Chapter 7 Bargaining

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Chapter 7 Bargaining

• Necessity never made a good bargain

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Economic Markets

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

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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

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Bargaining Game Theory

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

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Outline

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

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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

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Take-it-or-leave-it Offers
accept
1-p , p
p
reject
0 , 0

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

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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

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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

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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?

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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-?

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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!

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First or Second Mover Advantage?

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

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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

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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

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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

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Dont Waste Cake

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

COMMANDMENT In any bargaining setting, strike a
deal as early as possible!
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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.

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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.

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von Neumann/Morganstern Utility over wealth

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

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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

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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

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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

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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)

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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!

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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.

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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!

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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

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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) 150K
• 140 (170-140)p 140 30p 150
• if p 1/3 , you win

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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

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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.

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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.

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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.

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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?

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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!

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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.

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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?

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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

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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.

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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.

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Ultimatum Bargaining with Incomplete Information
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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.

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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.

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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.

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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?

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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.

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Ali Babas Pie Woes
Baba may as well accept first offer. It never
really gets better for him.
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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.
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Suppose the discount is 25
If Ali offered 50, Baba would have no reason to
question! He never gets more.
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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?

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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
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The Edgeworth Box for Amy and Betty
Amys apples
Measure Amys bundle from here
Amys bananas
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The Edgeworth Box for Amy and Betty
Measure Bettys bundle from here
Bettys bananas
Bettys apples
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The Endowment bundle initial amounts
15 bananas
OB
1 apple
10 apples
OA
2 bananas
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The allocation where Betty gets all the apples
and Amy gets all the bananas
OB
11 apples
OA
17 bananas
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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

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Amys indifference curves
We can draw Amys indifference curves Then put
them in the Edgeworth Box
10 apples
OA
2 bananas
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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
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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
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Bettys indifference curves
15 bananas
OB
1 apple
And we can put Bettys indifference curves in the
Edgeworth box
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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
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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.

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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
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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
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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
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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.

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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
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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?

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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.

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• 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.

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• 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.

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Look at characteristics of set of utility
allocations
Symmetric (about diagonal), but non convex.
symmetric, compact, and convex
non-symmetric
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• 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.

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• 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.

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Consider the maximizer curves tangent to S
Symmetric (about diagonal), but non convex.
symmetric, compact, and convex unique maximizer
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xy c
y
maximizer curve
x
77

• 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) )

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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.

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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.

80
Original bargaining
(d1,d2)
d1 increases player1 gets more player1 gets less
(d1,d2)
81

• 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

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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
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What would we consider to be more fair?

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

84
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.
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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.
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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.
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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
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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.

90
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.

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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!

92
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

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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.

94

• 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.
95
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
96
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

97
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.

98

• 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.

99

• 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.

100
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.

101
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

102

• 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 (cost of grand coalition)
103
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.

104
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 )
105
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.

106
Added value Jenny and George

• Divide a dollar. If cant agree, both get
  nothing.

Added Value George
Added Value Jenny

100 (0 0) 100

107
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.
108
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
109
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

110
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

111
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.)

112
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

113
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 is no price that both will accept!

114
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

115
The price divides the added value
Value captured by...
Willingness-to-Pay
Buyer (consumer surplus)
Added Value
Price
Seller (producer surplus)
Willingness-to-Sell
116
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.
117
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).

118
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

119
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

120
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?

121
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.

122
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.

123
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.

124
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

125
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

126
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
127
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.

128
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
• Let n be the number of expected landings.

129

• 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.

130

• 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.

131
If we actually ran the numbers we would get
Computationally complex, so book shows shortcuts.
132

• 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.

133

• 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

134

• 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
135

• 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