Английская Википедия: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 Шаблон:Mvar and Шаблон:Mvar:
- <math>V_i(X_i) = V_j(X_j)</math>
Where:
- <math>X_i</math> is the part of the resource allocated to partner Шаблон:Mvar;
- <math>V_i</math> is the value function of partner Шаблон:Mvar. Usually these functions are normalized such that <math>V_i(\emptyset)=0</math> and <math>V_i(EntireCake)=1</math> for every Шаблон:Mvar.
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? | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
Шаблон:Ya | Шаблон:Ya | Шаблон:Ya | |||||||
|
Шаблон:Ya | Шаблон:Ya | Шаблон:Na (Alice and Bob don't agree on the values of the pieces). | |||||||
|
Шаблон:Ya | Шаблон:Na (Alice and Bob envy each other's share). |
Шаблон:Na | |||||||
|
Шаблон:Na (Alice enjoys her share more than Bob enjoys his share). |
Шаблон:Ya | Шаблон:Na | |||||||
|
Шаблон:Na | Шаблон:Na (Bob envies Alice). |
Шаблон:Ya | |||||||
|
Шаблон:Na | Шаблон:Na | Шаблон:Na |
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
