# Envy-freeness

Envy-freeness (EF) is a criterion of fair division. In an envy-free division, every agent feels that their share is at least as good as the share of any other agent, and thus no agent feels envy.

## Definitions

A resource is divided among several agents such that every agent $i$ receives a share $X_{i}$ . Every agent $i$ has a subjective preference relation $\succeq _{i}$ over different possible shares. The division is called envy-free if for all $i$ and $j$ :

$X_{i}\succeq _{i}X_{j}$ If the preference of the agents are represented by a value functions $V_{i}$ , then this definition is equivalent to:

$V_{i}(X_{i})\geq V_{i}(X_{j})$ Put another way: we say that agent $i$ envies agent $j$ if $i$ prefers the piece of $j$ over his own piece, i.e.:

$X_{i}\prec _{i}X_{j}$ $V_{i}(X_{i}) A division is called envy-free if no agent envies another agent.

## History

EF was introduced to the problem of fair cake-cutting by George Gamow and Marvin Stern in 1958. In the context of fair cake-cutting, EF means that each agent believes that their share is at least as large as any other share. In the context of chore division, EF means that each agent believes their share is at least as small as any other share. The crucial issue is that no agent would wish to swap their share with any other agent.

See:

• Envy-free cake-cutting - a detailed survey of procedures and results related to the envy-free criterion in cake-cutting.
• Group-envy-free - a strengthening of the envy-free criterion from individuals to coalitions.

Later, EF was introduced to the economics problem of resource allocation by Duncan Foley in 1967. It became the dominant fairness criterion in economics. See, for example:

• Envy-free item assignment - a detailed survey of procedures and results related to the EF criterion in allocation of indivisible items.
• Rental harmony - an assignment problem in which envy-freeness is the dominant fairness criterion.

## Relations to other fairness criteria

### Implications between proportionality and envy-freeness

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:

• With two partners, PR and EF are equivalent;
• With three or more partners, EF implies PR but not vice versa. For example, it is possible that each of three partners receives 1/3 in his subjective opinion, but in Alice's opinion, Bob's share is worth 2/3.

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. The implications are summarized in the following table:

Valuations2 partners3+ partners
Additive$EF\implies PR$ $PR\implies EF$ $EF\implies PR$ Subadditive$EF\implies PR$ $EF\implies PR$ Superadditive$PR\implies EF$ -
General--