Mathematically, that means that for all partners i and j: Where: See the page on equitability for examples and comparison to other fairness criteria.
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).
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.
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.
[4] The minimum number of cuts required for a PE-EF-EQ division depends on the valuations of the partners.
In these cases, it is possible to both find the optimal number of cuts and their exact locations.
The algorithm requires full knowledge of the partners' valuations.
Therefore, they cannot be carried out in a finite number of discrete steps.
This infinity property is characteristic of division problems which require an exact result.
A near-equitable division for two partners can be found in finite time and with a single cut.
It gives each partner a piece with a subjective value of exactly
There is another procedure, using n-1 moving knives, that can be used to find a connected equitable allocation for any ordering of the agents.
They first prove that, for 3 partners that must get connected pieces, an EQ+EF division may not exist.
[3] They do this by describing 3 specific value measures on a 1-dimensional cake, in which every EQ allocation with 2 cuts is not EF.
Then they prove that, for 3 or more partners, a PE+EF+EQ division may not exist even with disconnected pieces.
Barbanel, Brams and Stromquist study the existence of divisions of a pie, which are both EQ and EF.
An equitable cake allocation cannot be found using a finite protocol in the Robertson–Webb query model, even for 2 agents.
[8] Moreover, for any ε > 0: The max-equitable division rule is a rule that selects, from among all equitable cake allocations, the one in which the common value of the agents is maximum.
It has two variants: There always exists a connected max-equitable alloation (both absolute and relative), and it can be found using a generalized moving-knives procedure.