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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 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 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

- 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

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 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

- 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

- 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

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

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

- 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?

- 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 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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 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 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

- 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

- 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

- 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

- 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

- 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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