Envy‐free Division Using Mapping Degree

Abstract

In this paper we study envy‐free division problems. The classical approach to such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions for this map to hit the center of the simplex. The mere continuity of the map is not sufficient for reaching such a conclusion. Classically, one makes additional assumptions on the behavior of the map on the boundary of the simplex (e.g., in the Knaster–Kuratowski–Mazurkiewicz and the Gale theorem). We follow Erel Segal‐Halevi, Frédéric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the meaning in economy as the possibility for a player to prefer an empty part in the segment partition problem. We solve the problem positively when n, the number of players that divide the segment, is a prime power, and we provide counterexamples for every n which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy‐free division problems when n is odd and not a prime power.

Envy-free division of multi-layered cakes

“…Let us comment briefly the special case when m = 1. Since the agents might prefer zero-length pieces in our setting, our theorem boils down then to a recent result of Avvakumov and Karasev [2]. They showed that when n is a prime power, there always exists an envy-free division, even if we do not assume that the agents are hungry, but that this is not true anymore if n is not a prime power.…”

“…Here, some agents may find that a part of the cake is unappetizing and prefer nothing, while others may find it tasty. Even in such cases, an envy-free division only using n − 1 cuts has been shown to exist for a particular number n of agents under the assumption of closed preferences [2,14,15,16]. The most general result obtained so far is the one by Avvakumov and Karasev [2], who showed the existence of an envy-free division for the case when n is a prime power.…”

“…Even in such cases, an envy-free division only using n − 1 cuts has been shown to exist for a particular number n of agents under the assumption of closed preferences [2,14,15,16]. The most general result obtained so far is the one by Avvakumov and Karasev [2], who showed the existence of an envy-free division for the case when n is a prime power. Panina and Živaljević [15] gave an alternative proof of the result of Avvakumov and Karasev, by using Volovikov's theorem [21].…”

“…For the number n of agents not equal to a prime power, it is known that Volovikov's theorem does not hold. Further, Avvakumov and Karasev [2] and Panina and Živaljević [15] provided counterexamples of a cake-cutting instance with choice functions for which no envy-free division exists, for every choice of n that is not a prime power. The counterexamples show some limitations of the approach of equivariant topology, but do not prohibit the existence of an envy-free division in the standard valuation function model.…”

Fair distributions for more participants than allocations

“…Many variations of the cake-splitting problem have been studied [Wel85, BT96, Bar05, Pro15] , including partitions of multiple cakes or pieces per participant [CNS10, NSZ20, ABB + 20, SH21], partitions with secretive guests [AFP + 18], and partitions weakening the hungry condition if there are empty pieces [MZ19,AK21].…”