Transcript The Simple Mathematics of Oligopoly

OLIGOPOLY
A market structure in which there are few firms, each of which is large relative to the
total industry.
Key idea: Decision of firms are interdependent.
THE SIMPLE MATHEMATICS OF OLIGOPOLY
Given:
P = 200 – 2Q
TC = 500 + 40Q + 2Q2
What is the profit maximizing price? $160
If this firm’s current price was $170 and it lowered its price,
how would its competition respond?
IF YOU DO NOT KNOW THE COMPETITOR’S RESPONSE, IT IS
DIFFICULT TO PREDICT WHAT THE NEW DEMAND CURVE WILL
BE!!! THEREFORE OUR SIMPLE PROBLEM HAS BECOME A BIT
MORE COMPLICATED!!
GAME THEORY
Game Theory – the study of how individuals make
decisions when they are aware that their actions
affect each other and when each individual takes this
into account.
History: Introduced in 1944 by John von Neumann and
Oskar Morgenstern in “The Theory of Games and
Economic Behavior.”
The work of von Neuman and Morgenstern was
expanded upon by John Nash.
INTRODUCTION TO GAME THEORY
A game is a situation in which a decision-maker must take into
account the actions of other decision-makers. Interdependency
between decision-makers is the essence of a game.
In games people must make strategic decisions. Strategic decisions
are decision that have implications for other people.
Strategy – a decision rule that describes that 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 from alternative
strategies.
COOPERATIVE AND NON-COOPERATIVE GAMES
Non-Cooperative Games are games in which
players cannot enter binding agreements
with each other before the play of the game.
Cooperative Games are games in which
players can enter binding agreements with
each other before the play of the game.
In class we only review non-cooperative
games.
TWO TYPES OF GAMES
Simultaneous move game – Game in which each
player makes decisions without knowledge of the
other players’ decision.
 Examples: Pitching in baseball, Calling plays in football
Sequential move game – Game in which one player
makes a move after observing the other player’s
move.
 Example: Chess
ELEMENTS OF A GAME
1. Set of Players.
2. Order of Play.
3. Description of the information available to any
player at any point during the game.
4. Set of actions available to each player when
making a decision.
5. Outcomes that result from every possible
sequence of actions by the players.
6. A payoff from the outcomes.
7. Strategic situations with the above elements is
considered to be well defined.
ACTIONS, STRATEGIES, AND PAYOFFS
Actions – The set of choices available at each decision in a game.
Pure strategy – a rule that tells the player what action to take at each of
her information sets in the game.
Mixed strategy – when players can choose randomly between the
actions available to them at every information set.
 Example: Play calling in sports is a mixed strategy.
Payoffs, for our purposes, consist of either profits to firms, or
income to individuals. Payoffs can also be characterized in
terms of utility.
SOLVING GAMES:
NASH EQUILIBRIUM
Solution Concept – a methodology for
predicting player behavior.
Nash Equilibrium - a collection of strategies
one for each player, such that every player's
strategy is optimal given that the other players
use their equilibrium strategy.
DOMINANT AND DOMINATED STRATEGIES
Payoff matrix – a matrix that displays the payoffs to each player for
every possible combination of strategies the players could choose.
Dominant Strategy – a strategy that is always strictly better than every
other strategy for that player regardless of the strategies chosen by
the other players.
Dominated Strategy – a strategy that is always strictly worse than
some other strategy for that player regardless of the strategies
chosen by the other players.
WEAKLY DOMINATE STRATEGIES
Weakly dominant strategy - a strategy that is always equal to or
better than every other strategy for that player regardless of the
strategies chosen by the other players.
Weakly Dominated Strategy – a strategy that is always equal to or
worse than some other strategy for that player regardless of the
strategies chosen by the other players.
Prisoner’s Dilemma
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Scenario: Two people are arrested for a
crime
The elements of the game:
The players: Prisoner One, Prisoner Two
The strategies: Confess, Don’t Confess
The payoffs:
– Are on the following slide
– Payoffs read Prisoner 1, Prisoner 2
Prisoner’s Dilemma, cont.
Prisoner 2
Confess
Prisoner 1
Don’t Confess
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Confess
6 years, 6 years
Don’t Confess
1 year, 10 years
10 year, 1 year
3 years, 3 years
Dominant strategy equilibrium: In this game, the dominant
strategy for each prisoner is to confess. So the outcome of
the game is that they each get six years.
This illustrates the prisoner’s dilemma: games in which the
equilibrium of the game is not the outcome the players
would choose if they could perfectly cooperate.
The Advertising Game
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Scenario: Two firms are determining how
much to advertise.
The elements of the game:
The players: Firm 1, Firm 2
The strategies:
– High advertising, low advertising
Advertising Game, Cont.
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The payoffs are as follows: (payoffs read 1,2)
Firm 1
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Firm 2
High
High
40,40
Low
10, 100
Low
100, 10
60,60
Dominant strategy equilibrium: In this game, the
dominant strategy for firm 1 and firm 2 is high. So the
outcome of the game is 40,40.
Again, this is an example of the prisoner’s dilemma. The
equilibrium of the game is not the outcome the players
would choose if they could cooperate.
More Prisoner Dilemmas
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Industrial Organization Examples
– Cruise Ship Lines and the move towards ‘glorious excess’.
Royal Caribbean offers a cruise with an 18 hole miniature golf
course. Princess Cruises has a ship with three lounges, a
wedding chapel, and a virtual reality theater.
– Owners of professional sports teams and the bidding on
professional athletes.
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Non-IO Examples
– Politicians and spending on campaigns.
– Worker effort in teams. The incentive exists to shirk, a
strategy that if followed by all workers, reduces the
productivity of the team.
Iterated Dominant Strategies
What if a dominant strategy does not
exist?
 We can still solve the game by
iterating towards a solution.
 The solution is reached by
eliminating all strategies that are
strictly dominated.
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Example of Iterated Dominance
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Down is Firm 1, Across is Firm 2
F1,F2
High
Medium
Low
High
100,80
95,85
80,100
Medium
85,95
110,105
110,100
Low
80,100
130,110
120,115
Alternative Solution Strategies
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Nash Equilibrium - a strategy combination in which no
player has an incentive to change his strategy, holding
constant the strategies of the other players.
Joint Profit Maximization: This is the objective of a cartel.
Cut-Throat: A strategy where one seeks to minimize the
return to her/his opponent.
Secure Strategy: A strategy that guarantees the highest
payoff given the worst possible scenario.
How does the previous game change when we change the
objectives of the players?
This is one of the advantages of game theory. We do not
have to assume profit maximization. We still need to be
able to identify the objectives of the players.
Sequential Move Game
High
Low
High
40,40
10,100
• Return to the advertising game. What if Firm 1 has firstmover advantage?
• What is the solution?
• Backwards induction: Start at the end of the game and
work backwards.
• If Firm 1 chooses High, Firm 2's best strategy is to pick
High.
• If Firm 1 chooses Low, Firm 2's best strategy is to pick
High.
• Firm 1, knows that Firm 2 will always pick High.
Knowing this, Firm 1 picks High also. Thus the solution
is High, High.
Low
100,10
60,60
More sequential move games
FIRM 1/FIRM 2
Raise price
Keep price
Lower price
Raise price
8,9
10,6
11,5
Keep price
6,12
12,10
9,8
Lower price
7,8
8,11
6,7
• What if firm 1 has first mover advantage? How does this change
the game?
• Solution concept:
Profit max
• If firm 1 were to raise its price, firm 2 would keep
• If firm 1 were to keep its price, firm 2 would lower
• If firm 1 were to lower its price, firm 2 would keep.
• Given this, firm 1 should lower its price and the Nash equilibrium
is lower, keep
More sequential move games
FIRM 1/FIRM 2
Raise price
Keep price
Lower price
Raise price
8,9
10,6
11,5
Keep price
6,12
12,10
9,8
Lower price
7,8
8,11
6,7
Solution concept:
Cut-throat (firm 1 moves first)
If firm 1 were to raise its price, firm 2 would keep
If firm 1 were to keep its price, firm 2 would lower
If firm 1 were to lower its price, firm 2 would lower.
Given this, if firm 1 plays cut-throat, the solution strategy is
lower, lower.
• if firm 1 plays profit-max, the solution strategy is keep, lower.
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Firm and Industry characteristics
that impact collusion
• Number of firms
• More firms increase monitoring costs
• Size of firms
• Smaller firms cannot afford monitoring
• History of the markets
• Tacit collusion cannot work if punishing is ineffective.
• Punishment mechanisms
• Can the punishing firm price discriminate?
• Price discrimination lowers the cost of punishing.
• UNDERSTAND THE SIMPLE CARTEL MODEL AND WHY CARTELS
ARE UNSTABLE
Mixed Strategy
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Pure Strategy is a rule that tells the player what action to
take at each information set in the game.
Mixed strategy allows players to choose randomly between
the actions available to the player at every information set.
Thus a player consists of a probability distribution over the
set of pure strategies.
Examples of mixed strategy games:
 Play calling in sports
 To shirk or not to shirk
The Shirking Game
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Scenario:
A worker is hired but does not wish to work. The firm
will not pay the worker if there is no work, but the firm cannot
directly observe the workers effort level or output.
Players:
The worker, the firm
Strategy:
Work or not work, monitor or not monitor
Payoffs:
Work pays $100, but the worker’s reservation wage is
$40.
Worker can produce $200 in revenue, but it costs $80 to monitor.
The Shirking Game, Cont.
Monitor
Don’t
Monitor
Work
60, 20
60, 100
Shirk
0, -80
100, -100
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There is no dominant
strategy, or iterated
dominant strategy.
There is also no clear
Nash Equilibrium. In
other words, no
combination of actions
makes both sides
happy given what the
other side has chosen.
The Shirking Game, cont.
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There are many mixed strategies. The worker could work with
probability (p) of 0.7, 0.6. 0.25, etc... The same is true for the
firm. Which mixed strategy should they choose?
If the worker is most likely to shirk, the firm should monitor.
Likewise, if the firm is more likely to monitor, the worker should
work. In any scenario, no Nash equilibrium will be found. The
key is to find a strategy that makes the opponent indifferent to
his/her potential choices.
A person is indifferent when the expected return from action A
equals the expected return form action B.
The Firm’s Solution
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How much should the firm monitor?
E(work) = 60p + 60(1-p) = 60
E(shirk) = 0p + 100(1-p) = 100 - 100p
100 - 100p = 60
40 = 100p
p = .40
The worker is indifferent when the probability of
monitoring is 40% and the probability of not
monitoring is 60%.
The Worker’s Solution
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How much should the worker work?
E(monitor) = 20p + -80(1-p) = 100p - 80
E(Not monitor) = 100p + -100(1-p) = 200p - 100
100p -80 = 200p - 100
20 = 100p
p = .2
The firm is indifferent when the probability of working is
20% and the probability of not working is 80%.
How does the cost of monitoring and the worker’s
reservation wage impact behavior?