# Fair in the Eyes of Others

Envy-freeness is a widely studied notion in resource allocation, capturing some aspects of fairness. The notion of envy being inherently subjective though, it might be the case that an agent envies another agent, but that she objectively has no reason to do so. The difficulty here is to define the notion of objectivity, since no ground-truth can properly serve as a basis of this definition. A natural approach is to consider the judgement of the other agents as a proxy for objectivity. Building on previous work by Parijs (who introduced "unanimous envy") we propose the notion of approval envy: an agent a_i experiences approval envy towards a_j if she is envious of a_j, and sufficiently many agents agree that this should be the case, from their own perspectives. Some interesting properties of this notion are put forward. Computing the minimal threshold guaranteeing approval envy clearly inherits well-known intractable results from envy-freeness, but (i) we identify some tractable cases such as house allocation; and (ii) we provide a general method based on a mixed integer programming encoding of the problem, which proves to be efficient in practice. This allows us in particular to show experimentally that existence of such allocations, with a rather small threshold, is very often observed.

## Authors

• 3 publications
• 4 publications
• 6 publications
• 8 publications
• ### Approximating Maximin Shares with Mixed Manna

We initiate the study of fair allocations of a mixed manna under the pop...
07/17/2020 ∙ by Rucha Kulkarni, et al. ∙ 0

• ### Fair Allocation of Indivisible Items With Externalities

One of the important yet insufficiently studied subjects in fair allocat...
05/16/2018 ∙ by Mohammad Ghodsi, et al. ∙ 0

• ### Fair-Share Allocations for Agents with Arbitrary Entitlements

We consider the problem of fair allocation of indivisible goods to n age...
03/07/2021 ∙ by Moshe Babaioff, et al. ∙ 0

• ### Fair Resource Sharing with Externailities

We study a fair resource sharing problem, where a set of resources are t...
12/08/2020 ∙ by Jiarui Gan, et al. ∙ 0

• ### Fair and Truthful Mechanism with Limited Subsidy

The notion of envy-freeness is a natural and intuitive fairness requirem...
05/05/2021 ∙ by Hiromichi Goko, et al. ∙ 0

• ### Fair Division through Information Withholding

Envy-freeness up to one good (EF1) is a well-studied fairness notion for...
07/04/2019 ∙ by Hadi Hosseini, et al. ∙ 0

• ### Achieving Proportionality up to the Maximin Item with Indivisible Goods

We study the problem of fairly allocating indivisible goods and focus on...
09/20/2020 ∙ by Artem Baklanov, et al. ∙ 0

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## 1 Introduction

Fair division is an ubiquituous problem in multiagent systems, economics [28, 20, 29], with applications ranging from allocation of schools, courses or rooms to students [1, 21], division of goods in inheritance or divorce settlement [7]. Envy-freeness (EF), is one of the prominent notions studied in fair division [13, 6, 18, 11, 26]. An allocation of items among a set of agents is said to be envy-free if no agent prefers the share of another agent to her own share. Unfortunately, envy-freeness is a pretty demanding notion and an envy-free allocation may not exist.

Now consider a given problem where no envy-free allocation can be returned, but suppose instead that two allocations make a single agent (say, ) envious of some other agent (for simplicity). Now assume that in allocation , is the only agent to prefer the bundle of over her own, while in allocation all the other agents agree on the fact that should indeed envy . According to Parijs [22], exhibits unanimous envy, and there seems to be no situation where should be returned in place of . Inspired by this notion, we introduce in this paper the notion of -approval envy, as a way to retrieve a continuum between envy-freeness and unanimous envy. As may be clear from the name, the idea is simply to ask agents to express their own view about envy relations expressed by other agents. The objective will thus be to seek allocations minimizing social support for the expressed envy relations, i.e. minimizing the number of agents approving the envy. Of course, this approach may be controversial: after all, the notion of preference is inherently subjective. Introducing this flavour of objectivity may lead to undesirable consequences. At the extreme, one may simply replace individual preferences by some unanimous “mean” profile, thus profoundly changing the very nature of the notion. We believe there are several justifications to investigate this new approach:

• First, note that we only seek the approval of other agents in the case the agent herself explicitly expresses envy: absence of envy thus remains completely subjective. While a symmetrical treatment may also be justifiable in some situations, there is an obvious reason which motivates us to start with the proposed definition, namely the fact that the notion would no longer be a relaxation of envy-freeness.

• Secondly, all other things being equal, we believe an allocation minimizing is socially more desirable. We do not necessarily regard this notion as a compelling choice, but we think this can enrich the picture of fallback allocations when no envy-free allocation exists, as other relaxations do [2].

• Finally, one further motivation of our work that we would like to emphasize is that our approach can be seen as providing guidance regarding agents and more specifically agents’ preferences which could be focused on, in order to progress towards envy-freeness. In particular, if we envision systems integrating deliberative phases in the collective decision-making process, our model could be used to set the agenda of such deliberations. If a vast majority of agents contradict an agent on her envy towards another agent, it may indicate for instance that she lacks information regarding the actual value of (some items of) her share. Initiating a discussion might help to solve such “objectively unjustified” envies when they occur.

#### Outline of the paper.

The remainder of this paper is as follows. Section 2 recalls some basic notions in fair division. Our notion of -approval envy is presented in Section 3. Some properties of this notion are then studied in Section 4: it is shown in particular, that if the hypothetical situation of allocation described at the beginning of the introduction occurs, then an EF allocation must also exist. We also show that our notion inherits from the complexity of related problems. This motivates the MIP formulation that we detail in Section 5. We next turn to the House Allocation setting and we show that if each agent exactly holds a single item, then an efficient algorithm allows for returning an allocation minimizing the value of . One caveat of our notion is that (unlike other relaxations) it is not guaranteed to exist, as intuitively observed in the case of unanimous envy. We thus consider greatly important to provide empirical evidence showing that both in different synthetic cultures as well as with real datasets, allocations with reasonable values of exist.

## 2 Model and Definitions

We consider MultiAgent Resource Allocation problems (MARA) where we aim at fairly dividing a set of indivisible goods (also called items or objects) among a set of agents. A MARA instance is defined as a finite set of objects , a finite set of agents and a profile of preferences representing the interest of each agent towards the objects. An allocation is a mapping of the objects in to the agents in . In the following, will denote the set of objects (the share) held by agent . An allocation is such that with (a given object cannot be allocated to more than one agent) and (all the objects from are allocated).

In this paper, we consider cardinal preference profiles so, the preferences of an agent over bundles of objects is defined by a utility function measuring her satisfaction when she obtains share . We make the assumption that utility functions are additive i.e. the utility of an agent over a share is defined as the sum of the utilities over the objects forming :

 ui(πi)\tiny{def}=∑ok∈πiu(i,k),

where is the utility given by agent to object . This assumption is commonly considered in MARA [18, 24, 12, 9, for instance] as additive utility functions provide a compact but yet expressive way to represent the preferences of the agents. MARA instances with additive utility functions are called add-MARA instances for short.

Different notions have been proposed in the literature to value the fairness of an allocation. When the agents can compare their shares, the absence of envy [13, 18, 10] is a particularly relevant notion of fairness. An agent would envy another agent if she prefers the share of over her own share. More formally, an agent envies an agent iff

 ui(πj)>ui(πi)

A completely fair allocation would thus be an envy-free allocation i.e. an allocation where no agent envies another agent. Formally:

 ∀ai,aj∈N,ui(πi)≥ui(πj)

The notion of envy-freeness conveys a natural concept of fairness viewed as social stability: agents are happy with their bundle and hence would not want to swap it with any other agent’s (regarding their own preferences). However, as soon as it is required to allocate all the objects in , an envy-free allocation may not exist. An alternative objective may be to minimize a degree of envy of the society [18, 10], based on the notion of paiwise envy.

###### Definition 1 (Pairwise envy).

Let be an allocation. The pairwise envy of an agent towards an agent in is defined as follows:

 pe(i,j,π)\tiny{def}=max{0,ui(πj)−ui(πi)}.

The pairwise envy can be interpreted as how much agent envies agent ’s share (this envy being 0 if does not envy ). We can derive from this notion a collective measure of envy:

###### Definition 2 (Degree of envy of the society).

The degree of envy of the society for an allocation is defined as follows:

 de(π)\tiny{def}=∑ai∈N∑aj∈Npe(i,j,π)

Note that an allocation is envy-free if and only if .

To cope with the possible inexistence of an envy-free allocation, another approach is to alleviate the requirements of the fairness notion. Recently, several relaxations of envy-freeness have been proposed such as envy-freeness up to one good (EF1) [8], envy-freeness up to any good (EFX) [9]. An allocation is said to be envy-free up to one good (resp. up to any good) if no agent envies the share of another agent after removing from one (resp. any) item. Existence for EF1 is guaranteed, but this is still to the best of our knowledge an open question for EFX. Amanitidis et al. [2] studied the relations between some fairness notions and their relaxations.

## 3 K-Approval Envy

The notion of envy being inherently subjective, it might be the case that an agent envies another agent, but that she objectively has no reason to do so. The difficulty here is to define the notion of objectivity, since no ground-truth can properly serve as a basis of this definition. In her book, Guibet-Lafaye [17] recalls the notion of unanimous envy, that was initially discussed in the book by Parijs [22], and that can be defined as follows: an agent unanimously envies another agent , if all the agents think that indeed envies . Here, unanimity is used as a proxy for objectivity.

As we can easily imagine, looking for allocations that are free of unanimous envy will be too weak to be interesting: as soon as one agent disagrees with the fact that envies , this potential envy will not be taken into account. Here, we propose an intermediate notion between envy-freeness and (unanimous envy)-freeness:

###### Definition 3 (K-approval envy).

Let be an allocation, be two different agents, and be an integer. We say that -approval envies (-app envies for short) if there is a subset of agents including such that:

 ∀ak∈NK,uk(πi)

In other words, at least agents amongst agree with on the fact that she should actually envy agent .

###### Example 1.

Let us consider the following add-MARA instance with 3 agents and 6 objects:

 o1o2o3o4o5o6a10\framebox3313\framebox2a2\framebox207\framebox2\framebox10a303\framebox5013

Note that there is no envy-free allocation for this instance. In the squared allocation, is not envious, envies and envies . Concerning the envy of towards , disagrees with being envious of whereas agent agrees. Hence, agent 2-app envies agent . Concerning the envy of towards , agent agrees with being envious of whereas agent does not. Hence, 2-app envies .

Note that in the definition, as soon as does not envy , then, does not -app envy , no matter what the value of is or how many agents think that should actually envy .

###### Observation 1.

Given an allocation of an add-MARA instance, if -app envies in , then -app envies in .

Moreover, if -app envies , we will say that unanimously envies . Finally, we can observe that -app envies if and only if envies .

We can naturally derive from Definition 3 the counterpart of envy-freeness:

###### Definition 4 ((K-approval envy)-free allocation).

An allocation is said to be (-app envy)-free if and only if does not -app envy for all pairs of agents .

###### Definition 5 ((K-approval envy)-free instance).

An add-MARA instance will be said to be (-app envy)-free if and only if it accepts a (-app envy)-free allocation.

###### Example 2.

Going back to Example 1, the squared allocation is (-app envy)-free so the instance is (-app envy)-free.

A threshold of special interest is obviously , since it requires a strict majority to approve the envy under inspection. A Strict Majority approval envy-free (SM-app-EF) allocation is a (-app envy)-free allocation such that , translating the fact that every time envy occurs, there is a strict majority of agents that do not agree with that envy.

Going further, it is important to notice that (-app envy)-freeness is not guaranteed to exist. Indeed, for all number of agents and all number of objects , there exist instances for which no (-app envy)-free allocation exists, no matter what is. Suppose for instance that all the agents rank the same object (say ) first, and that for all , . Then obviously, everyone agrees that all the agents envy the one that will receive . Such instances will be called unanimous envy instances:

###### Definition 6 (Unanimous envy instance).

An add-MARA instance will be said to exhibit unanimous envy if is not (-app envy)-free for any value of .

Observe that for an allocation to be (-app envy)-free, for all pairs of agents , , either or at least agents have to think that does not envy . Notice that it is different from requiring that at least agents think that this allocation is envy-free. This explains the parenthesis around (-app envy): this notion means “free of -app envy”, which is different from “-app-(envy-free)”.

A useful representation, for a given allocation, is the induced envy graph: vertices are agents, and there is a directed edge from to if and only if envies [18]. An allocation is envy-free if and only if the envy graph has no arc. In our context, we can define a weighted notion of the envy graph.

###### Definition 7 (Weighted envy graph).

The weighted envy graph of an allocation is defined as the weighted graph where nodes are agents, such that there is an edge if envies , with the weight corresponding to the number of agents (including ) approving this pairwise envy in .

Our notion of -approval envy can be interpreted as a vote on envy, that works as follows. For each pair of agents , if declares to envy , we ask the rest of the agents to vote on whether they think that indeed envies . Then, a voting procedure is used to determine whether envies according to the society of agents. Several voting procedures can be used. However, since there are only two candidates (yes / no), the most reasonable voting rules are based on quotas: envies if and only if there is a minimum quota of agents that think so.111More precisely, these rules exactly characterize the set of anonymous and monotonic voting rules [23].

## 4 Some Properties of K-App Envy

There are natural relations between the different notions of (-app envy)-freeness, for different values of . The following observation is a direct consequence of Observation 1.

###### Observation 2.

Let be an allocation, and be an integer. If is (-app envy)-free, then is also (-app envy)-free.

However, the converse does not hold. More precisely, the following proposition shows that the implication stated in Observation 1 may be strict.

###### Proposition 1.

Let be an allocation, and be an integer. If is (-app envy)-free, is not necessarily (-app envy)-free.

###### Proof.

Let be an integer, and let us consider the instance with agents and objects defined as follows:

• ;

• for ;

• for ;

• and for ;

and for other pairs with .

Consider the allocation where each agent gets item . Obviously, the only envy in this allocation concerns towards . Moreover, only agree on this envy. Therefore, -app envies , but does not -app envy her. Moreover, is (-app envy)-free, but not (-app envy)-free. ∎

###### Example 3.

In order to illustrate the previous proof, let us consider the following instance with 4 agents, 4 objects (and =3) and the squared allocation :

 o1o2o3o4a1\framebox$1$εεεa212\framebox$12$εεa3εε\framebox$1$εa4251515\framebox$15$

In this allocation, the only envy concerns towards . Moreover, only and agree with on her envy. Hence, is (4-app envy)-free but is obviously not (3-app envy)-free as we can find 3 agents (, and ) agreeing on the envy of towards (in other words 3-app envies ).

###### Proposition 2.

For any , there exist instances which are (-app envy)-free but not (-app envy)-free.

###### Proof.

Consider the same instance as in Proposition 1. We have already shown that we have an allocation that is (-app envy)-free which means that the instance is (-app envy)-free. We just have to show that there is no (-app envy)-free allocation in order to conclude. In that purpose, we first note that each agent has to get one and exactly one object. Indeed, if it is not the case at least one agent will have no object and will thus be envious of any agent that has an object. Moreover, as all agents value the empty bundle with utility 0 and every object is valued with a strictly positive utility, this envy will be unanimous. Hence, each agent has to get one and exactly one object in order to minimize the (-app envy)-freeness. Now consider objects for . The agents that receive an object and that are envious will -app envy the agent that received . Indeed, agents for value objects with a utility higher than (or equal to) the one of (and thus do not approve the envy) while it is the opposite for the other agents who are exactly hence the -app envy. So if we want to avoid that envy, we have to give the objects to agents so that they do not experience envy at all but it is not possible as such agents are agents for . It means that we have agents that have to receive one of the objects which is obviously impossible. This means that we cannot avoid -app envy which implies that no allocation can be (-app envy)-free. ∎

Proposition 2 proves that the hierarchy of -app envy instances is strict for . Rather surprisingly, we will see that it is not the case for .

In order to show this result, we will resort to a tool that has been proved to be really useful and powerful in many contexts dealing with envy [5, 3, 4]: the “bundle reallocation cycle technique”. This technique, originating from the seminal work of Lipton et al. [18], consists in performing a cyclic reallocation of bundles so that every agent is strictly better in the new allocation. Thus, such a reallocation corresponds to a cycle in the opposite direction of the edges in the — weighted — envy graph introduced in Definition 7. It is known that performing a reallocation cycle decreases the degree of envy [18]. Unfortunately, our first remark is that it does not necessarily decrease the level of -app envy. Worse than that, it can actually increase it:

###### Proposition 3.

Let be a (-app envy)-free allocation, for . After performing an improving bundle reallocation cycle (even between two agents), the resulting allocation may be (-app envy)-free (and not (-app envy)-free) such that .

###### Proof.

Let be an integer, and let us consider the instance with agents and objects defined by the following utility functions:

• : ,,;

• : ,;

• : ;

• for : for ;

• for : for ;

and for other pairs.

Consider the allocation where each agent gets item . Obviously, the only envy in this allocation concerns towards (approved by and agents ) and (approved by , and agents ), and the envy of towards (approved by and agents ). Hence the allocation is (-app envy)-free. We now consider the allocation resulting from the improving bundle reallocation cycle between and . We note that the only envy in is the one of towards . Moreover, this envy is approved by herself, and agents and . The allocation is thus (-app envy)-free and not (-app envy)-free as if then . ∎

Now consider a slight generalization of Lipton’s cycles, weakly improving cycles, that correspond to a reallocation of bundles where all the agents in the cycle receive a bundle they like at least as much as the one they held, with one agent at least being strictly happier. Of course, our example of Proposition 3 still applies. On the other hand, this notion suffices to guarantee the decrease of the degree of envy (note that identifying the cycles themselves may not be easy any longer, but this is irrelevant for our purpose). The proof, omitted, follows directly from the arguments of Lipton [18].

###### Observation 3.

Let be an allocation, and the allocation obtained after performing a weakly improving cycle. It holds that .

We now show that (-app envy)-freeness exhibits a special behaviour: in contrast with Proposition 3, improving cycles (in fact, even weakly improving cycles) enjoy the property of preserving the (-app envy)-freeness level of an allocation. We provide this result for swaps (cycles involving two agents only) as this is sufficient to establish our main result.

###### Lemma 1.

Let be a (-app envy)-free allocation. After performing a weakly improving bundle reallocation cycle between two agents, the resulting allocation is (-app envy)-free, with .

###### Proof.

Let us consider the two agents that are involved in the weakly improving bundle reallocation cycle. We respectively call and the initial and the resulting allocations (hence ,). First note that apart from and , the approval envy of the agents does not change as their bundle remains the same from to . So, we just have to show that it is not possible for (or ) to experience -app envy with in (w.l.o.g. we can focus on only as the same proof holds for ). Indeed, if such -app envy exists in , it means that this allocation would be at least (-app envy)-free. Let us consider for the sake of contradiction that does indeed experience -app envy with . This means that there is an agent that -app envies, i.e. and that there is some agent that approves this envy such that . Note that as can obviously be neither neither so the bundle of remained the same. However, we know (otherwise would have -app envied in thus contradicting the (-app envy)-freeness of ) and (otherwise would have -app envied in thus contradicting the (-app envy)-freeness of ) which obviously implies but also equals to and . So which is impossible because otherwise would have -app envied in thus contradicting the (-app envy)-freeness of . ∎

Putting together Lemma 1 and Observation 3 allows us to prove that (2-app envy)-freeness is essentially a vacuous notion, in the sense that any instance enjoying an allocation with this property will have an EF allocation as well.

###### Proposition 4.

If an add-MARA instance is (-app envy)-free then it is also envy-free.

###### Proof.

Take as being an arbitrary (-app envy)-free allocation. First note that if there is no envious agent in then, by definition, is envy-free and the proposition holds. Suppose that envies . Observe that cannot agree with , because otherwise would not be (2-app envy)-free. Hence, we can perform a weakly improving bundle reallocation cycle between those two agents and call the resulting allocation . If is envy-free then we are done. Otherwise, thanks to Lemma 1, we know that is still (2-app envy)-free, and by Observation 3 that the degree of envy has strictly decreased. We can apply the same argument as above with two agents envying each other and swap their bundles. The process stops when the current allocation is envy-free. The process is guaranteed to stop because the degree of envy of the society is bounded below by zero and the degree of envy of the society decreases at each step until it equals zero (which corresponds to an envy-free allocation). ∎

Another consequence is that, for two agents, instances fall either in the envy-free or unanimous envy category:

###### Corollary 1.

In the special case of 2 agents, if there is no envy-free allocation in then is a unanimous envy instance.

#### Complexity.

We conclude with a few considerations on the computational complexity of the problems mentioned so far. First of all, as envy-freeness is (-app envy)-freeness, the problem of finding the minimum for which there exists a (-app envy)-free allocation is at least as hard as determining whether an envy-free allocation exists.

One may also wonder how hard the problem of determining whether a given instance exhibits unanimous envy or not, i.e. whether a (-app envy)-free allocation exists for some value of . For this question, instances where agents all have the same preferences provide insights.

###### Proposition 5.

For any add-MARA instance, if all the agents have the same preferences then the notions of (-app envy)-freeness and (-app envy)-freeness coincide.

###### Proof.

We already know from Observation 2 that (-app envy)-freeness implies (-app envy)-freeness for any add-MARA instance. So we just have to prove that if all the agents have the same preferences then (-app envy)-freeness implies (-app envy)-freeness. If an allocation is (-app envy)-free then it means that for any pair of agents, does not envy or there is at least one agent that disagrees on the envy of towards . Obviously, if for every pair of agents , we have envy-free towards then the allocation is envy-free and the proof concludes. Besides, for every pair of envious/envied agents there is at least one agent disagreeing on the envy. But all the agents have the same preferences so it means that every agent should agree with each other. Hence, no envied agent can exist and we have (-app envy)-freeness of allocation . ∎

From Proposition 5 we get that the problem of deciding the existence of unanimous envy is at least as hard as deciding the existence of an EF allocation when agents have similar preferences. As membership in NP is direct, we thus get as a corollary that:

###### Corollary 2.

Deciding whether an allocation exhibits unanimous envy is NP-Complete.

## 5 A Mip Formulation for K-App Envy

We have seen in the previous section that the problem of determining, for a given instance , the minimal value of such that a (-app envy)-free allocation exists inherited from the high complexity of determining whether an envy-free allocation exists.

To address this problem, we present in this section a Mixed Integer linear Program that returns, for a given add-MARA instance

, a (-app envy)-free allocation with the minimal and no solution when is a unanimous envy instance. In this MIP, we use Boolean variables (we use bold letters to denote variables) to encode an allocation: if and only if gets item . We also introduce Boolean variables such that if and only if according to ’s preferences envies agent . We also need to add Boolean variables used to linearize the constraints on . Finally, we use an integer variable corresponding to the -app envy we seek to minimize.

In this section, we assume that all the utilities are integers. If they are not (recall that they are still in ) we can transform the instance at stake into a new one only involving integral utilities by multiplying them by the least common multiple of their denominators.

We first need to write the constraints preventing an item from being allocated to several agents:

 n∑i=1zji=1∀j∈⟦1,m⟧ (1)

By adding these constraints we also guarantee completeness of the returned allocation (all the items have to be allocated to an agent).

Secondly, we have to write the constraints that link the variables with the allocation variables :

 ∀k,i,h∈⟦1,n⟧,m∑j=1u(k,j)(zjh−zji)>0⟹ekih=1

As the utilities are integers, we can replace by . In order to linearize this implication between two constraints we introduce a number that can be arbitrarily chosen such that :

 Mekih≥m∑j=1u(k,j)(zjh−zji) ∀k,i,h∈⟦1,n⟧ (2) m∑j=1u(k,j)(zjh−zji)≥1−M(1−ekih) ∀k,i,h∈⟦1,n⟧ (3)

Finally, we have to write the constraints that convey the fact that the allocation we look for is (-app envy)-free:

 eiih=0∨n∑k=1ekih≤K−1∀i,h∈⟦1,n⟧

Since are Boolean variables, we can replace by . Now, this logical constraint is linearized as follows:

 eiih≤xih ∀i,h∈⟦1,n⟧ (4) n∑k=1ekih≤K−1+n(1−xih) ∀i,h∈⟦1,n⟧ (5)

We can now put things together. Let be an instance. Then, we will denote by the MIP defined as:

 minimizeKsuch thatzji,ekih,xih∈{0,1}∀k,i,h∈⟦1,n⟧,j∈⟦1,m⟧K∈⟦1,N⟧+ Constraints (???, ???, ???, ???, ???)
###### Proposition 6.

Let be an instance. Then, there is an optimal solution with to if and only if is an (-app envy)-free instance and not an (( envy)-free one. Moreover, does not admit any solution if and only if is an unanimous envy instance.

The proof of this proposition is not very involved and will thus be omitted. The key here is to show that there is a solution to the MIP such that iff the corresponding allocation such that if and only if is (-app envy)-free. The most critical point is to show that Constraints 2 and 3 are indeed a valid translation of the logical implication, and that Constraints 4 and 5 correctly encode the logical or. The rest follows easily.

As the problem is difficult in the general case, it is natural to seek special cases that could be solved efficiently.

## 6 House Allocation

The House Allocation Problem (HAP for short) is a standard problem where there are exactly as many items as agents, and each agent receives exactly one resource. This setting is relevant in many situations and has been extensively studied [27, 25, 1, to cite few of them]. In House Allocation Problems, computing an envy-free allocation comes down to solving a matching problem, since an envy-free allocation exists if and only if all the agents get (one of) their top item(s). It is therefore natural to wonder whether an allocation minimizing -app envy could also be computed efficiently.

Our first observation hints in that direction. Indeed, characterizing unanimous envy becomes easy in house allocation problems.

###### Proposition 7.

Let be an instance of HAP. is an unanimous envy instance if and only if there exists at least a pair of items such that all agents strictly prefer over .

###### Corollary 3.

Checking whether an instance of HAP is a unanimous envy instance or not can be done in .

From this characterization we can also derive a result on the likelihood that unanimous envy exists when the utilities are uniformly ditributed (that is, for each agent and object

, utilities are drawn i.i.d. following the uniform distribution on some interval

).

###### Proposition 8.

Under uniformly distributed preferences, the probability of unanimous envy is upper bounded by

.

###### Proof.

Wlog. suppose agent has preferences . The probability of the event is strictly preferred to by one agent is if preferences are strict. As preferences are not strict, this probability becomes an upper bound (think for instance if the agent values all the objects the same then the probability to have strict preference between two objects is zero). Hence, the probability of the event is strictly preferred to by all agents is upper bounded by as the preferences between the agents are independent. Assuming, for all pairs of items, these events to be independent (which is not the case, hence an upper bound of the upper bound), we derive our result by summing up over the possible pairs. ∎

Note that this value quickly tends towards : unanimous envy is thus already unlikely to occur for 10 agents.

We will now show here that finding an allocation that minimizes (-app envy)-freeness can be done in polynomial time. Before introducing the idea, we need an additional notation. For any pair , let denote the number of agents strictly preferring to . For any agent and object , we will also define as follows:

 maxEnvy[i][j]=maxoj′ s.t. u(i,j′)>u(i,j)#≺(j,j′)

In other words, denotes the maximal value of among the objects that are strictly preferred to by . As we can imagine, this will exactly be the value of the -app envy experienced by if she gets item (note that if is among ’s top objects, this value will be 0).

The key to the algorithm is to see that for a given , determining whether a (-app envy)-free allocation exists can be done in polynomial time by solving a matching problem. Namely, for each , we build the following bipartite graph: is the set of nodes, and we add an edge if and only if is lower than or equal to . We can observe that any perfect matching in this graph corresponds to a (-app envy)-free allocation. The only thing that remains to do is to run through all possible values of , which can be done by dichotomous search between and . This is formalized in Algorithm 1.

###### Proposition 9.

For any HAP instance, we can find (one of) its optimal (-app envy)-free allocations in .

###### Proof.

First, the computation of the matrix maxEnvy runs in . Indeed, to compute we first need to compute which already runs in as we have to ask for each couple of objects ( in total) the point of view of all the agents ( in total). From that, as , we can compute in . As there are different pairs we have the final complexity of computing maxEnvy.

Due to the dichotomous search, the algorithm needs to solve perfect matching problems, that can be solved in [16]. The global complexity of Algorithm 1 is thus . ∎

## 7 Experimental Results

We present here the results of the numerical tests we have conducted. These experiments serve two purposes: (i) evaluate the behaviour of the MIP we presented in Section 5 and of the polynomial algorithm described in Section 6, and (ii) observe how our notion of -app envy depends on the number of agents, of items, and on the type of preferences. All the tests presented in this section have been run on an Intel(R) Core(TM) i7-2600K CPU with 16GB of RAM and using the Gurobi solver to solve the Mixed Integer Program. We have tested our methods on three types of instances: Spliddit instances [15], instances under uniformly distributed preferences and instances under an adaptation of Mallows distributions to cardinal utilities [14].

### 7.1 Spliddit instances

We have first experimented our MIP on real-world data from the fair division website Spliddit [15]. There is a total of 3535 instances from 2 agents to 15 agents and up to 93 items. Note that 1849 of these instances involve 3 agents and 6 objects. By running the MIP with a timeout of 10 minutes (after this duration the best current solution, if it exists, is returned) we were able to solve all the instances but 6. Among these 6 instances, 3 of them were HAP instances that we managed to solve optimally with Algorithm 1. This only leaves us with 3 instances, for which the solver did return a solution but did not prove that it is optimal (within a timeout of 10 minutes). Besides, 65% of the instances are EF while 23% of the instances exhibit unanimous envy. Moreover, 28% of the remaining instances with more than 5 agents are SM-app EF.

### 7.2 Uniformly distributed preferences

#### General setting

We also ran tests on instances under uniformly distributed preferences, with varying from 3 to 10 and such that we produce settings where few EF allocations exist [12]. For each problem size, we kept 60 instances that admit no EF allocation as we wanted to measure the behaviour of our notion when no such allocation exists (we know that if an EF allocation exists it will be returned by our methods). As we are in the general setting we solved the instances via the MIP with a timeout of 60 seconds.

The first three rows of Table 1 respectively represent the percentage of instances that have been solved to optimal (a solution has been returned before the timeout), the percentage of unanimous envy instances and the percentage of SM-app-EF instances. We then have the mean value of . Finally, we store the mean computation time (in seconds) of the instances (solved to optimal).

First note that considering 2 agents is a special case as shown in Corollary 1. Indeed, as we have removed the EF instances, all the remaining instances are unanimous envy ones. Moreover, we observe that the percentage of SM-app-EF allocations is zero for 4 agents. Indeed, an allocation is SM-app-EF for 4 agents if there exists a (-app envy)-free allocation such that . As we have removed all the EF instances, we know (from Proposition 4) that we cannot find an SM-app-EF allocation. The same holds for 3 agents.

We can notice that the mean seems to be stabilising around 0.6. Besides, without any surprise, the computation time rapidly increases while the percentage of instances solved to optimal (under a timeout of 60 seconds) starts decreasing for 7 agents. Finally, positive results can be pinpointed: the very low percentage of unanimous envy instances, and the pretty high percentage of SM-app-EF ones.

#### House allocation

We have also tested our polynomial algorithm on HAP instances under uniformly distributed preferences. We have generated 20 instances for each number of agents from 5 to 100 agents (and objects) by steps of 5.

First note that we have only found 5 unanimous envy instances and all of them involved 5 agents. This supports the probability of unanimous envy instance showed in Proposition 8 and the fact that it decreases very quickly towards 0. Moreover, like for the general setting, we notice a convergence of the values towards 0.6. The algorithm runs, without any surprise (in light of Proposition 9) much faster than our MIP. Indeed, the mean runtime for 100 objects and agents is still around 2 seconds only whereas we already observed that our MIP cannot solve easier problems within 10 minutes.

### 7.3 Correlated preferences

In strict ordinal settings, a classical way to capture correlated preferences is to use Mallows distributions [19] allowing us to measure the impact of the similarity of the preferences between agents. In these experiments, we used a generalization of the Mallows distribution to cardinal preferences presented in [14] based on Von Mises–Fisher distributions. Similarly to the dispersion parameter in Mallows distributions, the similarity between the preferences of the agents is tuned by the concentration parameter: when it is zero agents’ preferences are uniformly distributed, whereas when it is infinite agents have the same preferences.

We expected that the more similar the preferences between the agents are, the higher the degree of -app envy would be and the more likely unanimous envy would occur. The results of our experiments both in the general setting and in HAP support this: the number of EF instances is decreasing along with the concentration value, and from a given threshold, all the instances exhibit unanimous envy. However, the exact correlation between the level of (-app) envy-freeness and the concentration deserves further study, especially for very low values of . Intuitively, in some circumstances, correlation of preferences may indeed help to find large majorities of agents that contradict an agent envy, while this situation is unlikely under uniformly distributed preferences.

## 8 Conclusion

In this paper, we have introduced a new relaxation of envy-freeness. This relaxation uses a consensus notion, approval envy, as a proxy for objective envy between pairs of agents. We have proposed algorithms to compute an allocation minimizing the -app envy, and we have experimentally shown that this notion makes sense in practice in situations where no envy-free allocation exists.

This work also opens to a more general study of consensus-based notions of envy. For instance, instead of focusing on approval envy between agents, one could also be interested in using consensus to determine whether a given agent should be envious in general or not. More generally, one could also look for allocations that are judged envy-free by a given quota of agents. We leave the study of these notions for future work.

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