it’s due on 4/2 noon
Problem 1: [20 pts]
Part a: [10 pts] Consider the following normal form game, which we call game G1:
D E
A 1, 1 0, 0
B 2, 1 −1,−1
Consider also a different normal form game in which player 1 has the additional option of playing
strategy C, which we call game G2:
D E
A 1, 1 0, 0
B 2, 1 −1,−1
C x, y a, b
Are there some possible values of x, y, a, b for which all of the Nash equilibria of game G2 give
strictly worse payoff to player 1 (row player) than the Nash equilibrium of game G1? If so, provide
an example of possible values of x, y, a, b. If not, justify why not.
Part b: [10 pts] Consider the following normal form game.
L M R
U 1, 0 1, 2 0, 1
D 0, 3 0, 1 2, 0
K −1, 0 2, 0 3,−1
Find all the rationalizable strategies for players 1 (row player) and 2 (column player). Justify your
answer, showing each step of elimination.
Part c: [10 pts, Extra Credit] Suppose that in a two player normal form game player 1 has strategies
A1, B1, C1, D1 and player 2 has strategies A2, B2, C2, D2. Suppose we know that in this game,
1. A1 is a best response to A2, B1 is a best response to B2;
2. and A2 is a best response to B1, B2 is a best response to A1.
Can we be sure that A1, B1, A2, B2 are rationalizable? If yes, justify. If no, provide an explicit
example.
Problem 2: [20 pts] Consider the following n player game. Each player has two pure strategies,
{0, 1}. The utilities of each player is as follows. Player 1 obtains 1 utility if he chooses a strategy
different from that chosen by player 2 and otherwise obtains a utility of 0.1 Similarly, any player
i < n obtains a utility of 1 if he chooses a strategy different from that chosen by player i + 1,
and otherwise obtains a utility of 0. Finally player n obtains a utility of 1 if he chooses a strategy
different from that chosen by player 1, and otherwise obtains a utility of 0.
Part a: [10 points] Find all Nash equilibria in pure strategies of this game when n = 3. Justify both
why the Nash equilibria you found are Nash equilibria and why there are no other Nash equilibria.
Part b: [10 points] Find all Nash equilibria in pure strategies of this game when n = 4. Justify both
why the Nash equilibria you found are Nash equilibria and why there are no other Nash equilibria.
1Note that the strategy of player 3 does not affect player 1’s utility.
Problem 3: [30 pts] Consider two firms that compete via Bertrand competition. The demand of
consumers is given by:
Q(p) = 150−p.
Firms 1 and 2 both have marginal costs of production of 15. Each firm 1 and 2 choose prices p1 ≥ 0
and p2 ≥ 0 simultaneously. Consumers buy from the firm with the lowest price. If the firms choose
the same price, then the firms share the consumers equally.
For example, if both firms choose a price of p1 = p2 = 20, then each firm obtains profits of
150−20
2
(20−15).
Part a: [5 pts] Compute the best response functions of each firm in this game.
Part b: [5 pts] What are all of the Nash equilibria of this game? Justify your answer.
Part c: [5 pts] Are any of the Nash equilibria that you found in part b Pareto efficient? If yes,
explain why. If not, provide an example of a strategy profile that Pareto dominates each such Nash
equilibria.
Part d: [5 pts] Now suppose that each firm can only set prices in increments of one dollar.2 What
are the Nash equilibria of this game?
Part e: [10 pts] Go back to the original game where firms can set any price that is a positive real
number. Now suppose that another firm, firm 3, with marginal cost 20 enters the market. All three
firms compete via Bertrand competition. Give an example of a Nash equilibrium, and justify your
answer.
Problem 4: [30 pts] Two companies compete via Cournot competition. Each firm simultaneously
chooses a quantity q1, q2 ≥ 0 to produce. Given quantities of q1, q2 chosen by the two firms, each firm
can sell each of the units of the good at a price (in dollars) given by the inverse demand function:
P(q1 + q2) = 150−q1 −q2.
Suppose that firm 1 has a marginal cost of production of 10 dollars and firm 2 has a marginal cost
of production of 20 dollars. As a result, each firm’s utility (profit) is given by:
u1(q1, q2) = (150−q1 −q2)q1 −10q1
u2(q1, q2) = (150−q1 −q2)q2 −20q2.
Part a: [10 pts] Compute and write the best response function for each firm. Show your work.
Part b: [10 pts] Solve for all of the Nash equilibria of this game. What is the price that is charged
by the firms in this Nash equilibrium? Show your work.
Part c: [10 pts] Suppose now that the government thinks that prices are too high because the firms
are producing too little. To incentivize more production, the government implements a subsidy. For
each unit of the good sold, the government promises to pay each firm an additional 5 dollars. What
are the Nash equilibria of this new game? Is the government successful in lowering the prevailing
price?
2Prices must be some positive integer.
Rationalizability and Iterated Dominance
Economics 402 (Spring 2020)
Yuhta Ishii
Pennsylvania State University
Game
Guess a whole number between 1 and 100.
The person whose guess is closest to 2/3 of the average in the classroom wins 10 dollars.
Write your name and your guess on a sheet and hand it in.
We will discuss this later in the class.
Introduction
We previously talked about the idea of strict dominance.
Rational behavior involves at least avoiding strictly dominated strategies.
But sophisticated players reason beyond this.
Reading: Ch. 7
Example 1:
Suppose player 1 is rational.
Can we say which strategy he will choose?
No, for each of his strategies there is a belief that makes it optimal.
Example 1:
What about player 2?
Can we say which strategy he will choose?
We don’t know for sure, but now we know that he will choose only between and .
Example 1:
We as an analyst came to the conclusion that 2 will not play or .
But since player 1 also knows that 2 is rational, and 2 has the payoffs in the above normal form, he should also conclude that will never play .
Example 1:
We as an analyst came to the conclusion that 2 will not play .
But since player 1 also knows that 2 is rational, and 2 has the payoffs in the above normal form, he should also conclude that will never play .
Example 1:
We as an analyst came to the conclusion that 2 will not play .
But since player 1 also knows that 2 is rational, and 2 has the payoffs in the above normal form, he should also conclude that will never play .
Example 1:
But now if 1 knows that 2 will never play , then player 1 should never play .
is strictly dominated in this smaller game.
So if player 1 knows that player 2 is rational, and player 1 is rational, then player 1 will never play .
Example 1:
Player 2 will then reason similarly: Player 2 knows
that player 1 knows that player 2 is rational.
that player 1 is rational.
So player 2 will realize that player 1 will never play .
Example 1:
Given this, player 2 will realize that he should play .
The only surviving strategy for player 1 is and the surviving strategy for player 2 is .
This general procedure is what is called the iterated elimination/deletion of strictly dominated strategies.
Iterated Elimination/Deletion of Strictly Dominated Strategies
Step 1: Delete all strictly dominated strategies for each player. Call the set of strategies that remain for player .
Step 2: Consider the reduced game, where each player only plays strategies in . Eliminate all strictly dominated strategies for each player in this reduced game. Call the remaining strategies for player , .
Repeat this process until no more strategies can be eliminated.
Call the surviving set of strategies for player , .
This set will be called the set of rationalizable strategies for player i.
More on this name in the following slide.
Rationalizability
An alternative way to describe this iteration procedure is what is called rationalizability.
Remember last time, we saw that:
a strategy is in if and only if it is also in .
In words, a strategy is not strictly dominated if and only if it is a best response for some belief.
So the iterated elimination of strictly dominated strategies is the same as the following procedure:
Rationalizability Procedure
Step 1: Keep all strategies that are best responses for some belief. Call the set of strategies that are kept for player .
Step 2: Consider the reduced game, where each player only plays strategies in . Keep for each player the set of strategies that are best responses for some belief in the reduced game. Call the remaining strategies for player , .
Repeat this process until no more strategies can be eliminated.
Call the surviving set of strategies for player , .
This set will be called the set of rationalizable strategies for player i.
These are called rationalizable, because these are the strategies that can be rationalized as best responses with some belief over opponents’ rationalizable strategies.
Rationalizability and Iterated Elimination of Strictly Dominated Strategies
As I said before, these two are exactly the same procedure, because of the fact that .
So in order to find the rationalizable strategies, you can just perform the iterated elimination of strictly dominated strategies.
Rationalizable Strategy Profiles
A strategy profile is rationalizable if each player plays a rationalizable strategy.
We denote this set as .
For example, the rationalizable strategy profile in the above game was: (B,Z).
Some Examples
Example 1
What are the rationalizable strategies for each player:
Example
What are the rationalizable strategies for each player:
Example 2
What are the rationalizable strategies for each player:
Example 2
What are the rationalizable strategies for each player:
Example 2
What are the rationalizable strategy profiles?
Example 2
What are the rationalizable strategy profiles?
Example 3:
What are the rationalizable strategy profiles of this game?
All strategy profiles
{(A,X), (A,Y), (B,X),(B,Y)}
{(A,X), (A,Y)}
{(A,X), (B,Y)}
Example 3:
What are the rationalizable strategy profiles of this game?
All strategy profiles
{(A,X), (A,Y), (B,X),(B,Y)}
{(A,X), (A,Y)}
{(A,X), (B,Y)}
What are the rationalizable strategies for each player?
Example: Eisner-Katzenberg game
Battle of the Sexes
What are the rationalizable strategies for each player?
,
,
Battle of the Sexes
What are the rationalizable strategies for each player?
,
,
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Common Knowledge
Introduction
The idea of common knowledge is at the heart of iterated elimination/deletion of strictly dominated strategies (and thus also rationalizability).
Some fact is common knowledge if
each player knows that fact;
each player knows that all players know the fact;
each player knows that all players know that all players know the fact;
Etc.
Common Knowledge in Iterated Elimination
What we are assuming is that both payoffs and rationality are common knowledge:
For example, we were able to delete in this game because we assumed:
Player 2 knows both
that player 1 knows that player 2 is rational.
that player 1 is rational.
Common Knowledge Assumptions in Game Theory
Common knowledge assumption allow us to sharpen our predictions:
We implicitly use it in iterated elimination of strictly dominated strategies/rationalizability.
However, this is oftentimes a simplifying assumption:
Many economic contexts don’t feature common knowledge of rationality or of payoffs.
We assume it for simplicity for now.
Much of modern game theory explores how to relax this unrealistic assumption.
Hat Game
To illustrate the importance of common knowledge assumptions, let’s play a game.
3 players.
Each player is given a hat to wear, which is either red or blue.
Each player cannot see the color of his own hat, but can see the color of the other players’ hats.
Hat game: Analysis
Hat Game
There were three red hats: I denote this as (R,R,R)
I first asked each of you, “Do you know the color of your own hat?’’
Each one answered no.
Now when I announced that “at least one of you is wearing a red hat,’’ player 3 is able to conclude the color of his/her hat to be red.
Why?
Hat Game
What’s going on? Consider the situation after I made the announcement.
What is player 3 thinking?
He sees two red hats and so he knows that either the color configuration is (R,R,R) or (R,R,B).
Hat Game
Suppose hypothetically that the true color configuration were (R,R,B).
Let’s replay this game.
Player 1 would still be uncertain about his/her hat.
Claim: Player 2 would be able to tell that his/her hat is red.
Player 2 is uncertain in this situation between (R,R,B) and (R,B,B)
Player 2 knows that because player 1 was uncertain, his hat must be red.
Here player 2 knows that player 1 knows that there is at least one red hat.
Hat Game
Given this, player 3 once (s)he sees that player 2 is uncertain, should conclude that the true configuration is (R,R,R)!
What is going on?
Common Knowledge
The idea of common knowledge is much stronger than just the idea of knowledge.
Notice that when I revealed to the players that “there is at least one person with a red hat,’’ I revealed some thing that everyone already knew.
Yet, it had an effect on what player 3 was able to conclude!
Before the announcement, every player was uncertain about the color of his/her hat.
Common Knowledge
What happened?
The revelation that “at least one person is wearing a red hat’’ made this fact common knowledge.
Everyone already knew this fact.
But now everyone knows that everyone knows it.
Everyone knows that everyone knows that everyone knows it, etc.
Common Knowledge
More precisely, consider the following statement:
“Player 2 knows that player 1 knows that there is at least one red hat.”
This fact is uncertain to player 3 before the announcement.
The announcement made this statement certain to player 3.
This example illustrates the importance of not just knowledge but common knowledge.
Back to the “Choose 1-100 game.”
Why didn’t the prediction of the game coincide with what actually happens in the classroom?
One reason why iterated deletion of strictly dominated strategies may not be a good prediction is that typically in many real world scenarios, rationality and the normal form are not common knowledge.
Common Knowledge
Nash Equilibrium
Yuhta Ishii
Econ 402
Pennsylvania State University
Introduction
Reading: Ch. 9, Ch. 10
So far we have restricted attention to the concept of rationalizability (iterated elimination of strictly dominated strategies) in analyzing
This however, has limitations for certain games:
Introduction (Cont.)
Recall that in the Battle of the Sexes,
The rationalizable strategy profiles are:
Note that at each rationalizable strategy profile, since each strategy is rationalizable, there is a belief about opponent’s play that rationalizes that strategy:
But at a strategy profile like , these beliefs are incorrect.
Introduction (Cont.)
The Nash equilibrium concept additionally imposes that indeed these beliefs are correct.
John Nash in seminal work proposed the idea of a Nash equilibrium.
Later won a Nobel prize in economics for this contribution.
Nowadays, Nash equilibrium is ubiquitous in economics.
Definition of Nash Equilibrium
A strategy profile, is a Nash equilibrium if for each player is a best response against . In other words, for each , and all pure strategies ,
In words, a strategy profile is a Nash equilibrium if every player is playing a best response against the (correct) belief that the opponents are playing the Nash equilibrium strategy profile.
Examples
Example 1:
Consider again the Battle of Sexes game.
Here the Nash equilibria are:
Notice that each of these strategy profiles are a rationalizable strategy profile, but not all rationalizable strategy profiles are Nash equilibria.
E.g. is not a Nash equilibrium.
Discussion
The last point is general and extremely important.
Nash equilibria are always rationalizable strategy profiles!
There may be rationalizable strategy profiles that are not Nash equilibria.
You may think about why the above statement is true, the proof is not too hard.
The above is useful because in looking for Nash equilibria, it is oftentimes helpful to first compute the rationalizable strategies.
Then look for Nash equilibria within these rationalizable strategies.
What is the number of Nash equilibria in these games?
1,2,1,1
2,2,1,1
2,2,2,1
2,2,1,2
1
2
3
4
What is the number of Nash equilibria in these games?
1,2,1,1
2,2,1,1
2,2,2,1
2,2,1,2
1
2
3
4
Midterm 1 Exam Info
Mean: 76
Standard Deviation: 23
Highest Score: 117
Will hand out the exams later.
If you’re really curious about your score, send me an email and I can email you your score.
Midterm 1 Exam Info
How many Nash equilibria are there?
0
1
2
3
4
Another example
How many Nash equilibria are there?
0
1
2
3
4
Another example
Relationship between Rationalizability and NE (Nash equilibria)
Introduction
We discussed before that all Nash equilibria are rationalizable strategy profiles.
So this means that:
In looking for NE, we can first find all the rationalizable strategy profiles, and look for NE within the set of all rationalizable strategy profiles.
Why are all NE rationalizable strategy profiles?
The idea is simple: suppose that is a NE.
By definition, is a best response against and similarly, is a best response against .
This means that and are both not strictly dominated.
Why are all NE rationalizable strategy profiles?
As a result, both and survive the first round of elimination of strictly dominated strategies.
But in the reduced game, neither nor are strictly dominated.
So again and survive the second round of elimination, etc.
So and survive every step of elimination: is a rationalizable strategy profile.
Example:
The following game is a game to illustrate the above point concretely.
The unique NE is: (B,Z).
Example:
We first eliminated X because it is strictly dominated by Z.
Notice that we don’t eliminate B nor Z.
Why?
Because B is a best response against Z, Z is a best response against B.
Best responses to pure strategies are never eliminated.
Example:
In the second round, we eliminated A.
Notice that we don’t eliminate B nor Z again.
Why?
B and Z both survived first round.
In the reduced game, B is a best response against Z, Z is a best response against B.
Best responses are not strictly dominated.
Example:
In the third round, we eliminate Y.
Notice that we don’t eliminate B nor Z again.
Why?
Same reasons as before.
Example:
In the third round, we eliminate Y.
Notice that we don’t eliminate B nor Z again.
Why?
Same reasons as before.
Nash Equilibria in Games with more players
So far, we have studied Nash equilibria in games with two players.
Of course, Nash equilibria also applies to games with three or more players.
Idea is exactly the same: it is strategy profile in which all players are best responding against the Nash equilibrium strategy profile of opponents.
Example: a simple partnership game
Three friends simultaneously decide whether to exert effort or not. If all three friends exert effort, then they win a prize that generates 100 utils for each friend. Otherwise, the friends do not win any prize. Exerting effort is costly and decreases utility by 1 util.
How many Nash equilibria does this game have?
More Applications of Nash Equilibrium
Cournot Duopoly
Recall the Cournot duopoly example that we analyzed using rationalizability in the last lecture notes.
Let’s solve for the Nash equilibrium of this game.
Note that we already know that we’ll find (40,40):
We already know that the unique rationalizable strategy profile was (40,40).
Finding this was a bit difficult.
But we can find the Nash equilibrium in a straightforward way.
Cournot Duopoly: Nash Equilibrium
Remember the definition of a Nash equilibrium.
is a Nash equilibrium if
is a best response against ;
And is a best response against .
Cournot Duopoly: Nash equilibrium
The fact that is a best response against means that:
solves the following maximization problem:
If firm 2 chooses , we solve for the best response by equating the derivative of the above with respect to equal to zero:
Cournot Duopoly: Nash equilibrium
Solving for we get:
Similarly, we also get:
Cournot Duopoly: Nash equilibrium
If is a Nash equilibrium, then
;
.
Plugging in from before we get:
Cournot Duopoly: Nash Equilibrium
Solving for the simultaneously, we get:
One can also see Nash equilibrium as the intersection points of the best response functions.
See graphs drawn in lecture.
Cournot Oligopoly: Nash Equilibrium
Lets extend this example a bit further.
This is additional material not covered in the book.
Suppose that the demand curve is the same as before, but now there are three firms that are competing with each other.
Firm ’s utility function:
What are the Nash equilibria of this game?
Cournot Oligopoly: Nash Equilibrium
A Nash equilibrium is a triple such that
is a best response against ,
is a best response against
is a best response against .
As in the duopoly case, we solve for the best response of each firm given an arbitrary choice of quantities by the opposing firms.
Cournot Oligopoly: Nash Equilibrium
Suppose that firms and choose quantities and respectively.
What is the best response of firm 1?
Firm 1 wants to solve the following maximization problem:
To solve this, we set the derivative of the above with respect to equal to zero:
Cournot Oligopoly: Nash Equilibrium
Solving for , we get the best response function:
Similarly, we get best response functions for 2 and 3:
Cournot Oligopoly: Nash Equilibrium
If is a Nash equilibrium, then we have to have the following three equations satisfied simultaneously:
Solving for the Nash equilibrium is a matter of solving these equations for three variables.
Cournot Oligopoly: Nash Equilibrium
Here we will solve for a symmetric Nash equilibrium.
A Nash equilibrium in which all firms choose the same quantity.
Turns out that all Nash equilibria will be symmetric, but don’t worry about this.
If you’re curious, you can try to show that all Nash equilibria must be symmetric.
Notice also that this was the case in the duopoly.
So if is a symmetric Nash equilibrium, then we must have:
Cournot Oligopoly: Nash Equilibrium
We then find that .
The unique symmetric Nash equilibrium is .
Compared to the duopoly case, each firm produces less.
However, the total supply with duopoly is 90.
The price is 30.
The price has lowered as a result of more competition!
This is consistent with the previous finding that duopoly reduces the price relative to monopoly.
Cournot Oligopoly: Nash Equilibrium
We can extend this example even further.
Suppose that there are n firms.
Now solve for the symmetric Nash equilibria of this game.
You will find that the unique symmetric Nash equilibrium satisfies:
So the unique symmetric Nash equilibrium is for every firm to produce: .
Cournot Oligopoly: Nash Equilibrium
As a result, the total supply in the Nash equilibrium is:
The price in the Nash equilibrium is:
The price gets driven down to zero as n becomes arbitrarily large.
More competition drives down the profits that each firm can obtain in Nash equilibrium!
As becomes huge, these profits are driven down to zero!
Bertrand Duopoly
Another model of competition between two firms that is very important is the Bertrand duopoly model.
In this model, firms decide on prices rather than quantities.
In a strategic situation involving multiple firms, this matters.
Remember with a monopolist, this didn’t matter.
Bertrand Duopoly
There are two firms.
Each firm simultaneously decides a price.
Both firms produce the same product, so consumers buy from the firm with the lowest price:
If the firms choose the same price, half consumers go to 1 and half buy from 2.
If the lowest price is , consumers’ demand:
Bertrand Duopoly
Suppose that each firm faces a marginal cost of production of 10.
So, the utility functions of the firm are:
What are all of the Nash equilibria of this game?
As before, we want to calculate the best responses for each firm.
Given , the price of firm 2, what is the best response of firm 1?
Bertrand Duopoly: Monopoly Benchmark
Before beginning the analysis, lets solve again the monopoly benchmark.
Suppose there’s only one firm selling to consumers who have an inverse demand curve:
The firm has a marginal cost of 10.
Bertrand Duopoly: Monopoly Benchmark
The monopoly wants to maximize profits:
We solve this by differentiating the above with respect to and setting it equal to zero:
Bertrand Duopoly: Monopoly Benchmark
So the monopolist
Charge a price of .
Supplies units.
Obtains a profit of
Bertrand Duopoly
Back to Bertrand duopoly, we need to calculate the best response of firm 1 against the prices chosen by firm 2.
Case 1:
What is the best response?
Best response is . Why?
If the other firm is charging price above the monopoly price, then I can obtain monopoly profits even if firm 1 charges monopoly profits.
This is the best possible profits that firm 1 can obtain, so this is the best response.
Bertrand Duopoly
Case 2:
What is the best response?
There is none! Why?
Bertrand Duopoly
Case 3:
In these cases, there is no best response for the same reason as in Case 2.
Case 4:
Best responses here are any price 10 or above.
Why?
Case 5:
Best responses here are any price strictly above .
At such prices, firm 1 does not produce.
Bertrand Duopoly
So, the best response of firm 1 can be summarized as follows:
Best response of firm 2 can be summarized similarly.
Bertrand Duopoly: Nash Equilibrium
Having solved for the best responses of each firm, it is now easy to see what the Nash equilibria are.
We can see it two ways:
Graphically, by seeing where the best response correspondences cross.
Analytically, determined when a strategy profile is a mutual best response.
Bertrand Duopoly: Nash Equilibrium
Given the previous we see that the unique Nash equilibrium is:
Each firm sets a price of 10, which is exactly equal to marginal cost.
Bertrand Duopoly: Key Takeaways
Each firm chooses price at exactly marginal cost, 10.
The total supply is given by 110.
Profits for both firms are zero!
Bertrand Duopoly: Key Takeaways
Unlike in Cournot competition, introducing one more firm with Bertrand competition drives the profits of all firms down to 0.
This is great for consumers since they now face lower prices.
Intuitively, with two firms, under Bertrand competition, as long as the competitor is pricing above marginal cost, each firm has an incentive to deviate to a price just below the price of the competitor to capture the whole market.
In Cournot however, firms have to raise output (which leads to a significant decrease in price) substantially in order to capture more market share.
Bertrand Duopoly
Lets modify the example slightly.
Recall that in the example before, marginal costs were exactly the same.
Fierce competition drove the prices offered by the firms to marginal cost.
What if marginal costs differed across the two firms?
There are no Nash equilibria.
This seems problematic from an applied standpoint.
Applied analysis in Bertrand oftentimes assume that prices are chosen from a discrete set.
E.g. prices must be set in increments of 1 dollar.
E.g. can’t set a price of .
Bertrand Duopoly with Different Marginal Costs
Suppose that firm 1 has marginal cost of 10 and firm 2 has marginal cost of 20.
Demand remains the same:
Firm 1’s monopoly price is unchanged: .
Firm 2’s monopoly price is:
Bertrand Duopoly with Different Marginal Costs
The best response function of firm 1 is unchanged:
Bertrand Duopoly with Different Marginal Costs
The best response function of firm 2 is different since he has a different marginal cost:
Best Response Curves
As you can see from the picture, there are no Nash equilibria.
This seems problematic for applications.
We can’t make a prediction of how firms will price.
Typically applications will discretize the price space as in the following example.
Bertrand with Discrete Prices
There are two firms. Demand and firms’ marginal costs are the same as before.
Suppose now that firm 1 can choose any positive integer price:
0,1,2,…
Similarly firm 2 can choose any positive integer price:
0,1,2,…
Neither firm can choose a price that involves decimals, e.g. 3.14.
Bertrand with Discrete Prices
With discrete prices, there is now many Nash equilibria.
Bertrand with Discrete Prices
With discrete prices, there is now many Nash equilibria.
Bertrand with Discrete Prices
As one can see from the picture of the best response curves, note that there are many Nash equilibria.
The set of all Nash equilibria are:
Note that in all of these Nash equilibria, firm 1 (the firm with the lower marginal cost) chooses the lower price and supplies the whole market.
The firm with the lower marginal cost wins the whole market.
Bertrand with Multiple Firms
The Bertrand model, like the Cournot model can be extended to arbitrarily many firms.
Suppose we have three firms all with the same marginal cost of 10.
What are the Nash equilibria of this game?
Cournot vs. Bertrand
Which one is more appropriate for applications?
This is important because the predictions substantially differ.
Typically depends on the market:
In industries such as automobiles, pharmaceuticals, etc., firms cannot adjust quantities easily.
Cournot is better here.
Quantity is adjusted today and fixed throughout the production cycle, and prices adjust to meet demand.
On the other hand, in markets where quantities can be adjusted easily,
Bertrand is better.
Prices are set and quantities can be adjusted very easily to meet demand.
Pareto Efficiency
Introduction
So far, we’ve talked about how we think players will play when they play a game.
Main solution concepts: Rationalizability, Nash equilibrium.
We haven’t talked about whether what they actually play is efficient.
Efficiency from a societal perspective.
In other words, what should players do to maximize “societal efficiency’’?
In contrast to Nash equilibrium, Pareto efficiency is a normative concept.
NE: positive concept.
Pareto Efficiency: Definition
We say that a strategy profile Pareto dominates a strategy profile if
for all player ;
for at least one player .
For example, in the Prisoner’s Dilemma, (Cooperate,Cooperate) Pareto dominates (Defect, Defect).
Examples
Pareto Efficiency: Definition
We say that a strategy profile is Pareto efficient if there is no that Pareto dominates .
It seems natural that we would not want the players to play a strategy profile that is Pareto dominated.
So we want players to play (at the bare minimum) Pareto efficient strategy profiles.
Of course, players do play Pareto dominated strategy profiles in Nash equilibria, rationalizable strategy profiles, etc.
Examples
Pareto Efficiency vs. Nash equilibrium
Typically our predictions (Nash equilibrium, etc.) are not Pareto efficient.
For example, Prisoner’s dilemma.
Pareto Efficiency vs. Nash equilibrium
Why are Nash equilibria sometimes not Pareto efficient?
Players in a Nash equilibria just maximize their own utility (best responding to others’ behavior) without thinking about how it impacts others’ utilities.
Your best response might actually really negatively impact others’ utilities.
On the other hand, to determine Pareto efficiency, we think about the possibility of making everyone weakly better off (and at least one person strictly better off).
Pareto Efficiency vs. Nash equilibria
The discrepancy between Pareto efficiency and Nash equilibria:
Suggests an important role for policy interventions in game settings.
Example via Prisoner’s Dilemma
Consider the Prisoner’s dilemma.
Now suppose that we are able to tax an individual for taking the strategy .
In this case, the normal form of the game changes.
We are now able to sustain a new equilibrium of which Pareto dominates the original equilibrium without the tax.
Interventions in a Coordination Game
Similarly consider the coordination game.
There are two Nash equilibria.
One of these is bad.
By imposing a tax on the bad action, we can change the game to get rid of the bad Nash equilibrium.
In this new game, we expect players to always play the good Nash equilibrium while in the original game, players may be stuck in a bad Nash equilibrium.
Similar interventions can be done via subsidies or combination of both taxes and subsidies.
Efficiency
Introduction
We have so far focused on: “what do we think players will do in a game?’’
We should probably also ask: “what would be efficient for players to play (from a societal perspective?’’
Mixed Strategy Nash Equilibrium
Chapter 11
Yuhta Ishii
Economics 402
Introduction
So far, we have focused on Nash equilibria in pure strategies.
Each player plays a strategy with probability one.
Remember that some normal form games don’t have any Nash equilibria in pure strategies.
E.g. matching pennies.
In contrast, there will always be a Nash equilibrium in mixed strategies.
Matching Pennies
There is no NE in pure strategies
Game is a zero-sum game where whatever player 1 wins, player 2 loses.
The sum of payoffs in each cell is 0.
In many such games, typically there is no NE in pure strategies.
Once I know what the other player will play (which is the case in NE), I never have an incentive to stay the “loser’’.
Matching Pennies
In such games, randomization overcomes this problem.
There will always exist some NE in mixed strategies (could be in pure strategies) in every finite normal form game.
In zero-sum games, this is natural.
For example, in matching pennies, you wouldn’t want to be predictable in your play.
So you randomize.
Matching Pennies
Matching Pennies
Matching Pennies
Definition
Property 1: Relationship to (pure strategy) NE
As you might guess, any pure strategy NE (from previous lectures) is still a mixed strategy NE.
A pure strategy NE is just a mixed strategy NE in which each player plays a mixed strategy that assigns probability one to exactly one strategy.
So mixed strategy NE are more general.
Property 2: Best Response Condition
Property 2: Best Response Condition
Property 3: Rationalizability
General Procedure for finding mixed strategy NE
Step 1: Find the set of rationalizable pure strategies by performing iterated elimination of strictly dominated strategies.
Step 2: In the reduced game where each player only plays rationalizable strategies, write equations for each players’ indifference conditions.
Step 3: Solve these equations to determine equilibrium randomization probabilities.
Nash’s Theorem
The important discovery of Nash was the following theorem.
Nash’s theorem: Every finite normal form game (finite number of players and finite number of strategies for each player) has at least one Nash equilibrium in either pure strategies or mixed strategies.
Example: Lobbying Game
Two firms simultaneously and independently decide whether to lobby (L) or not (N) the government in hopes of trying to generate favorable legislation.
Example 2: Tennis serves
Example 2: Tennis Serves
Example 3:
Compute all of the mixed strategy Nash equilibria of the above game.
Applications 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.
Applications 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.
Applications 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.