It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent.
The notion of envy-freeness was introduced by George Gamow and Marvin Stern in 1958.
[1] They asked whether it is always possible to divide a cake (a heterogeneous resource) among n children with different tastes, such that no child envies another one.
For n=2 children this can be done by the Divide and choose algorithm, but for n>2 the problem is much harder.
Envy-freeness was introduced to the economics problem of resource allocation by Duncan Foley in 1967.
Envy-freeness by its own is easy to attain by just giving each person 1/n of each resource.
The challenge was first defined by David Schmeidler and Menahem Yaari.
It requires envy-freeness to hold with respect to all additive valuations that are compatible with the ordinal ranking.
In other words, each agent should believe that his/her bundle is at least as good as the bundle of any other agent, according to the responsive set extension of his/her ordinal ranking of the items.
No justified envy is a weakening of no-envy for two-sided markets, in which both the agents and the "items" have preferences over the opposite side, e.g., the market of matching students to schools.
Student A feels justified envy towards student B, if A prefers the school allocated to B, and at the same time, the school allocated to B prefers A. Ex-ante envy-freeness is a weakening of envy-freeness used in the setting of fair random assignment.
Envy minimization is an optimization problem in which the objective is to minimize the amount of envy (which can be defined in various ways), even in cases in which envy-freeness is impossible.
Proportionality (PR) and envy-freeness (EF) are two independent properties, but in some cases one of them may imply the other.
When all valuations are additive set functions and the entire cake is divided, the following implications hold: When the valuations are only subadditive, EF still implies PR, but PR no longer implies EF even with two partners: it is possible that Alice's share is worth 1/2 in her eyes, but Bob's share is worth even more.
On the contrary, when the valuations are only superadditive, PR still implies EF with two partners, but EF no longer implies PR even with two partners: it is possible that Alice's share is worth 1/4 in her eyes, but Bob's is worth even less.
Similarly, when not all cake is divided, EF no longer implies PR.