1. Introduction
Division of resources among interested agents is wellexplored by researchers. Economists have proposed various fairness and efficiency notions widely applicable in realworld settings, such as division of inheritance, land, task, vaccines, etc. (Brams and Taylor, 1996; Moulin, 2004; SegalHalevi, 2019; STEIHAUS, 1948; Su, 2000). Computer Scientists, along with these notions, also explored computational aspects of some widely accepted fairness notions (Caragiannis et al., 2016; Aziz et al., 2019a, 2020; Barman et al., 2018b; de Keijzer et al., 2009; Freeman et al., 2019b; Procaccia and Wang, 2014). Such endeavors have led to webbased applications that offer readily available solutions, such as Spliddit ^{1}^{1}1www.spliddit.org, The Fair Proposals System ^{2}^{2}2www.fairproposals.com, Coursematch ^{3}^{3}3www.coursematch.io, Divide Your Rent Fairly ^{4}^{4}4https://www.nytimes.com/interactive/2014/science/rentdivisioncalculator.html, etc. In these applications, all the agents provide their valuations towards each item as input. The underlying algorithm aggregates the valuations and provides allocations that are fair and/or efficient as required. The algorithms used focus on resource allocation without externalities, i.e., an agent’s utility for not receiving a resource is zero. We believe this is inadequate when modeling resource allocation for necessary commodities.
The resources can be goods, chores, or a combination of both. Agents would want goods, i.e., they obtain a positive valuation for good for the resource while avoiding chores (negative valuation). Some resources could be goods for one and chore for another, which we call combination. Consider the situation during the Covid pandemic; there is a sudden and steep requirement for lifesupporting resources like hospital beds, ventilators, and vaccines. Getting a free covid vaccination affects an agent positively. Even if someone else gets the free vaccination instead of the agent, the agent values it positively, possibly less. The more vaccinated people, the better the situation. If an agent does not receive resources like a ventilator bed can incur a loss. Thus, the valuation of an agent may not only depend on obtaining the resource but also on not getting the resource. Such an effect of external factors on agents’ utility is captured via externality.
Externality in valuation signifies the effect of external factors on agents’ utility. The agent may incur a benefit – positive externalities or cost – negative externalities for an unassigned resource. In the most general form of externalities, the utility of not receiving an item depends on which other agent receives it. With agents, each agent’s valuations are dimensional; hence we refer to this as a ND domain. In the absence of externalities, it is a 1D domain. In a ND domain, the agent’s utility becomes complex, and in (Branzei et al., 2013), the authors focus only on positive externalities. In this work, which we motivate towards the pandemic, considering both positive and negative externalities is important. We consider externalities such that the agents incur a cost/benefit for not receiving a resource, yet it is independent of which other agent receives it. We refer to such a valuation domain as 2D domain – valuation for a bundle of items if an agent receives it, and valuation otherwise. If is positive, it is a positive externality, and if is negative, it is a negative externality. In 1D domain, is zero. In this work, we focus on fair and efficient resource allocation in a 2D domain for indivisible goods or/and chores. There are widely studied fairness and efficiency notions in a 1D domain, which we adapt to 2D.
Fairness notions. Fair allocation of goods and chores is well studied for 1D domain (Aziz et al., 2018b, 2019a, 2020; Caragiannis et al., 2016; Freeman et al., 2019a, 2020). The most desirable notions of fairness we consider are Envyfreeness (EF), which ensures that no agent has higher utility for a bundle received by another agent (Foley, 1967). Equitability (EQ) ensures that agents have equal utilities for their shares (Dubins and Spanier, 1961), Proportionality (PROP) ensures that every agent receives at least of its valuation of the entire bundle (STEIHAUS, 1948), and Maximin Share(MMS) ensures that each agent receive at least its MMS value (Budish, 2011).
We cannot ignore externalities in a 2D domain if we want to ensure the above properties. For example, consider two agents , and two goods . Agent 1 has valuations, represented as , and Agent 2 has . If we consider 1D valuation, allocating to 1 and to 2 is EF, which is not EF in a 2D domain because agent 1 envies agent 2. Similarly, it is crucial to note that the existing definition of PROP does not capture the essence of proportional allocation in the 2D domain; for example, it does not consider the disutility of not receiving good in case of a negative externality. For the above example, if we only consider the proportionality definition, each agent should receive goods worth at least . Guaranteeing this amount is impossible in a 2D domain. We define PROPE, an extension of proportionality for resource allocation with externalities. PROPE reduces to the Averageshare definition proposed in (Seddighin et al., 2019) for additive valuations, and both these definition reduces to proportionality under a 1D domain(Section 2). We conduct a study for the possibility and challenges of ensuring these fairness notions and their approximations for a 2D domain.
Efficiency notions. Efficiency is another essential aspect of resource allocation. Common efficiency notions like Pareto Optimality (PO), Maximum Utilitarian Welfare (MUW), Maximum Nash Welfare (MNW), and Maximum Egalitarian Welfare (MEW) also require further analysis in a 2D domain.
1.1. Our Contributions.
In this paper, we study the 2D domain for indivisible goods or chores with positive and negative externalities for different fairness (EF, PROPE, EQ, and MMS) and efficiency notions (PO MUW, MNW, and MEW).
We aim to adapt the algorithms designed for obtaining the above properties in a 1D domain. Towards this, our first approach proposes a reduction from a 2D domain to 1D through certain transformations such that ensuring fairness or efficiency in the transformed valuations also ensures them in the original 2D domain. In 2D domain, for general valuations we propose (Definition 3.1), and for additive valuations, we propose (Definition 3.2). Our second approach is to verify if an algorithm proposed for 1D domain can be directly used without transformation. In particular,

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EF and its relaxations like EF1, EFX can be retained through the transformation; hence we can adapt all the existing algorithms with appropriate transformation (Proposition 4.1).

Lastly, we study MMS and show that while we can retain MMS using transformation, we cannot retain MMS under a specific setting. The algorithm proposed in (Ghodsi et al., 2018; Garg et al., 2018; Garg and Taki, 2020; Aziz et al., 2017; Huang and Lu, 2019) cannot be applied directly in 2D, leaving scope to explore more in this domain. (Propositions [4.6,4.7,4.10,4.12,4.14])
1.2. Related Work
There are few papers that consider externalities in resource allocation. Branzie, Procaccia, and Zhang (Branzei et al., 2013) and Seddighin, Saleh, and Ghodsi (Seddighin et al., 2019) study allocating divisible and indivisible goods with positive externalities. Authors in (Branzei et al., 2013) generalize notions of proportionality and envyfreeness indivisible goods allocation with positive externalities. Authors in (Seddighin et al., 2019) explore MMS allocation for goods allocation with positive externalities; it generalizes maximin share as EMMS and further provides approximation algorithms for calculating EMMS and EMMS allocations. (Li et al., 2019) analyze facility location games with externalities. To the best of our knowledge, there is no relevant work that considers negative externalities.
We summarize the existing algorithms for 1D valuations available for each of the fairness notions.
Envyfreeness. EF may not exist for indivisible items, for example, two agents and one good. Hence we consider two popular relaxations of EF, Envyfreeness up to one item (EF1) (Budish, 2011; Lipton et al., 2004) and Envyfreeness up to any item(EFX) (Caragiannis et al., 2016). We have polytime 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., 2016) in goods or chores, and Double Round Robin (Aziz et al., 2018a) in combination. When the valuations are general, nonmonotone, and nonidentical, a polytime algorithm is given by (Bérczi et al., 2020) for two agents. Authors in (Plaut and Roughgarden, 2017) present an algorithm to find EFX allocation under identical general valuations for goods.
Equitability. Relaxation to EQ are EQ1 and EQX (Budish, 2011; Caragiannis et al., 2016). Authors in (Gourvès et al., 2014) present a polynomialtime algorithm to find EQX allocation. They present a pseudopolynomial time algorithm to find EQ1 and PO allocation in goods and chores in (Freeman et al., 2019b, 2020).
Proportionality. We have PROP1 and PROPX as the relaxation to PROP. For additive valuations, EF1 implies PROP1, and EFX implies PROPX.
MMS. MMS allocations do not always exist (Procaccia and Wang, 2014; Kurokawa et al., 2016). The papers (Procaccia and Wang, 2014; Amanatidis et al., 2017; Barman et al., 2018a; Garg et al., 2018) showed that 2/3MMS for goods always exists. (Ghodsi et al., 2018; Garg and Taki, 2020) showed that 3/4MMS for goods always exists. Authors in (Aziz et al., 2017) presented a polynomialtime algorithm for 2MMS for chores. The algorithm presented in (Barman et al., 2018a) gives 4/3MMS for chores. Authors in (Huang and Lu, 2019) showed that 11/9MMS for chores always exists. Authors in (Kulkarni et al., 2021) explored MMS for a combination of goods and chores.
Fair and Efficient. In (Caragiannis et al., 2016), the authors showed that MNW allocation is EF1 and PO for indivisible goods and gave a pseudopolynomial time algorithm (Barman et al., 2018b). For a combination of resources, the authors in (Aziz et al., 2018a) presented a polynomialtime algorithm to find EF1 and PO for two agents. An Algorithm to find PROP1 and PO was proposed by (Aziz et al., 2019b) for a combination of resources. Authors in (Aziz et al., 2020) proposed a pseudopolynomial time for finding Utilitarian maximizing among EF1 or PROP1 in goods.
Organization. In Section 2, we define all the notions and definitions related to fairness and efficiency used in this paper. In Section 4, we present results related to preserving fairness under transformation along with algorithmic analysis. In Section 5, we study efficiency preservation upon transformation and its corresponding algorithmic analysis.
2. Preliminaries
We consider a resource allocation problem for determining an allocation of indivisible items among interested agents, .

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An allocation is an way partition of . Here, is the bundle assigned to agent and and . We consider a complete allocation of items, i.e., . We represent as .

Typically in a 1D domain, agents have valuation function for items. In 2D domain, , , and for subset , . For goods, , and for chores, . For positive externality, , and for negative externality, .

The valuation naturally induces utility structure to the agents, i.e., the utility of an agent for a bundle in a 2D domain as, . Also, . In the case of 1D domain and , and , and utility .

For additive valuations, .
Note that, in 2D, additive valuations does not imply additive utility, i.e., , and . However, . Also when agents have identical valuations,

We assume monotonicity for goods, i.e., , . Similarly, We assume antimonotonicity for chores, i.e, , . As we increase goods in an agent’s bundle, the utility increases, and as we increase chores, the utility decreases. Note that we use the notation or to represent the valuation of agent for item .
We now formally define the fairness and efficiency criteria that we analyze for the 2D domain.
Definition 0 (Envyfree (EF) and relaxations (Aziz et al., 2018a; Budish, 2011; Bérczi et al., 2020; Caragiannis et al., 2016; Foley, 1967)).
An allocation that satisfies,
(1)  
(2)  
(3) 
Informally, is EF when no agent envies another. It is EF1, i.e. Envyfree up to one item, if each agent’s envy can be eliminated by either removing a good from the envied agent’s allocation or removing a chore from its allocation. For EFX, i.e., Envyfree up to any item, the envy is removed when any good or chore is removed. Hence EF EFX EF1.
Definition 0 (Equitable (EQ) and relaxations (Dubins and Spanier, 1961; Freeman et al., 2019a, 2020)).
An allocation is said to be equitable, when , . An allocation is said to be EQ1, i.e., Equitable up to one item, . An allocation is said to be EQX, i.e., Equitable up to any item, and , and , and .
Informally, in EQ, all agents are treated equally, i.e., everyone values their bundles equally. It is EQ1, i.e., Equitable up to one good, if the utility of an agent is less than other agents. There exists some good in other agents’ bundle or some chore in its bundle; on removing it, the utility of agent is at least the utility of other agents. Similarly, for EQX, the above satisfies all goods in other’s bundles and for all chores in their bundles.
Definition 0 (Proportionality (PROP) (Steihaus, 1948)).
An allocation is said to be proportional, if , .
For 2D domain, achieving PROP is impossible (e.g., given in Section 1). Hence, we adapt the definition and propose PROPE for valuations with externalities as follows,
Definition 0 (Proportionality with externality (PROPE)).
An allocation is said to PROPE if
(4) 
We now define the relaxations for PROPE as below, similar to the relaxations of PROP.
Definition 0 (PROPE relaxations).
An allocation that satisfies,
(5)  
(6) 
Definition 0 (Maxmin Share MMS (Budish, 2011)).
An allocation is said to be MMS if , where
An allocation is said to be MMS if it guarantees for , and when , where
We consider the following efficiency criteria,
Definition 0 (ParetoOptimal (PO)).
An allocation is PO if s.t., , and , .
We consider Utilitarian welfare, for which an allocation is the sum of agents utilities. Similarly, Nash welfare corresponds to the product of agents’ utilities, and Egalitarian welfare corresponds to the minimum individual agents’ utility. Formally, the optimal welfare is,
Definition 0 (Maximum Welfare).
The optimal welfare is,
(7)  
(8)  
(9) 
3. Reduction from 2D to 1D
We define two transformations that convert 2D utilities or valuations to corresponding 1D valuations. We then represent the transformed valuations in 1D as . We denote the utility transformation as and valuation transformation as .
Definition 0 (Transformation ).
Given a resource allocation problem where represents 2D valuations we obtain the 1D valuations denoted by as follows, ,
(10) 
Each agent , has utility function for goods and for chores.
Given that is monotone, is normalized (i.e. ), monotone and nonnegative for goods and accordingly for chores, i.e., antimonotone and negative.
transforms utility from to utility in . Now, in terms of valuations, the transformation that can reduce valuations from to for additive settings is known and is given below. However, such a transformation is unknown in general valuations.
Definition 0 (Transformation ).
We consider the following valuation transformation from 2D domain to a 1D domain . Each agent , has valuation function for goods and for chores. We define utility
(11) 
Proposition 0.
For additive valuations, utility transformation is equivalent to valuation transform.
Proof Sketch.
Similarly, we can proof vice versa. ∎
4. Fairness in 2D Domain
In this section, we analyze the challenges in adapting the existing fairness criteria and algorithms for the 2D domain. We will explore how transformation can help us retain certain fairness properties, and thus we can transform into 1D and apply the existing approaches. However, in the cases where transformation doesn’t work, we analyze their corresponding algorithms.
4.1. Analysis for EF and its relaxation
Proposition 0.
Given a resource allocation problem , an allocation is EF/EFX/EF1 in , iff is EF/EFX/EF1 in
Proof Sketch. The utility transformation retains these properties.
Similarly, the proof follows for EFX and EF1. Accordingly in additive setting retains these properties by Proposition 3.3. ∎
Given the above proposition, we make the following remarks for algorithms to find EF1 and EFX in additive and general 2D domain.
Remark 1.
EF1 for Additive Valuations: Round Robin for goods or chores and Double Round Robin for a combination (Aziz et al., 2018a; Caragiannis et al., 2016) find EF1 in polynomial time in 1D domain. But is not additive; hence we cannot directly use the algorithms. We can apply the transformation , i.e. we can transform 2D domain to 1D, , and find EF1 allocations due to Proposition 4.1.
Remark 2.
EF1 for General Valuations: Envycycle elimination algorithm (Lipton et al., 2004; Bhaskar et al., 2020) finds EF1 allocations in 1D general valuations. We can apply the utility transformation , and apply envycycle elimination algorithm to to obtained EF1 allocation in due to Proposition 4.1.
Proof Sketch. Envycycle elimination algorithm is applicable for indivisible goods with general 1D valuations. The assumption for the algorithm is that , is nonnegative, normalized, and monotone. In 1D domain , , i.e., the utility function is nonnegative, normalized, and monotone. Now, is nonnegative, normalized, and monotone. We can apply the envy cycle elimination algorithm to to find EF1 allocation in . Similarly, for chores presented in paper (Bhaskar et al., 2020). ∎
Remark 3.
EFX for Identical valuations: Leximin++ finds EFX allocations for a 1D domain with identical general valuations (Plaut and Roughgarden, 2017). Leximin++ selects the allocation which maximizes the minimum individual utility. If there are multiple such allocations, it will further maximize the size of the bundle of the minimum utility, and then go on to maximizing the second minimum, and so on.

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Note that Leximin/Leximin++ allocation in 2D domain and 1D domain might not be same.
Example 0.
Consider two agents and two goods . Agents have additive valuations. , , , . , and .

Similarly, we can adapt the cut and choose algorithm for two agents where an agent makes a cut using Leximin++ solution, which gives EFX and PO.
For EF and its relaxation, we show how few of the current algorithms result in ; however, given that the transformations retain this notion, we can adapt any algorithm via .
4.2. Analysis for EQ and its relaxations
Proposition 0.
Given a resource allocation problem ,

With identical general valuations, an allocation is EQ/EQX/EQ1 in iff is EQ/EQX/EQ1 in .

For nonidentical valuations an allocation is EQ/EQX/EQ1 in does not necessarily mean is EQ/EQX/EQ1 in , and vice versa, even for additive valuations.
Proof Sketch a. When the valuations are identical and additive EF/EFX/EF1 EQ/EQX/EQ1 (Freeman et al., 2019a). Hence, from Proposition 4.1, we can prove the first statement.
Proof Sketch b. Consider the example, where and . The 2D additive valuations for agent for , , , and . For agent 2, the additive valuations for , , , and .

is EQ in , but is not even EQ1 in .

is EQ in but is not even EQ1
∎
Given the above proposition, applying to adapt algorithms proposed for 1D is not possible unless identical valuations. Despite this, we make the following observation to ensure EQX.
Remark 4.
EQX for Additive Valuations: Authors in (Gourvès et al., 2014) proposed a greedy algorithm to find EQX for goods when agents have 1D additive valuations. In each round, the agent with the least utility picks its favourite good. We can apply this algorithm in .
Proof Sketch. , and an allocation , assume that , and consider the last good added to is , then before adding , suppose the bundles were and for , and respectively. Since was added to ’s bundle, , We know that ,
∎
Even though the transformation doesn’t retain EQX, the algorithm in (Gourvès et al., 2014) can be directly applied in .
4.3. Analysis for PROPE
Ensuring PROP given by Eq. 2.3 is impossible in a 2D domain; we use the definition given in Eq. 4. In (Seddighin et al., 2019) the authors have also modified PROP to include externalities when valuations are additive. An allocation is proportional when it ensures each agent the Averageshare. In the domain , the average value of item for agent , denoted by
(12) 
The averageshare of agent ,
(13) 
Proposition 0.
For additive valuations, PROPE reduces to Averageshare, and both reduce to proportionality definition in a 1D domain, i.e., no externalities.
Proof Sketch.
Similarly, we can prove vice versa. ∎
Proposition 0.
Given a resource allocation problem , an allocation is PROPE/PROPXE/PROP1E in , iff is PROP/ PROPX/PROP1 in for additive valuations.
Proof Sketch. Consider an allocation is PROP in , i.e., . For additive valuations, is equal to . Now, , and we get . Hence is PROPE in as given in Eq.4. The proof for PROPXE and PROP1E will follow in similar fashion. ∎

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 :

It is well know that in a 1D domain, EF PROP for subadditive valuation. Whereas in a 2D domain, EF is always PROPE, even for general arbitrary valuations, , . Also EF1 PROP1E, and EFX PROPXE in additive setting.
4.4. Analysis for MMS
Proposition 0.
Given a resource allocation problem , an allocation is MMS in , iff is MMS in
MMS allocation doesn’t always exists, and even if it does computing it is NPhard, but a PTAS exists (Woeginger, 1997). There are various algorithms for approximate MMS or MMS (Procaccia and Wang, 2014; Amanatidis et al., 2017; Barman et al., 2018a; Garg et al., 2018; Ghodsi et al., 2018; Garg and Taki, 2020; Aziz et al., 2017; Huang and Lu, 2019). Unfortunately, MMS is not retained through .
We know that is positive for goods and negative for chores in a 1D domain. However, we can no longer guarantee that for goods with negative externalities and chores with positive externalities. For a resource allocation problem, some agents can have , while others can have . We consider the generalized definition of MMS as defined in Def. 2.6.
Proposition 0.
Given a resource allocation problem ,

If an allocation is MMS in then is MMS in , for positive externalities in goods, and negative externalities in chores but not vice versa.

If an allocation is MMS in then is MMS in , for negative externalities in goods, and for positive externalities in chores but not vice versa.
Proof Sketch [a]. If allocation is MMS in , then
So in the case of positive externalities for goods, , and , and in the case of negative externalities for chores, , and , and so
However, if is MMS in , does not mean it is in .
Example 0.
Consider and , both having additive identical valuations, given by, , , , , , and . and . In , is MMS, while it is not in .
Proof Sketch [b]. If allocation is MMS in , then there are two conditions. Consider in the case of goods in ,  i) ii) . We will denote for . Note that in .
If allocation is MMS in , then

, i.e.
Now,

, i.e. The proof follows similarly.
However, if allocation is MMS in , does not mean it is MMS in .
Example 0.
Consider and , both having additive identical valuations, given by, , , , , , and . and Allocation is in , while it is not in .
The proof follows similarly for chores. ∎
Considering point (a) from Proposition 4.7, we conclude that we can apply transformation and adapt algorithms when we have goods with positive externalities or chores with negative externalities. Yet, due to point (b) from the same proposition, we cannot apply transformation when we have goods with negative externalities and chores with positive. Hence for the latter setting, we explore if we can directly use the algorithms in the 2D domain to obtain MMS. We make the following remarks towards some negative results along these lines.
Remark 6.
Proposition 0.
Bag Filling does not guarantee MMS allocation for goods with negative externalities when applied in .
Proof.
Example 0.
Consider 3 agents  , having additive valuations for 12 goods  provided in Table 1. The valuation profile in Table 1 is based on the MMS doesn’t exist example in the paper (Kurokawa et al., 2018). For calculating MMS Share, Agent 1 divides items as , Agent 2 divides items as , and Agent 3 divides items as .
(256,124)  (256,124)  (256,124)  380  380  380  
(274,75)  (274,75)  (275,75)  349  349  350  
(255,75)  (255,75)  (254,75)  330  330  329  
(245,75)  (245,75)  (245,75)  320  320  320  
(235,75)  (235,75)  (235,75)  310  310  310  
(198,75)  (198,75)  (198,75)  273  273  273  
(144,75)  (145,75)  (144,75)  219  220  219  
(135,75)  (134,75)  (135,75)  210  209  210  
(55,75)  (55,75)  (54,75)  130  130  129  
(45,75)  (44,75)  (45,75)  120  119  120  
(34,75)  (35,75)  (35,75)  109  110  110  
(25,75)  (25,75)  (25,75)  100  100  100 
In Bag Filling, similar to the famous moving knife algorithm, for goods valued , we keep adding goods in a bag until some agent claims it, i.e., the agent values it at least , and so on. We set , i.e., MMS allocation. In Table 1, for , , , and hence we can apply bag filling to obtain MMS allocation. Note that in , we start filling the bag. till , no one claims it. As the bag reaches , all three claim it, and we break ties lexicographically and assign it to agent , , and . We again start filling, till , no one claims it. As the bag reaches , both agents and claims it. We assign the bag to agent , , and . and we assign remaining items to agent , , and . This allocation is not MMS in ,i.e., in 2D domain proving our proposition. ∎
Further in , the allocation found using bag filling is MMS in .
Remark 7.
In the 1D domain, Bag Filling works for any additive valuations; however, algorithms in (Garg et al., 2018; Garg and Taki, 2020) first converted these valuations into ordered instances and then apply bag filling for lowvalued items. However, in 2D domain, Bag filling doesn’t work even for IDO (Identical Ordinal Preference) instance
Remark 8.
Proposition 0.
For chores with positive externalities in , Round Robin allocation doesn’t guarantee 2MMS allocation.
Proof.
Consider the following example,
Example 0.
Consider 2 agents  and 3 chores  , both having additive identical valuations, given by, , , and . , and . In Round Robin, the agents arrive in a fixed order and pick the item with the maximum utility of the remaining items sequentially. Agent choose first, and Round Robin allocation is . is 2MMS in , but not in
∎
Proposition 0.
For chores with positive externalities in , the algorithm in (Huang and Lu, 2019) doesn’t guarantee 11/9MMS allocation.
Proof.
Consider the following example,
Example 0.
Consider 3 agents  , having additive valuations for 12 chores  . In Table 1, we negated all the valuations, i.e., , and . Now , and . We set a threshold value for each agent, and then iterate times to create bundles of chores to allocate among agents. In each iteration, we add chores from largest (more negative) to lowest until, , we allocate the bundle to and repeat the process with remaining agents and chores. We set the threshold value to of MMS for each agent, , and .
In , we add chores to a bundle; it doesn’t violate any agent’s constraint. Now, as we add chore ; it violates all three agents’ threshold value. Similarly, adding a chore from to this bundle will again violate. On adding , the bundle satisfies , and further adding any chore will only violate. We break ties lexicographically and assign the bundle to agent 1, , . Similarly, we continue with the algorithm, and we get , , leaving chores unallocated. Assigning them to any agent will violate MMS. On apply this in , we get which is MMS in .
∎
We cannot retain MMS via transformation for goods with negative externality and chores with positive externality, and we cannot apply the algorithm presented in (Ghodsi et al., 2018; Garg et al., 2018; Garg and Taki, 2020; Aziz et al., 2017; Huang and Lu, 2019) in . It leaves further scope to explore the algorithms for the same.
5. Efficiency in 2D Domain
We will now analyze if the transformation defined in Equation 10 also helps us find efficient allocations in the 2D domain.
We find utility transformation holds for PO and MUW and does not hold for MEW and MNW. MNW cannot be defined in as some agents might have nonnegative utilities while some might have negative utilities.
Proposition 0.
Given a resource allocation problem , an allocation is PO in , iff is PO in
Proposition 0.
Given a resource allocation problem , an allocation is MUW in , iff is MUW in
Proposition 0.
Given a resource allocation problem an allocation is MEW in does not necessarily mean is MEW in , even with additive valuations.