Envy-freeness

Envy-freeness is a criterion of fair division. An envy-free division is a division in which every partner feels that his or her allocated share is at least as good as any other share.

Definitions

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

${\displaystyle X_{i}\succeq _{i}X_{j}}$

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

${\displaystyle V_{i}(X_{i})\geq V_{i}(X_{j})}$

Put another way: we say that agent $i$ envies agent $j$ if $i$ prefers the piece of $j$ over his own piece, i.e.:

${\displaystyle X_{i}\prec _{i}X_{j}}$

${\displaystyle V_{i}(X_{i})<V_{i}(X_{j})}$

A division is called envy-free if no agent envies another agent.

History

The envy-freeness concept was introduced to the problem of fair cake-cutting by George Gamow and Marvin Stern in 1958.^[1] In the context of fair cake-cutting, envy-freeness means that each partner believes that their share is at least as large as any other share. In the context of chore division, envy-freeness means that each partner believes their share is at least as small as any other share. The crucial issue is that no partner would wish to swap their share with any other partner.

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, the envy-freeness concept was introduced to the economics problem of resource allocation by Duncan Foley in 1967.^[2] It became the dominant fairness criterion in economics. See, for example:

• Varian's theorems

Envy-freeness was also studied in the context of fair item assignment.^[3] See:

• 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:

Valuations	2 partners	3+ partners
Additive	${\displaystyle EF\implies PR}$ ${\displaystyle PR\implies EF}$	${\displaystyle EF\implies PR}$
Subadditive	${\displaystyle EF\implies PR}$	${\displaystyle EF\implies PR}$
Superadditive	${\displaystyle PR\implies EF}$	-
General	-	-

See also

• Inequity aversion

References

1. Gamow, George; Stern, Marvin (1958). Puzzle-math. Viking Press. ISBN 0670583359.
2. Foley, Duncan (1967). "Resource allocation and the public sector". Yale Econ Essays. 7 (1): 45–98.
3. Herreiner, Dorothea K.; Puppe, Clemens D. (2007). "Envy Freeness in Experimental Fair Division Problems". Theory and Decision. 67: 65. doi:10.1007/s11238-007-9069-8.