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

Cake-cutting protocols aim at dividing a ``cake'' (i.e., a divisible resource) and assigning the resulting portions to several players in a way that each of the players 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 \emph{finite bounded} envy-free cake-cutting protocol for an arbitrary number of players, and even for four players. We introduce the notion of degree of guaranteed envy-freeness (DGEF) as a measure of how good a cake-cutting protocol can approximate the ideal of envy-freeness while keeping the protocol finite bounded (trading being disregarded). We propose a new finite bounded proportional protocol for any number n \geq 3 of players, and show that this protocol has a DGEF of 1 + \lceil (n^2)/2 \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.