# Fair division

Fair division is the problem of dividing a set of goods or resources between several people who have an entitlement to them, such that each person receives his/her due share. This problem arises in various real-world settings: auctions, divorce settlements, electronic spectrum and frequency allocation, airport traffic management, or exploitation of Earth Observation Satellites. This is an active research area in Mathematics, Economics (especially Social choice theory), Game theory, Dispute resolution, and more. 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.

There are many different kinds of fair division problems, depending on the nature of goods to divide, the criteria for fairness, the nature of the players and their preferences, and other criteria for evaluating the quality of the division.

The mathematical fair division problem is an idealization of those real life problems. 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.

## Definitions

There is a set $X$ and a group of $n$ players. A division is a partition of $X$ to $n$ disjoint subsets: $X = X_1 \sqcup X_2 \sqcup\cdots \sqcup X_n$, one subset per player.

### What is divided?

The set $X$ can be of several types:

• X may be a finite set of indivisible items, for example: $X = \{piano, car, apartment\}$, such that each item should be given entirely to a single person.
• X 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:

• desirable - such as a car or a cake, or -
• undesirable - such as house chores.

The problem of dividing a set of indivisible and heterogeneous items is called fair item assignment.

The problem of dividing a set of divisible and homogeneous items is called fair resource allocation.

The problem of dividing a divisible, heterogeneous and desirable resource is also called fair cake-cutting.

The problem of dividing a set of heterogeneous and undesirable items is also called fair Chore division (if the chores are divisible) or chore assignment (if they are not).

Combinations are also possible, for example:

• When dividing inheritance, or dividing household property during divorce, it is common to have both desirable indivisible heterogeneous items, desirable divisible heterogeneous property such as land, and desirable divisible homogeneous property such as money.
• In the housemates problem, several friends rent a house together, and they have to both allocate the rooms in the apartment (a set of indivisible, heterogeneous, desirable goods), and divide the rent to pay (divisible, homogeneous, undesirable good). This problem is also called room assignment-rent division.

### What is fair?

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,[1] 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.

According to the Subjective theory of value, there cannot be an objective measure of the value of each item. Therefore, objective fairness is not possible, as different people may assign different values to each item. Empirical experiments on how people define the concept of fairness [2] lead to inconclusive results.

Therefore, most current research on fairness focuses on concepts of subjective fairness. Each of the $n$ people is assumed to have a personal, subjective utility function or value function, $V_i$, which assigns a numerical value to each subset of $X$. Usually the functions are assumed to be normalized, so that every person values the empty set as 0 ($V_i (\empty) = 0$ for all i), and the entire set of items as 1 ($V_i (X) = 1$ for all i) if the items are desirable, and -1 if the items are undesirable. Examples are:

• If $X$ is the set of indivisible items {piano, car, apartment}, then Alice may assign a value of 1/3 to each item, which means that each item is important to her just the same as any other item. Bob may assign the value of 1 to the set {car, apartment}, and the value 0 to all other sets except X; this means that he wants to get only the car and the apartment together; the car alone or the apartment alone, or each of them together with the piano, is worthless to him.
• If $X$ is a long narrow cake (modeled as the interval [0,1]), then, Alice may assign each subset a value proportional to its length, which means that she wants as much cake as possible, regardless of the icings. Bob may assign value only to subsets of [0.4, 0.6], for example, because this part of the cake contains cherries and Bob only cares about cherries.

Based on these subjective value functions, 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 criteria described here are only for when each player is entitled to the same amount:

• A proportional division, also called simple fair division, means that every person gets at least his due share according to his own value function. For instance if three people divide up a cake each gets at least a third by their own valuation, i.e. each of the n people gets a subset of X which he values as at least 1/n:
• $V_i(X_i) \ge 1/n$ for all i.
• An envy-free division guarantees that no-one will want somebody else's share more than their own, i.e. every person gets a share that he values at least as much as all other shares:
• $V_i(X_i) \ge V_i(X_j)$ for all i and j.
• An exact division is one where every player thinks everyone received exactly their fair share, no more and no less:
• $V_i(X_i) = 1/n$ for all i.
• An equitable division means every person feels exactly the same happiness, i.e. the proportion of the cake a player receives by their own valuation is the same for every player. This is a difficult aim as players need not be truthful if asked their valuation:
• $V_i(X_i) = V_j(X_j)$ for all i and j.

When the recipients have different measures of value of the parts of the resource, it is possible to have super fair divisions - divisions in which each person receives strictly more than his due share. For example, 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 for many challenging questions and directions of further research.

In addition to fairness, it is sometimes desired that the division be Pareto optimal, i.e., no other allocation would make someone better off without making someone else worse off. The term efficiency comes from the economics idea of the efficient market. A division where one player gets everything is optimal by this definition so on its own this does not guarantee even a fair share.

Berlin divided by the Potsdam Conference

Note that 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 other's 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.

## Procedures

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:

• Agree on their criteria for a fair division
• Select a valid procedure and follow its rules

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.

### Two players

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 guarantees an envy-free division. If the valuations of the players are sigma additive, then an envy-free division is also proportional. 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[3] gives an exact division for two players. The first player positions a knife over the left side of the cake. He moves the knife to the right and when either player says to stop, they receive the left piece of cake. This produces an envy-free division.

The surplus procedure (SP) achieves a form of equitability called proportional equitability. This procedure is strategy proof and can be generalized to more than two people.[4]

### Many players

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(Selfridge–Conway discrete procedure) uses at most 5 cuts.

The Brams-Taylor procedure was the first cake-cutting procedure for four or more players that produced an envy-free division of cake for any number of persons and was published by Steven Brams and Alan Taylor in 1995.[5] 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.

A generalization of the surplus procedure called the equitable procedure (EP) achieves a form of equitability. Equitability and envy-freeness can be incompatible for 3 or more players.[4]

Most of the existing fair-division procedures outlined above assume players' utility to be additive. In other words, if a player derives a certain amount of utility from 25 g of chocolate cake, then the player is assumed to derive exactly twice as much utility from 50 g of the same chocolate cake.

In 2013, Rishi S. Mirchandani showed that most existing fair-division algorithms are incompatible with nonadditive utility functions.[6] Further, he proved that instances of the fair-division problem in which players have nonadditive utility functions may have no proportional solution.

Mirchandani suggested that the fair-division problem can be solved using techniques of nonlinear optimization. However, it remains as an open question whether there exist more efficient algorithms for specific subsets of nonadditive utility functions.

## The Price of Fairness

While satisfying fairness criteria is important in fair-division algorithms, it is not the only consideration. Even if an allocation satisfies a given fairness criterion, it may still be undesirable if the aggregate societal welfare resulting from the allocation is significantly lower than the maximum possible value.

Recent research has used the concept of the price of fairness to quantify the loss of aggregate societal welfare that results from satisfying a fairness criterion. Suppose each player $i$ has a corresponding utility function $V_i$. Further suppose that, in the optimal allocation (the one where aggregate societal welfare is maximized), each player is given a subset $X_i^*$ of the cake, and in the "fair" allocation (the one where some fairness criterion $C$ is satisfied), each player is given a subset $X_i$ of the cake. Then the price of fairness $P$ is: $P = \frac{\sum_i V_i(X_i^*)}{\sum_i V_i(X_i)}$.

Caragiannis et. al (2012)[7] find upper bounds on the price of proportionality and equitability for low numbers of players. However, Mirchandani (2013)[8] shows that the prices of envy-freeness and equitability are both unbounded with nonadditive utility functions.

## 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.:[9] Suppose n housemates in an n-bedroom house seek to decide who gets which room and for what part of the total rent. Also, suppose that the following conditions hold: (1) (Good House) In any partition of the rent, each person finds some room acceptable. (2) (Miserly Tenants) Each person always prefers a free room (one that costs no rent) to a non-free room. (3) (Closed Preference Sets) A person who prefers a room for a convergent sequence of prices prefers that room at the limiting price. Then there exists a partition of the rent so that each person prefers a different room.

An interesting application of the Rental Harmony Theorem can be found in the international trade theory.[10]

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.[11][12][13]

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.[14]

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.

## History

According to Sol Garfunkel, the cake-cutting problem had been one of the most important open problems in 20th century mathematics,[15] when the most important variant of the problem was finally solved with the Brams-Taylor procedure by Steven Brams and Alan Taylor in 1995.

Divide and choose's origins are undocumented. 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 1995.

A major advance on equitable division was made in 2006 by Steven J. Brams, Michael A. Jones, and Christian Klamler.[4]

## In popular culture

• 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 three-person version of fair division was devised by G. Krochmainy in Berdechów in 1944 and another by Mrs L Kott.[16]
• Martin Gardner and Ian Stewart have both published books with sections about the problem.[17][18] 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.
• A Dinosaur Comics strip is based on the cake-cutting problem.[19]

## References

1. ^ Game Theoretic Analysis of a bankruptcy Problem from the Talmud Robert J. Aumann and Michael Maschler. Journal of Economic Theory 36, 195-213 (1985)
2. ^ Yaari, M. E.; Bar-Hillel, M. (1984). "On dividing justly". Social Choice and Welfare 1: 1. doi:10.1007/BF00297056. edit
3. ^ A.K. Austin. Sharing a Cake. Mathematical Gazette 66 1982
4. ^ a b c Brams, Steven J.; Michael A. Jones and Christian Klamler (December 2006). "Better Ways to Cut a Cake" (PDF). Notices of the American Mathematical Society 53 (11): pp.1314–1321. Retrieved 2008-01-16.
5. ^ Steven J. Brams; Alan D. Taylor (January 1995). "An Envy-Free Cake Division Protocol". The American Mathematical Monthly (Mathematical Association of America) 102 (1): 9–18. doi:10.2307/2974850. JSTOR 2974850.
6. ^ Mirchandani, Rishi (August 2013). "Superadditivity and Subadditivity in Fair Division". Journal of Mathematics Research 5 (3): 78–91. doi:10.5539/jmr.v5n3p78. Retrieved 17 August 2014.
7. ^ Caragiannis, I., Kaklamanis, C., Kanellopoulos, P., & Kyropoulou, M. (2012). "The Efficiency of Fair Division". Theory of Computing Systems 50 (4): 589–610. doi:10.1007/s00224-011-9359-y.
8. ^ Mirchandani, Rishi (August 2013). "Superadditivity and Subadditivity in Fair Division". Journal of Mathematics Research 5 (3): 78–91. doi:10.5539/jmr.v5n3p78. Retrieved 17 August 2014.
9. ^ Francis Edward Su (1999). "Rental Harmony: Sperner's Lemma in Fair Division". Amer. Math. Monthly 106 (10): 930–942. doi:10.2307/2589747.
10. ^ Shiozawa, Y. A (2007). "New Construction of a Ricardian Trade Theory". Evolutionary and Institutional Economics Review 3 (2): 141–187. doi:10.14441/eier.3.141.
11. ^ Francis Edward Su. Cited above. (based on work by Forest Simmons 1980)
12. ^
13. ^ Ivars Peterson (March 13, 2000). "A Fair Deal for Housemates". MathTrek.
14. ^ Steven J. Brams; Alan D. Taylor (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. p. 38. ISBN 978-0-521-55644-6.
15. ^ Sol Garfunkel. More Equal than Others: Weighted Voting. For All Practical Purposes. COMAP. 1988
16. ^ Mathematical Snapshots. H.Steinhaus. 1950, 1969 ISBN 0-19-503267-5
17. ^ aha! Insight. Martin. Gardner, 1978. ISBN ISBN 978-0-7167-1017-2
18. ^ How to cut a cake and other mathematical conundrums. Ian Stewart. 2006. ISBN 978-0-19-920590-5
19. ^ http://www.qwantz.com/archive/001345.html