Firm behavior in oligopoly markets
“Oligopoly” is a market dominated by a few sellers,
at least some of which are large enough relative to the
entire market to be able to affect the market price.
An important feature of such market is that each firm’s
optimal actions depend on what other firms do.
(Recall this isn’t the case in monopoly or perfect
competition.)
Interdependence of actions of parties involved is not
unique for markets.
Consider wars, card games, tic-tac-toe, chess, etc.
Game theory is a field of math that combines various
tools useful for analyzing strategic behavior
(“strategic” refers to the case when each “player” has to
take the others’ actions into account).
Glossary:
Game – a situation in which in order to achieve desired
result one has to take the others’ actions into account.
Player – a personage involved in a game.
Strategy – a plan of actions to be undertaken.
“Pure strategy” – “Play X”
“Mixed strategy” – Randomization between two or more
pure strategies;
“Play X with a 40% probability and Y with a 60%
probability”
Outcome – a terminal situation resulting from the
players’ actions.
Payoff – the “score” each player gets at the end of the
game.
Equilibrium – a situation (outcome) in which none of the
players wants to change their actions.
Cops catch Billy and Bob breaking into someone’s car. They
also suspect that a week earlier the same two individuals
threatened a convenience store clerk with a gun.
Cops put them in separate cells so they can’t talk to each
other, and offer each the following deal:
• “If you confess while your partner remains silent, we’ll make
sure he serves 5 years while you’ll be set free.”
• “If none of you confesses, we put both of you in jail for the car
break-in, and each of you will serve one year.”
• “If both of you confess, then each of you spends three years
in jail.”
Clearly, for each of the suspects the final outcome depends on
his own choice as well as the choice of the other player.
Therefore it is a game (known as “Prisoners’ dilemma”).
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
The numbers in the table are payoffs.
(First – Billy, Second – Bob)
The payoffs are negative because they denote years
potentially spent in jail.
Let’s start with the situation when the players cannot
communicate. In other words, players know the structure
of the game and the payoffs, but not the actions chosen
by the opponent. However, they can infer each other’s
actions by assuming rationality.
What is the best strategy for Billy?
We can split this question into two:
- If Bob confesses, what should Billy do?
AND
- If Bob doesn’t confess, what should Billy do?
If Bob confesses, what should Billy do?
Billy should confess because (-3) is better than (-5)
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
If Bob doesn’t confess…
Billy should confess because 0 > -1
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
When one strategy leads to a better outcome that any
other strategy regardless of what the other player(s)
do(es), such strategy is called a DOMINANT strategy.
In the above example,
the dominant strategy for Billy is to confess.
What is the best decision for Bob?
Same logic:
If Billy confesses…
Bob should also confess because -3 > -5
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Same logic:
If Billy does not confess…
Bob should confess because 0 > -1
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Conclusion: the dominant strategy for Bob is to confess.
Thus, we naturally arrive at an equilibrium, or the
outcome when none of the players wants to change
his action. (The full name is “Nash equilibrium”.)
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
The right bottom corner would make both players better off,
but playing “not confess” is not safe, therefore that outcome is
NOT an equilibrium, therefore it is not likely to be played.
Techniques of solving for an equilibrium
•Elimination of dominated strategies.
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Techniques of solving for an equilibrium
•Elimination of dominated strategies.
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Techniques of solving for an equilibrium
•Elimination of dominated strategies.
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Techniques of solving for an equilibrium
•“Best response” technique.
If Bob’s strategy…
…then Billy’s best response is
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Techniques of solving for an equilibrium
•“Best response” technique.
If Bob’s strategy…
…then Billy’s best response
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Techniques of solving for an equilibrium
•“Best response” technique.
If Billy’s strategy…
then Bob’s best response
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Techniques of solving for an equilibrium
•“Best response” technique.
If Billy’s strategy…
then Bob’s best response
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Techniques of solving for an equilibrium
•“Best response” technique.
Bob
Confess
Not confess
Confess
-3, -3
0, -5
Not confess
-5, 0
-1, -1
Billy
Any cell with both entries circled is an equilibrium.
Application – airline market
(the numbers in the cells are profits!)
Firm 2
Low price
High price
Low price
4, 3
8, 1
High price
2, 7
6, 5
Firm 1
Application – airline market
(the numbers in the cells are profits!)
Firm 2
Low price
High price
Low price
4, 3
8, 1
High price
2, 7
6, 5
Firm 1
Application – airline market
(the numbers in the cells are profits!)
Firm 2
Low price
High price
Low price
4, 3
8, 1
High price
2, 7
6, 5
Firm 1
Summary of the pricing game:
1. If each firm cares only about its own profit, then the
game arrives at the non-cooperative equilibrium outcome.
2. If firms are able to coordinate, then they can arrive at an
outcome that is better for both firms. This “coordination”
may be explicit or implicit.
The explicit form (a cartel) is illegal.
3. Even if firms form a cartel, each of them has an
incentive to defect (drop the price unilaterally).
This is one of the reasons why any form of collusion is
unstable.
The number of equilibria:
A game may have more than one equilibrium in pure
strategies.
Consider the following game:
Firm 2’s price
High
Firm 1’s Medium
price
Low
High
Medium
Low
6, 7
3, 8
1, 6
7, 4
5, 6
2, 5
8, 0
4, 2
3, 3
Find all the equilibria in pure strategies.
The number of equilibria:
A game may have more than one equilibrium in pure
strategies.
Consider the following game:
Firm 2’s price
High
Firm 1’s Medium
price
Low
High
Medium
Low
6, 7
3, 8
1, 6
7, 4
5, 6
2, 5
8, 0
4, 2
3, 3
Two equilibria: (Medium, Medium) and (Low, Low).
What is the best strategy to play?
• If there is a dominant strategy, play it.
• If you don’t have a dominant strategy, but the game
has a unique Nash equilibrium, play the equilibrium
strategy.
• If you don’t have a dominant strategy, and there is
more than one (or none) equilibria, then you may want
to play a secure strategy.
Secure strategy – a strategy that guarantees the highest payoff
in the worst possible scenario.
You are firm 1. What is the best strategy to play?
Firm 2
Firm 1
U
D
L
3, 2
1, 6
R
8, 1
6, 2
Firm 1’s dominant strategy is “Up” therefore
it should play “Up”.
You are firm 1. What is the best strategy to play?
Firm 2
Firm 1
U
D
L
3, 3
1, 4
R
4, 1
6, 3
Firm 1 doesn’t have a dominant strategy but
the game has a unique equilibria, (Up, Left).
You are firm 1. What is the best strategy to play?
Firm 2
Firm 1
U
D
L
3, 3
1, 4
R
4, 1
6, 3
Firm 1 doesn’t have a dominant strategy but
the game has a unique equilibrium, (Up, Left).
Firm 1 should play “Up”.
You are firm 1. What is the best strategy to play?
Firm 2
Firm 1
U
D
L
3, 3
1, 4
R
4, 1
6, 6
None of the first two rules applies therefore we use
rule #3 (find the secure strategy).
The worst possible payoff from playing “Up” is 3.
The worst possible payoff from playing “Down” is 1.
“Up” is the secure strategy. Firm 1 should play “Up” .
Sequential games
So far, we have dealt with games in which players’
choices were simultaneous.
But what if in the game on slide 212 firm 1 gets to
pick the price first, and then commits to its decision?
In that case, it is more convenient to present the
game as a tree (the “extensive form”):
•Each node of the tree corresponds to the point when
one of the players chooses an action.
•A branch of the tree is an action.
•The pairs of numbers at the nine terminal nodes
represent the payoffs of players 1 and 2, respectively.
1
H
M
2
H
6,7
M
3,8
L
2
L
H
1,6
7,4
M
5,6
2
L
H M
2,5 8,0
4,2
L
3,3
“Backward induction”:
If Firm 2 knows Firm 1 has chosen H…
…then Firm 2 will choose M.
1
H
M
2
H
6,7
M
3,8
L
2
L
H
1,6
7,4
M
5,6
2
L
H M
2,5 8,0
4,2
L
3,3
“Backward induction”:
If Firm 2 knows Firm 1 has chosen M…
…then Firm 2 will choose M.
1
H
M
2
H
6,7
M
3,8
L
2
L
H
1,6
7,4
M
5,6
2
L
H M
2,5 8,0
4,2
L
3,3
“Backward induction”:
If Firm 2 knows Firm 1 has chosen L…
…then Firm 2 will choose L.
1
H
M
2
H
6,7
M
3,8
L
2
L
H
1,6
7,4
M
5,6
2
L
H M
2,5 8,0
4,2
L
3,3
Now, Firm 1 knows what to expect of Firm 2 in each case and can
choose the best course of action for itself.
Now, Firm 1 knows what to expect of Firm 2 in each case and can
choose the best course of action for itself.
1
H
M
2
H
6,7
M
3,8
L
2
L
H
1,6
7,4
M
5,6
2
L
H M
2,5 8,0
4,2
L
3,3
Firm 1 should choose M.
This is the most likely combination of choices the players will make.
Market entry decisions.
Preventing competitive entry
Two firms consider entering the market.
Market can accommodate only one firm.
Firms make decisions simultaneously:
Firm 2
Firm 1
Do Not Enter
Enter
Do Not Enter
0, 0
0, 1
Enter
1, 0
-1, -1
Market entry decisions.
Preventing competitive entry
Two firms consider entering the market.
Market can accommodate only one firm.
Firms make decisions simultaneously:
Firm 2
Firm 1
Do Not Enter
Enter
Do Not Enter
0, 0
0, 1
Enter
1, 0
-1, -1
Two equilibria:
(Enter, Do Not Enter) and (Do Not Enter, Enter).
Sequential decisions,
Firm 1 moves first:
Firm 1
Enter
Do Not Enter
Firm 2
Enter
-1,-1
Firm 2
Do Not Enter
1, 0
Enter
0, 1
Do Not Enter
0, 0
Sequential decisions,
Firm 1 moves first:
Firm 1
Enter
Do Not Enter
Firm 2
Enter
-1,-1
Firm 2
Do Not Enter
1, 0
Enter
0, 1
Do Not Enter
0, 0
The first firm that has the opportunity to enter – enters;
the second firm doesn’t.
When the market is too small for two firms, it is
important to enter it first.
A less dramatic example:
Market can accommodate both firms.
Firm 2
Firm 1
Do Not Enter
Enter
Do Not Enter
0, 0
0, 5
Enter
5, 0
1, 1
In this case, each firm has a dominant strategy, “Enter”
so the sequence of moves does not matter.
The unique equilibrium is (Enter, Enter)
If Firm 1 could buy the exclusive rights to
serve this market, up to what amount
would it be willing to pay for those rights?
If the firm is alone, Profit = 5
If it shares the market with another firm, Profit = 1
For the firm, the exclusive rights are worth 5 – 1 = 4
It would agree to pay any amount that is less than
(and maybe equal to) 4.
Entry decisions and government policy:
Airbus and Boeing are both building a new cargo plane.
Airbus
Enter
Do not enter
Enter
10, - 40
250, 0
Do not
enter
0, 200
0, 0
Boeing
The most likely outcome of the game?
Entry decisions and government policy:
Airbus and Boeing are both building a new cargo plane.
Airbus
Enter
Do not enter
Enter
10, - 40
250, 0
Do not
enter
0, 200
0, 0
Boeing
The most likely outcome of the game?
Boeing enters, Airbus doesn’t.
Profits are 250 and 0, respectively.
What if the EU government promises to give Airbus
a $50m subsidy if Airbus builds such a plane?
Airbus
Enter
Do not enter
Enter
10, 10
250, 0
Do not
enter
0, 250
0, 0
Boeing
What is the most likely outcome now?
What if the EU government promises to give Airbus
a $50m subsidy if Airbus builds such a plane?
Airbus
Enter
Do not enter
Enter
10, 10
250, 0
Do not
enter
0, 250
0, 0
Boeing
What is the most likely outcome now?
Both firms enter; Profits are 10 and 10.
A relatively small subsidy by the government
may affect the outcome substantially!
To be continued…