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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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