Degrees of Guaranteed Envy-Freeness in Finite Bounded Cake-Cutting Protocols

Abstract

Fair allocation of goods or resources among various agents is a central task in multiagent systems and other fields. The specific setting where just one divisible resource is to be divided fairly is commonly referred to as cake-cutting, and agents are called players in this setting. Cake-cutting protocols aim at dividing a cake and assigning the resulting portions to several players in a way that each of the players, according to his or her valuation of these portions, feels to have received a “fair” amount of the cake. An important notion of fairness is envy-freeness: No player wishes to switch the portion of the cake received with another player’s portion. Despite intense efforts in the past, it is still an open question whether there is a finite bounded envy-free cake-cutting protocol for an arbitrary number of players, and even for four players. In this paper, we introduce the notion of degree of guaranteed envy-freeness (DGEF, for short) as a measure of how good a cake-cutting protocol can approximate the ideal of envy-freeness while keeping the protocol finite bounded. We propose a new finite bounded proportional protocol for any number n ≥ 3 of players, and show that this protocol has a DGEF of $1 + \left\lceil{n^2}/{2} \right\rceil$. This is the currently best DGEF among known finite bounded cake-cutting protocols for an arbitrary number of players. We will make the case that improving the DGEF even further is a tough challenge, and determine, for comparison, the DGEF of selected known finite bounded cake-cutting protocols, among which the Last Diminisher protocol turned out to have the best DGEF, namely, 2 + n(n − 1)/2. Thus, the Last Diminisher protocol has $\left\lceil {n}/{2} \right\rceil - 1$ fewer guaranteed envy-free-relations than our protocol.