key: cord-0225515-ao6xrj84
authors: Mishra, Shaily; Padala, Manisha; Gujar, Sujit
title: Fair Allocation with Special Externalities
date: 2021-08-29
journal: nan
DOI: nan
sha: d167d140490752cc52dc86d33ab7498b1c35b2a4
doc_id: 225515
cord_uid: ao6xrj84

Most of the existing algorithms for fair division do not consider externalities. Under externalities, the utility an agent obtains depends not only on its allocation but also on the allocation of other agents. An agent has a positive (negative) value for the assigned goods (chores). This work focuses on a special case of externality, i.e., an agent receives positive or negative value for unassigned items independent of which other agent gets it. We show that it is possible to adapt existing algorithms using a transformation to ensure certain fairness and efficiency notions in this setting. Despite the positive results, fairness notions like proportionality need to be re-defined. Further, we prove that maximin share (MMS) may not have any multiplicative approximation in this setting. Studying this domain is a stepping stone towards full externalities where ensuring fairness is much more challenging.

We consider the problem of allocating m indivisible items among n agents who report their valuations for the items. The objective is to ensure fair allocation for a desirable notion of fairness. These scenarios often arise in the division of inheritance among family members, divorce settlements and distribution of tasks among workers Brams et al. [1996] , Moulin [2004] , Segal-Halevi [2019] , Steihaus [1948] , Su [2000] . Economists have proposed many fairness and efficiency notions widely applicable in such real-world settings. Researchers also explore the computational aspects of some widely accepted fairness notions Caragiannis et al. [2019] , Barman et al. [2018a ], De Keijzer et al. [2009 , Freeman et al. [2019] , Procaccia and Wang [2014] . Such endeavours have led to web-based applications like Spliddit 1 , The Fair Proposals System 2 , Coursematch 3 , Divide Your Rent Fairly 4 , etc. However, most approaches do not consider agents with externalities, which we believe is restrictive.

In general, externality implies that the agent's utility depends not only on their bundle but also on the bundles allocated to other agents. Such a scenario is relatively common, mainly in allocating necessary commodities. For example,

While fair resource division has an extremely rich literature, externalities in fair division is less explored. Velez [2016] extend the notion of EF in externalities and explored EF allocation of indivisible goods and money among interested agents in presence of externalities. Brânzei et al. [2013] generalize PROP and EF for divisible goods allocation with positive externalities. Treibich [2019] study egalitarian social welfare in presence of average externalities for divisible goods. Further, Seddighin et al. [2019] propose average-share definition, an extension of PROP, and study (EMMS) MMS allocation for indivisible goods with positive externalities. Note that in our setting MMS share is equivalent to EMS. Authors in Aziz et al. [2021] explore EF1/EFX for the specific setting of two and three agents. For two agents, their setting is equivalent to 2-D, hence existing algorithms for EF, PROP and their additive relaxations Caragiannis et al. [2019] , Plaut and Roughgarden [2020] , Aziz et al. [2020a] suffice. Beyond two agents, the setting is more general and Aziz et al. [2021] prove the non-existence of EFX for three agents. In contrast, for the special case of 2-D, EFX always exists for three agents since it exists in 1-D Chaudhury et al. [2020] . Further Aziz et al. [2021] , provide extension of PROP for additive valuations with full externalities.

We breifly summarize the existing algorithms for 1-D valuations available for each of the fairness notions.

Envy-freeness. EF may not exist for indivisible items. Hence we consider two prominent relaxations of EF, Envy-freeness up to one item (EF1) Budish [2011] , Lipton et al. [2004] and Envy-freeness up to any item(EFX) Caragiannis et al. [2019] . We have poly-time algorithms to find EF1 in general monotone valuations for goods Lipton et al. [2004] and chores Bhaskar et al. [2020] . For additive valuations, EF1 can be found using Round Robin Caragiannis et al. [2019] in goods or chores, and Double Round Robin Aziz et al. [2018] in combination. Authors in Plaut and Roughgarden [2020] present an algorithm to find EFX allocation under identical general valuations for goods. Researchers have also studied fair division in presence of strategic agents Barman et al. [2019] , Bei et al. [2020] , Padala and Gujar [2021] .

Proportionality. PROP1 and PROPX are popular relaxation of PROP. For additive valuations, EF1 implies PROP1, and EFX implies PROPX. Unfortunately, in paper Aziz et al. [2020a] , the authors showed the PROPX for goods may not always exists. Authors in Li et al. [2021] explored (weighted) PROPX showed that a (weighted) PROPX allocation always exists and can be computed efficiently. Procaccia and Wang [2014] , Kurokawa et al. [2016] . The papers Procaccia and Wang [2014] , Amanatidis et al. [2017] , Barman et al. [2018b] , Garg et al. [2019] showed that 2/3-MMS for goods always exists. Paper Ghodsi et al. [2018] , Garg and Taki [2021] showed that 3/4-MMS for goods always exists. Authors in Garg and Taki [2021] provides an algorithm that guarantees 3/4 + 1/12n-MMS for goods. Authors in Aziz et al. [2017] presented a polynomial-time algorithm for 2-MMS for chores. The algorithm presented in Barman et al. [2018b] gives 4/3-MMS for chores. Authors in Huang and Lu [2021] showed that 11/9-MMS for chores always exists. Authors in Kulkarni et al. [2021] explored α−MMS for a combination of goods and chores.

Fair and Efficient. In Caragiannis et al. [2019] , the authors showed that MNW allocation is EF1 and PO for indivisible goods and Barman et al. [2018a] gave a pseudo-polynomial time algorithm. For a combination of resources, the authors in Aziz et al. [2018] presented a polynomial-time algorithm to find EF1 and PO for two agents. An Algorithm to find PROP1 and fractional PO which is stronger than PO was proposed by Aziz et al. [2020a] for a combination of resources. Authors in Aziz et al. [2020b] proposed a pseudo-polynomial time for finding utilitarian maximizing among EF1 or PROP1 in goods.

We consider a resource allocation problem (N, M, V) for determining an allocation A of M = [m] indivisible items among N = [n] interested agents, m, n ∈ N. We only allow complete allocation and no two agents can receive the same item. That is,

We denote the allocation for all the agents except i as A −i .

where v i (S) denotes the value for receiving bundle S and v ′ i (S) for not receiving S. The value an agent i has for item k in 2-D is given by

. The utility an agent i ∈ N obtains for a bundle S ⊆ M is,

and utilties in 2-D are not normalized 5 . When agents have additive valuations, u i (S) = k∈S v ik + k / ∈S v ′ ik . We assume monotonicity of utility for goods, i.e., ∀S ⊆ T ⊆ M , u i (S) ≤ u i (T ) and anti-monotonicity of utility for chores, i.e, u i (S) ≥ u i (T ). We use the term full externalities to represent complete externalities, i.e., each agent has n-dimensional vector for its valuation for an item.

Given the notations, we next define fairness notions considered in this paper. Important Definitions. Since Envy-freeness (EF) may not exist for indivisible items, we consider EF1 and EFX. For goods, an allocation is EF1 when the agent values its own bundle no less than it values any other agent's bundle with the most valued item removed. EFX is stronger than EF1 and requires that the agent values its own bundle no less than the other agent's bundle with the least valued item removed. For chores, similar definition applies but unlike in goods, a chore is removed from the agent's own bundle and then compared with the other agents'. A common definition for is as follows, Definition 1 (Envy-free (EF) and relaxations Aziz et al. [2018] , Budish [2011] , Caragiannis et al. [2019 ], Foley [1966 , Velez [2016] ). For the items (chores or goods) an allocation A that satisfies ∀i, j ∈ N ,

Note that beyond 2-D, one must include the concept of swapping bundles to generalize the above definition as in Velez [2016] , Aziz et al. [2021] . We next state the definition of proportionality for 1-D. Definition 2 (Proportionality (PROP) Steihaus [1948] ). An allocation A is said to be proportional, if ∀i ∈ N ,

. For 2-D, achieving PROP is impossible as discussed in Section 1. To capture proportional allocations under externalities, we now introduce Proportionality with externality (PROP-E). Informally, while PROP guarantees 1/n share of the entire bundle, PROP-E guarantees 1/n share of the sum of utilities for all bundles. Note that, PROP-E is not limited to 2-D and applies to a general externality setting. Formally, Definition 3 (Proportionality with externality (PROP-E)). An allocation A satisfies PROP-E if, ∀i ∈ N ,

Analogous to EFX/EF1, we now define the relaxations for PROP-E for the combination of goods and chores, Definition 4 (PROP-E relaxations). An allocation A ∀i, ∀j ∈ N , satisfies PROPX-E if it is PROP-E up to any item, i.e.,

Finally, we state the definition of MMS and its multiplicative approximation. Definition 5 (Maxmin Share MMS Budish [2011] ).

Since it is common to consider efficiency with notions, we next define Pareto-optimality, a popular efficiency notions.

We also consider efficiency notions like Maximum Utilitarian Welfare (MUW), that maximizes the sum of agent utilities. Likewise Maximium Nash Welfare (MNW) maximizes the product of agent utilties and Maximum Egalitarian Welfare (MEW) maximizes the minimum agent utility.

In the next section, we define a transformation from 2-D to 1-D that plays a major role in adaptation of existing algorithms for ensuring desirable properties.

3 Reduction from 2-D to 1-D

When valuations are additive, we obtain

Example. Consider two goods {g 1 , g 2 } and an agent with 2-D additive valuations given as: {g 1 : (6, −1), g 2 : (5, −100)}. We apply T and obtain w 1 ({g 1 }) = 7 and w 1 ({g 2 }) = 105.

For an allocation A, the utility obtained by an agent in 2-D is u i (A i ) and the corresponding utility in 1-D is w i (A i ).

Note that utility is equal to the valuation in 1-D.

Proof. We assume monotonicity of utility for goods in 2-D, i.e., ∀i ∈ N, u i (·) is monotone. Therefore, for an

is monotone and normalized, it is non negative for goods. Similarly we can prove that w i (·) is normalized, anti-monotonic and non-negative for chores.

Starting with EF allocation in 2-D, we can prove it is EF in 1-D similarly. The complete proof of Theorem 3 for other notions is provided in Appendix Section A.

From Lemma 1 and Theorem 3, we obtain the following. Note that applying any algorithm on the 2-D utility values directly without transformation may not work. We state few examples are below. Modified leximin algorithm to find PROP1 and PO for chores for 3 or 4 agents given in Chen and Liu [2020] does not find PROP1-E (or PROP1) and PO in 2-D when applied on utilities. The following example demonstrates the same, Example 1. Consider 3 agents {1, 2, 3} and 4 chores {c 1 , c 2 , c 3 , c 4 } with positive externality. The 2-D valuation profile is as follows, V 1c1 = (−30, 1), V 1c2 = (−20, 1), V 1c3 = (−30, 1), V 1c4 = (−30, 1), V 3c1 = (−1, 40), V 3c2 = (−1, 40), V 3c3 = (−1, 40), and V 3c4 = (−1, 40). The valuation profile of agent 2 is the same as that of agent 1. Allocation {∅, ∅, (c 1 , c 2 , c 3 , c 4 )} is the leximin allocation, which is not PROP1-E. However, allocation {c 3 , (c 2 , c 4 ), (c 1 )} is leximin allocation on transformed valuations; it is PROP1 and PO in W and it is PROP1-E and PO in V.

In the same way, for chores, paper Li et al. [2021] showed that any PROPX allocation ensures 2-MMS for symmetric agents doesn't extend to 2-D. For example, consider two agents {1, 2} having additive identical valuations for six chores {c 1 , c 2 , c 3 , c 4 , c 5 , c 6 }, given as V 1c1 = (−9, 1), 

We remark that ensuring PROP (Def. 2) is too strict in 2-D. As a result, we introduce PROP-E and its additive relaxations in Defs. 3 and 4 for general valuations. Proposition 1. For additive 2-D, We can adapt the existing algorithms of PROP and its relaxations to 2-D using T.

Proof. In the absence of externalities, for additive valuations, PROP-E is equivalent to PROP as ∀i, j∈N u i (A j ) = v i (M ). From Theorem 3, we know that T retains PROP-E and its relaxations, and hence all existing algorithms of 1-D is applicable using T.

It is known that EF =⇒ PROP for sub-additive valuation in 1-D. As formally presented in Corollary 2, in the case of PROP-E also, ∀i, j ∈ N ,

We now compare PROP-E with existing PROP extensions for capturing externalities. We consider two definitions stated in literature from Seddighin et al. [2019] (Average Share) and Aziz et al. [2021] (General Fair Share). Note that both these definitions are applicable when agents have additive valuations, while PROP-E applies for any general arbitrary valuations. In Aziz et al. [2021] , the authors proved that Average Share =⇒ General Fair Share, i.e., if an allocation guarantees all agents their average share value, it also guarantees general fair share value. With that, we state the definition of Average Share (in 2-D) and compare it with PROP-E. Definition 8 (Average Share Seddighin et al. [2019] ). In V, the average value of item k for agent i, denoted by

The average share of agent

We can prove the reverse implication in a similar way.

Next, we briefly state the relation of EF, PROP-E, and Average Share beyond 2-D and give the proofs in the Appendix Section B. Remark 1. In case of full externality, EF =⇒ Average Share Aziz et al. [2021] . Proposition 3. Beyond 2-D, PROP-E =⇒ Average Share and Average Share =⇒ PROP-E.

To conclude this section, we state that for the special case of 2-D externalities with additive valuations, we can adapt existing algorithms to 2-D, and further analysis is required for the general setting.

Apart from the additive relaxations, the most commonly considered relaxation to PROP is maximin share (MMS) allocations. We provide analysis of MMS for 2-D valuations in the next section.

From Theorem 3, we showed that transformation T retains MMS property, i.e., an allocation A guarantees MMS in 1-D iff A guarantees MMS in 2-D. We draw attention to the point that,

where µ W i and µ i are the MMS value of agent i in 1-D and 2-D, respectively. Kurokawa et al. [2018] proved that MMS allocation may not exist even for additive valuations, but multiplicative approximation of MMS always exists in 1-D. The current best approximation results on MMS allocation are 3/4 + 1/(12n)-MMS for goods Garg and Taki [2021] and 11/9-MMS for chores Huang and Lu [2021] for additive valuations. We are interested in finding multiplicative approximation to MMS in 2-D. Note that we only study α-MMS for complete goods or chores in 2-D, as paper Kulkarni et al. [2021] proves the non-existence of α-MMS in the case of combination of goods and chores in 1-D. From Eq. 5 of α-MMS, if µ i is positive, we consider α-MMS allocation with α ∈ [0, 1], and if it is negative, 1/α-MMS with α ∈ (0, 1].

We categorize externalities in two ways for better analysis 1) Correlated Externality 2) Inverse Externality. In the correlated setting, we study goods with positive externality and chores with negative externality. And in the inverse externality, we study goods with negative externality and chores with positive externality. In the following sections, we show that α-MMS exists for correlated, while it may not exist for inverse externality.

In this section, we investigate approximate MMS guarantees for correlated externality.

Proof. In the first part of this proof, we prove that A is α-MMS in W, A is α-MMS in V, and then in the second part, we provide a counter-example such that A is α-MMS in V but not in W.

In the case of goods with positive externalities, µ i is positive, α ∈ (0, 1], and ∀S ⊆ M , v ′ (S) ≥ 0. From this, we derive that v ′ i (M ) ≤ αv ′ i (M ), and hence it is valid to say that u i (A i ) ≥ αµ i . In the case of chores with negative externalities, µ i is negative, 1/α ≥ 1, and ∀S ⊆ M, v ′ (S) ≤ 0. Similarly to the previous point, we derive that v ′ (M ) ≤ 1 α v ′ (M ) and thus u i (A i ) ≥ 1 α µ i . Part-2. We provide the following counter-example for goods to prove A is α-MMS in V but not in W.

Example. Consider N = {1, 2} both having additive identical valuations for 5 goods {g 1 , g 2 , g 3 , g 4 , g 5 , g 6 } given by, V ig1 = (0.5, 0.1), V ig2 = (0.5, 0.1), V ig3 = (0.3, 0.1), V ig4 = (0.5, 0.1), V ig5 = (0.5, 0.1), and V ig6 = (0.5, 0.1). After transformation, we get µ W i = 1 and in 2-D µ i = 1.6. Allocation, A = {{g 1 }, {g 2 , g 3 , g 4 , g 5 , g 6 }} is 1/2-MMS in V, but not in W. We provide the following counter-example for chores with negative externality to prove A is 1/α-MMS in V but not in W. Corollary 3. We can adapt the existing α-MMS algorithms using T for correlated externality for general valuations. Corollary 4. For correlated 2-D externality, we can always obtain 3/4 + 1/(12n)-MMS for goods and 11/9-MMS for chores for additive.

Motivated by the example given in Kurokawa et al. [2018] for non-existence of MMS allocation for 1-D valuations, we ingeniously adapted it to construct the following instance in 2-D to prove the impossibility of α-MMS in 2-D. We show that for any α ∈ (0, 1], an α-MMS or 1/α-MMS allocation may not exist for inverse externality. In this section, we present an instance for goods with inverse externality, such that ∀i, µ i is positive, we show that for α > 0, there is no α-MMS allocation. Further, we present an instance for chores with positive externality, such that ∀i, µ i is negative, and we prove that there is no 1/α MMS allocation. In order to prove the non-existence results for goods we consider the 1-D valuations W where MMS does not exist. Further we use this example to construct V g where W = T(V g ), in 2-D such that α-MMS exists in V g only if MMS allocation exists in W . Hence the contradiction.

Non-existence of α-MMS in Goods. Consider the following example. Example 3. We consider a problem of allocating 12 goods among three agents, and represent valuation profile as V g . The valuation profile V g is equivalent to 10 3 × V given in Table 1 . We set ǫ 1 = 10 −4 and ǫ 2 = 10 −3 . We transform these valuations in 1-D using T, and the valuation profile T(V g ) is the same as the instance in Kurokawa et al. [2018] that proves the non-existence of MMS for goods. Note that ∀i, v ′ i (M ) = −4055000 + 10 3 ǫ 1 The MMS value of every agent in T(V g ) is 4055000 and from Eq. 4, the MMS value of every agent in V g is 10 3 ǫ 1 .

Recall that T retains MMS property (Theorem 3) and thus we can say that MMS allocation doesn't exist in V g .

Lemma 2. There is no α-MMS allocation for the valuation profile V g of Example 3 for any α ∈ [0, 1].

, which gives us w i (A i ) ≥ 4055000 − 0.1. For this to be true, we need w i (A i ) ≥ 4055000 since T(V g ) has all integral values. We know that such an allocation doesn't exist Kurokawa et al. [2018] . Hence for any α ∈ [0, 1], α-MMS does not exist for V g .

Non-existence of 1/α-MMS in Chores. Consider the following example.

Example 4. We consider a problem of allocating 12 chores among three agents. The valuation profile V c is equivalent to −10 3 V given in Table 1 . We set ǫ 2 = −10 −3 . We transform these valuations in 1-D, and T(V c ) is the same as the instance in Aziz et al. [2017] that proves the non-existence of MMS for chores. Note that v ′ i (M ) = 4055000 − 10 3 ǫ 1 . The MMS value of every agent in T(V c ) amd V c is -4055000 and −10 3 ǫ 1 , respectively. Lemma 3. There is no 1/α-MMS allocation for the valuation profile V c of Example 4 with ǫ 1 ∈ (0, 10 −4 ] for any α > 0.

Note that 0 < 10 3 ǫ 1 ≤ 0.1 and since w i (A i ) has only integral values, we need ∀i, w i (A i ) ≥ −4055000. Such A does not exist Aziz et al. [2017] . As ǫ 1 decreases, 1/ǫ 1 increases, and even though approximation guarantees weakens, it still does not exist for V c .

From Lemma 2 and 3 we conclude the following theorem, Theorem 2. There may not exist α-MMS for any α ∈ [0, 1] for µ i > 0 or 1/α-MMS allocation for any α ∈ (0, 1] for µ i < 0 in the presence of externalities. 

Interestingly, in 1-D, α-MMS's non-existence is known for α value close to 1 Kurokawa et al. [2018] , Feige et al. [2021] , while in 2-D, it need not exist even for α = 0. It follows because α-MMS may not be lead to any relaxation in the presence of inverse externalities.

Consider the situation of goods having negative externalities, where MMS share µ i comprises of the positive value from the assigned bundle A i and negative value from the unassigned bundles A −i . We re-write µ i as follows, µ i = µ + i + µ − i where µ + i corresponds to utility from assigned goods/unassigned chores and µ − i corresponds to utility from unassigned goods/assigned chores. When µ i ≥ 0, applying αµ i is not only relaxing positive value αµ + i , but also requires αµ − i which is stricter than µ − i since µ − i < 0. Hence, the impossibility of α-MMS in 2-D. Similar argument holds for chores with positive externalities.

In the next section, we explore relaxing MMS such that it is guaranteed to exist in 2-D.

In this section, we define Shifted α-MMS that guarantees a fraction of MMS share shifted by certain value, such that it always exist in 2-D. We also considered intuitive ways of approximating MMS in 2-D. These ways are based on relaxing the positive value obtained from MMS allocation µ + and the negative value µ − , µ = µ + + µ − . In other words, we look for allocations that guarantee αµ + and (1 + α) or 1/α of µ − . Unfortunately, such approximations may not always exist. We provide detailed explanation in the Appendix Section C.

Definition 9 (Shifted α-MMS). An allocation A guarantees shifted α-MMS if ∀i ∈ N, α ∈ (0, 1]

Similarly we can prove vice versa. Corollary 5. We can adapt all the existing algorithms for α-MMS in W to get shifted α-MMS in V.

We use T and apply the existing algorithms for additive valuations as well as general valuations and obtain the corresponding shifted multiplicative approximations. Since a direct multiplicative approximation of MMS need not exist in presence of externalities, we consider additive relaxation of MMS in the next section.

Definition 10 (MMS relaxations). An allocation A that satisfies, ∀i, j ∈ N , MMSX i.e., MMS upto any item,

Proposition 6. From Theorem 3, we conclude that MMS1 and MMSX retains after transformation.

EF1 is a stronger fairness notion than MMS1 and can be computed in polynomial time. On the other hand, PROPX might not exist for goods Aziz et al. [2020a] . Since PROPX implies MMSX, it is interesting to settle the existence of MMSX for goods. Note that MMSX and Shifted α-MMS are not related. It is interesting to study these relaxations further, even in full externalities.

In this paper, we conducted a study on indivisible item allocation with special externalities -2-D externalities. We proposed a simple yet compelling transformation from 2-D to 1-D to employ existing algorithms to ensure many fairness and efficiency notions. We can adapt existing fair division algorithms via the transformation in such settings. We proposed proportionality extension in the presence of externalities and studied its relation with other fairness notions. For MMS fairness, we proved the impossibility of multiplicative approximation of MMS in 2-D, and we proposed Shifted α-MMS instead. There are many exciting questions here which we leave for future works. (i) It might be impossible to have fairness-preserving valuation transformation for general externalities. However, what are some interesting domains where such transformations exist? (ii) What are interesting approximations to MMS in 2-D as well as in general externalities?

Considering relaxation of PROP-E, we will prove it for PROP1-E, and the proof for PROPX-E follows in a similar fashion. An allocation is said to be PROP1-E if it satisfies,

i.e., item k is chore for agent i. Let allocation A be PROP1-E in W then, ∀i ∈ N, ∃k ∈ {M \ A i }, i.e., k is a good for agent i,

Next, we show the prove for MMS allocation. An allocation is said to be MMS, if each agent gets at least its maximin share value, i.e., ∀i ∈ N,

We now consider EQ for which T does not retain. First, we define EQ and its relaxations. An allocation A is said to be equitable, when ∀i, j ∈ N , u i (A i ) = u j (A j ). An allocation A is said to be EQ1, i.e., Equitable up to one item,

An allocation A is said to be EQX, i.e., Equitable up to any item, • A = {(g 1 , g 2 ), (g 3 , g 4 )} is EQ in W, but is not even EQ1 in V.

• A = {(g 3 , g 4 ), (g 1 , g 2 )} is EQ in V but is not even EQ1 in W.

Thus, among fairness notions, T retains EF, PROP-E, MMS and their additive relaxations.

Efficiency notions. We will discuss efficiency notions such as PO, MUW, MNW, and MEW.

We

We can re-write that,

It is easy to verify that MUW allocation is also retained under transformation T.

We cannot define MNW in presence of externalities, as for goods, agents can have positive as well as negative utility.

Note MEW is not retained using T. Consider two agents {1, 2} and two goods {1, 2}. Agents have additive valuations. Among efficiency notions, we can retain PO and MUW using transformation T.

We compare PROP-E and Average Share beyond 2-D. First we define valuation space in the presence of full externalities. The valuation function for n agents is denoted by V = {V 1 , V 2 , . . . , V n }. For each i ∈ N , V i : 2 M → R n , i.e., V i ∈ R n 2 M . Further, for any bundle S ⊆ M , V i (S) = (v i1 (S), v i2 (S), . . . , v in (S)), where v ij (S) denotes the value agent i receives when bundle S is assigned to agent j. Proposition 7. Beyond 2-D, PROP-E =⇒ Average Share and Average Share =⇒ PROP-E.

Proof. We will show that there is no relation between PROP-E and average share beyond 2-D, for that we will consider n = 3.

An Allocation A is said to be PROP-E, u i (A i ) ≥ 1/n · j∈N u i (A j ). Let i = {1} u 1 (A 1 ) ≥ 1/3 · u 1 (A 1 ) + u 1 (A 2 ) + u 1 (A 3 )

An Allocation A is said to be average share, u i (A i ) ≥ 1/n · k∈M j∈N v ij (k). Let i = {1}

From Eq. 7 and 8, we conclude there is no relation between PROP-E and average share beyond 2-D.

We re-write µ i as follows, µ i = µ + i + µ − i where µ + i corresponds to utility from assigned goods/unassigned chores and µ − i corresponds to utility from unassigned goods/assigned chores. We propose two more approximate MMS definitions, such that it relaxes both utility and dis-utility obtained. The first two definition Def. 11 and 12 is based on relaxing µ + and µ − simultaneously. Unfortunately we show that they need not exist in Lemma 4 and 5. Example 5. In order to prove this, we make few changes in the valuation profile V g of Example 3 and represent it as V G . We set V G 1k10 = (1000 − ǫ 1 + ǫ 3 , −ǫ 1 ), V G 2k10 = (1000 − ǫ 1 + ǫ 3 , −ǫ 1 ), and V G 1k4 = (1001 − ǫ 1 + ǫ 3 , −ǫ 1 ). We set ∀i, V G ik8 = (1028 − ǫ 1 + ǫ 3 , −ǫ 1 ). We multiply 10 to V G . We set ǫ 1 ≤ 10 −5 ǫ 2 = 10 −3 and ǫ 3 = 10 −4 in the valuation profile V G . We consider ǫ 3 = 10 −4 so that agents have unique MMS bundle, for example, agent 1 unique MMS bundle is {k 1 , k 2 , k 3 , k 4 }. We transform these valuations in 1-D using T, and the valuation profile T(V G ) is similar to the instance in Kurokawa et al. [2018] , and it is easy to verify that MMS allocation doesn't exist. The MMS value of every agent in T(V G ) and V G is 40550000 and 10 4 ǫ 1 , respectively. Note that µ i = µ + + µ − , µ + i = 9 · 10 4 ǫ 1 , and µ − i = −8 · 10 4 ǫ 1 . Also v ′ i (M ) = −40550000 + 10 4 ǫ 1 . Definition 11 (α-MMS (I.)). An allocation A is said to be α-MMS if it guarantees ∀i ∈ N, u i (A i ) ≥ α · µ + i + (1 + α) · µ − i where α ∈ [0, 1]. Definition 12 (α-MMS (II.)). An allocation A is said to be α-MMS if it guarantees ∀i ∈ N, u i (A i ) ≥ α · µ + i + (1/α) · µ − i where α ∈ (0, 1].

Unfortunately we cannot ensure α-MMS according to definition 11 and 12 in 2-D. Lemma 4. There is no α-MMS (Def. 11) for the valuation profile V G for any α ∈ [0, 1].

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The unreasonable fairness of maximum nash welfare

Finding fair and efficient allocations

On the complexity of efficiency and envyfreeness in fair division of indivisible goods with additive preferences

Equitable allocations of indivisible goods

Fair enough: Guaranteeing approximate maximin shares

Resource allocation and the public sector

An improved approximation algorithm for maximin shares

An algorithmic framework for approximating maximin share allocation of chores

Externalities in cake cutting

Welfare egalitarianism with other-regarding preferences

Externalities and fairness

Fairness concepts for indivisible items with externalities

Almost envy-freeness with general valuations

A polynomial-time algorithm for computing a pareto optimal and almost proportional allocation

Efx exists for three agents

The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes

On approximately fair allocations of indivisible goods

On approximate envy-freeness for indivisible chores and mixed resources

Fair allocation of combinations of indivisible goods and chores. CoRR, abs/1807.10684

International Foundation for Autonomous Agents and Multiagent Systems

Truthful fair division without free disposal

Mechanism design without money for fair allocations

Almost (weighted) proportional allocations for indivisible chores

When can the maximin share guarantee be guaranteed?

Approximation algorithms for computing maximin share allocations

Groupwise maximin fair allocation of indivisible goods

Approximating maximin share allocations. Open access series in informatics

Fair allocation of indivisible goods: Improvements and generalizations

Algorithms for max-min share fair allocation of indivisible chores

Indivisible mixed manna: On the computability of mms+ po allocations

Computing fair utilitarian allocations of indivisible goods

The fairness of leximin in allocation of indivisible chores

Fair enough: Guaranteeing approximate maximin shares

A tight negative example for mms fair allocations

2. We introduce PROP-E for general valuations in the presence of full externalities (Section 2) and derive relation with existing proportionality extensions (Section 4).3. We prove that multiplicative approximation of MMS may not exist in 2-D (Theorem 2).4. We propose Shifted α-MMS which is a novel way of approximating MMS in 2-D (Section 5.3).

Proof. Fairness notions.When k is good for agent i, then k ∈ A j and when k is chore to agent i, then k ∈ A i . Let allocation A be EF1 in W then, ∀i, ∀j ∈ N, ∃k ∈ A j , i.e., k is a good for agent i in A j bundleWhen k ∈ A i , i.e., k is a chore for agent i,The reverse implication follows similarly. The proof of EFX is similar to that of EF1., which gives us w i (A i ) ≥ 40550000 − 0.8. For this to be true, we need w i (A i ) ≥ 40550000 since T(V G ) has all integral values. We know that such an allocation doesn't exist. Hence for any α ∈ [0, 1], α-MMS does not exist for V G .Lemma 5. An α-MMS (Def. 12) allocation may not exist.Proof. Consider α = 8 · 10 4 ǫ 1 and since ǫ 1 ≤ 10 −5 , we obtain ∀i,.72−1+40550000−10 4 ǫ 1 . Since 0 < 10 4 ǫ 1 ≤ 0.1 and for this to be true, we need w i (A i ) ≥ 40550000 since T(V G ) has all integral values. Note that as α ≥ 8 · 10 4 , u i (A i ) ≥ 0.72 − 1, i.e., approximation guarantees strengthens. As we decrease ǫ 1 ; we decrease α which is 8 · 10 4 ǫ 1 and even though we weaker the approximation guarantees, α-MMS still doesn't exist.