Equitability

Equitability is a criterion for fair division. A division is called equitable if the subjective value of all partners is the same, i.e., each partner is equally happy with his/her share. Mathematically, that means that for all partners i and j:

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

Where:

• ${\displaystyle X_{i}}$ is the part of the resource allocated to partner i;
• ${\displaystyle V_{i}}$ is the value function of partner i. Usually these functions are normalized such that ${\displaystyle V_{i}(\emptyset )=0}$ and ${\displaystyle V_{i}(EntireCake)=1}$ for every i.

Comparison to other criteria

• Equitability (EQ) compares values of different people to different pieces;
• Envy-freeness (EF) compares values of the same person to different pieces;
• Exact division (EX) compares values of different people to the same pieces.

The following table illustrates the difference. In all examples there are two partners, Alice and Bob. Alice receives the left part and Bob receives the right part.

Division		EQ?	EF?	EX?
A: 50% 50% B: 50% 50%
A: 60% 40% B: 40% 60%				(Alice and Bob don't agree on the values of the pieces).
A: 40% 60% B: 60% 40%			(Alice and Bob envy each other's share).
A: 70% 30% B: 40% 60%		(Alice enjoys her share more than Bob enjoys his share).
A: 60% 40% B: 60% 40%			(Bob envies Alice).
A: 60% 40% B: 70% 30%

Note that the table has only 6 rows, because 2 combinations are impossible: an EX+EF division must be EQ, and an EX+EQ division must be EF.

Existence and computation

Equitability has been mainly applied in the division of a heterogeneous continuous resource; see Equitable cake-cutting.

It has also been applied in the division of homogeneous resources; see Adjusted winner procedure.

Recently, it has also been studied in the context of fair item allocation. With indivisible items, an equitable allocation might not exist, but it can be approximated in several ways. For example, an allocation is called EQ1 if the difference between subjective valuations is at most a single item. It was studied for goods, ^[1] for chores, ^[2] for a goods on a path, ^[3] and in conjunction with utilitarian optimality. ^[4]

References

1. Freeman, Rupert; Sikdar, Sujoy; Vaish, Rohit; Xia, Lirong (2019-05-25). "Equitable Allocations of Indivisible Goods". arXiv: 1905.10656 [cs.GT].
2. Freeman, Rupert; Sikdar, Sujoy; Vaish, Rohit; Xia, Lirong (2020-02-24). "Equitable Allocations of Indivisible Chores". arXiv: 2002.11504 [cs.GT].
3. Misra, Neeldhara; Sonar, Chinmay; Vaidyanathan, P. R.; Vaish, Rohit (2021-01-26). "Equitable Division of a Path". arXiv: 2101.09794 [cs.GT].
4. Aziz, Haris; Huang, Xin; Mattei, Nicholas; Segal-Halevi, Erel (2023). "Computing welfare-Maximizing fair allocations of indivisible goods". European Journal of Operational Research. 307 (2): 773–784. arXiv: 2012.03979 . doi:10.1016/j.ejor.2022.10.013.