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Lecture8.pptxNash 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?’’
