Hua Yinglei, Fan Xin, Hu Zhihao, Si Peiyu
Professor Chi-Kwong Li
Mathematics in Daily Life
June 2014
Fair Division: the art of cutting a cake
Abstract
Fair division is the problem of dividing a set of goods between several people,
such that each person receives his/her due share. In this paper, we show how
mathematics can illuminate the study of cake-cutting in ways that have practical
implications. Specifically, we analyze cake-cutting algorithms where a cake is a
metaphor for a heterogeneous, divisible good, whose parts may be valued differently
by different people.
Introduction
There are many different kinds of fair division problems, depending on the
nature of goods to divide, the criteria for fairness, the nature of the players and their
preferences, and other criteria for evaluating the quality of the division. For example,
proportional division, also called simple fair division means, that every person gets at
least his due share, according to his own value function, i.e., each of the n people gets
a subset of X which he values as at least 1/n, while envy-free division means that
every person gets a share that he values at least as much as all other shares. The
question of fair division becomes interesting when the goods to divide are
heterogeneous and an envy-free division is desired.
The cake discussed in the paper is a metaphor. And the algorithms in the paper
are mainly used to deal with other divisions in daily life such as the distribution of
social welfare, divorce settlements, airport traffic management, to name just a few.
Cut-and choose method—2-people situation
Let’s start from a simple, but famous 2-person, 1-cut cake-cutting procedure.
The classical method ‘divide-and-choose’ is the key. It means that A divides
this cake and B chooses in advance of A. For A’s own benefit, he must divide this
cake, and he will divide it into two equal parts because he does not know which part B
will choose. For B, he chooses cake in advance of A, so he has the chance to choose
the bigger one. As a result, it’s a fair method.
However ‘divide-and-choose’ is not totally fair. Because if A still focuses the
volume of this cake, then he can get half value of this cake, but things change if left
half of this cake is with chocolate on it, and the right half is with strawberries on it. In
fact, I prefer chocolate, and then if I choose the left half, then I can get more than half
value of this cake. Obviously, it’s not fair. A more fair division is that A gets all the
strawberries and small part of chocolate and B gets the rest of it.
In order to achieve this ideal division, A and B should know the other people’s
subjective valuation. To tell the truth, it’s hard to realize in modern society. Therefore,
we have to turn to ‘divide-and-choose’ method. The principle behind this method is
called ‘proportional division’, which means that if n people share this cake ,then
everyone can get at least one-nth value of this cake according their own valuations.
Proportional division
There are two solutions to proportional division with n people. The goal of
proportional division is that each of the n people gets a subset of X which he values as
at least 1/n.

Using the idea of induction
In this algorithm, we start from the well-known 2-person, 1-cut cake-cutting
procedure, “I cut, you choose”. Let’s assume the players are P1 and P2. Then the
cake can be divided into two equal halves with the cut-and-choose method. Next,
P1 and P2 cut their half of cake into three smaller pieces. So we now get 6 small
pieces of cake in total. Then P3 comes and he chooses one piece of cake from P1
and another piece from P2. So each of them now get 2 pieces of cake. Obviously,
P3 will choose the largest 2 pieces. So if P1 or P2 doesn’t divide the cake fairly,
the small piece will be left to himself and he will get less than 1/n of the cake,
making others get more than he does. As rational people, P1 and P2 will try to
avoid such situation and divide the cake in three equal pieces. Since each of the
players get 2 pieces of cake and the pieces are equal in size according to our
analysis, now P1, P2 and P3 each has the same amount of cake and a three-person
proportional division is done.
What if we have 4 people? Using the idea of induction, we just need to repeat
the process above. P1, P2 and P3 divide their own share of cake, at this time, into
4 small pieces. So we have 12 pieces in total. P4 comes and choose one piece
from each person. 12 pieces can be divided into 4 groups and each group has 3
pieces. Now each one gets 3 pieces of the cake and the pieces will have the same
size according to the previous analysis. And 4-person proportional division can
also be done. Here comes the induction. When (n-1)-person proportional division
is done, n-person proportional division can also be done. And n people can cut
the cake fairly according to the standard of proportional division.

The Final Reduction Algorithm
Here is the other different way called “the final reduction algorithm” to cut the
cake for proportional division.
If there are n (>0) people in total, the first person cuts the cake in n parts
according to his valuation and gives the cut-cake to the next person. Then the second
person has two choices. One of them is giving the cut-cake to the next one directly if
he thinks the first cut is fair enough. Otherwise, he must cut down the size of cake
into 1/n to modify it according to his valuation. And so on, everyone has a chance to
modify the cake into 1/n and give it to the next person. We have a rule that the person
who is the last one to cut the 1/n part will get this part of cake. Until now, we finish
the first step. Other n-1 people will repeat this step until everyone gets the 1/n part of
cake.
The cleverness of this way is that everyone must cut and modify the cake into
1/n according to their valuation consciously. No one can dare to cheat because if
anyone cuts the cake into piece which is smaller than 1/n of the cake deliberately,
people who follow him will not modify it and as the rule he must get this piece of
cake in the end. If anyone gives a piece of cake which is bigger than 1/n of the cake to
someone else, what he get in the final will probably less than 1/n.
This is the perfect solution to fair division. But sometimes this is far from the
satisfaction of people in daily life.
Selfridge-Conway Algorithm—solution to envy-free division
Now let’s make the situation more complex and taking each player’s subjective
valuation into consideration.
In problems of envy-free division, the Selfridge–Conway discrete procedure
presents a solution for three players. It is named after John Selfridge and John Horton
Conway. Selfridge discovered it in 1960, and told it to Richard Guy, who told it to
many people, but Selfridge did not publish it. John Conway discovered it
independently in 1993, and also never published it, but the result is attributed to them
in a number of books. This procedure was the first envy-free discrete procedure
devised for three players. Solutions for n players were later found by Steven Brams
and Alan Taylor.
A procedure is envy-free if each recipient believes that (according to its
measure) no other recipient has received more than he has. In their solution, the
maximal number of cuts in the procedure is five. The pieces are not always
contiguous.

Selfridge–Conway division
Suppose we have three players P1, P2 and P3. Where the procedure gives a
criterion for a decision it means that criterion gives an optimum choice for the
player.
1.
P1 divides the cake into three pieces he considers of equal size.
2.
Let's call A the largest piece according to P2.
3.
P2 cuts off a bit of A to make it the same size as the second largest. Now
A is divided into: the trimmed piece A1 and the trimmings A2. Leave the
trimmings A2 to the side for now.

If P2 thinks that the two largest parts are equal, then each player
chooses a part in this order: P3, P2 and finally P1.
4.
P3 chooses a piece among A1 and the two other pieces.
5.
P2 chooses a piece with the limitation that if P3 didn't choose A1, P2 must
choose it.
6.
P1 chooses the last piece leaving just the trimmings A2 to be divided.
Now, the cake minus the trimmings A2 has been divided in an envy free
manner. The trimmed piece A1 has been chosen by either P2 or P3. Let's call the
player who chose it PA and the other one Player PB.
1.
PB cuts A2 into three equal pieces.
2.
PA chooses a piece of A2 - we name it A21.
3.
P1 chooses a piece of A2 - we name it A22.
4.
PB chooses the last remaining piece of A2 - we name it A23.

Analysis
Let's see why the procedure is envy-free. It must be shown that each player
believes that no other player received more than he had. Without loss of
generality, we can write (see illustration above):

PA received: A1 + A21.

PB received: B + A23.

P1 received: C + A22.
In the following analysis "largest" means "largest according to the player":

PA received A1 + A21. For him, A1≥B and A1≥C. And he considers his
choice A21 to be the largest piece of A2. So no other player received more
than he did: A1 + A21 ≥B + A23, C + A22.

PB received B + A23. For him, B≥A1 and B≥C since he chose B. Also, he is
the one that cut A2 in 3 pieces, so for him all those pieces are equal.

P1 received C + A22. For him, C≥A1 and C=B.
o
P1 believes that PB didn't receive more than he did. In other words: C
+ A22 ≥B + A23. Remember that P1 chose his piece of A2 before PB,
thus A22 ≥A23 in his view.
o
P1 believes that PA didn't receive more than he had. In other words: C
+ A22 ≥A1 + A21. Remember that for P1, C is equal to A since he cut
the cake in the first round. Also,
A = A1 + A2 = A1 + (A21 + A22 + A23); therefore C ≥A1 + A21.
(Even if PA took the whole A2 and P1 did not receive A22, P1 would
not envy PA.)
Conclusions and applications
Most of the algorithms require us to divide the cake into small pieces and
combine them together again into different larger sizes. Actually we rarely need to cut
a real cake with such accuracy when sharing the happiness of eating a cake.
Nevertheless, the algorithms are really useful in areas such as auctions, divorce
settlements, electronic spectrum and frequency allocation, airport traffic management,
or exploitation of Earth Observation Satellites.
As we can see, the cake cutting algorithms becomes more and more complex
with more and more players and more and more complicated situations. However, the
more complex the situation is, the closer it is to our real life. To solve the problem of
fair division, we still have a long way to go.
Moreover, fair division is an active research area in mathematics or game
theory and it also appears in popular culture.

Example 1: In Numb3rs season 3 episode "One Hour", Charlie talks about
the cake-cutting problem as applied to the amount of money a kidnapper
was demanding.

Example 2: A Dinosaur Comics strip is based on the cake-cutting
problem.
References

“Fair Division.” N.p., n.d. Web. 22 June.2014
< http://en.wikipedia.org/wiki/Fair_division>.

Brams S J, Jones M A, Klamler C. Better ways to cut a cake[J]. Notices of the
AMS, 2006, 53(11): 1314-1321.

顾森. 思考的乐趣：Matrix67 数学笔记. 北京：人民邮电出版社，2012
Appendix—Self-evaluation of the presentation

Preparation for the presentation
 To tailor the information to our needs and make things clear for a better time
management.

During the presentation
 To take control of the time, and adjust the content according to the time limit
flexibly.
 To use more pictures and clearer pictures, making the logic clear and easy to
understand when talking about Selfridge-Conway algorithm.
 To add more interactions and make more classmates involved when
explaining the algorithms of cutting a cake if we had more time.

Things we learnt from other groups
 To be prepared for possible accidents and make a final check on all the
materials to be displayed such as PPT and videos.
 To talk without a script and speak confidently.