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Equitable cake-cutting
Equitable (EQ) cake-cutting is a kind of a fair cake-cutting problem, in which the fairness criterion is equitability. It is a cake-allocation 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:
See the page on equitability for examples and comparison to other fairness criteria.
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)
The same procedure can be used for dividing chores (with negative utility).
The full revelation procedure has a variant which satisfies a weaker kind of equitability and a stronger kind of truthfulness. The procedure first finds the median points of each partner. Suppose the median point of partner A is and of partner B is , with . Then, A receives and B receives . Now there is a surplus - . The surplus is divided between the partners in equal proportions. So, for example, if A values the surplus as 0.4 and B values the surplus as 0.2, then A will receive twice more value from than B. So this protocol is not equitable, but it is still EF. It is weakly-truthful in the following sense: a risk-averse player has an incentive to report his true valuation, because reporting an untrue valuation might leave him with a smaller value.
![{\displaystyle [0,a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63d050b0ffe6cc6f635808b9a013366a60e6d0c0)
![{\displaystyle [b,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba2db3c6c4b4ecf26b0165005c40bb18eb119ce)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
Austin moving-knife procedure gives each of the two partners a piece with a subjective value of exactly 1/2. Thus the division is EQ, EX and EF. It requires 2 cuts, and gives one of the partners two disconnected pieces.
When more than two cuts are allowed, it is possible to achieve a division which is not only EQ but also EF and PE. Some authors call such a division "perfect".
Equitable cake-cutting
Equitable (EQ) cake-cutting is a kind of a fair cake-cutting problem, in which the fairness criterion is equitability. It is a cake-allocation 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:
See the page on equitability for examples and comparison to other fairness criteria.
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:
The same procedure can be used for dividing chores (with negative utility).
The full revelation procedure has a variant which satisfies a weaker kind of equitability and a stronger kind of truthfulness. The procedure first finds the median points of each partner. Suppose the median point of partner A is and of partner B is , with . Then, A receives and B receives . Now there is a surplus - . The surplus is divided between the partners in equal proportions. So, for example, if A values the surplus as 0.4 and B values the surplus as 0.2, then A will receive twice more value from than B. So this protocol is not equitable, but it is still EF. It is weakly-truthful in the following sense: a risk-averse player has an incentive to report his true valuation, because reporting an untrue valuation might leave him with a smaller value.
Austin moving-knife procedure gives each of the two partners a piece with a subjective value of exactly 1/2. Thus the division is EQ, EX and EF. It requires 2 cuts, and gives one of the partners two disconnected pieces.
When more than two cuts are allowed, it is possible to achieve a division which is not only EQ but also EF and PE. Some authors call such a division "perfect".