Applications of Rationalizability and Iterated DominanceEcon 4.docx

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
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.

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 .

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 ?
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 ,
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.
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A few years back, Equifax Inc. experienced a system breach that
compromised personally identifying data belonging to millions
of customers. For a couple of years,
the Equifax organization had been the leading organization
when it comes to credit reporting. Notably, its ability to assess
the financial health of multiple clients
established a strong background that influenced its rapid
growth. Despite knowing the increasing cyberattacks attributed
to the advancement of the technology,
Equifax faced a lax security posture that exposed multiple data
to attackers who were able to take hold of the security system
and extract terabytes of critical data
(Fruhlinger, 2020). The top executives in the organization were
blamed for this occurrence because it would have been
prevented had they taken the necessary
safeguards. The deep crisis happened for several months,
whereby it was initiated in March of 2017 and was discovered
in July, the same year.
The organization first experienced a hack via a consumer
complaint web portal where the attackers utilized a well- known
vulnerability to move from web portal to
other servers. This was facilitated by the fact that the systems
had not been adequately segmented, an aspect that made hackers
obtain the unencrypted usernames
and passwords, and eventually accessed the full policy. The
vulnerability could have been patched, but the organization had
not put such efforts. Hackers successfully
extracted the encrypted essential information for several months
bearing in mind that Equifax had not renewed encryption
certificate (Whittaker, 2018). It took only a

few months for the attackers to access the multiple Equifax
databases that possessed hundreds of millions of data belonging
to customers. Personally identifiable
information, including social security numbers and driver's
license numbers, was compromised and could be later used to
impersonate the customers. This aspect
could easily wreak havoc to one's records. This situation was
poorly handled because the management took the time to report
the incident, and this led to the
suspicion of whether the top executive was capable of managing
the crisis. If at all Equifax installed the upgraded Apache Struts
Software, this system breach could
have been prevented.
Link Used
1. https://techcrunch.com/2018/12/10/equifax-breach-
preventable-house-oversight-report/
2. https://www.csoonline.com/article/3444488/equifax-data-
breach-faq-what-happened-who-was-affected-what-was-the-
impact.html
References
Fruhlinger, J. (2020, February 12). Equifax data breach FAQ:
What happened, who was affected, what was the impact?
Retrieved from
https://www.csoonline.com/article/3444488/equifax-data-
breach-faq-what-happened-who-was-affected-what-was-the-
impact.html
Whittaker, Z. (2018, December 10). Equifax breach was
‘entirely preventable’ had it used basic security measures, says
house report – TechCrunch.

Retrieved from https://techcrunch.com/2018/12/10/equifax-
breach-preventable-house-oversight-report/
1
2
3 4 5
6 7 6
1
Student paper
https://techcrunch.com/2018/12/10/equif
ax-breach-preventable-house-oversight-
report/
Original source
report/
2
Student paper
https://www.csoonline.com/article/34444
88/equifax-data-breach-faq-what-
happened-who-was-affected-what-was-
the-impact.html
Original source

the-impact.html
3
Student paper
(2020, February 12).
Original source
February 12, 2020
4
Student paper
Equifax data breach FAQ: What
happened, who was affected, what was
the impact?
Original source
Equifax data breach FAQ What
happened, who was affected, what was
the impact
5
Student paper
Retrieved from

the-impact.html
Original source
Retrieved from
the-impact.html
6
Student paper
(2018, December 10).
Original source
(2018, December 10)
anith 100%
Student paper 100%
7

Student paper
Equifax breach was ‘entirely preventable’
had it used basic security measures, says
house report – TechCrunch.
Original source
Equifax breach was ‘entirely preventable’
had it used basic security measures, says
House report – TechCrunch
6
Student paper
Retrieved from
report/
Original source
Retrieved from
report/
Econ 402
Yuhta Ishii
Introduction

Key take aways:
Reading: Chapter 8
Cournot Duopoly
Cournot Duopoly
by:
Benchmark
monopolist:

The Monopoly Case
marginal cost.
Back to Duopoly
original game.

Best Responses
Best Responses
strategy of firm 2.

mixed belief.
dominated.
dominated.

Similarly, .
Iterated Dominance
pure strategies.
So .
What about ?

Pattern
right.
left.
right.
one point.
far less:

pay lower prices!
intuition why.
Econ 402
Yuhta Ishii
Introduction
Key take aways:
Reading: Chapter 8
Cournot Duopoly

Cournot Duopoly
by:
Benchmark
monopolist:
The Monopoly Case
marginal cost.
Back to Duopoly

original game.
Best Responses
Best Responses

strategy of firm 2.
mixed belief.
dominated.

dominated.
Similarly, .
Iterated Dominance
pure strategies.
So .

What about ?
Pattern
right.
left.
right.
one point.

far less:
pay lower prices!
intuition why.
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.

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.
Why?
Same reasons as before.
Example:
In the third round, we eliminate Y.
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:
Solving for we get:
Similarly, we also get:
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?
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.
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:
Solving for , we get the best response function:
Similarly, we get best response functions for 2 and 3:
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.
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:
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.
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: .
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.
The monopoly wants to maximize profits:

We solve this by differentiating the above with respect to and
setting it equal to zero:
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:
The best response function of firm 1 is unchanged:

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.
With discrete prices, there is now many Nash equilibria.

With discrete prices, there is now many Nash equilibria.
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?’’
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 .
will never play .
Example 1:
We as an analyst came to the conclusion that 2 will not play .
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
Example 2

Example 2
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
,
,

Battle of the Sexes
,
,
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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
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.

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