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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
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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?
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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
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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
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
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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