards


electric current.
traffic laws.
A Coordination Game in
Normal Form
Player 1
Player 2
Strategy
1
2
3
A
0,0
$10,$10
0,0
B
0,0
0,0
$10,$10
C
$10,$10
0,0
0,0
A Coordination Problem:
Three Nash Equilibria!
Player 1
Player 2
Strategy
1
2
3
A
0,0
$10,$10
0,0
B
0,0
0,0
$10, $10
C
$10,$10
0,0
0,0
More than one Nash equilibrium!!
Anytime we do the same…we maximize our
profits.
Key Insights:
• Not all games are games of conflict.
• Communication can help solve coordination
problems.
• Sequential moves can help solve coordination
problems.
An Advertising Game
• Two firms (Kellogg’s & General Mills) managers
want to maximize profits.
• Strategies consist of advertising campaigns.
• Simultaneous moves.


One-shot interaction.
Repeated interaction.
Infinitely Repeated Games
• It appears that Collusion is not possible in a one-shot
game, but what if we can build trust???
• Need to look at the STREAM of payoffs when they make
their current decision

Cheat or not cheat
• What was the formula for PV??
PV   o 
1
1 i

2
1  i 
2
 ... 
t
1  i 
t
If profits are the same in every
period
PVfirm
1 i 


 i 
Can collusion work if firms play the
game each year, forever?
• Consider the following “trigger strategy” by each
firm:

“Don’t advertise, provided the rival has not advertised in the past.
If the rival ever advertises, “punish” it by engaging in a high level
of advertising forever after.”
• In effect, each firm agrees to “cooperate” so long
as the rival hasn’t “cheated” in the past.
“Cheating” triggers punishment in all future
periods.

Trigger causes a FUTURE cost of cheating
• Remember the deal is  Don’t advertise!!
Suppose General Mills adopts this
trigger strategy. Kellogg’s profits?
Cooperate = 12 +12/(1+i) + 12/(1+i)2 + 12/(1+i)3 + …
Value of a perpetuity of $12 paid
= 12 + 12/i
at the end of every year
Cheat = 20 +2/(1+i) + 2/(1+i)2 + 2/(1+i)3 + …
= 20 + 2/I
Gain 20 the first period but 2 after cheating episode
Kellogg’s
General Mills
Strategy
None
Moderate
High
None
12,12
20, 1
15, -1
Moderate
1, 20
6, 6
9, 0
High
-1, 15
0, 9
2, 2
Kellogg’s Gain to Cheating:
• Cheat - Cooperate = 20 + 2/i - (12 + 12/i) = 8 - 10/i

Suppose i = .05
• Cheat - Cooperate = 8 - 10/.05 = 8 - 200 = -192
• It doesn’t pay to deviate.

Collusion is a Nash equilibrium in the infinitely repeated
game!
• If Immediate Benefit - PV of Future Cost > 0

Pays to “cheat”.
• If Immediate Benefit - PV of Future Cost  0

Doesn’t pay to “cheat”.
Key Insight
• Collusion can be sustained as a Nash equilibrium
when there is no certain “end” to a game.
• Doing so requires:



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Ability to monitor actions of rivals.
Ability (and reputation for) punishing defectors.
Low interest rate.
High probability of future interaction.
Real World Examples of
Collusion
•
•
•
•
Garbage Collection Industry
OPEC
NASDAQ
Airlines
Can we do it? (demonstration
problem 10-6)
• Suppose firm A and firm B
repeatedly face the situation
presented below and the interest
rate is 40 percent. The firms agree
to charge a high price each period
provided neither firm has cheated
on this agreement in the past.



What are firm A’s profits if it cheats on
the collusive agreement?
What are firm A’s profits if it does not
cheat on the collusive agreement?
Does an equilibrium result where the
firms charge the high price each period?
Price
Firm B
Low
High
Firm A
Low
0,0
50,-40
High
-40,50
10,10
• Cheat  get 50 now and zero after that
• Not Cheat  get 10 forever
 1  .40 
PV  
10  35
 .40 
• Since 50 (cheat) > 35 (not cheat we would end up CHEAT
Factors affecting collusion
• Collusion is easier when firms know…





Who their rivals are (so then they know who to punish)
Who their rivals’ customers are (so they can steal them by
charging lower prices if they have to punish the rival)
When their rivals deviate from the agreement
Be able to punish successfully for deviations from agreement
The end of the game is not known
• If there is no tomorrow  there is no punishment  everyone will
cheat
Chapter 10 Homework
Numbers 2, 4, and 14