Proportional division

Last updated June 06, 2021

A proportional division is a kind of fair division in which a resource is divided among n partners with subjective valuations, giving each partner at least 1/n of the resource by his/her own subjective valuation.

Example

Consider a land asset that has to be divided among 3 heirs: Alice and Bob who think that it's worth 3 million dollars, and George who thinks that it's worth $4.5M. In a proportional division, Alice receives a land-plot that she believes to be worth at least $1M, Bob receives a land-plot that he believes to be worth at least $1M (even though Alice may think it is worth less), and George receives a land-plot that he believes to be worth at least $1.5M.

Existence

A proportional division does not always exist. For example, if the resource contains several indivisible items and the number of people is larger than the number of items, then some people will get no item at all and their value will be zero. Nevertheless, such a division exists with high probability for indivisible items under certain assumptions on the valuations of the agents.^[2]

Moreover, a proportional division is guaranteed to exist if the following conditions hold:

The valuations of the players are non-atomic, i.e., there are no indivisible elements with positive value.
The valuations of the players are additive, i.e., when a piece is divided, the value of the piece is equal to the sum of its parts.

Hence, proportional division is usually studied in the context of fair cake-cutting. See proportional cake-cutting for detailed information about procedures for achieving a proportional division in the context of cake-cutting.

A more lenient fairness criterion is partial proportionality, in which each partner receives a certain fraction f(n) of the total value, where f(n) ≤ 1/n. Partially proportional divisions exist (under certain conditions) even for indivisible items.

Variants

Super-proportional division

A super-proportional division is a division in which each partner receives strictly more than 1/n of the resource by their own subjective valuation.

Of course such a division does not always exist: when all partners have exactly the same value functions, the best we can do is give each partner exactly 1/n. So a necessary condition for the existence of a super-proportional division is that not all partners have the same value measure.

The surprising fact is that, when the valuations are additive and non-atomic, this condition is also sufficient. I.e., when there are at least two partners whose value function is even slightly different, then there is a super-proportional division in which all partners receive more than 1/n. See super-proportional division for details.

Relations to other fairness criteria

Implications between proportionality and envy-freeness

Proportionality (PR) and envy-freeness (EF) are two independent properties, but in some cases one of them may imply the other.

When all valuations are additive set functions and the entire cake is divided, the following implications hold:

With two partners, PR and EF are equivalent;
With three or more partners, EF implies PR but not vice versa. For example, it is possible that each of three partners receives 1/3 in his subjective opinion, but in Alice's opinion, Bob's share is worth 2/3.

When the valuations are only subadditive, EF still implies PR, but PR no longer implies EF even with two partners: it is possible that Alice's share is worth 1/2 in her eyes, but Bob's share is worth even more. On the contrary, when the valuations are only superadditive, PR still implies EF with two partners, but EF no longer implies PR even with two partners: it is possible that Alice's share is worth 1/4 in her eyes, but Bob's is worth even less. Similarly, when not all cake is divided, EF no longer implies PR. The implications are summarized in the following table:

Valuations	2 partners	3+ partners
Additive	$EF\implies PR$ $PR\implies EF$	$EF\implies PR$
Subadditive	$EF\implies PR$	$EF\implies PR$
Superadditive	$PR\implies EF$	-
General	-	-

Stability to voluntary exchanges

One advantage of the proportionality criterion over envy-freeness and similar criteria is that it is stable with regards to voluntary exchanges.

As an example, assume that a certain land is divided among 3 partners: Alice, Bob and George, in a division that is both proportional and envy-free. Several months later, Alice and George decide to merge their land-plots and re-divide them in a way that is more profitable for them. From Bob's point of view, the division is still proportional, since he still holds a subjective value of at least 1/3 of the total, regardless of what Alice and George do with their plots. On the other hand, the new division might not be envy free. For example, it is possible that initially both Alice and George received a land-plot which Bob subjectively values as 1/3, but now after the re-division George got all the value (in Bob's eyes) so now Bob envies George.

Hence, using envy-freeness as the fairness criterion implies that we must constrain the right of people to voluntary exchanges after the division. Using proportionality as the fairness criterion has no such negative implications.

Individual rationality

An additional advantage of proportionality is that it is compatible with individual rationality in the following sense. Suppose n partners own a resource in common. In many practical scenarios (though not always), the partners have the option to sell the resource in the market and split the revenues such that each partner receives exactly 1/n. Hence, a rational partner will agree to participate in a division procedure, only if the procedure guarantees that he receives at least 1/n of his total value.

Additionally, there should be at least a possibility (if not a guarantee) that the partner receives more than 1/n; this explains the importance of the existence theorems of super-proportional division.

Related Research Articles

In game theory, fair division is the problem of dividing a set of resources among several people who have an entitlement to them, such that each person receives their due share. This problem arises in various real-world settings, such as: division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. This is an active research area in mathematics, economics, 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.

An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation.

Divide and choose is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource and two partners who have different preferences over parts of the cake. The protocol proceeds as follows: one person cuts the cake into two pieces; the other person selects one of the pieces; the cutter receives the remaining piece.

Fair cake-cutting is a kind of fair division problem. The problem involves a heterogeneous resource, such as a cake with different toppings, that is assumed to be divisible – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be unanimously fair - each person should receive a piece that he or she believes to be a fair share.

The last diminisher procedure is a procedure for fair cake-cutting. It involves a certain heterogenous and divisible resource, such as a birthday cake, and n partners with different preferences over different parts of the cake. It allows the n people to achieve a proportional division, i.e., divide the cake among them such that each person receives a piece with a value of at least 1/n of the total value according to his own subjective valuation. For example, if Alice values the entire cake as $100 and there are 5 partners then Alice can receive a piece that she values as at least $20, regardless of what the other partners think or do.

Efficient cake-cutting is a problem in economics and computer science. It involves a heterogeneous resource, such as a cake with different toppings or a land with different coverings, that is assumed to be divisible - it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible, etc. The allocation should be economically efficient. Several notions of efficiency have been studied:

In the theory of fair division, the price of fairness (POF) is the ratio of the largest economic welfare attainable by a division to the economic welfare attained by a fair division. The POF is a quantitative measure of the loss of welfare that society has to take in order to guarantee fairness.

In the context of fair cake-cutting, a super-proportional division is a division in which each partner receives strictly more than 1/n of the resource by their own subjective valuation.

The fair pie-cutting problem is a variation of the fair cake-cutting problem, in which the resource to be divided is circular.

An equitable division (EQ) is a division of a heterogeneous object, in which the subjective value of all partners is the same, i.e., each partner is equally happy with his/her share. Mathematically, that means that for all partners $i$ and $j$ :

Fair item allocation is a kind of a fair division problem in which the items to divide are discrete rather than continuous. The items have to be divided among several partners who value them differently, and each item has to be given as a whole to a single person. This situation arises in various real-life scenarios:

Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in his or her eyes, at least as good as the share received by any other agent. In other words, no person should feel envy.

A proportional cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the proportionality criterion, namely, that every partner feels that his allocated share is worth at least 1/n of the total.

Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among n partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a cake fairly without compromising on economic efficiency.

Utilitarian cake-cutting is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions, such that the sum of the utilities of the partners is as large as possible. It is a special case of the utilitarian social choice rule. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with fair cake-cutting.

The lone divider procedure is a procedure for proportional cake-cutting. It involves a heterogenous and divisible resource, such as a birthday cake, and n partners with different preferences over different parts of the cake. It allows the n people to divide the cake among them such that each person receives a piece with a value of at least 1/n of the total value according to his own subjective valuation.

Symmetric fair cake-cutting is a variant of the fair cake-cutting problem, in which fairness is applied not only to the final outcome, but also to the assignment of roles in the division procedure.

Truthful cake-cutting is the study of algorithms for fair cake-cutting that are also truthful mechanisms, i.e., they incentivize the participants to reveal their true valuations to the various parts of the cake.

Maximin share (MMS) is a criterion of fair item allocation. Given a set of items with different values, the 1-out-of-n maximin-share is the maximum value that can be gained by partitioning the items into n parts and taking the part with the minimum value.

References

↑ Steinhaus, Hugo (1948). "The problem of fair division". Econometrica. 16 (1): 101–104. JSTOR 1914289.
↑ Suksompong, Warut (2016). "Asymptotic existence of proportionally fair allocations". Mathematical Social Sciences. 81: 62–65. arXiv: 1806.00218 . doi:10.1016/j.mathsocsci.2016.03.007.

A summary of proportional and other division procedures appears in: Austin, A. K. (1982). "Sharing a Cake". The Mathematical Gazette. 66 (437): 212. doi:10.2307/3616548. JSTOR 3616548.

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[1] Steinhaus, Hugo (1948). "The problem of fair division". Econometrica. 16 (1): 101–104. JSTOR 1914289.

[2] Suksompong, Warut (2016). "Asymptotic existence of proportionally fair allocations". Mathematical Social Sciences. 81: 62–65. arXiv: 1806.00218 . doi:10.1016/j.mathsocsci.2016.03.007.

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