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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

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game theory

Game Theory

“Necessity Never Made a Good Bargain.”

- Benjamin Franklin

Mike Shor

Lecture 11

the bargaining problem
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

the bargaining problem1
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

Game Theory - Mike Shor

the bargaining problem2
The Bargaining Problem
  • Importance of rules:

The structure of the game determines the outcome

  • Diminishing pies

The importance of patience

  • Screening and bargaining

Game Theory - Mike Shor

take it or leave it offers
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

Game Theory - Mike Shor

take it or leave it offers1
Take-it-or-leave-it Offers

p-1000 , 1500-p

accept

p

reject

0 , 0

Game Theory - Mike Shor

rollback
Rollback
  • Consider the subgame:
      • Accept: p-1000 , 1500-p
      • Reject: 0 , 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-or-leave-it offer?

Game Theory - Mike Shor

take it or leave it offers2
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?

Game Theory - Mike Shor

counteroffers and diminishing pies
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.

Game Theory - Mike Shor

two stage bargaining
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)

Game Theory - Mike Shor

rollback1
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

rollback2
Rollback
  • Give you at least as much surplus
  • Your surplus if you accept in the first period is 1500-p
  • Accept if: Your surplus in first period

 Your surplus in second period

1500-p  500  p  1500-500

  • p = 1500-500
  • Note: the more that you value the future, the less you pay now!

Game Theory - Mike Shor

example
Example
  • If =4/5
  • Period 2: You offer a price of 1000
      • You get (4/5) (1500-1000) = 400
      • I get 0 = 0
  • In the first period, I offer 1100
      • You get (1500-1100) = 400
      • I get (1100-1000) = 100

Game Theory - Mike Shor

first or second mover advantage
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 (2/5)(1500-1000) = 200
      • I get 0 = 0
  • In the first period, I offer 1300
      • You get (1500-1300) = 200
      • I get (1300-1000) = 300

Game Theory - Mike Shor

first or second mover advantage1
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

bargaining games with diminishing pies
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!

Game Theory - Mike Shor

information
Information
  • Why doesn’t this happen?
      • “Time has no meaning”
      • Lack of information about values!
      • Reputation-building in repeated settings!

COMMANDMENT

In any bargaining setting, strike a deal as early as possible!

Game Theory - Mike Shor

examples
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

Game Theory - Mike Shor

uncertainty i civil trial
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: $500K-$100K = $ 400K
      • Defendant: -$500K-$100K = $-600K
  • If simply agree on the expected winnings, $500K, each is better off

Game Theory - Mike Shor

civil trial
Civil Trial
  • What if both parties are too optimistic?
  • Each thinks that their side has a ¾ chance of winning:
      • Plaintiff: $750K-$100K = $ 650K
      • Defendant: -$250K-$100K = $-350K
  • No way to agree on a settlement!
  • “Delicate Disclosure Game”

Game Theory - Mike Shor

uncertainty ii non monetary utility
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

Game Theory - Mike Shor

non monetary utility
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

Game Theory - Mike Shor

example uncertain company value
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

Game Theory - Mike Shor

take it or leave it offer
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)

Game Theory - Mike Shor

take it or leave it offer example i
Take-it-or-leave-it OfferExample 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!

Game Theory - Mike Shor

take it or leave it offer example ii
Take-it-or-leave-it OfferExample 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

Game Theory - Mike Shor

two period bargaining
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

Game Theory - Mike Shor

screening in bargaining
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

Game Theory - Mike Shor

separating equilibrium
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)

Game Theory - Mike Shor

what s happening
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

Game Theory - Mike Shor

lessons
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

Game Theory - Mike Shor

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