A group-envy-free division is a division of a resource among several partners such that every group of partners feel that their allocated share is at least as good as the share of any other group with the same size.
Group-envy-freeness is a very strong fairness requirement: a group-envy-free allocation is both envy-free and Pareto efficient, but the opposite is not true.
Each agent i receives a certain allocation Xi (e.g. a piece of cake or a bundle of resources).
Each agent i has a certain subjective preference relation
Consider a group G of the agents, with its current allocation
We say that group G prefers a piece Y to its current allocation, if there exists a partition of Y to the members of G:
An allocation {X1, ..., Xn} is called group-envy-free if there is no group of agents that envies another group with the same number of agents.
A group-envy-free allocation is also envy-free, since G and H may be groups with a single agent.
A group-envy-free allocation is also Pareto efficient, since G and H may be the entire group of all n agents.
Group-envy-freeness is stronger than the combination of these two criteria, since it applies also to groups of 2, 3, ..., n-1 agents.
Moreover, it can be attained as a competitive equilibrium with equal initial endowments.
[3][4][2] In fair cake-cutting settings, a group-envy-free allocation exists if the preference relations are represented by positive continuous value measures.
I.e., each agent i has a certain function Vi representing the value of each piece of cake, and all such functions are additive and non-atomic.
[1] Moreover, a group-envy-free allocation exists if the preference relations are represented by preferences over finite vector measures.
I.e., each agent i has a certain vector-function Vi, representing the values of different characteristics of each piece of cake, and all components in each such vector-function are additive and non-atomic, and additionally the preference relation over vectors is continuous, monotone and convex.
Aleksandrov and Walsh[6] use the term "group envy-freeness" in a weaker sense.
They assume that each group G evaluates its combined allocation as the arithmetic mean of its members' utilities, i.e.:
and evaluates the combined allocation of every other group H as the arithmetic mean of valuations, i.e.:
By their definition, an allocation is g,h-group-envy-free (GEFg,h) if for all groups G of size g and all groups H of size h:
By this definition, group-envy-freeness does not imply Pareto-efficiency.
They define an allocation X as k-group-Pareto-efficient (GPEk) if there is no other allocation Y that is at least as good for all groups of size k, and strictly better for at least one group of size k, i.e., all groups G of size k:
GPEn is equivalent to utilitarian-maximal allocation, since for the grand group G of size n, the utility uG is equivalent to the sum of all agents' utilities.
The inverse implication is not true even with two agents.