Game Theory
“Necessity Never Made
a Good Bargain.”
- Benjamin Franklin
Mike Shor
Lecture 11
The Bargaining Problem




If an owner of some object values it less
than a potential buyer, there are
gains from trade  A surplus is created
Example: I value a car that I own at $1000.
If you value the same car at $1500, there is
a $500 gain from trade
Well-established market prices often control
the division of surplus
If such cars are priced at $1200:
$200 to the seller $300 to the buyer
Game Theory - Mike Shor
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The Bargaining Problem
In the absence of markets  bargaining

Bargaining Problem
Determining the actual sale price or surplus
distribution in the absence of markets


Home sales
“Comps” are rarely truly comparable
Labor/management negotiations
Surplus comes from production
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The Bargaining Problem

Importance of rules:
The structure of the game
determines the outcome

Diminishing pies
The importance of patience

Screening and bargaining
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Take-it-or-leave-it Offers
Consider the following bargaining
game for the used car:
 I name a take-it-or-leave-it price.
 If you accept, we trade
 If you reject, we walk away


Under perfect information, there is a
simple rollback equilibrium
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Take-it-or-leave-it Offers
accept
p-1000 , 1500-p
reject
0,0
p
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Rollback

Consider the subgame:
•Accept: p-1000 ,
•Reject: 0 ,
1500-p
0
You will reject if p>1500,
accept otherwise
 Rollback: I will offer highest
acceptable price of 1500


What if you make the take-it-orleave-it offer?
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Take-it-or-leave-it Offers
Simple to solve
 Unique outcome
 Unrealistic

Ignore “real” bargaining
 Assume perfect information

• We do not necessarily know each other’s
values for the car

Not credible
• If you reject my offer,
will I really just walk away?
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Counteroffers and
Diminishing Pies
In general, bargaining takes on a
“take-it-or-counteroffer” procedure
 Multiple-round bargaining games
 If time has value, both parties prefer
trade earlier to trade later
 E.g. Labor negotiations –
later agreements come at a price
of strikes, work stoppages, etc.

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Two-stage Bargaining
Value of car: $1000 me, $1500 you
 I make an offer in period 1
 You can accept the offer or reject it
 If you reject, you can make a
counteroffer in the second period.
 Payoffs

• In first period: p-1000,1500-p
• In second period: (p-1000) , (1500-p)
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Rollback
What happens in period 2?
 In the final period, this is just like a
leave-it-or-take-it offer:
You will offer me the lowest price
that I will accept, p=1000
 This leaves you with 500

• (1500-p)= (1500-1000)
and leaves me with 0
 What do I do in the first period?
Game Theory - Mike Shor
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Rollback

Give you at least as much surplus
Your surplus if you accept
in the first period is 1500-p

Accept if:

p = 1500-500
Note: the more that you value the future,
the less you pay now!


Your surplus in first period
 Your surplus in second period
1500-p  500

p  1500-500
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Example
If =4/5
 Period 2: You offer a price of 1000

• You get
• I get

(4/5) (1500-1000)
0
= 400
=0
In the first period, I offer 1100
• You get
• I get
(1500-1100)
(1100-1000)
Game Theory - Mike Shor
= 400
= 100
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First or Second Mover
Advantage?
In the previous example,
second mover gets more surplus
 What if =2/5?
 Period 2: You offer a price of 1000

• You get
• I get

(2/5)(1500-1000)
0
= 200
=0
In the first period, I offer 1300
• You get
• I get
(1500-1300)
(1300-1000)
Game Theory - Mike Shor
= 200
= 300
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First or Second Mover
Advantage?
Who has the advantage?
 Depends on the value of the future!
 If players are patient:

• Second mover is better off!
• Power to counteroffer is stronger than
power to offer

If players are impatient
• First mover is better off!
• Power to offer is stronger than
power to counteroffer
Game Theory - Mike Shor
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Bargaining Games With
Diminishing Pies
More periods with diminishing pies
 Suppose the same player makes an
offer in each period
 Infinite number of periods


Same point: if players are fully
informed, a deal should occur in
the first round!
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Information
COMMANDMENT
In any bargaining setting,
strike a deal as early as possible!

Why doesn’t this happen?
• “Time has no meaning”
• Lack of information about values!
• Reputation-building in repeated settings!
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Examples

British Pubs and American Bars

Civil Lawsuits
• If both parties can predict the future jury
award, can settle for same outcome and
save litigation fees and time
• If both parties are sufficiently optimistic,
they do not envision gains from trade
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Uncertainty I:
Civil Trial
Plaintiff sues defendant for $1M
 Legal fees cost each side $100,000
 If each agrees that the chance of
the plaintiff winning is ½:

• Plaintiff:
• Defendant:

$500K-$100K = $ 400K
-$500K-$100K = $-600K
If simply agree on the expected
winnings, $500K, each is better off
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Civil Trial
What if both parties are too
optimistic?
 Each thinks that their side has a ¾
chance of winning:

• Plaintiff:
• Defendant:
$750K-$100K = $ 650K
-$250K-$100K = $-350K
No way to agree on a settlement!
 “Delicate Disclosure Game”

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Uncertainty II:
Non-monetary Utility

Labor negotiations are often a simple
game of splitting a known surplus

Company will profit $200K –
how much of this goes to labor?

Rules of the bargaining game uniquely
determine the outcome if money is the
only consideration
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Non-monetary Utility

Each side has a reservation price
• Like in civil suit: expectation of winning
The reservation price is unknown
 One must:

• Consider non-monetary payoffs
• Probabilistically determine best offer
• But – probability implies a chance that no
bargain will be made
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Example: Uncertain
Company Value
Company annual profits are either
$150K or $200K per employee
 Two types of bargaining:

• Union makes a take-it-or-leave-it offer
• Union makes an offer today.
If it is rejected, the Union strikes,
then makes another offer

A strike costs the company 20% of
annual profits
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Take-it-or-leave-it Offer
Probability that the company is
“highly profitable,” i.e. $200K is p
 If offer wage of $150

• Definitely accepted
• Expected wage = $150K

If offer wage of $200K
• Accepted with probability p
• Expected wage = $200K(p)
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Take-it-or-leave-it Offer
Example I

p=9/10
• 90% chance company is highly profitable
Best offer: Ask for $200K wage
 Expected value of offer:
(.9)$200K = $180K


But: 10% chance of No Deal!
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Take-it-or-leave-it Offer
Example II

p=1/10
• 10% chance company is highly profitable
Best offer: Ask for $150K wage
 If ask for $200K
Expected value of offer:
(.1)$200K = $20K


If ask for $150K, get $150K
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Two-period Bargaining


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If first-period offer is rejected:
A strike costs the company 20% of
annual profits
Note: strike costs a high-value company
more than a low-value company!
Use this fact to screen!
Assume (for simplicity):
A strike doesn’t cost the Union anything
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Screening in Bargaining
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What if the Union asks for $170K in the
first period?
Low-profit firm ($150K) rejects
High-profit firm must guess what will
happen if it rejects:
• Best case –
Union strikes and then asks for only $150K
• In the mean time –
Strike cost the company $20K

High-profit firm accepts
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Separating Equilibrium
Only high-profit firms accept in the
first period
 If offer is rejected, Union knows that
it is facing a low-profit firm
 Ask for $150K in second period


Expected Wage:
• $170K (p) + $150K (1-p)
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What’s Happening

Union lowers price after a rejection
• Looks like “Giving in”
• Looks like Negotiating

Actually, the Union is screening
its bargaining partner
• Different “types” of firms have different
values for the future
• Use these different values to screen
• Time is used as a screening device
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Lessons

Rules of the game uniquely
determine the bargaining outcome

Which rules are better for you
depends on patience, information

Delays are always less profitable

But may be necessary to screen
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