Download Game Theory

Game Theory
Introduction
• Behavior of competitors, or impact of own
actions, cannot be ignored in oligopoly
• Managers maximize profit or market share by
outguessing competitors
• Insight into oligopolistic markets by using
GAME THEORY (Von Neumann and
Morgenstern in 1950): designed to evaluate
situations with conflicting objectives or
bargaining processes between at least two
parties.
Types of Games
• Normal Form
• Simultaneous
(act without knowing
other player’s strategy)
• One Shot
• Zero Sum
(market share)
• Single Stage
vs. Extensive Form
vs. Sequential
(one player moves
after observing others)
vs. Repeated
(infinite and finite with
uncertain and certain
final period)
vs. Non-zero Sum
(profit maximization)
vs. Multistage
A Normal Form Game
Elements of the game:
Player 2
Player 1
Planed decision or actions
Strategy
a
b
c
Players
Strategies or feasible actions
Payoff matrix
A
B
C
12,11
11,10
10,15
11,12
10,11
10,13
14,13
12,12
13,14
Results from strategy dependent on
the strategies of all the players
Dominant Strategy
• Regardless of whether Player 2 chooses A, B, or
C, Player 1 is better off choosing “a”!
(Indiana Jones and the Holy Grail)
• “a” is Player 1’s Dominant Strategy!
Player 1
Player 2
Strategy
a
b
c
1’s best
strategy
A
12,11
11,10
10,15
a
B
11,12
10,11
10,13
a
C
14,13
12,12
13,14
a
2’s best
strategy
c
c
a
The Outcome
• What should player 2 do?
• 2 has no dominant strategy, but should reason that 1 will play “a”.
• Therefore 2 should choose “C”.
Player 1
Player 2
Strategy
a
b
c
A
B
C
12,11
11,12
14,13
14,13*
11,10
10,15
10,11
10,13
12,12
13,14
• This outcome is called a Nash equilibrium (set of strategies
were no player can improve payoffs by unilaterally changing
own strategy given other player’s strategy)
•“a” 1’s best response to “C” and “C” is 2’s best response to “a”.
Best Response Strategy
•
•
•
•
•
Try to predict the likely action of competitor
to identify your best response:
Conjecture choice of rival
Select your own best response
Was conjecture reasonable
or
Look for dominant strategies
Put yourself in your rival’s shoes
Market-Share Game Equilibrium
• Two managers want to maximize market share
(zero-sum game)
• Strategies are pricing decisions
• Simultaneous moves
• One-shot game
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*
Dominated Strategy
• Dominance exception rather than rule
• In absence of dominance it might be possible to simplify the game
by eliminating dominated strategy (never played: lowest payoff
regardless of other player’s strategy)
• Steelers trial by 2, have the ball & enough time for 2 plays
• Payoff matrix in yards gained by offense: no dominant strategy
• Pass dominant offense without Blitz (dominated defense)
Offense
Defense
Strategy
Run
Pass
Best Offense
Defend
Run
Defend
Pass
Blitz
2
6
14
Best
Defense
Run
8
7*
10
Pass
Pass
Pass
Run
Maximin or Secure Strategy
In absence of dominant strategy risk averse players may
abandon Nash or best response (*) and seek maximin
option (^) that maximizes the minimum possible payoff.
This is not design to maximize payoff but rather to avoid
highly unfavorable outcomes (choose the best of all worst).
Firm 1
Firm 2
Strategy
None
New Product
Best for 1
Min for 2
None
4, 4^
6, 3*
New 6
None 3
Best
New Product for 2
New 6
3, 6*
None 3
2, 2
None 3
New 2
Min
for 1
New 3
None 2
Board of Getty Oil agreed to sell 40% stake to Pennzoil @ $128.5 in Jan 1984.
Board of Getty Oil subsequently accepted Texaco’s offer for 100% @ $128.
Pennzoil sued Texaco for breach of contract & received $10 bill jury award in 1985.
Texaco appealed. Before Supreme Court’s decision, they settled for $3 bill in 1987 .
Examples of Coordination
Games
• Industry standards
• size of floppy disks
• size of CDs
• etc.
• National standards
• electric current
• traffic laws
• etc.
A Coordination Problem:
Three Nash Equilibria!
Player 1
Player 2
Strategy
1
2
3
A
B
C
0,0
0,0
$10,$10*
$10,$10*
0,0
0,0
0,0
$10, $10*
0,0
Key Insights:
• In some cases one-shot, non-cooperative games result in
undesirable outcome for individuals (prisoner’s dilemma)
and some times for society (advertisement).
• Communication can help solve coordination problems.
• Sequential moves can help solve coordination problems.
• Time in jail, Nash (*) and Maximin (^) equilibrium in
Prisoner’s dilemma.
Suspect 1
Suspect 2
Strategy
Confess
Do Not
Best for 1
Max for 2
Confess
5, 5*^
15, 2
Confess
Confess
Do Not
2, 15
0, 0*
Do Not
Confess
Best
for 2
Confess
Do Not
Max
for 1
Confess
Confess
One-Shot Advertising Game Equilibrium
• Kellogg’s & General Mills want to maximize profits
• Strategies consist of advertising campaigns
• Simultaneous moves
• One-shot interaction
• Repeated interaction
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*
Nash Equilibrium
Repeating the game 2 times will not improve outcome
• In the last period the game is a one-shot game, so equilibrium entails
High Advertising.
• Period 1 is “really” the last period, since everyone knows what will
happen in period 2.
• Equilibrium entails High Advertising by each firm in both periods.
Kellogg’s
• The same holds true if we repeat the game any known, finite number
of times.
General Mills
Strategy
None
Moderate
High
None
Moderate
12,12 *
1, 20
20, 1
6, 6
15, -1
9, 0
High
-1, 15
0, 9
2, 2*
Nash Equilibrium
Can collusion work if firms play the
game each year, forever?
• Consider the “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.”
• Each firm agrees to “cooperate” so long as the rival hasn’t
“cheated”, which triggers punishment in all future periods.
• “Tit-for-tat strategy” of copying opponents move from the
previous period dominates “trigger strategy” for:
• Simple to understand
• Never invites nor rewards cheating
• Forgiving: allows cheater to restore cooperation by reversing actions
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
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!
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
Benefits & Costs of Cheating
• Cheat - Cooperate = 8 - 10/i
• 8 = Immediate Benefit (20 - 12 today)
• 10/i = PV of Future Cost (12 - 2 forever after)
• If Immediate Benefit > PV of Future Cost
•
Pays to “cheat”.
• If Immediate Benefit  PV of Future Cost
• Doesn’t pay to “cheat”.
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
Key Insight
• Collusion can be sustained as a Nash
equilibrium when game lasts infinitely
many periods or finitely many periods
with uncertain “end”.
• Doing so requires:
•
•
•
•
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
Homogeneous products
Bertrand oligopoly
Known identity of customers
Known identity of competitors
Firm 2
One-Shot Bertrand
(Nash) Equilibrium
Strategy Low Price High Price
0,0*
20,-1
Firm 1 Low Price
High Price -1, 20
15, 15
Firm 2
Repeated Game
Equilibrium
Strategy Low Price High Price
0,0
20,-1
Firm 1 Low Price
High Price -1, 20
15, 15*
Real World Examples of Collusion:
OPEC
One-Shot Cournot
(Nash) Equilibrium
Repeated Game
Equilibrium
Assuming a Low
Interest Rate
Saudi Arabia Saudi Arabia
• Cartel founded in 1960 by Iran, Iraq, Kuwait, Saudis and
Venezuela: “to co-ordinate and unify petroleum policies
among Members in order to secure fair and stable prices”
• Absent collusion: PCompetition < PCournot (OPEC) < PMonopoly
Venezuela
Strategy
High Q
Med Q
Low Q
High Q
5, 3
6, 7
8, 1
Med Q
9,4
12,10*
10, 18
Low Q
3, 6
20, 8
18, 15
Venezuela
Strategy
High Q
Med Q
Low Q
High Q
5, 3
6, 7
8, 1
Med Q
9,4
12,10
10, 18
Low Q
3, 6
20, 8
18, 15*
OPEC’s Demise
40
35
Low Interest
Rates
High Interest
Rates
30
25
20
15
10
5
0
1970
-5
1972
1974
1976
1978
Real Interest Rate
1980
1982
1984
Price of Oil
1986
Simultaneous-Move Bargaining
•
•
•
•
Management and a union are negotiating a wage increase.
Strategies are wage offers & wage demands.
Simultaneous, one-shot move at making a deal.
Successful negotiations lead to $600 million in surplus (to be split
among the parties), failure results in a $100 million loss to the firm
and a $3 million loss to the union.
• Experiments suggests that, in the absence of any “history,” real
players typically coordinate on the “fair outcome”
• When there is a “bargaining history,” other outcomes may prevail
Three Nash Equilibria
in Normal Form
Management
Union
Strategy W = $10 W = $5
W = $10 100, 500* -100, -3
W = $5 -100, -3 300, 300*
W = $1 -100, -3 -100, -3
W = $1
-100, -3
-100, -3
500, 100*
Single Offer Bargaining
• Now suppose the game is sequential in nature, and
management gets to make the union a “take-it-orleave-it” offer.
• Analysis Tool: Write the game in extensive form
•
•
•
•
•
Summarize the players
Their potential actions
Their information at each decision point
The sequence of moves and
Each player’s payoff
To get The Game in Extensive Form
Step 1: Management’s Move
Step 2: Add the Union’s Move
Step 3: Add the Payoffs
Accept
Union
Reject
100, 500
-100, -3
10
Firm
5
Union
Accept
Reject
1
Accept
Union
Reject
300, 300
-100, -3
500, 100
-100, -3
Step 4: Identify Nash Equilibriums
Outcomes such that neither player has an incentive
to change its strategy, given the strategy of the other
Accept
100, 500
Union
Reject
-100, -3
Accept
300, 300
10
Firm
5
Union
Reject
1
Accept
-100, -3
500, 100
Union
Reject
-100, -3
Step 5: Find the Subgame
Perfect Nash Equilibriums
Outcomes where no player has an incentive to change its strategy,
given the strategy of the rival, that are based on “credible actions”:
not the result of “empty threats” (not in its “best self interest”).
Accept
100, 500
Reject
-100, -3
Union
10
Firm
5
Union
Accept
Reject
1
Accept
Union
Reject
300, 300
-100, -3
500, 100
-100, -3
Re-Cap
• In take-it-or-leave-it bargaining, there is a
first-mover advantage.
• Management can gain by making a take-it
or leave-it offer to the union. But...
• Management should be careful, however;
real world evidence suggests that people
sometimes reject offers on the the basis of
“principle” instead of cash considerations.
Pricing to Prevent Entry:
An Application of Game Theory
Two firms: an incumbent and potential entrant.
Identify Nash and then Subgame Perfect Equilibria.
-1, 1
Hard
Incumbent
Enter
Soft
Entrant
Out
5, 5*
0, 10
Establishing a reputation for being unkind to entrants
can enhance long-term profits.
It is costly to do so in the short-term, so much so that it
isn’t optimal to do so in a one-shot game.
The Value of a Bad Reputation:
Price Retaliation
• In early 1970s General Foods’ Maxwell House
dominated the non-instant coffee market in the Eastern
USA, while Proctor & Gamble’s Folgers dominate
Western USA.
• In 1971 P&G started advertising & distributing Folgers
in Cleveland and Pittsburgh.
• GF’s immediately increased advertisement & lowered
prices (sometimes below cost) in these regions &
midwestern cities (Kansas City) where both marketed.
• GF’s profit dropped from 30% in 1970 to –30% in
1974. When P&G reduced its promotional activities,
GF’s price increased and profits were restored.