A Mathematical Approach to Division

Mathematics now may be used to solve a problem that has tormented people
since the dawn of humanity:  how to divide things fairly.

Mathematician Alan Taylor and political scientist Steven Brams say they
have devised a system based on "preference points" that can split just
about anything -- from the spoils of war to a child's birthday cake --
into "envy-free pieces."

Not only do all parties get what they think is fair, they say, each
thinks it got the better of the other guys.

Taylor and Brams' work is only a small part of a rapidly surfacing
trend.  Mathematics is invading political science in attempts to find
rational approaches to complex, often highly emotional questions.

The California Institute of Technology, for example, recently received
an $800,000 grant from the National Science Foundation to apply
mathematics to finding fair ways to divide society's scarce resources.
It's the biggest grant the foundation has given in the social sciences.

Among the questions Caltech is tackling:  how to spread the cost of
cleaning up Los Angeles' polluted skies.

"Philosophers have argued about fairness for thousands of years," said
John Ledyard, who heads Caltech's social sciences division.  "What's
different now is we have a formal mathematical structure.  That takes it
out of ideological debate.  There's science here."

One of the earliest notions of fairness is the Biblical story of King
Solomon, who had to decide which of two women claiming the same baby was
the mother.  He ordered the child cut in half, which prompted the mother
to give up her claim to save her infant's life.

Solomon's story illustrates the idea that fair division means more than
cutting things into equal pieces.  It involves the value the parties --
and society -- place on what is to be divided.

Finding out how much people really value things often is difficult.
Most people try to take whatever they can get rather than honestly state
their choices.

Trying to encode the wisdom of Solomon into equations hasn't been easy,
but scientists are making progress.

Having their cake

Mathematicians do most of their research on fairness on a simple but
versatile model called the cake-cutting problem.

Suppose two people want to share a small cake.  The fairest way to
divide it is to let one person cut the cake and let the other choose a
piece first.

The cutter reveals her true preferences in the way she cuts the cake.
For example, if she values icing, she might cut one piece smaller but
with more icing, hoping her friend will go for the bigger slice.

Either way, both people can feel they are winners; one gets a bigger
piece and the other gets more icing.

Perception is as important as mathematics in making the solution work.
In this example, each person gets a role in choosing a slice.

The cake-cutting model works well for two.  In more complicated
situations, more players often are involved.

In 1992, in response to a challenge posed in the Sciences magazine,
Brams came up with a solution that worked for three players.  The first
divides the cake into three; the second is allowed to trim one piece he
thinks is bigger than the others.  The third player gets to choose
first.  In the end, everyone has a say, and everyone gets to choose a
piece.  Ergo, Brams says, the solution is envy-free.

But when he tried to expand his system to four people, it didn't work.
So he called his friend Taylor, a mathematician at Union College in
Schenectady, N.Y. Taylor, who had never worked on fair division, thinks
that freshness helped him come up with a radical approach.

Essentially, his new method involved cutting a cake into an extra piece
-- four for three players, and so forth.  This allowed everyone to take
a role in both trimming and choosing.

Taylor's breakthrough won kudos in the mathematics community, because it
showed how to divide anything into any possible number of "envy-free"
pieces.

And now, divorce

In practice, though, the method is too unwieldy to use in everyday life,
because eventually the number of extra pieces required increases much
faster than the number of players.  Besides, there are things one can't
divide or trim, such as the family dog in a divorce.

So this past year, Brams and Taylor turned their attention to a more
workable method.  Instead of viewing the goods to be divided as a cake,
in the new system -- called Adjusted Winner -- each player gets 100
points to distribute based on value preferences.

In a divorce, one spouse might assign 90 points to retaining custody of
the children; the other spouse might care more about the house and put
70 points there.

In the first step of this process, each person wins whatever he placed
more points on.  In this case, the spouse with 70 getting, say, a life
insurance policy rated 30 points; then the other spouse might get the
family computer, worth 10.  The rest is split according to a
mathematical formula that Taylor and Brams guarantee is envy-free.

"The big problem we can help solve is divorce," said Brams, who teaches
at New York University.  He also sees immediate applications in issues
ranging from labor disputes to the federal budget.