CHAPTER 10
Game Theory:
Inside Oligopoly
Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
Chapter Outline
Chapter Overview
• Overview of games and strategic thinking
• Simultaneous-move, one-shot games
– Theory
– Application of one-shot games
• Infinitely repeated games
– Theory
– Factors affecting collusion in pricing games
– Application of infinitely repeated games
• Finitely repeated games
– Games with an uncertain final period
– Games with a known final period: the end-of-period problem
• Multistage games
– Theory
– Applications of multistage games
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Introduction
Chapter Overview
• Chapter 9 examined market environments
when only a few firms compete in a market,
and determined that the actions of one firm
will impact its rivals. As a consequence, a
manager must consider the impact of her
behavior on her rivals.
• This chapter focuses on additional manager
decisions that arise in the presence of
interdependence. The general tool developed
to analyze strategic thinking is called game
theory.
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Overview of Games and Strategic Thinking
Game Theory Framework
• Game theory is a general framework to aid
decision making when agents’ payoffs depends
on the actions taken by other players.
• Games consist of the following components:
–
–
–
–
–
Players or agents who make decisions.
Planned actions of players, called strategies.
Payoff of players under different strategy scenarios.
A description of the order of play.
A description of the frequency of play or interaction.
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Overview of Games and Strategic Thinking
Order of Decisions in Games
• Simultaneous-move game
– Game in which each player makes decisions
without the knowledge of the other players’
decisions.
• Sequential-move game
– Game in which one player makes a move after
observing the other player’s move.
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Overview of Games and Strategic Thinking
Frequency of Interaction in Games
• One-shot game
– Game in which players interact to make decisions
only once.
• Repeated game
– Game in which the same players interact to make
decisions more than once.
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Simultaneous-Move, One-Shot Games
One-Shot Games: Theory
• Strategy
– A decision rule that describes the actions a player
will take at each decision point.
• Normal-form game
– A representation of a game indicating the players,
their possible strategies, and the payoffs resulting
from alternative strategies.
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Simultaneous-Move, One-Shot Games
Normal-Form Game
Set of players
Player B’s strategies
Player B
Strategy
Player A
Player A’s strategies
Left
Right
Up
10, 20
15, 8
Down
-10 , 7
10, 10
Player B’s
possible
payoffs
from
strategy
“right”
Player A’s possible payoffs
from strategy “down”
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Simultaneous-Move, One-Shot Games
Possible Strategies
• Dominant strategy
– A strategy that results in the highest payoff to a
player regardless of the opponent’s action.
• Secure strategy
– A strategy that guarantees the highest payoff given
the worst possible scenario.
• Nash equilibrium strategy
– A condition describing a set of strategies in which
no player can improve her payoff by unilaterally
changing her own strategy, given the other players’
strategies.
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Simultaneous-Move, One-Shot Games
Dominant Strategy
Player B
Strategy
Player A
Left
Right
Up
10, 20
15, 8
Down
-10 , 7
10, 10
Player A has a dominant strategy: Up
Player B has no dominant strategy
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Simultaneous-Move, One-Shot Games
Secure Strategy
Player B
Strategy
Strategy
Player A
Left
Right
Up
10, 20
15, 8
Down
-10 , 7
10, 10
Player A’s secure strategy: Up … guarantees at least a $10 payoff
Player B’s secure strategy: Right … guarantees at least an $8 payoff
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Simultaneous-Move, One-Shot Games
Nash Equilibrium Strategy
Player B
Strategy
Player A
Left
Left
Right
Right
Up
10, 20
15, 8
Down
-10 , 7
10, 10
A Nash equilibrium results when Player A’s plays “Up”
and Player B plays “Left”
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Simultaneous-Move, One-Shot Games
Application of One-Shot Games:
Pricing Decisions
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -10
High price
-10 , 50
10, 10
A Nash equilibrium results when both players charge “Low price”
Payoffs associated with the Nash equilibrium is inferior from the
firms’ viewpoint compared to both “agreeing” to charge
“High price”: hence, a dilemma.
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Simultaneous-Move, One-Shot Games
Application of One-Shot Games:
Coordination Decisions
Firm B
Strategy
Firm A
120-Volt Outlets
90-Volt Outlets
120-Volt Outlets
$100, $100
$0, $0
90-Volt Outlets
$0 , $0
$100, $100
There are two Nash equilibrium outcomes associated with this game:
Equilibrium strategy 1: Both players choose 120-volt outlets
Equilibrium strategy 2: Both players choose 90-volt outlets
Ways to coordinate on one equilibrium:
1) permit player communication 2) government set standard
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Simultaneous-Move, One-Shot Games
Application of One-Shot Games:
Monitoring Employees
Worker
Strategy
Manager
Work
Shirk
Monitor
-1, 1
1, -1
Don’t Monitor
1, -1
-1, 1
There are no Nash equilibrium outcomes associated with this game.
Q: How should the agents play this type of game?
A: Play a mixed (randomized) strategy, whereby a player randomizes
over two or more available actions in order to keep rivals from
being able to predict his or her actions.
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Simultaneous-Move, One-Shot Games
Application of One-Shot Games:
Nash Bargaining
Union
Management
Strategy
0
50
100
0
0, 0
0, 50
0, 100
50
50 , 0
50, 50
-1, -1
100
100, 0
-1, -1
-1, -1
There three Nash equilibrium outcomes associated with this game:
Equilibrium strategy 1: Management chooses 100, union chooses 0
Equilibrium strategy 2: Both players choose 50
Equilibrium strategy 3: Management chooses 0, Union chooses 100
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Infinitely Repeated Games
Infinitely Repeated Games: Theory
• An infinitely repeated game is a game that is
played over and over again forever, and in
which players receive payoffs during each play
of the game.
• Disconnect between current decisions and
future payoffs suggest that payoffs must be
appropriately discounted.
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Infinitely Repeated Games
Present Value Analysis Review
• When a firm earns the same profit, 𝜋, in each
period over an infinite time horizon, the
present value of the firm is:
1+𝑖
𝑃𝑉𝐹𝑖𝑟𝑚 =
𝜋
𝑖
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Infinitely Repeated Games
Supporting Collusion with Trigger Strategies
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -40
High price
-40 , 50
10, 10
The Nash equilibrium to the one-shot, simultaneous-move
pricing game is: Low, Low
When this game is repeatedly played, it is possible for firms to
collude without fear of being cheated on using trigger strategies.
Trigger strategy: a strategy that is contingent on the past play of a
game and in which some particular past action “triggers” a different
action by a player.
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Infinitely Repeated Games
Supporting Collusion with Trigger Strategies
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -40
High price
-40 , 50
10, 10
Trigger strategy example: Both firms charge the high price, provided
neither of us has ever “cheated” in the past (charge low price).
If one firm cheats by charging the low price, the other player will
punish the deviator by charging the low price forever after.
When both firms adopt such a trigger strategy, there are conditions
under which neither firm has an incentive to cheat on the collusive
outcome.
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Infinitely Repeated Games
Trigger Strategy Conditions
to Support Collusion
• Suppose a one-shot game is infinitely repeated and the
interest rate is 𝑖. Further, suppose the “cooperative”
one-shot payoff to a player is 𝜋 𝐶𝑜𝑜𝑝 , the maximum oneshot payoff if the player cheats on the collusive outcome
is 𝜋 𝐶ℎ𝑒𝑎𝑡 , the one-shot Nash equilibrium payoff is 𝜋 𝑁 ,
𝜋𝐶ℎ𝑒𝑎𝑡 −𝜋𝐶𝑜𝑜𝑝
1
and 𝐶𝑜𝑜𝑝 𝑁 ≤ .
𝜋
−𝜋
𝑖
Then the cooperative (collusive) outcome can be
sustained in the infinitely repeated game with the
following trigger strategy: “Cooperate provided that no
player has ever cheated in the past. If any player cheats,
“punish” the player by choosing the one-shot Nash
equilibrium strategy forever after.
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Infinitely Repeated Games
Supporting Collusion with Trigger Strategies
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -40
High price
-40 , 50
10, 10
Suppose firm A and B repeatedly play the game above, and the
interest rate is 40 percent. Firms agree to charge a high price in
each period, provided neither has cheated in the past.
Q: What are firm A’s profits if it cheats on the collusive agreement?
A: If firm B lives up to the collusive agreement but firm A cheats,
firm A will earn $50 today and zero forever after.
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Infinitely Repeated Games
Supporting Collusion with Trigger Strategies
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -40
High price
-40 , 50
10, 10
Q: What are firm A’s profits if it does not cheat on the collusive
agreement?
A: 10 +
10
1+0.4
+
10
1+0.4 2
+⋯=
10 1+0.4
0.4
= $35
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Infinitely Repeated Games
Supporting Collusion with Trigger Strategies
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -40
High price
-40 , 50
10, 10
Q: Does an equilibrium result where the firms charge the high price
in each period?
A: Since $50 > $35, the present value of firm A’s profits are higher
if A cheats on the collusive agreement. In equilibrium both firms
will charge low price and earn zero profit each period.
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Infinitely Repeated Games
Factors Affecting Collusion in
Pricing Games
• Sustaining collusion via trigger strategies is
easier when firms know:
– who their rivals are, so they know whom to
punish, if needed.
– who their rival’s customers are, so they can “steal”
those customers with lower prices.
– when their rivals deviate, so they know when to
begin punishment.
– be able to successfully punish rival.
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Infinitely Repeated Games
Factors Affecting Collusion in
Pricing Games
•
•
•
•
Number of firms in the market
Firm size
History of the market
Punishment mechanisms
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Finitely Repeated Games
Finitely Repeated Games: Theory
• Finitely repeated games are games in which a
one-shot game is repeated a finite number of
times.
• Variations of finitely repeated games: games in
which players
– do not know when the game will end;
– know when the game will end.
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Finitely Repeated Games
Games with Uncertain Final Period
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -40
High price
-40 , 50
10, 10
Suppose the probability that the game will end after a given play is
𝜃, where 0 < 𝜃 < 1.
An uncertain final period mirrors the analysis of infinitely repeated
games. Use the same trigger strategy.
No incentive to cheat on the collusive outcome associated with a
finitely repeated game with an unknown end point above, provided:
10
𝐶ℎ𝑒𝑎𝑡
Π𝐴
= 50 ≤
= Π𝐴 𝐶𝑜𝑜𝑝
𝜃
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Finitely Repeated Games
Repeated Games with a Known Final
Period: End-of-Period Problem
Firm B
Strategy
Firm A
Low price
High price
Low price
0, 0
50, -40
High price
-40 , 50
10, 10
When this game is repeated some known, finite number of times
and there is only one Nash equilibrium, then collusion cannot work.
The only equilibrium is the single-shot, simultaneous-move Nash
equilibrium; in the game above, both firms charge low price.
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Multistage Games
Multistage Games: Theory
• Multistage games differ from the previously
examined games by examining the timing of
decisions in games.
– Players make sequential, rather than simultaneous,
decisions.
– Represented by an extensive-form game.
• Extensive form game
– A representation of a game that summarizes the
players, the information available to them at each
stage, the strategies available to them, the sequence
of moves, and the payoffs resulting from alternative
strategies.
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Multistage Games
Theory: Sequential-Move Game in
Extension Form
Player A payoff
Player B payoff
(10,15)
Decision node for player A
denoting the beginning
of the game
B
A
(5,5)
(0,0)
Player A feasible strategies:
B
Up
Down
Player B’s decision nodes
Player B feasible strategies:
Up, if player A plays Down and Down, if player A plays Down
Up, if player A plays Up and Down, if player A plays Up
(6,20)
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Multistage Games
Equilibrium Characterization
(10,15)
B
A
(5,5)
(0,0)
Nash Equilibrium
B
Player A: Down
Player B: Down, if player A chooses Up,
and Down if Player A chooses Down
Is this Nash equilibrium reasonable?
(6,20)
No! Player B’s strategy involves a non-credible threat since if A plays Up,
B’s best response is Up too!
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Multistage Games
Subgame Perfect Equilibrium
• A condition describing a set of strategies that
constitutes a Nash equilibrium and allows no
player to improve his own payoff at any stage
of the game by changing strategies.
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Multistage Games
Equilibrium Characterization
(10,15)
B
A
B
Subgame Perfect Equilibrium
Player A: Up
Player B: Up, if player A chooses Up,
and Down if Player A chooses Down
(5,5)
(0,0)
(6,20)
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Multistage Games
Application of Multistage Games:
The Entry Game
Nash Equilibrium I:
Player A: Out
Player B: Hard, if player A chooses In
Non-credible, threat since if A plays
In, B’s best response is Soft
(−1,1)
B
A
(5,5)
Nash Equilibrium II:
Player A: In
Player B: Soft, if player A chooses In
Credible. This is subgame perfect equilibrium.
(0,10)
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Conclusion
• In one-shot games resulting payoffs are sometimes lower
than they would be if players could collude
– Players play according to their dominant strategy, which often
corresponds to the “cheating” strategy
• In infinitely repeated games players can collude
by using trigger strategies given a sufficiently low
interest rate
• In finitely repeated games players can only use
trigger strategies to collude if the players are
uncertain as to when the game ends.
– Collusion cannot be supported when players know
when the game ends due to unraveling
– In sequential, or multi-stage, games collusion can only
be supported when threats are credible
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