## 1 Introduction

The fair allocation of scarce resources to interested parties is a central issue in economics.
A fundamental fairness criterion is *envy-freeness*, which requires that all agents find their assigned bundle to be the best among all assigned bundles.
We address the problem of allocating indivisible goods to agents in an (approximately) envy-free manner when the agents have fixed, positive weights that determine their entitlement of the goods.

There are several ways to interpret these weights and associated envy concepts, depending on the problem domain.
In some situations, the weights may represent some publicly known and accepted measure of entitlement such as *eligibility* or *merit*; weighted envy-freeness then stipulates that agents think that the welfare they achieve from their bundle (to the welfare they achieve from another’s bundle) corresponds to the ratio of their respective eligibility scores. Another possible interpretation is that a weight specifies the objective *cost* of participation in the resource allocation “game”—no player should find that a strictly higher *bang per buck* than her own could be obtained by investing or paying as much as any other agent.
Alternatively, Benabbou et al. (2018) (and subsequent work by Benabbou et al. (2019)) study a model where agents represent groups of members/individuals; each member has her own utilities for the items, and the group valuation can be computed from the utilities of its members. In such settings, the *size* of a group can be considered as its weight; thus, group has weighted envy towards group if and only if the average value members of group derive from their bundle is less than the value held by according to ’s valuation function, averaged over the members of ; in other words, members of group believe that, according to their own subjective utilities, agents in group are, on average, better off than they are. In the setting studied by Benabbou et al. (2018), groups correspond to ethnic groups (namely, the major ethnic groups in Singapore); maintaining provable fairness guarantees amongst the ethnic groups is highly desirable: it is one of the major tenants of Singaporean society.

To conclude, the goal of this work is to extend notions of envy-freeness to the weighted case, and explore their relationship to other important welfare notions, such as Pareto optimality.

### 1.1 Our contributions

In Section 2, we propose two extensions of the popular fairness concept *envy-freeness up to one item (EF1)* (Lipton et al., 2004; Budish, 2011) to the weighted setting: (strong) weighted envy-freeness up to one item () and weak weighted envy-freeness up to one item (). In addition to some negative results, we show in Section 3 that a allocation always exists and can be computed efficiently for additive valuation functions using a weight-based picking sequence — this generalizes a well-known result from the unweighted setting.
Moreover, for two agents, we show that a weighted variant of the classic adjusted winner protocol allows us to efficiently compute an allocation that is both and Pareto optimal.
In Section 4, we prove that an allocation with maximum weighted Nash welfare is both Pareto optimal and , thereby generalizing an important result of Caragiannis et al. (2016); we also show that such an allocation may fail to satisfy .
We conclude in Section 5 with some promising directions on extensions of these ideas to settings with non-additive valuation functions.

### 1.2 Related work

There is a vast literature on fair division of indivisible items; see, e.g., Bouveret et al. (2016) for an overview.
The majority of this literature assumes that the agents are symmetric in that they have equal entitlements.
Prior work on the fair allocation of indivisible items to asymmetric agents has tackled fairness concepts that are not based on envy. Farhadi et al. (2019) introduce and study *weighted maxmin share (WMMS)* fairness, a generalization of an earlier fairness notion of Budish (2011). Aziz et al. (2019a) explore WMMS fairness in the allocation of indivisible *chores* — items that, in contrast to goods, are valued negatively by the agents — where agents’ weights can be interpreted as their shares of the workload. Recently, Aziz et al. (2019b) propose a polynomial-time algorithm for computing an allocation of a pool of goods and chores that satisfies both Pareto optimality and weighted *proportionality up to one item (PROP1)* for agents with asymmetric weights.
Unequal entitlements have also been considered in the context of divisible goods (Zeng, 2000; Cseh and Fleiner, 2018).
Unlike all of these previous works, we focus on fairness concepts based on weighted envy for the indivisible goods scenario.

## 2 Model and definitions

Throughout the paper, denote by the set for any positive integer . We are given a set of agents, and a set of items or goods. Subsets of are referred to as bundles, and each agent has a valuation function over bundles; the valuation function for every is normalized (i.e., ) and monotone (i.e., whenever ). We will denote simply by for any and .

An allocation of the items to the agents is a collection of disjoint bundles such that ; the bundle is allocated to agent and is agent ’s realized valuation under . Given an allocation , we denote by the set , and its elements are referred to as withheld items. An allocation is said to be *complete* if and *incomplete* otherwise.

In our asymmetric setting, each agent has a fixed weight : these weights regulate how agents value their own allocated bundles relative to those of other agents, and hence bear on the overall (subjective) fairness of an allocation. More precisely, we define the *weighted envy* of agent towards agent under an allocation as . An allocation is *weighted envy-free* () if no agent has positive weighted envy towards another agent. Weighted envy-freeness reduces to traditional envy-freeness when , for some positive real constant . Since a complete envy-free allocation may not always exist (see, e.g., (Bouveret et al., 2016)), it follows trivially that a complete allocation may not exist in general.

We now state the main definitions of our paper, which are extensions of *envy-freeness up to one item (EF1)* (Lipton et al., 2004; Budish, 2011) to the weighted setting.

###### Definition 2.1.

An allocation is said to be *(strongly) weighted envy-free up to one item* () if for any pair of agents with , there exists an item such that

More generally, is said to be *weighted envy-free up to items* () for an integer if for any pair of agents , there exists a subset of size at most such that

###### Definition 2.2.

An allocation is said to be *weakly weighted envy-free up to one item* () if for any pair of agents with , there exists an item such that

A valuation function is said to be *additive* if for every . If all agents have additive valuation functions, both and reduce to EF1 in the unweighted setting. Moreover, one can check that under additive valuations, an allocation satisfies if and only if for any pair of agents with , there exists an item such that

In addition to fairness, we also often want our allocation to satisfy an efficiency criterion. One important such criterion is Pareto optimality. An allocation is said to *Pareto dominate* an allocation if for all agents and for some agent .
An allocation is *Pareto-optimal* (or for short) if it is not Pareto dominated by any other allocation.

A property of an allocation that is known to have strong guarantees with respect to both fairness and efficiency in the unweighted setting is the *Nash welfare*, defined as (Caragiannis et al., 2016).
For our weighted setting, we define a natural extension called *weighted Nash welfare*: . Since it is possible that the maximum attainable is , we define the *maximum weighted Nash welfare* or allocation along the lines of Caragiannis et al. (2016) as follows: given a problem instance, we find a maximal subset of agents, say , to which we can allocate bundles of positive value, and compute an allocation to agents in that maximizes .

To see why the notion of makes some intuitive sense, consider a setting where agents have a value of for each item they receive; furthermore, assume that the number of items is exactly . In this case, one can verify (using standard calculus) that an allocation maximizing assigns agent exactly items. Indeed, following the interpretation of as the number of members of group , the expression can be thought of as each member of group deriving the same value from the set ; the group’s overall Nash welfare is thus .

### 2.1 Discussion of weak weighted envy

The criterion can be criticized as being too demanding in certain circumstances, when the weight of the envied agent is much larger than that of the envying agent. To illustrate this, consider a problem instance where agent has an additive valuation function and is indifferent among all items taken individually, e.g., for every . Now, if and , then eliminating one item from agent ’s bundle reduces agent ’s weighted valuation of this bundle by merely . As such, we might trigger a substantial adverse effect on the welfare/efficiency of the allocation by aiming to (approximately) eliminate ’s weighted envy towards . This line of thinking was our motivation for introducing the weak weighted envy-freeness concept.

The property turns out to have the following interesting interpretation for additive valuations. Let us call an agent with a higher (resp., lower) weight a *bigger* (resp., *smaller*) agent.
If , then the property for agents and reduces to having for some , which is identical to the (strong) condition — a bigger agent is weakly weighted envy-free up to one item towards a smaller agent if the latter’s bundle has an item which can be eliminated to make the bigger agent prefer (in terms of weight-adjusted valuation) her own allocated bundle to the smaller agent’s reduced bundle.
If , then the property requires , i.e. for some — a smaller agent is weakly weighted envy-free up to one item towards a bigger agent if the latter’s bundle has an item such that augmenting the smaller agent’s bundle with (a replica of) this item makes her prefer (again, in a weight-adjusted manner) this augmented bundle to the bigger agent’s original bundle.

Both of the above statements can be viewed as stronger versions along different lines of what we can call *transfer weighted envy-freeness*: agent is transfer weighted envy-free up to one item towards agent under the allocation if there is an item which would eliminate the weighted envy of towards upon being transferred from to , i.e., . The first and second conditions in the previous paragraph relax transfer to elimination from only and replication in only, respectively.

## 3 allocations

Although it is known that for arbitrary monotone valuation functions, a complete (unweighted) allocation always exists and can be efficiently computed assuming polynomial-time oracle access to the valuation functions (Lipton et al., 2004), this fact does not imply the existence of the more general complete allocations for arbitrary weights and valuation functions.

As far as envy in the traditional sense is concerned, what an agent actually “envies” is an allocated bundle regardless of who owns that bundle. However, both the subjective valuations of allocated bundles and the relative weights interact in non-trivial ways to decide weighted envy. It is easy to see that weighted envy of towards does not imply (traditional) envy of towards , and vice versa. A crucial implication is that even if agent ’s bundle is replaced with the bundle of an agent towards whom has weighted envy, ’s realized valuation (and hence the ratio of its realized valuation to its weight) may decrease as a result. Indeed, consider a problem instance with , ; weights and ; and identical, additive valuation functions such that , , . Under the complete allocation with , agent has weighted envy towards agent since , although agent would not prefer to replace with since that reduces her realized valuation from to . On the other hand, agent could benefit from replacing with even though she does not have weighted envy towards agent . As such, the natural extension of Lipton et al. (2004)’s seminal envy cycle elimination algorithm does not guarantee a complete allocation except in special cases. One such special case is when the agents all have identical valuations.

###### Proposition 3.1.

The weighted version of Lipton et al. (2004)’s envy cycle elimination algorithm (where an edge exists from agent to agent if and only if has weighted envy towards ) produces a complete allocation whenever agents have identical valuations, i.e., for some , , .

###### Proof.

From the construction of the algorithm (Lipton et al., 2004), it is evident that the (partial) allocation at the end of each iteration is guaranteed to be as long as we can find an agent, say , towards whom no other agent has weighted envy at the beginning of the iteration: we give the item under consideration to this agent and thus any resulting weighted envy towards can be eliminated by removing this item. The only way we would not find such an agent is if the weighted envy graph consisted of cycles only; we will show that this cannot be the case. Suppose that agents form a cycle (in that order) for some . Then, under identical valuations, we must have , which is a contradiction. Hence, the weighted envy graph can never have cycles, ensuring there is always an agent we can give an item to, while maintaining the property. ∎

###### Proposition 3.2.

If agents do not have identical valuation functions, then the weighted version of Lipton et al. (2004)’s envy cycle elimination algorithm may not produce a complete allocation, even in a problem instance with two agents having additive valuations.

###### Proof.

Consider a problem instance with and ; weights and ; and valuation functions such that:

Suppose that the weighted envy cycle elimination algorithm iterates over , and starts by allocating to agent due to, say, lexicographic tie-breaking. At this point, agent has weighted envy towards agent and not vice versa; moreover, this condition persists until items have all been allocated to agent . At this point, item also goes to agent , resulting in valuations and . Agent still has weighted envy towards agent since ; on the other hand, agent also develops weighted envy towards agent since . Thus, there is a cycle in the induced envy graph. For an unweighted envy graph, we would “de-cycle” the graph at this point by swapping bundles over the cycle and that would still maintain the invariant that the allocation is . However, if we swap bundles in this example, agent will end up having weighted envy up to more than one item towards agent since and for every .^{1}^{1}1In fact, the weighted envy is up to eight items. Interestingly, in this example the allocation up to is actually *without swapping*. However, the approach inspired by (Lipton et al., 2004) cannot even proceed to .^{2}^{2}2Another interesting feature of this example is that the two agents have *commensurable* valuations: .
∎

### 3.1 and Allocations for Two Agents via the Adjusted Winner Protocol

We now show that for two agents with additive valuations, we can achieve both and by adapting the classic adjusted winner protocol to the weighted setting. We first address instances where and for every , and then comment on what happens otherwise.

###### Theorem 3.3.

For two agents with additive valuations and arbitrary positive real weights, a complete, and allocation always exists, and can be computed in polynomial time.

###### Proof.

Assume first that both agents have positive value for all items. Our proof is constructive: we claim that the Weighted Adjusted Winner algorithm as delineated in Algorithm 1 produces an allocation satisfying the theorem statement.

First note that the left-hand side of the while loop condition is strictly increasing in and trivially exceeds the right-hand side for ;^{3}^{3}3For , we set the right-hand side to zero. hence, there always exists a which satisfies the stop criterion and the loop terminates at the smallest such .

#### property:

If the while loop ends with , let us denote by . Then, by construction, , which implies that agent is weighted envy-free towards agent up to one item (specifically this item ).

On the other hand, by construction, we also get that

(1) |

Moreover, due to the ordering of the ratios, we have

Combining with Inequality (1), dividing through by , and simplifying, we get

Thus, agent is also weighted envy-free towards agent up to one item (specifically ).

#### Pareto optimality:

First note that no incomplete allocation can be Pareto optimal since the realized valuation of either agent could be strictly improved by augmenting its bundle with a withheld item (under our assumption that each agent values each item positively). Since the allocation produced by Algorithm 1 is complete, it suffices to show that it cannot be Pareto dominated by an alternative complete allocation . Any such complete allocation can be thought of as being generated by transferring items between and .

Suppose that for an arbitrary complete allocation . Since , the above inequality implies that

(2) |

If , then (since ). Hence, so that cannot Pareto dominate . As such, we will assume that . Then, due to the ratio ordering and how are constructed,

By similar reasoning,

Combining with Inequality 2, we get

which contradicts the necessary condition for to Pareto dominate : . Assuming leads us to an analogous contradiction. Hence, must be Pareto optimal.

#### Complexity:

The ratios can be sorted in time. The while loop condition can be checked in time, so the total time taken by the while loop is . Hence, the algorithm runs in time.

Let us now address the scenario where there are items that are of zero value to an agent. Of course, items valued at zero by both agents can be safely assumed away. We will initialize the bundle with items valued positively by agent only, i.e., and . Then we run Algorithm 1 on the remaining items and use its output to augment the respective bundles. We will now show that the resulting allocation is and .

By the construction of the algorithm, there is an such that

since valuations are additive and . Thus, agent is weighted envy-free towards agent up to one item; the converse is also true by similar arguments.

Since a Pareto optimal allocation cannot be incomplete, it suffices to show that the (complete) allocation under consideration is not Pareto dominated by any complete allocation. Again, any complete allocation can be obtained from by swapping items between agents. It is evident that any allocation in which an item (resp., an item ) belongs to agent (resp., agent ) is Pareto dominated by the allocation wherein this item is given to agent (resp., agent ), everything else remaining the same. Hence, it suffices to show that a Pareto improvement cannot be achieved by swapping items in between the agents — but we already know this from the earlier part of the proof. ∎

### 3.2 Picking Sequence Protocols for Allocations with Agents

When all agents have equal weight, it is well-known that a round-robin algorithm, wherein the agents take turns picking an item, produces an EF1 allocation (see, e.g., Caragiannis et al. (2016)). We show next that in the general case we can construct a weight-dependent picking sequence, which guarantees for any number of agents and arbitrary weights.

###### Theorem 3.4.

For any number of agents with additive valuations and arbitrary positive real weights, a complete allocation always exists and can be computed in polynomial time.

###### Proof.

Our proof is constructive: we construct a picking sequence such that at each turn, an agent with the lowest weight-adjusted picking frequency picks the next item (Algorithm 2). We claim that after the allocation of each item, for any agent , every other agent is weighted envy-free towards up to the item that picked first.

To this end, first note that picking the agent who has had the minimum (weight-adjusted) number of picks thus far ensures that the first picks are a round robin over the agents; in this phase, the allocation is obviously since each agent has at most one item at any point. We will show that, after this phase, the algorithm generates a picking sequence over the agents with the following property:

###### Lemma 3.5.

Consider an agent chosen by Algorithm 2 to pick an item at some iteration , and suppose that this is not her first pick. Let and be the numbers of times agent and some other agent appear in the prefix of iteration in the sequence respectively, not including iteration itself. Then .

We will then show that this property is sufficient to ensure that the latest picker does not attract weighted envy up to more than one item towards herself after her latest pick:

###### Lemma 3.6.

Suppose that, for *every* iteration in which agent receives an item *after* her first pick, the numbers of times that agent and some other agent appear in the prefix of the iteration in the sequence— and respectively—satisfy the relation . Then, in the partial allocation up to and including ’s latest pick, agent is weighted envy-free towards up to the first item picked.

Obviously, after its latest picking, an agent’s envy towards others cannot get any worse. Since the partial allocation after the initial, round-robin phase is and every agent is weighted envy-free up to one item towards every picker due to Lemmas 3.5 and 3.6, the allocation is at every iteration. Hence, for the proof of correctness, it suffices to prove the two lemmas.

###### Proof of Lemma 3.5.

Since agent is picked at iteration , it must be the case that . This means that , i.e., since . ∎

###### Proof of Lemma 3.6.

Let . Consider any iteration in which agent is chosen. Let agent ’s values for the items allocated to agent in the latter’s second, third, , picks (the last one occurring at the iteration under consideration) be , , , respectively. If is the first item picked by agent and the partial allocation up to and including iteration , then clearly . Let the number of times agent appears in the prefix of agent ’s second pick be ; that between agent ’s second and third picks be ; ; that between agent ’s and picks be . Let agent ’s values for the items she herself picked during phase be respectively. Then, . Now, since and are the numbers of times agents and appear in the prefix of the latter’s pick respectively, the condition of the lemma dictates that

(3) |

Note that ; but can be zero for some — this corresponds to the scenario that agent picked more than once successively. Moreover, every time agent was chosen, she picked one of the items she values the most among those available, including the items picked by agent later. Hence, if , then

(4) |

Note that Inequality (4) holds trivially if since both sides are zero; hence it holds for every .

We claim that for each ,

To prove the claim, we proceed by induction on . For the base case , we have . Hence, from Inequality (4),

where we use due to Inequality (3). For the inductive step, assume that the claim holds for ; we will prove it for . If , We have

where the first inequality follows directly from the inductive hypothesis, the second from Inequality (4), and the third from Inequality (3) as well as the fact that . This proves the claim.

Now, taking in the claim, we get

where we use Inequality (3) again for the second inequality. This implies that , i.e., agent is weighted envy-free towards agent up to one item, concluding the proof. ∎

For the time-complexity, note that there are iterations of the while loop. In each iteration, determining the next picker takes time, while letting the picker pick her favorite item takes time. Since we may assume that (otherwise it suffices to allocate at most one item to every agent), the algorithm runs in time . ∎

If , a positive constant for every , then Algorithm 2 degenerates into the traditional round-robin procedure which is guaranteed to return an allocation for additive valuations, but may not be ; as such, Algorithm 2 may not produce a allocation either. This is easily seen in the following example: , ; ; , , , . With lexicographic tie-breaking for both agents and items, our algorithm will give us and , which is Pareto dominated by and .

## 4 allocations

In this section, we show that the allocation satisfies and , thereby generalizing Caragiannis et al. (2016)’s result from the unweighted setting. We also show that for any constant , this allocation may fail to be .

###### Theorem 4.1.

For any number of agents with additive valuations and arbitrary positive real weights, the allocation is always and Pareto optimal.

###### Proof.

Let be a allocation, with being the subset of agents having strictly positive realized valuations under . If it were not PO, there would exist an allocation such that for some and for every . If , we would have for every , which contradicts the assumption that is a largest subset of agents to which it is possible to give positive valuations simultaneously. If , then , which violates the optimality of the right-hand side. This proves that is PO.

As in Caragiannis et al. (2016), we will start by proving that is

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