Transcript Chapter Title

3 The Mathematics of Sharing
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser
Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminsher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
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Basic Elements of a Fair-Division Game
The underlying elements of every fairdivision game are as follows:
The goods (or “booty”).
This is the informal name we will give to the
item or items being divided. Typically, these
items are tangible physical objects with a
positive value, such as candy, cake, pizza,
jewelry, art, land, and so on.
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Basic Elements of a Fair-Division Game
In some situations the items being divided
may be intangible things such as rights–
water rights, drilling rights, broadcast
licenses, and so on– or things with a
negative value such as chores, liabilities,
obligations, and so on. Regardless of the
nature of the items being divided, we will
use the symbol S throughout this chapter to
denote the booty.
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Basic Elements of a Fair-Division Game
The players.
In every fair-division game there is a set of
parties with the right (or in some cases the
duty) to share S. They are the players in the
game. Most of the time the players in a fairdivision game are individuals, but it is worth
noting that some of the most significant
applications of fair division occur when the
players are institutions (ethnic groups,
political parties, states, and even nations).
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Basic Elements of a Fair-Division Game
The value systems.
The fundamental assumption we will make
is that each player has an internalized value
system that gives the player the ability to
quantify the value of the booty or any of its
parts. Specifically, this means that each
player can look at the set S or any subset of
S and assign to it a value – either in
absolute terms (“to me, that’s worth
$147.50”) or in relative terms (“to me, that
piece is worth 30% of the total value of S”).
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Basic Assumptions
Like most games, fair-division games are
predicated on certain assumptions about
the players. For the purposes of our
discussion, we will make the following four
assumptions:
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Basic Assumptions
Rationality
Each of the players is a thinking, rational
entity seeking to maximize his or her share
of the booty S. We will further assume that
in the pursuit of this goal, a player’s moves
are based on reason alone (we are taking
emotion, psychology, mind games, and all
other non-rational elements out of the
picture.)
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Basic Assumptions
Cooperation
The players are willing participants and
accept the rules of the game as binding.
The rules are such that after a finite number
of moves by the players, the game
terminates with a division of S. (There are
no outsiders such as judges or referees
involved in these games – just the players
and the rules.)
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Basic Assumptions
Privacy
Players have no useful information on the
other players’ value systems and thus of
what kinds of moves they are going to make
in the game. (This assumption does not
always hold in real life, especially if the
players are siblings or friends.)
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Basic Assumptions
Symmetry
Players have equal rights in sharing the set
S. A consequence of this assumption is
that, at a minimum, each player is entitled to
a proportional share of S – then there are
two players, each is entitled to at least onehalf of S, with three players each is entitled
to at least one-third of S, and so on.
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Fair Share
Given the booty S and players P1, P2, P3,…,
PN, each with his or her own value system,
the ultimate goal of the game is to end up
with a fair division of S, that is, to divide S
into N shares and assign shares to players
in such a way that each player gets a fair
share. This leads us to what is perhaps the
most important definition of this section.
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Fair Share
FAIR SHARE
Suppose that s denotes a share of the
booty S and that P is one of the players
in a fair-division game with N players.
We will say that s is a fair share to player
P if s is worth at least 1/Nth of the total
value of S in the opinion of P. (Such a
share is often called a proportional fair
share, but for simplicity we will refer to it
just as a fair share.)
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Fair Division Methods
We can think of a fair-division method as
the set of rules that define how the game is
to be played. Thus, in a fair-division game
we must consider not only the booty S and
the players P1, P2, P3,…, PN (each with his
or her own opinions about how S should be
divided), but also a specific method by
which we plan to accomplish the fair
division.
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Fair Division Methods
There are many different fair-division
methods known, but in this chapter we will
only discuss a few of the classic ones.
Depending on the nature of the set S, a fairdivision game can be classified as one of
three types: continuous, discrete, or mixed,
and the fair-division methods used depend
on which of these types we are facing.
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Types of Fair Division Games
Continuous
In a continuous fair-division game the set
S is divisible in infinitely many ways, and
shares can be increased or decreased by
arbitrarily small amounts. Typical examples
of continuous fair-division games involve the
division of land, a cake, a pizza, and so
forth.
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Types of Fair Division Games
Discrete
A fair-division game is discrete when the
set S is made up of objects that are
indivisible like paintings, houses, cars,
boats, jewelry, and so on. (One might argue
that with a sharp enough knife a piece of
candy could be chopped up into smaller and
smaller pieces. As a semantic convenience
let’s agree that candy is indivisible, and
therefore dividing candy is a discrete fairdivision game.)
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Types of Fair Division Games
Mixed
A mixed fair-division game is one in which
some of the components are continuous
and some are discrete. Dividing an estate
consisting of jewelry, a house, and a parcel
of land is a mixed fair-division game.
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Types of Fair Division Games
Fair-division methods are classified
according to the nature of the problem
involved. Thus, there are discrete fairdivision methods (used when the set S is
made up of indivisible, discrete objects),
and there are continuous fair-division
methods (used when the set S is an
infinitely divisible, continuous set). Mixed
fair-division games can usually be solved by
dividing the continuous and discrete parts
separately, so we will not discuss them in
this chapter.
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3 The Mathematics of Sharing
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser
Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminsher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
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Divider-Chooser Method
The divider-chooser method is undoubtedly
the best known of all continuous fair-division
methods. This method can be used anytime
the fair-division game involves two players
and a continuous set S (a cake, a pizza, a
plot of land, etc.). Most of us have unwittingly
used it at some time or another, and
informally it is best known as the you cut–I
choose method. As this name suggests, one
player, called the divider, divides S into two
shares, and the second player, called the
chooser, picks the share he or she wants,
leaving the other share to the divider.
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Divider-Chooser Method
Under the rationality and privacy
assumptions we introduced in the previous
section, this method guarantees that both
divider and chooser will get a fair share (with
two players, this means a share worth 50%
or more of the total value of S). Not knowing
the chooser’s likes and dislikes (privacy
assumption), the divider can only guarantee
himself a 50% share by dividing S into two
halves of equal value (rationality
assumption); the chooser is guaranteed a
50% or better share by choosing the piece he
or she likes best.
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Example 3.1 Damian and Cleo Divide
a Cheesecake
On their first date, Damian and Cleo go to the
county fair. They buy jointly a raffle ticket, and
as luck would have it, they win a half
chocolate–half strawberry cheesecake.
Damian likes chocolate and strawberry
equally well, so in his eyes the chocolate and
strawberry
halves are
equal in
value.
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Example 3.1 Damian and Cleo Divide
a Cheesecake
On the other hand, Cleo hates chocolate–she
is allergic to it and gets sick if she eats any–
so in her eyes the value of the cake is 0% for
the chocolate
half, 100%
for the
strawberry
part.
Once again, to ensure a fair division, we will
assume that neither of them knows anything
about the other’s likes and dislikes.
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Example 3.1 Damian and Cleo Divide
a Cheesecake
Damian volunteers to go first (the divider). He
cuts the cake in a perfectly rational division of
the cake based on his value system–each
piece is half of the cake and to him worth onehalf of the total value of the cake. It is now
Cleo’s turn to choose, and her choice is
obvious she
will pick the
piece having
the larger
strawberry
part.
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Example 3.1 Damian and Cleo Divide
a Cheesecake
The final outcome of this division is that
Damian gets a piece that in his own eyes is
worth exactly half of the cake, but Cleo ends
up with a much sweeter deal–a piece that in
her own eyes is worth about two-thirds of the
cake. This is, nonetheless, a fair division of the
cake–both players get pieces worth 50% or
more.
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Better to be the Chooser
Example 3.1 illustrates why, given a choice,
it is always better to be the chooser than the
divider–the divider is guaranteed a share
worth exactly 50% of the total value of S,
but with just a little luck the chooser can end
up with a share worth more than 50%. Since
a fair-division method should treat all
players equally, both players should have
an equal chance of being the chooser. This
is best done by means of a coin toss, with
the winner of the coin toss getting the
privilege of making the choice.
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3 The Mathematics of Sharing
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser
Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminsher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
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Lone-Divider Method
The first important breakthrough in the
mathematics of fair division came in 1943,
when Steinhaus came up with a clever way
to extend some of the ideas in the dividerchooser method to the case of three players,
one of whom plays the role of the divider and
the other two who play the role of choosers.
Steinhaus’ approach was subsequently
generalized to any number of players N (one
divider and N – 1 choosers) by Princeton
mathematician Harold Kuhn.
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Lone-Divider Method for Three Players
Preliminaries
One of the three players will be the divider;
the other two players will be choosers. Since
it is better to be a chooser than a divider, the
decision of who is what is made by a random
draw (rolling dice, drawing cards from a deck,
etc.). We’ll call the divider D and the
choosers C1 and C2.
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Lone-Divider Method for Three Players
Step 1 (Division)
The divider D divides the cake into three
shares (s1, s2, and s3). D will get one of these
shares, but at this point does not know which
one. (Not knowing which share will be his is
critical – it forces D to divide the cake into
three shares of equal value.)
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Lone-Divider Method for Three Players
Step 2 (Bidding)
C1 declares (usually by writing on a slip of
paper) which of the three pieces are fair
shares to her. Independently, C2 does the
same. These are the bids. A chooser’s bid
must list every single piece that he or she
considers to be a fair share (i.e.,worth onethird or more of the cake)–it may be tempting
to bid only for the very best piece,but this is a
strategy that can easily backfire. To preserve
the privacy requirement, it is important that
the bids be made independently, without the
choosers being privy to each other’s bids.
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Lone-Divider Method for Three Players
Step 3 (Distribution)
Who gets which piece? The answer, of
course, depends on which pieces are listed in
the bids. For convenience, we will separate
the pieces into two types: C-pieces (these
are pieces chosen by either one or both of
the choosers) and U-pieces (these are
unwanted pieces that did not appear in either
of the bids).
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Lone-Divider Method for Three Players
Step 3 (Distribution) continued
Expressed in terms of value, a U-piece is a
piece that both choosers value at less than
33 1/3% of the cake, and a C-piece is a piece
that at least one of the choosers (maybe
both) value at 33 1/3% or more. Depending
on the number of C-pieces, there are two
separate cases to consider.
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Lone-Divider Method for Three Players
Case 1
When there are two or more C-pieces, there
is always a way to give each chooser a
different piece from among the pieces listed
in her bid. (The details will be covered in
Examples 3.2 and 3.3.) Once each chooser
gets her piece, the divider gets the last
remaining piece. At this point every player
has received a fair share, and a fair division
has been accomplished.
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Lone-Divider Method for Three Players
Case 1
(Sometimes we might end up in a situation in
which C1 likes C2’s piece better than her own
and vice versa. In that case it is perfectly
reasonable to add a final, informal step and
let them swap pieces–this would make each
of them happier than they already were, and
who could be against that?)
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Lone-Divider Method for Three Players
Case 2
When there is only one Cpiece, we have a bit of a
problem because it means that
both choosers are bidding for
the very same piece.
The solution requires a little
more creativity. First, we take
care of the divider D–to whom
all pieces are equal in value–
by giving him one of the pieces
that neither chooser wants.
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Lone-Divider Method for Three Players
Case 2
After D gets his piece,the two
pieces left (the C-piece and the
remaining U-piece) are
recombined into one piece that
we call the B-piece.
We can revert to the dividerchooser method to finish the
fair division: one player cuts the
B-piece into two pieces; the
other player chooses the piece
she likes better.
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Lone-Divider Method for Three Players
Case 2
This process results in a fair division of the cake
because it guarantees fair shares for all players.
We know that D ends up with a fair share by the
very fact that D did the original division, but what
about C1 and C2? The key observation is that in
the eyes of both C1 and C2 the B-piece is worth
more than two-thirds of the value of the original
cake (think of the B-piece as 100% of the original
cake minus a U-piece worth less than 33 1/3%),
so when we divide it fairly into two shares, each
party is guaranteed more than one-third of the
original cake.
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Example 3.2 Lone Divider with 3
Players (Case 1, Version 1)
Dale, Cindy, and Cher are dividing a cake
using Steinhaus’s lone-divider method. They
draw cards from a well-shuffled deck of cards,
and Dale draws the low card (bad luck!) and
has to be the divider.
Step 1 (Division)
Dale divides the cake into three pieces s1, s2,
and s3. Table 3-1 shows the values of the
three pieces in the eyes of each of the
players.
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Example 3.2 Lone Divider with 3
Players (Case 1, Version 1)
Step 2 (Bidding)
We can assume that Cindy’s bid list is {s1, s3}
and Cher’s bid list is also {s1, s3}.
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Example 3.2 Lone Divider with 3
Players (Case 1, Version 1)
Step 3 (Distribution)
The C-pieces are s1 and s3. There are two
possible distributions. One distribution would
be: Cindy gets s1, Cher gets s3, and Dale gets
s2. An even better distribution (the optimal
distribution) would be: Cindy gets s3, Cher
gets s1, and Dale gets s2. In the case of the
first distribution, both Cindy and Cher would
benefit by swapping pieces, and there is no
rational reason why they would not do so.
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Example 3.2 Lone Divider with 3
Players (Case 1, Version 1)
Step 3 (Distribution) continued
Thus, using the rationality assumption, we
can conclude that in either case the final
result will be the same: Cindy gets s3, Cher
gets s1, and Dale gets s2.
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Example 3.3 Lone Divider with 3
Players (Case 1, Version 2)
We’ll use the same setup as in Example 3.2–
Dale is the divider, Cindy and Cher are the
choosers.
Step 1 (Division)
Dale divides the cake into three pieces s1, s2,
and s3. Table 3-2 shows the values of the
three pieces in the eyes of each of the
players.
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Example 3.3 Lone Divider with 3
Players (Case 1, Version 2)
Step 2 (Bidding)
Here Cindy’s bid list is {s2} only, and Cher’s
bid list is {s1} only.
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Example 3.3 Lone Divider with 3
Players (Case 1, Version 2)
Step 3 (Distribution)
This is the simplest of all situations, as there
is only one possible distribution of the pieces:
Cindy gets s2, Cher gets s1, and Dale gets s3.
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Example 3.4 Lone Divider with 3
Players (Case 2)
The gang of Examples 3.2 and 3.3 are back
at it again.
Step 1 (Division)
Dale divides the cake into three pieces s1, s2,
and s3. Table 3-3 shows the values of the
three pieces in the eyes of each of the
players.
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Example 3.4 Lone Divider with 3
Players (Case 2)
Step 2 (Bidding)
Here Cindy’s and Cher’s bid list consists of
just {s3}.
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Example 3.4 Lone Divider with 3
Players (Case 2)
Step 3 (Distribution)
The only C-piece is s3. Cindy and Cher talk it
over, and without giving away any other
information agree that of the two U-pieces, s1
is the least desirable, so they all agree that
Dale gets s1. (Dale doesn’t care which of the
three pieces he gets, so he has no rational
objection.) The remaining pieces (s2 and s3)
are then recombined to form the B-piece, to
be divided between Cindy and Cher using the
divider-chooser method (one of them divides
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Example 3.2 Lone Divider with 3
Players (Case 1, Version 1)
Step 3 (Distribution) continued
the B-piece into two shares, the other one
chooses the share she likes better).
Regardless of how this plays out, both of
them will get a very healthy share of the cake:
Cindy will end up with a piece worth at least
40% of the original cake (the B-piece is worth
80% of the original cake to Cindy), and Cher
will end up with a piece worth at least 45% of
the original cake (the B-piece is worth 90% of
the original cake to Cher).
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The Lone-Divider Method for More Than
Three Players
The first two steps of Kuhn’s method are a
straightforward generalization of Steinhaus’s
lone-divider method for three players, but the
distribution step requires some fairly
sophisticated mathematical ideas and is
rather difficult to describe in full generality, so
we will only give an outline here and will
illustrate the details with a couple of
examples for N = 4 players.
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The Lone-Divider Method for More Than
Three Players
Preliminaries
One of the players is chosen to be the divider
D, and the remaining N – 1 players are all
going to be choosers. As always, it’s better to
be a chooser than a divider, so the decision
should be made by a random draw.
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The Lone-Divider Method for More Than
Three Players
Step 1 (Division)
The divider D divides the set S into N shares
s1, s2, s3, ..., sN. D is guaranteed of getting
one of these shares, but doesn’t know which
one.
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The Lone-Divider Method for More Than
Three Players
Step 2 (Bidding)
Each of the N – 1 choosers independently
submits a bid list consisting of every share
that he or she considers to be a fair share
(i.e., worth 1/Nth or more of the booty S).
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The Lone-Divider Method for More Than
Three Players
Step 3 (Distribution)
The bid lists are opened. Much as we did
with three players, we will have to consider
two separate cases, depending on how these
bid lists turn out.
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The Lone-Divider Method for More Than
Three Players
Case 1.
If there is a way to assign a different share to
each of the N – 1 choosers, then that should
be done. (Needless to say, the share
assigned to a chooser should be from his or
her bid list.) The divider, to whom all shares
are presumed to be of equal value, gets the
last unassigned share. At the end, players
may choose to swap pieces if they want.
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The Lone-Divider Method for More Than
Three Players
Case 2.
There is a standoff–in other words, there are
two choosers both bidding for just one share,
or three choosers bidding for just two shares,
or K choosers bidding for less than K shares.
This is a much more complicated case, and
what follows is a rough sketch of what to do.
To resolve a standoff, we first set aside the
shares involved in the standoff from the
remaining shares.
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The Lone-Divider Method for More Than
Three Players
Case 2.
Likewise, the players involved in the standoff
are temporarily separated from the rest. Each
of the remaining players (including the
divider) can be assigned a fair share from
among the remaining shares and sent
packing. All the shares left are recombined
into a new booty S to be divided among the
players involved in the standoff, and the
process starts all over again.
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Example 3.5 Lone Divider with 4
Players (Case 1)
We have one divider, Demi, and three
choosers, Chan, Chloe, and Chris.
Step 1 (Division)
Demi divides the cake into four shares s1, s2,
s3, and s4. Table 3-4 shows how each of the
players values each of the four shares.
Remember that the information on each row
of Table 3-4 is private and known only to that
player.
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Example 3.5 Lone Divider with 4
Players (Case 1)
Step 2 (Bidding)
Chan’s bid list is {s1, s3}; Chloe’s bid list is {s3}
only; Chris’s bid list is {s1, s4}.
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Example 3.5 Lone Divider with 4
Players (Case 1)
Step 3 (Distribution)
The bid lists are opened. It is clear that for
starters Chloe must get s3 – there is no other
option. This forces the rest of the distribution:
s1 must then go to Chan, and s4 goes to
Chris. Finally, we give the last remaining
piece, s2, to Demi.
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Example 3.5 Lone Divider with 4
Players (Case 1)
This distribution results in a fair division of the
cake, although it is not entirely “envy-free”
Chan wishes he had Chloe’s piece (35% is
better than 30%) but Chloe is not about to
trade pieces with him, so he is stuck with s1.
(From a strictly rational point of view, Chan
has no reason to gripe–he did not get the best
piece, but got a piece worth 30% of the total,
better than the 25% he is entitled to.)
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Excursions in Modern Mathematics, 7e: 3.1 - 61
Example 3.6 Lone Divider with 4
Players (Case 2)
Once again, we will let Demi be the divider
and Chan, Chloe, and Chris be the three
choosers (same players, different game).
Step 1 (Division)
Demi divides the cake into four shares s1, s2,
s3, and s4. Table 3-5 shows how each of the
players values each of the four shares.
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Excursions in Modern Mathematics, 7e: 3.1 - 62
Example 3.6 Lone Divider with 4
Players (Case 2)
Step 2 (Bidding)
Chan’s bid list is {s4}; Chloe’s bid list is {s2, s3}
only; Chris’s bid list is {s4}.
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Example 3.6 Lone Divider with 4
Players (Case 2)
Step 3 (Distribution)
The bid lists are opened, and the players can
see that there is a standoff brewing on the
horizon–Chan and Chris are both bidding for
s4. The first step is to set aside and assign
Chloe and Demi a fair share from s1, s2, and
s4. Chloe could be given either s2 or s3. (She
would rather have s2,of course, but it’s not for
her to decide.)
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Example 3.6 Lone Divider with 4
Players (Case 2)
Step 3 (Distribution)
A coin toss is used to determine which one.
Let’s say Chloe ends up with s3 (bad luck!).
Demi could be now given either s1 or s2.
Another coin toss, and Demi ends up with s1.
The final move is ... you guessed it!–
recombine s2 and s4 into a single piece to be
divided between Chan and Chris using the
divider-chooser method.
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Excursions in Modern Mathematics, 7e: 3.1 - 65
Example 3.6 Lone Divider with 4
Players (Case 2)
Step 3 (Distribution)
Since (s2 + s4) is worth 60% to Chan and 58%
to Chris (you can check it out in Table 3-5),
regardless of how this final division plays out
they are both guaranteed a final share worth
more than 25% of the cake.
Mission accomplished! We have produced a
fair division of the cake.
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Excursions in Modern Mathematics, 7e: 3.1 - 66
3 The Mathematics of Sharing
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser
Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminsher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
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Excursions in Modern Mathematics, 7e: 3.1 - 67
The Lone-Chooser Method
A completely different approach for extending
the divider-chooser method was proposed in
1964 by A.M.Fink, a mathematician at Iowa
State University. In this method one player
plays the role of chooser, all the other players
start out playing the role of dividers. For this
reason, the method is known as the lonechooser method. Once again, we will start
with a description of the method for the case
of three players.
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The Lone-Chooser Method for 3 Players
Preliminaries
We have one chooser and two dividers. Let’s
call the chooser C and the dividers D1 and
D2. As usual, we decide who is what by a
random draw.
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The Lone-Chooser Method for 3 Players
Step 1 (Division)
D1 and D2 divide S between themselves into
two fair shares. To do this, they use the
divider-chooser method. Let’s say that D1
ends up s1 with D2 and ends up with s2.
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The Lone-Chooser Method for 3 Players
Step 2 (Subdivision)
Each divider divides his share into three
subshares. Thus, D1 divides s1 into three
subshares, which we will call s1a, s1b, and s1c.
Likewise, D2 divides s2 into three subshares,
which we will call s2a, s2b, and s2c.
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The Lone-Chooser Method for 3 Players
Step 3 (Selection)
The chooser C now selects one of D1’s three
subshares and one of D2’s three subshares
(whichever she likes best). These two
subshares make up C’s final share. D1 then
keeps the remaining two subshares from s1,
and D2 keeps the
remaining two
subshares from
s2.
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The Lone-Chooser Method for 3 Players
Why is this a fair division of S? D1 ends up
with two-thirds of s1. To D1, s1 is worth at
least one-half of the total value of S, so twothirds of s1 is at least one- third–a fair share.
The same argument applies to D2. What
about the chooser’s share? We don’t know
what s1 and s2 are each worth to C, but it
really doesn’t matter–a one-third or better
share of s1 plus a one-third or better share of
s2 equals a one-third or better share of
(s1 + s2) and thus a fair share of the cake.
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Excursions in Modern Mathematics, 7e: 3.1 - 73
Example 3.7 Lone Chooser with 3
Players
David, Dinah, and Cher are dividing an
orange-pineapple cake using the lonechooser method. The cake is valued by each
of them at $27, so each of them expects to
end up with a share worth at least $9. Their
individual value systems (not known to one
another, but available to us as outside
observers) are as follows:
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Example 3.7 Lone Chooser with 3
Players
■
■
■
David likes pineapple and orange the same.
Dinah likes orange but hates pineapple.
Cher likes pineapple twice as much as she
likes orange.
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Example 3.7 Lone Chooser with 3
Players
After a random selection, Cher gets to be the
chooser and thus gets to sit out Steps 1 & 2.
Step 1 (Division)
David and Dinah start by dividing the cake
between themselves using the divider-chooser
method. After a coin flip, David cuts the cake
into two pieces.
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Example 3.7 Lone Chooser with 3
Players
Step 1 (Division) continued
Since Dinah doesn’t like
pineapple, she will take
the share with the most
orange.
Step 2 (Subdivision)
David divides his share
into three subshares that
in his opinion are of equal
value (all the same size).
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Example 3.7 Lone Chooser with 3
Players
Step 2 (Subdivision) continued
Dinah also divides her share
into three smaller subshares
that in her opinion are of equal
value. (Remember that Dinah
hates pineapple. Thus, she
has made her cuts in such a
way as to have one-third of the
orange in each of the
subshares.)
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Example 3.7 Lone Chooser with 3
Players
Step 3 (Selection)
It’s now Cher’s turn to choose one sub-share
from David’s three and one subshare from
Dinah’s three. She will choose one of the two
pineapple wedges
from David’s
subshares and the
big orangepineapple wedge
from Dinah’s
subshares.
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Example 3.7 Lone Chooser with 3
Players
Step 3 (Selection)
The final fair division of the cake is shown.
David gets a final share worth $9, Dinah gets
a final share worth $12, and Cher gets a final
share worth $14. David is satisfied, Dinah is
happy, and Cher is ecstatic.
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Excursions in Modern Mathematics, 7e: 3.1 - 80
The Lone-Chooser Method for N
Players
In the general case of N players, the lonechooser method involves one chooser C and
N – 1 dividers D1, D2,…, DN–1. As always, it is
preferable to be a chooser than a divider, so
the chooser is determined by a random draw.
The method is based on an inductive
strategy. If you can do it for three players,
then you can do it for four players; if you can
do it for four, then you can do it for five; and
we can assume that we can use the lonechooser method with N – 1 players.
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The Lone-Chooser Method for N
Players
Step 1 (Division)
D1, D2,…, DN–1 divide fairly the set S among
themselves, as if C didn’t exist. This is a fair
division among N – 1 players, so each one
gets a share he or she considers worth at
least of 1/(N –1)th of S.
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The Lone-Chooser Method for N
Players
Step 2 (Subdivision)
Each divider subdivides his or her share into
N sub-shares.
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The Lone-Chooser Method for N
Players
Step 3 (Selection)
The chooser C finally gets to play. C selects
one sub-share from each divider – one
subshare from D1, one from D2, and so on. At
the end, C ends up with N – 1 subshares,
which make up C’s final share, and each
divider gets to keep the remaining N – 1
subshares in his or her subdivision.
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The Lone-Chooser Method for N
Players
When properly played, the lone-chooser
method guarantees that everyone, dividers
and chooser alike, ends up with a fair share
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3 The Mathematics of Sharing
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser
Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminsher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
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The Last-Diminisher Method
The last-diminisher method was proposed by
Polish mathematicians Stefan Banach and
Bronislaw Knaster in the 1940s. The basic
idea behind this method is that throughout
the game, the set S is divided into two
pieces–a piece currently “owned” by one of
the players (we will call that piece the Cpiece and the player claiming it the
“claimant”) and the rest of S, “owned” jointly
by all the other players.
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The Last-Diminisher Method
We will call this latter piece the R-piece and
the remaining players the “non-claimants.”
The tricky part of this method is that the
entire arrangement is temporary – as the
game progresses, each non-claimant has a
chance to become the current claimant (and
bump the old claimant back into the nonclaimant group) by making changes to the Cpiece and consequently to the R-piece. Thus,
the claimant, the non-claimants, the C-piece,
and the R-piece all keep changing throughout
the game.
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The Last-Diminisher Method
Preliminaries
Before the game starts the players are
randomly assigned an order of play (like in a
game of Monopoly this can be done by rolling
a pair of dice). We will assume that P1 plays
first, P2 second,…, PN last, and the players
will play in this order throughout the game.
The game is played in rounds, and at the end
of each round there is one fewer player and a
smaller S to be divided.
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The Last-Diminisher Method
Round 1
P1 kicks the game off by “cutting” for herself a
1/Nth share of S (i.e., a share whose value
equals 1/Nth of the value of S). This will be
the current C-piece, and P1 is its claimant. P1
does not know whether or not she will end up
with this share, so she must be careful that
her claim is neither too small (in case she
does) nor too large (in case someone else
does).
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The Last-Diminisher Method
Round 1
P2 comes next and has a choice: pass
(remain a nonclaimant) or diminish the Cpiece into a share that is a 1/Nth share of S.
Obviously, P2 can be a diminisher only if he
thinks that the value of the current C-piece is
more than 1/Nth the value of S. If P2
diminishes, several changes take place: P2
becomes the new claimant; P1 is bumped
back into the nonclaimant group; the
diminished C-piece becomes the new current
C-piece; and the “trimmed” piece is added to
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The Last-Diminisher Method
Round 1
the old R-piece to form a new, larger R-piece
(there is a lot going on here and it’s best to
visualize it).
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The Last-Diminisher Method
Round 1
P3 comes next and has exactly the same
opportunity as P2: pass or diminish the
current C-piece. If P3 passes, then there are
no changes and we move on to the next
player. If P3 diminishes (only because in her
value system the current C-piece is worth
more than 1/Nth of S), she does so by
trimming the C-piece to a 1/Nth share of S.
The rest of the routine is always the same:
The trimmed piece is added to the R-piece,
and the previous claimant (P1 or P2) is
bumped back into the nonclaimant group.
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The Last-Diminisher Method
Round 1
The round continues this way, each player in
turn having an opportunity to pass or
diminish. The last player PN can also pass or
diminish, but if he chooses to diminish, he
has a certain advantage over the previous
players–knowing that there is no player
behind him who could further diminish his
claim. In this situation if PN chooses to be a
diminisher, the logical move would be to trim
the tiniest possible sliver from the current Cpiece–a sliver so small that for all practical
purposes it has zero value.
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The Last-Diminisher Method
Round 1
(Remember that a player’s goal is to
maximize the size of his or her share.) We
will describe this move as “trimming by 0%,”
although in practice there has to be
something trimmed, however small it may be.
At the end of Round 1, the current claimant,
or last diminisher, gets to keep the C-piece
(it’s her fair share) and is out of the game.
The remaining players (the nonclaimants)
move on to the next round, to divide the Rpiece among themselves.
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The Last-Diminisher Method
Round 1
At this point everyone is happy–the last
diminisher got his or her claimed piece, and
the non-claimants are happy to move to the
next round, where they will have a chance to
divide the R-piece among themselves.
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The Last-Diminisher Method
Round 2
The R-piece becomes the new S, and a new
version of the game is played with the new S
and the N – 1 remaining players [this means
that the new standard for a fair share is a
value of 1/(N–1)th or more of the new S]. At
the end of this round, the last diminisher gets
to keep the current C-piece and is out of the
game.
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The Last-Diminisher Method
Rounds 3, 4, and so on
Repeat the process, each time with one
fewer player and a smaller S, until there are
just two players left. At this point, divide the
remaining piece between the final two
players using the divider-chooser method.
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Example 3.8 The Castaways
A new reality TV show called The Castaways
is making its debut this season. In the show
five contestants (let’s call them P1, P2, P3, P4,
and P5) are dropped off on a deserted tropical
island in the middle of nowhere and left there
for a year to manage on their own. After one
year, the player who has succeeded and
prospered the most wins the million-dollar
prize. (The producers are counting on the
quarreling, double-crossing, and backbiting
among the players to make for great reality
TV!)
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Example 3.8 The Castaways
In the first episode of the show, the players
are instructed to divide up the island among
themselves any way they see fit. Instead of
quarreling and double-dealing as the
producers were hoping for, these five players
choose to divide the island using the lastdiminisher method. This being reality TV,
pictures speak louder than words, and the
whole episode unfolds in Figs.3-11 through
3-15.
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Example 3.8 The Castaways
Round 1
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Example 3.8 The Castaways
Round 1
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Example 3.8 The Castaways
Round 1
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Example 3.8 The Castaways
Round 2
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Example 3.8 The Castaways
Round 2
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Example 3.8 The Castaways
Round 3
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Example 3.8 The Castaways
Round 3
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Example 3.8 The Castaways
Round 4
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Example 3.8 The Castaways
The Final Division of the Island
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Continuous versus Discrete
In the next two sections we will discuss
discrete fair-division methods – methods for
dividing a booty S consisting of indivisible
objects such as art, jewels, or candy. As a
general rule of thumb, discrete fair division
is harder to achieve than continuous fair
division because there is a lot less flexibility
in the division process, and discrete fair
divisions that are truly fair are only possible
under a limited set of conditions. Thus, it is
important to keep in mind that while discrete
methods have limitations, they still are the
best methods we have available.
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Excursions in Modern Mathematics, 7e: 3.1 - 110
3 The Mathematics of Sharing
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser
Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminsher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
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Discrete versus Continuous
As a general rule of thumb, discrete fair
division is harder to achieve than
continuous fair division because there is a
lot less flexibility in the division process, and
discrete fair divisions that are truly fair are
only possible under a limited set of
conditions. Thus, it is important to keep in
mind that while both of the methods we will
discuss in the next two sections have
limitations, they still are the best methods we
have available. Moreover, when they work,
both methods work remarkably well and
produce surprisingly good fair divisions.
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The Method of Sealed Bids
The method of sealed bids was originally
proposed by Hugo Steinhaus and Bronislaw
Knaster around 1948. The best way to
illustrate how this method works is by means
of an example.
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Example 3.9 Settling Grandma’s Estate
In her last will and testament, Grandma plays
a little joke on her four grandchildren (Art,
Betty, Carla, and Dave) by leaving just three
items–a cabin in the mountains, a vintage
1955 Rolls Royce, and a Picasso painting–
with the stipulation that the items must remain
with the grandchildren (not sold to outsiders)
and must be divided fairly in equal shares
among them. How can we possibly resolve
this conundrum? The method of sealed bids
will give an ingenious and elegant solution.
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Example 3.9 Settling Grandma’s Estate
Step 1 (Bidding)
Each of the players makes a bid (in dollars)
for each of the items in the estate, giving his
or her honest assessment of the actual value
of each item. To satisfy the privacy
assumption, it is important that the bids are
done independently, and no player should be
privy to another player’s bids before making
his or her own. The easiest way to
accomplish this is for each player to submit
his or her bid in a sealed envelope. When all
the bids are in, they are opened.
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Example 3.9 Settling Grandma’s Estate
Step 1 (Bidding)
Table 3-6 shows each player’s bid on each
item in the estate.
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Example 3.9 Settling Grandma’s Estate
Step 2 (Allocation)
Each item will go to the highest bidder for that
item. (If there is a tie, the tie can be broken
with a coin flip.) In this example the cabin
goes to Betty, the vintage Rolls Royce goes
to Dave, and the Picasso painting goes to Art.
Notice that Carla gets nothing. Not to worry–it
all works out at the end! (In this method it is
possible for one player to get none of the
items and another player to get many or all of
the items. Much like in a silent auction, it’s a
matter of who bids the highest.)
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Example 3.9 Settling Grandma’s Estate
Step 3 (First Settlement)
It’s now time to settle things up. Depending
on what items (if any) a player gets in Step 2,
he or she will owe money to or be owed
money by the estate. To determine how much
a player owes or is owed, we first calculate
each player’s fair-dollar share of the estate. A
player’s fair-dollar share is found by adding
that player’s bids and dividing the total by the
number of players.
The last row of Table 3-7 shows the fair-dollar
share of each player.
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Example 3.9 Settling Grandma’s Estate
Step 3 (First Settlement)
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Example 3.9 Settling Grandma’s Estate
Step 3 (First Settlement)
For example, Art’s bids on the three items
add up to $540,000. Since there are four
equal heirs, Art realizes he is only entitled to
one-fourth of that–his fair-dollar share is
therefore $135,000.
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Example 3.9 Settling Grandma’s Estate
Step 3 (First Settlement)
The fair-dollar shares are the baseline for the
settlements–if the total value of the items that
the player gets in Step 2 is more than his or
her fair-dollar share, then the player pays the
estate the difference. If the total value of the
items that the player gets is less than his or
her fair-dollar share, then the player gets the
difference in cash.
Here are the details of how the settlement
works out for each of our four players.
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Example 3.9 Settling Grandma’s Estate
Art
As we have seen, Art’s fair dollar share is
$135,000. At the same time, Art is getting a
Picasso painting worth (to him) $280,000, so
Art must pay the estate the difference of
$145,000 ($280,000 – $135,000). The
combination of getting the $280,000 Picasso
painting but paying $145,000 for it in cash
results in Art getting his fair share of the
estate.
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Example 3.9 Settling Grandma’s Estate
Betty
Betty’s fair-dollar share is $130,000. Since
she is getting the cabin, which she values at
$250,000, she must pay the estate the
difference of $120,000. By virtue of getting
the $250,000 house for $120,000, Betty ends
up with her fair share of the estate.
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Example 3.9 Settling Grandma’s Estate
Carla
Carla’s fair-dollar share is $123,000. Since
she is getting no items from the estate, she
receives her full $123,000 in cash. Clearly,
she is getting her fair share of the estate.
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Example 3.9 Settling Grandma’s Estate
Dave
Dave’s fair-dollar share is $110,000. Dave is
getting the vintage Rolls, which he values at
$52,000, so he has an additional $58,000
coming to him in cash. The Rolls plus the
cash constitute Dave’s fair share of the
estate.
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Example 3.9 Settling Grandma’s Estate
At this point each of the four heirs has
received a fair share, and we might consider
our job done, but this is not the end of the
story–there is more to come (good news
mostly!). If we add Art and Betty’s payments
to the estate and subtract the payments made
by the estate to Carla and Dave, we discover
that there is a surplus of $84,000! ($145,000
and $120,000 came in from Art and Betty;
$123,000 and $58,000 went out to Carla and
Dave.)
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Example 3.9 Settling Grandma’s Estate
Step 4 (Division of the Surplus)
The surplus is common money that belongs
to the estate, and thus to be divided equally
among the players. In our example each
player’s share of the $84,000 surplus is
$21,000.
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Example 3.9 Settling Grandma’s Estate
Step 4 (Division of the Surplus)
The surplus is common money that belongs
to the estate, and thus to be divided equally
among the players. In our example each
player’s share of the $84,000 surplus is
$21,000.
Step 5 (Final Settlement)
The final settlement is obtained by adding the
surplus money to the first settlement obtained
in Step 3.
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Example 3.9 Settling Grandma’s Estate
Step 5 (Final Settlement)
Art:
Gets the Picasso painting and pays the estate
$124,000–the original $145,000 he had to pay
in Step 3 minus the $21,000 he gets from his
share of the surplus.
(Everything done up to this point could be
done on paper, but now, finally, real money
needs to change hands!)
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Example 3.9 Settling Grandma’s Estate
Step 5 (Final Settlement)
Betty:
Gets the cabin and has to pay the estate only
$99,000 ($120,000 – $21,000).
Carla:
Gets $144,000 in cash ($123,000 + $21,000).
Dave:
Gets the vintage Rolls Royce plus $79,000
($58,000 + $21,000).
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The Method of Sealed Bids
The method of sealed bids works so well
because it tweaks a basic idea in economics.
In most ordinary transactions there is a buyer
and a seller, and the buyer knows the other
party is the seller and vice versa. In the
method of sealed bids, each player is
simultaneously a buyer and a seller, without
actually knowing which one until all the bids
are opened. This keeps the players honest
and, in the long run, works out to everyone’s
advantage.
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The Method of Sealed Bids
At the same time, the method of sealed bids
will not work unless the following two
important conditions are satisfied.
• Each player must have enough money to
play the game. If a player is going to make
honest bids on the items, he or she must
be prepared to buy some or all of them,
which means that he or she may have to
pay the estate certain sums of money. If the
player does not have this money available,
he or she is at a definite disadvantage in
playing the game.
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The Method of Sealed Bids
• Each player must accept money (if it is a
sufficiently large amount) as a substitute for
any item. This means that no player can
consider any of the items priceless.
The method of sealed bids takes a
particularly simple form in the case of two
players and one item.
Consider the following example.
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Example 3.10 Splitting Up the House
Al and Betty are getting a divorce. The only
joint property of any value is their house.
Rather than hiring attorneys and going to
court to figure out how to split up the house,
they agree to give the method of sealed bids
a try. Al’s bid on the house is $340,000;
Betty’s bid is $364,000. Their fair-dollar
shares of the “estate” are $170,000 and
$182,000, respectively. Since Betty is the
higher bidder, she gets to keep the house and
must pay Al cash for his share.
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Example 3.10 Splitting Up the House
The computation of how much cash Betty
pays Al can be done in two steps. In the first
settlement, Betty owes the estate $182,000.
Of this money, $170,000 pays for Al’s fair
share, leaving a surplus of $12,000 to be split
equally between them.
The bottom line is that Betty ends up paying
$176,000 to Al for his share of the house, and
both come out $6000 ahead.
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The Method of Sealed Bids
The method of sealed bids can provide an
excellent solution not only to settlements of
property in a divorce but also to the equally
difficult and often contentious issue of splitting
up a partnership. The catch is that in these
kinds of splits we can rarely count on the
rationality assumption to hold. A divorce or
partnership split devoid of emotion, spite, and
hard feelings is a rare thing indeed!
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3 The Mathematics of Sharing
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser
Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminsher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
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The Method of Markers
The method of markers is a discrete fairdivision method proposed in 1975 by William
F. Lucas, a mathematician at the Claremont
Graduate School. The method has the great
virtue that it does not require the players to
put up any of their own money. On the other
hand, unlike the method of sealed bids, this
method cannot be used effectively unless (1)
there are many more items to be divided than
there are players in the game and (2) the
items are reasonably close in value.
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The Method of Markers
In this method we start with the items lined
up in a random but fixed sequence called an
array. Each of the players then gets to make
an independent bid on the items in the array.
A player’s bid consists of dividing the array
into segments of consecutive items (as many
segments as there are players) so that each
of the segments represents a fair share of the
entire set of items.
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The Method of Markers
For convenience, we might think of the array
as a string. Each player then cuts” the string
into N segments, each of which he or she
considers an acceptable share. (Notice that
to cut a string into N sections, we need N – 1
cuts.) In practice, one way to make the “cuts”
is to lay markers in the places where the cuts
are made. Thus, each player can make his or
her bids by placing markers so that they
divide the array into N segments.
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The Method of Markers
To ensure privacy, no player should see the
markers of another player before laying down
his or her own.
The final step is to give to each player one of
the segments in his or her bid.
The easiest way to explain how this can be
done is with an example.
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Example 3.11 Dividing the Halloween
Leftovers
Alice, Bianca, Carla, and Dana want to divide
the Halloween leftovers shown in Fig. 3-16
among themselves. There are 20 pieces, but
having each randomly choose 5 pieces is not
likely to work well–the pieces are
too varied for that. Their teacher,
Mrs. Jones, offers to divide the
candy for them, but the children
reply that they just learned about
a cool fair-division game they
want to try, and they can do it
themselves, thank you.
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Example 3.11 Dividing the Halloween
Leftovers
Arrange the 20 pieces randomly in an array.
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Example 3.11 Dividing the Halloween
Leftovers
Step 1 (Bidding)
Each child writes down independently on a
piece of paper exactly where she wants to
place her three markers. (Three markers
divide the array into four sections.) The bids
are opened, and the results are shown on the
next slide.
The A-labels indicate the position of Alice’s
markers (A1 denotes her first marker, A2 her
second marker, and A3 her third and last
marker).
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Example 3.11 Dividing the Halloween
Leftovers
Step 1 (Bidding)
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Excursions in Modern Mathematics, 7e: 3.1 - 145
Example 3.11 Dividing the Halloween
Leftovers
Step 1 (Bidding)
Alice’s bid means that she is willing to accept
one of the following as a fair share of the
candy: (1) pieces 1 through 5 (first segment),
(2) pieces 6 through 11 (second segment), (3)
pieces 12 through 16 (third segment), or (4)
pieces 17 through 20 (last segment). Bianca’s
bid is shown by the B-markers and indicates
how she would break up the array into four
segments that are fair shares; Carla’s bid (Cmarkers) and Dana’s bid (D-markers).
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
This is the tricky part, where we are going to
give to each child one of the segments in her
bid. Scan the array from left to right until the
first first marker comes up. Here the first first
marker is Bianca’s B1.
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
This means that Bianca will be the first player
to get her fair share consisting of the first
segment in her bid (pieces 1 through 4).
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
Bianca is done now, and her markers can be
removed since they are no longer needed.
Continue scanning from left to right looking
for the first second marker. Here the first
second marker is
Carla’s C2, so
Carla will be the
second player
taken care of.
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
Carla gets the second segment in her bid
(pieces 7 through 9). Carla’s remaining
markers can now be removed.
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
Continue scanning from left to right looking
for the first third marker. Here there is a tie
between Alice’s A3 and Dana’s D3.
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
As usual, a coin toss is used to break the tie
and Alice will be the third player to go–she
will get the third segment in her bid (pieces 12
through 16).
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
Dana is the last player and gets the last
segment in her bid (pieces 17 through 20,).
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Example 3.11 Dividing the Halloween
Leftovers
Step 2 (Allocations)
At this point each player has gotten a fair
share of the 20 pieces of candy. The amazing
part is that there is leftover candy!
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Example 3.11 Dividing the Halloween
Leftovers
Step 3 (Dividing the Surplus)
The easiest way to divide the surplus is to
randomly draw lots and let the players take
turns choosing one piece at a time until there
are no more pieces left. Here the leftover
pieces are 5, 6, 10, and 11 The players now
draw lots; Carla gets to choose first and takes
piece 11. Dana chooses next and takes piece
5. Bianca and Alice
receive pieces 6 and 10,
respectively.
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The Method of Markers Generalized
The ideas behind Example 3.11 can be easily
generalized to any number of players. We
now give the general description of the
method of markers with N players and M
discrete items.
Preliminaries
The items are arranged randomly into an
array. For convenience, label the items 1
through M, going from left to right.
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The Method of Markers Generalized
Step 1 (Bidding)
Each player independently divides the array
into N segments (segments 1, 2, . . . , N) by
placing N – 1 markers along the array. These
segments are assumed to represent the fair
shares of the array in the opinion of that
player.
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The Method of Markers Generalized
Step 2 (Allocations)
Scan the array from left to right until the first
first marker is located. The player owning that
marker (let’s call him P1) goes first and gets
the first segment in his bid. (In case of a tie,
break the tie randomly.) P1’s markers are
removed, and we continue scanning from left
to right, looking for the first second marker.
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The Method of Markers Generalized
Step 2 (Allocations)
The player owning that marker (let’s call her
P2) goes second and gets the second
segment in her bid. Continue this process,
assigning to each player in turn one of the
segments in her bid. The last player gets the
last segment in her bid.
Step 3 (Dividing the Surplus)
The players get to go in some random order
and pick one item at a time until all the
surplus items are given out.
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The Method of Markers: Limitation
Despite its simple elegance, the method of
markers can be used only under some fairly
restrictive conditions: it assumes that every
player is able to divide the array of items into
segments in such a way that each of the
segments has approximately equal value. This
is usually possible when the items are of small
and homogeneous value, but almost
impossible to accomplish when there is a
combination of expensive and inexpensive
items (good luck trying to divide fairly 19 candy
bars plus an iPod using the method of
markers!).
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