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 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 3 of players, and show that this protocol has a DGEF of 1 + (n^2) 2 . 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.
Another approach is due to @cite_9 , who define various metrics for the evaluation of envy in order to classify the degree of envy in a society,'' and they use the term degree of envy'' in the quite different setting of multiagent allocation of resources.
@cite_9
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Mechanisms for dividing a set of goods amongst a number of autonomous agents need to balance efficiency and fairness requirements. A common interpretation of fairness is envy-freeness, while efficiency is usually understood as yielding maximal overall utility. We show how to set up a distributed negotiation framework that will allow a group of agents to reach an allocation of goods that is both efficient and envy-free.