Fair Division - Variants

Variants

Some cake-cutting procedures are discrete, whereby players make cuts with a knife (usually in a sequence of steps). Moving-knife procedures, on the other hand, allow continuous movement and can let players call "stop" at any point.

A variant of the fair division problem is chore division: this is the "dual" to the cake-cutting problem in which an undesirable object is to be distributed amongst the players. The canonical example is a set of chores that the players between them must do. Note that "I cut, you choose" works for chore division. A basic theorem for many person problems is the Rental Harmony Theorem by Francis Su. An interesting application of the Rental Harmony Theorem can be found in the international trade theory.

Sperner's Lemma can be used to get as close an approximation as desired to an envy-free solutions for many players. The algorithm gives a fast and practical way of solving some fair division problems.

The division of property, as happens for example in divorce or inheritance, normally contains indivisible items which must be fairly distributed between players, possibly with cash adjustments (such pieces are referred to as atoms).

A common requirement for the division of land is that the pieces be connected, i.e. only whole pieces and not fragments are allowed. For example the division of Berlin after World War 2 resulted in four connected parts. A consensus halving is where a number of people agree that a resource has been evenly split in two, this is described in exact division.