Editing Fair division (section)

== Things that can be divided ==

Formally, a fair division problem is defined by a set <math>C</math> (often called "the cake") and a group of <math>n</math> players. A division is a partition of <math>C</math> into <math>n</math> disjoint subsets: <math>C = X_1 \sqcup X_2 \sqcup\cdots \sqcup X_n</math>, one subset per player.

The set <math>C</math> can be of various types:
* <math>C</math> may be a finite set of indivisible items, for example: <math>C = \{\text{piano}, \text{car}, \text{apartment}\}</math>, such that each item should be given entirely to a single person.
* <math>C</math> may be an infinite set representing a divisible resource, for example: money, or a cake. Mathematically, a divisible resource is often modeled as a subset of a real space, for example, the section [0,1] may represent a long narrow cake, that has to be cut into parallel pieces. The [[unit disk]] may represent an apple pie.

Additionally, the set to be divided may be:
* homogeneous – such as money, where only the amount matters, or
* heterogeneous – such as a cake that may have different ingredients, different icings, etc.

Finally, it is common to make some assumptions about whether the items to be divided are:
* goods – such as a car or a cake, or
* bads – such as house chores.

Based on these distinctions, several general types of fair division problems have been studied:
* [[Fair item assignment]] – dividing a set of ''indivisible and heterogeneous'' goods.
* Fair resource allocation – dividing a set of ''divisible and homogeneous'' goods. A special case is [[fair division of a single homogeneous resource]].
* [[Fair cake-cutting]] – dividing a ''divisible, heterogeneous good''. A special case is when the cake is a circle; then the problem is called [[fair pie-cutting]].
* Fair [[chore division]] – dividing a ''divisible, heterogeneous bad.''

Combinations and special cases are also common:
* [[housemates problem|Rental harmony]] (aka the housemates problem) – dividing a set of ''indivisible heterogeneous goods'' (e.g., rooms in an apartment), and simultaneously a ''homogeneous divisible bad'' (the rent on the apartment).
* [[Fair river sharing]] – dividing waters flowing in an international river among the countries along its stream.
* [[Fair random assignment]] – dividing lotteries over divisions – is especially common when allocating indivisible goods.