# Fair Allocation of Indivisible Goods to Asymmetric Agents

###### Abstract

We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang [20] wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We show that, in some cases with agents, no allocation can guarantee better than approximation of a fair allocation when the entitlements are not necessarily equal. Furthermore, we devise a simple algorithm that ensures a approximation guarantee.

Our second result is for a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. Although this assumption might seem w.l.o.g, we show it enables us to find a approximation fair allocation via a greedy algorithm. Finally, we run some experiments on real-world data and show that, in practice, a fair allocation is likely to exist. We also support our experiments by showing positive results for two stochastic variants of the problem, namely stochastic agents and stochastic items.

## 1 Introduction

In this work, we conduct a study of fairly allocating indivisible goods among agents with unequal claims on the goods. Fair allocation is a very fundamental problem that has received attention in both Computer Science and Economics. This problem dates back to 1948 when Steinhaus [23] introduced the cake cutting problem as follows: given agents with different valuation functions for a cake, is it possible to divide the cake between them in such a way that every agent receives a piece whose value to him is at least of the whole cake? Steinhaus answered this question in the affirmative by proposing a simple and elegant algorithm which is called moving knife. Although this problem admits a straightforward solution, several ramifications of the cake cutting problem have been studied since then, many of which have not been settled after decades [7, 21, 11, 19, 14, 12, 24, 10, 1, 6, 3]. For instance, a natural generalization of the problem in which we discriminate the agents based on their entitlements is still open. In this problem, every agent claims an entitlement to the cake such that , and the goal is to cut the cake into disproportional pieces and allocate them to the agents such that every agent ’s valuation for his piece is at least fraction of his valuation for the entire cake. For two agents, Brams et al. [5] showed that at least two cuts are necessary to divide the cake between the agents. Furthermore, Robertson et al. [22] proposed a modified version of cut and choose method to divide the cake between two agents with portions , where and are real numbers. McAvaney, Robertson, and Web [18] considered the case when the entitlements are rational numbers. They used Ramsey partitions to show that when the entitlements are rational, one can make a proper division via cuts.

Recently, a new line of research is focused on the fair allocation of indivisible goods. In contrast to the conventional cake cutting problem, in this problem instead of a heterogeneous cake, we have a set of indivisible goods and we wish to distribute them among agents. Indeed, due to trivial counterexamples in this setting^{1}^{1}1For instance if there is only one item, at most one agent has a non-zero profit in any allocation., the previous guarantee, that is every agent should obtain of his valuation for all items from his allocated set, is impossible to deliver. To alleviate this problem, Budish [8] proposed a concept of fairness for the allocation of indivisible goods namely the maxmin share. Suppose we ask an agent to divide the items between the agents in a way that he thinks is fair to everybody. Of course, agent does not take into account other agents’ valuations and only incorporates his valuation function in the allocation. Based on this, we define equal to the minimum profit that any agent receives in this allocation, according to agent ’s valuation function. Obviously, in order to maximize , agent chooses an allocation that maximizes the minimum profit of the agents. We call an allocation fair (approximately fair), if every agent receives a set of items that is worth at least (a fraction of ) to him.

It is easy to see that is the best possible guarantee that one can hope to obtain in this setting. If all agents have the same valuation function, then at least one of the agents receives a collection of items that are worth no more than to him. A natural question that emerges here is whether a fair allocation with respect to ’s is always possible? Although the experiments are in favor of this conjecture, Procaccia and Wang [20] (EC’14) refuted this by an elegant and delicate counterexample. They show such a fair allocation is impossible in some cases, even when the number of agents is limited to 3. On the positive side however, they show an approximately fair allocation can be guaranteed. More precisely, they show that there always exists an allocation in which every agent’s profit is at least . Such an allocation is called a - allocation. Amanatidis, Markakis, Nikzad, and Saberi [2] later provided a proof for the existence of an allocation for the case, when there are large enough items and the value of each agent for every items is drawn independently from a uniform distribution. A generalized form of this result was later proposed by Kurokawa et al. [17] for arbitrary distributions. Caragiannis et al. [9] later proved that the maximum Nash welfare (MNW) solution, which selects an allocation that maximizes the product of utilities, for each agent guarantees a fraction of her . In a recent work, Ghodsi et al. [15] provided a proof for existence of a - allocation.

Although it is natural to assume the agents have equal entitlements on the items, in most real-world applications, agents have unequal entitlements on the goods. For instance, in various religions, cultures, and regulations, the distribution of the inherited wealth is often unequal. Furthermore, the division of mineral resources of a land or international waters between the neighboring countries is often made unequally based on the geographic, economic, and political status of the countries.

For fairly allocating indivisible items to agents with different entitlements, two procedures are proposed in [7]. The first one is based on Knaster’s procedure of sealed bids. In this method, we have an auction for selling each item. Therefore, for using it all the agents should have an adequate reserve of money which is the main issue of the procedure. The second procedure mentioned in [7] is based on method of markers developed by William F. Lucas which is spiritually similar to the moving knife procedure. In this method, first we line up the items, and then the agents place some markers for dividing the items. This method suffers from high dependency of its final allocation to the order of the items in the line.

Agent duplication is another idea to deal with unequal entitlements. More precisely, when all of the entitlements are fractional numbers, we can duplicate each agent to some agents with similar valuation functions to . The goal of this duplication is to reduce the problem to the case of equal entitlements. After the allocation, every agent owns all of the allocated items to her duplicated agents. For instance, assume that we have three agents with entitlements , , and , respectively. In this case, we duplicate the first agent to five agents and the second agent to four agents each having an entitlement of . This way, we can reduce our problem to the case of equal entitlements. Although agent duplication may be practical when the items are divisible, in the indivisible case, this method does not apply to the indivisible setting. For instance, if the number of the agents is higher than the number of available items, we cannot allocate anything to some agents. Another issue with this method is that it works only for fractional entitlements.

In this paper, we study fair allocation of indivisible items with different entitlements using a model which resolves the mentioned issues. Our fairness criterion mimics the general idea of Budish for defining maxmin shares. Similar to Budish’s proposal, in order to define a maxmin share for an agent , we ask the following question: how much benefit does agent expect to receive from a fair allocation, if we were to divide the goods only based on his valuation function? If agent expects to receive a profit of from the allocation, then he should also recognize a minimum profit of for any other agent , so that his own profit per entitlement is a lower bound for all agents. Therefore, a fair answer to this question is the maximum value of for which there exists an allocation such that agent ’s profit-per-entitlement can be guaranteed to all other agents (according to his own valuation function). We define the maxmin shares of the agents based on this intuition.

Recall that we denote the number of agents with and the entitlement of every agent with . We assume the entitlements always add up to 1. For every agent , we define the weighted maxmin share denote by , to be the highest value of for which there exists an allocation of the goods to the agents in which every agent receives a profit of at least based on agent ’s valuation function. Similarly, we call an allocation -, if every agent obtains an fraction of from his allocated goods. Notice that in case for all agents, this definition is identical to Budish’s definition. Since our model is a generalization of the Budish’s model, it is known that a fair allocation is not guaranteed to exist for every scenario. However, whether a approximation or in general a constant approximation allocation exists remains an open question.

Our main result is in contrast to that of Procaccia and Wang. We settle the above question by giving a hardness result for this problem. In other words, we show no algorithm can guarantee any allocation which is better than - in general. We further complement this result by providing a simple algorithm that guarantees a - allocation to all agents. As we show in Section 2, this hardness is a direct consequence of unreasonably high valuation of agents with low entitlements for some items. Moreover, in Section 3 we discuss that not only are such valuation functions unrealistic, but also an agent with such a valuation function has an incentive to misrepresent his valuations (Observation 3.1). Therefore, a natural limitation that one can add to the setting is to assume no item is worth more than for any agent . We also study the problem in this mildly restricted setting and show in this case a - guarantee can be delivered via a greedy algorithm.

In contrast to our theoretical results, we show in practice a fair allocation is likely to exist by providing experimental results on real-world data. The source of our experiments is a publicly available collection of bids for eBay goods and services^{2}^{2}2http://cims.nyu.edu/ munoz/data/. Note that since those auctions are truthful^{3}^{3}3An action is called truthful, if no bidder has any incentive to misrepresent his valuation, it is the users’ best interest to bid their actual valuations for the items and thus the market is transparent. More details about the experiments can be found in Section 4. We also support our claim by presenting theoretical analysis for the stochastic variants of the problem in which the valuation of every agent for a good is drawn from a given distribution.

### 1.1 Our Model

Let be a set of agents, and be a set of items. Each agent has an additive valuation function for the items. In addition, every agent has an entitlement to the items, namely . The entitlements add up to , i.e., .

Since our model is a generalization of maxmin share, we begin with a formal definition of the maxmin shares for equal entitlements, proposed by Budish [8]. In this case, we assume all of the entitlements are equal to . Let be the set of -partitionings of the items. Define the maxmin share of agent () of player as

(1) |

One can interpret the maxmin share of an agent as his outcome as a divider in a divide-and-choose procedure against adversaries [8]. Consider a situation that a cautious agent knows his own valuation on the items, but the valuations of other agents are unknown to him. If we ask the agent to run a divide-and-choose procedure, he tries to split the items in a way that the least valuable bundle is as attractive as possible.

When the agents have different entitlements, the above interpretation is no longer valid. The problem is that the agents have different entitlements and this discrepancy must somehow be considered in the divide-and-choose procedure. Thus, we need an interpretation of the maxmin share that takes the entitlements into account.

Let us get back to the case with the equal entitlements. Another way to interpret maxmin share is this: suppose that we ask agent to fairly distribute the items in between agents of , based on his own valuation function. In an ideal situation (e.g., if the goods are completely divisible), we expect to allocate a share with value to every agent. However, since the goods are indivisible, some sort of unfairness is inevitable. For this case, we wish that does his best to retain fairness. is in fact, a parameter that reveals how much fairness can guarantee, regarding his valuation function.

Formally, to measure the fairness of an allocation by , define a value for any allocation as

In fact, we wish to make sure reports an allocation such that is as close to as possible. The maxmin share of is therefore defined as

(2) |

It is easy to observe that Equations (1) and (2) are equivalent, since the fairest allocation in the absence of different entitlements is an allocation that maximizes value of the minimum bundle:

Now, consider the case with different entitlements. Let be the entitlement of agent . Similar to the second interpretation for , ask agent to fairly distribute the items between the agents, but this time, considers the entitlements. In an ideal situation (e.g., a completely divisible resource), we expect the allocation to be proportional to the entitlements, i.e. allocates a share to agent with value exactly (note that when the entitlements are equal, this value equals to for every agent). But again, such an ideal situation is very rare to happen and thus we allow some unfairness. In the same way, define the fairness of an allocation as

(3) |

Let be an allocation by that maximizes . The weighted maxmin share of agent is defined in the same way as , that is:

In summery, the value for every agent is defined as follows:

For more intuition, consider the following example:

###### Example 1.1

Assume that we have two agents with and . Furthermore, suppose that there are 5 items with the following valuations for : and . For the allocation , we have which means . Moreover, for allocation , we have which means . Thus, is a fairer allocation than . In addition, is the fairest possible allocation and hence, .

Example 1.1 also gives an insight about why agent duplication (as introduced in the Introduction) is not a good idea. For this example, if we duplicate agent , we have three agents with the same entitlements. But any partitioning of the items into three bundles, results in a bundle with value at most to .

Finally, an allocation of the items in to the agents in is said to be , if the total value of the share allocated to each agent is worth at least to him.

## 2 A Tight Bound on the Optimal Allocation

In spite of the fact that there exists a - guarantee when all the entitlements are equal, in our general setting surprisingly we provide a counterexample which proves that there is no guarantee better than -. We complement this result by showing that a - always exists. Thus, these two theorems make a tight bound for the problem.

The main property of our counterexample is a large gap between the value of items for different agents. We provide a counterexample according to this property in Theorem 2.1.

###### Theorem 2.1

There exists no guarantee better than - when the entitlements to the items may differ.

Proof. We propose an example that admits no allocation better than -. To this end, consider an instance with agents and items and let for all and . The valuation functions of the first agents are the same. For every agent with , is as follows:

Also, for agent we have:

First, note that for the first agents and . For the first agents, the optimal partitioning is to allocate to for all . Furthermore, the optimal partitioning for is to allocate items to the first agents and keep the first items for himself. In this case, , and for . Therefore, .

On the other hand, in any allocation that guarantees a non-zero fraction of for every agent, at most one of the items is allocated to , since the rest of the items have value for the first agents. Therefore, the items allocated to are worth at most

to him. Thus, the best fraction of that can be guaranteed is

(4) |

Equation (4) can be made arbitrarily close to , by choosing sufficiently small . Thus, no allocation can guarantee an approximation better than for this example.

Theorem 2.1 gives a - upper-bound. In Theorem 2.2, we show that the provided upper-bound is tight. Algorithm 1 uses a simple greedy procedure, which is spiritually similar to an algorithm in [2], guaranteeing - allocation as follow: sort the agents in descending order of entitlements. Starting from the first agent, ask every person to collect the most valuable item from the remaining set, one by one. Repeat the process until no more item is left.

###### Theorem 2.2

Algorithm 1 guarantees a - allocation.

Proof. If the number of items is smaller than the number of agents then the proof is trivial. Therefore, from this point on, we assume . Without loss of generality we assume agents are sorted in descending order of their entitlements, that is The goal is to prove that for each agent he receives at least - using Algorithm 1. Suppose that the optimal allocation for is . Without loss of generality suppose items are sorted according to their value for in descending order that is:

We call items , , …, and heavy items for . Let be the set of heavy items for . Since the entitlements of agents are sorted from to , for agents and when we have:

(5) |

Now, the goal is to prove and use it to the guarantee of the algorithm. Since , if , clearly holds. In case , consider agent where and . According to Inequality (5), has a greater entitlement than . Therefore, we have: which yields that is worth at least for where all the items of are in . Therefore, we can imply that The items are sorted in the descending order of their value to from to which implies

The -th assigned item to by the algorithm is not worth less than . Hence, the assigned items to is worth at least for him which is not less than -.

## 3 Allocation for the Restricted Case

In Section 2, we gave a tight - guarantee for the fair allocation problem with unequal entitlements . In this section, we consider a reasonable restriction of the problem which gives a - guarantee. In this restricted setting the value of each item to each agent is no more than . Observation 3.1 shows that how this assumption can be invaluable.

###### Observation 3.1

If , it is in the best interest of to report his valuation for equal to infinity. Because his may increase in this way, and he achieves more items after the allocation of the algorithm, because his may increase in this way, and he will be satisfied even if he receives only this item.

In this section, we provide an algorithm with - guarantee for the restricted case of the problem. For a case that the entitlements are equal, an algorithm, namely bag filling guarantees - for all the agents. In the bag filling algorithm, we start with an empty bag. In each step, we add a remaining item to the bag. After each addition, if the total value of items in the bag becomes more than - for an unsatisfied agent , we allocate all the items in the bag to him and repeat the procedure with an empty bag. After running this simple procedure, each agent receives at least -.

Our algorithm allocates the items to agents in a more clever way. We allocate an item to an agent in each step of the algorithm until each agent receives at least - by the allocation. To this end, in each step, first the algorithm for each agent creates a candidate set of items where is in the candidate set of if be the maximum number among all of the unsatisfied agents. Then, the algorithm chooses unsatisfied agent and item in its candidate set which maximize among all of the unsatisfied agents and items in their candidate sets, and assigns this item to the agent.

Before proving Algorithm 2 guarantees a - allocation, we prove an auxiliary lemma which argues that using Algorithm 2 no agent receives more than his .

###### Lemma 3.1

Algorithm 2 does not allocate more than for any agent .

Proof. The algorithm does not assign anymore items to a satisfied agent. Hence, any satisfied agent has less than - value of items in before he receives the last item. For the sake of contradiction, suppose that the algorithm allocates more than to . Without loss of generality, suppose that the last item allocated to is . Since before allocating to he had less than - value of items in , the value of to is more than -. Since , before allocating to we have . Since is more than the value of all the other allocated items to , was not in the candidate set of when we were assigning the other items to . Hence, was in the candidate set of an other agent . Therefore, . Since this value is greater than the values of all other items in , the algorithm first allocate to unsatisfied agent .

Now, using Lemma 3.1, we prove the approximation guarantee of the algorithm.

###### Theorem 3.2

Algorithm 2 ensures a - guarantee when for each agent and item ,

Proof. It is clear that if the algorithm satisfies all the agents, it ensures the approximation guarantee. Now, for the sake of contradiction assume that there exists an unsatisfied agent at the end of the algorithm. For each item we define

where is the recipient of in the allocation. Since is maximal according to the algorithm, and is not a satisfied agent, we have:

(6) |

Lemma 3.1 implies that , and since we have at least one unsatisfied agent we can write:

(7) |

Inequality (6) along with Inequality (7) implies

(8) |

Finally Inequality (8) yields that which is a contradiction.

## 4 Empirical Results

As we discussed in Section 2, in extreme cases, making a allocation or even an approximately allocation is theoretically impossible. However, our counter-example is extremely delicate and thus very unlikely to happen in real-world. Here, we show in practice fair allocations w.h.p exist, especially when the number of items is large.

We draw the valuation of the agents for the goods based on a collection of bids for eBay items publicly available at http://cims.nyu.edu/ munoz/data/. More precisely, for items, we randomly choose different categories of goods from the dataset. Moreover, for every agent and item , we set to a submitted bid for the corresponding category of item chosen uniformly at random. The bids vary from 0.01 to 113.63 and their mean is 6.57901. Moreover, the expected variance of the bids in every category is 200.513.

For an instance of the problem with agents and items, we run the experiments with 1000 different vector of entitlements drawn from the uniform distribution (and scaled up to satisfy ). For every and , we take the minimum guarantee obtained all 1000 runs, and show it in Figures 1 and 2. We used heuristic algorithms to compute the maxmin shares and maxmin guarantees. Thus, our results are only lower bounds to the actual guarantees. Nonetheless, the optimal guarantees are very close to the estimated ones.

## 5 Stochastic Setting

In Section 2 we presented a counterexample to show that no allocation better than - can be guaranteed. However, the construction described in the counterexample is very unlikely to happen in the real settings. Here, we show that allocation exists with high probability when a small randomness is allowed in the setting.

Considering stochastic settings is common in the fair allocation problems since many real-world instances can be modeled with random distributions [8, 17, 2, 13]. The general probabilistic model used in previous works is as follows: every agent has a probability distribution over and for every item , the value for is randomly sampled from . In [2], the existence of an allocation is proved for the special case of , where is the standard uniform distribution with minimum and maximum . Kurokawa, Procaccia, and Wang [17] considered the problem for arbitrary random distribution with the condition that for a positive constant . A considerable part of the proof for the existence of an allocation in [17] is referred to [13], where the authors proved the existence of an envy-free allocation in the stochastic settings with arbitrary random distributions.

In this section, we consider two different probabilistic models. Our first model is the same as [17], with the exception that we omit the restriction . We name this model as Stochastic Agents model. In the second model, every item has a probability distribution and for every agent , value of is randomly drawn from . We choose the name Stochastic Items for the second model. We believe that Stochastic Items model is more realistic since the first model does not make any distinguish between the items. None of the previous works mentioned above considered this model.

We leverage Hoeffding inequality to prove the existence of allocation. Theorem 5.1 states the general form of this inequality [16].

###### Theorem 5.1 (General Form of Hoeffding (1963))

Let be random variables bounded by the interval . We define the empirical mean of these variables by . Then, the following inequality holds:

(9) |

Regarding Theorem 5.1, let and let . By Inequality (9), we have:

(10) |

By setting , we rewrite Equation (10) as:

(11) |

### 5.1 Model I: Stochastic Agents

As mentioned before, in the first model we assume that every agent has a probability distribution and for every item , the value of is randomly sampled from . Furthermore, we suppose . Throughout this section, we assume that . This is w.l.o.g , because for the case , for every agent equals to zero. For the Stochastic Agents model, we state Theorem 5.2.

###### Theorem 5.2

Consider an instance of the fair allocation problem with unequal entitlements, such that the value of every item for every agent is randomly drawn from distribution . Furthermore, let and . Then, for every , there exists a value (which means is a function of ) such that if , then almost surely a - allocation exists.

In the rest of this section, we prove Theorem 5.2. Consider the algorithm that allocates items to every agent . We know that the value of item for agent is randomly sampled from . From the point of view of the algorithm, it trivially does not matter whether the value of items are sampled after the allocation or before the allocation. Thus, we can suppose that the value of every item is sampled after the allocation of items.

For now, we know that number of items are assigned to . Argue that for every , there exists a value , such that for every , . In Lemma 5.3 we bound the value of in terms of and .

###### Lemma 5.3

For every , .

Proof. We know . If , then . This, completes the proof.

For the rest of the proof, suppose that . Let be the variable indicating total value of items allocated to . Note that . Regarding Equation (11), we have:

We want to choose such that . We have:

Regarding the facts that and , we have:

Therefore, it’s enough to choose such that

(12) |

Now, let be the number of items that are not assigned to . Since , regarding the fact that , we have , which means . On the other hand, . Also, let be the variable indicating total value of the items that are not allocated to . By the same deduction as for we have:

Let be the value that . We have:

(13) |

Thus, it’s enough to choose in a way that Inequality (13) holds. Regarding the facts that and ,

Therefore, it’s enough to choose in a way that

(14) |

Now, suppose that both Inequalities (12) and (14) are held. Considering as the set of items assigned to , with the probability of at least we have:

It is easy to show that, there always exist an such that for all both Inequalities (12) and (14) hold for and . Regarding this, we have:

Therefore, with the probability at least we have:

(15) |

Now, suppose that the Inequality 15 holds for every agent , with probability at least . Considering all the agents, with the probability at least , value of for every agent is at least . Regarding the fact that , we have

This completes the proof.

### 5.2 Model II: Stochastic Items

As mentioned, in Stochastic Items model, every item has a probability distribution and the value of every agent for item is randomly chosen from . For this model, we prove Theorem 5.4. The theorem states that for large enough , almost surely a - allocation exists.

###### Theorem 5.4

Suppose that for every agent , for a non-negative constant . Then for all , there exists such that if , then, almost surely, - allocation exists.

In the rest of the section, we prove Theorem 5.4. First, in Lemma 5.6 we prove the existence of an allocation that assigns to every agent , a set of items with value at least , where . The idea to prove this fact is inspired by [4]. Argue that we can formulate the allocation problem with unequal entitlements as the following Integer program:

(16) |

In IP(16), variable determines whether is assigned to agent or not. Considering the fact that is a trivial upper bound on , any solution to IP16 is a feasible solution to the assignment problem with unequal entitlements. By relaxing the second and the third condition, we can convert IP16 to LP17: