Group envy free

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 ∈ 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 ∈ 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 ( ∪ i ∈ 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.