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 receivers whereeach receiver has to be allocated a ﬁxed, prespeciﬁed fraction of all items, whileensuring that each receiver 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 receivers, under the constraint that each receiver i must beallocated a prespeciﬁed 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 receiver.
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 receiver’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, we allow thecentral planner to specify the level of envy that is tolerated, and ﬁnd the most efﬁcient allocationgiven 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 signiﬁcantly without paying thefull 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 [18].
Revised: To concretely motivate our model, let us consider the blood donor matching problem that was ﬁrststudied in [22].

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 [18].
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 [22].

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 thatreceiver A envies receiver 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 eachreceiver 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 fastcomputation of a match given an item.
Revised: This is particularly attractive for large-scale allocation problems, due to the fast 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 ) .3.
Revised: has a provable convergence rate of O (1 / p T ) .

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  stochastic optimization algorithm.
Revised: Section 5 develops a practical stochastic optimizationalgorithm.

Original: In Section 6, we show that an ✏ -approximate solution can be found with high probability given ˜ O ( n ✏ 2 )samples, where n is the number of receivers.
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 the effectivenessof 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: [18] 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: [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.

Original: The model formulation there is complex, as it takes into account the factthat not every blood bank is in need of blood every day, a dynamic which we do not consider here.

Original: Furthermore, their matching policy is rather cumbersome, requiring a separate parameter for each(donor, receiver) 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 their paper, we are able to provide better structural insights to theproblem by utilizing a simpler model that 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 thatof online 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 [14,11, 12].
Revised: or random permutation input models [18, 13,14, 2, 1, 6].

Original: Agrawal, Wang, and Ye [2] studied the setting with linear objective and gave competitiveratio bounds.

Original: Then, Agrawal and Devanur [1] generalized the results to concave objectives andconvex constraints.

Original: Later, Devanur et al.

Original: [12] improved the approximation ratio bounds and relaxedthe input assumptions on the budgets.

Original: Balseiro, Lu, and Mirrokni [5] show that online mirrordescent on the dual multpliers does well under both i.i.d.

Original: adversarial, and certain non-stationaryinput settings.

Original: However, none of theses papers study the envy-free criterion.
Revised: However, none of theses papers study the concept of envy.

Original: Recently, Balseiro, 
Revised: Recently, Balseiro, Lu, and

Original: Lu, and Mirrokni [4] studied an online resource allocation problem with fairness regularization.
Revised: Mirrokni [5] studied an online resource allocation problem with fairness regularization.

Original: Although the authors did not explicitly study envy regularization, their regularization frameworkcan be modiﬁed to accommodate envy regularization. However, like all the other papers mentionedin this paragraph, the ofﬂine solution is used as the benchmark to measure regret, but no explicitsolution is given to the ofﬂine problem.
Revised: Althoughenvy regularization can be formulated in their framework, they did not provide an explicit solution tothe ofﬂine problem.

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

Original: In another recent paper, Sinclair, Banerjee, and Yu [23] studiedthe trade-off between minimizing envy and minimizing waste, which refers to un-allocated resources.

Original: Despite close similarity between our titles, their ofﬂine benchmark is the standard Eisenberg-Galeprogram, which is envy-free but does not address the welfare cost of achieving envy-freenes.

Original: Fair Division.
Revised: Fair Division.

Original: Ofﬂine resource allocation, or commonly known as fair division, isalso a popularresearch area.
Revised: Envy is a popular concept studied in the fair division literature.

Original: A large body of these papers are formulated as a cake-cutting problem ([21, 9, 20])where the resources are modeled as an interval and the agents’ valuations are represented as functionson this interval. 
Revised: 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. Caragianniset al.

Original: However most existing results are based on  relatively simple (e.g., piece wiseuniform, or piece wise linear) valuation functions, where as our problem can be thought of  as a cakecutting problem with arbitrarily complex valuation functions.
Revised: [9] provide a analysis on the worst case efﬁciency loss due to the envy-freeness constraint. Later Cohler et al. [12] design algorithms for computing optimal envy-free cake cutting allocationsunder different relatively simple classes of valuation functions. 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: Other fair division literature that studiesenvy-freeness as a fairness criterea usually treat it as a hard constraint, and tries to maximize socialwelfare subject to that constraint ([10, 25]).
Revised: In indivisible settings, there are some related conceptsof envy that does not require a strict zero envy. For instance, Budish [8] proposes an approximatecompetitive equilibrium from equal incomes approach that achieves envy free up to 1 item (EF1).

Original: Others have also characterized the worst-case loss insocial welfare due to the requirement of envy-freeness [8], and other fairness notions [7]. Unlikethese papers that focus on achieving zero envy, we treat the allowable envy as a parameter, and ﬁndthe most efﬁcient solutions subject to  the desired amount of envy.
Revised: Caragiannis et al. [10] introduce the concept of envy free up to the Least Valued Good, which is a stronger version of EF1. 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: 2.1 Background on Optimal Transport
Revised: Finally, Donahue and Kleinberg [15]also studies a setting where the central planner can set his own tolerance of (un)fairness. 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. Optimal Transport. OT has been applied before in resource allocation settings in the economicsliterature (see [17] for a survey).

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.

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].

Revised: Scetbon et al.

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

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

Original: [Equation]
Revised: [Equation]

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

Original:  If  is a discrete measure, i.e. Y is ﬁnite, then it is known [3] that the dual to (1)can be written as (here we abuse the notation  to also represent the vector of probability masses,where  i is the probability mass on point y i ):
Revised: ( x, y ) = ⇡ ( y | x ) d ↵ ( x ) . 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 receivers 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 receivers’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 receiver y i is matched p
Revised: The goal is to maximize the expected matched utilities of the recipients, whilemaintaining 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 receivers’ importance.
Revised: Here{ p ⇤ i } ni =1 is called the target matching distribution, which intuitively represents recipients’ importance.

Original: 118 A matching policy ⇡ takes a valuation vector and maps it (potentially with randomness) to one of then receivers.
Revised: 121 A matching policy ⇡ takes a valuation vector and maps it (potentially with randomness) to one of then 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 problem formulation is to solve the following optimization problem:

Original: [Equation]
Revised: [Equation]

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

Original: This is WLOG, because we can always pretend that a receiverdoes not exist, if the target fraction for that receiver is 0 .

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 intoblue and orange regions (given to A and B respectively) such that each region covers the desired p ⇤ A ,p ⇤ B probability mass.
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 orange regions (given to A and B respectively) 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 that the 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 (probabilitistic partition), and thatthe boundary does not have to be linear as illustrated in the ﬁgure.

Original: As we willshow later in Section 4, despite the large design space permitted by the formulation in (4), we can infact focus on a much smaller 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 (maximizingsocial welfare, or the sum of all receivers’ 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 mostcommonly studied fairness criteria 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 ivalues the allocation given to y 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 formallydeﬁne 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]

Original: [Equation]
Revised: [Equation]

Original: In existing fair resource allocation literature, people focus on ﬁnding allocations such that Envy ( y i )is at most 0 for every y 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 reducingefﬁciency by a signiﬁcant 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 a different 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.

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 matched receiver c ( x, y i ) =
Revised: In this case, theproblem can be stated in the form of (1) as follows: the cost function is the negative utility ofthe matched recipient c ( x, y i ) =

Original:  x i , the  measure is the discrete measure
Revised:  x i , the  measure is the discrete measure

Original: P ni =

Revised: P ni =

Original: p
Revised: p

Original: ⇤ i  y
Revised: ⇤ i  y

Original: i , and the 147 matching policy ⇡ ( y | x ) in (6) is exactly the conditional probability of the joint distribution in (1).
Revised: 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  153 x k + g k } .
Revised: \ k { x : x i + g i  156 x k + g k } .

Original: To visualize this better, consider the simple setting with two receivers A, B where their valuations 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 receiver 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: ,c ( x, y 0 )  .

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: Theorem 1 shows that the optimal solution to ourallocation problem  in (6) is represented by theoptimal dual solution from (8).
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: To solve this optimization problem, ﬁrst note that the objectivefunction E ( g,  ) is concave. To see this, we can rewrite the objectivefunction as follows:
Revised: In this section we present a practical algorithm for actuallycomputing the optimal Laguerre cells. 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]

Original: [Equation]

Original: [Equation]

Original: 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 ⇠ ↵ :

Original: [Equation]

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 +g t + ⌘ ˆ r g E ( g,  )
Revised: Sample x t ⇠ ↵ g t +

Original:   t + ⇣  t + ⌘ ˆ r  E ( g,  ) ⌘
Revised: [Equation] g,  )  t + [Equation] + ⌘

Original: + 6 return
Revised: [Equation]

Original: P
Revised: r

Original: Tt =1 g t /T,
Revised: [Equation] [Equation] ,  [Equation]

Original: P
Revised: 6 return [Equation] Calculating this gradient is hard, as it involves integration over an arbitrary measure ↵ . However, anunbiased, stochastic version of the gradient can be easily obtained from a single sample x ⇠ ↵ :

Original: Tt =1  t /T
Revised: [Equation] [Equation] [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[15]) tells us that Algorithm 1 converges at the rate E ( g ⇤ ,  ⇤ )
Revised: Standard projected SGD analysis (see for example[19]) 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 considered the setting where the true underlying distribution is known, and assumedthat we can freely draw independent samples from that distribution.
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 haveaccess to ↵ in the form of ﬁnite number of i.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: In this section, we focus only on theassignment cost function c ( x, y i ) =

Original:  x i , which models our original resource allocation problemproposed in Section 3.

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

Original: Let S
Revised: This complements the result in the previous section by providing an algorithm agnosticbound on the sample complexity. 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. 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. In this section, we focus only on the assignment cost function c ( x, y i )

Original: =
Revised: =

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

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

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

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

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 are interested in boundingthe number of samples needed so that E ( g ⇤ ,  ⇤ )
Revised: We want to bound the numberof 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’sintroduce some new notations to facilitate our later discussions.
Revised: Let’s introduce somenotations to facilitate our later discussions.

Original: Deﬁne the following hypothesis classfor 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: and
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

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

Original: [Equation]
Revised: [Equation]

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 minimum over n afﬁne hypothesis classesis the key observation to prove the sample complexity bound.

Original: We prove our main result under thefollowing boundedness assumption:
Revised: We prove our main result under thefollowing 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 207 imply that F i and F are uniformly bounded by R ¯ x + R .
Revised: In particular, these assumptions 215 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]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: We prove the result via uniform convergence:
Revised: We prove the result via uniform convergence:

Original: [Equation]
Revised: [Equation]

Original:  Clearly, it sufﬁces to show that E S ( · ) converges uniformly to E ( · ) .
Revised: [Equation] Clearly, it sufﬁces to show that E S ( · ) converges uniformly to E ( · ) .

Original: For a given g,  , the dual objective function  can be written asan expectation over their corresponding f 2 F , [Equation] and similarly for the empirical objective function [Equation]
Revised: For a given g,  , the dual objective function and its’ empirical version can be written as

Original: m

Original: X

Original: [Equation]

Original: f ( X

Original: [Equation]
Revised: [Equation]

Original: Then we can rewrite the supremum in (21) as:
Revised: Then we can rewrite the supremum in (21) as:

Original: [Equation]
Revised: [Equation]

Original: Since | f ( X ) |  ( R ¯ x + R ) for all f 2 F, X 2 X , it follows from Theorem 26.5 in [22] that with 216 probability  ,
Revised: 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  ,

Original: [Equation]
Revised: [Equation]

Original: and the same also holds by replacing F with  F .
Revised: and the same also holds by replacing F with  F .

Original: Here
Revised: Here

Original: [Equation]
Revised: [Equation]

Original:  is the standard deﬁnition of Rademacher complexity of the set F  S .
Revised: 5 is the standard deﬁnition of Rademacher complexity of the set F  S .

Original: Since  i are i.i.d.
Revised: Since  i are i.i.d.

Original: Rademacherrandom variables, it is easy to see that Rad m ( F  S ) =
Revised: Rademacherrandom variables, it is easy to see that Rad m ( F  S ) =

Original: Rad m (  F  S ) .
Revised: Rad m (  F  S ) .

Original: Therefore we can use aunion bound to obtain that with probability 1   , 
Revised: Therefore we can use aunion bound to obtain that with probability 1   , [Equation]

Original: [Equation]
Revised: [Equation]

Original: It remains to bound the Rademacher complexity of the F  S .
Revised: It remains to bound the Rademacher complexity of the F  S .

Original: To do so, we use tools from learningtheory, and give the following bound on the fat-shattering dimension ([6]) of the hypothesis class F .
Revised: To do so, we use tools from learningtheory, and give the following bound on the fat-shattering dimension ([7]) of the hypothesis class F .

Original: Lemma 1.
Revised: Lemma 1.

Original: Under Assumption 1, F has ⇣ -fat-shattering dimension of at most c 0 ( R ¯ x + R )
Revised: Under Assumption 1, F has ⇣ -fat-shattering dimension of at most c 0 ( R ¯ x + R )

Original: 2 ⇣n log( n ) ,where c 
Revised: 2 ⇣n log( n ) ,where c 0 is some universal constant.

Original: The non-monotonicity in the plot for the simulator data is due to the stochasticity in the SGD algorithm Proof. Theorem 3 in [17] shows that fat ⇣ ( F min )  c 0 ( R ¯ x + R ) 2 ⇣n log n .
Revised: The proof of Lemma 1 can be found in the Appendix.

Original: Since the shattering dimension is monotone in the size of the set, we are done.

Original: The above bound on the fat-shattering dimension can be used to bound the covering number (see Deﬁnition 27.1 of [22]) of F  S .
Revised: The above bound on the fat-shattering dimensioncan be used to bound the covering number (see Deﬁnition 27.1 of [27]) of F  S .

Original: Theorem 1 from [19] states that
Revised: Theorem 1 from[23] states that

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where B is a uniform bound on the absolute value of any f 2 F .
Revised: where B is a uniform bound on the absolute value of any f 2 F .

Original: Let B =
Revised: Let B =

Original: ( R ¯ x + R ) , we have that
Revised: ( R ¯ x + R ) , we have that

Original: [Equation]
Revised: [Equation]

Original: Wherein the ﬁrst, second, and last step we used Dudley’s chaining integral [24, 13], Lemma 1 and(25), and setting  [-
-] = 1 p m respectively.
Revised: Where we used Dudley’s chaining integral [28, 16], Lemma 1 and (25), and setting  [-
-]0 = 1 p mrespectively.

Original: Plugging the above back to (24) and (21), we see that with probability 1   
Revised: Plugging the above back to (24) and (21), we see that with probability 1   ,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: 1  m

Original: A 
Revised: A .

Original: [Equation]

Original: Conversely, ignoring the log terms, m needs to be at most on the order of ˜ O  n ✏ in order for
Revised: Conversely, ignoring the log terms, m needs to be at most on the order of ˜ O  n ✏ in order for

Original: E ( g ⇤ ,  ⇤ )
Revised: E ( g ⇤ ,  ⇤ )

Original:  E (ˆ g S , ˆ  S ) to be bounded by ✏ with high probability.
Revised:  E (ˆ g S , ˆ  S ) to be bounded by ✏ with high probability.

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 [18].
Revised: We test our solution with both artiﬁcial data, and simulated data from a realistic simulator for blooddonor matching developed by [22].

Original: The artiﬁcial data contains two receivers, 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 receivers.
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 receiver: 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, 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 receivers.
Revised: 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.

Original: Next we investigate the tradeoff 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 priceof 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 receivers’ utilities are highly correlated,but one receiver has larger variance than others, that receiver receives almost all the good items(which results in large envy for other receivers), 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 receiver 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 receiver.
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 receivers 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 receivers 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. Checklist

Revised: For all authors...

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

Revised: contributions and scope?

Revised: [Yes]

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

Revised: [Yes] See Section

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

Revised: [Yes] See

Revised: Section

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

Revised: them?

Revised: [Yes]

Revised: If you are including theoretical results...

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

Revised: [Yes]

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

Revised: [Yes]

Revised: If you ran experiments...

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

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

Revised: [Yes] Included as

Revised: supplemental material.

Revised: (b) Did you specify all the training details (e

Revised: g., data splits, hyperparameters, how theywere chosen)?

Revised: [Yes] See Appendix A.

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

Revised: ments multiple times)?

Revised: [Yes] See Figure 3,

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)?

Revised: [Yes] See Appendix A.

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

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

Revised: [Yes] Yes, this cited paper

Revised: is this one: [22]

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

Revised: [Yes] See Appendix A.

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

Revised: [Yes]

Revised: Our code is provided as supplemental material.

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

Revised: using/curating?

Revised: [N/A]

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

Revised: information or offensive content?

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

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

Revised: There is no personal information in

Revised: simulator data.

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

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

Revised: [N/A]

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

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

Revised: Board (IRB) approvals, if applicable?

Revised: [N/A]

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

Revised: spent on participant compensation?

Revised: [N/A]