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September 29th, 2018
Economics: Notes on Chapter 3

Chapter 3 Notes: Game Theory
Essential Idea:
The payoff of an action often depends not only on the action itself, but also on how it relates to
actions taken by others in the same environment
...
It recognizes that your satisfaction may depend on what others do
...
Moreover, you need to recognize that others will try to
second guess you
...

When stating this rule, we implicitly focused on the benefit and cost that directly accrue to the
person doing the activity
...

Example 1: Burn or Haul? (Two Dominant Strategies)
1
...
They
now each face a choice of what to do with the dead leaves they have collected
...

2
...

3
...
Their houses are close together, so if Jack burns his leaves it imposes an external
cost on Jill
...
The
external cost of each fire amounts to $15 for both Jack and Jill
...
If Jack and Jill burn their leaves they each pay a cost of $31—the $1 labor cost and the
$30 damage from two fires
...
Similarly, if
Jill burns her leaves while Jack hauls his, then Jill pays $16 and Jack pays $35
...
These payoffs can be
summarized as follows:

September 29th, 2018
Economics: Notes on Chapter 3

Jill
Jack

Burn

Haul

Burn
$31 Loss for Both

$16 Loss for Jill;
$35 Loss for Jack

Haul
$16 Loss for
Jack;
$35 Loss for Jill
$20 Loss for
Both

The dominant strategy for both Jack and Jill
is to burn, for by so doing they earn a higher
payoff no matter what option the other
chooses
...


To find out what is best for Jack we need to "read down columns"
...
For example, if Jill burns leaves:
a
...
he loses $31 rather than $35
...
Similarly, if Jill hauls leaves
a
...
he loses $16 instead of $20
...
Note that, no matter what Jill does, Jack does best to burn
...

1
...
then reading across the Burn row we see that Jill's payoff is also higher if she also
burns her leaves
...
Similarly, if Jack hauls his leaves
a
...

i
...

Explanation of Example 1
Some games, like the one above, have a dominant strategy
...

• The dominant strategy for both Jack and Jill is to burn
...
However, the
problem is that this is collectively the worse that they could do—it is much better if they both
hauled leaves
...
This scenario is an example of a social dilemma
...
A social dilemma is a game in which individual and collective interests do not
coincide
...
If every
player were to cooperate then everyone's payoff would increase
...

Dominant strategy:
The strategy in a game that produces better results irrespective of the strategy chosen by one’s
opponent
...
The
prosecutor has only enough hard evidence to convict them of a minor offence, for which the
penalty is 1 year in prison
...
If they both confess, they will get
confessing
...

of 5 years
...

Reginald
Confess
Remain Silent
Ronald
Confess
5 Years for Both
0 Years for
Ronald;
20 Years for
Reginald
Remain
20Years for
1 Year for Both
Silent
Ronald;
0 Year for
Reginald
Example 3: School Project
1
...
Each hour that a student spends working
increases the group mark by 1
...
And they
value the cost of working an extra $30
...
Consider the individual incentives of a student called Andy
...
Two
hours on the project would cost him $60 and only yield him $20
...
The same logic applies to the other
students
...
Could the students do better?

September 29th, 2018
Economics: Notes on Chapter 3

3
...
Specifically, an
hour of work costs him $10 but yields a total benefit of $40 because it increases the
marks of all four students
...
For instance, suppose each
student spends 2 hours working on the project
...

4
...
Each student has a dominant strategy to not work on the project
even if they would collectively do better to spend time on the project
...
In social dilemmas, it is often convenient to refer to one of the strategies as cooperate and
the other as defect
...
To cooperate is to do what is in the collective interest
...
So, hauling leaves, remaining silent and working on the group project are
examples of cooperative strategy
...
To defect is to do what is in a person's best interest
...
So, burning leaves, confessing, and not working on the group projects are
examples of defect
...
Pareto efficiency requires all players to cooperate
...
However, each individual has a personal incentive to defect
...

Commitment Problems
Commitment problem:
• People would do better if they could collectively commit themselves to behave in a way
that is inconsistent with their own material interests
Social dilemmas like the one above are an example of a broader class of problems called
commitment problems
...

Players in a social dilemma would do better to cooperate despite incentive to defect
...

• wants to set his victim free but is afraid he will go to the police
...

• BUT both realize it will no longer be in the victim's interest to keep this promise once he
is free
...


September 29th, 2018
Economics: Notes on Chapter 3

•
•

The kidnapper’s belief that the victim will act in a rational, self-interested way spells
doom for the victim
...

o The blackmailable act serves here as a commitment device
...

The Invisible Hand vs
...

• Self-interested consumers and producers are driven by an invisible hand to produce
greater social good
...

For instance, competition among seller’s fosters attempts to develop better products and cheaper
ways of producing them
...
As others follow new products and methods, their offerings also put
downward pressure on prices
...

In chapter 2, we concluded that free markets result in pareto efficiency
...
Put differently, no individual buyer or seller can influence price
...
Similarly, the worker may have to decide whether or not to accept a
job without being able to influence the wage she would be paid
...
Apple, for example,
has influence on the prices it charges
...
To
illustrate, consider again the example above
...
But,
if Jack hauls his leaves then the "price" to Jill of limiting the smoke damage to $15 is only $1
because she can burn her leaves
...
By the same logic, Jill can influence the price for Jack
...
Jack may want to take account of Jill's
likely action when deciding what he should do with his leaves
...

It is important to recognize that the invisible hand may go awry
...


Repeated Games: Tit-for-Tat
Failing to cooperate in a social dilemma is costly—if someone defects, it removes the
opportunity for pareto efficiency
...
New possibilities emerge if the players interact
repeatedly
...

An agent using this strategy will first cooperate, then subsequently replicate an opponent's
previous action
...
If not, the
agent is not
...

Conditions:
1
...
each of whom can remember what other players have done in previous
interactions
2
...

1
...
if players know exactly how many times they will interact, then mutual
cooperation breaks down
...
However, there are many games where not every player has a dominant strategy
...

Example 4: Burn or Haul? (One Dominant Strategy)
Consider again the scenario in example 1
...
What is predicted to
happen?
Jill
Jack

Burn

Haul

Burn
$31 Loss for Jill;
$41 Loss for Jack
$16 Loss for Jill;
$35 Loss for Jack

Haul
$35 loss for Jill;
$16 Loss for
Jack;
$20 Loss for
Both

If Jill burns, Jack should haul
...
There is no dominant
strategy for Jack—only for Jill (to burn)
...



A good rule for assessing whether a particular combination of strategies constitutes a Nash
equilibrium is to ask whether either player has any incentive to choose a different strategy
...

• However, the example above shows that a Nash equilibrium does not require all players
to have a dominant strategy
...
But, suppose that now both Jack and Jill value the cost
from the damage of one fire as equal to $15 and that of two fires equal
In this game, neither Jack nor Jill has a
to $40
...
If Jack burns then Jill
Jill
Jack

Burn

Burn
$41 Loss for Both

Haul
$35 loss for Jill;
$16 Loss for
Jack;

should haul
...
The same logic applies to Jill
...
This combination of strategies is a Nash equilibrium
because neither Jack nor Jill would have any incentive to change strategy
...
But, suppose that Jack hauls and Jill burns
...
This game has
more than one Nash equilibrium
...
Jack and Jill have to adjust their strategies
depending on each other
...
It is not clear which player this should be—this creates a source of conflict
...
However, there
are alternatives
...
This involves choosing the option
that makes the lowest payoff one can receive as large as possible
...
They have to decide whether to stay out of the Dutch market or enter it
...
Firm 2 has a dominant strategy to enter the Dutch market
...
This is the Nash Equilibrium outcome
...
However, firm 1 may not be sure that firm 2 will choose rationally and enter the market
...
This is far worse than the 90 it would get from entering
...

3
...


September 29th, 2018
Economics: Notes on Chapter 3

a
...
The maximin strategy is one way of dealing with
uncertainty
...
A player that uses the maximin strategy knows with certainty the worst that can
happen
...

Sequential Games
In the examples that we have considered, both players have had to choose their strategies
simultaneously
...

Such games are usually illustrated by a game tree
...

Game of Incomplete information:
• A game where some players have private information about their payoffs or possible
moves in the game
Example:
Suppose you are the owner of a company and are looking to employ a new software
programmer
...
You then publicize the vacancy and receive 10 applications
...
Of the 10 applicants, there are only a few who have the attributes that you want
...
All 10 applicants want the job and so they all have
the incentive to look like a perfect employee
...
You want to find the best applicant; the best applicants want to signal that she is the
best applicant; but the other applicants want you to believe that they are the best
applicant
...
This is a game of incomplete information because you do not know whether the
applicants have the attributes that you are looking for
...
What you can do is distinguish between possible types of applicants
...
These types are sometimes called
Harsanyi types
...
Having distinguished possible Harsanyi types you can then form a belief about the
likelihood of each applicant
...
For instance, you
might think that 50% of applicants are likely to be good and 50% are likely to be bad
...
For instance, you
might think that anyone who gets top marks at university must be a good applicant
...

5
...
First, you need to have correct beliefs and so you need to
correctly interpret the information you have about applicants
...


September 29th, 2018
Economics: Notes on Chapter 3

Types of Games:
1
...
No mutually best response
b
...
My win is your loss
c
...
It is a discoordination game
2
...
A coordination game is one where both players have optimal strategies which are
contingent on the other player’s actions
i
...
This gives rise to two possible outcomes which are mutual best responses
...
Battle of the Sexes

a
...
Neither player has a dominant strategy
ii
...
But this time each player has a different preferred outcome

September 29th, 2018
Economics: Notes on Chapter 3

4
...
Even when one of the equilibria is clearly worse than the other for everybody, we
can still get stuck in the bad one
...

Newton-second conversion is another
...
Coordination failure
• Coordination failure – this is when two players in a coordination game do not manage to
coordinate
• Miscoordination – this is when the two do manage to coordinate, but they coordinate on
the wrong thing:
5
...
Similar to the Pareto Coordination game
b
...
Prisoners’ Dilemma
a
...
And the equilibrium is a bad one! (From the
prisoners’ perspective
...
Ascending bid, or English auctions
a
...

b
...
g
...
Bidding
starts at the reserve price and rises in regular increments (e
...
$5k)
...
The price keeps rising until only one
bidder remains
...

c
...
The seller also pays commission of
around 10%
...
(The
auctioneers much be creating a lot of value for customers to be willing to pay so
much commission
...
Price
i
...
Optimal behavior
i
...
Descending bid, or Dutch auctions
a
...
2m 𝑚P )
...
This represents about $3bn of the
$40bn global cut flower market
...
Structure – Flowers are trucked or flown in from around the world and inspected
by experts
...
) are put on the clocks along with a starting price
...

3
...
Size – In the US alone, government infrastructure spending is about $300 billion
per year
...


September 29th, 2018
Economics: Notes on Chapter 3

b
...
g
...
The firm with the lowest bid wins, and is paid
the amount that it bid
...
We know that a bidder won’t bid all the way up to their true valuation – they will
shade their bid
d
...
We can guess that a bidder with a higher valuation should bid more than a bidder
with a low valuation
4
...
Size – Google’s auctions earn $19 billion per year
...
Structure – Advertisers bid on a ‘cost per click’ basis for AdWords (possibly
combinations of words)
...
The
highest bid wins, but pays a price based on the second highest bid
...
01
...
Quality score – The truth is a bit more complicated because the nominal bid is
adjusted by a Quality Score which reflects Google’s estimates of the number of
clicks, but this doesn’t change the basic idea
...
Optimal behaviour
i
...
Why?
ii
...
You cannot benefit by bidding above your true valuation because if this
helps you win, it is only because someone else also bid above your true
valuation, and this means that you will pay more than your true valuation
and you will lose money
e
...
Given that the bids can be ranked 𝑣N > 𝑣P > 𝑣Q > ⋯ > 𝑣S , this means
that the expected selling price is again 𝑣P
ii
...
An advantage of Vickrey is that the seller gets more information (e
...
all
of the bidders’ valuations)
iv

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