Equitable division
An equitable division (EQ) is a division of a heterogeneous object (e.g. a cake), in which 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:
Where:
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.
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.
When there are 2 partners, it is possible to get an EQ division with a single cut, but it requires full knowledge of the partners' valuations. Assume that the cake is the interval [0,1]. For each , calculate and and plot them on the same graph. Note that the first graph is increasing from 0 to 1 and the second graph is decreasing from 1 to 0, so they have an intersection point. Cutting the cake at that point yields an equitable division. This division has several additional properties:
![{\displaystyle x\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64a15936df283add394ab909aa7a5e24e7fb6bb2)
![{\displaystyle u_{1}([0,x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86904330ee3001610736e6f33db89337f9f49cd9)
![{\displaystyle u_{2}([x,1])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1c2ac1d52ecbd4b48b72b72c502f23ec423a1c)
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