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 theory’s 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 can’t 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
reject
0,0
p
• Second period: Accept if p > 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
strikes, work stoppages, etc.
of
• 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
• I get 0
= 0.8
=0
• In the first period, I offer 80%
• You get 80% of whole cake
• I get 20%
of whole cake
= 0.8
= 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
• I get 0
= 0.2
=0
• In the first period, I offer 20%
• You get 20% of whole cake
• I get 80%
of whole cake
= 0.2
= 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
Don’t Waste Cake
COMMANDMENT
In any bargaining setting, strike a deal as early as
possible!
• Why doesn’t this happen?
– Reputation building
– Lack of information
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:
• Defendant:
-
$500K - $100K = $ 400K
$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:
• Defendant:
-
$750K - $100K = $ 650K
$250K - $100K = $-350K
• No way to agree on a settlement! Defendant
would be willing to give plaintiff $350, but plaintiff
won’t 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?
intuition
Trust your
• 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 Offer
Example I
• p=9/10
• 90% chance company is highly profitable
• Best offer: Ask for $200K wage
• Expected value of offer:
(.9)$200K = $180K
• But: 10% chance of No Deal!
Take-it-or-leave-it Offer
Example II
• p=1/10
• 10% chance company is highly profitable
• Best offer: Ask for $150K wage
• If ask for $200K
Expected value of offer:
(.1)$200K = $20K
• If ask for $150K, get $150K
• 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 can’t 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) > 150K
140 +(170-140)p = 140+ 30p >150
if p > 1/3 , you win
What’s 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 one’s choice set is
defined irrespective of other people’s 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:
1.
2.
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:
1.
2.
The total payoff created through cooperation must be greater
than the sum of each party’s 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 player’s ‘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 can’t 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
1. %j ≥ 0;
2. %g ≥ 0 and;
3. %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 can’t 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
1. %j ≥ 0;
2. %g ≥ 50 and;
3. %j + %g = %100.
•
Note by changing George’s ‘next best alternative’ to
agreeing with Jenny, we change the potential
bargaining outcomes.
Ultimatum Games:
Basic Experimental Results
• In a review of numerous ultimatum experiments
Camerer (2003) found:
– The results reported…are 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 with
Incomplete 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:
1) How much should Player 1 offer Player 2?
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.
Composition of Urn:
0 - 30
5 - 25
10 - 20
Mean % of Pie Offered to Receiver:
31.2%
34.2%
42.4%
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 & Baba’s 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 & Baba’s Pie Woes
Offerer Ali
Round 1
Pie Size 100
Baba
2
80
Ali
3
60
Baba
4
40
Ali
5
20
Pie Split
Ali
60
Baba
40
<40
>40
>40
<20
<20
>20
10
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: (5+1) / (2*5) = 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
Ali
75
25
50
0
Baba
25
50
0
25
Pie Split
If Ali offered 50%, Baba would have no reason to question! He never gets more.
Model for Bargaining – no shrinking pie
Example – 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
Box is 11
units high
– to
represent
the 11
apples 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
The Edgeworth Box for Amy and Betty
Amy’s
apples
Measure
Amy’s
bundle from
here
Amy’s bananas
The Edgeworth Box for Amy and Betty
Betty’s bananas
Measure
Betty’s
bundle from
here
Betty’s
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
Amy’s indifference curves
We can draw Amy’s
indifference curves
Then put them in the
Edgeworth Box
10 apples
OA
2 bananas
This is Amy’s 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
Betty’s indifference curves
15 bananas
OB
1 apple
And we can put Betty’s 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 Amy’s and Betty’s 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:
1. Pareto Optimal
2. independence of irrelevant alternatives
3. 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
y
x*y = c
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 ( pi*u1(si),  pi*u2(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 < A1+A2
u2 K
A2
A1+A2
fair allocation
Original debts are equal
Nash solution picks equal
division along line of distribution
A1
u1
K
A2
A1
K
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
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 doesn’t satisfy monotonicity.
That is, when the set of possible solutions is larger,
a person can actually get less.
Kalai-Smorodinsky solution rule
for 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.
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
yes
yes
yes
Z
Value
yes
0
yes
200
yes
yes
220
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 water
Assume in our models, MUST be better to all work together.
Amalga Benson Cove
Joint
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.
Cost
30
yes
yes
yes
yes
yes
yes
yes
yes
30
yes
30
yes
40
yes
40
40
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 applications 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
$a+$b, $a+$c and $b+$c can each be no more than $40m
$a+$b+$c = $66m
But this is impossible!
To see this:
$a + $b
$a +
$40m
$c  $40m
$b + $c  $40m
Add up:
$2a + $2b + $2c  $120
So:
$a + $b + $c  $60
One of the coalitions
of 2 towns will block
the grand coalition
unless this is satisfied.
But this is impossible!
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 nperson bargaining game is the function v:N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 player’s 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} }
For S  B define e S   n with eiS  1 if i  S and 0 otherwise
• B is called a balanced collection if there exist weights lS (S
S
N
element of B) such that
l e
SB
• Example: l = {0.5, 0.5, 0.5, 1}
S
e
1
1
0 0 1
 
 
    
1 1 1 0 1 1 0 1


1 
2 0 2 1 2 1 0 1
 
 
    
0
0
 
 
0 1 1
• Definition: A game is balanced if for every balanced collection B
with corresponding weights lS:
 l c( S )  c( N )
SB
S
• 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} {BC}{AC}{ABC}
• We could find weights (so collection is balanced)
1
1
0 1
 
 
  
1 1 1 0 1 1 1



2 0 2 1 2 1 1
 
 
  
0 
0 
0 1
But when we apply those weights to the costs of coalitions
½(40) + ½(40) + ½(40) = 120 < 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 can’t agree, both get nothing.
Added
Value
George
=
Added
Value
Jenny
= $100 – ($0 +$0)
= $100
Added value – Jenny and George
• Divide a dollar. If can’t agree, George gets $50.
Added
Value
George
Added
Value
Jenny
=
$100 – ($0 +$50)
$50
=
$100 – ($50 +$0)
$50
Note – these
are the same.
This is a
general result
for TWO people
bargaining (but
ONLY for two
people)
Note, both add $50 as if they don’t work together, the best the two of them
can do is $50, but earn $100 together.
disagree
ment
point
(George)
Total payoff if
cooperate
disagree
ment
poing
(Jenny)
George’s Best
Alternative To
Negotiated
Agreement –
disagreement
point
Jenny’s Best
Alternative To
Negotiated
Agreement –
disagreement
point
George’s added value = Total Payoff – George’s BATNA – Jenny’s BATNA
But this clearly equals Jenny’s added value
So with 2 people: Total surplus from agreement = each person’s 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 person’s 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 doesn’t 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 doesn’t 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 < 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 buyer’s economic profit
= WTP – Price
The value to the seller is the seller’s economic profit
= Price – WTS
The price divides the added value
Value
captured by...
Willingness-to-Pay
Added
Value
Price
Willingness-to-Sell
Buyer
(consumer
surplus)
Seller
(producer
surplus)
Multi-party bargaining
1. Each individual or sub group should never get
less than their outside option
– Because they can always ‘split off’ and go their own way
2. 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(C+i) – 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 S (, a) 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:
1
Sh( A, a) 
(v(S (, a)  a)  v( S (, a)))

A! 
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 – doesn’t depend on labeling
linear - (au+bv) = a(u) + b(v)
irrelevant to dummy players: If i is a dummy player
i(v) = 0
v(C  {i})  v(C ) is called the
• The value
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
yes
yes
yes
Greens 16
yes
1
yes
yes
Value
1
yes
yes
yes
yes
yes
Browns 8
1
0
yes
0
yes
0
yes
yes
yes
yes
yes
1
yes
yes
1
yes
1
yes
1
yes
yes
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
1
Sh( A, a) 
(v(S (, a)  a)  v( S (, a)))

A! 
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?
• Let’s assume ki is the cost needed to land plane
of type i.
• Order the plane types so 0 < k1 < k2 … < 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 Number
Runway 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:
Charge
per landing
Plane Type
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 plane’s 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
k
i (v)   ( Kl  1  Kl ) /( t l }Nt}
T
l 1
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/(5+2+1+1+1) =
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