Chapter 14: Fair Division
Part 4 – Divide and Choose
for more than two players
Divide and Choose for More than Two Players
• Question: Can we use divide and choose when there are more than two
people dividing some object?
• The answer is - not exactly.
• For example, suppose that three people (Abe, Barry and Clyde) want to
divide a cake. Suppose Abe cuts the cake into what he considers to be
three equal pieces (x, y and z). Now we have two people (Barry and
Clyde) who will choose from the three pieces. Who chooses first? This
could be a problem if our interest is in fair division. That is, suppose
Barry and Clyde both consider piece x to be a fair share, but do not
consider either y or z to be fair. If Barry chooses first, the Clyde is left
with no choice he considers to be fair.
• Later, when we define mathematically what we mean by “fair”, we
might want to consider a division to be fair if everyone involved
considers it fair.
Divide and Choose for More than Two Players
• Remember that in divide and choose with two players, if
both followed the appropriate strategy, each could expect a
fair share.
– With two players:
• the first player divides to make sure he considers
both pieces fair
• the second player choose the piece that he considers
fair.
• We have seen if we apply divide and choose directly to a
situation with three people, one player may not receive
what he or she considers to be a fair share of the division.
Divide and Choose for More than Two Players
• We will consider three methods of fair division that are based on divide
and choose but work for more than two people:
– Lone Divider (a.k.a the Steinhaus Procedure)
– Last Diminisher (a.k.a the Banach-Knaster Procedure)
– Selfridge-Conway Method
• Later we can consider a mathematical definition fair and discuss these
methods in terms of our definition of fairness.
• All of these variations of divide and choose assume that the object to
be divided is continuous – that is, it can be divided many times and
retains its value. For example, the object could be a cake and the knife
is so sharp that there are no crumbs when it is cut.
• Another example of a continuous object would be a piece of land –
divisions of land become imaginary property lines.
Lone Divider Method (a.k.a the Steinhaus Procedure)
• This method is called the “lone divider method” because there is one
divider and the other players are choosers.
• We only consider this method in the case of 3 players.
• Suppose we have a cake and three people (A, B, and C) who wish to
divide the cake fairly.
• We would use the lone divider method as follows …
• One person is the divider (say A) and cuts the cake into three pieces (x,
y and z) that he considers to be fair. That is, the divider would be
happy with any of the three pieces because in his estimation they are
all fair.
• Next, players B and C will be choosers – but three different things can
happen…
Lone Divider Method – three things can happen
1.
2.
3.
One of the choosers ( B or C ) considers more than one piece as fair.
For example, suppose B would be happy with piece x or y. Then we
would let C pick any piece and then B can pick either x or y and finally
A can pick the remaining piece (remember A liked them all.)
Both of the choosers (B and C) like exactly one piece and those pieces
are different. Then the choosers each take the piece they want and A
takes the remaining piece.
Both of the choosers (B and C) like exactly one piece and those pieces
are the same. For example, suppose B and C both like only piece x and
they don’t like the other pieces. To handle this situation, we give one
of the other pieces (say z) to A. Remember A considered them all fair
so he takes z and leaves. Now we put x and y back together into one
piece, lets call this new piece xy. Notice that both B and C consider
this new piece xy to be more than two-thirds of the cake because they
considered z to be less than one third. Now we divide xy using simple
divide and choose with players B and C.
Silly ?
• This may sound complicated and silly when we discuss
dividing a cake but remember the example of the
Convention of the Law of the Sea.
• We may be dividing valuable sections of seabed for mining.
(Of course, you might consider the Convention of the Law
of the Sea as silly also!)
• Regardless of opinion, these methods are attempts at fair
division. We can discuss how or why they may be fair
when we define fairness later…
Last Diminisher Method (a.k.a. Banach-Knaster Procedure)
• This method works for three or more people. It is called “last diminisher”
because there are no designated dividers or choosers. Every player can
be a divider and chooser. In this method one player begins by cutting a
piece. Other players might cut the original piece and they are called
diminishers. The last diminisher in this method takes the piece he or she
cut and leaves.
• The way this method of division works is by first randomly assigning an
order for the players. The dividing and choosing is always done in the
order that was established at the beginning of the procedure.
•
For example, suppose that five cast members from the TV show Lost
are stranded on a deserted tropical island and decided to divide up the
island using the last diminisher method.
• We begin by randomly assigning an order in which the division is to be
done.
Last Diminisher Method (a.k.a. Banach-Knaster Procedure)
• Suppose we have randomly assigned each of the five players a number so
that they will participate in the division in the following order:
(P1, P2, P3, P4, P5). They will keep this order throughout the division
process.
• The division of the island will take place in rounds (or stages). The
number of rounds is always one less than the number of participants (in
this case there will be four rounds). The last round is always a simple
divide and choose between the remaining two players.
• The process of division is as follows:
– We begin when P1 cuts a piece. The other players examine the piece. They
can cut it or pass. If all pass, then P1 keeps that piece. If someone cuts it,
that player has claim to the new piece. The remains from the new piece are
put back with the rest of the island and P1 now has claim (with the others
except the last diminisher) to the rest of island. The last to cut, following the
assigned order, keeps the piece they cut. When that player leaves another
round begins. We continue until two remain and then use divide and choose.
Last Diminisher Method – Round 1
Round 1: P1 cuts a piece. It is in P1’s interest to do this as fairly as possible.
If the piece is too small, he may be stuck with it and if it’s too big, he might
lose it. (If he loses it, he’ll get a chance at another piece later.)
– Each player (in order) examines the piece that P1 cut. For example, P2
examines the piece and must decide to diminish (cut) it or pass.
– If P2 passes then he doesn’t want it and thinks it is less than or equal to a
fair share. That is, if he passes that means he thinks that piece is not more
than a fair share. If every player passes on that piece, they all think it is not
more than a fair share and P1 keeps the piece he cut and leaves. In that
case, when all passed, P1 was the last diminisher.
– If P2 (or any of the other players) decides to diminish it, he must be careful
to cut it fairly – he may end up with what he cuts. If he does cut that initial
piece then he temporarily has claim to the new piece that he cut and loses
rights to the rest of the island. The remains of the initial piece that P1 cut
(what P2 or the last diminisher didn’t need) are put back with rest of the
island and P1 again has a fair share claim - with the others except the last
diminisher - to the rest of the island.
Last Diminisher Method – Round 1 - continued
Round 1 (continued): After we have gone through and let each player
examine the piece cut by P1 the one of the following could have
happened…
If all the players passed on the piece that P1 cut, then P1 keeps that piece and
exits the game.
If some player cuts the piece, the last to cut it takes that piece and exits the
game. Notice only one player will cut the initial piece but each have an
opportunity to examine it. After going in sequence from P1 to P5 (or
however many players there are) the last to diminish the piece must keep
that piece. (As everyone knows: You can’t have your cake and eat it too!)
Once we go through all the players and someone takes a piece, we begin a new
round.
Last Diminisher Method – Round 2
Round 2: We begin again. Now there is one less player in the game but we
continue as in round 1. We have a smaller amount of land now for which all
of the remaining players have equal claim. Note that all of these players
believe the remaining land represents at least 4/5 of the total island because
they approved of the first cut and considered it less or equal to a fair share.
The first to cut in the next round is the first of the remaining players from the
ordered list of players established at the outset of the procedure.
That player cuts a piece, just as in round 1, and each player, in turn, examines
the piece. The process is repeated exactly as it was in round 1 with the
remaining players.
After every player has examined the most recently cut piece of land, the last
diminisher keeps the piece that he or she cut and exits the negotiation.
Last Diminisher Method – Round 3
Round 3: We begin again.
This example started with five people dividing up an island and now two have
taken a piece of the island and left the decision process. The three people
that remain believe that the total land remaining is at least 3/5 of the original
island because they approved of the pieces that the others took as being
either fair or less than a fair share.
We start this round as before: with the first person remaining, from the original
ordered list of players, cutting a piece of the island. The remaining two
examine the piece also in the originally established order. Either could
make an additional cut but then he or she may finish with what they cut for
themselves. The last to cut (the last diminisher) then takes the piece that
was cut. As before, if, after the first to cut in this round, we find that the
other two pass (approve of the piece) then the first to cut in this round takes
that piece and exits the negotiations.
Last Diminisher Method – Round 4
Round 4: In this example, we originally had five players and so this is the last
round. Whenever this method is used we would continue one round after
another until two players remain. The final round is always completed with
divide and choose among the two remaining players.
This completes the last diminisher method.
Each party to the negotiations had an opportunity to divide or choose what he
or she considered to be a fair share of the island.
Last Diminisher Method – Summary
• The last diminisher method works with objects that are continuous – like a piece
of cake, a bottle of wine, or maybe a keg of beer, some land, a section of the
seabed, or perhaps a political district.
• This method works with 3 or more players.
• For example, if there were 10 players, there wound be 9 rounds in the last
diminisher procedure and the 9th round would be divide and choose between the
last two players remaining.
• Would could use last diminisher with 3 players. There would be just two
rounds. The first round would involve one player cutting and the others
examining the piece and possibly being a diminisher. After the first round, in
the case of 3 players, the second and final round would be divide and choose
between the two remaining players.
Selfridge-Conway Method of Fair Division
• This method can be applied for 3 or more players.
• We will study how this method is applied in the case of 3 players only.
•
After studying this method of fair division we will finally define
“fairness” and compare all of the methods in terms of this definition.
Selfridge-Conway Method of Fair Division
• To explain this method, again we use an example. Suppose Aaron (A),
Bill (B) and Chris (C) will share a pizza. They bought it together and
each claims a fair share of the pizza. Suppose the pizza is not yet cut
and they have a razor sharp knife and any pieces that are cut can be put
back in the box and will have not lost any value in the process (no
cheese slips off!)
• First we need to randomly assign an order, but, unlike the last
diminisher method, we do not keep this order throughout the entire
division of the pizza. The Selfridge-Conway method is always
completed in two rounds and the order of selection is different in the
second round from what it was in the first round.
Selfridge-Conway Method – Order of Division and Selection
• The rule for deciding the order of division in the Selfridge-Conway
method is this: Given three players P1, P2 and P3
– Suppose the round 1 order of selection is P1, P2 and then P3.
– Assuming there are trimmings, then the second round order of selection is
determined as follows:
• If P2 received a trimmed piece then P3 cuts the trimmings and the
order of selection is P2, P1 then P3.
• If P3 received a trimmed piece then P2 cuts the trimmings and the
order of selection is P3, P1 then P2.
• Suppose, for example, with Aaron, Bill and Chris, the initial order of
division (for round 1) is as follows: A, B and then C. Then suppose C
selects a piece trimmed by B. Then round 2 proceeds as follows: B
cuts the trimmings and the selection of the trimmings is done in the
order C, A and B.
Selfridge-Conway Method - Round 1
• Round 1 - Because A is initially first (in round 1), A will cut the pizza
into three pieces that he thinks are fair (so he would be happy with any
one of the three pieces he cut.)
• A then passes all three pieces to B. At this point B may decide to cut
one or pass them all to C.
– If B decides to cut one of the pieces, he may only cut one. If B decides to
cut a piece he will do so to create at least a two-way tie for best piece (so
B would be happy with either of those two of the three.)
– If B cuts a piece he needs to be prepared to accept the piece he cut after C
examines them all. If B did cut a piece then he puts the trimmings from
that piece back into the box for later. We can call the trimmings that are
put back into the box T.
– Whether B trimmed any of the pieces or not, B now passes three pieces to
C. (Remember one of those pieces might have been trimmed by B.)
Selfridge-Conway Method – Round 1
• We continue with round 1 –
– C now has three pieces to examine. One of those pieces may have
been trimmed by B, however two of them were not trimmed.
– C can choose whichever of the three pieces he wants.
– If C chooses the piece that was trimmed by B then C considered
that piece to be a fair share. However C will still be entitled to
some of the trimmings in the box.
– However, suppose that C considers the piece trimmed by B to be
less than one fair share. Then C will consider the other two pieces
– plus the trimmings still in the box - are worth more than a fair
share. In a sense, C would consider the other two pieces, plus the
trimmings, to be more than two fair shares and would claim a fair
portion of that. In this case, C can choose whichever of the two
untrimmed pieces that he wants and will then expect a fair share of
the trimmings in the box.
Selfridge-Conway Method – Round 1
• We continue with round 1 –
– After C chooses, the two remaining pieces go back to B.
– If B made any cuts previously and the piece B trimmed is still there,
he is obliged to take that piece.
– Otherwise, B takes either of the remaining two pieces – one of
which he considered was in a two-way tie for best piece, since C
could have taken at most one of those two that B coveted.
– If B had made no cuts previously then he considered all pieces
equally fair and must now choose one of the two remaining pieces.
– At this point A takes the last remaining piece.
– This is the end of round 1.
– If there are no trimmings, then the division is complete and the
process is done
• Otherwise, if there were some trimmings, we now begin the 2nd and last
round - dividing up the trimmings.
Selfridge-Conway Method – Round 2
• Round 2 – Dividing up the trimmings:
– The may be no round 2 if no trimmings where cut in round 1.
– The order of selection in round 2 is critical to what will make this
method meet certain criteria of fairness which we define later. That
order is based on the order of division and selection in round 1 as
described previously.
– In round 2, there is only one cut (into three pieces) and then the
players will choose a piece in the specified order. There are no
more trimmings created in this round. The second round only
divides up the trimmings in a way so that everyone will consider
the final result fair.
– This concludes the Selfridge-Conway method of fair division.
Is it worth the trouble?
• So far we have considered three methods of fair division, based on
divide and choose, to use when a continuous object is to be divided
among 3 or more people: Lone divider, last diminisher and SelfridgeConway.
• The last diminisher method can easily be extended to work for any
number of people greater than or equal to 3. For example, the previous
example was given with 5 people dividing an island.
• However, if we wish to use either the lone divider or Selfridge-Conway
methods, some changes would need to be made to maintain their
fairness. Why? To answer this question we will finally need to define
what we mean by “fair.”
• Any of these methods may be worth the trouble depending on how you
define fairness.
Is it worth the trouble?
• Thus, to answer the question Are these methods of fair division worth
the trouble? We might ask: What do we consider fair?
• We may be willing to consider the effort involved if we knew the
method guaranteed a certain kind of fairness.
• That is, with a definition of fairness, we can look again at each method
and decide if each method satisfies our definition of fairness.
• Of course, in the case of dividing a cake, a bottle of wine or a pizza,
because the value of the item is not that high, the effort doesn’t seem
worth the trouble. They are just simple examples of a theoretical ideal.
• However, if the object to be divided is extremely valuable, for example
in either financial or political terms, perhaps one of these methods is
worth the trouble if its can satisfy our definition of fairness.