As an example, consider a birthday cake shaped as a disk.
A possible application of the pie model might be for dividing an island’s shoreline into connected lots.
[1][2] Every procedure for fair cake-cutting can also be applied to cutting a pie by just ignoring the fact that the two endpoints are identified.
For example, if the cake-cutting procedure yielded a division in which Alice receives [0,1/3] and the George receives [1/3,1], then we would give Alice a circular sector of 120 degrees and George the remaining sector with 240 degrees.
Pie cutting becomes more interesting when we consider questions of efficiency, since in pie-cutting more divisions are possible.
An EF division of a pie can always be found using divide and choose: one partner cuts the pie into two sectors he considers equal, and the other partner chooses the sector that he considers better.
Often, Pareto efficiency is evaluated only with relation to a subset of all possible divisions.
When the valuations of the partners are (additive) measures, the following moving-knife procedure guarantees a division which is EF, and PE relative to divisions to two contiguous sectors.
The other partner (the Chooser) observes this process during an entire cycle.
Then, in the next cycle, he identifies the position that, in his view, gives the maximum value to one of the two sectors so determined.
If her value is not additive, then the division would still be envy-free but not necessarily Pareto-efficient.
Moreover, when the valuations of both partners are not additive (so none of them can play as the Rotator), a PEEF division does not always exist.
Suppose the sum of all weights is 1, and the value of the pie for all agents is normalized to 1 too.
A division is called proportional if each of two partners receives a value of at least 1/2.
are pre-specified weights representing the different entitlements of the partners to the cake.
The above procedure shows that in a pie, a WPR division with connected pieces always exists.
This is in contrast to a non-circular cake (an interval), in which a WPR with connected pieces might not exist.
If the valuations of the partners are absolutely continuous with respect to each other, then there exists a WPR division which is also weighted-envy-free (WEF) and Pareto efficient (PE), and the ratio between the values of the partners is exactly w1/w2.
The partition is WEF because the value of each partner is exactly his due share.
There always exists a partition of a pie to two partners, which is both envy-free and equitable.
A division rule is called dictatorial if it allocates the entire cake to a single, pre-specified partner.
[4] Iyer and Huhns[6] were the first to present a specialised protocol for dividing a pie.
In their protocol, each agent marks (n+1) disjoint pieces on the pie.
But there is a simpler algorithm, that takes advantage of the circularity of the pie.
[7][3] Partner A rotates three radial knives continuously around the pie, maintaining what s/he believes to be 1/3-1/3-1/3 sectors.
Hence, by the intermediate value theorem, there must be a position in the rotation when partner B views two sectors as tied for largest.
For 3 partners, there exist a pie and corresponding measures for which no allocation is PEEF.
Since the measures are nearly uniform, it is easy to find allocations of the pie that are almost envy-free and almost undominated.
When there are 3 or more partners with different entitlements, a weighted-proportional (WPR) division is needed.
[5] This is analogous to an impossibility result for 1-dimensional interval cake and 2 partners (see proportional cake-cutting with different entitlements).