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

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 envyfreeness: 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 + ⌈ n/2 ⌉ . This is the currently best DGEF among known finite bounded protocols for an arbitrary number of players. For comparison, we determine 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 ⌈n/2⌉− 1 fewer guaranteed envy-free-relations than our protocol.