On Democratic Fairness for Groups of Agents

09/08/2017 ∙ by Erel Segal-Halevi, et al. ∙ 0

We study the problem of allocating indivisible goods to groups of interested agents in a fair manner. The agents in the same group share the same set of goods even though they may have different preferences from one another. Previous work on this model has shown positive results that hold either asymptotically or for groups with a small number of agents. Using the concept of democratic fairness, which aims to satisfy a certain fraction of the agents in each group, we provide fairness guarantees that hold for general instances. In particular, we show that for two groups with any number of agents, there exists an allocation that is envy-free up to one good and yields at least half of the maximin share for at least half of the agents in each group.

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1 Introduction

Fair division is the study of how to allocate resources among agents with different preferences so that agents perceive the resulting allocation as fair. This problem occurs in a wide range of situations, from negotiating over international interests and reaching divorce settlements [[6]] to dividing household tasks and sharing apartment rent [[10]].

Two kinds of fairness criteria are common in the literature. The first, envy-freeness (EF), means that each agent finds her share at least as good as the share of any other agent. When allocating indivisible goods, envy-freeness is sometimes unattainable (consider two agents quarreling over a single good), so it is often relaxed to envy-freeness up to one good (EF1), which is always attainable (LiptonMaMo04, ; Budish11, ). The second kind, maximin share fairness, means that each agent finds his share at least as good as his maximin share (MMS), which is the best share he can secure by dividing the goods into parts and getting the worst part, where is the number of agents. Again, with indivisible goods MMS fairness cannot be guaranteed, but at least a constant fraction of the MMS can (ProcacciaWa14, ).

Most works on fair division involve individual agents, each of whom has individual preferences. But in reality, resources often have to be allocated among groups of agents, such as families or states. A good allocated to a group is shared among the group members and all of them derive full utility from the good. For example, when dividing real estate among families, all members of a family enjoy their allocated house and backyard. In international negotiations, the divided rights and settled outcomes are enjoyed by all citizens of a country. When resources are allocated between different buildings of a university, all occupants of a building benefit from the whiteboards and open space allocated to their building. However, different group members may have different preferences. The same share can be perceived as fair by one member and unfair by another member of the same group. Ideally, we would like to find an allocation considered fair by all agents in all groups. However, two recent works show that this “unanimous fairness” might be too strong to be practical.

(a) Suksompong Suksompong18 shows that when allocating indivisible goods among groups, there might be no allocation that is unanimously EF1. Moreover, there might be no division that gives all agents a positive fraction of their MMS. This impossibility occurs even for two groups of three agents.

(b) Segal-Halevi and Nitzan SegalhaleviNi15 show that when allocating a divisible good (“cake”) among groups, there might be no division that is unanimously envy-free and gives each group a single connected piece, or even a constant number of connected pieces. In contrast, with individual agents a connected envy-free division always exists (Stromquist80, ).

What do groups do when they cannot attain unanimity? In democratic societies, they use some kind of voting. The premise of voting is that it is impossible to satisfy everyone, so we should try to satisfy as many members as possible. Based on this observation, we say that a division is -democratic fair, for some fairness notion and for some , if at least a fraction of the agents in each group believe it is fair. In this paper we focus on allocating indivisible goods. We would like , the fraction of appy agents, to be as large as possible. We thus pose the following question:

Given a fairness notion, what is the largest such that an -democratic fair allocation of indivisible goods can always be found?

We study democratic fairness under three different assumptions on the agents’ valuations. In the most general case, the agents can have arbitrary monotonic valuations on bundles of goods. A more common assumption in the literature is that agents’ valuations are additive (the value of a bundle is the sum of the values of the goods in the bundle). We also study a special case of additive valuations in which agents’ valuations are binary (each agent has a set of desired goods and her utility equals the number of desired goods allocated to her group).

1.1 Overview of Our Results

Initially (Section 3) we consider two groups with binary agents. We study a relaxation of envy-freeness that we call envy-freeness up to goods (EF), a generalization of EF1. One might expect to have a trade-off curve where a larger corresponds to a larger . However, we find that the actual trade-off curve is degenerate: for every constant , it is possible to guarantee -democratic EF and the is tight. The same holds for MMS fairness. To get a more flexible trade-off curve, we study a generalization of MMS called -out-of--MMS, which is the best share an agent can secure by dividing the goods into subsets and receiving the worst one. We prove that for every integer , -out-of- MMS can be guaranteed to at least and at most of the agents in both groups.

Our positive results are attained by an efficient round-robin protocol where each group in turn picks a good using weighted approval voting with carefully calculated weights. The weights of agents who lose in early votes are increased in later votes. We believe this weighted voting scheme can be interesting in its own right as a way to make fair group decisions.

Next (Section 4) we consider two groups whose agents have arbitrary monotonic valuations. We present an efficient protocol that guarantees EF1 to at least of the agents in each group (which is tight even for binary agents). When all agents are additive, this protocol guarantees of the MMS to of the agents. This is tight: one cannot guarantee more than of the MMS to more than of the agents. Moreover, a positive fraction of the MMS can be guaranteed to at least and at most of the agents in both groups. If we are instead interested in relaxing envy-freeness, it is possible to guarantee unanimous EF when agents have additive valuations, where denotes the total number of agents in the two groups.

Finally (Section 5), we present two generalizations of our results to groups. The first generalization has stronger fairness guarantees: when all valuations are binary, it guarantees to of the agents in all groups both EF1 and MMS (the is tight for EF1). When valuations are additive, it guarantees an additive approximation to EF and MMS. However, the run-time of the protocol might be exponential. The second generalization uses a polynomial-time protocol but has weaker guarantees: when all valuations are binary, it guarantees MMS to of the agents, and when valuations are additive, it guarantees an additive approximation to MMS. We also show that for any number of groups, a generalized version of our round-robin protocol from Section 3 gives a positive fraction of the MMS to at least of the agents in each group.

Some of our results and open questions are summarized in Table 1.

Happy Share Positive
Yes (Thm. 4.4) Yes (Cor. 4.2) Bin: Yes (Thm. 3.7), Add: ?
Bin: Yes (), Add: No ()
? Bin: ?, Add: No (Prop. 4.3)
?
No (Prop. 3.1)
Happy EF for any constant
Yes (Thm. 4.1)
No (Prop. 3.5)
Table 1: Summary of results for two groups with Binary and Additive agents. For each range of , the table shows whether there always exists an allocation that gives at least a fraction of the agents in each group at least a fraction of their maximin share. For EF, the results hold for monotonic valuations too. The arrows refer to the directions pointed to in the table. The table is clearer when viewed in color.

1.2 Related Work

The group resource allocation problem is relatively new. We already mentioned the impossibility result of Suksompong Suksompong18

, which is for worst-case agents’ utilities. On the other hand, if the agents’ utilities are drawn at random from probability distributions, Manurangsi and Suksompong 

ManurangsiSu17 showed that a unanimously envy-free allocation exists with high probability as the number of agents and goods grows. In our terminology, unanimous fairness is called -democratic-fairness.

The term democratic fairness appears in the work of Segal-Halevi and Nitzan SegalhaleviNi15 ; however, they use it in the narrower sense that at least of the agents in each group must be satisfied. In our terminology this is called -democratic fairness. Hence, our democratic fairness notion generalizes existing notions of group fairness.

A related model, in which a subset of public goods is allocated to a single group of agents but the rest of the goods remain unallocated, has also been studied ManurangsiSu17-2 ; Suksompong16 .

MMS fairness was introduced by Budish Budish11 based on earlier concepts by Moulin Moulin90 . Budish also considered its relaxation to -out-of- MMS. The notion -out-of- MMS is a special case of -out-of- MMS, recently defined by Babaioff et al. babaioff2017competitive .

Our group-fairness notions differ from those defined, e.g., by Berliant et al. Berliant1992Fair , Husseinov Husseinov2011Theory , and Todo et al. TodoLiHu11 . In their setting, goods are divided among individuals. The challenge comes from the requirement to eliminate envy, not only between individuals, but also between subsets of agents. In our setting, the challenge is that the goods are divided among groups. A share that is desirable for some group members might be undesirable for other members of the same group. This motivates the use of social choice techniques such as having each group vote on which good to pick.

Group preferences are important in matching markets, too. For example, when matching doctors to hospitals, usually a husband and a wife want to be matched to the same hospital. This issue poses a substantial challenge to stable-matching mechanisms Klaus2005Stable ; Klaus2007Paths ; Kojima2013Matching .

2 Preliminaries

There is a set of goods. A bundle is a subset of . There is a set of agents. The agents are partitioned into groups with agents, respectively. Let denote the th agent in group . Each agent has a nonnegative utility for each . For any agent , denote by the maximum utility of the agent for any single good. Denote by

the utility vector of agent

for single goods. The agents’ utility functions are monotonic, i.e., for every and every agent . A subclass of monotonic utilities is the class of additive utilities, i.e., for every bundle and every agent , we have .

Sometimes we will study a special case of additive utilities in which utilities are binary, i.e., each agent either approves or disapproves each good. Since we will not engage in interpersonal comparison of utilities, we may assume without loss of generality that in this case for each .

We allocate a bundle to each group . All goods should be allocated. The goods are treated as public goods within each group, i.e., for every group , the utility of every agent is . We refer to a setting with agents partitioned into groups, goods and utility functions as an instance.

We now define the fairness notions considered in this paper. We begin by defining what it means for an allocation to be fair for a specific agent. We start with envy-freeness.

Definition 2.1.

Given an agent and an integer , an allocation is called envy-free up to goods (EF) for if for every there is a set with such that

In other words, one can remove the envy of toward group by removing at most goods from the group’s bundle.

An EF0 allocation is also known as envy-free.

Next, we define the maximin share concepts.

Definition 2.2.

Given an agent and an integer , the -out-of- maximin share (MMS) of is defined as the maximum, over all partitions of into sets, of the minimum of the agent’s utilities for the sets in the partition:

where is a partition of . When (the number of groups), the -out-of- MMS of an agent is simply called his MMS and denoted by . An allocation is said to be:

  • -out-of- MMS-fair for , if .

  • MMS-fair for , if .

  • -MMS-fair for , for some fraction , if .

  • positive-MMS-fair if every agent with positive MMS gets positive value: .

Note that MMS-fairness implies -MMS-fairness (for any ), which implies positive-MMS-fairness. The next lemma shows an interesting link between EF1 and MMS-fairness.

Lemma 2.1.

If an allocation is EF1 for an agent with an additive utility function, then (a) it is also -MMS-fair for that agent—the is tight; (b) if the agent’s utility function is binary, then the allocation is also MMS-fair for that agent.

Proof.

Denote by the utility function of the agent and assume without loss of generality that the agent is in group .

(a) EF1 implies that in each bundle (for ) there exists a subset with such that . Summing over all groups gives that . Now, in any partition of into bundles, there is at least one bundle that does not contain any good in . This bundle is contained in . Therefore, the MMS is at most which is at most . Therefore, is at least of the MMS.

To show that the factor is tight, assume that there are goods with and . If the agent’s group (say, group ) gets and group gets , the agent gets utility and finds the allocation EF1. However, the MMS is , as can be seen from the partition .

(b) Suppose the agent’s group wins of the agent’s desired goods. EF1 implies that each of the other groups wins at most of the agent’s desired goods. Hence the agent has at most desired goods. Therefore the agent’s MMS is at most , so the allocation is MMS-fair for her. ∎

Now we are ready to define our main group fairness notion:

Definition 2.3.

For any given fairness notion, an allocation is said to be -democratic fair if it is fair for at least agents in group , for all .

We also refer to -democratic fairness as unanimous fairness.

3 Two Groups with Binary Valuations

This section considers the special case in which there are two groups, the agents have additive valuations, and each agent either desires a good (in which case her utility for the good is ) or does not desire it (in which case her utility is ). Even in this special case, some fairness guarantees are unattainable.

Proposition 3.1.

For any , there is a binary instance in which no allocation is -democratic positive-MMS-fair.

Proof.

There are 3 goods. In each group there are 3 members, each of whom has utility 0 for a unique good and utility 1 for each of the other two goods. Each agent has a positive MMS (), but no allocation gives all agents a positive utility. ∎

We next leverage a combinatorial construction of Erdős to show the limitations of -out-of- MMS-fairness.

Lemma 3.2.

For any integer , there is an instance with two groups consisting of agents with binary valuations in each group, such that each agent desires goods but no allocation gives all agents a positive utility.

Proof.

Erdős Erdos64 proved that for any positive integer , there exists a collection of subsets of size of a base set that does not have “property B”.111 We are grateful to Fedor Petrov for suggesting the connection with Property B. This means that no matter how we partition into two subsets and , some subset in has an empty intersection with or .

Take the elements of to be our goods. Each group consists of agents, each of whom desires all goods in a unique subset of goods in . Then every agent desires goods, but no allocation gives all agents a positive utility. ∎

Proposition 3.3.

For any integer and , there is a binary instance with two groups in which no allocation is -democratic -out-of- MMS-fair.

Proof.

Consider the instance from Lemma 3.2. The -out-of- MMS of an agent who desires goods is positive (1), but no allocation gives all agents a positive utility. ∎

As Lemma 3.2 shows, in certain instances it might be impossible to give every agent a positive utility. Interestingly, deciding whether an instance admits an allocation that leaves no agent with zero utility is an NP-complete problem.

Proposition 3.4.

Deciding whether an instance with two groups with binary agents admits an allocation that gives every agent a positive utility is NP-complete.

Proof.

For any allocation, we can clearly verify in polynomial time whether it yields a positive utility to every agent. To show that the problem is NP-hard, we reduce from Monotone SAT, a variant of the classical satisfiability problem where each clause contains either only positive literals or only negative literals. Monotone SAT is known to be NP-hard (GareyJo79, , p. 259).

Given a Monotone SAT formula with variables , let there be items corresponding to the variables. For each clause that contains only positive literals, we construct an agent in the first group who values exactly the items contained in this clause. Similarly, for each clause that contains only negative literals, we construct an agent in the second group who values exactly the items contained in this clause. Any assignment that satisfies gives rise to an allocation where the items corresponding to true variables in the assignment are allocated to the first group and those corresponding to false variables in the assignment are allocated to the second group; this allocation gives every agent nonzero utility. Likewise, any allocation that gives every agent nonzero utility yields a satisfying assignment of . Hence the reduction is valid. ∎

If we change the fairness requirement from MMS to EF, then we can satisfy no more than half of the agents in each group.

Proposition 3.5.

For any constant integer and , there is an instance with two groups with binary agents in which no allocation is -democratic EF.

Proof.

Consider an instance with goods and agents in each group, for some . Each agent desires a unique subset of goods. An allocation is EF for an agent iff her group receives at least of her desired goods.

The symmetry between the groups implies that the best fairness guarantee can be attained by giving exactly goods to each group; the symmetry between the goods implies that it does not matter which goods are given to which group. In each such allocation, the number of a group’s members who receive exactly desired goods is . Therefore, the number of a group’s members who receive at least desired goods is: where the equality follows from expanding the central binomial coefficient . The fraction of a group’s members who think the division is EF is attained by dividing this expression by . This fraction is:

Using Stirling’s approximation, we find that , so

As , the fraction of agents in each group who think that the allocation is EF approaches , as claimed. ∎

Our positive results are attained with a protocol we call Round-robin with Weighted Approval Voting (RWAV). The groups take turns picking a single good until all goods are taken. Each group picks its good using a weighted approval voting scheme. The members’ weights are determined using fiat money. Initially, each group has an account that starts at 0, and each agent also starts with 0. Each agent can pay money to the group account or receive money from the group account. RWAV proceeds as follows.

(a) Initially, each member pays to his group account some amount of fiat money to be calculated later.

(b) Whenever it is the group’s turn to pick, each member is assigned a positive weight to be calculated later.

(c) For each good, the total weight is calculated as the sum of the weights of the members who desire this good. The group picks a good with a maximal total weight.

(d) Every member whose desired good was picked by the group pays his weight to the group account.

(e) When it is the other group’s turn to pick, each member whose desired good was picked by the other group receives his weight from the group account.

We now calculate the weights such that, when the protocol ends, the net payment paid by each happy agent is and by each unhappy agent is (i.e., each unhappy agent got all his money back). An agent’s weight will be a function of the number of his desired goods that emain untaken, and the number of desired goods that he is mising to be happy. Both and of an agent weakly decrease as the protocol runs. An agent becomes happy when and unhappy when . Let be the weight of such an agent and the net amount paid by such an agent. Then, :

by step (d)
by step (e)

This implies the following recurrence relation for :

Its solution is (**):
and (***):

Some values of are shown in Table 2.

We make several observations. First, when is fixed, decreases when increases, since the sum in (**) has fewer elements. Second, when is fixed, increases when increases, since is an average of with the larger term . Now, (***) implies that all weights are positive, as required by the protocol.

To complete the specification of the protocol, we have to calculate the initial payments in step (a). We define a generalized fairness criterion called -fairness, where is some integer function. An allocation is -fair for an agent if the agent’s group receives at least desired goods whenever the agent has desired goods. Note that EF1 and MMS-fairness are both equivalent to -fairness. Suppose we are interested in -fairness for some function . Then, in each group, an agent with desired goods has initially and . Hence the initial payment of each such agent should be .

0 1 2 3 4 5 6 7 8 9
0 1* 0
1 1 .50* 0
2 1 .75* .25 0
3 1 .87 .50* .12 0
4 1 .93 .68* .31 .06 0
5 1 .96 .81 .50* .18 .03 0
6 1 .98 .89 .65* .34 .10 .01 0
7 1 .99 .93 .77 .50* .22 .06 .00 0
8 1 .99 .96 .85 .63* .36 .14 .03 .00 0
9 1 .99 .98 .91 .74 .50* .25 .08 .01 .00
Table 2: Some values of . The starred cells are the cells of where the function corresponding to EF1 and MMS-fairness. Each such cell represents the initial payment that should be paid by an agent of the second group who starts with desired goods. All empty cells have the value 0. The values are rounded to two decimal digits.

We are now ready to state the main lemma:

Lemma 3.6.

For every integer function , it is possible to select weights such that the RWAV protocol attains -democratic -fairness, where:

Proof.

We have to prove that, when RWAV ends, in each group there are at least happy agents. By (*), the final balance of each unhappy agent is and of each happy agent . The total balance of the group plus the agents is 0. Therefore, it is sufficient to prove that the final balance of group is at least .

In each step (e), the group pays the total weight of a good, while in step (d), the group receives the total weight of another good. Since the good picked in step (c) has a maximal total weight, the outgoing payment is at most the incoming payment. Therefore the balance of each group weakly increases since the first time it picks a good in step (c) until the end of the protocol. Now it is sufficient to prove that the initial balance of group , when its first turn arrives, is at least .

In group 1, in step (a), each member with desired goods pays which is larger than so at least . Therefore its initial balance is .

As for group 2, before its first turn it might have to pay to members whose desired good was picked by group 1. To each member with desired goods, it has to pay at most , so the new net payment of this member is . It is still at least so the initial balance of group 2 is too. ∎

Note that each group can choose its own and get its own . As an example, suppose our group follows the egalitarian philosophy and we want to ensure that as many members as possible receive a positive utility. Then we can let . For each , the initial voting weight of a member with desired goods is . As the protocol progresses, the weight of a member whose desired good is taken by our group drops to zero, and the weight of a member whose desired good was taken by the other group is multiplied by 2. Thus the interests of poor agents are prioritized.

Lemma 3.6 implies the following positive result.

Theorem 3.7.

For every integer , it is possible to select weights such that RWAV attains -democratic -out-of- MMS-fairness.

Proof.

Suppose an agent wants goods, for some integers and . So her -out-of--MMS is . We can arbitrarily ignore of her desired goods and aim to give her a utility of . Applying Lemma 3.6 with gives . It remains to prove that for every , . That is:

We prove the latter inequality using combinatorial arguments.222 We are grateful to Y. Forman for suggesting these arguments in https://math.stackexchange.com/
a/2598269/29780.
Consider a row of light bulbs, each of which can be either on or off. The bulbs are divided to groups with bulbs in each group. There are switches. Of these, are “normal” switches, each of which toggles a single distinct bulb. The remaining are “special” switches: special switch number toggles all the bulbs in groups and . Initially all lights are off. We are allowed to push at most switches. How many light patterns can we make?

Clearly, the left-hand side is the number of choices of at most switches, and the right-hand side is the number of different light patterns. Therefore, to prove the inequality it is sufficient to prove that any two distinct switch choices generate different light patterns.

Consider two distinct switch choices and let be their set of special switches. If then the two choices must contain different sets of normal switches and therefore generate different light patterns. Suppose . Any set of special switches leaves an even number of groups of lights on, so the light patterns resulting only from the special switches differ in at least two groups, i.e., positions. However, both choices must contain at most switches overall, and at least one of them already contains at least one special switch so only normal switches remain. Therefore the total number of normal switches in both groups is at most . The difference cannot be canceled using normal switches, so the patterns must be different. ∎

Our results raise the following open question: what is the maximum fraction of agents that can be guaranteed 1-out-of--fairness? By Theorem 3.7 it is at least ; by Proposition 3.3 it is at most .

An interesting special case of Theorem 3.7 is when , since with two groups -out-of- MMS-fairness is equivalent to MMS-fairness, and with binary agents this is also equivalent to EF1. Hence Theorem 3.7 implies that -democratic EF1 and MMS-fairness are attainable. The fraction exactly matches the upper bound in Proposition 3.5.

It is worth noting that our RWAV protocol gives the following unanimous guarantee that holds for all agents.

Corollary 3.8.

For two groups each containing at most agents with binary valuations, there exists a unanimous -out-of- MMS-fair allocation. The rate is asymptotically tight.

Proof.

Let , so . By Theorem 3.7, we can attain -democratic -out-of- MMS-fairness. However, -democratic fairness and unanimous fairness are equivalent when the number of agents in each group is at most . The tightness of the rate follows from the remark after Theorem 3.7. ∎

4 Two Groups with General Valuations

In this section we assume that there are two groups and each agent can have an arbitrary monotonic utility function.

We start with a positive result: it is always possible to efficiently allocate goods so that at least half of the agents in each group believe the division is EF1. The protocol mirrors the well-known “cut-and-choose” protocol for dividing a cake between two agents. Despite the simplicity of the protocol, we find the result important since, unlike previous results in this setting ManurangsiSu17 ; Suksompong18 , our result holds for worst-case instances with any number of agents in the groups and very general utility functions.

Theorem 4.1.

For two groups of agents with monotonic valuations, -democratic EF1 is attainable.

Proof.

We arrange the goods in a line and process them from left to right. Starting from an empty block, we add one good at a time until the current block is EF1 for at least half of the agents in at least one group. We allocate the current block to one such group, and the remaining goods to the other group.

Since the whole set of goods is EF1 for both groups, the protocol terminates. Assume without loss of generality that the left block is allocated to the first group , and the right block to the second group . By the description of the protocol, the allocation is EF1 for at least half of the agents in , so it remains to show that the same holds for . Let be the last good added to the left block. More than half of the agents in think that is not EF1, so for these agents, is worth less than for any . Taking , we find that these agents value less than . But this implies that the agents find to be EF1, completing the proof. ∎

Theorem 4.1 shows that if the goods lie on a line, we can find a -democratic EF1 allocation that moreover gives each group a contiguous block on the line. This may be important, for example, if the goods are houses on a street and each group wants to have all its houses in a contiguous block (BouveretCeEl17, ; Suksompong17, ).

If agents have additive valuations, Lemma 2.1 implies:

Corollary 4.2.

For two groups with additive agents, -democratic -MMS-fairness is attainable.

For EF1, the factor in Theorem 4.1 is tight even for binary valuations, as shown in Proposition 3.5. For -MMS-fairness, the factor in Corollary 4.2 is “almost” tight:

Proposition 4.3.

For any and , there is an additive instance with two groups in which no allocation is -democratic -MMS-fair.

Proof.

Consider an instance with goods and agents in each group, with utility vectors: , , and for . The MMS of every agent is 2. In any allocation, one group receives at most one good, so at most one of its three agents receives utility more than 1. In that group, at most of the agents receive more than of their MMS. ∎

A corollary of Proposition 4.3 is that, for every , the maximum fraction such that there always exists an -democratic -MMS-fair allocation is .

What fraction of the agents can be satisfied if we are only interested in positive-MMS-fairness? The upper bound on in Proposition 3.1 is even for binary agents. Below we show a lower bound of that holds for additive agents.

Theorem 4.4.

For any two groups with additive agents, there exists a -democratic positive-MMS allocation.

Proof.

An agent’s maximin share is positive only if the agent has at least two goods with a positive utility. Therefore, for positive-MMS it is sufficient to give an agent at least one of his two best goods. We show that this can be attained for at least of the agents in each group. First, convert all valuations to binary by assuming each agent desires only his two most valuable goods (breaking ties arbitrarily).

If, in one of the groups, at least of the agents desire the same good , then give to that group and give all other goods to the other group. The allocation is obviously -democratic positive-MMS.

Otherwise, run RWAV as usual. As in the proof of Lemma 3.6, we have to prove that, for each group , its balance when it first picks an item is at least .

We have for all , so the initial payment of each agent is . Therefore the initial balance of group 1 is .

As for group 2, before its first turn it might have to pay to members whose good was picked by group 1. There are less than such members and the weight of each is . Therefore the initial balance of group 2 is above . ∎

Remark 4.5.

While the present paper focuses on deterministic algorithms, it is interesting that with a single bit of randomness we can improve the guarantee of Theorem 4.4 to in expectation. If, in one of the groups, at least of the agents desire the same good , then give to that group and give all other goods to the other group. Otherwise, run RWAV, selecting the first group by tossing a fair coin. In group 1, the initial balance is so at least of its members are happy; in group 2, the initial balance is at least so at least of its members are happy. Therefore the expected fraction of happy agents in each group is .

The example in Proposition 3.1 implies an upper bound of for randomized algorithms.

Proposition 3.5 shows that the factor in Theorem 4.1 cannot be improved even for binary agents and even if we relax EF1 to EF for any constant . Nevertheless, if we let the relaxation of envy-freeness depend on the number of agents, it is possible to obtain a unanimous fairness guarantee.

Theorem 4.6.

For any two groups of agents with additive valuations, there exists an allocation that is EF for all agents, where is the total number of agents in both groups.

Proof.

Choose an arbitrary agent in one of the groups. We will partition the goods into two parts and let the agent choose the part that she prefers. The resulting allocation is envy-free and therefore for this agent. It therefore suffices to show that there exists a partition in which each bundle is EF (with respect to the other bundle) for all of the remaining agents.

To this end, assume that there is a divisible good (“cake”) represented by the half-open interval . The value-density functions of the agents over the cake are piecewise-constant: for every , the value-density in the half-open interval equals .

It is known that there exists a partition of the cake into two parts, using at most cuts, in which every agent has equal value for both parts Alon87 . Starting with two empty bundles, for each , we add good to the bundle corresponding to the part that contains at least half of the interval . (If both parts contain exactly half of the interval, we add to an arbitrary bundle.)

We claim that every agent finds either bundle to be EF. Fix an agent and a bundle . From our partitioning choice, we have that for some set of size at most . This implies that the agent finds to be EF with respect to , as claimed. ∎

5 Three or More Groups

In this section we study the most general setting where we allocate goods among any number of groups. When there are two groups, the protocol in Theorem 4.1 is computationally efficient and yields an allocation that is both approximately envy-free and approximately MMS-fair. We present two ways of generalizing the result to multiple groups: one keeps the approximate envy-freeness guarantee but loses computational efficiency, while the other keeps only the approximate MMS-fairness guarantee but also retains computational efficiency.

5.1 Approximate Envy-freeness

The main theorem in this subsection is:

Theorem 5.1.

When all agents have binary valuations, there exists an allocation that is -democratic EF1 and -democratic MMS-fair. The factor is tight for EF1.

To establish this theorem, we prove two lemmas that may be of independent interest—one on cake-cutting and the other on group allocation for agents with additive valuations.

The result on cake-cutting generalizes the theorems of Stromquist Stromquist80 and Su Su99 , who prove the existence of contiguous envy-free cake allocations for individual agents. Since these results are well-known, we present the model and proof quite briefly, focusing on the changes required to generalize from individuals to groups.

We consider a “cake” modeled as the interval . Each agent has a value-density function . The value of an agent for a piece is . Denoting by the allocation to group , an allocation is envy-free for an agent if for every group . A contiguous allocation is an allocation of the cake in which each group gets a contiguous interval.

Lemma 5.2.

There always exists a contiguous cake allocation that is -democratic envy-free. The factor is tight.

Proof.

The space of all contiguous partitions corresponds to the standard simplex in . Triangulate that simplex and assign each vertex of the triangulation to one of the groups. In each vertex, ask the group owning that vertex to select one of the pieces using plurality voting among its members, breaking ties arbitrarily. Label that vertex with the group’s selection. The resulting labeling satisfies the conditions of Sperner’s lemma (see Su Su99 ). Therefore, the triangulation has a Sperner subsimplex—a subsimplex all of whose labels are different. We can repeat this process with finer and finer triangulations. This gives an infinite sequence of smaller and smaller Sperner subsimplices. This sequence has a subsequence that converges to a single point. By the continuity of preferences, this limit point corresponds to a partition in which each group selects a different piece. Since the selection is by plurality, at least of the agents in each group prefer their group’s piece over all other pieces.

The tightness of the factor follows from Lemma 6 of Segal Halevi and Nitzan SegalhaleviNi15 . It shows an example with groups and agents in each group with the property that in order to give a positive value to out of agents in each group, we need to cut the cake into at least intervals. In a contiguous partition there are exactly intervals. Therefore, the fraction of agents in each group that can be guaranteed a positive value is . Since can be arbitrarily large, the largest fraction that can be guaranteed is . ∎

The next lemma presents a reduction from approximate envy-free allocation of indivisible goods to envy-free cake-cutting. We call this approximation “EF-minus-2”. An allocation is EF-minus-2 for agent if for every group , . The reduction generalizes Theorem 5 of Suksompong Suksompong17 ; a similar reduction was used in Theorem 3 of Barrera et al. BarreraNyRu15 .

Lemma 5.3.

When agents have additive valuations, there always exists a contiguous allocation of indivisible goods that is -democratic EF-minus-2.

Proof.

We create an instance of the cake-cutting problem in the following way.

  • The cake is the half-open interval .

  • The value-density functions are piecewise constant: for every , the value-density in the half-open interval equals .

By Lemma 5.2, there exists a contiguous cake allocation that is envy-free for at least of the agents in each group. From this allocation we construct an allocation of goods as follows.

  • If point of the cake is in the interior of a piece, then good is given to the group owning that piece.

  • If point of the cake is at the boundary between two pieces, then good is given to the group owning the piece to its left.

A group gets good only if it owns a positive fraction of the interval . Hence, in the allocation, each group loses strictly less than the value of a good and gains strictly less than the value of a good (relative to its value in the cake division). This means that every agent who believes that the cake allocation is envy-free also believes that the goods allocation is EF-minus-2. ∎

We are now ready to prove Theorem 5.1.

Proof of Theorem 5.1.

Suppose an allocation is EF-minus-2 for some agent . This means that the agent’s envy towards any other group is less than . Since the agent has binary valuations, the envy is at most , meaning that the allocation is EF1 for that agent. Hence any -democratic EF-minus-2 allocation, which is guaranteed to exist by Lemma 5.3, is also -democratic EF1. By Lemma 2.1 it is also -democratic MMS-fair.

We next show that the factor is tight. Assume that there are goods for some large positive integer . Each group consists of agents, each of whom values a distinct combination of the goods. Consider first an allocation that gives exactly goods to each group, and fix a group. We claim that the fraction of the agents in the group whose utilities for some two bundles differ by at most 1 converges to 0 for large

. Indeed, this follows from the central limit theorem: Fix two bundles and consider a random agent from the group; let

be the random variable denoting the (possibly negative) difference between the agent’s utilities for the two bundles. Then

is a sum of independent and identically distributed random variables with mean 0. The central limit theorem implies that for any fixed , there exists a constant such that for any sufficiently large . Taking the union bound over all pairs of bundles, we find that the fraction of agents in the group who value some two bundles within 1 of each other approaches 0 as goes to infinity. This means that all but a negligible fraction of the agents find only one bundle to be EF1. By symmetry, of these agents find the bundle allocated to the group to be EF1. It follows that the fraction of agents in the group for whom the allocation is EF1 converges to .

It remains to consider the case where the allocation does not give the same number of goods to all groups. In this case, let denote the set of bundles with the smallest number of goods, which must be strictly smaller than goods. If we move goods from bundles with more than goods to bundles in in such a way that the number of goods in each bundle in increases by exactly one, the fraction of agents in an arbitrary group that receives a bundle in who finds the allocation to be EF1 can only increase. We can repeat this process, at each step possibly adding bundles to , until all bundles contain the same number of goods, which is the case we have already handled. Since the fraction of agents for whom the allocation is EF1 is bounded above by for large in the latter allocation, and this fraction only increases during our process of moving goods, the same is true for the original allocation. ∎

The cake-cutting protocol of Lemma 5.2 might take infinitely many steps to converge. In fact, there is no finite protocol for contiguous envy-free cake-cutting even for individuals (Stromquist2008Envyfree, ). However, the division guaranteed by Lemma 5.3 and Theorem 5.1 can be found in finite time (exponential in the input size) by checking all possible allocations. An interesting open question is whether a faster algorithm exists.

5.2 Approximate MMS

In this subsection, we show that if we weaken our fairness requirement to approximate MMS, it is possible to compute a fair allocation in time polynomial in the input size.

Lemma 5.4.

When agents have additive valuations, there always exists an allocation such that at least of the agents in each group receive utility at least , and such an allocation can be computed efficiently.

This lemma generalizes the corresponding result for the setting with one agent per group by Suksompong Suksompong17 . The factor is tight even for individual agents.

Proof.

We arrange the goods in a line and process them from left to right. Starting from an empty block, we add one good at a time until the current block yields utility at least for at least of the agents in at least one group. We allocate the current block to one such group and repeat the process with the remaining groups. It is clear that this algorithm can be implemented efficiently. Any group that receives a block from the algorithm meets the requirement, so it suffices to show that the algorithm allocates a block to every group. We claim that if groups are yet to receive a block, at least of the agents in each of these groups have utility at least for the remaining goods. This would imply the desired result because for the last group, at least of the agents have utility , which is exactly our requirement.

To show the claim, we proceed by backward induction on . The claim trivially holds when . Suppose that the statement holds when there are groups left, and consider a group that is not the next one to receive a block. At least of the agents in the group have utility at least for the remaining goods. Since the group does not receive the next block, less than of the agents in the group have utility at least for the block excluding the last good. Hence, less than of the agents have utility at least for the whole block. This means that at least of the agents have utility at least , completing the induction. ∎

It is clear by definition that the MMS of any agent is at most . Lemma 5.4 therefore implies the following:

Theorem 5.5.

When agents have additive valuations, there always exists an allocation such that at least of the agents in each group receive utility at least , and such an allocation can be computed efficiently.

For binary valuations, if we change the stopping condition in Lemma 5.4 to be when the current block yields the MMS for at least of the agents in some group, we get:

Theorem 5.6.

When agents have binary valuations, there always exists a -democratic MMS-fair allocation, and such an allocation can be computed efficiently.

5.3 Generalizing the RWAV protocol

The RWAV protocol of Section 3 can be generalized to groups. The main change is that each member whose desired good was picked by its group should pay times its weight to the group account, since there are rounds in which the group might have to pay to losing agents.

Therefore the recurrence relation on (the net payment of an member to its group) becomes:

by step (d)
by step (e)

This implies the following recurrence relation for :

Its solution is (**):
and (***):

Some values of for the case are shown in Table 3.

0 1 2 3 4 5 6 7 8 9
0 1 0
1 1 .33 0
2 1 .55 .11 0
3 1 .70 .25 .03 0
4 1 .80 .40 .11 .01 0
5 1 .86 .53 .20 .04 .00 0
6 1 .91 .64 .31 .10 .01 .00 0
7 1 .94 .73 .42 .17 .04 .00 .00 0
8 1 .96 .80 .53 .25 .08 .01 .00 .00 0
9 1 .97 .85 .62 .34 .14 .04 .00 .00 .00
Table 3: Some values of when there are groups. All empty cells have the value 0. The values are rounded to two decimal digits.

This allows us to prove the following positive result.

Theorem 5.7.

For any groups with additive agents, there exists a -democratic positive-MMS allocation, where .

Proof.

An agent’s maximin share is positive only if the agent has at least goods with a positive utility. Therefore, for positive-MMS it is sufficient to give an agent at least one of his best goods. Moreover, we can convert all valuations to binary by assuming each agent desires only his most valuable goods (breaking ties arbitrarily). Now, we prove that it is possible to give to at least of the agents in each group at least one desired good.

The proof is by induction on . For we already proved in Theorem 4.4 that it is possible to satisfy of the agents in each group, which is larger than .

Assume the claim is true up until ; we will prove it for .

If in some group at least members desire the same good , give them good and divide the remaining goods among the remaining groups recursively. Note that in each remaining group, each agent now desires at least goods, so by the inductive hypothesis, it is possible to satisfy at least an fraction of each group.

Otherwise, run RWAV modified for groups as explained above. As in the proof of Lemma 3.6, it is sufficient to prove that, for each group , its balance when it first picks an item is at least .

We have for all , so the initial payment of each member is , and the initial amount paid to each group is . This is also the balance of group 1 when it first picks an item.

The balance of groups is smaller since they have to pay to their members whose desired goods were picked. Obviously group is in the worst situation since it has to pay times, so we focus on this group. Each time a good is picked, the group has to pay to at most members. It has to pay to each member with remaining goods. Note that which is larger when is smaller. Therefore, the worst case for group is when it has to pay again and again to the same members. In this case it has to pay . The total balance remaining in group ’s account when it first picks an item is thus at least: . ∎

We next show an upper bound of for positive-MMS. In particular, when the number of groups is large, it is not possible to satisfy more than about half of the agents. The following proposition generalizes Proposition 3.1.

Proposition 5.8.

For any and any , there is a binary instance in which no allocation is -democratic positive-MMS-fair.

Proof.

There are goods placed in a circle. In each group there are members. Each member of a group values a unique block of consecutive goods on the circle; the member has utility 1 for each of the goods and utility 0 for the remaining goods. Each agent has a positive MMS (1). However, in any allocation some group gets at most one good, and only members of the group get positive utility. ∎

6 Discussion and Future Work

For two groups, we have a comprehensive understanding of possible democratic fairness guarantees. We have a complete characterization of possible envy-freeness approximations, and upper and lower bounds for maximin-share-fairness approximations. Some remaining gaps are shown in Table 1; closing them raises interesting combinatorial challenges.

For groups, the challenges are much greater. Currently all our fairness guarantees are to no more than of the agents in each group. From a practical perspective, it may be important in some settings to give fairness guarantees to at least half of the agents in all groups. Finding protocols that provide such guarantees is an avenue for future work. From an algorithmic perspective, it is interesting whether there exists a polynomial-time algorithm that guarantees EF1 to any positive fraction of the agents.

A possible concern about democratic fairness is that it completely leaves aside a fraction of the agents in each group. As Lemma 3.2 shows, it might be inevitable to leave some agents with zero utility. In these cases, the goal of an egalitarianist is to minimize the fraction of such poor agents. While the weighting scheme used by our RWAV protocol indeed prioritizes the interests of poor agents (see the example after Lemma 3.6), it may be interesting to develop an algorithm that directly minimizes the maximum fraction of poor agents across all groups.

Acknowledgments

We are grateful to Fedor Petrov, Arnaud Mortier, Darij Grinberg, Michael Korn, Kevin P. Costello, Nick Gill, Jack D’Aurizio, Leon Bloy and anonymous referees of IJCAI-ECAI 2018 and COMSOC 2018 for their helpful comments.

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