Achieving Proportionality up to the Maximin Item with Indivisible Goods
Artem Baklanov, Pranav Garimidi, Vasilis Gkatzelis, Daniel Schoepflin
aa r X i v : . [ c s . G T ] S e p Achieving Proportionality up to the Maximin Itemwith Indivisible Goods
Artem Baklanov a , Pranav Garimidi b , Vasilis Gkatzelis c , and Daniel Schoepflin ca Higher School of Economics, St. Petersburg b Conestoga High School c Drexel University
Abstract
We study the problem of fairly allocating indivisible goods and focus on the classic fairness notion of proportion-ality. The indivisibility of the goods is long known to pose highly non-trivial obstacles to achieving fairness, and avery vibrant line of research has aimed to circumvent them using appropriate notions of approximate fairness. Recentwork has established that even approximate versions of proportionality (PROPx) may be impossible to achieve evenfor small instances, while the best known achievable approximations (PROP1) are much weaker. We introduce thenotion of proportionality up to the maximin item (PROPm) and show how to reach an allocation satisfying this notionfor any instance involving up to five agents with additive valuations. PROPm provides a well-motivated middle-ground between PROP1 and PROPx, while also capturing some elements of the well-studied maximin share (MMS)benchmark: another relaxation of proportionality that has attracted a lot of attention.
We consider the well-studied problem of fairly distributing a set of scarce resources among a group of n agents. Thisproblem is at the heart of the long literature on fair division, initiated by Steinhaus [20], which has recently receivedrenewed interest, partly due to the proliferation of automated resource allocation processes. To reach a fair outcome,such processes need to take into consideration the preferences of the agents, i.e., how much each agent values eachof the resources. The most common modelling assumption regarding these preferences is that they are additive : eachagent i has a value v ij ≥ for each resource j , and her value for a set S of resources is v i ( S ) = P j ∈ S v ij . But, whatwould constitute a “fair” outcome given such preferences?One of the classic notions of fairness is proportionality . An outcome satisfies proportionality if the value of everyagent for the resources that were allocated to them is at least a /n fraction of her total value for all of the resources.For the case of additive valuations, if M is the set of all the resources, then every agent i should receive a value ofat least n P j ∈ M v ij . This captures fairness in a very intuitive way: since there are n agents in total, if they were tosomehow divide the total value equally among them, then each of them should be receiving at least a /n fraction ofit; in fact they could potentially all receive more than that if they each value different resources. However, it is well-known that achieving proportionality may be impossible when the resources are indivisible , i.e., cannot be dividedin smaller parts and shared among the agents. This can be readily verified with the simple example involving only asingle indivisible resource and at least two agents competing for it. In this case, whoever is allocated that resource willreceive all of her value but all other agents will receive none of it, violating proportionality.In light of this impossibility to achieve proportionality in the presence of indivisible resources, the literature hasturned to relaxations of this property. A natural candidate would be a multiplicative approximation of proportionality,aiming to guarantee that every agent receives at least a λ/n fraction of their total value, for some λ ∈ [0 , . However,the single resource example provided above directly implies that no λ > is small enough to guarantee the existence ofsuch an approximation. As a result, research has instead considered additive approximations, leading to two interestingnotions: PROP1 and PROPx. These relaxations allow the value of each agent i to be less than a /n fraction of hertotal value, but by no more than some additive difference d i . For the case of PROP1, d i corresponds to the maximum value of agent i over all the items that were allocated to some other agent [9]. For the case of PROPx, d i corresponds to1he minimum value of agent i over all the items that were allocated to some other agent [3]. On one hand, PROP1 is abit too forgiving, and is known to be easy to satisfy, while on the other PROPx is too demanding and is not guaranteedto exist even for instances with just three agents.In a parallel line of work, an alternative relaxation that has received a lot of attention is the maximin share (MMS) [6]. According to this notion, every agent’s “fair share” is defined as the value that the agent could secureif she could choose any feasible partition of the resources into n bundles, but was then allocated her least preferredbundle among them. It is not hard to verify that this benchmark is weakly smaller than the one imposed by pro-portionality, yet prior work has shown that this too may be impossible to achieve, even for instances with just threeagents.In this paper we propose PROPm, a new notion that provides a middle-ground between PROP1 and PROPx, whilealso capturing the “maximin flavor” of the MMS benchmark, and we prove that there always exists an allocationsatisfying PROPm for any instance involving up to five agents. The proportionality up to the most valued item (PROP1) notion is a relaxation of proportionality that was introducedby Conitzer et al. [9], who observed that there always exists a Pareto optimal allocation that satisfies PROP1. Azizet al. [2] later extended this notion to settings where the objects being allocated are chores, i.e., the valuations arenegative, and very recently Aziz et al. [3] provided a strongly polynomial time algorithm for computing allocationsthat are Pareto optimal and PROP1 for both goods and chores. On the other extreme, it is known that the notionof proportionality up to the least valued item (PROPx) may not be achievable even for small instances with threeagents [18, 10, 3].The PROP1 and PROPx notions are analogs of relaxations that have been proposed and studied for another veryimportant notion of fairness: envy-freeness (EF). An allocation is said to be envy-free if no agent would prefer tobe allocated some other agent’s bundle over her own. The example with the single indivisible item discussed in theintroduction shows that envy-free outcomes may not exist, which motivated the approximate fairness notions of envy-freeness up to the most valued item (EF1) [6] and envy-freeness up to the least valued item (EFx) [7]. These two notionspermit each agent i some additive amount of envy toward some other agent j , but this is at most i ’s highest value foran item in j ’s bundle in EF1, and at most i ’s lowest value for an item in j ’s bundle in EFx.The existence of EF1 allocations was implied by an older, and classic, argument by Lipton et al. [17]. Caragianniset al. [7] demonstrated that the allocation maximizing the Nash social welfare (the geometric mean of the agents’valuations) satisfies both EF1 and Pareto optimality. But, computing this allocation is APX-hard [16], so Barman et al.[5] went a step further by designing a pseudo-polynomial time algorithm that computes an EF1 and Pareto optimalallocation. On the other hand, the progress on the EFx notion has been much more limited. Plaut and Roughgarden[19] proved that EFx allocations always exist in two-agent instances, even for general valuations beyond additive, anda recent breakthrough by Chaudhury et al. [8] showed that EFx allocations always exist in all instances with threeadditive agents. Even though this result applies only to instances with three agents, its proof required a very carefuland cumbersome case analysis in order to show how an EFx allocation can be produced for all possible scenarios.Whether an EFx allocation always exists or not for instances of four or more agents is a major open question in fairdivision.The maximin share (MMS), originally defined by Budish [6], is an alternative relaxation of proportionality thatuses a “maximin” argument to define the minimum amount of utility that each agent “deserves”. However, similarlyto PROPx, an allocation satisfying this notion of fairness may not always exist, even for three-agent instances [15].To circumvent this issue, a vibrant line of work has instead aimed to guarantee that every agent’s value is always atleast λ times their MMS benchmark, for some λ ∈ [0 , . The first result along this direction showed that an allocationguaranteeing an approximation of λ = 2 / can be computed in polynomial time [1]. Subsequent work by Barmanand Krishnamurthy [4] and Garg et al. [12] also provided simpler algorithms achieving the same guarantee. Ghodsiet al. [13] then provided a non-polynomial time algorithm producing an allocation guaranteeing λ = 3 / and furtherdeveloped this into a polynomial-time approximation scheme guaranteeing λ = 3 / − ǫ . The most recent update inthis line of work further improved the existence bound to / / n , while also providing a strongly polynomialtime algorithm to compute an allocation guaranteeing the / approximation [11].2 Our Results
We propose a relaxation of proportionality which we call proportionality up to the maximin item (PROPm). Just likePROP1 and PROPx, our notion allows the value of each agent i to be less than a /n fraction of her total value, butby no more than some additive difference d i which is a function of agent i ’s value for items allocated to other agents.Rather than going with the most valued item (like PROP1) or the least valued item (like PROPx), our definition of d i is equal to max i ′ = i min j ∈ X i ′ { v ij } , where X i ′ is the bundle of items allocated to agent i ′ . In other words, we considerthe least valued item (from i ’s perspective) in each of the other agent’s bundles, and we take the highest value amongthem. It is easy to verify that this notion lies between the two extremes of PROP1 and PROPx, and it also captures themaximin element that is used to define the MMS benchmark.Our main result is a constructive argument proving the existence of a PROPm allocation for any instance withup to five agents. This is in contrast to the PROPx and MMS notions for which existence fails even for three-agentinstances. Similarly to the breakthrough by Chaudhury et al. [8] proving the existence of EFx allocations for three-agent instances, our proof requires a careful case analysis to reach PROPm allocations for each scenario.What significantly complicates the arguments for the existence of allocations that satisfy EFx or PROPm is that,according to these notions, the satisfaction of each agent depends not only on what they are allocated, but also on howall the remaining items are distributed among the other agents. This leads to non-trivial interdependence which pre-cludes the use of greedy-like algorithms. To streamline our arguments we introduce a notion of close-to-proportional bundles, which allow us to decouple the allocation of one subset of agents from another, and reduce the required caseanalysis. Although we prove the existence for up to five agent instances, this is not due to a hard limit to our approach,other than the fact that the case analysis becomes more complicated and does not provide much more intuition. In fact,we suspect the PROPm property can be satisfied even for instances with an arbitrary number of agents. We study the problem of allocating a set M of m indivisible items (or goods) to a set of n agents N = { , , . . . , n } .Each agent i has a value v ij ≥ for each good j and her value for receiving some subset of goods S ⊆ M is additive,i.e., v i ( S ) = P j ∈ S v ij . For ease of presentation, we normalize the valuations so that v i ( M ) = 1 for all i ∈ N . Givena bundle of goods S ⊆ M , we let m i ( S ) = min j ∈ S { v ij } denote the least valuable good for agent i in bundle S .An allocation X = ( X , X , . . . X n ) is a partition of the goods into bundles such that X i is the bundle allocatedto agent i . Given an allocation X , we use d i ( X ) = max i ′ = i { m i ( X i ′ ) } to denote the maximin good of agent i in X ,and we say that an agent i is PROPm satisfied by X if v i ( X i ) + d i ( X ) ≥ /n . In turn, an allocation X is PROPm ifevery agent is PROPm satisfied by it.Given a positive integer k ≤ n and a set of goods S ⊆ M , the close-to-proportional (CP) bundle for agent i ,denoted CP i ( k, S ) , is the most valuable subset of goods B ⊂ S from agent i ’s perspective for which v i ( B ) ≤ k v i ( S ) .In other words, if i was one of k agents that need to be allocated the set of goods S , then CP i ( k, S ) is the most valuablesubset of these goods that agent i could receive without exceeding her proportional share. When there are multiplebundles that satisfy this definition, then we let CP i ( k, S ) be one with the maximum cardinality, breaking ties arbitrarilyamong them. We begin by first making some initial observations regarding the construction of PROPm allocations and CP i sets.Our first observation provides us with a sufficient condition under which “locally” satisfying PROPm can lead to a“globally” PROPm allocation. Given an allocation of a subset of items to a subset of agents, we say that this partialallocation is PROPm if the agents involved would be PROPm satisfied if no other agents or items were present. Observation 1.
Let N , N be two disjoint sets of agents, let M and M = M \ M be a partition of the items intotwo sets, and let X be an allocation of the items in M to agents in N and items in M to agents in N . Then, if someagent i ∈ N is PROPm satisfied with respect to the partial allocation of the items in M to the agents in N , and v i ( M ) ≥ | N || N + N | , then i is PROPm satisfied by X regardless of how the items in M are allocated to agents in N .Proof. This follows from the definition of PROPm. For all i ∈ N we have d i ( X ) ≥ max i ′ ∈ N \{ i } { min j ∈ X i ′ { v ij }} .Then, if v i ( X i ) + max i ′ ∈ N \{ i } { min j ∈ X i ′ { v ij }} ≥ v i ( M ) | N | (i.e., i is PROPm satisfied by X with respect to the agents3n N and items in M ) and v i ( M ) ≥ | N || N + N | , it must be that v i ( X i ) + d i ( X ) ≥ n so i is also PROPm satisfied by X in the overall allocation of the items in M to N ∪ N .We now observe that we may, without loss of generality, assume that v ij ≤ /n for every agent i and item j . Lemma 2.
If there exists some agent i ∈ N and item j ∈ M such that v ij > /n , we may allocate item j to agent i and reduce the problem to finding a PROPm partial allocation of the items in M \ { j } to agents in N \ { i } .Proof. Let X be an allocation which gives j to agent i and is a PROPm allocation with respect to items in M \ { j } and agents in N \ { i } . Observe that agent i is clearly PROPm satisfied by X (she is, in fact, proportionally satisfied).If any other agent i ′ = i also has value v i ′ j > /n for this item, then d i ′ ( X ) ≥ /n (since j is the only item in X i ).This implies that i ′ is PROPm satisfied since v i ′ ( X i ′ ) + d i ′ ( X ) ≥ d i ′ ( X ) ≥ /n . Finally, all remaining agents i ′ = i have v i ′ j ≤ /n implying that v i ′ ( M \ { j } ) ≥ n − n and since i ′ is PROPm satisfied by X with respect to the items in M \ { j } she must be PROPm satisfied with respect to the entire allocation by Observation 1 substituting N = N \ { i } and M = M \ { j } .Our next observation provides some initial intuition regarding why CP i sets play a central role in this paper. Observation 3.
If agent i is allocated her CP i ( n, M ) set, then i is guaranteed to be PROPm satisfied regardless ofhow the other items are allocated.Proof. Let S be the CP i ( n, M ) set of agent i and consider an arbitrary allocation of M \ S among the remaining n − agents. By definition v i ( S ) + min j ∈ M \ S v ij ≥ /n , so it must be that if i is allocated S , she is PROPm satisfied.We now provide a “recursive” construction of CP i ( k, S ) sets which gives us even stronger guarantees. Suppose weask some agent i to first define the bundle S n = CP i ( n, M ) , then the bundle S n − = CP i ( n − , M \ S n ) , then thebundle S n − = CP i ( n − , M \ ( S n ∪ S n − )) , and so on. We show that as long as i receives one of these bundles,then we have some flexibility over how to allocate the remaining items. Theorem 4.
Let S n , S n − , . . . , S be the recursively defined CP i sets for some agent i , as above. If this agent receivesany bundle S ℓ and no item from S n ∪ S n − ∪ · · · ∪ S ℓ +1 is allocated to the same agent as an item from S ℓ − ∪ S ℓ − ∪· · · ∪ S , then agent i will be PROPm satisfied.Proof. For all k ∈ [ n ] , we have v i ( S k ) ≤ k v i ( M \ ( S n ∪ S n − ∪ · · · ∪ S k +1 )) by definition of S k . Applying this upperbound on v i ( S k ) for k = n , because v i ( M ) = 1 we have that v i ( M \ S n ) ≥ − n = n − n . By applying the upper boundon v i ( S k ) for k = n − and our lower bound on v i ( M \ S n ) we get v i ( M \ ( S n ∪ S n − )) ≥ n − n − n − · n − n ≥ n − n .Iteratively repeating this process, we obtain that for all k ∈ [ n ] we know that v i ( M \ ( S n ∪ S n − ∪ · · · ∪ S k )) ≥ k − n .Also by definition, each S k is a CP i ( k, M \ ( S n ∪ S n − ∪ · · · ∪ S k +1 )) set for M \ ( S n ∪ S n − ∪ · · · ∪ S k +1 ) , sowe have that v i ( S ℓ ) + min j ∈ M \ ( S n ∪ S n − ∪···∪ S ℓ +1 ) { v ij } ≥ ℓ · v i ( M \ ( S n ∪ S n − ∪ · · · ∪ S ℓ +1 )) ≥ ℓ · ℓn = n . Butfinally, as long as the items from S n ∪ S n − ∪ · · · ∪ S ℓ +1 are not included in any of the bundles containing the itemsin M \ ( S n ∪ S n − ∪ · · · ∪ S ℓ ) in the complete allocation X , we have that d i ( X ) ≥ min j ∈ M \ ( S n ∪ S n − ∪···∪ S ℓ ) { v ij } so i is PROPm satisfied when allocated set S ℓ . In this section, we demonstrate that PROPm allocations can be found for any instance with 4 agents. The constructionof the allocation proceeds by finding an appropriate initial partition of the items into bundles (based on our notionof CP i bundles) for some arbitrary agent i . Given these bundles, we then show that we have enough freedom inreallocating items to PROPm satisfy each agent. We note that our proof is constructive, but finding the initial bundlesis computationally demanding (as determining if there is some CP i ( n, M ) set with value /n is an instance of subsetsum).Whenever we say that a set of two or three agents split a bundle ˜ M , we mean that we find a PROPm allocation ofthe items in ˜ M for these agents. Note that Chaudhury et al. [8] show how to compute EFx allocations for up to threeagent instances, and it is easy to verify that EFx outcomes that allocated all the items are also PROPm. But, sincethe arguments for these results are quite complicated and require additional machinery, for completeness we providemuch simpler arguments for reaching PROPm outcomes with up to three agents using only tools defined in this paper(deferred to the appendix). 4 heorem 5. In every instance involving 4 agents with additive valuations there always exists a PROPm allocation.Proof.
We index the agents arbitrarily and begin by recursively constructing CP i sets from the perspective of agent .We construct 4 bundles of items A, B, C, D as follows: • C = CP (4 , M ) • B = CP (3 , M \ C ) • A = CP (2 , M \ ( C ∪ B )) • D = CP (1 , M \ ( C ∪ B ∪ A )) = M \ ( A ∪ B ∪ C ) By Observation 3, we know that if agent is allocated bundle C , she satisfies PROPm. However, we can alsoobserve that she would be satisfied if she is allocated bundle D because v ( D ) ≥ / (which follows by the repeatedapplication of the definition of CP i sets as in Theorem 4).We next want to find bounds on the total value of items in some bundles for agent . This will allow us to recursivelydivide the problem into instances with a smaller number of agents. Lemma 6.
With agent and sets A, B, C, D as defined above, v ( A ∪ D ) ≥ / Proof.
By the definition of an CP i set, we have initial upper bounds on the total value agent has for the generatedsets. • v ( C ) ≤ / • v ( B ) ≤ / − v ( C )) • v ( A ) ≤ / − v ( C ) − v ( B )) By combining these upper bounds, we may obtain lower bounds on v ( A ∪ D ) as follows v ( A ∪ D ) = 1 − ( v ( B ) + v ( C )) ≥ − (1 / / v ( C ))) ≥ − (1 / / ≥ / From here we proceed with case analysis based on the value other agents have for A ∪ D . We present each case asa separate lemma for ease of presentation. Lemma 7.
If no agents in { , , } have value weakly greater than / for the items in A ∪ D we can construct anallocation satisfying PROPm.Proof. If there is no agent i ∈ { , , } for which v i ( D ) ≥ then we can give D to agent 1 and split the remainingitems between the remaining three agents to produce a PROPm allocation by Observation 1. Otherwise there must besome agent i = 1 where v i ( D ) ≥ . Then we can give D to agent i , give A to agent 1 and split B ∪ C between theremaining two agents to arrive at a PROPm allocation by Observation 1 (since for any agent k if v k ( A ∪ D ) < , then v k ( B ∪ C ) ≥ ) and Theorem 4. Lemma 8.
If one agent in { , , } has value weakly greater than / for the items in A ∪ D we can construct anallocation satisfying PROPm.Proof. Without loss of generality let this be agent 2. Split A ∪ D between agents 1 and 2 and split B ∪ C betweenagents 3 and 4 to generate a PROPm allocation by Observation 1. Lemma 9.
If exactly two agents in { , , } have value weakly greater than / for the items in A ∪ D we canconstruct an allocation satisfying PROPm. roof. Without loss of generality, let agent 2 be the agent who has v ( A ∪ D ) < / . For agent 2 it must be that v ( B ) > or v ( C ) > since v ( B ∪ C ) ≥ . But then, we can split A ∪ D between the agents 3 and 4, giveagent 2 her favorite bundle among B and C and give agent 1 the remaining bundle to arrive at a PROPm allocation byObservation 1 and Theorem 4. Lemma 10.
If all three agents in { , , } have value weakly greater than / for the items in A ∪ D we can constructan allocation satisfying PROPm.Proof. If for one of the agents i ∈ { , , } we have that either v i ( B ) ≥ or v i ( C ) ≥ then the allocation followsthe same from the previous lemma. Otherwise, we have that all three agents i = 1 have v i ( C ) < and we can give C to agent who is PROPm satisfied by Observation 3 and split the remaining items between the remaining agentswhich yields a PROPm allocation by Observation 1.Since in each case, we have demonstrated how one may construct a PROPm allocation, for any set of four agentswith additive valuations, a PROPm allocation exists. In this section, we demonstrate that PROPm allocations can be found for any instance with 5 agents. The proofproceeds similarly to the four agent case but requires a closer analysis of various cases. As above, whenever we saythat a set of fewer than five agents “split” a bundle ˜ M , we mean that we find a PROPm allocation of the items in ˜ M for these agents. Theorem 11.
In every instance involving 5 agents with additive valuations there always exists a PROPm allocation.Proof.
We index the agents arbitrarily and begin by recursively constructing CP i sets from the perspective of agent .We construct 5 bundles of items A, B, C, D, E as follows: • D = CP (5 , M ) • C = CP (4 , M \ D ) • B = CP (3 , M \ ( C ∪ D )) • A = CP (2 , M \ ( B ∪ C ∪ D )) • E = CP (1 , M \ ( A ∪ B ∪ C ∪ D )) = M \ ( A ∪ B ∪ C ∪ D ) By Observation 3, we know that if agent is allocated bundle D , she satisfies PROPm. However, we can alsoobserve that she would be satisfied if she is allocated bundle E because v ( E ) ≥ / (which follows by the repeatedapplication of the definition of CP i sets as in Theorem 4).We next want to find bounds on the total value of items in some bundles for agent . This will allow us to recursivelydivide the problem into instances with a smaller number of agents. Lemma 12.
With agent and sets A, B, C, D, E as defined above, v ( A ∪ E ) ≥ / and v ( A ∪ B ∪ E ) ≥ / .Proof. By the definition of an CP i set, we have initial upper bounds on the total value agent has for the generatedsets. • v ( D ) ≤ / • v ( C ) ≤ / − v ( D )) • v ( B ) ≤ / − v ( D ) − v ( C )) • v ( A ) ≤ / − v ( D ) − v ( C ) − v ( B ))
6y combining these upper bounds, we may obtain lower bounds on v ( A ∪ E ) as follows v ( A ∪ E ) = 1 − ( v ( B ) + v ( C ) + v ( D )) ≥ − (1 / / v ( C ) + v ( D ))) ≥ − (1 / / / v ( D )) ≥ − (1 / / ≥ / . Similarly, we can lower bound v ( A ∪ B ∪ E ) as v ( A ∪ B ∪ E ) = 1 − ( v ( C ) + v ( D )) ≥ − (1 / / v ( D )) ≥ − (1 / / ≥ / With Lemma 12 in hand, we proceed with case analysis on the value that the other agents have for A ∪ E and A ∪ B ∪ E . We present each case as a separate lemma for ease of presentation. Lemma 13.
If all four agents { , , , } have value weakly greater than / for the items in A ∪ B ∪ E we canconstruct an allocation satisfying PROPm.Proof. Suppose that at least one of the agents i ∈ { , , , } has v i ( C ) ≥ / or v i ( D ) ≥ / . Without loss ofgenerality, let this be agent . Then, we may give agent either C or D , respectively and is satisfied. We can givethe other of these two sets to agent and then then find a PROPm allocation of A ∪ B ∪ E for agents { , , } . ByObservation 1 and the assumption that agents , , and have value at least / for A ∪ B ∪ E , we know that theywill also be satisfied. Finally, since we have only repartitioned A ∪ B ∪ E , we know by Theorem 4 that agent is alsosatisfied.Now suppose that all of the agents i ∈ { , , , } have value v i ( C ) < / and v i ( D ) < / . Then, by Theorem4, we know that we may give D to agent and reallocate A ∪ B ∪ C ∪ E to the remaining agents and satisfy agent .But since all four remaining agents have value at least / for A ∪ B ∪ C ∪ E , by Observation 1 we can then find anallocation PROPm satisfying these agents as well. Lemma 14.
If exactly three of the agents in { , , , } have value weakly greater than / for the items in A ∪ B ∪ E we can construct a PROPm allocation.Proof. Without loss of generality suppose agent is the agent who has value v ( A ∪ B ∪ E ) < / . We can thengive agent her preferred bundle among C and D and agent the other bundle. Agent must be satisfied since shereceives value at least / and agent is satisfied regardless of how the items in A ∪ B ∪ E are distributed by Theorem4. But then, since all i ∈ { , , } have v i ( A ∪ B ∪ E ) ≥ / we can split A ∪ B ∪ E between these agents to obtaina PROPm allocation by Observation 1. Lemma 15.
If exactly two of the agents in { , , , } have value weakly greater than / for the items in A ∪ B ∪ E we can construct a PROPm allocation.Proof. Without loss of generality, let agents and be the agents with value weakly greater than / for the items in A ∪ B ∪ E . By Lemma 12 we know that agent also has value greater than / for A ∪ B ∪ E . Further, we knowthen that agents and each have value greater than / for the items in C ∪ D . By Observation 1, we can then finda PROPm allocation of these items by splitting A ∪ B ∪ E between agents , , and and splitting C ∪ D betweenagents and .We now move to consider the number of agents which have value greater than / for A ∪ E . Lemma 16.
If exactly two of the agents in { , , , } have value weakly greater than / for the items in A ∪ E wecan construct a PROPm allocation. roof. Without loss of generality let agents and have value weakly greater than / for the items in A ∪ E . Welet these two agents split A ∪ E and move to allocate the remaining bundles among agents , , and . We perform asmall case analysis on the number of bundles that agent or agent values greater than / .Suppose that agents and collectively value at least two distinct bundles in { B, C, D } greater than or equal to / (i.e., they both value exactly one bundle more than / but these bundles are distinct or at least one of the twoagents values more than one bundle more than / ). Then, we may give both of these agents a bundle which they valueat least / and agent the remaining bundle to arrive at a PROPm allocation by Observation 1 and Theorem 4.Now suppose that agents and collectively value exactly one bundle in { B, C, D } at least / . If this bundle is B or C , we know that v ( B ∪ C ) ≥ / and v ( B ∪ C ) ≥ / (since v ( D ) < / and v ( D ) < / ). We can thenallocate D to agent and split B ∪ C between agents and to arrive at a PROPm allocation. If the bundle that and value more than / is D then we know that v ( C ∪ D ) ≥ / and v ( C ∪ D ) ≥ / so we may allocate B to agent and split C ∪ D between agents and to arrive at a PROPm allocation by Observation 1 and Theorem 4. Lemma 17.
If exactly one agent in { , , , } has value weakly greater than / for the items in A ∪ E we canconstruct a PROPm allocation.Proof. Without loss of generality, let agent have value weakly greater than / for the items in A ∪ E . By Lemma12, we know that agent also has value at least / for these items, and by assumption it must be that agents , , and have value at least / for the items in B ∪ C ∪ D . But then, by Observation 1, we can find a PROPm allocation forall the items by reallocating items in A ∪ E to agents and and reallocating items in B ∪ C ∪ D to agents , , and . Lemma 18.
If no agents in { , , , } have value weakly greater than / for the items in A ∪ E we can construct aPROPm allocation.Proof. If this is the case, then it must be that all four of these agents have value more than / for items in B ∪ C ∪ D .If none of these agents have value more than / for E , then we can allocate E to agent and allocate A ∪ B ∪ C ∪ D to agents , , , and to arrive at a PROPm allocation. Suppose, on the other hand, that at least one of these agents,say agent , has v ( E ) ≥ / , we can allocate E to agent , A to agent and repartition B ∪ C ∪ D to agents , ,and to find an allocation that remains PROPm for all agents by Observation 1 and Theorem 4.We now proceed to analyze the four remaining cases which are more elaborate. Lemma 19.
If all four of the agents in { , , , } have value weakly greater than / for A ∪ E and value less than / for A ∪ B ∪ E we can construct a PROPm allocation.Proof. Observe that by assumption all agents , , , and have value weakly greater than / for C ∪ D . We canthen allocate B to agent and split A ∪ E among agents and and C ∪ D among and . Note that agent issatisfied by Theorem 4 and since agents and split value at least / and agents and split value at least / byObservation 1 we construct a PROPm allocation.We then immediately resolve the case when agents , , , and all have value weakly greater than / for A ∪ E and exactly one agent, (without loss of generality) say agent , has value greater than / for A ∪ B ∪ E by followingthe exact same allocation described in the previous lemma. Lemma 20.
If all four of the agents in { , , , } have value weakly greater than / for A ∪ E and exactly one ofthese agents has value weakly greater than / for A ∪ B ∪ E we can construct a PROPm allocation. The final two cases we examine occur when all but one agent have value at least / for A ∪ E . Lemma 21.
If exactly three of the agents in { , , , } have value weakly greater than / for A ∪ E and all of theseagents have value less than / for A ∪ B ∪ E we can construct a PROPm allocation.Proof. Without loss of generality, suppose that v ( A ∪ E ) < / . Since the remaining agents i ∈ { , , } have v i ( A ∪ E ) ≥ / , we can split the set A ∪ E between agents and and they will be satisfied by Observation 1.Moreover, since we have that v ( C ∪ D ) ≥ / and v ( C ∪ D ) ≥ / we can split the set C ∪ D between agents and and they will be satisfied by Observation 1. Finally, by assigning B to agent we construct a PROPm allocationby Theorem 4. 8 emma 22. If exactly three of the agents in { , , , } have value weakly greater than / for A ∪ E and exactly oneof these agents has value weakly greater than / for A ∪ B ∪ E we can construct a PROPm allocation.Proof. First suppose that the agent with value less than / for A ∪ E is the agent with value weakly greater than / for A ∪ B ∪ E . Without loss of generality, let this be agent . By additivity, it must be that v ( B ) > / so agent is satisfied by bundle B . We have that v ( C ∪ D ) ≥ / and v ( C ∪ D ) ≥ / so we can split the set C ∪ D between these agents and they will be satisfied by Observation 1. Finally, we know that v ( A ∪ E ) ≥ / by Lemma12 and v ( A ∪ E ) ≥ / so we may split the set A ∪ E between these agents to complete the PROPm allocation byObservation 1.On the other hand, suppose that the agent with value less than / for A ∪ E is not the agent with value weaklygreater than / for A ∪ B ∪ E . Without loss of generality, suppose v ( A ∪ E ) < / and v ( A ∪ B ∪ E ) ≥ / . Weknow that v ( C ∪ D ) ≥ / and v ( C ∪ D ) ≥ / so we again can split this set between agents and and they willbe satisfied by Observation 1. Since v ( A ∪ E ) ≥ / and v ( A ∪ E ) ≥ / we can split A ∪ E between and andsatisfy both by Observation 1. Finally, we can give B to agent to produce a PROPm allocation by Theorem 4.Since in each case, we have demonstrated how one may construct a PROPm allocation, for any set of five agentswith additive valuations, a PROPm allocation exists. Even though, according to our definition, an agent is PROPm satisfied if v i ( X i ) + d i ( X ) ≥ /n , our argumentsactually prove a slightly stronger bound: v i ( X i ) + n − n d i ( X ) > n . (1)This reduces the gap between the value of each agent i and her proportional share by d i ( X ) /n , which is not negligiblefor n ≤ that we focus on here. But, more importantly, this observation also provides a connection between ourresults and average EFx (a-EFx), an interesting fairness notion that we propose in this section as the natural next stepin light of the results in this paper.Note that if an allocation X is EFx is for every pair of agents i, k ∈ N we have v i ( X i ) + m i ( X k ) ≥ v i ( X k ) .This means that the value of agent i for the bundle X k of some other agent k can be greater than her value for herown bundle, but by no more than m i ( X k ) , i.e., the smallest value of agent i for an item in X k . Summing up theseinequalities over all the k ∈ N \ { i } we get: X k ∈ N \{ i } ( v i ( X i ) + m i ( X k )) ≥ X k ∈ N \{ i } v i ( X k ) ⇒ ( n − v i ( X i ) + X k ∈ N \{ i } m i ( X k ) ≥ − v i ( X i ) ⇒ nv i ( X i ) + X k ∈ N \{ i } m i ( X k ) ≥ ⇒ v i ( X i ) + 1 n X k ∈ N \{ i } m i ( X k ) ≥ n . (2)We say that an allocation X satisfies a-EFx if Inequality (2) is satisfied for every agent i ∈ N . Clearly, the argumentabove verifies that EFx implies a-EFx, but the inverse is not true. Specifically, for an agent i to satisfy EFx she needs tonot envy any other agent k more than m i ( X k ) . On the other hand, agent i could still satisfy a-EFx if she envies someagent k more than m i ( X k ) , as long as this extra envy “vanishes” after averaging over all agents k = i , i.e., it satisfiesEFx “on average”, hence the name.An interesting thing to point out is the similarity between Inequality (1), which we satisfy in this paper, andInequality (2) which is required by a-EFx. Specifically, the gap between v i ( X i ) and n allowed by the former is atmost /n times n − maximin item values of i , whereas the latter is at most /n times the sum of the n − minimumitem values of i on for each X k bundle. Drawing from this connection, we believe that an interesting open problem is to9tudy the existence of a-EFx allocations in instances with more than 3 agents. Since the PROPm notion is a relaxationof a-EFx, and a-EFx is a relaxation of EFx, this provides an interesting path toward the exciting open problem ofwhether EFx solutions always exist for instances with 4 or more agents. Our work defines a new notion of approximate proportionality called PROPm. In contrast to similar notions of fairnesssuch as PROPx and MMS we show that PROPm does exist in the cases of four and five agents with additive valuations.After constructing particular subsets of items for an arbitrary agent (i.e., the close-to-proportional sets), we are thenable to carefully assign these subsets to agents or unions of these subsets to a group of agents to recursively constructPROPm allocations. Although we do not extend our results to cases beyond five agents, we have no reason to believethat a similar methodology would not work for higher number of agents. The main barrier seems to be the increasinglycomplex casework that arises from our approach as the number of agents increases.However, while we prove the existence of PROPm for up to five agents and conjecture that a similar approachwould work for any number of agents, we note that finding a CP i set is at least as hard as subset sum (as one needsto check if some subset gives an agent exactly proportional value), a known NP-hard problem [14], so our approachdoes not provide an efficient way to calculate a PROPm allocation. Finding a polynomial time algorithm producinga PROPm allocation for any number of items (and any number of agents) via an alternative method is an interestingpossible avenue of future research. Another question we do not explore in this work is achieving PROPm and Paretoefficiency simultaneously. Aziz et al. [3] provide an algorithm which simultaneously achieves Pareto optimality andPROP1, so an analogous result combining PROPm and Pareto optimality (or proof that no such allocation exists)would complement their work and ours. References [1] Georgios Amanatidis, Evangelos Markakis, Afshin Nikzad, and Amin Saberi. Approximation algorithms forcomputing maximin share allocations.
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PROPm allocations for the 2 and 3 agent cases
For completion we include a brief proof of the existence of PROPm allocations in the case of two and three agents.We note that the proof of the existence of PROPm allocations for two agents is essentially the same as the proofof Theorem 4.3 in Plaut and Roughgarden [19] showing the existence of EFx allocations for two agents. They employa “cut-and-choose” approach where one agent partitions bundles according to the leximin++ solution and the otheragent selects the preferred bundle. We use the technique except ask the dividing agent i to split the items based on ourdefinition of CP i bundles. Theorem 23.
For agents with additive valuations one can always find a PROPm allocation.Proof. Index the two agents arbitrarily and let agent split M into two bundles A and B where A is her CP (2 , M ) set and B = M \ A . By Observation 3, if agent receives A she is satisfied. Furthermore, by definition v ( B ) ≥ / ,so agent is satisfied regardless of which bundle she receives. We can then allow agent to select her favorite bundlebetween A and B and by additivity she must obtain value at least / (and is therefore PROPm satisfied).In the construction of PROPm allocations for agents when we say that a set of two agents “split” a bundle ˜ M ,we mean that we find a PROPm allocation of the items in ˜ M for these agents. Theorem 24.
For agents with additive valuations one can always find a PROPm allocation.Proof. We index the agents arbitrarily and begin by recursively constructing CP i sets from the perspective of agent .We construct 3 bundles of goods A, B, C as follows: • B = CP (3 , M ) • A = CP (2 , M \ B ) • C = CP (1 , M \ ( A ∪ B )) = M \ ( A ∪ B ) By Theorem 4, we know that if agent is allocated bundle B or C she will be satisfied no matter how the rest isallocated (since v ( C ) > / ) and if she is allocated A she will be satisfied provided that we can assign bundles B and C to the remaining agents without redistributing items. We proceed via case analysis on the values agents and have for bundles A , B , and C . Lemma 25.
If agents or collectively value at least two distinct bundles among A , B , and C greater than or equalto / , we can construct a PROPm allocation.Proof. If agents and collectively value at least two distinct bundles greater than / then we may assign both agent and agent a bundle for which they receive value at least / . We can then assign agent the remaining bundle toarrive at a PROPm allocation. Lemma 26.
If agents and collectively value exactly one bundle among A , B , and C greater than or equal to / ,we can construct a PROPm allocation.Proof. Suppose first that this is either bundle B or A . If so, then it must be that v ( B ∪ A ) ≥ / and v ( B ∪ A ) ≥ / (since v ( C ) < / and v ( C ) < / ). But then we can split B ∪ A between agents and and allocate C to agent to obtain a PROPm allocation by Observation 1.On the other hand, if this bundle is C , then it must be that v ( A ∪ C ) ≥ / and v ( A ∪ C ) ≥ / (since v ( B ) < / and v ( B ) < / ). But then we can split A ∪ C between agents and and allocate B to agent1