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What is "Fair Division"?

The theory of fair division is concerned with finding ways to divide
an object among *n* people *fairly*. There are many possible
notions of fairness. The notion that has interested me recently is finding
*envy-free* divisions, i.e., one in which each person feels she received
the *best* piece, in her own estimation.

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Exact vs. Approximate Algorithms

Existence of such divisions has been known for some time. Of greater interest
are algorithms for achieving such divisions. For instance, in dividing
a desirable object, such as cakes, a 2-person algorithm has been known
since antiquity, a 3-person algorithm since at least 1960, but an exact
algorithm for achieving *n*-person envy-free cake divisions was not
produced until Brams and Taylor's solution in 1995. (See the references
below.) Elisha Peterson and I have recently extended these techniques to
provide an exact envy-free
chore-division algorithm for *n* people. However, such methods
tend to be quite complicated as *n* grows large.
By contrast, I have been interested in developing *approximate*
envy-free schemes, in particular, through the use of arguments from
combinatorial topology (Sperner, Tucker, and generalizations).
Such methods will find envy-free divisions up to
a specified tolerance for error (such as at the level of crumbs), and have
the advantage of (1) being easily generalized for arbitrary numbers of
players, (2) using the minimal number of cuts,
(3) for a fixed number of players and a fixed error, having
a bounded number of steps in which the algorithm halts. My main inspiration
and partner in this endeavor has been Forest Simmons. More recently, my
students Elisha Peterson,
Patrick Vinograd,
and Karl Mahlburg have also been working with me.

To explore such algorithms, we have developed: