Fair Cake-Cutting among Families This research was funded in part by the following institutions: The Doctoral Fellowships of Excellence Program at Bar-Ilan University, the Mordechai and Monique Katz Graduate Fellowship Program, and the Israel Science Foundation grant 1083/13. We are grateful to Galya Segal-Halevi, Yonatan Aumann, Avinatan Hassidim, Noga Alon, Christian Klamler, Ulle Endriss, Neill Clift and Sophie Bade for helpful discussions. This paper started with a discussion in the MathOverflow website at http://mathoverflow.net/questions/203060/fair-cake-cutting-between-groups . We are grateful to the members who participated in the discussion: Pietro Majer, Tony Huynh and Manfred Weis. Other members of the StackExchange network contributed useful answers and ideas: Alex Ravsky, Andrew D. Hwang, BKay, Christian Elsholtz, Daniel Fischer, David K, D.W., Hurkyl, Ittay Weiss, Kittsil, Michael Albanese, Raphael Reitzig, Real, Babou, Domotor Palvolgyi (domotorp), Ian Turton (iant) and ivancho.

Fair Cake-Cutting among Familiesthanks: This research was funded in part by the following institutions: The Doctoral Fellowships of Excellence Program at Bar-Ilan University, the Mordechai and Monique Katz Graduate Fellowship Program, and the Israel Science Foundation grant 1083/13.

We are grateful to Galya Segal-Halevi, Yonatan Aumann, Avinatan Hassidim, Noga Alon, Christian Klamler, Ulle Endriss, Neill Clift and Sophie Bade for helpful discussions.

This paper started with a discussion in the MathOverflow website at http://mathoverflow.net/questions/203060/fair-cake-cutting-between-groups . We are grateful to the members who participated in the discussion: Pietro Majer, Tony Huynh and Manfred Weis. Other members of the StackExchange network contributed useful answers and ideas: Alex Ravsky, Andrew D. Hwang, BKay, Christian Elsholtz, Daniel Fischer, David K, D.W., Hurkyl, Ittay Weiss, Kittsil, Michael Albanese, Raphael Reitzig, Real, Babou, Domotor Palvolgyi (domotorp), Ian Turton (iant) and ivancho.

Erel Segal-Halevi Bar-Ilan University, Ramat-Gan 5290002, Israel,
11email: erelsgl@gmail.com,nitzans@biu.ac.il
   Shmuel Nitzan Bar-Ilan University, Ramat-Gan 5290002, Israel,
11email: erelsgl@gmail.com,nitzans@biu.ac.il
Received: date / Accepted: date
Abstract

We study the fair division of a continuous resource, such as a land-estate or a time-interval, among pre-specified groups of agents, such as families. Each family is given a piece of the resource and this piece is used simultaneously by all family members, while different members may have different value functions. Three ways to assess the fairness of such a division are examined. (a) *Average fairness* means that each family’s share is fair according to the ”family value function”, defined as the arithmetic mean of the value functions of the family members. (b) *Unanimous fairness* means that all members in all families feel that their family received a fair share according to their personal value function. (c) *Democratic fairness* means that in each family, at least half the members feel that their family’s share is fair. We compare these criteria based on the number of connected components in the resulting division, and based on their compatibility with Pareto-efficiency.

Keywords:
fair division, cake-cutting, public good, club good, fair-share, no-envy

1 Introduction

Fair division of heterogeneous resources among agents with different preferences has been an important issue since Biblical times. Today it is an active area of research in the interface of computer science (Robertson and Webb, 1998; Procaccia, 2015) and economics (Moulin, 2004). Its applications range from politics (Brams and Taylor, 1996) to multi-agent systems (Chevaleyre et al, 2006).

In most fair division problems, the resource is divided among individual agents, and the fairness of a division is assessed based on their individual preferences. A common fairness criterion is the fair share (FS). It requires that each agent receives a share that is at least as good as of the total endowment, according to the agent’s individual preferences.111 The condition of receiving at least of the total endowment was introduced by Steinhaus (1948). Economists often call it fair-share guarantee (Bogomolnaia et al, 2017). Computer scientists often call it proportionality (Robertson and Webb, 1998).

In practice, however, goods are often owned and used by groups. As an example, consider a land-estate inherited by families, a river that has to be divided among states, or the usage-time of a conference room that has to be divided among meeting groups. The resource (whether land or time) should be divided to pieces, one piece per group. Each group’s share is then used by all its members simultaneously. The land-plot allotted to a family is inhabited by the entire family. The share of the river allotted to a state becomes a national park open to all its citizens. In the time-slot allotted to a group, the conference room is used by all group members.222 In economic terms, the allotted piece becomes a ”club good” (Buchanan, 1965).

The happiness of each group member depends on his/her valuation of the entire share of the group. But, in each group there are different members with different valuations. The group’s share can be valued by some of its members as at least of the total and by others as less than of the total. How, then, should the fairness of a division be assessed?

The present paper studies this question in the classic setting of cake-cutting, introduced by Steinhaus (1948). In this setting, there is a measurable space (e.g. an interval or a polygon) called the cake. The preferences of each agent are represented by a value-measure on the cake. We study three ways to assess the fairness of a division.

First, it is possible to aggregate the valuations in each family to a single family valuation. Following the utilitarian tradition (Bentham, 1789), the family-valuation can be defined as the sum or (equivalently) the arithmetic average of the valuations of all family members. We call a division average-fair if it is fair according to these family valuations. In particular, a division is average/FS if every family receives a share with an average value (averaged over all family members) of at least of its average value of the entire endowment.

By this definition, the family-division problem is easy to solve. Since the average of measures is itself a measure, each family can be represented by a single agent, and the problem reduces to fair division among the representatives. Classic results imply that average/FS allocations exist (Section 3).

Average fairness makes sense only when the numeric values of the agents’ valuations are meaningful and they are all measured in the same units, e.g. in dollars (see chapter 3 of Moulin (2004) for some real-life examples of such situations). However, if the valuations represent individual happiness measures that cannot be put on a common scale, then their sum is meaningless, and other fairness criteria should be used.

A second option is to require that all members of every family agree that the division is fair. We call a division unanimous fair if it is fair according to every individual valuation. In particular, a division is unanimous/FS if every agent values his/her family’s share as at least of the total value. The advantage of this definition is that it does not need to assume that all valuations share a common scale. Even though it is a very strong requirement, we prove that unanimous/FS allocations exist (Section 4).

A disadvantage of unanimous fairness, compared to average fairness, is that unanimously-fair divisions might be highly fractioned. As an illustration, when an interval is divided, there always exists an average/FS division that is also connected — the share of each family is a single interval (Section 3). However, we prove that there might not exist connected unanimous/FS divisions. Moreover, in some cases, the number of intervals in any unanimous/FS division might be at least — the number of individual agents (Section 4). When the number of agents is large, as in the case of dividing land among states, such divisions might be impractical.

In democratic societies, decisions are almost never accepted unanimously. In fact, when the number of citizens is large, it may be impossible to attain unanimity on even the most trivial issue. The simplest decision rule in such societies is the majority rule. Inspired by this rule, we suggest a third fairness criterion. We call a division democratic fair if at least half the citizens in each family consider it fair. In particular, a division is democratic/FS if at least half the agents in each family value their family’s share as at least of the total.

Democratic fairness can be justified by the following process. After a division is proposed, each group conducts a referendum in which each citizen approves the division if he/she feels that the division is fair. The division is implemented only if, in every group, at least half of its members approve it.

Democratic/fairness combines some advantages of average/fairness and unanimous/fairness. It is similar to unanimous/fairness in that it does not need to assume that all valuations share a common scale. When there are families, it is similar to average/fairness in that it can be satisfied with connected pieces — there always exists a democratic/FS division in which each family receives a single connected piece. An additional advantage of democratic fairness in this case is that it can be computed efficiently (Section 5).333In contrast, average-fair and unanimous-fair allocations cannot be computed by any finite protocol. See Remark 1.

Although democratic/fairness might leave up to half the citizens unhappy, this may be unavoidable in real-life situations. This is understandable in light of Winston Churchil’s dictum: “democracy is the worst form of government, except all the others that have been tried”.444 A fourth fairness criterion that could be considered is individual fairness. In particular, an allocation is individually-FS if the allocation admits a refinement , where for each family , , such that for each agent , . Individually-fair allocations always exist and can be found by using any classic fair division procedure on the individual agents, disregarding their families. Individual-fairness makes sense if, after the division of the land among the families, each family intends to further divide its share among its members. However, often this is not the case. When an inherited land-estate is divided between two families, the members of each family intend to live and use their entire share together, rather than dividing it among them. Therefore, the happiness of each family member depends on the entire value of his family’s share, rather than on the value of a potential private share he would get in a hypothetic sub-division.

While the geometric requirement of having a connected division is practically important, an even more important requirement from an economic perspective is Pareto-efficiency. We prove that all three variants of fair-share are compatible with Pareto-efficiency (Section 6).

A second fairness criterion that is very common in economics is no envy (NE). In the context of individual agents, it means that each agent receives a share that is at least as good as the share of any other agent, according to the first agent’s individual valuation.555 The condition of receiving at least as much as any other agent was introduced by Gamow and Stern (1958) and Foley (1967). Economists often call it no envy (Bogomolnaia et al, 2017). Computer scientists often call it envy-freeness (Robertson and Webb, 1998). In the context of families, three variants of NE can be defined analogously to the three variants of fair-share: average/NE, unanimous/NE and democratic/NE (Section 7).

From a geometric perspective, these three variants behave similarly to their FS counterparts, that is:

  • Connected average/NE allocations always exist;

  • Connected unanimous/NE allocations are not guaranteed to exist even for two families;

  • Connected democratic/NE allocations are guaranteed to exist for two but not for three or more families.

However, from an economic perspective, NE behaves differently:

  • Pareto-efficient average/NE allocations always exist;

  • Pareto-efficient unanimous/NE allocations are guaranteed to exist for two but not for three or more families;

  • Pareto-efficient democratic/NE allocations are guaranteed to exist for two but not for five or more families (we do not know whether they always exist with three or four families).

The paper is organized as follows. Most of the paper focuses on the fair-share criterion. Section 2 formally presents the model. Sections 3, 4 and 5 study average, unanimous and democratic fair-share divisions respectively. We study this criterion both for families with equal entitlements and for families that have different entitlements to the resource.

Section 6 studies the three variants of fair-share in combination with Pareto-efficiency. Section 7 studies family fairness based on the no-envy criterion, explaining the differences between the results for fair-share and for no-envy. Finally, Section 8 compares our work to previous and ongoing related work.

2 Model and Notation

2.1 Resource and agents

In the usual cake-cutting setting, there is a resource (“cake”) that has to be divided. For simplicity it is assumed that is an interval in . A realistic example of such a resource is time: consider a conference room that can host a single meeting at a time. It is active between 8:00 and 20:00, and this time-interval must be divided among all those who want to use the room. Another realistic example is the shoreline of a sea or a river: while usually not a straight line, it can be easily mapped to an interval.

There is a set of agents . Each agent has a value measure , defined on the Borel subsets of . The are assumed to be nonatomic, so that all singular points have a value of 0 to all agents. As the term measure implies, the are additive — the value of a union of two disjoint pieces is the sum of the values of the pieces. Such value functions can be viewed as having a ”constant marginal utility” property (Chambers, 2005). The value measures are normalized such that .

2.2 Families and entitlements

In our setting, there is a set of families . We use the term “family” to emphasize that the partition of agents to groups is fixed in advance and cannot be modified during the division process.

The number of agents in is denoted . Each agent is a member of exactly one family , so .

For each family , there is a positive weight representing the entitlement of this family. The sum of all weights is one: .

In the simplest setting, the families have equal entitlements, i.e, for each : . Equal entitlements make sense, for example, when siblings inherit their parents’ estate. While an heir will probably like to take his family’s preferences into account when selecting a share, each heir is entitled to of the estate regardless of the size of his/her family.

In general, each family may have a different entitlement. The entitlement of a family may depend on its size but may also depend on other factors. For example, consider several families who jointly buy a vacation apartment. The apartment can host one family at a time, so the families have to divide the year (a time-interval) among them. The entitlement of each family naturally depends on the amount of money it contributed to the purchase, rather than on the family’s size.666 See Cseh and Fleiner (2017) for a recent account of fair division among individual agents with different entitlements.

2.3 Allocations and components

An allocation is a vector of pieces, , one piece per family, such that the are pairwise-disjoint and .

Each piece is a finite union of intervals. We denote by the number of connected components (intervals) in the piece , and by the total number of components in the allocation X, i.e:

Ideally, we would like that each piece be connected, i.e, and . This requirement is especially meaningful when the divided resource is a time-interval or a land-resource (e.g. a river-bank), since a contiguous piece of time or land is much easier to use than a collection of disconnected patches.

However, we will show that a fair division with connected pieces is not always possible.777 This impossibility appears not only in our one-dimensional theoretic model but also in practical, two-dimensional land division situations. A striking example was the India-Bangladesh border. According to Wikipedia page India–Bangladesh enclaves, up to 2015, “Within the main body of Bangladesh were 102 enclaves of Indian territory, which in turn contained 21 Bangladeshi counter-enclaves, one of which contained an Indian counter-counter-enclave… within the Indian mainland were 71 Bangladeshi enclaves, containing 3 Indian counter-enclaves”. Another example is Baarle-Hertog — a Belgian municipality made of 24 separate parcels of land, most of which are exclaves in the Netherlands. For more details and examples see the Wikipedia page List of enclaves and exclaves. We are grateful to Ian Turton for the references. In case a division with connected pieces is not possible, it is still desirable that the number of connectivity components — — be as small as possible. When dividing an interval, the components are sub-intervals and their number is one plus the number of cuts. Hence, the number of components is minimized by minimizing the number of cuts (Robertson and Webb, 1995; Webb, 1997; Shishido and Zeng, 1999; Barbanel and Brams, 2004, 2014). In a realistic, 3-dimensional world, the additional dimensions can be used to connect the components, e.g, by bridges or tunnels. Still, it is desirable to minimize the number of components in the original division in order to reduce the number of required bridges/tunnels.888 The goal of minimizing the number of components is pursued not only in cake-cutting papers but also in real-life politics. Going back to India and Bangladesh, after many years of negotiations they finally started to exchange most of their enclaves during the years 2015-2016. This reduced the number of components from 200 to a more reasonable number.

2.4 Fairness criteria

We first define the family-valuation functions:

Now, an allocation is called:

average/FS
unanimous/FS
democratic/FS

A property of an allocation is called feasible if for every families and agents there exists an allocation satisfying this property. Otherwise, the property is called infeasible. In the following sections we will study the feasibility of the above fairness criteria.

Note that unanimous/FS obviously implies both average/FS and democratic/FS. The other two do not imply each other, as shown in the following example.

Example 1

Consider an interval consisting of four sub-intervals. It has to be divided between two families: (1) {Alice,Bob,Chana} and (2) {David,Esther,Frank}. The families have equal entitlements, i.e, . Each member’s valuation of each sub-interval is shown in the table below:

Alice 60 30 3 3
Bob 50 40 3 3
Chana 10 80 3 3
David 3 3 60 30
Esther 3 3 60 30
Frank 3 3 0 90

Note that the value of the entire interval is 96 according to all agents. Therefore, FS implies that each family should get a value of at least 48.

If the two leftmost subintervals are given to family 1 and the two rightmost subintervals are given to family 2, then the division is unanimous/FS, since each member of each family feels that his family’s share is worth 90. This division is also, of course, average/FS and democratic/FS.

If only the single leftmost subinterval is given to family 1 and the other three are given to family 2, then the division is still democratic/FS, since Alice and Bob feel that their family received more than 48. However, Chana feels that her family received only 10, so the division is not unanimous/FS. Moreover, the division is not average/FS since the average valuation of family 1 is only (60+50+10)/3=40.

If the three leftmost subintervals are given to family 1 and only the rightmost one is given to family 2, then the division is average/FS, since family 2’s average valuation of its share is (30+30+90)/3=50. However, it is not unanimous/FS nor even democratic/FS, since David and Esther feel that their share is worth only 30. ∎

3 Average fairness

With average fairness, the family cake-cutting problem can be reduced to the classic problem of cake-cutting among individuals. This gives the following results.

Theorem 3.1

(a) When families have equal entitlements, average/FS with connected pieces (and components) is feasible.

(b) When families have different entitlements, average/FS with connected pieces is infeasible. Moreover, at least components may be required for an average/FS allocation.

(c) When families have different entitlements, average/FS with at most components is feasible.

Proof

The positive results — parts (a) and (c) — are based on the following reduction. For each family , define a representative agent whose valuation is the function defined in Subsection 2.4 above. Note that, since the are all nonatomic measures, the family-valuations are nonatomic measures too. By classic results (Steinhaus, 1948; Even and Paz, 1984), when there are agents with equal entitlements, there always exists a connected FS division. As shown in a recent technical report (Segal-Halevi, 2018), when there are agents with different entitlements, there always exists a FS allocation with at most cuts, where rounded up to the nearest power of two. These cuts create components. By definition, such a division is an average/FS division among the families.

The negative result (b) follows immediately from an identical negative result for individual agents (Segal-Halevi, 2018), by considering one-member families. ∎

Remark 1

Fairness for individuals and average-fairness for families are equivalent only from an existential perspective; from a computational perspective they are quite different. FS division among individual agents with equal entitlements can be found by asking the agents queries (Even and Paz, 1984). However, average-FS division cannot be found using a finite number of queries even when there are families. To see this, suppose there are two identical families, each of which has two different members with valuations and . Then, a division is average-FS if and only if . Therefore, finding an average-FS allocation is equivalent to finding a subset for which the sum equals 1. However, queries can only be sent to individual agents, and it might be impossible to find such a subset using a finite number of queries to the agents. We omit the details here since our focus is on existence. See the accompanying technical report (Segal-Halevi and Nitzan, 2016) for more details.

4 Unanimous fairness

Before presenting our results, we note that unanimous/FS, like average/FS, can also be defined using family-valuation functions. Define:

Then, a division is unanimous/FS if and only if:

However, in contrast to the functions defined in Section 3, the functions are in general not additive. For example, suppose is an interval with three subintervals and a family has the following valuations:

Alice 1 1 1
Bob 0 2 1
Chana 0 1 2
0 1 1

While the individual valuations are additive, is not additive (it is not even subadditive). Therefore, the classic results we used in Theorem 3.1 are inapplicable here, and different techniques are needed.

4.1 Exact division

Initially, we assume that the entitlements are equal, i.e: for all . We relate unanimous/FS to the problem of finding an exact division:999The definition uses capital and to distinguish the parameters of exact division from the parameters of unanimous-fair division..

Definition 1

Exact is the following problem. Given agents and an integer , divide to pieces, such that each of the agents assigns exactly the same value to all pieces:

From an economic perspective, there is little intrinsic value in the concept of exact division. However, in this section we will prove that it is closely linked to the concept of unanimously-fair division. In fact, we will prove that the existence a solution to each of these problems implies a solution to the other problem.

Denote by UnanimousFS the problem of finding a unanimous/FS division when there are agents grouped in families with equal entitlements.

4.2 UnanimousFS Exact

Lemma 1

For every pair of integers , a solution to UnanimousFS implies a solution to Exact .

Proof

Given an instance of Exact ( agents and a number of required pieces), create families. Each of the first families contains agents with the same valuations as the given agents. The -th family contains a single agent whose valuation is the average of the given valuations:

The total number of agents in all families is . Use UnanimousFS to find a unanimous/FS division, . By definition of unanimous fairness, for each agent in family : .

By construction, each of the first families has an agent with valuation . Hence, all agents value each of the first pieces as at least and:

Hence, by additivity, every agent values the -th piece as at most :

The piece is given to the agent with value measure , so by fair-share: . By construction, is the average of the . Hence:

Again by additivity:

Hence, necessarily:

So we have found an exact division and solved Exact as required. ∎

Alon (1987) proved that for every and , an Exact division might require at least components. Combining this result with the above lemma implies the following negative result:

Theorem 4.1

For every , let . A unanimous/FS division for agents grouped into families might require at least components.

In particular, unanimous/FS with connected pieces is infeasible.

4.3 Exact UnanimousFS

Lemma 2

For every pair of integers , a solution to Exact implies a solution to UnanimousFS for any grouping of the agents to families.

Proof

Suppose we are given an instance of UnanimousFS, i.e, agents in families. Select agents arbitrarily. Use Exact to find a partition of to pieces, such that each of the agents values each of these pieces as exactly . Ask the -th agent to choose a favorite piece; by the pigeonhole principle, this value is worth at least for that agent. Give that piece to the family of the -th agent. Give the other pieces arbitrarily to the remaining families. The resulting division is unanimous/FS. ∎

Alon (1987) proved that for every and , Exact has a solution with at most components (at most cuts). Combining this result with the above lemma implies the following positive result:

Theorem 4.2

Given agents in families with equal entitlements, a unanimous/FS division with components is feasible.

For families, the number of components in Theorem 4.2 is , which matches the lower bound of Theorem 4.1. For families, the number of components can be made smaller, as explained below.

4.4 Less components: equal entitlements

The purpose of this subsection is to find a unanimous/FS allocation with less components than the guarantee of Theorem 4.2, when all families have equal entitlements.

We start with an example. Assume there are families. By Theorem 4.2, using cuts, can be divided to 4 subsets which are considered equal by all members. But for a unanimous/FS division, it is not required that all members think that all pieces are equal, it is only required that all members believe that their family’s share is worth at least . This can be achieved as follows:

  • Divide to two subsets which all agents value as exactly . This is equivalent to solving Exact, which by Alon (1987), can be done with at most cuts. Call the two resulting subsets West and East.

  • Assign arbitrary two families to West and the other two families to East. Mark by the total number of members in the families assigned to West and by the total number of members assigned to East.

  • Divide the West to two pieces which all agents value as exactly ; this can be done with cuts. Give a piece to each family. Divide the East similarly using cuts.

The first step requires cuts and the second step requires cuts too. Hence the total number of cuts required is only , rather than .

In fact, two cuts can be saved in each step by excluding two members (from two different families) from the exact division. These members will not think that the division is equal, but they will be allowed to choose the favorite piece for their family. Thus only cuts are required. A simple inductive argument shows that whenever is a power of 2, cuts are required.

When is not a power of 2, a result by Stromquist and Woodall (1985) can be used. They prove that, for every fraction , it is possible to cut a piece of such that all agents agree that its value is exactly using at most cuts.101010They prove that, if is a circle, the number of connected components is . Hence, the number of cuts is . This is also true when is an interval, although the number of connected components in this case is . This can be used as follows:

  • Select integers such that .

  • Apply Stromquist and Woodall (1985) with : using cuts, cut a piece that agents value as exactly . This means that these agents value the other piece, , as exactly .

  • Let the -th agent choose a piece for his family; assign the other families arbitrarily such that families are assigned to piece and the other families to piece .

  • Recursively divide piece to its families and piece to its families.

After a finite number of recursion steps, the number of families assigned to each piece becomes 1 and the procedure ends. The number of cuts in each level of the recursion is at most . The depth of recursion can be bounded by by dividing to halves (if it is even) or to almost-halves (if it is odd; i.e. take and ). Hence:

Theorem 4.3

Given agents in families with equal entitlements, a unanimous/FS division with components is feasible.

Note that Theorem 4.2 and Theorem 4.3 both give upper bounds on the number of components required for unanimous/FS. The bound of Theorem 4.2 is stronger when is small and the bound of Theorem 4.3 is stronger when is large.

4.5 Less components: different entitlements

The purpose of this subsection is to find a unanimous/FS allocation with less components than the guarantee of Theorem 4.2, when families may have different entitlements.

When families have different entitlements, the procedure of the previous subsection cannot be used. We cannot let the -th agent select a piece for his family, since the pieces are different. For example, suppose there are two families with entitlements . We can divide to two pieces such that agents value as 1/3 and as 2/3. So all of them agree that should be given to family 1 and should be given to family 2. But, the -th agent might select the wrong piece for his family. Therefore, the procedure should be modified as follows.

  • Select an integer .

  • Divide the families to two subsets: and .

  • Apply Stromquist and Woodall (1985) with : using cuts, cut a piece which all agents value as exactly . This means that all agents value the other piece, , as exactly .

  • Recursively divide piece to and piece to .

Here, the number of cuts in each level of the recursion is at most . The depth of recursion can be bounded by by choosing (if is even) or (if is odd). Hence:

Theorem 4.4

Given agents in families with different entitlements, a unanimous/FS division with components is feasible.

In concluding the analysis of unanimous/FS, recall that, even for families, unanimous/FS is as difficult as exact division and might require the same number of components — . In the worst case, we might need to give a disjoint component to each member, which negates the concept of division to families. Therefore we now turn to the analysis of an alternative fairness criterion that yields more useful results.

5 Democratic fairness

Like unanimous/FS (Section 4), democratic/FS can also be defined using family-valuation functions. Define:

A division is democratic/FS if and only if:

However, the functions are not additive,111111See the example in the beginning of Section 4. In that example is identical to . so again the classic results referred to in Theorem 3.1 are inapplicable.

5.1 Two families: a division procedure

We start with a positive result for two families with equal entitlements, which shows that democratic/FS is substantially easier than unanimous/FS.

Theorem 5.1

When there are families with equal entitlements, democratic/FS with connected pieces is feasible. Moreover, it can be found by an efficient algorithm.

Proof

The proof is given by Algorithm 1. It finds a democratic/FS division between two families. For each family, a location is calculated such that, if is cut at , half the members value the interval as at least and the other half value the interval as at least . Then, is cut between the two family medians, and each family receives the piece containing its own median. By construction, at least half the members in each family value their family’s share as at least 1/2, so the division is democratic/FS. Each family receives a single connected piece.

INPUT:

- , which is assumed to be the unit interval .

- additive agents, all of whom value as 1.

- A grouping of the agents to families, .

OUTPUT:

A democratic/NE division of to pieces.

ALGORITHM:

- Each agent marks an such that .

- For each family , find the median of its members’ marks: . Find the median of the family medians: .

- If then give to and to .
Otherwise give to and to .

Algorithm 1 Finding a democratic/NE division for two families

Unfortunately, this positive result is not applicable when there are more than two families, as shown in the following subsection.

5.2 Three or more families: an impossibility result

This subsection presents a lower bound on the number of components required for a democratic-fair division. The lower bound holds not only for FS but even for a much weaker fairness notion called positivity.

Given a specific division of among families, define a zero agent as an agent who values his family’s share as 0 and a positive agent as an agent who believes his family received a share with a positive value. Note that FS implies positivity but not vice-versa. The following lower bound holds even for positivity, hence it also holds for FS.

Lemma 3

Assume there are agents, divided into families with members in each family. To guarantee that at least members in each family are positive, the total number of components may need to be at least:

Proof

Number the families by and the members in each family by . Assume that is the interval . In each family , each member wants only the following interval: . Thus there is no overlap between desired intervals of different members. The table below illustrates the construction for . The families are {Alice,Bob,Chana} and {David,Esther,Frank}:

Alice 1 0 0 0 0 0
Bob 0 0 1 0 0 0
Chana 0 0 0 0 1 0
David 0 1 0 0 0 0
Esther 0 0 0 1 0 0
Frank 0 0 0 0 0 1

Suppose the piece (the piece given to family ) is made of components. We can make members of positive using intervals of positive length inside their desired areas. However, if , we also have to make the remaining members positive. For this, we have to extend intervals to length . Each such extension totally covers the desired area of one member in each of the other families. Overall, each family creates zero members in each of the other families. The number of zero members in each family is thus . Adding the members which must be positive in each family, we get the following necessary condition: . This is equivalent to:

The total number of components is , which is at least equal to the expression stated in the Lemma. ∎

In other words, if we want at least a fraction of the members in each family to have a non-zero utility, the number of components might have to be at least:

In a unanimous/FS division , so the number of components is at least , which coincides with the lower bound of Theorem 4.1. In a democratic/FS so we get the following negative result:

Theorem 5.2

In a democratic/FS division with agents grouped into families, the number of components may need to be at least

Note that for the lower bound is 0, and indeed we already saw that in this case a connected allocation is feasible.

Lemma 3 has another interesting corollary. Suppose we have but still insist that the division be connected. We already know that we cannot guarantee that 1/2 the agents in each group be positive. But there is an even stronger impossibility result.

Theorem 5.3

When dividing a cake among families, for every constant fraction , it may be impossible to find a connected division where at least a fraction of the agents in each group are positive.

Proof

By Lemma 3 the number of components should be at least . Since , for sufficiently large the number of components is larger than , so the division cannot be connected.∎

The fraction in Theorem 5.3 is tight:

Theorem 5.4

For every integer , there exists a connected division among families, that is FS for at least of the members in each family.

Proof

The Dubins-Spanier moving-knife protocol (Dubins and Spanier, 1961) can be adapted to families as follows. A knife moves continuously over the cake from left to right. Whenever in a certain family at least of its members believe that the cake to the left of the knife is worth at least , they shout “stop”, the cake is cut at the knife location, and the shouting family receives the cake to its left (the division is now FS for the members in this family). In the other families, at least of the members believe the remaining cake is worth at least of its original value; by dividing the remaining cake recursively using the same procedure, they division is FS for of their members too. ∎

5.3 Three or more families: positive results

Suppose we do want a democratic/FS division for three or more families. How many components are sufficient?

As a first positive result, we can use Theorem 4.4, substituting instead of : select half of the members in each family arbitrarily, then find a division which is unanimous/FS for them while ignoring all other members. This leads to:

Theorem 5.5

Given agents in families with different entitlements, democratic/FS with components is feasible.

However, for families with equal entitlements we can do much better. Algorithm 2 generalizes Algorithm 1 for any number of families:

INPUT:

- , which is assumed to be the unit interval .

- additive agents, all of whom value as 1.

- A grouping of the agents to families, .

OUTPUT:

A democratic/FS division of to pieces.

ALGORITHM:

Step 1: Halving

- Each agent selects an such that (this means if is even and if is odd). Note: .

- For each family , find the median of its members’ selections: .

- Order the families in increasing order of their medians. Find the median of the family-medians: . Cut at .

Step 2: Sub-division

- Define the western families as the with . Let be the total number of members in these families. Divide the interval among the western families using UnanimousFS.

- Similarly, define the eastern families as the with . There are such families. Let be their total number of members. Divide the interval among the eastern families using UnanimousFS.

Algorithm 2 Finding a democratic/FS division for families.

The algorithm works in two steps.

Step 1: Halving. For each family, a location is calculated such that, if is cut at , half the family members value the interval as at least and the other half value the interval as at least . Then, is cut in — the median of the family medians. The “western families” — for which — are assigned to the western interval of . By construction, at least half the members in each of the western families value as at least . We say that these members are “happy”. Similarly, the eastern families — for which — are assigned to the eastern interval ; at least half the members in each of these families are “happy”, i.e, value the interval as at least .

If there are only two families (), then we are done: there is exactly one western family and one eastern family ( ). For each family , at least half the members of each family value their family’s share as at least . Hence, the allocation of to family is democratic/FS.

If there are more than two families (), an additional step is required.

Step 2: Sub-division. Each of the two sub-intervals should be further divided to the families assigned to it. In each family , at least members are happy. So for each , select exactly members who are happy. Our goal now is to make sure that these agents remain happy. This can be done using a unanimous/FS allocation, where only happy members in each family (hence members overall) are counted.

The unanimous/FS allocation guarantees that every western-happy-member believes that his family’s share is worth at least . Similarly, every eastern-happy-member believes that his family’s share is worth at least . Hence, the resulting division is democratic/FS.

We now calculate the number of components in the resulting division. One cut is required for the halving step. For the unanimous/FS division of the western interval, the number of required cuts is at most by Theorem 4.2, and at most by Theorem 4.3. Similarly, for the eastern interval the number of required cuts is at most the minimum of and . The total number of cuts is thus at most and at most . The total number of components is larger by one. To conclude:

Theorem 5.6

Given agents in families with equal entitlements, democratic/FS is feasible, and the number of required components is at most:

5.4 Comparison and Open Questions

Table 1 compares the three variants of FS, focusing on families with equal entitlements. Recall that is the total number of agents in all families.

Fairness #Families () #Connectivity Components
Lower Upper
average/FS (Sec. 3) (connected)
unanimous/FS 2
(Sec. 4)
democratic/FS 2 2 2 (connected)
(Sec. 5)
Table 1: Number of components required for a fair-share division in various situations.

The case of families is well-understood. The results for all fairness criteria are tight: by all fairness definitions, we know that a fair division exists with the smallest possible number of connectivity components.

The case of families opens some questions:

  • Is unanimous/FS with components feasible for all ? (particularly, with families, is the number of required components as in the lower bound, or as in the upper bound?).

  • Is democratic/FS with components feasible for all ? (particularly, with families, is the number of required components as in the lower bound, or as in the upper bound?).

The case of different entitlements is much less understood even for individual agents (Segal-Halevi, 2018), let alone for families.

What fairness notion is the most practical? The table shows that it depends on the total number of agents (). When is small (as is common when dividing an estate among heirs), it is reasonable to try to attain a unanimously-fair division. However, when is large (as is common when dividing disputed lands among states), unanimous fairness quickly becomes impractical, as the number of components might grow linearly with . In this case, we must settle for a weaker fairness criterion. When , we can find a democratically-fair allocation that is also connected. When , democratic fairness too might be impractical, and we may have to settle for average-fairness.

6 Pareto-efficiency

So far, we studied the compatibility of fairness criteria with a geometric requirement — reducing the number of connectivity components. In this section we replace the geometric requirement with an economic requirement — Pareto efficiency. An allocation is called Pareto-efficient (PE) if no other allocation is weakly better for all individual agents and strictly better for some individual agents. Fortunately, PE is compatible even with the strongest variant of the fair-share criterion:

Theorem 6.1

There always exists an allocation that is both PE and unanimous/FS (hence also average/FS and democratic/FS).

Proof

We use a famous theorem of Dubins and Spanier (1961), which is a special case of a measure-theoretic theorem by Dvoretzky et al (1951).

For every partition of to pieces, let be its value-matrix — an -by- matrix where . Let be the set of all matrices that correspond to such partitions:

Theorem 1 of Dubins and Spanier (1961) implies that the set is compact.

Define a second set of matrices representing the unanimous/FS condition:

Finally, define . This set represents all value-matrices of allocations of that are unanimous/FS. By Theorem 4.2, is non-empty. Since is compact and is closed, their intersection is compact.

Define the following function :

This is a continuous function, so it has a maximum point in ; let’s call it . This matrix corresponds to an allocation that maximizes, among all unanimous/FS allocations, the product of valuations of all agents: . This product is strictly increasing with the value of each agent , so the allocation is Pareto-efficient in the set . Since every Pareto-improvement of an allocation in is also in , the allocation is also Pareto-efficient in general. ∎

7 No Envy

So far, we used fair-share (FS) as our individual fairness criterion. Another criterion that is very common in economics is no-envy (NE). We study this criterion for families with equal entitlements.

Analogously to the definitions in subsection 2.4, we call an allocation :

average/NE
unanimous/NE
democratic/NE

With individual agents, it is well known that NE implies FS (with equal entitlements). With two individual agents, NE and FS are equivalent. The same implications are true with families. Each variant of NE implies the corresponding variant of FS.121212 Suppose an agent thinks the division is not FS. This means that . But, the sum of all weights equals 1 which equals the sum of the values of all pieces. Therefore there must be some for which: . Hence, , so agent thinks the division is not NE. When there are families, each variant of NE is equivalent to the corresponding variant of FS. 131313 Suppose an agent thinks the division is FS. This means that . By additivity, for the other family , . Hence , so agent thinks the division is NE.

Most of our results for FS with equal entitlements are also valid for NE. For average/NE, we can use classic results proving the existence of NE allocations with connected pieces among individual agents (Stromquist, 1980; Su, 1999). Applying the same reduction as in Theorem 3.1 we get:

Theorem 3.1

For any families, average/NE with connected pieces is feasible.

Since NE implies FS, the negative results are still valid:

Theorem 4.1

For every , let . A unanimous/NE division for agents grouped into families might require at least components.

Some positive results remain valid too. Lemma 2 is based on an exact division. Therefore it holds, with the same proof, even if we replace unanimous/FS with unanimous/NE. Therefore:

Theorem 4.2

Given agents in families, a unanimous/NE division with components is feasible.

However, the recursive-halving procedure of Theorem 4.3 cannot be used here. Suppose we divide to two subsets, West and East, which all agents value as exactly . Then, we assign arbitrary families to West and the other families to East. We find an exact division of the West among the western families and an exact division of the East among the eastern families. While this division satisfies FS, it does not satisfy NE, since the agents in the west might envy families in the east and vice versa. Therefore, while the number of components required for unanimous/FS division is in , the best we can currently say about the number of components required for unanimous/NE is that it is in .

With two families FS implies NE, so the following positive result holds:

Theorem 5.1

When there are families, democratic/NE with connected pieces is feasible. Moreover, it can be found by an efficient algorithm.

Similarly, the negative results for democratic/FS in Theorems 5.2 and 5.3 are equally valid for democratic/NE. The positive result of Theorem 5.4 holds for NE too:

Theorem 5.4

For every integer , there exists a connected division among families, that is NE for at least of the members in each family.

Proof

Su (1999) presents a procedure (attributed to Simmons) for finding a connected NE division among individual agents. It is based on presenting various connected partitions to the agents, and asking each agent which of the pieces is the best. He proves that there exists a partition in which each agent gives a different answer; that partition corresponds to a no-envy allocation. He also shows a procedure for finding a sequence of partitions that converges (after possibly infinitely many steps) to that no-envy allocation.

The Simmons-Su procedure can be adapted to families in the following way. Whenever a family is asked “which of the pieces is better?”, it answers by doing a plurality voting among its members. Then, in the final division, each family receives a piece that is considered the best by a plurality of its members, which is at least a fraction of its members. Therefore, at least of each family’s members feel that the allocation has no envy. ∎

Theorem 5.6 about democratic/FS does not hold as-is for democratic/NE, but it can be adapted by adapting Algorithm 2. Step 1 — the halving step — remains the same. Step 2 — the subdivision step — should be modified to use an exact division, as follows:

- Using Exact , find an exact division of the interval into pieces, such that all happy agents find the pieces equal. Assign these pieces to the western families — the with .
- Using Exact , find an exact division of the interval into pieces, such that all happy agents find the pieces equal. Assign the pieces to the eastern families with .

The halving step requires a single cut. The two exact divisions require cuts. Therefore the total number of components is :

Theorem 5.6

Given agents in families with equal entitlements, democratic/NE with at most components is feasible.

Table 2 summarizes our results for no-envy division and shows some remaining gaps.


Fairness
#Families () #Connectivity Components
Lower bound Upper bound
average/NE Any (connected)
2
unanimous/NE 3
(Sec. 4) 4
Any
2 2 2 (connected)
democratic/NE 3
(Sec. 5) 4
Any
Table 2: Number of components required for a no-envy division in various situations.

We now consider the combination of no-envy with Pareto-efficiency. Some of our positive results from Section 6 are still valid:

Theorem 6.1

(a) With families, there always exists an allocation that is both PE and unanimous/NE (hence also average/NE and democratic/NE).

(b) With any number of families, there always exists an allocation that is PE and average/NE.

Proof

(a) With families, NE and FS are equivalent, so this follows directly from Theorem 6.1.

(b) We use the same reduction as in Theorem 3.1 and the same compactness argument as in Theorem 6.1. For each family , define a representative agent whose valuation is . There exists an allocation that maximizes the product of valuations of the representatives: .

Segal-Halevi and Sziklai (2018, Section 5) prove, in the setting of cake-cutting among individuals, that every allocation maximizing the product of values has no envy. Therefore, in the allocation , there is no envy among the representatives. By definition of average/NE, is an average/NE allocation among the families.

The product is strictly increasing with the value of each individual agent . Therefore, the allocation maximizing this product is Pareto-efficient. ∎

In contrast to these positive results, Pareto-efficiency is incompatible with unanimous/NE and democratic/NE.

The incompatibility between PE and unanimous/NE appears even when we take a minimal step forward from the case of two families: there are three families, only one of which is a couple and the other two are singles.

Theorem 7.1

With three or more families, there might be no allocation that is both PE and unanimous/NE.

Proof

The proof is based on an example used by Bade and Segal-Halevi (2018) in the context of fair division of homogeneous goods. is an interval composed of two sub-intervals and of length 1. has to be divided among three families — a couple and two singles — with the following valuations:

Y Z
Alice 1 1
George 7 1
Bob 2 1
Esther 5 1

Suppose that we have a unanimous/NE allocation of among the three families. Denote by the lengths of given to Alice+George, and similarly are the lengths given to Bob and Esther. Then:

(George does not envy Bob)
(Bob does not envy George)
(from the above inequalities)
(Alice does not envy Bob)
(Bob does not envy Alice)
(from the above inequalities)
(from * and **)
(Bob and Alice+George do not envy)

We proved that, in any unanimous/NE allocation, the share given to Alice+George is identical to the share given to Bob (i.e, the same lengths of both subintervals). The proof does not depend on the exact valuation functions — it only depends on the fact that , i.e, Bob’s valuation of is strictly between Alice’s and George’s valuations. Hence, exactly the same proof works for Esther, i.e: and . Therefore, the shares given to all three families are identical.

We now prove that this allocation cannot be PE. We consider three cases.

  • Case 1: . Then also so remains unallocated and the allocation is not PE.

  • Case 2: . Then also so remains unallocated and the allocation is not PE.

  • Case 3: and are positive. Let . Suppose that Bob gives of his to Esther, and gets in exchange of her . Then, Bob’s value increases by ; Esther’s value increases by ; and the values of Alice and George are unchanged. This means that the original allocation was not Pareto-efficient. ∎

Weakening unanimous/NE to democratic/NE does not help when there are 5 or more families.

Theorem 7.2

With five or more families, there might be no allocation that is both PE and democratic/NE.

Proof

The proof is based on an extension of the example of Theorem 7.1, where there are five families — one triplet and four singles — with the valuations:

Y Z
Alice 1/4 1
Dina