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Lecture7.pptxApplications of Rationalizability and Iterated Dominance
Econ 402
Yuhta Ishii
Introduction
Last lecture: provided basic definitions of rationalizability/iterated dominance in an abstract setting.
Key take aways:
rationalizability and iterated dominance are equivalent.
Sometimes rationalizability leads to unique prediction.
Sometimes, it doesn’t.
Today: Apply rationalizability in some concrete examples.
Reading: Chapter 8
Cournot Duopoly
Suppose that two firms produce the same product.
Firm 1 and 2 simultaneously choose respective quantities: .
No cost of production (for simplicity).
Given these quantities, the total quantity is and the price is determined by the demand curve:
Cournot Duopoly
Since costs are zero, each firm’s utility function is just given by:
How do firms choose quantity in this market?
Benchmark
Before we solve the duopoly model, suppose we just have a monopolist:
One monopolist who faces a demand curve:
Monopolist does not face any competition.
How does such a monopolist choose quantity?
The Monopoly Case
The optimal quantity, , here satisfies:
Notice the usual tradeoff in the monopoly problem:
Higher quantity entails higher amount sold, but lower price.
The monopolist at optimum equates marginal revenue to marginal cost.
Back to Duopoly
Now suppose that there are two firms in the market.
How does the amount supplied change?
To solve for the supply decisions of each firm, we use iterated dominance/rationalizability.
Iterated Dominance/Rationalizability
We first look for the set :
The pure strategies that are not strictly dominated in the original game.
To compute this set, we first look at pure strategies that are best responses to a pure strategy of the opponent firm.
Best Responses
Suppose that you are firm 1 and that firm 2 produces .
Then what is firm 2’s best response?
To solve this, we need to solve the following problem:
The optimal quantity then is given by:
Best Responses
Suppose that you are firm 1 and that firm 2 produces .
Then what is firm 2’s best response?
Similarly, if firm 1 produces , the firm 2’s best response is:
Best Responses in Cournot Duopoly
Higher the quantity of the opponent, the less I want to produce:
Intuitively, higher quantity of the opponent decreases the price lower marginal revenue.
Best Responses in Cournot Duopoly
Now using the above, we know that the quantities are all in .
These pure strategies are all best responses to some pure strategy of firm 2.
What about the quantities ?
We know that such pure strategies are not best responses against any pure strategy.
But we do not know whether they are best responses against a mixed belief.
We now show that they are not, and in fact they are strictly dominated.
Strictly Dominated Strategies in Cournot Duopoly
Claim: For any is strictly dominated by .
To prove this, we need to show that for all ,
To see this, note that if we take the derivative of the left hand side with respect to , we get:
Strictly Dominated Strategies in Cournot
But for all and any .
This means that the utility function
Is strictly decreasing in in the interval for all values of .
Strictly Dominated Strategies in Cournot
This then means that: whenever for all ,
This shows that any pure strategy strictly above 60 is strictly dominated.
Combined with our first observation, we see that
Similarly, .
Iterated Dominance
But with iterated dominance, we shouldn’t stop there.
Now we are faced with a smaller game where each player chooses strategies in .
We look for strictly dominated strategies in this reduced game.
Strictly Dominated Strategies in Reduced Game
We first determine what pure strategies are the best responses to pure strategies in this reduced game:
Now notice that all strategies in [30,60] are best responses to pure strategies.
So .
What about ?
Strictly Dominated Strategies in Reduced Game
Claim: All are strictly dominated by 30.
The idea is exactly the same as in the previous analysis.
We need to show that for all ,
Strictly Dominated Strategies in Reduced Game
To see this, for every , we show that
is strictly increasing in in the interval [0,30)
Take the derivative of the above with respect to :
As a result, all strategies are strictly dominated.
Pattern
Now a pattern emerges:
In the first step, we eliminated half of the strategies on the right.
In the second step, we eliminated half of the strategies on the left.
In the third step, we will eliminate half of the strategies on the right.
We can keep doing this indefinitely, until we eliminate all but one point.
What strategy remains?
Comparison with Benchmark
Notice that the prediction according to iterated strict dominance is that each firm chooses .
So total supply is: 80.
Price under duopoly is 40.
In comparison with the monopoly model, a monopolist supplies far less:
Monopolist only supplies .
Price under the monopolist is 60.
The monopolist’s price is far higher than under duopoly.
Consumers like more competition by the sellers because they pay lower prices!
We will come back to this in a couple lectures to see the intuition why.
