It is the mirror-image of the fair cake-cutting problem, in which the divided resource is desirable so that each participant wants to get as much as possible.
The chore division problem was introduced by Martin Gardner in 1978.
In a situation of dividing inheritance, this yard would be considered good, since each heir would like to have as much land as possible, so it is a cake-cutting problem.
But in a situation of dividing house-chores such as lawn-mowing, this yard would be considered bad, since each child would probably like to have as little land as possible to mow, so it is a chore-cutting problem.
Some results from fair cake-cutting can be easily translated to the chore-cutting scenario.
For example, the divide and choose procedure works equally well in both problems: one of the partners divides the resource to two parts that are equal in his eyes, and the other partner chooses the part that is "better" in his eyes.
is the total number of partners): Most protocols for proportional cake-cutting can be easily translated to the chore-cutting.
An example is the Austin moving-knife procedure, which guarantees each partner a piece that he values as exactly 1/n of the total.
that is worth, according to his own personal disutility function, at most as much as any other piece: For two partners, divide and choose produces an envy-free chore-cutting.
Alice cuts the chore to three pieces equal in her eyes (this is also the first step in the Selfidge-conway protocol).
The easy case is that they disagree, since then we can give each partner a smallest piece and we are done.
I.e, if Bob trims X2 to X2' and X3 to X3', such that both X2' and X3' are for him as small as X1, then Carl thinks X1 is still a smallest piece – weakly smaller than X2' and X3'.
For each trimming, the following is done: Carl is not envious since he chose first; Bob is not envious since he cut; Alice is not envious since she had a (negative) advantage over Carl: in the first step, Carl took X1, while Alice took a piece that is smaller than X1 by max(E2,E3), while in the last step, Alice took two pieces that are worth at most (E2+E3)/2.
I.e, if Carl trims X2 to X2' and X3 to X3', such that both X2' and X3' are for him as small as X1, then Bob thinks X1 is still a smallest piece – weakly smaller than X2' and X3'.
Peterson and Su gave a moving knife procedure for 4-person chore division.
Dehghani et al.[5] provided the first discrete and bounded envy-free protocol for chore division among any number of agents.
The following procedures can be adapted to divide a bad cake with disconnected pieces: Heydrich and van Stee[6] calculate the price of fairness in chore division when the pieces have to be connected.
It may be possible to use chore division procedures to divide up the work and cost of reducing climate change among nations.
However, using chore division procedures reduces the need for a supra-national authority to partition and oversee work by those nations.