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Chapter 7 Bargaining

- Necessity never made a good bargain

Economic Markets

- Allocation of scarce resources
- Many buyers many sellers
- ? traditional markets
- Many buyers one seller
- ? auctions
- One buyer one seller
- ? bargaining

The Move to Game-Theoretic Bargaining

- Baseball
- Each side submits an offer to an arbitrator who

must chose one of the proposed results - Meet-in-the-Middle
- Each side proposes its worst acceptable offer

and a deal is struck in the middle, if possible - Forced Final
- If an agreement is not reached by some deadline,

one party makes a final take-it-or-leave-it offer

Bargaining Game Theory

- Art Negotiation
- Science Bargaining
- Game theorys contribution
- to the rules of the encounter

Outline

- Importance of rules
- The rules of the game determine the outcome
- Diminishing pies
- The importance of patience
- Estimating payoffs
- Trust your intuition

Take-it-or-leave-it Offers

- Consider the following bargaining game (over a

cake) - I name a take-it-or-leave-it split.
- If you accept, we trade
- If you reject, no one eats!
- Faculty senate if we cant agree on a

recommendation (for premiums in health care) the

administration will say, There is no consensus

and do what they want. We will have no vote. - Under perfect information, there is a simple

rollback equilibrium

Take-it-or-leave-it Offers

accept

1-p , p

p

reject

0 , 0

- Second period Accept if p gt 0
- First period Offer smallest possible p
- The offerer keeps all profits

Counteroffers and Diminishing Pies

- In general, bargaining takes on a

take-it-or-counteroffer procedure - If time has value, both parties prefer to trade

earlier to trade later - E.g. Labor negotiations
- Later agreements come at a price

of strikes, work stoppages, etc. - Delays imply less surplus left to be shared among

the parties

Two Stage Bargaining

- Bargaining over division of a cake
- I offer a proportion, p, of the cake to you
- If rejected, you may counteroffer (and ? of the

cake melts) - Payoffs
- In first period 1-p , p
- In second period (1-?)(1-p) , (1-?)p

Rollback

- Since period 2 is the final period, this is just

like a take-it-or-leave-it offer - You will offer me the smallest piece that I will

accept, leaving you with all of 1-? and leaving

me with almost 0 - What do I do in the first period?

Rollback

- Give 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

period - p ? 1-?

Rollback

- If 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!

First or Second Mover Advantage?

- Are you better off being the first to make an

offer, or the second?

Example Cold Day

- If ?1/5 (20 melts)
- Period 2 You offer a division of 1,0
- You get all of remaining cake 0.8
- I get 0 0
- In the first period, I offer 80
- You get 80 of whole cake 0.8
- I get 20 of whole cake 0.2

Example Hot Day

- If ?4/5 (80 melts)
- Period 2 You offer a division of 1,0
- You get all of remaining cake 0.2
- I get 0 0
- In the first period, I offer 20
- You get 20 of whole cake 0.2
- I get 80 of whole cake 0.8

First or Second Mover Advantage?

- When players are impatient (hot day)
- First mover is better off
- Rejecting my offer is less credible since we both

lose a lot - When players are patient (cold day)
- Second mover better off
- Low cost to rejecting first offer
- Either way if both players think through it,

deal struck in period 1

Dont Waste Cake

- Why doesnt this happen?
- Reputation building
- Lack of information

COMMANDMENT In any bargaining setting, strike a

deal as early as possible!

Uncertainty in Civil Trials

- 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 - 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 settling out of court. - Defendant should just give the plaintiff 400K as

he saves 200K.

Uncertainty in Civil Trials

- What if both parties are too optimistic?
- Each thinks that his or her side has a ¾ chance

of winning - Plaintiff 750K - 100K 650K
- Defendant - 250K - 100K -350K
- No way to agree on a settlement! Defendant would

be willing to give plaintiff 350, but plaintiff

wont accept.

von Neumann/Morganstern Utility over wealth

- How big is the cake?
- Is something really better than nothing?

Lessons

- Rules of the bargaining game uniquely determine

the bargaining outcome - Which rules are better for you depends on

patience, information - What is the smallest acceptable piece? Trust

your intuition - Delays are always less profitable Someone must

be wrong

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

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 10 of annual profits

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)

Take-it-or-leave-it OfferExample I

- p9/10
- 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!

Take-it-or-leave-it OfferExample II

- p1/10
- 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
- Not worth the risk to ask for more.

Two-period Bargaining

- If first-period offer is rejected A strike

costs the company 10 of annual profits - Note strike costs a high-value company more than

a low-value company! - Use this fact to screen!

Screening in Bargaining

- What if the Union asks for 160K in the first

period? - Low-profit firm (150K) rejects as cant afford

to take. - High-profit firm must guess what will happen if

it rejects - Best case
- Union strikes and then asks for only 140K
- (willing to pay for some cost of strike, but

not all) - In the mean time
- Strike cost the company 20K
- High-profit firm accepts

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 140K in second period
- Expected Wage
- 170K (p) 140K (1-p)
- In order for this to be profitable
- 170K (p) 140K (1-p) gt 150K
- 140 (170-140)p 140 30p gt150
- if p gt 1/3 , you win

Whats Happening

- Union lowers price after a rejection
- Looks like Giving in
- Looks like Bargaining
- 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

Bargaining

- The non cooperative games miss something

essential people can make deals - then can agree

to behave in a way that is better for both.

Economics is based on the fact that there are

many opportunities to "gain from trade. - With the opportunities, however comes the

possibility of being exploited. Human beings have

developed a systems of contracts and agreements,

as well as institutions that enforce those

agreements. - Cooperative game theory is about games with

enforceable contracts.

Strategic Decisions

- Non-strategic decisions are those in which ones

choice set is defined irrespective of other

peoples choices. - Strategic decisions are those in which the choice

set that one faces and/or the outcomes of such

choices depend on what other people do. These

decisions can be characterised in two general

ways - Cooperative games
- Where the outcome is agreed upon through joint

action and enforced by some outside arbitrator. - Non-cooperative games
- The outcome arises through separate action, and

thus does not rely on outside arbitration.

Cooperative Bargaining

- A bargaining situation can be approached as a

cooperative game. All bargaining situations have

two things in common - The total payoff created through cooperation must

be greater than the sum of each partys

individual payoff that they could achieve

separately. - The bargaining is thus over the surplus payoff.

As no bargaining party would agree to getting

less than what they get on their own. - A players outside option is also known as a

BATNA - (Best Alternative To Negotiated Agreement) or

disagreement value.

Two people dividing cash

- CONSIDER THE FOLLOWING BARGAINING GAME
- Jenny and George have to divide candy bar
- They have to agree how to divide up the candy
- If they do not agree they each get nothing
- They cant divide up more than the whole thing
- They could leave some candy on the table
- What is the range of likely bargaining outcomes?

Likely range of outcomes

- Clearly neither Jenny nor George can individually

get more than 100 - Further, neither of them can get less than zero

either could veto and avoid the loss - Finally, it would be silly to agree on something

that does not divide up the whole 100 they

could both agree to something better - But that is about as far as our prediction can go!

Likely range of outcomes

- So our prediction is that Jenny will get j and

George will get g where - j 0
- g 0 and
- j g 100.

Modified bargaining game

- Jenny and George still have to divide 100
- They must agree to any split
- If they do not agree then Jenny gets nothing and

George gets 50 - They cant divide up more than 100
- They could leave some on the table
- Now, what is the range of likely bargaining

outcomes?

Likely range of outcomes in modified game

- Clearly neither Jenny nor George can individually

get more than 100 - Further, Jenny would veto anything where she gets

less than 0 - George will veto anything where he gets less than

50 - And it would be silly to agree on something that

does not divide up the whole 100

Likely range of outcomes for modified game

- So our prediction is that Jenny will get j and

George will get g where - j 0
- g 50 and
- j g 100.
- Note by changing Georges next best alternative

to agreeing with Jenny, we change the potential

bargaining outcomes.

Ultimatum GamesBasic Experimental Results

- In a review of numerous ultimatum experiments

Camerer (2003) found - The results reportedare very regular. Modal and

median ultimatum offers are usually 40-50 percent

and means are 30-40 percent. There are hardly

any offers in the outlying categories of 0,

1-10, and the hyper-fair category 51-100.

Offers of 40-50 percent are rarely rejected.

Offers below 20 percent or so are rejected about

half the time.

Ultimatum Bargaining with Incomplete Information

Ultimatum Bargaining withIncomplete Information

- Player 1 begins the game by drawing a chip from

the bag. Inside the bag are 30 chips ranging in

value from 1.00 to 30.00. Player 1 then makes

an offer to Player 2. The offer can be any amount

in the range from 0.00 up to the value of the

chip. - Player 2 can either accept or reject the offer.

If accepted,Player 1 pays Player 2 the amount of

the offer and keeps the rest. If rejected, both

players get nothing.

Experimental Results

- Questions
- How much should Player 1 offer Player 2?
- Does the amount of the offer depend on the size

of the chip? - 2) What should Player 2 do?
- Should Player 2 accept all offers or only offers

above a specified amount? - Explain.

How should Ali Baba split the pie?

- Ali and Baba have to decide how to split up an

ice cream pie. - The rules specify that Ali begins by making an

offer on how to split the pie. Baba can then

either accept or reject the offer. - If Baba accepts the offer, the pie is split as

specified and the game is over. - If Baba rejects the offer, the pie shrinks, since

it is ice cream, and Baba must then make an offer

to Ali on how to split the pie. - Ali can either accept or reject this offer.
- If rejected, the pie shrinks again and Ali must

then make another offer to Baba. - This procedure is repeated until and offer is

accepted or the pie is gone.

How should Ali Baba split the pie?

- 1. How much should Ali offer Baba in the first

round? - 2. Should Baba accept this offer? Why or why not?

Ali Babas Pie Woes

- Initial Pie Size 100
- Pie decreases by 20 each time an offer is

rejected. - Question What is the optimal split of this pie?

That is, how much should Ali offer Baba in the

first round so that Baba will accept the offer.

Ali Babas Pie Woes

Offerer Ali Baba Ali Baba Ali

Round 1 2 3 4 5

Pie Size 100 80 60 40 20

Pie Split

Ali 60 lt40 gt40 lt20 10

Baba 40 gt40 lt20 gt20 10

Baba may as well accept first offer. It never

really gets better for him.

Formulas If the number of rounds in the game is

even, the pie should be split 50/50. If the

number of rounds in the game is odd, then the

proportion of the pie for each player is (n

1)/2n for Ali (initial offer) first person

advantage! (n-1)/2n for Baba. For example, in

this game n 5, so Ali gets (51) / (25)

6/10. 60 of 100 is 60.

Suppose the discount is 25

Offerer Ali Baba Ali Baba

Round 1 2 3 4

Pie Size 100 75 50 25

Pie Split

Ali 75 25 50 0

Baba 25 50 0 25

If Ali offered 50, Baba would have no reason to

question! He never gets more.

Model for Bargaining no shrinking pieExample

two people bargaining over goods

- Amy has 10 apples and 2 banana
- Betty has 1 apple and 15 bananas
- Before eating their fruit, they meet together
- Questions
- Can Amy and Betty agree to exchange some fruit?
- If so, how do we characterize the likely set of

possible trades between Amy and Betty?

The Edgeworth Box for Amy and Betty

Box is 17 units wide to represent the 17

bananas in total

First what are they trading over? Amy has 10

apples and 2 banana Betty has 1 apple and 15

bananas So in total they are bargaining over the

division of 11 apples and 17 bananas So we

can represent ALL possible trading outcomes by

points in a rectangle called an Edgeworth Box

Box is 11 units high to represent the 11 apples

in total

The Edgeworth Box for Amy and Betty

Amys apples

Measure Amys bundle from here

Amys bananas

The Edgeworth Box for Amy and Betty

Measure Bettys bundle from here

Bettys bananas

Bettys apples

The Endowment bundle initial amounts

15 bananas

OB

1 apple

10 apples

OA

2 bananas

The allocation where Betty gets all the apples

and Amy gets all the bananas

OB

11 apples

OA

17 bananas

Bargaining and the Edgeworth box

- An allocation is only a feasible outcome of trade

between Betty and Amy if it cannot be blocked - This means that Betty must be at least as well

off with the trade as she is with her endowment - Also Amy must be at least as well off with the

trade as he is with his endowment - And the allocation must be Pareto optimal for

Betty and Amy so that they BOTH cannot do better

Amys indifference curves

We can draw Amys indifference curves Then put

them in the Edgeworth Box

10 apples

OA

2 bananas

This is Amys indifference curve through his

endowment bundle. She will block any allocation

that puts her on a lower indifference curve

10 apples

OA

2 bananas

So ANY bargaining outcome must be in the shaded

region of the Edgeworth Box otherwise Amy will

block the allocation.

10 apples

OA

2 bananas

Bettys indifference curves

15 bananas

OB

1 apple

And we can put Bettys indifference curves in the

Edgeworth box

ANY outcome of bargaining between Betty and Amy

must lead to an allocation that is inside the

shaded area below. This area is called the lens

of trade.

15 bananas

OB

1 apple

10 apples

OA

2 bananas

Definition the lens of trade

- When two people bargain over allocating goods,

any agreed outcome must lie in the lens of trade. - The lens of trade is the area in the Edgeworth

box bounded by the indifference curves for each

person through the endowment bundle - Any allocation outside the lens of trade will be

blocked by one of the people. - We call this non blocked set of choices the

core.

Note that we can move to an allocation that is

better for BOTH Betty and Amy, like the green

bundle. This bundle puts both Amy and Betty on

higher (better) indifference curves. So the brown

bundle cannot be Pareto optimal and will be

blocked.

OB

OA

The ONLY situation where we cannot find another

bundle that makes both people better off is when

we are at the tangency of Amys and Bettys

indifference curves like the black bundle

below. So this bundle is Pareto optimal.

OB

OA

The contract curve

15 bananas

OB

1 apple

10 apples

OA

2 bananas

The red curve joins all Pareto optimal bundles

for Amy and Betty. This is the contract curve. An

agreed allocation must lie on this curve

So

- From co-operative game theory we know that an

acceptable allocation must be in the Core - It must lie in the lens of trade or else either

Amy or Betty will block the allocation - It must lie on the contract curve or else another

coalition of both Amy and Betty would block the

allocation - So the core allocations are the contract curve

inside the lens of trade.

The red line (the contract curve inside the lens

of trade) is the core. It gives the likely

bargaining outcomes for Amy and Betty

15 bananas

OB

1 apple

10 apples

OA

2 bananas

Summary so far

- The Edgeworth box can be used to model bargaining

outcomes for two people over bundles of goods - The core is the set of bundles on the contract

curve inside the lens of trade - We predict that any trade will most likely lead

to a core allocation - But which allocation?

Bargaining (Chapter 7)

- Feasible alternatives each person does better

than disagreement point (d1, d2) - S is set of alternatives
- s is agreement point
- U (u1(s), u2(s)), s ? S) is set of utility

allocations - Goals of a solution rule
- Pareto Optimal
- independence of irrelevant alternatives
- independence of linear transformations
- (if utilities are transformed by vi ai biui,

solution is the same) - Similarly disagreement points are transformed by

same function - Notice the multipliers and adders can be

different for each person. - Point is that relatively speaking the values have

same relationship.

- Nash rule
- maximize (u1(s)-d1)(u2(s)-d2)
- Nash rule gives solution which satisfies the

three goals listed! - Would be nice if there was only one set of values

that were maximizers.

- compact
- bounded can be contained in circle or box
- closed contains its boundary points
- Continuous functions on compact sets always

attain their maximum. - If f is continuous on a compact set X, then there

exists x1 and x2 in X such that - f(x1) ?f(x) ? f(x2) for all x in X.
- Theorem 7.3 The Nash rule is pareto optimal,

independent of irrelevant alternatives and

independent of linear transformations.

Look at characteristics of set of utility

allocations

Symmetric (about diagonal), but non convex.

symmetric, compact, and convex

non-symmetric

- Defn 7.4 A set of utility allocations U of a

bargaining game is said to be convex if it

contains every point on the line segment joining

any two vertices. - A set of utility allocations U of a bargaining

game is said to be symmetric if (u1,u2) ? U

implies (u2,u1) ? U - A solution rule is symmetric if for every

symmetric bargaining game u1(s) u2(s) for each

s. Both get same utility from a deal.

- Thm 7.6 In a convex bargaining game, there

exists exactly one utility allocation in the Nash

solution. - If the game is symmetric, then the utilities in a

Nash soluiton are equal.

Consider the maximizer curves tangent to S

Symmetric (about diagonal), but non convex.

symmetric, compact, and convex unique maximizer

xy c

y

maximizer curve

x

- In a strategic game (without cooperation), such

as Bach or Stravinsky, either Bach/Bach or

Stravinsky/Stravinky is best, but they are not

equal, so we pick a mixed strategy. Here, you

lose when Bach/Stravinsky or Stravinsky/Bach is

picked. - In a correlated system, specific options are

selected with certain probabilities. Thus, you

could pick each of the good choices 50 of the

time (or whatever is fair) - Defn 7.10 A correlated utility allocation with

probability distribution with probability

distributions (p1,p2,pn) the utility is (?

piu1(si), ? piu2(si) )

Assymetric bargaining games

- Many bargaining games are essentially asymmetric

either because of - differing attitudes towards risk between players
- difference in payoffs in case of a disagreement
- asymmetry in the set of utility allocations.

Monotonicity in Bargaining

- The Nash solution works well when there are

asymmetries due to risk aversion or even in

disagreement points. - When a disagreement point increases (due, say, to

an outside option), the amount going to a person

increases. - maximize (u1(s) d1)(u2(s) d2). We agree to a

certain distribution, but if my outside options

increase, I expect more. In water example, may

agree to split the costs down the middle. When

my costs for working alone go down, I expect you

to pick up more of the costs of working together. - Changes in risk affect the utility function, so

the Nash solution still works quite well.

Original bargaining

(d1,d2)

d1 increases player1 gets more player1 gets less

(d1,d2)

- Nash solution may not work well in terms of other

asymmetric situations. - Example. Bankruptcy. Assets are less than

debts. Nash solution provides an equal division

of remaining assets. Unfair, if sizes of

outstanding debt are different. - Example. Have K dollars to use to pay debts.

Owed A1 and A2 to two people. - K lt A1A2

K

A1A2

A2

fair allocation Original debts are equal Nash

solution picks equal division along line of

distribution

A1

K

unfair allocation Original debts are unequal Nash

solution picks player 2 to get complete payoff,

while player 1 (who invested more) gets less

than full payment

A2

A1

K

K

What would we consider to be more fair?

- Each person loses same amount?
- Each person gets same percent of debt repaid?

Notice the two overlapping solution sets. The

larger one actually gives player one a smaller

payoff. This violates monotincity, which states

that as the solution set increases, your utility

does not decrease.

We also see that Nash doesnt satisfy

monotonicity. That is, when the set of possible

solutions is larger, a person can actually get

less.

Kalai-Smorodinsky solution rulefor dealing with

assymetries

- Take the furthest point on a line from (0,0) to

u1_max u2_max.

KS utility allocation

KS line

KS solution is independent of linear

transformations, but not of irrelevant

alternatives. If B is a convex and symmetric

bargaining game, then KS and Nash are the same.

Kalai-Smorodinsky solution rule

- Take the furthest point on a line from (0,0) to

u1_max u2_max.

KS utility allocation

KS line

KS solution is independent of linear

transformations, but not of irrelevant

alternatives. Notice, how if an unchosen part is

added, I can earn less.

(No Transcript)

7.3 The Core minimal requirements that any

reasonable agreement must have.

- Consider the coalition of all players
- An allocation just refers to a split of the total

payoff available to all players. - An allocation is blocked if some coalition (an

individual or subgroup) is better off separating

and going their own way (i.e. the allocation does

not give them their outside option). Thus, the

allocation will never be agreed to. - An allocation is in the core if it cannot be

blocked by any coalition including the grand

coalition (the coalition of all players).

The core is the range of reasonable bargaining

outcomes

Example

- Three firms, x, y and z are negotiating a joint

venture (JV). - If any firm does not join the JV then it receives

nothing. - Firm y is critical to the JV. If x and z work

together then they get 0m. - Neither x nor z is critical to the JV. If x and y

work together then they get 200m. Similarly if z

and y work together then they get 220m. - But if all three work together then they get

300m in total.

X Y Z Value

yes yes 0

yes yes 200

yes yes 220

yes yes yes 300

Example

- What is the range of likely bargaining outcomes

(i.e. the core)? - Is an equal split blocked? Yes! Under an equal

split, x, y and z each get 100m. So y and z

together get 200m. But if y and z leave x out of

the JV, then they get 220m. So the coalition of

y and z will block an even split. - To be in the core we need a split so that each

player gets a positive payoff x and y together

get at least 200m y and z together get at least

220m and the total 300m is divided up. - e.g. x gets 50m, y gets 160m, z gets 90m.
- e.g. x gets 80m, y gets 120m, z gets 100m.

Properties of the core

- The core represents stable outcomes in the sense

that no individual or subgroup can do better by

themselves. - Allocations in the core are Pareto Efficient

(i.e. they involve no waste otherwise the

allocation would be blocked by the grand

coalition of all players) - But the core may not exist!

Core existence sharing the cost of water

- Three towns, Amalga, Benson and Cove are

bargaining over new water supplies - Each town pays 30m if it builds its own supply
- Any two towns together pay only 40m
- All three together pay 66m
- So to be in the core, an allocation cannot

involve any town paying more than 30m, or any

two towns paying more than 40m, but all three

towns in total pay 66m

Core existence sharing the cost of waterAssume

in our models, MUST be better to all work

together.

- But this cannot hold for any allocation there

is no core for this bargaining problem! As 40m

is an average of 20m per town and 66m is an

average of 22m per town, so no one will agree to

grand coalition. - If any two try to combine, the left out one

will offer a better deal.

Amalga Benson Cove Joint Cost

yes 30

yes 30

yes 30

yes yes 40

yes yes 40

yes yes 40

yes yes yes 66

- See instability.
- No one will agree to the grand coalition as it is

worse that the pairs. Once the grand coalition

was formed, a pair would splinter off as it would

be better off.

The core focuses on stability of coalitions.

However, in many appli- cations it is empty.

Core Existence

Say Amalga pays a, Benson pays b and Cove pays

c. Then a, b and c must be no more than

30m each ab, ac and bc can each be no

more than 40m abc 66m But this is

impossible! To see this a b

?40m a c ? 40m b c

? 40m

One of the coalitions of 2 towns will block the

grand coalition unless this is satisfied. But

this is impossible!

Add up 2a 2b 2c ? 120 So a b c

? 60

Summary

- For multi-person bargaining
- We expect that the outcome will be in the Core
- These are the stable outcomes
- But the Core does not always exist

Section 7.3

- The characteristic (or the coalition function) of

an n-person bargaining game is the function

vN?P(Rn) - where N is the set of all subsets of N.
- It maps each coalition to its value (for each

agent). - v(c) is also known as the worth of the coaltion

C. - Any output of an n-person bargaining game that

cannot be blocked is called a core-outcome. - Important issue is whether it has a non-empty

core. - Balancedness ensures a non-empty core.
- Balanced contributions (what I contribute is

equivalent to what you contribute) require a

unique sharing.

- Xc is the indicator function of C which is

defined by Xc(k) 1 if k ?C and 0 otherwise. - Def 7.19 A family of coalitions is said to be

balanced if we can assign weighting factors to

each so that when we multiply by the weights and

add up, we get the grand coalition. - A set is comprehensive, if for any vector x in

the set of utilities, any vectors where each

component is smaller is in the set.

- In essence, the weights in a balanced collection

indicate a players presence and importance in

the coalitions. - A side payment game indicates that utility can be

transferred.

Bondareva-Shapley theorem

- Different ways to prove non-emptiness- use the

definition of the core and construct a core

element- use the following well-known theorem - Bondareva-Shapley theorem (Bondareva (1963) and

Shapley (1967))The core of a cooperative game

is non-empty if and only if the game is balanced.

Definition balancedness

- Let B be a collection of the set 2N Example n

4, B 1, 2, 1, 3, 2, 3, 4 - B is called a balanced collection if there exist

weights lS (S element of B) such that - Example l 0.5, 0.5, 0.5, 1
- Definition A game is balanced if for every

balanced collection B with corresponding weights

lS

- In other words, it must be more costly to work

separately than to work together. - In the Amalga, Benson, Cove water example
- A B C AB BCACABC
- We could find weights (so collection is balanced)

But when we apply those weights to the costs of

coalitions ½(40) ½(40) ½(40) 120 lt 122

(cost of grand coalition)

Definition - Added value

Case Study Several bands exist and would like

you to join them. Which do you join and what is

your share of the profits?

- We can consider any group of players and ask,
- what do you bring to the group?
- The answer is your added value.
- Helps one to estimate what share of the whole

belongs to each person in the group.

Added Value

Your added value (surplus make possible by you of

joining the group) equals Value of group (with

you as a member) minus ( Value of group without

you plus your value alone )

Added Value - example

- You have an assignment due and you are allowed to

work in groups of four people if you choose - Without you, the other three members of your

group will be able to get 75 marks (out of 100)

each. - If you work alone then you can get 80 marks
- But if you work with your group, then each of you

will get 85 marks - So your Added Value
- (85 4) (75 3) 80
- 340 305
- 35 marks! (5 points for you and 10 for each

of the others) - Thus, it is a measure of what your presence is

worth, above the minimum you would require for

your services.

Added value Jenny and George

- Divide a dollar. If cant agree, both get

nothing.

Added Value George

Added Value Jenny

100 (0 0) 100

Added value Jenny and George

- Divide a dollar. If cant agree, George gets

50.

Added Value George

100 (0 50) 50

Note these are the same. This is a general

result for TWO people bargaining (but ONLY for

two people)

Added Value Jenny

100 (50 0) 50

Note, both add 50 as if they dont work

together, the best the two of them can do is

50, but earn 100 together.

Georges Best Alternative To Negotiated Agreement

disagreement point

disagreement point (George)

Total payoff if cooperate

Jennys Best Alternative To Negotiated Agreement

disagreement point

disagreement poing (Jenny)

Georges added value Total Payoff Georges

BATNA Jennys BATNA But this clearly equals

Jennys added value

So with 2 people Total surplus from agreement

each persons added value

Predicted outcome for two person bargaining

- For two person bargaining, the bargaining is over

the added value from agreement. - Each person gets a share of the added value.
- Each persons TOTAL payoff is their disagreement

point PLUS their share of the added value. - So the least anyone will get is their

disagreement point (their BATNA best alternative

to negotiation agreement) - The most anyone will get is their outside option

PLUS all the added value - In general, get in between - as added value is

shared

Application to a buyer and seller

- So far just looked at two people dividing money
- But the same ideas apply to two people bargaining

over a good - The trick is to find
- The outside options disagreement points
- The Added Value

Definition Willingness-to-pay

- Willingness-to-Pay (WTP) is the highest price

that a buyer will agree to pay for a good or

service. - In other words
- WTP is the price at which the buyer doesnt care

if he buys or walks away - WTP is the price at which the economic profit

from buying is zero - (So it is like the regular price - you could

get that price anytime, so no benefit to buy now.

Or, it is like you will use this item in

production and just break even what you sell

the item for equals what you paid for the raw

goods plus labor.)

Definition Willingness-to-sell

- Willingness-to-Sell (WTS) is the lowest price

that a seller will agree to accept in return for

a good or service. - In other words
- WTS is the price at which the seller doesnt care

if she sells or walks away - WTS is the price at which the economic profit

from selling is zero

When is trade possible?

WTP

Buyer will accept a price below their WTP

Seller will accept a price Above their WTS

WTS

- If WTP ? WTS, then trade is possible
- But if WTP lt WTS, no trade is possible there

is no price that both will accept!

What is the added value created by trade?

- If the buyer and seller agree to a deal then the

added value is just the WTP WTS. - The value to the buyer is the buyers economic

profit - WTP Price
- The value to the seller is the sellers economic

profit - Price WTS

The price divides the added value

Value captured by...

Willingness-to-Pay

Buyer (consumer surplus)

Added Value

Price

Seller (producer surplus)

Willingness-to-Sell

Multi-party bargaining

- Each individual or sub group should never get

less than their outside option - Because they can always split off and go their

own way - No individual or subgroup can get more than their

added value their outside option. - Because all others can always throw you out!

The key here is the extension to subgroups of

individuals.

7.4 Shapley

- In many cases, the outcomes in the core are not

unique or are confusingly large. Which

allocation do we pick? - In other cases, the core may be empty.
- The Shapley value provides an appealing method of

deciding the share of each individual in an

n-person game. - Concept is that of added value.
- You look at all permutations and figure if you

were added to the group in the order represented

by the permutation, what would you bring to the

group. - The reason all orders are used is this. Suppose

Ali and Ben can get 10 together, but 1 and 3

individually. There is a total of 6 surplus to

divide. - The shapley value works with what I brings to the

group V(Ci) V(C). The difference in the

coalition value with i and without i. This value

is called the marginal worth of player i when she

joins coalition C. - Ali could say, I add 7 when I join you. When I

join an empty coaltion I add 1. The average I

add is 4. Ben could say, I add 9 when I join

you and 3 when I join an empty coaltion. The

average is 6. - Each person gets their average value.
- Notice that this is the same as splitting the

added value (over disagreement point).

The Shapley Value (Cont.)

- A well know value division scheme
- Aims to distribute the gains in a fair manner
- A value division that conforms to the set of the

following axioms - Dummy players get nothing
- Equivalent players get the same
- If a game v can be decomposed into two sub games,

an agent gets the sum of values in the two games

The Shapley Value

- Given an ordering of the agents in A, we

define to be the set of

agents of A that appear before a in - The Shapley value is defined as the marginal

contribution of an agent to its set of

predecessors, averaged on all possible

permutations of the agents

A Simple Way to Compute The Shapley Value

- Simply go over all the possible permutations of

the agents and get the marginal contribution of

the agent, sum these up, and divide by A! - Extremely slow
- Can we use the fact that a game may be decomposed

to sub games, each concerning only a few of the

agents?

Defn 7.25 Shapley value ?(v) satisfies these

properties

- efficient everything is allocated
- symmetric doesnt depend on labeling
- linear - ?(aubv) a?(u) b?(v)
- irrelevant to dummy players If i is a dummy

player ?i(v) 0 - The value

is called the marginal worth of player i

when she joins coalition C. - The Shapley value is best thought of as an

allocation rule which gives every player his

average or expected marginal worth.

The Shapley Value

- Grounded in set of axioms that a good solution

should satisfy. - It is the only concept that conforms to all these

axioms. - Are the axioms desirable? Are there other axioms

that are desirable? - The test is in actual predictive power. What

really happens in practice? - The Shapley value does pretty well in this regard.

Using Shapley Values Example (Shapley, Shubik,

and Banzhaf)

- Determine the power of a party in a multi-party

legislature. - Say Reds (43), Blues(33), Greens(16) and Browns

(8). - No party has a majority.
- The power of a party depends on how crucial it is

to the formation of a majority coalition.

Reds 43 Blues 33 Greens 16 Browns 8 Value

yes yes 1

yes yes 1

yes yes 1

yes yes 0

yes yes 0

yes yes 0

yes yes yes 1

yes yes yes 1

yes yes yes 1

yes yes yes 1

yes yes yes yes 1

Measuring Contributions

- Give value 1 to any majority coalition and 0

otherwise. - So a party makes the contribution 1 if by joining

a coalition gives the coalition a majority and 0

otherwise. This party is pivotal. - Total of 15 possible coalitions (2n-1)
- The majority coalitions
- 4 one party coalitions none earn points
- 6 two party coalitions 3 earn points R,B

R,G R,Br - Both members are pivotal
- 4 three party coalitions 3 where red is pivotal

R,B,G R,B,Br R,G,Br and 1 where B, G, Br

are each pivotal. - No party is pivotal in the Grand Coalition

Using the Formula

- The probability term corresponding to each two

party coalition is (4-2)!(2-1)!/4! Or 1/12. - The probability terms corresponding each three

party coalition is (4-3)!(3-1)!/(4)! 1/12

Credit?

red blue green brown blue

red blue brown green blue

red green blue brown green

red green brown blue green

red brown green blue brown

red brown blue green brown

green blue brown red brown

green blue red brown red

green red blue brown red

green red brown blue red

green brown red blue red

green brown blue red blue

blue red green brown red

blue red brown green red

blue green brown red brown

blue green red brown red

blue brown red green red

blue brown green red green

brown blue red green red

brown blue green red green

brown green red blue red

brown green blue red blue

brown red green blue red

brown red blue green red

Who gets credit for each of 24 orders red 12

value 1/2 blue 4 value 1/6 green

4 value 1/6 brown 4 value 1/6

The Shapley Values for Each Party

- For Red 1/12 x 3 1/12 x 3 ½
- For each of the other three it is 1/12 x 1

1/12 x1 1/6 - So Red has the most power.
- The other three have equal power even though they

are widely disparate in size. - Small parties matter
- Not to be used as a precise quantitative measure

because we have assumed that all coalitions are

equally likely and that all contributions are 0

or 1. - So if two of the larger parties are ideologically

completely opposed to each other (never in a

coalition) then the smaller parties may have even

greater power.

Example 7.27 Setting Landing Fees

- Airport fixed costs
- variable costs depending on types of planes

that use airport. Consider building one runway. - Who should pay what for its use?
- Lets assume ki is the cost needed to land plane

of type i. - Order the plane types so 0 lt k1 lt k2 lt kT
- Let n be the number of expected landings.

- In this case, values added to a coaltion are

non-positive as they represent costs. - We assume a runway of cost 10 can handle any

smaller needs.

Plane Type Cost of Runway Number of landings

1 1M 5K

2 2M 2K

3 3.5M 1K

4 7.5M 1K

5 10M 1K

- So, to accommodate everyone we need a 10M runway,

but what should each plane type pay for each

landing? - Consider the planes 1111122345
- Consider all possible orderings and make each of

them pay what they add to the cost of the needed

runway (on average). - So for example, the second 2 in an order would

never have to pay anything as the first two would

have paid it already. - The first two would only have to pay if it were

preceded by lesser numbers.

If we actually ran the numbers we would get

Plane Type Charge per landing Number of landings Total revenue

1 100 5 500

2 300 2 600

3 800 1 800

4 2800 1 2800

5 5300 1 5300

10000

Computationally complex, so book shows shortcuts.

- Only really care about first occurrence of each

plane type. So could simplify by looking at

ordering of each of five plane types. - Need to count all ways each order could occur so

get proper weight. - v(C union i) v(C) must be paid multiple times

depending how many times this pattern occurs - Notice that the cost is 0 if anything of equal or

higher cost already occus in C - Notice that the cost is the difference of this

planes cost and the cost of the highest cost

previous plane.

- Note, we get exactly the same costs in the

following cases - C is permuted in any order, followed by i,

followed by any permutation of remaining planes. - So order the C elements before i in C! ways
- Order the remaining elemements after I in (N -

C-1)! ways. - We then see the formula
- C! (N-C-1)!/N! v(C union i) - v(C)
- This is still pretty expensive to compute as

there are lots of choices for C

- In the text, they divide up the costs associated

with each element into costs for each level. So

a type 4 plane has a fee associated with it for

each level (1,2,3,4). - The formula they finally end up with is

Computation is a bit tricky but it is just the

Shapley value, computing using

- For our example this means
- Planes of type 1 pay 1M/(52111)
- 1000000/10000

100 - Planes of type 2 pay 1M/10K 1M/5K 300
- Planes of type 3 pay 1M/10K 1M/5K 1.5M/3K

800 - Planes of type 4 pay
- 1M/10K 1M/5K 1.5M/3K 4M/2K 2800
- Planes of type 5 pay
- 1M/10K 1M/5K 1.5M/3K 4M/2K 2.5M/1K

5300

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