Original: Optimal Efﬁciency-Envy Trade-Off via Optimal Transport
Revised: Optimal Efﬁciency-Envy Trade-Off via Optimal Transport

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AfﬁliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: We consider the problem of allocating a distribution of items to n recipients whereeach recipient has to be allocated a ﬁxed, prespeciﬁed fraction of all items, whileensuring that each recipient does not experience too much envy.
Revised: We consider the problem of allocating a distribution of items to n recipients whereeach recipient has to be allocated a ﬁxed, prespeciﬁed fraction of all items, whileensuring that each recipient does not experience too much envy.

Original: We show that thisproblem can be formulated as a variant of the semi-discrete optimal transport (OT)problem, whose solution structure in this case has a concise representation and asimple geometric interpretation.
Revised: We show that thisproblem can be formulated as a variant of the semi-discrete optimal transport (OT)problem, whose solution structure in this case has a concise representation and asimple geometric interpretation.

Original: Unlike existing literature that treats envy-freenessas a hard constraint, our formulation allows us to optimally trade off efﬁciencyand envy continuously.
Revised: Unlike existing literature that treats envy-freenessas a hard constraint, our formulation allows us to optimally trade off efﬁciencyand envy continuously.

Original: Additionally, we study the statistical properties of thespace of our OT based allocation policies by showing a polynomial bound onthe number of samples needed to approximate the optimal solution from samples.
Revised: Additionally, we study the statistical properties of thespace of our OT based allocation policies by showing a polynomial bound onthe number of samples needed to approximate the optimal solution from samples.

Original: Our approach is suitable for large-scale fair allocation problems such as the blooddonation matching problem, and we show numerically that it performs well on aprior realistic data simulator.
Revised: Our approach is suitable for large-scale fair allocation problems such as the blooddonation matching problem, and we show numerically that it performs well on aprior realistic data simulator.

Original:  Introduction
Revised:  Introduction

Original: In this work, we focus on the problem of ﬁnding an allocation policy that divides a pool of items,represented by a distribution D , to n recipients, under the constraint that each recipient i must beallocated a pre-speciﬁed fraction p
Revised: In this work, we focus on the problem of ﬁnding an allocation policy that divides a pool of items,represented by a distribution D , to n recipients, under the constraint that each recipient i must beallocated a pre-speciﬁed fraction p

Original: ⇤ i of the items, where p ⇤ i 2 (0 , 1) , 1 > p
Revised: ⇤ i of the items, where p ⇤ i 2 (0 , 1) , 1 > p

Original: ⇤ = 1 is an input to theproblem that characterizes the priority of each recipient.
Revised: ⇤ = 1 is an input to theproblem that characterizes the priority of each recipient.

Original: We refer to { p ⇤ i } ni =1 as the target matching 18 distribution.
Revised: We refer to { p ⇤ i } ni =1 as the target matching 18 distribution.

Original: In addition to this matching distribution constraint, we also require that the recipient’senvy, which we will deﬁne formally later, be bounded.
Revised: In addition to this matching distribution constraint, we also require that the recipient’senvy, which we will deﬁne formally later, be bounded.

Original: Unlike the existing resource allocationliterature, where envy-free is either treated as a hard constraint or not considered at all (in divisiblesettings), we allow the central planner to specify the level of envy that is tolerated, and ﬁnd the mostefﬁcient allocation given the amount of envy budget.
Revised: Unlike the existing resource allocationliterature, where envy-free is either treated as a hard constraint or not considered at all (in divisiblesettings), we allow the central planner to specify the level of envy that is tolerated, and ﬁnd the mostefﬁcient allocation given the amount of envy budget.

Original: This allows us to reduce the envy signiﬁcantlywithout paying the full price of fairness with respect to efﬁciency.
Revised: This allows us to reduce the envy signiﬁcantlywithout paying the full price of fairness with respect to efﬁciency.

Original: To concretely motivate our model, let us consider the blood donor matching problem that was ﬁrststudied in [22].
Revised: To concretely motivate our model, let us consider the blood donor matching problem that was ﬁrststudied in [21].

Original: The Meta platform has a tool called Facebook Blood Donations , where users whoopt in to receive notiﬁcations are notiﬁed about blood donation opportunities near them.
Revised: The Meta platform has a tool called Facebook Blood Donations , where users whoopt in to receive notiﬁcations are notiﬁed about blood donation opportunities near them.

Original: Dependingon the user’s and the blood bank’s speciﬁc characteristics (e.g., age, occupation for the user, andlocations, hours for the blood bank), notiﬁcations about different donation opportunities have differentprobabilities of resulting in an actual blood donation.
Revised: Dependingon the user’s and the blood bank’s speciﬁc characteristics (e.g., age, occupation for the user, andlocations, hours for the blood bank), notiﬁcations about different donation opportunities have differentprobabilities of resulting in an actual blood donation.

Original: The platform would like to send each userthe most relevant notiﬁcations (to maximize the total number of potential blood donations), whilemaintaining certain fairness criteria for all the blood banks that participate in this program.
Revised: The platform would like to send each userthe most relevant notiﬁcations (to maximize the total number of potential blood donations), whilemaintaining certain fairness criteria for all the blood banks that participate in this program.

Original: Althoughthe platform can theoretically send each user multiple notiﬁcations about multiple blood banks, foruser experience and other practical reasons, this is not done, at least in the model introduced in [22].
Revised: Althoughthe platform can theoretically send each user multiple notiﬁcations about multiple blood banks, foruser experience and other practical reasons, this is not done, at least in the model introduced in [21].

Original: Therefore in this problem, users’ attention is the scarce resource that the platforms needs to allocate todifferent blood banks.
Revised: Therefore in this problem, users’ attention is the scarce resource that the platforms needs to allocate todifferent blood banks.

Original: The most natural type of fairness criteria in this setting is perhaps the number of users that received notiﬁcations about each of the blood banks.
Revised: The most natural type of fairness criteria in this setting is perhaps the number of users that received notiﬁcations about each of the blood banks.

Original: For example, it is not desirable tomatch zero users to a particular, potentially inconveniently-located, blood bank, even if matchingzero users to this blood bank results in more blood donations in expectation.
Revised: For example, it is not desirable tomatch zero users to a particular, potentially inconveniently-located, blood bank, even if matchingzero users to this blood bank results in more blood donations in expectation.

Original: Another fairness desideratum commonly studied in the literature is called envy-freeness .
Revised: Another fairness desideratum commonly studied in the literature is called envy-freeness .

Original: We say thatrecipient A envies recipient B if A values B ’s allocation more than her own.
Revised: We say thatrecipient A envies recipient B if A values B ’s allocation more than her own.

Original: Intuitively, an allocationthat is envy-free–where no agent envies another agent–is perceived to be fair.
Revised: Intuitively, an allocationthat is envy-free–where no agent envies another agent–is perceived to be fair.

Original: This paper considers both of the two fairness criteria mentioned above: we study a setting where thegoal is to maximize social welfare under a matching distribution constraint, while ensuring that eachrecipient has bounded envy.
Revised: This paper considers both of the two fairness criteria mentioned above: we study a setting where thegoal is to maximize social welfare under a matching distribution constraint, while ensuring that eachrecipient has bounded envy.

Original: We make the following contributions in regards to this problem:
Revised: We make the following contributions in regards to this problem:

Original: We formulate it as a constrained version of a semi-discrete optimal transport problem and
Revised: We formulate it as a constrained version of a semi-discrete optimal transport problem and

Original: show that the optimal allocation policy has a concise representation and a simple geometric
Revised: show that the optimal allocation policy has a concise representation and a simple geometric

Original: structure.
Revised: structure.

Original: This is particularly attractive for large-scale allocation problems, due to the fast
Revised: This is particularly attractive for large-scale allocation problems, due to the fast

Original: computation of a match given an item.
Revised: computation of a match given an item.

Original: This insight also shines new light on the question of
Revised: This insight also shines new light on the question of

Original: when envy arises, and when the welfare price on envy-freeness is large.
Revised: when envy arises, and when the welfare price on envy-freeness is large.

Original: We propose an efﬁcient stochastic optimization algorithm for this problem and show that it
Revised: We propose an efﬁcient stochastic optimization algorithm for this problem and show that it

Original: has a provable convergence rate of O (1 / p T ) .
Revised: has a provable convergence rate of O (1 / p T ) .3.

Original: We investigate the statistical properties of the space of our optimal transport based allocation
Revised: We investigate the statistical properties of the space of our optimal transport based allocation

Original: policies by showing a Probably Approximately Correct (PAC)-like sample complexity bound
Revised: policies by showing a Probably Approximately Correct (PAC)-like sample complexity bound

Original: for approximating the optimal solution given ﬁnite samples.
Revised: for approximating the optimal solution given ﬁnite samples.

Original: In Section 3 we formally deﬁne the problem we are interested in.
Revised: In Section 3 we formally deﬁne the problem we are interested in.

Original: In Section 4, we show that thisproblem can be formulated as a semi-discrete optimal transport problem, whose solution has a simplestructure with a nice geometric interpretation.
Revised: In Section 4, we show that thisproblem can be formulated as a semi-discrete optimal transport problem, whose solution has a simplestructure with a nice geometric interpretation.

Original: Section 5 develops a practical stochastic optimizationalgorithm.
Revised: Section 5 develops a practical stochastic optimizationalgorithm.

Original: In Section 6, we show that an ✏ -approximate solution can be found with high probabilitygiven ˜ O ( n ✏ 2 ) samples, where n is the number of recipients.
Revised: In Section 6, we show that an ✏ -approximate solution can be found with high probabilitygiven ˜ O ( n ✏ 2 ) samples, where n is the number of recipients.

Original: Finally, in Section 7 we demonstrate theeffectiveness of our approach using both artiﬁcial and a semi-real data.
Revised: Finally, in Section 7 we demonstrate theeffectiveness of our approach using both artiﬁcial and a semi-real data.

Original:  Literature Review
Revised:  Literature Review

Original: Blood donation matching.
Revised: Blood donation matching.

Original: McElfresh et al.
Revised: McElfresh et al.

Original: [22] introduced this problem and modeled it as anonline matching problem, where the matching quality between an user and a blood bank is assumedto be known to the platform.
Revised: [21] introduced this problem and modeled it as anonline matching problem, where the matching quality between an user and a blood bank is assumedto be known to the platform.

Original: The model formulation there is complex and their matching policy israther cumbersome, requiring a separate parameter for each (donor, recipient) pair.
Revised: The model formulation there is complex and their matching policy israther cumbersome, requiring a separate parameter for each (donor, recipient) pair.

Original: Compared to theirpaper, we are able to provide better structural insights to the problem by utilizing a simpler modelthat still captures the most salient part of the problem.
Revised: Compared to theirpaper, we are able to provide better structural insights to the problem by utilizing a simpler modelthat still captures the most salient part of the problem.

Original: Online Resource Allocation.
Revised: Online Resource Allocation.

Original: Another strand of work that our paper is closely related to is that ofonline resource allocation, especially those with i.i.d.
Revised: Another strand of work that our paper is closely related to is that ofonline resource allocation, especially those with i.i.d.

Original: or random permutation input models [18, 13,14, 2, 1, 6].
Revised: or random permutation input models. Agrawal,

Original: However, none of theses papers study the concept of envy.
Revised: Wang, and Ye [2] studied the setting with linear objective and gave competitive ratio bounds. Then, Agrawal and Devanur [1] generalized the results to concave objectives and convex constraints. Later,

Original: Recently, Balseiro, Lu, and 
Revised: Devanur et al. [14] improved the approximation ratio bounds and relaxed the input assumptions onthe budgets. Balseiro, Lu, and Mirrokni [6] show that online mirror descent on the dual multipliersdoes well under both i.i.d. adversarial, and certain non-stationary input settings.

Revised: However, none oftheses papers study the envy-free criterion.

Original:  Mirrokni [5] studied an online resource allocation problem with fairness regularization.
Revised: Recently, Balseiro, Lu, and Mirrokni [5] studied an onlineresource allocation problem with fairness regularization.

Original: Althoughenvy regularization  can be formulated in their framework, they did not provide an explicit solution tothe ofﬂine problem.
Revised: Although the authors did not explicitlystudy envy regularization, their regularization framework can be modiﬁed to accommodate envyregularization.

Revised: However, like all the other papers mentioned in this paragraph, the ofﬂine solutionis used as the benchmark to measure regret, but no explicit solution is given to the ofﬂine problem.

Original: Our analysis focuses on the ofﬂine problem, and draws an explicit connection tooptimal transport, which allowed us to provide a novel PAC-like analysis on the sample complexityof the problem.
Revised: Our analysis focuses on the ofﬂine problem, and draws an explicit connection to optimal transport,which allowed us to provide a novel PAC-like analysis on the sample complexity of the problem.

Revised: Inanother recent paper, Sinclair, Banerjee, and Yu [27] studied the trade-off between minimizing envyand minimizing waste, which refers to un-allocated resources.

Revised: Despite close similarity between ourtitles, their ofﬂine

Original: Fair Division.
Revised: Fair Division.

Original: Envy is a popular concept studied in the fair division literature.
Revised: Envy is a popular concept studied in the fair division literature.

Original: A large body ofthese papers are formulated as a cake-cutting problem ([25, 11, 24]) where the resources are modeledas an interval and the agents’ valuations are represented as functions on this interval.
Revised: A large body ofthese papers are formulated as a cake-cutting problem ([24, 12, 23]) where the resources are modeledas an interval and the agents’ valuations are represented as functions on this interval.

Original: Caragianniset al.
Revised: Caragianniset al.

Original: [9] provide a analysis on the worst case efﬁciency loss due to the envy-freeness constraint.
Revised: [10] provide a analysis on the worst case efﬁciency loss due to the envy-freeness constraint.

Original: Later Cohler et al.
Revised: Later Cohler et al.

Original: [12] design algorithms for computing optimal envy-free cake cutting allocationsunder different relatively simple classes of valuation functions.
Revised: [13] design algorithms for computing optimal envy-free cake cutting allocationsunder different relatively simple classes of valuation functions.

Original: Unlike these papers that focus on thehard constraint of zero envy, we treat the allowable envy as a parameter, and ﬁnd the most efﬁcientsolutions subject to the desired amount of envy.
Revised: Unlike these papers that focus on thehard constraint of zero envy, we treat the allowable envy as a parameter, and ﬁnd the most efﬁcientsolutions subject to the desired amount of envy.

Original: In indivisible settings, there are some related conceptsof envy that does not require a strict zero envy.
Revised: In indivisible settings, there are some related conceptsof envy that does not require a strict zero envy.

Original: For instance, Budish [8] proposes an approximatecompetitive equilibrium from equal incomes approach that achieves envy free up to 1 item (EF1).
Revised: For instance, Budish [9] proposes an approximatecompetitive equilibrium from equal incomes approach that achieves envy free up to 1 item (EF1).

Original: Caragiannis et al.
Revised: Caragiannis et al.

Original: [10] introduce the concept of envy free up to the Least Valued Good, which is a stronger version of EF1.
Revised: [11] introduce the concept of envy free up to the Least Valued Good, which is astronger version of EF1.

Original: Although these papers do allow a small amount of envy, these conceptsare introduced mainly to circumvent the impossibilities introduced by the indivisible settings, not toallow central planners to control the level of envy tolerance.
Revised: Although these papers do allow a small amount of envy, these conceptsare introduced mainly to circumvent the impossibilities introduced by the indivisible settings, not toallow central planners to control the level of envy tolerance.

Original: Finally, Donahue and Kleinberg [15]also studies a setting where the central planner can set his own tolerance of (un)fairness.
Revised: Finally, Donahue and Kleinberg [15]also studies a setting where the central planner can set his own tolerance of (un)fairness.

Original: Howevertheir setting is quite different from ours, as they consider the allocation of ﬁxed amount of identicalresource, where as we assume that recipients have different valuations for the different items.
Revised: Howevertheir setting is quite different from ours, as they consider the allocation of ﬁxed amount of identicalresource, where as we assume that recipients have different valuations for the different items.

Original: Optimal Transport.
Revised: Optimal Transport.

Original: OT has been applied before in resource allocation settings in the economicsliterature (see [17] for a survey).
Revised: OT has been applied before in resource allocation settings in the economicsliterature (see [17] for a survey).

Original: For example, the Hotelling location model with a continuous massof consumers can be solved with the same assignment procedure as the one we consider for ourallocation problem without envy constraints.
Revised: For example, the Hotelling location model with a continuous massof consumers can be solved with the same assignment procedure as the one we consider for ourallocation problem without envy constraints.

Original: The more general method of assigning points in spaceto a ﬁnite set of sites via this procedure was developed by Aurenhammer, Hoffmann, and Aronov [4].
Revised: The more general method of assigning points in spaceto a ﬁnite set of sites via this procedure was developed by Aurenhammer, Hoffmann, and Aronov [4].

Original: Scetbon et al.
Revised: Scetbon et al.

Original: [26] consider an OT setting with equitability (every agent’s ﬁnal utility is the same),which is a different fairness criteria from envy, and also not adjustable like in our setting.
Revised: [25] consider an OT setting with equitability (every agent’s ﬁnal utility is the same),which is a different fairness criteria from envy, and also not adjustable like in our setting.

Original: To the bestof our knowledge, no existing application of OT models the envy constraints that we consider here.
Revised: To the bestof our knowledge, no existing application of OT models the envy constraints that we consider here.

Original: 2.1 Background on Optimal Transport
Revised: 2.1 Background on Optimal Transport

Original: Since our work draws an explicit connection to optimal transport (OT), we provide a summary ofkey
Revised: Since our work draws an explicit connection to optimal transport (OT), we provide a summary ofkey

Original: OT results here.
Revised: OT results here.

Original: Let ↵ ,  be two probability measures on the metric spaces
Revised: Let ↵ ,  be two probability measures on the metric spaces

Original: X , Y respectively.
Revised: X , Y respectively.

Original: We deﬁne ⇧ ( ↵ ,  ) as the set of joint probability measures on X ⇥ Y with marginals ↵ and  .
Revised: We deﬁne ⇧ ( ↵ ,  ) as the set of joint probability measures on X ⇥ Y with marginals ↵ and  .

Original: The
Revised: The

Original: Kantorovich formulation of the optimal transport problem [20] can be written as
Revised: Kantorovich formulation of the optimal transport problem [19] can be written as

Original: [Equation]
Revised: [Equation]

Original: (1) where c ( x, y ) is the cost associated with “moving” x to y .
Revised: where c ( x, y ) is the cost associated with “moving” x to y .

Original: This is called a transportation problembecause the conditional probability ⇡ ( y | x ) speciﬁes a transportation plan for moving probabilitymass from X to Y .
Revised: This is called a transportation problembecause the conditional probability ⇡ ( y | x ) speciﬁes a transportation plan for moving probabilitymass from X to Y .

Original: Note that ⇡
Revised: Note that ⇡

Original: ( x, y ) =
Revised: ( x, y ) =

Original: ⇡ ( y | x ) d ↵ ( x ) .
Revised: ⇡ ( y | x ) d ↵ ( x ) .

Original: If  is a discrete measure, i.e. Y is ﬁnite, thenit is known [3] that the dual to (1) can be written as (here we abuse the notation  to also representthe vector of probability masses, where  i is the probability mass on point y i ):
Revised: If  is a discrete measure, i.e. Y is ﬁnite, thenit is known [3] that the dual to (1) can be written as (here we abuse the notation  to also representthe vector of probability masses, where  i is the probability mass on point y i ):

Original: [Equation]
Revised: [Equation]

Original: where n = |Y| , and L y i is what is sometimes referred to as the Laguerre cell :
Revised: where n = |Y| , and L y i is what is sometimes referred to as the Laguerre cell :

Original: [Equation]
Revised: [Equation]

Original: Proposition 1 (Proposition 2.1 [3]) .
Revised: Proposition 1 (Proposition 2.1 [3]) .

Original: If ↵ is a continuous measure, and  a discrete measure, then
Revised: If ↵ is a continuous measure, and  a discrete measure, then

Original: L ( ↵ ,  ) =
Revised: L ( ↵ ,  ) =

Original: max g E ( g ) , and the optimal solution ⇡ of (1) is given by the partition { L y i ( g ⇤ ) , i 2 [ n ] } ,i.e. d ⇡ ( x, y i )
Revised: max g E ( g ) , and the optimal solution ⇡ of (1) is given by the partition { L y i ( g ⇤ ) , i 2 [ n ] } ,i.e. d ⇡ ( x, y i )

Original: =
Revised: =

Original: d ↵ ( x ) if x 2 L y i ( g ⇤ ) , 0 otherwise.
Revised: d ↵ ( x ) if x 2 L y i ( g ⇤ ) , 0 otherwise.

Original:  Problem Formulation
Revised:  Problem Formulation

Original: There is a set of n recipients Y .
Revised: There is a set of n recipients Y .

Original: There is a “pool” of items, represented by a distribution ↵ over
Revised: There is a “pool” of items, represented by a distribution ↵ over

Original: X ✓ [0 , ¯ x ] n .
Revised: X ✓ [0 , ¯ x ] n .

Original: Each random draw from this distribution X ⇠ ↵ is a vector representing the n recipients’valuations of this item.
Revised: Each random draw from this distribution X ⇠ ↵ is a vector representing the n recipients’valuations of this item.

Original: The goal is to maximize the expected matched utilities of the recipients, whilemaintaining the constraint that the recipient y i is matched p
Revised: The goal is to maximize the expected matched utilities of the recipients, while maintaining the constraint that the recipient y i is matched p

Original: ⇤ i fraction of the times in expectation.
Revised: ⇤ i fraction of the times in expectation.

Original: Here{ p ⇤ i } ni =1 is called the target matching distribution, which intuitively represents recipients’  importance.
Revised: Here { p ⇤ i } ni =1 is called the target matching distribution, which intuitively represents recipients’ 131 importance.

Revised: Note that the constrains here are satisﬁed in expectation , which are sometimes referredto as “ex-ante” guarantees.

Revised: The reason why we consider ex-ante guarantees has to do with the type ofapplication we’re interested in.

Revised: Our motivating example is concerned with recommending hundredsof millions of users to different blood banks.

Revised: In such settings, even if we want the constraints to holdex-post, the large-scale nature of the problem and the law of large numbers means that in-expectationguarantees translate into something that is very close to holding ex-post.

Revised: That is why for internetplatform problems, requiring constraints to hold in expectation is a standard setup (see for examplethe literature on budget constraints in ad auctions, where this is the case [7]).

Original: 121 A matching policy ⇡ takes a valuation vector and maps it (potentially with randomness) to one of then recipients.
Revised: A matching policy ⇡takes a valuation vector and maps it (potentially with randomness) to one of the n recipients.

Original: Let ⇡ ( y | x ) denote the probability of matching the item to y given valuation vector x .
Revised: Let⇡ ( y | x ) denote the probability of matching the item to y given valuation vector x .

Original: The basic problem formulation is to solve the following optimization problem:
Revised: The basic problemformulation is to solve the following optimization problem:

Original: [Equation]
Revised: [Equation]

Original: WLOG, we assume that p ⇤ i > 0 for all i .
Revised: WLOG, we assume that p ⇤ i > 0 for all i .

Original: An example of such problem can be see in Figure 1, where↵ is a distribution over the unit square, and the goal is to partition the square into blue and orange regions (given to A and B respectively)
Revised: An example of such problem can be see in Figure 1, where↵ is a distribution over the unit square, and the goal is to partition the square into blue and orangeregions (given to A and B respectively)

Original: such that each region covers the desired p ⇤ A , p ⇤ B probabilitymass.
Revised: such that each region covers the desired p ⇤ A , p ⇤ B probabilitymass.

Original: Note that the orange and blue regions are allowed to over lap (probabilitistic partition), and thatthe boundary does not have to be linear as illustrated in the ﬁgure.
Revised: Note that the orange and blue regions are allowed to over lap (probabilistic partition), and thatthe boundary does not have to be linear as illustrated in the ﬁgure.

Original: As we will show later in Section 4,despite the large design space permitted by the formulation in (4), we can in fact focus on a muchsmaller design space.
Revised: As we will show later in Section 4,despite the large design space permitted by the formulation in (4), we can in fact focus on a muchsmaller design space.

Original: In resource allocation problems, it is often the case that we care not just about efﬁciency (the sum ofall recipients’ utilities), but also other fairness criteria.
Revised: In resource allocation problems, it is often the case that we care not just about efﬁciency (the sum ofall recipients’ utilities), but also other fairness criteria.

Original: One of the most commonly studied fairnesscriteria is envy-freeness.
Revised: One of the most commonly studied fairnesscriteria is envy-freeness.

Original: Agent y
Revised: Agent y

Original: i envies another agent y j if agent y i values the allocation given toy j more (after adjusting for their priority weights).
Revised: i envies another agent y j if agent y i values the allocation given toy j more (after adjusting for their priority weights).

Original: We can formally deﬁne agent
Revised: We can formally deﬁne agent

Original: y i ’s envy as
Revised: y i ’s envy as

Original: [Equation]
Revised: [Equation]

Original: Instead of the vanilla formulation in (4), we consider the following more general formulation:
Revised: Instead of the vanilla formulation in (4), we consider the following more general formulation:

Original: [Equation]
Revised: [Equation]

Original: Most existing literature focuses on ﬁnding allocations such that Envy ( y i ) is at most 0 for everyy i .
Revised: Most existing literature focuses on ﬁnding allocations such that Envy ( y i ) is at most 0 for everyy i .

Original: This can be a very restrictive constraint, often satisﬁed at the cost of reducing efﬁciency by asigniﬁcant amount (This reduction is sometimes referred to as the Cost-of-Fairness).
Revised: This can be a very restrictive constraint, often satisﬁed at the cost of reducing efﬁciency by asigniﬁcant amount (This reduction is sometimes referred to as the Cost-of-Fairness).

Original: We take adifferent approach, and allow the central planner to set non-negative constraints on envy.
Revised: We take adifferent approach, and allow the central planner to set non-negative constraints on envy.

Original: Note thatsince we are motivated by internet-scale problems such as allocating hundreds of millions of users todonation centers, we focus on the ex-ante guarantees on the constraints.
Revised: Note thatsince we are motivated by internet-scale problems such as allocating hundreds of millions of users todonation centers, we focus on the ex-ante guarantees on the constraints.

Original:  Optimal Solution Structure
Revised:  Optimal Solution Structure

Original: The space of feasible solutions for (6) is large, which makes the problem difﬁcult to optimize directly.
Revised: The space of feasible solutions for (6) is large, which makes the problem difﬁcult to optimize directly.

Original: However we can use the tools from OT to reduce the search space to something with much morestructure.
Revised: However we can use the tools from OT to reduce the search space to something with much morestructure.

Original: The key observation is that (6) can be formulated as variation of the semi-discrete optimaltransport problem given in Equation (1).
Revised: The key observation is that (6) can be formulated as variation of the semi-discrete optimaltransport problem given in Equation (1).

Original: Let’s ﬁrst consider the simpler case in (4) where there are no envy constraints.
Revised: Let’s ﬁrst consider the simpler case in (4) where there are no envy constraints.

Original: In this case, theproblem can be stated in the form of (1) as follows: the cost function is the negative utility ofthe
Revised: In this case, theproblem can be stated in the form of (1) as follows: the cost function is the negative utility of

Original: matched recipient c

Original: ( x, y i ) =

Original:  x i , the  measure is the discrete measure

Original: [-P ni =
-]
Revised: [-P ni =
-]

Original: p

Original: ⇤ i  y

Original: i , and the 150 matching policy ⇡ ( y | x ) in (6) is exactly the conditional probability of the joint distribution in (1).

Original: From Theorem 1 it follows that the optimal matching policy is represented by Laguerre cells given in(3): x is matched to y i if i =
Revised: From Theorem 1 it follows that the optimal matching policy is represented by Laguerre cells given in(3): x is matched to y i if i =

Original: arg
Revised: arg

Original: min k
Revised: min k

Original:  x k  g k .
Revised:  x k  g k .

Original: Note that the dual variables g 2 R n serve as an“adjustment” over the agents’ reported utilities, and the resulting matching policy is simply a greedypolicy over this adjusted valuation vector.
Revised: Note that the dual variables g 2 R n serve as an“adjustment” over the agents’ reported utilities, and the resulting matching policy is simply a greedypolicy over this adjusted valuation vector.

Original: Geometrically, each Laguerre cell is simply the intersection of half-spaces: L i ( g ) =
Revised: Geometrically, each Laguerre cell is simply the intersection of half-spaces: L i ( g ) =

Original: \ k { x : x i + g i  156 x k + g k } .
Revised: \ k { x : x i + g i  174 x k + g k } .

Original: To visualize this better, consider the simple setting with two recipients A, B where theirvaluations for an item is a joint distribution supported on [0 , 1] 2 .
Revised: To visualize this better, consider the simple setting with two recipients A, B where theirvaluations for an item is a joint distribution supported on [0 , 1] 2 .

Original: Suppose we want to match p ⇤fraction of the items to recipient B .
Revised: Suppose we want to match p ⇤fraction of the items to recipient B .

Original: Figure 1 gives a proof-by-picture that the optimal strategy is todivide the space up with a slope- 1 diagonal line such that the probability mass lying above the line isequal to p ⇤ .
Revised: Figure 1 gives a proof-by-picture that the optimal strategy is todivide the space up with a slope- 1 diagonal line such that the probability mass lying above the line isequal to p ⇤ .

Original: This geometric interpretation of the matching policy plays a crucial role in getting us asample complexity bound later in Section 6.
Revised: This geometric interpretation of the matching policy plays a crucial role in getting us asample complexity bound later in Section 6.

Original: With this geometric interpretation of the solution space in mind, let us consider the more general casewith envy constraints as formulated in (6).
Revised: With this geometric interpretation of the solution space in mind, let us consider the more general casewith envy constraints as formulated in (6).

Original: The envy constraints can be added to the OT problem in(1) like so:
Revised: The envy constraints can be added to the OT problem in(1) like so:

Original: [Equation]
Revised: [Equation]

Original: Although envy constraints make the solution space more complicated, we show that it retains thegeometric structure of being the intersection of half-spaces.
Revised: Although envy constraints make the solution space more complicated, we show that it retains thegeometric structure of being the intersection of half-spaces.

Original: The dual of (7) can be derived using
Revised: The dual of (7) can be derived using

Original: Fenchel-Rockafellar’s theorem:
Revised: Fenchel-Rockafellar’s theorem:

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Theorem 1.
Revised: Theorem 1.

Original: If ↵ is a continuous measure, and  a discrete measure, then L ( ↵ ,  ,  ) = max g,  E ( g,  ) ,and the optimal solution ⇡ of (7) is given by the partition { L y i ( g ⇤ ,  ⇤ ) , i 2 [ n ] } :
Revised: If ↵ is a continuous measure, and  a discrete measure, then L ( ↵ ,  ,  ) = max g,  E ( g,  ) ,and the optimal solution ⇡ of (7) is given by the partition { L y i ( g ⇤ ,  ⇤ ) , i 2 [ n ] } :

Original: d ⇡ ( x, y i ) =
Revised: d ⇡ ( x, y i ) =

Original: d ↵ ( x )if x 2 L y i ( g ⇤ ,  ⇤ ) , 0 otherwise.
Revised: d ↵ ( x )if x 2 L y i ( g ⇤ ,  ⇤ ) , 0 otherwise.

Original: Note that when c ( x, y i ) =
Revised: Note that when c ( x, y i ) =

Original:  x i , ¯ g  ,c ( x, y j ) is linear in x , which means that the new Laguerre cells
Revised:  x i , ¯ g  ,c ( x, y j ) is linear in x , which means that the new Laguerre cells

Original: L y ( g, y ) given in Equation (10) are still intersections of half spaces (some examples are given laterin Figure 2).
Revised: L y ( g, y ) given in Equation (10) are still intersections of half spaces (some examples are given laterin Figure 2).

Original: Furthermore, the allocation policy can be interpreted as a greedy policy based on theadjusted utility given by (9), which contains additional interaction terms that take envy into account.
Revised: Furthermore, the allocation policy can be interpreted as a greedy policy based on theadjusted utility given by (9), which contains additional interaction terms that take envy into account.

Original:  Stochastic Optimization
Revised:  Stochastic Optimization

Original: In Section 4 we showed that the optimal allocation policy to our problem has a simple geometricstructure in the form of Laguerre cells.
Revised: In Section 4 we showed that the optimal allocation policy to our problem has a simple geometricstructure in the form of Laguerre cells.

Original: In this section we present a practical algorithm for actuallycomputing the optimal Laguerre cells.
Revised: In this section we present a practical algorithm for actuallycomputing the optimal Laguerre cells.

Original: First we show that the objective function E ( g,  ) in (6) isconcave by rewriting the objective as follows:
Revised: First we show that the objective function E ( g,  ) in (6) isconcave by rewriting the objective as follows:

Original: [Equation]
Revised: [Equation]

Original: Since ¯ g  ,c ( x, y j ) is linear in g and  and taking a minimum preserves concavity, the objective functionis concave.
Revised: Since ¯ g  ,c ( x, y j ) is linear in g and  and taking a minimum preserves concavity, the objective functionis concave.

Original: Therefore, the dual problem is a constrained convex optimization problem.
Revised: Therefore, the dual problem is a constrained convex optimization problem.

Original: The gradientof E ( g,  ) can be computed as follows:
Revised: The gradientof E ( g,  ) can be computed as follows:

Original: [Equation]
Revised: [Equation]

Original: Input: Distribution ↵ , target matching distribution p ⇤ , timesteps T .
Revised: Input: Distribution ↵ , target matching distribution p ⇤ , timesteps T .

Original: 1 Initialize g 0 = 0 ,  0 = 0 , ⌘ = 1 p T .
Revised: 1 Initialize g 0 = 0 ,  0 = 0 , ⌘ = 1 p T .

Original: 2 for t 0 , 1 , 2 , . . .
Revised: 2 for t 0 , 1 , 2 , . . .

Original: , T do
Revised: , T do

Original: Sample x t ⇠ ↵ g t +
Revised: Sample x t ⇠ ↵ g t +

Original: [Equation]
Revised: [Equation]

Original: g,  )  t +
Revised:  t +

Original: [Equation]
Revised: [Equation]

Original: + ⌘ 
Revised: + ⌘ ˆ r

Original: [Equation]
Revised: [Equation]

Original: r
Revised: ,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: ⌘ + 6 return

Original: , 
Revised: P

Original: [Equation]
Revised: [Equation]

Original: 6 return
Revised: P

Original: [Equation]
Revised: [Equation]

Original: Calculating this gradient is hard, as it involves integration over an arbitrary measure ↵ .
Revised: Calculating this gradient is hard, as it involves integration over an arbitrary measure ↵ .

Original: However, anunbiased, stochastic version of the gradient can be easily obtained from a single sample x ⇠ ↵ :
Revised: However, anunbiased, stochastic version of the gradient can be easily obtained from a single sample x ⇠ ↵ :

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: The details of the algorithm is given in Algorithm 1.
Revised: The details of the algorithm is given in Algorithm 1.

Original: Standard projected SGD analysis (see for example[19]) tells us that Algorithm 1 converges at the rate E ( g ⇤ ,  ⇤ )
Revised: Standard projected SGD analysis (see for example[18]) tells us that Algorithm 1 converges at the rate E ( g ⇤ ,  ⇤ )

Original:  E ( E [ g T ] , E [  T ])  O ⇣ 1 p T ⌘ .
Revised:  E ( E [ g T ] , E [  T ])  O ⇣ 1 p T ⌘ .

Original:  Learning from Samples
Revised:  Learning from Samples

Original: So far we have assumed that the true underlying distribution is known, and that we can freely drawindependent samples it.
Revised: So far we have assumed that the true underlying distribution is known, and that we can freely drawindependent samples it.

Original: In many settings, we only have access to ↵ in the form of ﬁnite number ofi.i.d.
Revised: In many settings, we only have access to ↵ in the form of ﬁnite number ofi.i.d.

Original: samples.
Revised: samples.

Original: The goal of this section is to establish a sample complexity bound for solving the dualproblem (8).
Revised: The goal of this section is to establish a sample complexity bound for solving the dualproblem (8).

Original: This complements the result in the previous section by providing an algorithm agnosticbound on the sample complexity.

Original: To build some intuitions for how we are able to derive the resultin this section, readers familiar with traditional PAC learning results might recall that hyperplaneshave a low VC-dimension.

Original: Similar to how low VC-dimension leads to low sample complexity inclassiﬁcation tasks, the fact the boundaries of our Laguerre cells are consisted of hyperplanes alsolead to low sample complexity in our setting.

Original: In this section, we focus only on the assignment cost function c ( x, y i )
Revised: In this section, we focus only on the assignment cost function c ( x, y i )

Original: =
Revised: =

Original:  x i , which models ouroriginal resource allocation problem proposed in Section 3.
Revised:  x i , which models ouroriginal resource allocation problem proposed in Section 3.

Original: Let S = { X 1 , X 2 , . . .
Revised: Let S = { X 1 , X 2 , . . .

Original: , X m } be mindependent samples from ↵ .
Revised: , X m } be mindependent samples from ↵ .

Original: The empirical version of the dual objective (11) is:
Revised: The empirical version of the dual objective (11) is:

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Let ˆ g S , ˆ  S be the empirical maximizer given the set of samples S : (ˆ g S , ˆ  S ) :
Revised: Let ˆ g S , ˆ  S be the empirical maximizer given the set of samples S : (ˆ g S , ˆ  S ) :

Original: =
Revised: =

Original: arg max E S ( g,  ) ,and g ⇤ ,  ⇤ be the population maximizer ( g ⇤ ,  ⇤ ) =
Revised: arg max E S ( g,  ) ,and g ⇤ ,  ⇤ be the population maximizer ( g ⇤ ,  ⇤ ) =

Original: arg max E ( g,  ) .
Revised: arg max E ( g,  ) .

Original: We want to bound the numberof samples needed so that E ( g ⇤ ,  ⇤ )
Revised: We want to bound the number of samples needed so that E ( g ⇤ ,  ⇤ )

Original:  E (ˆ g S , ˆ  S ) is small with high probability.
Revised:  E (ˆ g S , ˆ  S ) is small with high probability.

Original: Let’s introduce somenotations to facilitate our later discussions.
Revised: Let’s introduce somenotations to facilitate our later discussions.

Original: Deﬁne the following hypothesis class for each i :
Revised: Deﬁne the following hypothesis class for each i :

Original: [Equation]
Revised: [Equation]

Original: as well as the overall hypothesis class:
Revised: as well as the overall hypothesis class:

Original: [Equation]
Revised: [Equation]

Original: Plugging c ( x, y i ) =
Revised: Plugging c ( x, y i ) =

Original:  x i into the deﬁnition of ¯ g  ,c , we see that for a given g, and  , the correspondinghypothesis f i 2 F i can be written as f i ( x ) =
Revised:  x i into the deﬁnition of ¯ g  ,c , we see that for a given g, and  , the correspondinghypothesis f i 2 F i can be written as f i ( x ) =

Original: w > x + b , where
Revised: w > x + b , where

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: It also follows that
Revised: It also follows that

Original: [Equation]
Revised: [Equation]

Revised: Note that F deﬁned in (18) is the main object of interest, as it contains all the possible Laguerre cellparameters.

Revised: We showed in (21) that F is at most as complex as F min , a hypothsis class constructedfrom n afﬁne hypothesis classes.

Original: This interpretation of the original hypothesis class as the minimum over n afﬁne hypothesis classesis the key observation to prove the sample complexity bound.
Revised: This interpretation of the original hypothesis class as the minimumover n afﬁne hypothesis classes is the key observation to prove the sample complexity bound.

Original: We prove our main result under thefollowing boundedness assumption:
Revised: Weprove our main result under the following boundedness assumption:

Original: Assumption 1.
Revised: Assumption 1.

Original: The hypothesis f ( x ) = min
Revised: The hypothesis f ( x ) = min

Original: i f i ( x ) = min
Revised: i f i ( x ) = min

Original: i w i > x + b i corresponding to the optimaldual solution g ⇤ ,  ⇤ satisﬁes k w i k 1 _ | b i |  R for some R > 0 .
Revised: i w i > x + b i corresponding to the optimaldual solution g ⇤ ,  ⇤ satisﬁes k w i k 1 _ | b i |  R for some R > 0 .

Original: In particular, these assumptions 215 imply that F i and F are uniformly bounded by R ¯ x + R .
Revised: In particular, these assumptions 231 imply that F i and F are uniformly bounded by R ¯ x + R .

Original: From (19) and (20) we can see that this is essentially a bound on the optimal dual variables g ⇤ ,  ⇤ ,and a bound on the ratio  j /  i , both of which are determined by the input distributions ↵ ,  , and donot depend on the number of samples.
Revised: From (19) and (20) we can see that this is essentially a bound on the optimal dual variables g ⇤ ,  ⇤ ,and a bound on the ratio  j /  i , both of which are determined by the input distributions ↵ ,  , and donot depend on the number of samples.

Original: In other words, R is a problem dependent constant.
Revised: In other words, R is a problem dependent constant.

Original: Theorem 2.
Revised: Theorem 2.

Original: Under Assumption 1, for a given sample size m , with probability 1   , E ( g ⇤ ,  ⇤ ) 
Revised: Under Assumption 1, for a given sample size m , with probability 1   , E ( g ⇤ ,  ⇤ )

Original: [Equation] Proof. We prove the result via uniform convergence: [Equation]
Revised: E (ˆ g S , ˆ  S ) <

Original: [Equation] Clearly, it sufﬁces to show that E S ( · ) converges uniformly to E ( · ) . For a given g,  , the dual objective function and its’ empirical version can be written as
Revised: O

Original: [Equation]
Revised: [Equation]

Original: Then we can rewrite the supremum in (21) as: [Equation] Since | f ( X ) |  ( R ¯ x + R ) for all f 2 F, X 2 X , it follows from Theorem 26.5 in [27] that with 224 probability  , [Equation]
Revised: The proof of this theorem can be found in the appendix.

Original: and the same also holds by replacing F with  F . Here [Equation] 5 is the standard deﬁnition of Rademacher complexity of the set F  S .
Revised: We will provide some high level intuition here.

Original: Since  i are i.i.d. Rademacherrandom variables, it is easy to see that Rad m ( F  S ) = Rad m (  F  S ) . Therefore we can use aunion bound to obtain that with probability 1   ,
Revised: The proof uses the fat-shattering dimension, which is a concept that generalizes the

Original: [Equation] [Equation] It remains to bound the Rademacher complexity of the F  S .
Revised: Vapnik–Chervonenkis (VC) dimension to functions of real values.

Original: To do so, we use tools from learningtheory, and give the following bound on the fat-shattering dimension ([7]) of the hypothesis classF . Lemma 1. Under Assumption 1, F has ⇣ -fat-shattering dimension of at most c 0 ( R ¯ x + R )
Revised: Like the VC dimension, thefat-shattering dimension is a measure of capacity for a class of functions. The larger the fat-shatteringdimension, the more “complex” the function class is.

Original: 2 ⇣n log( n ) ,where c 0 is some universal constant. The proof of Lemma 1 can be found in the Appendix. The above bound on the fat-shattering dimensioncan be used to bound the covering number (see Deﬁnition 27.1 of [27]) of F  S .
Revised: Intuitively, the more complex the functionclass is, the more samples one needs to accurately identify the optimal function within that class. Recall that the object of interest in our paper is the Laguerre cell, whose boundaries are deﬁned by aset of hyperplanes.

Original: Theorem 1 from[23] states that 
Revised: Readers readers familiar with traditional PAC learning results might recall thathyperplanes have a low VC dimension. It turns out that the hypothesis class associated with Laguerrecells deﬁned above in (18) has a low complexity as well.

Original: [Equation]
Revised: Similar to how low VC dimension leadsto low sample complexity in classiﬁcation tasks, the fact the boundaries of our Laguerre cells areconsisted of hyperplanes also lead to low sample complexity in our setting. Relation to Section 5 Note that there is some connection between the convergence of a stochasticoptimization method (computational complexity), and the sample complexity bound of the sameproblem (learning complexity). Both say something about the number of steps/samples one needsto arrive at a good solution.

Original: where B is a uniform bound on the absolute value of any f 2 F .
Revised: However, in general, these two things are not the same.

Original: Let B = ( R ¯ x + R ) , we have that
Revised: In particular,the learning complexity result from this section is algorithm agnostic.

Original: [Equation]{+The proof of Theorem 
+}
Revised: {+The proof of Theorem 
+}

Original: Where we used Dudley’s chaining integral [28, 16], Lemma 1 and (25), and setting  0 = 1 p mrespectively. Plugging the above back to (24) and (21), we see that with probability 1   , [Equation] 1  m
Revised: The non-monotonicity in the plot for the simulator data is due to the stochasticity in the SGD algorithm actually implies uniform convergence: that no matter which hypothesis one considers, its’ empiricalobjective will be close to the expected objective.

Original: A .
Revised: However, Theorem 2 does not provide a way forpractitioners to actually compute the solution. The result from Section 5 on the other hand doesprovide an algorithm for practitioners to use and shows that the algorithm is computationally efﬁcient.

Original: Conversely, ignoring the log terms, m needs to be at most on the order of ˜ O  n ✏ in order for E ( g ⇤ ,  ⇤ )
Revised: However, the convergence result in Section 5 only works for the speciﬁc optimization method that weproposed.

Original:  E (ˆ g S , ˆ  S ) to be bounded by ✏ with high probability.
Revised: Therefore these two results are complements of each other and together paint a relativelycomplete picture on how difﬁcult the task it.

Original:  Experiments
Revised:  Experiments

Original: We test our solution with both artiﬁcial data, and simulated data from a realistic simulator for blooddonor matching developed by [22].
Revised: We test our solution with both artiﬁcial data, and simulated data from a realistic simulator for blooddonor matching developed by [21].

Original: The artiﬁcial data contains two recipients, and their valuationdistribution is a linearly transformed uniform distribution.
Revised: The artiﬁcial data contains two recipients and their valuationdistribution is a linearly transformed uniform distribution.

Original: This is to make visualization of the resultingallocation policy easier.
Revised: This is to make visualization of the resultingallocation policy easier.

Original: The simulator data is based on geographical and population informationfrom San Francisco, and contains 5 recipients.
Revised: The simulator data is based on geographical and population informationfrom San Francisco, and contains 5 recipients.

Original: To set the envy budgets  2 R n , we ﬁrst decide on aconstant ✏ 2 R + , and then multiply this by the target matching distribution p
Revised: To set the envy budgets  2 R n , we ﬁrst decide on aconstant ✏ 2 R + , and then multiply this by the target matching distribution p

Original: ⇤ 2 R n :  ij =
Revised: ⇤ 2 R n :  ij =

Original: ✏ p ⇤
Revised: ✏ p ⇤

Original: i 8 i, j .
Revised: i 8 i, j .

Original: With this setup ✏ is a bound on the normalized envy for each recipient: 1 p ⇤ i Envy ( i )  ✏ 8 i. Figure 2illustrates how the allocation policy changes as we change ✏ .
Revised: With this setup ✏ is a bound on the normalized envy for each recipient: 1 p ⇤ i Envy ( i )  ✏ 8 i. Figure 2illustrates how the allocation policy changes as we change ✏ .

Original: As the envy constraint tightens, the The non-monotonicity in the plot for the simulator data is due to the stochasticity in the SGD algorithm decision boundary tilts in the direction that split the “good” (items which both agents prefer) and“bad” (items which both agents dislike) items more evenly between the recipients.
Revised: As the envy constraint tightens, thedecision boundary tilts in the direction that split the “good” (items which both agents prefer) and“bad” (items which both agents dislike) items more evenly between the recipients.

Original: Next we investigate the trade-off between envy and social welfare by using SGD to compute approximately optimal allocations for varying ✏ .
Revised: Next we investigate the trade-off between envy and social welfare by using SGD to compute approximately optimal allocations for varying ✏ .

Original: We plot the percent welfare gap (difference between themaximum welfare without envy constraints, and the welfare with envy constraints, divided by theformer) with respect to realized, max normalized envy.
Revised: We plot the percent welfare gap (difference between themaximum welfare without envy constraints, and the welfare with envy constraints, divided by theformer) with respect to realized, max normalized envy.

Original: Figure 3 shows the result.
Revised: Figure 3 shows the result.

Original: For the simulatordata, the welfare gap is small even with a no-envy constraint, which means that aiming for envy freeallocations might make sense.
Revised: For the simulatordata, the welfare gap is small even with a no-envy constraint, which means that aiming for envy freeallocations might make sense.

Original: In the case of the artiﬁcial data however, paying 50% of the full priceof fairness reduces 65% of the envy.
Revised: In the case of the artiﬁcial data however, paying 50% of the full price of fairness reduces 65% of the envy.

Original: In such settings, one might want to sacriﬁce some envy forbetter welfare.
Revised: In such settings, one might want to sacriﬁce some envy forbetter welfare.

Original: These experiments also highlight when envy arises.
Revised: These experiments also highlight when envy arises.

Original: When recipients’ utilities are highly correlated,but one recipient has larger variance than others, that recipient receives almost all the good items(which results in large envy for other recipients), even though others value the items almost as much.
Revised: When recipients’ utilities are highly correlated,but one recipient has larger variance than others, that recipient receives almost all the good items(which results in large envy for other recipients), even though others value the items almost as much.

Original: In such cases, a small reduction in welfare can reduce a large amount of envy.
Revised: In such cases, a small reduction in welfare can reduce a large amount of envy.

Original: This seems to be thecase for the simulator data.
Revised: This seems to be thecase for the simulator data.

Original: On the other hand, if utilities are correlated, but only one recipient hasvery strong preferences, then allowing a small amount of envy can improve the welfare signiﬁcantly.
Revised: On the other hand, if utilities are correlated, but only one recipient hasvery strong preferences, then allowing a small amount of envy can improve the welfare signiﬁcantly.

Original: Finally, in Figure 4 we investigate the quality of the empirical solutions as the sample size increases.
Revised: Finally, in Figure 4 we investigate the quality of the empirical solutions as the sample size increases.

Original: It can been seen that the approximation gap decreases faster than the theoretical rate, conﬁrming oursample complexity bound in Theorem 2.
Revised: It can been seen that the approximation gap decreases faster than the theoretical rate, conﬁrming oursample complexity bound in Theorem 2.

Original:  Limitations and Future Directions
Revised:  Limitations and Future Directions

Original: Although we believe that the model proposed here is natural, and captures the most salient aspects ofsome of the resource allocation problems in real life, any implementation of our proposed strategy incritical applications such as blood donation should be prefaced with more rigorous backtesting inorder to minimize the risk of unintended consequences in application speciﬁc metrics not studied inthis paper.
Revised: Although we believe that the model proposed here is natural, and captures the most salient aspects ofsome of the resource allocation problems in real life, any implementation of our proposed strategy incritical applications such as blood donation should be prefaced with more rigorous backtesting inorder to minimize the risk of unintended consequences in application speciﬁc metrics not studied inthis paper.

Original: For future directions, one key property that we did not study in this paper is the problem of incentivecompatibility.
Revised: For future directions, one key property that we did not study in this paper is the problem of incentivecompatibility.

Original: For our motivating application of blood donation, this is not an issue because onlineplatforms such as Meta has proprietary models that can predict the matching quality between donorand recipient.
Revised: For our motivating application of blood donation, this is not an issue because onlineplatforms such as Meta has proprietary models that can predict the matching quality between donorand recipient.

Original: This means that the platform observes the value of matchings without having to relyon the recipients to self-report.
Revised: This means that the platform observes the value of matchings without having to relyon the recipients to self-report.

Original: This is also true in many other online matching problems such assponsored ads.
Revised: This is also true in many other online matching problems such assponsored ads.

Original: However, in settings where the central planner relies on the recipients to self-reporttheir valuations for each of the items, incentive compatibility becomes a crucial issue.
Revised: However, in settings where the central planner relies on the recipients to self-reporttheir valuations for each of the items, incentive compatibility becomes a crucial issue.

Original: We are excitedabout the potential of using Optimal Transport in fair-division, and plan on exploring the incentiveissues in future work.
Revised: We are excitedabout the potential of using Optimal Transport in fair-division, and plan on exploring the incentiveissues in future work.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] See Section
Revised: [Yes] See Section

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [Yes] See
Revised: [Yes] See

Original: Section
Revised: Section

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: If you ran experiments...
Revised: If you ran experiments...(

Original: (a) Didyou include the code, data, and instructions needed to reproduce the main exper-
Revised: a) Did include the code, data, and instructions needed to reproduce the main exper-

Original: imental results (either in the supplemental material or as a URL)?
Revised: imental results (either in the supplemental material or as a URL)?

Original: [Yes] Included as
Revised: [Yes] Included as

Original: supplemental material.
Revised: supplemental material

Original: (b) Did you specify all the training details (e g., data splits, hyperparameters, how theywere chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] See Appendix A.
Revised: [Yes] See Appendix A.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi-
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experi-

Original: ments multiple times)?
Revised: ments multiple times)?

Original: [Yes] See Figure 3,
Revised: [Yes] See Figure 3,

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?

Original: [Yes] See Appendix A.
Revised: [Yes] See Appendix A.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] Yes, this cited paper
Revised: [Yes] Yes, this cited paper

Original: is this one: [22]
Revised: is this one: [21]

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] See Appendix A.
Revised: [Yes] See Appendix A.

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes]
Revised: [Yes]

Original: Our code is provided as supplemental material.
Revised: Our code is provided as supplemental material.

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?
Revised: using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable

Original: information or offensive content?
Revised: information or offensive content?

Original: [N/A] We used the simulator developed, and made
Revised: [N/A] We used the simulator developed, and made

Original: public by the authors of a paper that we cite.
Revised: public by the authors of a paper that we cite.

Original: There is no personal information in
Revised: There is no personal information in

Original: simulator data.
Revised: simulator data.

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, ifapplicable?
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A]
Revised: [N/A]

Original: We did not crowdsource or conduct research with human subjects.
Revised: We did not crowdsource or conduct research with human subjects.

Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]