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Chapter 10
Game Theory: Inside Oligopoly
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Oligopoly
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Learning Objectives
Apply
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Apply normal form and extensive form representations of games to formulate decisions in
strategic environments that include pricing, advertising, coordination, bargaining, innovation,
product quality, monitoring employees, and entry.
Distinguish
Distinguish among dominant, secure, Nash, mixed, and subgame perfect equilibrium strategies,
and identify such strategies in various games.
Identify
Identify whether cooperative (collusive) outcomes may be supported as a Nash equilibrium in a
repeated game, and explain the roles of trigger strategies, the interest rate, and the presence of
an indefinite or uncertain final period in achieving such outcomes.
Prisoner's Dilemma
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Overview of Games and Strategic Thinking
Game theory is a general framework to aid decision making when
agents’ payoffs depend on the actions taken by other players.
Games consist of the following
components:
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•
•
•
•
•
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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Game Environments
• Players’ planned decisions are called strategies.
• Payoffs to players are the profits or losses resulting from strategies.
The order of Decisions in Games is Important
• Simultaneous-move game: A game in which each player makes decisions without the
knowledge of the other players’ decisions. aka Bertrand duopoly game.
• Sequential-move game: A game in which one player makes a move after observing the
other player’s move.
Frequency of Interaction in Games
• One-shot game: A game in which the underlying game is played only once.
• Repeated game: A game in which the underlying game is played more than once.
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Simultaneous-Move, One-Shot Games: Theory
Strategy
• 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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A Normal Form Game
Player 1
Player 2
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Strategy
a
b
c
A
B
C
12,11
11,12
14,13
11,10
10,11
12,12
10,15
10,13
13,14
Normal-Form Game (Table 10-1)
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Normal-Form Game (Table 10-1)
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Normal-Form Game (Table 10-1)
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Normal-Form Game (Table 10-1)
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Normal-Form Game (Table 10-1)
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Normal-Form Game (Table 10-1)
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Normal-Form Game (Table 10-1)
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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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Dominant Strategy (Table 10-1)
Dominant strategy
•
A strategy that results in the highest payoff to a player regardless of the opponent’s action.
Player A has a dominant strategy: Up.
Player B has no dominant strategy.
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Secure Strategy (Table 10-1)
Secure strategy
• A strategy that guarantees the highest payoff given the worst possible scenario.
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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Nash Equilibrium Strategy (Table 10-1)
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.
A Nash equilibrium results when Player A’s plays “Up” and Player B
plays “Left”.
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Normal-Form Game (Table 10-1)
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Normal-Form Game (Table 10-1)
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Application of One-Shot Games:
Pricing Decisions (Table 10-2)
• 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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Why not collude?
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Application of One-Shot Games:
Advertising and Quality Decisions (Table 10-3)
• A Nash equilibrium results when both firms “Advertise”.
• Collusion would not work because this is a one-shot game; if you and your rival “agreed” not to
advertise each of you would have an incentive to cheat on the agreement.
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Application of One-Shot Games:
Coordination Decisions (Table 10-4)
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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Application of One-Shot Games:
Monitoring Employees (Table 10-5)
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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Key Insights
Look for dominant strategies.
Put yourself in your rival’s shoes.
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A Market-Share Game
Two managers want to maximize market share: i ∈ {1,2}.
Strategies are pricing decisions
– S1 = {1, 5, 10}.
– S2 = {1, 5, 10}.
Simultaneous moves.
One-shot game.
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The Market-Share Game
in Normal Form
Manager 1
Manager 2
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Strategy
P=$10
P=$5
P=$1
P=$10
.5, .5
.8, .2
.9, .1
P=$5
.2, .8
.5, .5
.8, .2
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
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P = $1
.1, .9
.2, .8
.5, .5
Key Insight
• Game theory can be used to analyze situations where
“payoffs” are nonmonetary!
• We will, without loss of generality, focus on
environments where businesses want to maximize
profits.
–Hence, payoffs are measured in monetary units.
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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.
Due to the time value of money, payoffs must be appropriately
discounted. The value of a dollar earned during the first repetition of
the game is worth more than a dollar earned in later repetitions.
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Review of Present Value
When a firm earns the same profit, π , in each period over an infinite
time horizon, the present value of the firm is:
PVFirm
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 1 i 


 i 
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Supporting Collusion with Trigger Strategies
(Table 10-7) 1
• 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: 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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Supporting Collusion with Trigger Strategies
(Table 10-7) 2
• 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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Sustaining Cooperative Outcomes with Trigger
Strategies
Suppose a one-shot game is infinitely repeated and the interest rate is i.
Coop

, the maximum
Further, suppose the “cooperative” one-shot payoff to a player
is
one-shot payoff if the player cheats on the collusive
 Cheat , the oneCheat
Coop



1
outcome is
N
 .
shot Nash equilibrium payoff is , and Coop
N


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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Supporting Collusion with Trigger Strategies In Action
(Table 10-7) 1
Q: What are firm A’s profits if it
cheats on the collusive
agreement?
• 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.
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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Supporting Collusion with Trigger Strategies in Action
(Table 10-7) 2
Q: What are firm A’s profits if it does not cheat on the
collusive agreement?
10 1  0.4 
10
10
A: 10 +

  
 $35

0.4
1  0.4  1  0.4 
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Supporting Collusion with Trigger Strategies in Action
(Table 10-7) 3
Q: Does an equilibrium result where the firms charge the high price in each
period?
$50  $35, the present value of firm A’s profits is higher if A cheats on
A: Since
the
collusive agreement. In equilibrium, both firms will charge low price and earn
zero profit each period.
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Factors Affecting Collusion in Pricing Games
1
Sustaining collusion via punishment 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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Factors Affecting Collusion in Pricing Games
•
•
•
•
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Number of firms in the market.
Firm size.
History of the market.
Punishment mechanisms.
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Finitely Repeated Games
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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Games with an Uncertain Final Period (Table 10-9)
• Suppose the probability that the game will end after a given play is θ , where
0    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:
A
Cheat
 50 
10

  ACoop
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Repeated Games with a Known Final Period: Endof Period Problem (Table 10-9)
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.
Consequently, in the last period players will behave just as they would in a oneshot game.
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Applications of the End-of-Period Problem
Resignations and Quits.
The “Snake-Oil” Salesman.
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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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Sequential-Move Game in Extension Form (Figure 10-1)
If Player A chooses Up, then Player B
can choose either Up or Down.
If Player A chooses Down, then Player
B can choose either Up or Down.
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Equilibrium Characterization
(Figure 10-1)
Nash Equilibrium.
Player A: Down.
Player B: Down, if player A chooses Up,
and Down if Player A chooses Down.
Is this Nash equilibrium reasonable?
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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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.
• A subgame perfect equilibrium is a Nash equilibrium that
involves only credible threats.
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Equilibrium Characterization (Figure 10-1)
Subgame Perfect Equilibrium.
Player A: Up.
Player B: Up, if player A chooses Up,
and Down if Player A chooses
Down.
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Application of Multistage Games: The Entry Game
(Figure 10-2)
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.
Nash Equilibrium II:
Player A: In.
Player B: Soft, if player A chooses In.
Credible. This is subgame perfect
equilibrium.
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