games people play. 13: bargaining in this section we shall learn how to break down a bargain into...

Games People Play.Games People Play.

13: Bargaining13: Bargaining

In this section we shall learnIn this section we shall learnHow to break down a bargain into its fundamental determinants.

Four ways a bargain may be manipulatedVia the surplus to be split

Via our alternative option

Via our opponents alternative option

Via our bargaining skill


Manipulating a Bargaining Solution.Manipulating a Bargaining Solution.

Consider two bargainers A and B that have to agree on how to Consider two bargainers A and B that have to agree on how to split some amount X.split some amount X.

A

B

X

X


Acceptable Offers.Acceptable Offers.

Any acceptable bargain must provide a bargainer with at least the Any acceptable bargain must provide a bargainer with at least the payoff they could receive otherwise.payoff they could receive otherwise.

A

B

X

X

a

b

Solution must lie in here

A can getthiswithoutabargain

B can get this without a bargain


A SolutionA Solution

Given the skill of the bargainers we get a solutionGiven the skill of the bargainers we get a solution

A

B

X

X

a

b

A canget thiswithouta bargain

B can get this without a bargain

Solution

Slope = bargaining skill


Manipulating the Solution.Manipulating the Solution.

It may be possible to increase the surplus to be split.It may be possible to increase the surplus to be split.

A

B

X

X

a

b

X'

X'

S

S'

Your strategy raisesthe surplus


Manipulating the Solution.Manipulating the Solution.

It may be possible to improve your alternative, or, to convince your It may be possible to improve your alternative, or, to convince your

opponent your alternative has improved.opponent your alternative has improved. A

B

X

X

a

b

Sa'

S' You have asuperioralternative


Manipulating the SolutionManipulating the SolutionIt may be possible to lessen your opponents alternative, or, make It may be possible to lessen your opponents alternative, or, make them believe you can.them believe you can.

A

B

X

X

a

b

S

S'

b'

Opponent has an inferior alternative


Manipulating the SolutionManipulating the Solution

You may be able to improve your bargaining “skill”.You may be able to improve your bargaining “skill”.

A

B

X

X

a

S

b

S'

Improved bargaining skill


BargainingBargaining

The Ultimatum Game.The Ultimatum Game.You have $10 to divide between yourself and an anonymous partner.

Half of the players make an offer of a division of the $10, the other half have each to decide to accept or reject the offer.

If you accept the offer you both get whatever you agreed to.

If you reject neither you nor your partner gets anything.



The Ultimatum Game Again.The Ultimatum Game Again.You have $10 to divide between yourself and an known partner.

Each half of the players make an offer of a division of the $10, the other half have each to decide to accept or reject the offer.

If you accept the offer you both get whatever you agreed to.

If you reject neither you nor your partner gets anything.



The Ultimatum Game Equilibrium.The Ultimatum Game Equilibrium.Aware that the recipient of the offer will accept any positive amount the proposer should offer a split of $9.99 and $0.01.

By backwards induction the person receiving the offer should accept any positive amount. Something is better than nothing.


The Nash Bargaining Solution.The Nash Bargaining Solution.

A characterization of the Nash Bargaining A characterization of the Nash Bargaining Solution.Solution.Suppose two bargainers must agree to undertake a joint activity, if the activity is carried out x is generated to be shared between them.If the activity is not carried out the bargainers receive their “threat point payoffs” a and b respectively.Hence the surplus from the activity is s = x – a – b.Suppose each bargainer receives their threat point payoff plus a share ( h and j) of the surplus. That is

A = a + h(x – a – b), and B = b + j(x – a – b).



A characterization of the Nash Bargaining Solution.A characterization of the Nash Bargaining Solution.Obviously (A – a)/(B - b) = h/j(A – a)/(B - b) = h/j

A

B

a

b

Threat point

(A – a)/(B - b) = h/j

Nash Bargaining Solution

X

X



• This is just characterization of solution not an explanation of how it might be achieved.

• Nash viewed Bargaining as a process where individuals sat down together and cooperatively agreed on the properties that a solution to their problem should have.

• Having agreed on a solution it was immediately implemented, or, written into a contract.

A

B

a

b

Threat point

(A – a)/(B - b) = h/j

Nash Bargaining Solution

X

X


The Properties of the Nash Bargaining The Properties of the Nash Bargaining SolutionSolution

Efficiency Efficiency – No mutual gain should go unexploited.– No mutual gain should go unexploited.

Independence of Irrelevant Alternatives Independence of Irrelevant Alternatives –– If an option If an option that wouldn’t be chosen is taken away the outcome that wouldn’t be chosen is taken away the outcome should not change.should not change.Scale invariance Scale invariance – The outcome should not change – The outcome should not change with the units the outcome is measured in.with the units the outcome is measured in.

These properties are intuitively appealing and satisfied by These properties are intuitively appealing and satisfied by Nash’s solution. But, by what mechanism are they Nash’s solution. But, by what mechanism are they achieved?achieved?


Variable Threat Bargaining.Variable Threat Bargaining.

Suppose we take as given Nash’s solution to the Suppose we take as given Nash’s solution to the bargaining problem.bargaining problem.

But we allow the players to make strategic But we allow the players to make strategic moves prior to bargaining. What should they do?moves prior to bargaining. What should they do?


Variable Threat Bargaining.Variable Threat Bargaining.

Strategic moves for player BStrategic moves for player B Raise your own threat point To → T2

Or lower your opponents

To → T2

A

B

a

b

(A – a)/(B - b) = h/jj

X

X

T0

T1

T2

N

N'


Alternating Offers with Value Alternating Offers with Value DecayDecay

Often the longer it takes to reach an agreement the less there is to Often the longer it takes to reach an agreement the less there is to be divided between the bargainers. For example a delay gives a be divided between the bargainers. For example a delay gives a rival an opportunity to “claim” a share of a market, or a rival an opportunity to “claim” a share of a market, or a technological lead may be lost.technological lead may be lost.

With alternating offers the players take turns making moves. With alternating offers the players take turns making moves.

Each moves consists of either accepting their opponents last offer Each moves consists of either accepting their opponents last offer or making an offer of their own. or making an offer of their own.



The ticket-tout example.The ticket-tout example.Suppose a fan wishes to Suppose a fan wishes to watch football game. Each watch football game. Each quarter is worth $25. He has quarter is worth $25. He has no ticket.no ticket.Outside the stadium there is a ticket tout who Outside the stadium there is a ticket tout who has a ticket for sale, and has just the one fan to has a ticket for sale, and has just the one fan to sell it to. sell it to. The question is what is the price of the ticket?The question is what is the price of the ticket?The solution follows from backwards induction.The solution follows from backwards induction.



The ticket-tout example.The ticket-tout example.Suppose the fan has not bought the ticket from the tout and the Suppose the fan has not bought the ticket from the tout and the game is at the beginning of the final quarter.game is at the beginning of the final quarter.The fan knows that the tout’s ticket would then be worthless.The fan knows that the tout’s ticket would then be worthless.He would offer him an arbitrarily small amount for it and this would He would offer him an arbitrarily small amount for it and this would be accepted.be accepted.Consider now the situation at the beginning of the third period. The Consider now the situation at the beginning of the third period. The tout knows that if he waits until the beginning of the fourth period the tout knows that if he waits until the beginning of the fourth period the will get essentially nothing for the ticket, offer the ticket for slightly will get essentially nothing for the ticket, offer the ticket for slightly less than $25. The value to the fan of the third quarter. The fan less than $25. The value to the fan of the third quarter. The fan would accept this offer.would accept this offer.Now consider the beginning of the second quarter Now consider the beginning of the second quarter



The ticket-tout example continued.The ticket-tout example continued.Now consider the beginning of the second Now consider the beginning of the second quarter. The fan must make an offer to the tout. quarter. The fan must make an offer to the tout. He knows that if he waits until the third quarter He knows that if he waits until the third quarter he can get the ticket for $25, so he offers slightly he can get the ticket for $25, so he offers slightly more than $25. The tout would accept this.more than $25. The tout would accept this.Finally consider the beginning of the game the Finally consider the beginning of the game the tout can ask the fan for $50, the value of the first tout can ask the fan for $50, the value of the first quarter ($25) plus the value of the rest of the quarter ($25) plus the value of the rest of the game ($25).game ($25).



Consider any game lasting n stages, the value of which falls by xi per period. So the value of the game is

V = x1+ x2 +……. + xn

The last player to make an offer will ask for xn In the preceding period the other player knows she has to allow her opponent xn, so she asks for xn-1.The two players hence receive

x1+ x3 +……. + xn-1 and x2+ x4 +……. + xn

Hence as n becomes large the players end up splitting the surplus.


Alternating Offers withAlternating Offers with

Suppose Suppose Instead of the value decaying the players value a $1 today greater that $1 in the next period.Suppose also that the two players are equally impatient.Suppose that $1 today is worth $0.95 in the next period.

LetLetThe bargainers be A and B A makes the first offer. Suppose also that the two players are equally impatient.A know that B will receive x in the next rounds when it is B’s turn to make an offer.


Alternating Offers with Alternating Offers with ImpatienceImpatience

Equilibrium Equilibrium Since A knows B will get x in the next round, she know that she must offer 0.95x in this round. As there is a dollar to be split A must get 1 – 0.95x.But since A and B are equally impatient and the game is symmetric, it must be the case that A gets x today.Equilibrium then involves 1 – 0.95x = x, so x = 0.512.Notice that there is an advantage to making the first offer 0.512 > 0.5!!


Alternating offers when one player is Alternating offers when one player is

more impatient than the other.more impatient than the other. Suppose.Suppose.

For A $1 today is worth $0.95 tomorrow.

For B $1 today is worth $0.90 tomorrow.

We can use the same technique as before to figure out We can use the same technique as before to figure out the equilibrium.the equilibrium.

But we must think two steps ahead to solve the But we must think two steps ahead to solve the problem. problem.


Alternating offers when one player is Alternating offers when one player is

more impatient than the other.more impatient than the other.

Suppose A makes the first offer. Let x and y be the amounts the two bargainers get from the bargain.

We can use the same technique as before to figure out the equilibrium.

But we must think two steps ahead to solve the problem.


Alternating offers when one player is Alternating offers when one player is more impatient than the other.more impatient than the other.

Solution.Solution.For B y today is worth 0.90y tomorrow, so A must offer B at least 0.90y. If tomorrow is reached B must offer A 0.95x. So applying backwards induction, if we reach the second period y = 1 - 0.95x.But to reach the second period requires x = 1 – 0.9y.So since the value today to A is defines as x we have x = 1 – 0.9(1 - 0.95x) = 0.690 and y = 1 - 0.690 = 0.31.

We see that the greater impatience of B is We see that the greater impatience of B is quite costly.quite costly.


Rubinstein’s Bargaining Model.Rubinstein’s Bargaining Model.

Rubenstein’s bargaining model allows us to Rubenstein’s bargaining model allows us to find the solution to an alternating offer infinite find the solution to an alternating offer infinite bargaining game.bargaining game.

SupposeSupposeTwo players A and B bargain over a cake of size 1.

At time 0 A makes the offer xA to player B.

B may either accept of reject A’s offer. If B accepts they receive 1 – xA. If they reject they get to make the next offer.

The players discount the future at rates δA and δB.



Rubinstein’s reasoning that allows this problem to be Rubinstein’s reasoning that allows this problem to be solved involves two insights.solved involves two insights.

Each time a player gets the opportunity to make an offer they are in the same situation as previously, and will make the same offer again (Stationarity).

You never offer more to the other player than you need to get your offer accepted (Indifference).



Equilibrium.Equilibrium.Let xA

* be the equilibrium offer of A, and xB* be the

equilibrium offer of B.

Indifference requires that the players are both indifferent in each period between accepting the offer on the table and waiting to make the next offer themselves or and .

Solving these expressions gives

* *1 B A Ax x

* *1 A B Bx x

* 1

1B

AA B

x

* 1

1A

BA B

x



But we may find this too simplistic since it But we may find this too simplistic since it impliesimplies

No Delay – In equilibrium all offers are immediately accepted.

Stationarity – A player always makes the same offers in equilibrium.

Suggests we need to complicate matters.Suggests we need to complicate matters.


Manipulating Information in Manipulating Information in BargainingBargaining

The previous bargaining solutions make a lot of sense …. yet they appear naive. The reason is that the solutions make sense only after the process of information exchange has taken place.What the bargaining solutions do tell us, is that what matters for the outcome of the bargain, is the two sides beliefs about their respective threat points and levels of impatience. This takes us back to signaling and screening.


Manipulating Information in Manipulating Information in BargainingBargaining

If you are patient you can signal the fact by being patient. Wait sufficiently long that an impatient bargainer would not be willing to match your inaction.If you are impatient try to make the other side believe you are not.If you have a low threat point, try to find a way of making the other side believe it is higher.Look for strategic moves. For example a union bargainer may receive a mandate from her members to only accept certain offers.Look for ways to make threats credible.Engage in Brinkmanship.


Multiple parties and issues Multiple parties and issues bargaining.bargaining.

It might seem that bargaining with multiple parties and/or over multiple issues would make to process of reaching agreement more difficult.

This need not be the case.


Multiple parties and issues Multiple parties and issues bargaining.bargaining.

Multiple issues.Multiple issues.All deals depend on differences in valuations between the parties.The value to you of a packet of cornflakes exceeds the value of the $3 purchase price. To the store the $3 exceeds the value of the cornflakes.The bargain I strike with the store thus reflects our different valuations of $3 and cornflakes. In a sense all deals are multi-issue bargains based on divergent valuations.This simple intuition can be applied to all bargaining situations.


Multiple issue bargaining Multiple issue bargaining continued.continued.

When there are multiple issues on the table they When there are multiple issues on the table they may be traded off against each other to realize may be traded off against each other to realize mutually beneficial deals.mutually beneficial deals.

The more issues are brought to the table the The more issues are brought to the table the greater are the opportunities to find the required greater are the opportunities to find the required divergences of valuations.divergences of valuations.


Multiple party bargainingMultiple party bargaining

In the same vein as multiple issue bargaining, multi party bargaining again provides more opportunities for divergences in valuations.Multiple party bargaining allows options unavailable to bilateral bargainers.For example 3 bargainers A, B and C. Wish to exchange x, y and z.A has x and wants y, B has y and wants z, C has z and wants x. A three way bargain can facilitate this. Bilateral bargaining cannot.


Multiple party bargaining Multiple party bargaining continuedcontinued

The Drawback.The Drawback.With bilateral bargaining if I fail to complete my side of the deal you can respond in kind.

With multilateral bargaining I am supplying you with a good, but in return am receiving a good from someone else. Here there may be no simple form of retaliation if you renege on the deal.


Real Saddam


The Teaching Evaluation GameThe Teaching Evaluation GameObjective of the gameObjective of the game

Make sure that Prof. Ellis and GTF Hong receive the highest teaching evaluations in the Economics Department. Better than those nasty Blonigen, Singell and Davies guys. They said they didn’t like you, did you know that?

How the game is playedHow the game is playedGame in two stagesStage #1: Each student receives and completes a teaching evaluation form.Stage #2: Each student looks at the form completed by their neighbor on their right.

If the neighbor has given the professor and GTF a good evaluation you must say “Good Job” loudly for all to hear. These good students receive a payoff of $1mil.If the neighbor has given the professor and GTF a poor evaluation you must yell “Heretic” as loudly as you can. All students should then “punish” the offender, giving them a payoff of zero….or worse.


Groucho and Teaching Evaluations Groucho and Teaching Evaluations

I've had a wonderful time, I've had a wonderful time, but this wasn't it. but this wasn't it.

I’ve had a wonderful time, but I’ve had a wonderful time, but this wasn’t it – Groucho Marxthis wasn’t it – Groucho Marx


The EndThe End

I don't pretend we have all the answers. But the questions I don't pretend we have all the answers. But the questions

are certainly worth thinking about.are certainly worth thinking about. - Arthur C. Clarke- Arthur C. Clarke

games people play. 13: bargaining in this section we shall learn how to break down a bargain into...

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hjthe nash bargaining

characterization of

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acceptable bargain

anonymous partner

known partner