Group envy-freeness

A group-envy-free division (also known as: coalition-fair 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. The term is used especially in problems of fair division, such as resource allocation and fair cake-cutting.

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.

Definitions

Consider a set of n agents. Each agent i receives a certain allocation A_i (e.g. a piece of cake or a bundle of resources). Each agent i has a certain subjective preference relation <_i over pieces/bundles (i.e. $A<_{i}B$ means that agent i prefers piece B to piece A).

Consider a group X of the agents, with its current allocation $\{A_{i}\}_{i\in X}$. We say that group X prefers a piece B to its current allocation, if there exists a partition of B to the members of X: $\{B_{i}\}_{i\in X}$, such that at least one agent i prefers his new allocation over his previous allocation ($A_{i}<_{i}B_{i}$), and no agent prefers his previous allocation over his new allocation.

Consider two groups of agents, X and Y, each with the same number k of agents. We say that group X envies group Y if group X prefers the common allocation of group Y ($\cup _{i\in Y}{A_{i}}$) to its current allocation.

An allocation {A₁, ..., A_n} is called group-envy-free if there is no group of agents that envies another group with the same number of agents.

Relations to other criteria

A group-envy-free allocation is also envy-free, since X and Y can be groups with a single agent.

A group-envy-free allocation is also Pareto efficient, since X and Y can be the entire group of all n agents.

Group-envy-freeness is much stronger than the combination of these two criteria, since it applies also to groups of 2, 3, ..., n-1 agents.

Existence

In resource allocation settings, a group-envy-free allocation exists. Moreover, it can be attained as a competitive equilibrium with equal initial endowments.

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 V_i representing the value of each piece of cake, and all such functions are additive and non-atomic.

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 V_i, 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.