Fair division, also known as the cake cutting problem, is the problem of dividing a resource in such a way that all recipients believe that they have received a fair amount. The problem is easier when recipients have different measures of value of the parts of the resource: in the "cake cutting" version, one recipient may like marzipan, another prefers cherries, and so on—then, and only then, the n recipients may get even more than what would be one n-th of the value of the "cake" for each of them. On the other hand, the presence of different measures opens a vast potential (a Pandora box) for many challenging questions and directions of further research.
There are a number of variants of the problem. The definition of 'fair' may simply mean that they get at least their fair proportion, or harder requirements like envy-freeness may also need to be satisfied. The theoretical algorithms mainly deal with goods that can be divided without losing value. The division of indivisible goods, as in for instance a divorce, is a major practical problem. Chore division is a variant where the goods are undesirable.
Fair division is often used to refer to just the simplest variant. That version is referred to here as proportional division or simple fair division.
Most of what is normally called a fair division is not considered so by the theory because of the use of arbitration. This kind of situation happens quite often with mathematical theories named after real life problems. The decisions in the Talmud on entitlement when an estate is bankrupt reflect some quite complex ideas about fairness, and most people would consider them fair. However they are the result of legal debates by rabbis rather than divisions according to the valuations of the claimants.
Fair division is a mathematical theory based on an idealization of a real life problem. The real life problem is the one of dividing goods or resources fairly between people, the 'players', who have an entitlement to them. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods.
The theory of fair division provides explicit criteria for various different types of fairness. Its aim is to provide procedures (algorithms) to achieve a fair division, or prove their impossibility, and study the properties of such divisions both in theory and in real life.
The assumptions about the valuation of the goods or resources are:
Indivisible parts make the theory much more complex. An example of this would be where a car and a motorcycle have to be shared. This is also an example of where the values may not add up nicely, as either can be used as transport. The use of money can make such problems much easier.
The criteria of a fair division are stated in terms of a players valuations, their level of entitlement, and the results of a fair division procedure. The valuations of the other players are not involved in the criteria. Differing entitlements can normally be represented by having a different number of proxy players for each player but sometimes the criteria specify something different.
In the real world of course people sometimes have a very accurate idea of how the other players value the goods and they may care very much about it. The case where they have complete knowledge of each others valuations can be modeled by game theory. Partial knowledge is very hard to model. A major part of the practical side of fair division is the devising and study of procedures that work well despite such partial knowledge or small mistakes.
A fair division procedure lists actions to be performed by the players in terms of the visible data and their valuations. A valid procedure is one that guarantees a fair division for every player who acts rationally according to their valuation. Where an action depends on a player's valuation the procedure is describing the strategy a rational player will follow. A player may act as if a piece had a different value but must be consistent. For instance if a procedure says the first player cuts the cake in two equal parts then the second player chooses a piece, then the first player cannot claim that the second player got more.
What the players do is:
It is assumed the aim of each player is to maximize the minimum amount they might get, or in other words, to achieve the maximin.
Procedures can be divided into finite and continuous procedures. A finite procedure would for instance only involve one person at a time cutting or marking a cake. Continuous procedures involve things like one player moving a knife and the other saying stop. Another type of continuous procedure involves a person assigning a value to every part of the cake.
There are a number of widely used criteria for a fair division. Some of these conflict with each other but often they can be combined. The criterion are described here only for when each player is entitled to the same amount.
For two people there is a simple solution which is commonly employed. This is the so-called divide and choose method. One person divides the resource into what they believe are equal halves, and the other person chooses the "half" they prefer. Thus, the person making the division has an incentive to divide as fairly as possible: for if they do not, they will likely receive an undesirable portion. This solution satisfies gives a proportional and envy-free division.
The article on divide and choose describes why the procedure is not equitable. More complex procedures like the Adjusted Winner procedure are designed to cope with indivisible goods and to be more equitable in a practical context.
Austin's moving-knife procedure gives an exact division for two players. The first player places two knives over the cake such that one knife is at the left side of the cake, and one is further right; half of the cake lies between the knives. He then moves the knives right, always ensuring there is half the cake – by his valuation – between the knives. If he reaches the right side of the cake, the leftmost knife must be where the rightmost knife started off. The second player stops the knives when he thinks there is half the cake between the knives. There will always be a point at which this happens, because of the intermediate value theorem.
Fair division with three or more players is considerably more complex than the two player case.
Proportional division is the easiest and the article describes some procedures which can be applied with any number of players. Finding the minimum number of cuts needed is an interesting mathematical problem.
Envy-free division was first solved for the 3 player case in 1960 independently by John Selfridge of Northern Illinois University and John Horton Conway at Cambridge University. The best algorithm uses at most 5 cuts.
The first cake cutting procedure for 4 or more players that produced an envy-free division of cake for any number of persons was first published by Steven Brams and Alan Taylor in 1992. This number of cuts that might be required by this procedure is unbounded. A bounded moving knife procedure for 4 players was found in 1997.
There are no discrete algorithms for an exact division even for two players, a moving knife procedure is the best that can be done. There are no exact division algorithms for 3 or more players but there are 'near exact' algorithms which are also envy-free and can achieve any desired degree of accuracy.
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.
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.
According to Sol Garfunkel, the cake cutting problem had been one of the most important open problems in 20th century mathematics, when the most important variant of the problem was finally solved together by Steven Brams and Alan Taylor in 1995.
Divide and choose has probably been used since prehistory. The related activities of bargaining and barter are also ancient. Negotiations involving more than two people are also quite common, the Potsdam Conference is a notable recent example.
The theory of fair division dates back only to the end of the second world war. It was devised by a group of Polish mathematicians, Hugo Steinhaus, Bronisław Knaster and Stefan Banach, who used to meet in the Scottish Café in Lvov (then in Poland). A proportional (fair division) division for any number of players called 'last-diminisher' was devised in 1944. This was attributed to Banach and Knaster by Steinhaus when he made the problem public for the first time at a meeting of the Econometric Society in Washington D.C. on 17 September 1947. At that meeting he also proposed the problem of finding the smallest number of cuts necessary for such divisions.
Envy-free division was first solved for the 3 player case in 1960 independently by John Selfridge of Northern Illinois University and John Horton Conway at Cambridge University, the algorithm was first published in the 'Mathematical Games' column by Martin Gardner in Scientific American.
Envy-free division for 4 or more players was a difficult open problem of the twentieth century. The first cake cutting procedure that produced an envy-free division of cake for any number of persons was first published by Steven Brams and Alan Taylor in 1992.
In Numb3rs season 3 episode 'One Hour' Charlie talks about the cake-cutting problem as applied to the amount of money a kidnapper was demanding.
Hugo Steinhaus wrote about a number of variants of fair division in his book 'Mathematical Snapshots'. In his book he says a special 3 person version of fair division was devised by G. Krochmainy in Berdechów in 1944 and another by Mrs L Kott.
Martin Gardner and Ian Stewart have both published books with sections about the problem. Martin Gardner introduced the chore division form of the problem. Ian Stewart has popularized the fair division problem with his articles in Scientific American and New Scientist.