Fair distributions for more participants than allocations

Abstract

We study the existence of fair distributions when we have more players than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the players can be weakened from those of classic cake-cutting and rent-splitting results of Stromquist, Woodall, and Su. We extend existing variations of the cakesplitting problem with secretive players and those that resist the removal of any sufficiently small set of players.

A Sparse colorful polytopal KKM Theorem

“…, y k τ k } and k i=1 A i τi = ∅. Recently, Soberón [16] proved a beautiful generalization of the colorful KKM Theorem that we call here the sparse colorful KKM theorem.…”

“…The notion of m-weakly KKM cover was defined by Soberón in [16], where he proved: Theorem 1.3 (Soberón [16]). Let n ≥ k be positive integers.…”

“…Soberón [16] used Theorem 1.3 to prove the following generalization of the classical fair division theorem due to Stromquist [18] and Woodall [20].…”

“…

Recently Soberón [16] proved a far-reaching generalization of the colorful KKM Theorem due to Gale [6]: let n ≥ k, and assume that a family of closed sets) has the property that for every I ∈[n]n−k+1 , the familyWe prove a polytopal generalization of this result, answering a question of Soberón in the same note. We also discuss applications of our theorem to fair division of multiple cakes, d-interval piercing, and a generalization of the colorful Carathéodory theorem.

…”

“…Recently Soberón [16] proved a far-reaching generalization of the colorful KKM Theorem due to Gale [6]: let n ≥ k, and assume that a family of closed sets…”