Sperner's Lemma in Fair Division

Harshit Yadav

Nikhil Sumer

Rajat Agrawal

Fair division is the problem of dividing a set of goods or resources between several people who have an entitlement to them, such that each person receives his/her due share. This problem arises in various real-world settings:

auctions
electronic spectrum and frequency allocation
airport traffic management, etc.

The mathematical fair division problem is an idealization of those real life problems. The theory of fair division provides explicit criteria for various different types of fairness.
Its aim is to provide procedures to achieve a fair division, or prove their impossibility, and study the properties of such divisions both in theory and in real life.

Rental harmony

The problem of dividing a set of heterogeneous and undesirable items is called fair chore division (if the chores are divisible) or chore assignment (if they are not).
The problem of dividing a divisible, heterogeneous and desirable resource is called fair cake-cutting.

We have studied the proof of Sperner's lemma which is a combinatorial analogue of Brouwer's Fixed point Theorem.
We have also studied the problem of envy free division of a cake using Sperner's lemma.
Finally we studied the rental harmony theorem which can be seen as a generalization of fair chore division and fair cake cutting problems.

What is the rental harmony theorem?

Suppose n housemates in an n-bedroom house seek
to decide who gets which room and for what part of the total rent. Then assuming a few conditions, the rental harmony theorem states that there exists a partition of the rent so that each person prefers a different
room.

Sperner's Lemma

Sperner's lemma states that every Sperner labelling (described below) of a triangulation of an n-dimensional simplex contains a cell labelled with a complete set of labels.

Envy free cake cutting

We assume the cake to be a rectangular in shape with total length 1. Then for n players, we consider n-1 cuts to divide the cake into n parts whose sum is 1.
The space S of possible partitions naturally forms a standard (n − 1)-simplex in R^n.

Cake Cutting

Consider the example of division into 3 parts.
Label the triangle such that all sub-triangles have all the labels different.
Go to each vertex, and ask the owner of that vertex, “Which piece would you choose if the cake were cut with this cut-set?”

Cake Cutting

Label that vertex by the number of the piece that is desired.
This way we obtain a Sperner labelling.
By Sperner’s lemma, there must be a (1,2,3) - elementary simplex in the triangulation.

Cake Cutting

Since every such simplex arose from an ABC triangle, this means that we have found 3 very similar cut-sets in which different people choose different pieces of cake.
Carry out this procedure for a sequence of finer and finer triangulations, each time yielding smaller and smaller (1, 2, 3)-triangles.

Cake Cutting

By compactness of the triangle and decreasing size of the triangles, there must be a convergent subseque- -nce of triangles converging to a single point.
Such a point corresponds to a cut-set in which the players are satisfied with different pieces.

This procedure can be easily generalized for n-players case where we use Barycentric subdivision in order to achieve the triangulation.
A constructive e-approximate algorithm is obtained by choosing a small enough e (such as the level of the crumbs.
In this case, we start the the procedure mentioned before with triangulation mesh size less than e.

Chores and rent partitioning

Finding schemes for envy-free chore division has historically been a more complicated problem than cake-cutting. Most envy-free procedures for cake-cuttingdo not carry over to chore division without significant modifications.

We now give a simpler e-approximate algorithm for chore division, which falls out nicely as a special case
of the rent-partitioning problem.

We prove the following theorem using Sperner's Lemma.

Rental Harmony Theorem

In this case we obtain a different type of labelling as shown in the figure.
This is not a Sperner's labelling.
However, one can prove the existence of a fully labelled simplex on the interior by appealing directly to Sperner’s lemma.

The key idea is to dualize
the simplex S to form a new simplex S* .

Now, S* can be triangulated using the labelling inherited form S. The new labelling is a Sperner's labelling.
Hence there exists a fully labelled elementary simplex of S* , which corresponds to a fully labelled elementary simplex of S, as desired.
Now, we choose an e smaller than the rental difference for which housemates wouldn’t care.
We again follow the same procedure as before to obtain an approximate solution.

Thank You