Game Theory

Necessity Never Made a Good Bargain.- Benjamin Franklin

Mike Shor Lecture 13

Game Theory - Mike Shor

EconomicsAllocation of scarce resources

Many buyers & many sellers traditional markets

Many buyers & one seller auctions

One buyer & one seller bargaining

Game Theory - Mike Shor

Role of Game TheoryDesign non-traditional marketsMarket Design, Inc.

Charles River Associates

NERA Economic Consulting

Open Options

LECG

Game Theory - Mike Shor

The Move to Game-Theoretic BargainingBaseballEach side submits an offer to an arbitrator who must chose one of the proposed resultsMeet-in-the-MiddleEach side proposes its worst acceptable offer and a deal is struck in the middle, if possibleForced FinalIf an agreement is not reached by some deadline, one party makes a final take-it-or-leave-it offer

Game Theory - Mike Shor

Bargaining & Game Theory

Art:Negotiation

Science:BargainingCAVEAT: Limited Applicability

Game Theory - Mike Shor

The Contribution of Game TheoryImportance of rules:The rules of the game determine the outcome

Diminishing pies:The importance of patience

Estimating payoffs:Trust your intuition

Game Theory - Mike Shor

Take-it-or-leave-it OffersConsider the following bargaining game over a cake:I name a take-it-or-leave-it split.If you accept, we tradeIf you reject, no one eats!

Under perfect information, there is a simple rollback equilibrium

Game Theory - Mike Shor

Take-it-or-leave-it Offerspacceptreject1-p , p0 , 0

Game Theory - Mike Shor

RollbackConsider the subgame:Accept: 1-p , pReject: 0 , 0You will accept if p>0, reject otherwiseRollback: I will offer the smallest acceptable piece (i.e. almost none)

What if you make the take-it-or-leave-it offer?

Game Theory - Mike Shor

Take-it-or-leave-it OffersSimple to solveUnique outcomeUnrealisticIgnore real bargainingAssume perfect informationNot credible If you reject my offer, will I really just walk away?

Game Theory - Mike Shor

Counteroffers and Diminishing PiesIn general, bargaining takes on a take-it-or-counteroffer procedure

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

Multi-stage BargainingBargaining over division of a cake

I offer a proportion, p, of the cakeIf rejected, you counteroffer (and of the cake melts)Payoffs: In first period: 1-p , p In second period: (1-)(1-p) , (1-)p

Game Theory - Mike Shor

RollbackWhat happens in period 2?Since 2 is the final period, this is just like a take-it-or-leave-it offer:

You will offer me the lowest price that I will accept, leaving you with all of 1-and leaving me with almost 0

What do I do in the first period?

Game Theory - Mike Shor

RollbackGive you at least as much surplus Your surplus if you accept in the first period is p

Accept if: Your surplus in first period Your surplus in second periodp 1-

Game Theory - Mike Shor

EquilibriumIf there is a second stage, you get 1- and I get 0.

You will reject any offer in the first stage that does not offer you at least 1-.

In the first period, I offer you 1-.

Note: the more patient you are (the slower the cake melts) the more you receive now!

Game Theory - Mike Shor

First or Second Mover Advantage?

Are you better off being the first to make an offer, or the second?

Game Theory - Mike Shor

Example: Cold DayIf =1/5 (20% melts)Period 2: You offer a division of 1,0You getall of remaining cake = 0.8I get 0= 0In the first period, I offer 80%You get 80% of whole cake= 0.8I get 20%of whole cake= 0.2

Game Theory - Mike Shor

Example: Hot DayIf =4/5 (80% melts)Period 2: You offer a division of 1,0You getall of remaining cake = 0.2I get 0= 0In the first period, I offer 20%You get 20% of whole cake= 0.2I get 80%of whole cake= 0.8

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!If players are impatientFirst mover is better off!

Game Theory - Mike Shor

Information

Why doesnt this happen?Time has no meaningLack of information about values!Reputation-building in repeated settings!COMMANDMENTIn any bargaining setting, strike a deal as early as possible!

Game Theory - Mike Shor

ExamplesBritish Pubs and American Bars

Civil LawsuitsIf both parties can predict the future jury award, can settle for same outcome and save litigation fees and timeIf both parties are sufficiently optimistic, they do not envision gains from trade

Game Theory - Mike Shor

Uncertainty I:Civil TrialPlaintiff sues defendant for $1MLegal fees cost each side $100,000If each agrees that the chance of the plaintiff winning is :Plaintiff: $500K-$100K = $ 400KDefendant:-$500K-$100K = $-600KIf simply agree on the expected winnings, $500K, each is better off

Game Theory - Mike Shor

Civil TrialWhat if both parties are too optimistic?Each thinks that their side has a chance of winning:Plaintiff: $750K-$100K = $ 650KDefendant:-$250K-$100K = $-350KNo way to agree on a settlement!

Game Theory - Mike Shor

Uncertainty II:Non-monetary UtilityLabor 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 UtilityEach side has a reservation priceLike in civil suit: expectation of winningThe reservation price is unknownOne must:Consider non-monetary payoffsProbabilistically determine best offerBut probability implies a chance that no bargain will be made

Game Theory - Mike Shor

LessonsRules of the game uniquely determine the bargaining outcome

Which rules are better for you depends on patience, information

Delays are always less profitable:Someone must be wrong

Game Theory - Mike Shor