Exact division

An exact division, also called even division or consensus division, is a division of a heterogeneous resource ("cake") to several subsets such that each of n people with different tastes agree about the valuations of the pieces.

For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided to three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is a consensus division, as both Alice and George value the three pieces as 20%, 50% and 30% respectively.

As the example illustrates, a consensus division is not necessarily fair. For example, if the 20% piece is given to Alice and the 50% is given to George, this is obviously unfair to Alice. In the theory of cake, consensus divisions are often used as subroutines for creating fair divisions.

Consensus divisions always exist, but they cannot be found by discrete protocols (with a finite number of queries). In some cases, exact divisions can be found by moving-knife protocols. Near-exact divisions can be found by discrete protocols.

Let ${\displaystyle w_{1},w_{2},\ldots ,w_{k}}$ be k weights whose sum is 1. Assume that all n partners value the cake C as 1.

An exact division (aka consensus division) in the ratios ${\displaystyle w_{1},w_{2},\ldots ,w_{k}}$ is a partition of the cake to k pieces: ${\displaystyle C=X_{1}\sqcup \cdots \sqcup X_{k}}$, such that for every partner i and every piece j:

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