HISTORY AND PHILOSOPHY OF TEACHING FAIR DIVISION FOR SUSTAINABLE DEVELOPMENT

by

Joel R. Noche Department of Mathematics and Natural Sciences

Ateneo de Naga University Naga City, Camarines Sur, Philippines

jrnoche@adnu.edu.ph

Abstract

The field of study called Fair Division involves the division of objects among

players so that each player gets a share that she or he considers fair in her or

his own value system. This paper first defines some terms commonly used in

the fair division literature to represent specific fairness concepts. It discusses

five examples in detail to illustrate these concepts; all of them involve the

simplest case of dividing one object between two players. It then presents a

brief history of important discoveries of fair division procedures. Finally, it

mentions some recent fair division courses that attempt to show undergraduate

students the connection between mathematics and social justice and

sustainable development.

Key Words: fair division, undergraduate education, history, philosophy,

social justice, sustainable development

Introduction

The interdisciplinary field of study called Fair Division involves the division of one or more

objects among two or more players so that each player gets a share that she or he considers

fair in her or his own value system. Many factors complicate finding a solution: some

procedures cannot be easily extended to handle more players; the value systems of the players

may differ; the objects may be non-uniform or they may be indivisible; the players may be

distrusting or envious of one another; some players may be entitled to have a larger share

than the others; and some players might manipulate the procedure to get better shares.

(Pivato, 2007)

Some examples of fair division problems are: resolving a border dispute between two or

more warring states; dividing an inheritance among squabbling heirs; splitting the property in

a divorce settlement; allocating important government positions among the various factions

in a coalition government; defining ‘fair use’ of intrinsically common property (for example,

resource rights in international waters); partitioning chores among the members of a

household; and allocating military responsibilities to different member states in an alliance

(Pivato, 2007).1

The remainder of this paper: illustrates through examples more precise fairness concepts;

presents a brief history of the study of fair division; and describes current efforts in teaching

fair division to undergraduates.

A philosophical question: What is fair?

The word “fair” has many different meanings. To avoid ambiguity, the following words are

used in the fair division literature (Brams, Jones, & Klamler, 2006; Consortium for Mathe-

matics and Its Applications [COMAP], 1997). A fair division procedure is proportional if

each of the n players gets at least one-nth of the whole as each values it. It is envy-free if

each player thinks that no other player got a larger piece as each values it. (Procedures that

are envy-free are also proportional.) It is efficient if there is no other allocation that is better

for one player and at least as good for the other players. It is equitable if each player’s

subjective valuation of the piece received is the same as any other player’s subjective

valuation. It is strategy-proof if players may do worse if they misrepresent their subjective

valuation; it is strategy-vulnerable if players will do better by lying about their preferences.

Cut-and-choose

The simplest fair division problem is dividing one object between two players. One well-

known procedure is called cut-and-choose: one player divides the object into two parts in

any way that she or he desires, and the other player chooses whichever part she or he wants.

(COMAP, 1997)

1 The first five of these involve the division of something desirable (players want large shares); these are called cake-cutting problems. (Although some authors reserve this term for cases where there is only one object to be divided. See Brams, Jones, and Klamler (2006).) The last two involve the division of something undesirable (players want small shares); these are called chore-division problems. (“Chore division,” 2006)

This procedure has been known since at least the 8th century B.C. Hesiod’s Theogony

includes the following passage (Hesiod, 1914):

For when the gods and mortal men had a dispute at Mecone, even then Prometheus

was forward to cut up a great ox and set portions before them, trying to befool the

mind of Zeus. Before the rest he set flesh and inner parts thick with fat upon the hide,

covering them with an ox paunch; but for Zeus he put the white bones dressed up

with cunning art and covered with shining fat. Then the father of men and of gods

said to him:

“Son of Iapetus, most glorious of all lords, good sir, how unfairly you have divided

the portions!”

So said Zeus whose wisdom is everlasting, rebuking him. But wily Prometheus

answered him, smiling softly and not forgetting his cunning trick:

“Zeus, most glorious and greatest of the eternal gods, take which ever of these

portions your heart within you bids.”

One practical application of the cut-and-choose procedure as it applies to sustainable

development is found in the United Nations Convention on the Law of the Sea.2 Whenever a

developed country wants to mine a portion of the seabed, that country must propose a

division of the portion into two tracts. An international mining company called the

Enterprise, funded by the developed countries but representing the interests of the developing

countries through the International Seabed Authority, then chooses one of the two tracts to be

reserved for later use by the developing countries. (COMAP, 1997)

Fairly dividing even one object between two players is not as simple as it may at first seem.

The following three examples illustrate the problems that could be encountered when using

the cut-and-choose procedure.

Cut-and-choose: Example 1

For this first example, let there be a circular cake3, where one (semicircular) half has

strawberries and the other (semicircular) half has nuts. Let there be two players, A and B,

where A is allergic to strawberries but not to nuts and B is allergic to nuts but not to

2 This was signed on December 10, 1982 and was in force from November 16, 1994. For further details, see http://www.un.org/Depts/los/convention_agreements/convention_historical_perspective.htm. 3 If the fair division problem involves radial cuts made from the center of a disk, the term used is pie-cutting. (Brams, Jones, & Klamler, 2006, p. 1315)

strawberries. Now assume that each player has no information or beliefs about the other

player’s preferences. Say that A is the cutter and B is the chooser. A will not cut the cake so

that one piece is entirely strawberries and the other piece is entirely nuts; if she does, then B

might get the half with all the nuts.4 (Remember that she does not know B’s preferences.)

She will cut the cake so that half of each piece has strawberries and half of each piece has

nuts. In this way, she is sure to get a piece with nuts whichever piece B chooses. The result

would be the same if, instead, B is the cutter and A is the chooser. B will cut the cake so that

he is sure to get a piece with strawberries whichever piece A chooses.

Note that because each player does not know the other player’s preferences, the procedure in

this case is not efficient. It would have been better for both players if A had gotten the half

with all the nuts and B had gotten the half with all the strawberries.5

Cut-and-choose: Example 2

The cake in this second example is the same as in the first example. But this time, the players

report their preferences to each other. Say that player A is the cutter and that she really is

allergic to strawberries but not to nuts and tells this to player B. Now suppose B tells A that

he is allergic to nuts but not to strawberries. Believing what B has told her, A then cuts the

cake so that one piece is entirely strawberries and the other piece is entirely nuts. Expecting

B to choose the piece with all the strawberries, she is surprised when B chooses the piece with

all the nuts. It turns out that B has lied to her; B is actually allergic to strawberries but not to

nuts. A has been punished for trusting B, and B has been rewarded for being untruthful to A.

Note that because the players reported their preferences to each other, the procedure in this

case is strategy-vulnerable. B was able to benefit by lying to A about his preferences.

Cut-and-choose: Example 3

This last example is taken from Brams, Jones, and Klamler (2006). Each player has no

information or beliefs about the other player’s preferences. Now the cake is shaped like a

rectangular box, and the cut is perpendicular to the cake’s length and parallel to the cake’s

width and height. The left half is vanilla and the right half is chocolate. Player A values

4 Player A is being conservative or risk-averse. (Pivato, 2007, p. 169) 5 See also Pivato (2007, pp. 189–190).

vanilla twice as much as chocolate. Player B values vanilla and chocolate equally. Figure 1

shows the actual cake and how each player values it.

Figure 1. The actual cake, as it is seen by player A, and as it is seen by player B.

If A is the cutter (Figure 2), then she cuts at 3/8 so that both pieces are of size “1/2” (in her

value system).6 B is the chooser and he chooses the right half because in his value system it

is the larger piece. A gets “1/2” of the cake and B gets “5/8” of the cake.7

Each of the two players got at least “1/2” of the cake, so the procedure is proportional. But it

is not envy-free: A can see that B got a “larger” piece.

Figure 2. Cut-and-choose, with A as the cutter.

If B is the cutter (Figure 3), then he cuts at 1/2 so that both pieces are of size “1/2” (in his

value system). A is the chooser and she chooses the left half because in her value system it is

the larger piece. A gets “2/3” of the cake and B gets “1/2” of the cake.

6 The left piece as A sees it has an area of “4/3” × 3/8 = “1/2”. Since the whole cake has an area of 1, the right piece has an area of 1 − “1/2” = “1/2”. 7 The cut at 3/8 is for the actual cake, so the fraction is written as it is (that is, without scare quotes). The scare quotes around the fractions “1/2” and “5/8” indicate that the fractions are relative to the player’s value system. As such, the pieces of the cake, being perceived by players with different value systems, do not necessarily equal one whole (that is, 1/2 + 5/8 ≠ 1, although “1/2” + “5/8” = 1).

Again the procedure is proportional but not envy-free. Note that the cutter is at a

disadvantage.

Figure 3. Cut-and-choose, with B as the cutter.

Surplus procedure

Brams, Jones, and Klamler (2006) present the following procedure, which they call the

surplus procedure. Both players independently report their preferences to a referee. The

referee determines the cutting mark of each player (3/8 for A; 1/2 for B). The left and right

pieces are assigned (the left piece to A; the right piece to B). Each player now has “1/2” (see

Figure 4). In general, there will be a surplus (a middle piece).

Figure 4. Surplus procedure

The surplus may be divided in two ways: (1) each player gets the same proportions of it as

each values it (proportional equitability), or (2) each player gets the same value of it as each

values it (absolute equitability, or, simply, equitability).

Surplus procedure: Example 1

If proportional equitability is to be used, then the referee cuts at 7/16 (see Figure 5). A gets

“1/2” of the surplus and B gets “1/2” of the surplus. A gets “1/2” + “1/12” = “7/12” of the

cake and B gets “1/2” + “1/16” = “9/16” of the cake. The player that values the surplus more

gets more value from it.

Figure 5. Surplus procedure: Proportional equitability

Surplus procedure: Example 2

If equitability is to be used, then the referee cuts at 3/7 (see Figure 6). A gets “6/14” of the

surplus and B gets “8/14” of the surplus. A gets “1/2” + “1/14” = “8/14” of the cake and B

gets “1/2” + “1/14” = “8/14” of the cake. Both players get the same value from the surplus.

Figure 6. Surplus procedure: Equitability

The surplus procedure is proportional, envy-free, and efficient. The surplus procedure with

proportional equitability is strategy-proof. The surplus procedure with equitability is

strategy-vulnerable. Proofs of these statements are omitted here but can be found in Brams,

Jones, and Klamler (2006).

A brief history of fair division

Historically, the discovery of fair division procedures generally proceeds toward increasing

numbers of players and improved fairness properties.8 In 1944, Hugo Steinhaus found a

8 It also proceeds towards fewer cuts and lesser “moving knives.” See Brams, Jones, and Klamler (2006), COMAP (1997), and Pivato (2007).

procedure for three players that is proportional but not envy-free. Later, Stefan Banach and

Bronislaw Knaster found a proportional but not envy-free procedure for more than three

players. In 1960, John Selfridge and John Conway independently found an envy-free

procedure for three players. In 1992, Steven Brams and Alan Taylor found an envy-free

procedure for more than three players. (COMAP, 1997) And in December 2006, Steven

Brams, Michael Jones, and Christian Klamler (2006) found a strategy-proof, efficient, envy-

free, and proportionally equitable procedure for two players, and a strategy-proof, efficient,

and equitable procedure for more than two players.

Teaching fair division to undergraduates

Fair division is an interdisciplinary field, involving mathematics, political science, and

economics, among others. It is interesting to note that some universities are currently

offering undergraduate courses with it as a central topic. A quick internet search yields

information on three influential people who teach fair division concepts to undergraduates.

Steven Brams of the Wilf Family Department of Politics at New York University in New

York (http://www.nyu.edu/gsas/dept/politics/faculty/brams/brams_home.html)

has taught the course V53.0844 (Games, Strategy, and Politics) Fall 2004.

Joseph Malkevitch of the Department of Mathematics & Computer Studies at York College

(CUNY) in New York (http://york.cuny.edu/~malk/) has taught the courses Liberal

Studies 400 (Seminar in Fairness and Equity) Fall 2004, Math 484 (Seminar in Contemporary

Mathematics) Fall 2006, and Humanities 320 (Honors Seminar: Fairness and Equity) Spring

2007.

Francis Su of the Department of Mathematics at Harvey Mudd College in California

(http://www.math.hmc.edu/~su/) has taught the course Math 188 (Social Choice and

Decision Making) Fall 2004.

Last May 23–25, 2006, a course development workshop at Lafayette College in Pennsylvania

entitled “The Mathematics of Social Justice” (http://ww2.lafayette.edu/~math/Rob/

MathOfSJ/) stopped accepting participants when more than twice as many as that expected

applied.9 All of these seem to show that the teaching of fair division to undergraduates is

presently quite popular.

9 Here is a quote from the workshop announcement:

Conclusion

Mathematics is usually seen as a tool of the physical sciences. Seldom do students realize

that it also has a lot of applications in the social sciences. Fair division is a currently very

active interdisciplinary field of study that uses mathematics as a tool to promote social justice

and sustainable development. Perhaps because it is relatively novel, very practical, and uses

simple mathematical concepts, some universities are currently offering undergraduate courses

having it as a major component.10

References

Brams, S., Jones, M. & Klamler, C. (2006). Better ways to cut a cake. Notices of the AMS,

53(11), 1314–1321.

Chore division. (2006, June 6). In Wikipedia, The Free Encyclopedia. Retrieved August 12,

2007, from http://en.wikipedia.org/w/index.php?title=Chore_division&

oldid=57181666

Consortium for Mathematics and Its Applications. (1997). For all practical purposes:

Introduction to contemporary mathematics, 4th ed. New York: W.H. Freeman and

Company.

Hesiod. (1914). The Theogony. (H. G. Evelyn-White, Trans.). (Original work published c.

800 B.C.). Retrieved August 11, 2007, from http://www.sacred-texts.com/cla/

hesiod/theogony.htm

Philosophy of Mathematics Education Journal (2007), (20). Special Issue on Social Justice,

Part 1. Retrieved August 11, 2007, from http://www.people.ex.ac.uk/PErnest/

pome20/index.htm “Contemporary society is filled with political, economic, and cultural issues that arise from mathematical ideas. This workshop will aid faculty in mathematics and related disciplines to develop undergraduate general education courses for their home institutions, courses that engage students in understanding the connection between quantitative literacy and social justice. “Topics addressed will be set by the participants, however possible topics include voting rights, voting fraud, gerrymandering, and one person/one vote; the impact of opinion polls on the democratic process; financial exploitation of the quantitatively illiterate; statistical misconceptions and their consequences in politics and policy; mathematics education as a determinant of economic status; and statistics in public health, health care, and health policy.” 10 Those who would like to read further may be interested in two of Ian Stewart’s columns in Scientific American (Stewart, 1998; Stewart, 1999). Also of interest is the June 2007 issue of the Philosophy of Mathematics Education Journal (2007). This is the first special issue on social justice; the second is due before the year 2007 ends. It has papers on gender, race, social class, politics of mathematics education, equity and teacher education, and special students.

Pivato, M. (2007, March 10). Voting, arbitration, and fair division: The mathematics of

social choice. Retrieved June 24, 2007, from http://xaravve.trentu.ca/pivato/

Teaching/voting.pdf

Stewart, I. (1998). Mathematical recreations: Your half’s bigger than my half! Scientific

American, 279(6), 82–83.

Stewart, I. (1999). Mathematical recreations: Division without envy. Scientific American,

280(1), 110–111.