Original: Fair and Efficient Allocations Without Obvious Manipulations
Revised: Fair and Efficient Allocations Without Obvious Manipulations

Original: Abstract
Revised: Abstract

Original: We consider the fundamental problem of allocating a set of indivisible goods among strategic agents with additive valuation functions.
Revised: We consider the fundamental problem of allocating a set of indivisible goods among strategic agents with additive valuation functions.

Original: It is well known that, in the absence of monetary transfers, Pareto efficient and truthful rules are dictatorial, while there is no deterministic truthful mechanism that allocates all items and achieves envy-freeness up to one item (EF1), even for the case of two agents.
Revised: It is well known that, in the absence of monetary transfers, Pareto efficient and truthful rules are dictatorial, while there is no deterministic truthful mechanism that allocates all items and achieves envy-freeness up to one item (EF1), even for the case of two agents.

Original: In this paper, we investigate the interplay of fairness and efficiency under a relaxation of truthfulness called non-obvious manipulability (NOM), recently proposed by [TM20].
Revised: In this paper, we investigate the interplay of fairness and efficiency under a relaxation of truthfulness called non-obvious manipulability (NOM), recently proposed by [TM20].

Original: We show that this relaxation allows us to bypass the aforementioned negative results in a very strong sense.
Revised: We show that this relaxation allows us to bypass the aforementioned negative results in a very strong sense.

Original: Specifically, we prove that there are deterministic and EF algorithms that are not obviously manipulable, and the algorithm that maximizes utilitarian social welfare (the sum of agents’ utilities), which is Pareto efficient but not dictatorial, is not obviously manipulable for n ≥ 3 agents (but obviously manipulable for n = 2 agents).
Revised: Specifically, we prove that there are deterministic and EF algorithms that are not obviously manipulable, and the algorithm that maximizes utilitarian social welfare (the sum of agents’ utilities), which is Pareto efficient but not dictatorial, is not obviously manipulable for n ≥ 3 agents (but obviously manipulable for n = 2 agents).

Original: At the same time, maximizing the egalitarian social welfare (the minimum of agents’ utilities) or the Nash social welfare (the product of agents’ utilities) is obviously manipulable for any number of agents and items.
Revised: At the same time, maximizing the egalitarian social welfare (the minimum of agents’ utilities) or the Nash social welfare (the product of agents’ utilities) is obviously manipulable for any number of agents and items.

Original: Our main result is an approximation preserving black-box reduction from the problem of designing EF1 and NOM mechanisms to the problem of designing
Revised: Our main result is an approximation preserving black-box reduction from the problem of designing EF1 and NOM mechanisms to the problem of designing

Original: EF1 algorithms.
Revised: EF1 algorithms.

Original: En route, we prove an interesting structural result about EF allocations, as well as new “best-of-both-worlds” results (for the problem without incentives), that might be of independent interest.
Revised: En route, we prove an interesting structural result about EF allocations, as well as new “best-of-both-worlds” results (for the problem without incentives), that might be of independent interest.

Original:  Introduction
Revised:  Introduction

Original: We consider the fundamental problem of allocating a set of indivisible items among strategic agents with additive preferences.
Revised: We consider the fundamental problem of allocating a set of indivisible items among strategic agents with additive preferences.

Original: It is well-understood that, in the absence of monetary transfers, fairness, efficiency and truthfulness cannot be reconciled, in a very strong sense.
Revised: It is well-understood that, in the absence of monetary transfers, fairness, efficiency and truthfulness cannot be reconciled, in a very strong sense.

Original: For example, a serial dictatorship (arguably the most unfair rule) is the unique truthful and Pareto efficient mechanism [KM02, EK03], even for the case of two agents and randomized mechanisms (or, equivalently, divisible items) [Sch96].
Revised: For example, a serial dictatorship (arguably the most unfair rule) is the unique truthful and Pareto efficient mechanism [KM02, EK03], even for the case of two agents and randomized mechanisms (or, equivalently, divisible items) [Sch96].

Original: At the same time, achieving envy-freeness up to one item (EF1) [Bud11], a popular fairness notion for indivisible goods that is compatible with Pareto efficiency [CKM + 19], is impossible for truthful and deterministic mechanisms to achieve, even for two additive agents [ABCM17].
Revised: At the same time, achieving envy-freeness up to one item (EF1) [Bud11], a popular fairness notion for indivisible goods that is compatible with Pareto efficiency [CKM + 19], is impossible for truthful and deterministic mechanisms to achieve, even for two additive agents [ABCM17].

Original: The standard abstraction for a strategic agent assumes that, when presented with an algorithm, the agent will perfectly understand it and optimally respond to it.
Revised: The standard abstraction for a strategic agent assumes that, when presented with an algorithm, the agent will perfectly understand it and optimally respond to it.

Original: Therefore, a truthful mechanism should protect against all deviating strategies.
Revised: Therefore, a truthful mechanism should protect against all deviating strategies.

Original: This modeling is overly pessimistic since, for example, in practice agents do not have exact knowledge about others’ preferences.
Revised: This modeling is overly pessimistic since, for example, in practice agents do not have exact knowledge about others’ preferences.

Original: Since truthfulness rules out reasonable solutions, previous work typically ignores strategic issues altogether.
Revised: Since truthfulness rules out reasonable solutions, previous work typically ignores strategic issues altogether.

Original: In this paper we aim to escape the aforementioned impossibility results by relaxing truthfulness: our goal is to design mechanisms that protect only against obviously dominant deviating strategies.
Revised: In this paper we aim to escape the aforementioned impossibility results by relaxing truthfulness: our goal is to design mechanisms that protect only against obviously dominant deviating strategies.

Original: Informally, a strategy is obviously dominant if it guarantees an outcome better than any outcome of every other strategy [Li17].
Revised: Informally, a strategy is obviously dominant if it guarantees an outcome better than any outcome of every other strategy [Li17].

Original: Designing obviously truthful mechanisms, those where honest reporting is obviously dominant, is a problem that has attracted significant attention in recent years [AG18, KV19, FV19, AMN20].
Revised: Designing obviously truthful mechanisms, those where honest reporting is obviously dominant, is a problem that has attracted significant attention in recent years [AG18, KV19, FV19, AMN20].

Original: Since dishonesty is also a strategy, a deviation can be profitable but not obviously profitable.
Revised: Since dishonesty is also a strategy, a deviation can be profitable but not obviously profitable.

Original: A manipulation is obvious if it yields a higher utility than truth-telling in either the best or worst-case scenarios.
Revised: A manipulation is obvious if it yields a higher utility than truth-telling in either the best or worst-case scenarios.

Original: Seminal work of [TM20] defines this notion, and shows that relaxing truthfulness to non-obvious manipulability can strictly improve the designer’s objective in a number of domains (e.g. two-sided matching).
Revised: Seminal work of [TM20] defines this notion, and shows that relaxing truthfulness to non-obvious manipulability can strictly improve the designer’s objective in a number of domains (e.g. two-sided matching).

Original: In this paper we aim to provide a comprehensive study of what can and what cannot be achieved by non-obviously manipulable mechanisms in the fundamental domain of fair and efficient allocation of indivisible goods.In this domain,
Revised: In this paper we aim to provide a comprehensive study of what can and what cannot be achieved by non-obviously manipulable mechanisms in the fundamental domain of fair and efficient allocation of indivisible goods.

Original: Our Contribution.
Revised: Our Contribution.

Original: We first consider the problem of designing a non-obviously manipulable mechanism (NOM) which always outputs allocations that are envy-free up to one item (EF1), a task that is impossible for truthful mechanisms to achieve [ABCM17].
Revised: We first consider the problem of designing a non-obviously manipulable mechanism (NOM) which always outputs allocations that are envy-free up to one item (EF1), a task that is impossible for truthful mechanisms to achieve [ABCM17].

Original: We prove that the RoundRobin procedure (agents choose items one at a time, following a fixed order) which is known to always output EF1 allocations, is also not obviously manipulable (Theorem 1), giving a separation between truthfulness and non-obviously manipulability for our problem.
Revised: We prove that the RoundRobin procedure (agents choose items one at a time, following a fixed order) which is known to always output EF1 allocations, is also not obviously manipulable (Theorem 1), giving a separation between truthfulness and non-obviously manipulability for our problem.

Original: In fact, we show that NOM is compatible with even stronger notions of fairness.
Revised: In fact, we show that NOM is compatible with even stronger notions of fairness.

Original: Specifically, a randomized algorithm can be ex-post
Revised: Specifically, a randomized algorithm can be ex-post

Original: EF1 (i.e. the output allocation is always EF1), and simultaneously be envy-free in expectation (ex-ante
Revised: EF1 (i.e. the output allocation is always EF1), and simultaneously be envy-free in expectation (ex-ante

Original: EF).
Revised: EF).

Original: There are known algorithms that satisfy this “best-of-both-worlds” guarantee [FSV20, Azi20b]; we show that the PS-Lottery algorithm of [Azi20b] is not-obviously manipulable.
Revised: There are known algorithms that satisfy this “best-of-both-worlds” guarantee [FSV20, Azi20b]; we show that the PS-Lottery algorithm of [Azi20b] is not-obviously manipulable.

Original: This result uses a connection between ex-ante proportionality and not obvious manipulability; we later exploit this connection to establish new negative “best-of-both-worlds” results.
Revised: This result uses a connection between ex-ante proportionality and not obvious manipulability; we later exploit this connection to establish new negative “best-of-both-worlds” results.

Original: We proceed to study efficient algorithms.
Revised: We proceed to study efficient algorithms.

Original: We focus on the three most prominent notions of efficiency: utilitarian social welfare (the sum of agents’ utilities), egalitarian social welfare (the minimum of agents’ utilities), and Nash social welfare (the geometric mean of agents’ utilities).
Revised: We focus on the three most prominent notions of efficiency: utilitarian social welfare (the sum of agents’ utilities), egalitarian social welfare (the minimum of agents’ utilities), and Nash social welfare (the geometric mean of agents’ utilities).

Original: For the case of n = 2 agents all three notions are incompatible with non obvious manipulability, i.e. every (deterministic or randomized) algorithm that always outputs an integral solution that is optimal with respect to any of these objectives is obviously manipulable (Theorem 2).
Revised: For the case of n = 2 agents all three notions are incompatible with non obvious manipulability, i.e. every (deterministic or randomized) algorithm that always outputs an integral solution that is optimal with respect to any of these objectives is obviously manipulable (Theorem 2).

Original: This is also true for the case of more than three agents for egalitarian and Nash social welfare (Theorems 8 and 4).
Revised: This is also true for the case of more than three agents for egalitarian and Nash social welfare (Theorems 8 and 4).

Original: The high level intuition for these impossibilities is as follows.
Revised: The high level intuition for these impossibilities is as follows.

Original: There are instances where a specific agent 1 gets only her least favorite item, in the worst-case.
Revised: There are instances where a specific agent 1 gets only her least favorite item, in the worst-case.

Original: However, maximizing egalitarian or Nash social welfare ensures that the number of agents that get non-zero utility are maximized.
Revised: However, maximizing egalitarian or Nash social welfare ensures that the number of agents that get non-zero utility are maximized.

Original: Therefore, by reporting that she only values a single item (her favorite item) to the mechanism, this agent can force the mechanism to give her this item (or increase the probability that she gets it) in the worst-case.
Revised: Therefore, by reporting that she only values a single item (her favorite item) to the mechanism, this agent can force the mechanism to give her this item (or increase the probability that she gets it) in the worst-case.

Original: Note that lack of agents’ knowledge about each others’ preferences is used as a justification for why manipulations might not be a first-order concern in our domain [CKM + 19]; the fact that maximizing Nash social welfare is manipulable even when agents know nothing about each other further highlights the need for more nuanced formal models between “agents are always honest” and “agents are perfectly rational and all-knowing expected utility maximizers.”
Revised: Note that lack of agents’ knowledge about each others’ preferences is used as a justification for why manipulations might not be a first-order concern in our domain [CKM + 19]; the fact that maximizing Nash social welfare is manipulable even when agents know nothing about each other further highlights the need for more nuanced formal models between “agents are always honest” and “agents are perfectly rational and all-knowing expected utility maximizers.”

Original: Surprisingly, the same is not true for utilitarian social welfare: there is a NOM mechanism that always outputs an allocation that maximizes utilitarian social welfare for the case of n ≥ 3 agents (Theorem 3).
Revised: Surprisingly, the same is not true for utilitarian social welfare: there is a NOM mechanism that always outputs an allocation that maximizes utilitarian social welfare for the case of n ≥ 3 agents (Theorem 3).

Original: This gives an efficient and not obviously manipulable mechanism that is not dictatorial.
Revised: This gives an efficient and not obviously manipulable mechanism that is not dictatorial.

Original: Since maximizing Nash social welfare, an objective that simultaneously achieves Pareto efficiency and
Revised: Since maximizing Nash social welfare, an objective that simultaneously achieves Pareto efficiency and

Original: EF1, is obviously manipulable, the next natural question is whether there are any other Pareto efficient and EF1 algorithms that satisfy NOM.
Revised: EF1, is obviously manipulable, the next natural question is whether there are any other Pareto efficient and EF1 algorithms that satisfy NOM.

Original: Our main result answers this question in the affirmative.
Revised: Our main result answers this question in the affirmative.

Original: In fact, we prove that the problem of designing a Pareto efficient, EF1 and NOM mechanism is exactly as hard as designing a Pareto efficient and EF1 algorithm .
Revised: In fact, we prove that the problem of designing a Pareto efficient, EF1 and NOM mechanism is exactly as hard as designing a Pareto efficient and EF1 algorithm .

Original: We give a black-box reduction, that preserves
Revised: We give a black-box reduction, that preserves

Original: Pareto efficiency guarantees, which given an algorithm that always outputs (clean and non-wasteful)
Revised: Pareto efficiency guarantees, which given an algorithm that always outputs (clean and non-wasteful)

Original: EF1 allocations, outputs a mechanism that is not obviously manipulable and always outputs EF allocations.
Revised: EF1 allocations, outputs a mechanism that is not obviously manipulable and always outputs EF allocations.

Original: There are two crucial steps in our reduction.
Revised: There are two crucial steps in our reduction.

Original: Given the valuation function v i of agent i , let EF1( i, v i ) be the set of allocations that are clean, non-wasteful and EF1 for agent i .
Revised: Given the valuation function v i of agent i , let EF1( i, v i ) be the set of allocations that are clean, non-wasteful and EF1 for agent i .

Original: First, our reduction ensures that, when agent i reports her valuation to be b i , every single allocation in
Revised: First, our reduction ensures that, when agent i reports her valuation to be b i , every single allocation in

Original: EF1( i, b i ) is possible, i.e. for every allocation
Revised: EF1( i, b i ) is possible, i.e. for every allocation

Original: A ∈ EF1( i, b i ) there are reports for the other agents such that A is the output of the mechanism.
Revised: A ∈ EF1( i, b i ) there are reports for the other agents such that A is the output of the mechanism.

Original: Second, we prove the following intuitive structural result, that might be of independent interest.
Revised: Second, we prove the following intuitive structural result, that might be of independent interest.

Original: For an agent i having valuation function v i , the worst-case allocation in EF1( i, v i ) is better than the worst-case allocation in EF1( i, b i )
Revised: For an agent i having valuation function v i , the worst-case allocation in EF1( i, v i ) is better than the worst-case allocation in EF1( i, b i )

Original: (Lemma 4).
Revised: (Lemma 4).

Original: These facts
Revised: These facts

Original: [Table]
Revised: [Table]

Original: combined establish the “worst-case” part from the definition of NOM, which is the most challenging step.
Revised: combined establish the “worst-case” part from the definition of NOM, which is the most challenging step.

Original: As a direct application of our reduction, by giving as input the algorithmic results [GM21], we get a fractionally Pareto efficient, EF1 and NOM mechanism for additive agents; this last result cannot be improved by adding ex-ante fairness guarantees (due to a theorem of [FSV20]).
Revised: As a direct application of our reduction, by giving as input the algorithmic results [GM21], we get a fractionally Pareto efficient, EF1 and NOM mechanism for additive agents; this last result cannot be improved by adding ex-ante fairness guarantees (due to a theorem of [FSV20]).

Original: Finally, despite the aforementioned technical subtleties, the “code” of our reduction is fairly simple: it checks whether some allocations are EF1, and if not it calls the black-box algorithm.
Revised: Finally, despite the aforementioned technical subtleties, the “code” of our reduction is fairly simple: it checks whether some allocations are EF1, and if not it calls the black-box algorithm.

Original: This simplicity is a feature, with concrete practical implications: by adding a few lines of code to the implementation of any EF1 and Pareto
Revised: This simplicity is a feature, with concrete practical implications: by adding a few lines of code to the implementation of any EF1 and Pareto

Original: efficient algorithm
Revised: efficient algorithm

Original: (e.g. the algorithm that maximizes Nash social welfare, which is used by the popular website Spliddit [GP15]) one can maintain these guarantees, while provably protecting against certain types of manipulations.
Revised: (e.g. the algorithm that maximizes Nash social welfare, which is used by the popular website Spliddit [GP15]) one can maintain these guarantees, while provably protecting against certain types of manipulations.

Original: Our results show a connection between certain notions of fairness and not obvious manipulability: positive algorithmic results for EF1 and “best-of-both-worlds” guarantees can be used to get positive results for not obviously manipulable mechanisms.
Revised: Our results show a connection between certain notions of fairness and not obvious manipulability: positive algorithmic results for EF1 and “best-of-both-worlds” guarantees can be used to get positive results for not obviously manipulable mechanisms.

Original: In Appendix E we exploit this connection in the other direction, and show how negative results for not obviously manipulable mechanisms can be used to prove new negative results for “best-of-both-worlds” algorithms.
Revised: In Appendix E we exploit this connection in the other direction, and show how negative results for not obviously manipulable mechanisms can be used to prove new negative results for “best-of-both-worlds” algorithms.

Original: Specifically, we prove that it is impossible to achieve ex-ante proportionality, ex-post Pareto efficiency while ex-post maximizing the number of agents with positive utility.
Revised: Specifically, we prove that it is impossible to achieve ex-ante proportionality, ex-post Pareto efficiency while ex-post maximizing the number of agents with positive utility.

Original: As direct corollaries we recover a known result of [FSV20] that ex-ante proportional and ex-post MNW allocations do not exist, as well as new results: it is impossible to achieve ex-ante proportionality by randomizing over allocations that are (1) ex-post
Revised: As direct corollaries we recover a known result of [FSV20] that ex-ante proportional and ex-post MNW allocations do not exist, as well as new results: it is impossible to achieve ex-ante proportionality by randomizing over allocations that are (1) ex-post

Original: Pareto efficient and ex-post egalitarian, or (2) ex-post leximin.
Revised: Pareto efficient and ex-post egalitarian, or (2) ex-post leximin.

Original: Related work.
Revised: Related work.

Original: [TM20] define the notion of not obvious manipulability.
Revised: [TM20] define the notion of not obvious manipulability.

Original: They show that a number of known mechanisms that are not truthful satisfy not obviously manipulability, e.g. stable mechanisms in the context of two-sided matching are not obviously manipulable (in direct contrast to [Rot82], who shows that there exists no mechanism that is both stable and truthful).
Revised: They show that a number of known mechanisms that are not truthful satisfy not obviously manipulability, e.g. stable mechanisms in the context of two-sided matching are not obviously manipulable (in direct contrast to [Rot82], who shows that there exists no mechanism that is both stable and truthful).

Original: At the same time, many known mechanisms that are not truthful are obviously manipulable, e.g. the first-price auction, and more generally the pay-as-bid auction.
Revised: At the same time, many known mechanisms that are not truthful are obviously manipulable, e.g. the first-price auction, and more generally the pay-as-bid auction.

Original: The notion of non obvious manipulability has been explored in some very recent works [OSH19,
Revised: The notion of non obvious manipulability has been explored in some very recent works [OSH19,

Original: AL21].
Revised: AL21].

Original: [OSH19] study cake-cutting and show that, as opposed to truthfulness, NOM is compatible with proportionality: an adaptation of the well-known moving knife procedure satisfies both properties.
Revised: [OSH19] study cake-cutting and show that, as opposed to truthfulness, NOM is compatible with proportionality: an adaptation of the well-known moving knife procedure satisfies both properties.

Original: They also observe that every proportional (direct revelation) rule satisfies the worst-case guarantee required for NOM (we show a similar statement in Lemma 5), while every Pareto efficient (direct revelation) rule satisfies the best-case guarantee required for NOM.
Revised: They also observe that every proportional (direct revelation) rule satisfies the worst-case guarantee required for NOM (we show a similar statement in Lemma 5), while every Pareto efficient (direct revelation) rule satisfies the best-case guarantee required for NOM.

Original: [AL21] explore obvious manipulations in voting theory.
Revised: [AL21] explore obvious manipulations in voting theory.

Original: They give sufficient conditions for voting rules to be not obviously manipulable, and show that a number of voting rules escape the pessimistic conclusions of the
Revised: They give sufficient conditions for voting rules to be not obviously manipulable, and show that a number of voting rules escape the pessimistic conclusions of the

Original: Gibbard-Satterthwaite theorem, e.g., the Borda rule is not obviously manipulable.
Revised: Gibbard-Satterthwaite theorem, e.g., the Borda rule is not obviously manipulable.

Original: Other than relaxing the notion of incentive compatibility, one can escape the aforementioned impossibility results by restricting agents’ valuations, e.g. focus on dichotomous [HPPS20, BEF21,
Revised: Other than relaxing the notion of incentive compatibility, one can escape the aforementioned impossibility results by restricting agents’ valuations, e.g. focus on dichotomous [HPPS20, BEF21,

Original: BCIZ21, BV21] or Leontief valuations [GZH + 11, FGP14, PPS15], or
Revised: BCIZ21, BV21] or Leontief valuations [GZH + 11, FGP14, PPS15], or

Original:  Preliminaries
Revised:  Preliminaries

Original: We consider the problem of allocating a set M of m items among a set N of n agents with additive utilities.
Revised: We consider the problem of allocating a set M of m items among a set N of n agents with additive utilities.

Original: We use [ k ] to denote the set { 1 , 2 , . . .
Revised: We use [ k ] to denote the set { 1 , 2 , . . .

Original: , k } for any positive integer k ∈ Z + .
Revised: , k } for any positive integer k ∈ Z + .

Original: Proofs pertaining to randomized mechanisms are deferred to the appendix; see Appendix A for the corresponding preliminaries on randomized allocations/mechanisms.
Revised: Proofs pertaining to randomized mechanisms are deferred to the appendix; see Appendix A for the corresponding preliminaries on randomized allocations/mechanisms.

Original: Allocations.
Revised: Allocations.

Original: A fractional allocation A ∈ [0 , 1] n · m is a m × n matrix that defines for each agent i ∈ N and item j ∈ M the fraction
Revised: A fractional allocation A ∈ [0 , 1] n · m is a m × n matrix that defines for each agent i ∈ N and item j ∈ M the fraction

Original: A i,j of the item j that the agent i receives.
Revised: A i,j of the item j that the agent i receives.

Original: We represent a fractional allocation as A =
Revised: We represent a fractional allocation as A =

Original: ( A 1 , . . .
Revised: ( A 1 , . . .

Original: , A n ) where A i =
Revised: , A n ) where A i =

Original: ( A i, 1 , . . .
Revised: ( A i, 1 , . . .

Original: , A i,m ) ∈ [0 , 1] m denotes the fractions of all items allocated to agent i .
Revised: , A i,m ) ∈ [0 , 1] m denotes the fractions of all items allocated to agent i .

Original: A feasible allocation satisfies (cid:80) i ∈
Revised: A feasible allocation satisfies (cid:80) i ∈

Original: N A
Revised: N A

Original: i,j ≤ 1 , for all j ∈ M .
Revised: i,j ≤ 1 , for all j ∈ M .

Original: A fractional allocation A is integral if A i ∈ { 0 , 1 } m for all agents i ∈ N .
Revised: A fractional allocation A is integral if A i ∈ { 0 , 1 } m for all agents i ∈ N .

Original: An integral allocation can be equivalently defined as n disjoint subsets of the set of items M , this representation is often convenient.
Revised: An integral allocation can be equivalently defined as n disjoint subsets of the set of items M , this representation is often convenient.

Original: Let Π n (M) =
Revised: Let Π n (M) =

Original: { ( S 1 , . . .
Revised: { ( S 1 , . . .

Original: , S n ) | ∪ ni =1 S
Revised: , S n ) | ∪ ni =1 S

Original: i ⊆
Revised: i ⊆

Original: M and ∀ i ̸
Revised: M and ∀ i ̸

Original: = j, S i ∩ S j = ∅} denote the set of all n ordered disjoint subsets of the set of items M .
Revised: = j, S i ∩ S j = ∅} denote the set of all n ordered disjoint subsets of the set of items M .

Original: Given an integral allocation A
Revised: Given an integral allocation A

Original: =
Revised: =

Original: ( A 1 , . . .
Revised: ( A 1 , . . .

Original: , A n ) , we can interpret the binary vectors
Revised: , A n ) , we can interpret the binary vectors

Original: A i =
Revised: A i =

Original: ( A i, 1 , . . .
Revised: ( A i, 1 , . . .

Original: , A i,m ) as subsets of items
Revised: , A i,m ) as subsets of items

Original: A i := { j ∈ M |
Revised: A i := { j ∈ M |

Original: A
Revised: A

Original: i,j = 1 } .
Revised: i,j = 1 } .

Original: This allows us to view an integral allocation A as n ordered disjoint subsets of items, i.e., A ∈ Π n (M) .
Revised: This allows us to view an integral allocation A as n ordered disjoint subsets of items, i.e., A ∈ Π n (M) .

Original: An integral allocation A is complete if ∪ ni =
Revised: An integral allocation A is complete if ∪ ni =

Original: A i = M and partial if ∪ ni
Revised: A i = M and partial if ∪ ni

Original: =1 A i ⊂ M .
Revised: =1 A i ⊂ M .

Original: Unless stated otherwise, we will use the term allocation to refer to complete allocations.
Revised: Unless stated otherwise, we will use the term allocation to refer to complete allocations.

Original: We will use the term bundle to refer to any subset of items.
Revised: We will use the term bundle to refer to any subset of items.

Original: Preferences.
Revised: Preferences.

Original: Each agent i ∈ N has a private valuation function v i ( . ) that outputs the utility that agent i derives from a given set (or fractions) of items.
Revised: Each agent i ∈ N has a private valuation function v i ( . ) that outputs the utility that agent i derives from a given set (or fractions) of items.

Original: We use the notation v i ( X i ) (resp.
Revised: We use the notation v i ( X i ) (resp.

Original: v i
Revised: v i

Original: ( A i ) ) to denote the utility that agent i gets from the items allocated to her in a fractional allocation X (resp.
Revised: ( A i ) ) to denote the utility that agent i gets from the items allocated to her in a fractional allocation X (resp.

Original: integral allocation A ).
Revised: integral allocation A ).

Original: We consider agents with additive utilities.
Revised: We consider agents with additive utilities.

Original: An additive agent i ∈ N has a non-negative valuation v i,j for receiving the entirety of item j ; her utility for an allocation X is v i ( X i ) =
Revised: An additive agent i ∈ N has a non-negative valuation v i,j for receiving the entirety of item j ; her utility for an allocation X is v i ( X i ) =

Original: (cid:80) j ∈ M X i,j v
Revised: (cid:80) j ∈ M X i,j v

Original: i,j ; for an integral allocation A , the utility is simply v i
Revised: i,j ; for an integral allocation A , the utility is simply v i

Original: ( A i ) =
Revised: ( A i ) =

Original: (cid:80)
Revised: (cid:80)

Original: j ∈
Revised: j ∈

Original: A i v i,j .
Revised: A i v i,j .

Original: Mechanisms.
Revised: Mechanisms.

Original: A mechanism M elicits “bids” (i.e. reported valuations) b
Revised: A mechanism M elicits “bids” (i.e. reported valuations) b

Original: = ( b 1 , . . .
Revised: = ( b 1 , . . .

Original: , b n ) from every agent i ∈ N , and outputs a feasible allocation.
Revised: , b n ) from every agent i ∈ N , and outputs a feasible allocation.

Original: We write X i,j ( b ) for the fraction of item j allocated to agent i when each agent j ∈ N reports a valuation b j .
Revised: We write X i,j ( b ) for the fraction of item j allocated to agent i when each agent j ∈ N reports a valuation b j .

Original: We use the notation b (and b i ) to refer to the input to a mechanism, and v (and v i ) to refer to the true valuations of agents.
Revised: We use the notation b (and b i ) to refer to the input to a mechanism, and v (and v i ) to refer to the true valuations of agents.

Original: A mechanism M is deterministic if for every reported valuations b it deterministically outputs an integral allocation.
Revised: A mechanism M is deterministic if for every reported valuations b it deterministically outputs an integral allocation.

Original: We will use M ( b ) =
Revised: We will use M ( b ) =

Original: ( M 1 ( b ) , M 2 ( b ) , . . .
Revised: ( M 1 ( b ) , M 2 ( b ) , . . .

Original: , M n ( b )) to denote the integral allocation that a mechanism
Revised: , M n ( b )) to denote the integral allocation that a mechanism

Original: M outputs given reported valuations b
Revised: M outputs given reported valuations b

Original: = ( b 1 , . . .
Revised: = ( b 1 , . . .

Original: , b n ) as input, here M i ( b ) ⊆
Revised: , b n ) as input, here M i ( b ) ⊆

Original: M represents the bundle of items that agent i receives in the allocation M ( b ) .
Revised: M represents the bundle of items that agent i receives in the allocation M ( b ) .

Original: For a deterministic mechanism M , the value v i ( M i ( b )) denotes the utility of agent i for the allocation output by M on input b .
Revised: For a deterministic mechanism M , the value v i ( M i ( b )) denotes the utility of agent i for the allocation output by M on input b .

Original: Notions of incentive compatibility.
Revised: Notions of incentive compatibility.

Original: A mechanism M is truthful if agents cannot strictly improve their utility by misreporting their valuation, i.e. for all i ∈ N , valuations v i , b i , and reports of the other agents v − i , v i ( M i ( v i , v − i ))
Revised: A mechanism M is truthful if agents cannot strictly improve their utility by misreporting their valuation, i.e. for all i ∈ N , valuations v i , b i , and reports of the other agents v − i , v i ( M i ( v i , v − i ))

Original: ≥ v i
Revised: ≥ v i

Original: ( M i ( b i , v − i )) .
Revised: ( M i ( b i , v − i )) .

Original: Our work focuses on a notion of incentive compatibility that is a relaxation of truthfulness called not obvious manipulability .
Revised: Our work focuses on a notion of incentive compatibility that is a relaxation of truthfulness called not obvious manipulability .

Original: Definition 1 (Not Obviously Manipulable [TM20]) .
Revised: Definition 1 (Not Obviously Manipulable [TM20]) .

Original: A mechanism M is not obviously manipulable (NOM) if for every agent i ∈ N with valuation function v i , and every possible report
Revised: A mechanism M is not obviously manipulable (NOM) if for every agent i ∈ N with valuation function v i , and every possible report

Original: b i of agent i , the following two inequalities hold:
Revised: b i of agent i , the following two inequalities hold:

Original: (1) min v − i v i
Revised: (1) min v − i v i

Original: ( M i ( v i , v − i ))
Revised: ( M i ( v i , v − i ))

Original: ≥ min v − i v i
Revised: ≥ min v − i v i

Original: ( M i ( b i , v − i )) .
Revised: ( M i ( b i , v − i )) .

Original: (2) max v − i v i
Revised: (2) max v − i v i

Original: ( M i ( v i , v − i )) ≥ max v − i v i ( M i ( b i , v − i )) .
Revised: ( M i ( v i , v − i )) ≥ max v − i v i ( M i ( b i , v − i )) .

Original: Intuitively, if a mechanism is NOM then an agent cannot increase her worst-case utility or her best-case utility (computed with respect to the true valuation) by misreporting her valuation.
Revised: Intuitively, if a mechanism is NOM then an agent cannot increase her worst-case utility or her best-case utility (computed with respect to the true valuation) by misreporting her valuation.

Original: If either the worst-case or the best-case utility can be improved then the mechanism is obviously manipulable .
Revised: If either the worst-case or the best-case utility can be improved then the mechanism is obviously manipulable .

Original: Notions of efficiency.
Revised: Notions of efficiency.

Original: An integral allocation A =
Revised: An integral allocation A =

Original: ( A 1 , . . .
Revised: ( A 1 , . . .

Original: , A n ) is Pareto efficient (or PO)
Revised: , A n ) is Pareto efficient (or PO)

Original: iff there is no integral allocation
Revised: iff there is no integral allocation

Original: A ′
Revised: A ′

Original: =
Revised: =

Original: ( A ′ 1 , . . .
Revised: ( A ′ 1 , . . .

Original: , A ′ n )
Revised: , A ′ n )

Original: such that for all agents i ∈ N , v i ( A ′ i ) ≥ v i
Revised: such that for all agents i ∈ N , v i ( A ′ i ) ≥ v i

Original: ( A i ) , and for at least one agent this inequality is strict.
Revised: ( A i ) , and for at least one agent this inequality is strict.

Original: An (integral or fractional) allocation X =
Revised: An (integral or fractional) allocation X =

Original: ( X 1 , . . .
Revised: ( X 1 , . . .

Original: , X n ) is fractionally Pareto efficient (or fPO)
Revised: , X n ) is fractionally Pareto efficient (or fPO)

Original: iff there is no fractional allocation X ′ =
Revised: iff there is no fractional allocation X ′ =

Original: ( X ′ 1 , . . .
Revised: ( X ′ 1 , . . .

Original: , X ′ n ) such that for all agents i ∈ N , v i ( X ′ i ) ≥ v i ( X i ) , and for at least one agent this inequality is strict.
Revised: , X ′ n ) such that for all agents i ∈ N , v i ( X ′ i ) ≥ v i ( X i ) , and for at least one agent this inequality is strict.

Original: Note that fractional Pareto efficiency is a strictly stronger notion than Pareto efficiency.
Revised: Note that fractional Pareto efficiency is a strictly stronger notion than Pareto efficiency.

Original: An (integral or fractional) allocation X is α -approximately (resp.
Revised: An (integral or fractional) allocation X is α -approximately (resp.

Original: fractionally)
Revised: fractionally)

Original: Pareto efficient if there is no integral allocation (resp.
Revised: Pareto efficient if there is no integral allocation (resp.

Original: fractional allocation)
Revised: fractional allocation)

Original: X ′ =
Revised: X ′ =

Original: ( X ′ 1 , X ′ 2 , . . .
Revised: ( X ′ 1 , X ′ 2 , . . .

Original: , X ′ n ) such that α · v i ( X ′ i ) ≥ v i
Revised: , X ′ n ) such that α · v i ( X ′ i ) ≥ v i

Original: ( X i ) , with at least one of these inequalities strict [RF90, ILWM17, FGPS19, ZP20].
Revised: ( X i ) , with at least one of these inequalities strict [RF90, ILWM17, FGPS19, ZP20].

Original: Notice that for α = 1 we exactly recover Pareto efficiency (resp.
Revised: Notice that for α = 1 we exactly recover Pareto efficiency (resp.

Original: fractional Pareto efficiency).
Revised: fractional Pareto efficiency).

Original: A (partial) allocation A =
Revised: A (partial) allocation A =

Original: ( A 1 , A 2 , . . .
Revised: ( A 1 , A 2 , . . .

Original: , A n ) is non-wasteful iff for each i ∈ N , v i,j = 0 for every unallocated item j ∈ M \ ∪ k ∈ N
Revised: , A n ) is non-wasteful iff for each i ∈ N , v i,j = 0 for every unallocated item j ∈ M \ ∪ k ∈ N

Original: A k 2 .
Revised: A k 2 .

Original: A bundle S ⊆
Revised: A bundle S ⊆

Original: M is clean for wrt valuation v i if the items comprising S have positive value, i.e., v i ( g ) > 0 for all g ∈ S .
Revised: M is clean for wrt valuation v i if the items comprising S have positive value, i.e., v i ( g ) > 0 for all g ∈ S .

Original: Further, a (partial) allocation
Revised: Further, a (partial) allocation

Original: A =
Revised: A =

Original: ( A 1 , A 2 , . . .
Revised: ( A 1 , A 2 , . . .

Original: , A n ) is clean if for each i ∈ N the bundle
Revised: , A n ) is clean if for each i ∈ N the bundle

Original: A i is clean wrt valuation v i .
Revised: A i is clean wrt valuation v i .

Original: Often, we are interested in computing or approximating specific points (i.e., allocations) present on the Pareto frontier that have additional desirable properties.
Revised: Often, we are interested in computing or approximating specific points (i.e., allocations) present on the Pareto frontier that have additional desirable properties.

Original: The utilitarian social welfare of an allocation X , denoted by SW ( X ) , is defined as the sum of utilities that each agent gets in the allocation X , i.e., SW ( X ) =
Revised: The utilitarian social welfare of an allocation X , denoted by SW ( X ) , is defined as the sum of utilities that each agent gets in the allocation X , i.e., SW ( X ) =

Original: (cid:80) i ∈ N v i ( X i ) .
Revised: (cid:80) i ∈ N v i ( X i ) .

Original: The Nash social welfare of an (integral or fractional) allocation X , denoted by NSW ( X ) , is defined as the geometric mean of the utilities of agents in the allocation X , i.e., NSW ( X ) =
Revised: The Nash social welfare of an (integral or fractional) allocation X , denoted by NSW ( X ) , is defined as the geometric mean of the utilities of agents in the allocation X , i.e., NSW ( X ) =

Original: ( (cid:81)
Revised: ( (cid:81)

Original: i ∈ N v i ( x i ))
Revised: i ∈ N v i ( x i ))

Original: 1 n .
Revised: 1 n .

Original: An integral allocation that maximizes the Nash social welfare, among all integral allocations, is called a Nash social welfare maximizing (or MNW) allocation.
Revised: An integral allocation that maximizes the Nash social welfare, among all integral allocations, is called a Nash social welfare maximizing (or MNW) allocation.

Original: There are instances where every integral allocation A is such that NSW ( A ) = 0 , i.e., there is always an agent having zero utility.
Revised: There are instances where every integral allocation A is such that NSW ( A ) = 0 , i.e., there is always an agent having zero utility.

Original: To cover such an edge case, integral MNW allocations are defined as follows [CKM + 19].
Revised: To cover such an edge case, integral MNW allocations are defined as follows [CKM + 19].

Original: An integral allocation is MNW iff ( i ) it maximizes, among the set of all integral allocations, the number of agents having positive utility and ( ii ) for any such maximal set of agents S , it maximizes the geometric mean of the utilities of agents in S .
Revised: An integral allocation is MNW iff ( i ) it maximizes, among the set of all integral allocations, the number of agents having positive utility and ( ii ) for any such maximal set of agents S , it maximizes the geometric mean of the utilities of agents in S .

Original: Notions of fairness.
Revised: Notions of fairness.

Original: An (integral or fractional) allocation X is called envy-free (EF) if for every pair of agents i, j ∈ N , agent i values her allocation at least as much as the allocation of agent j , i.e., v i ( X i ) ≥ v i ( X j ) .
Revised: An (integral or fractional) allocation X is called envy-free (EF) if for every pair of agents i, j ∈ N , agent i values her allocation at least as much as the allocation of agent j , i.e., v i ( X i ) ≥ v i ( X j ) .

Original: Achieving envy-freeness is impossible for integral allocations (consider the case of a single item and two agents that both want it), so relaxation of envy-freeness are considered.
Revised: Achieving envy-freeness is impossible for integral allocations (consider the case of a single item and two agents that both want it), so relaxation of envy-freeness are considered.

Original: An integral allocation A is envy-free up to one item (EF1) if for every pair of agents i, j ∈ N , where
Revised: An integral allocation A is envy-free up to one item (EF1) if for every pair of agents i, j ∈ N , where

Original: A j ̸ = ∅ , agent i values her allocation at least as much as the allocation of agent j , subject to the removal of one item from agent j ’s bundle, i.e., v i ( A i ) ≥ v i
Revised: A j ̸ = ∅ , agent i values her allocation at least as much as the allocation of agent j , subject to the removal of one item from agent j ’s bundle, i.e., v i ( A i ) ≥ v i

Original: ( A j \ { g } ) for some item
Revised: ( A j \ { g } ) for some item

Original: g ∈
Revised: g ∈

Original: A j .
Revised: A j .

Original: Finally, we are often interested in, so called, best-of-both-worlds guarantees.
Revised: Finally, we are often interested in, so called, best-of-both-worlds guarantees.

Original: Let P be a fairness or efficiency notion for integral allocations and Q be a fairness or efficiency notion for fractional allocations.
Revised: Let P be a fairness or efficiency notion for integral allocations and Q be a fairness or efficiency notion for fractional allocations.

Original: A randomized allocation R (see Appendix A for definition), which outputs an integral allocation
Revised: A randomized allocation R (see Appendix A for definition), which outputs an integral allocation

Original: A z with probability p z , satisfies the notion P ex-post if each integral allocation
Revised: A z with probability p z , satisfies the notion P ex-post if each integral allocation

Original: A z in the support of R satisfies P .
Revised: A z in the support of R satisfies P .

Original: Additionally, the randomized allocation R satisfies the notion Q ex-ante if the expected fractional allocation Y =
Revised: Additionally, the randomized allocation R satisfies the notion Q ex-ante if the expected fractional allocation Y =

Original: (cid:80) kz
Revised: (cid:80) kz

Original: =1 p z · A z corresponding to R satisfies the notion Q .
Revised: =1 p z · A z corresponding to R satisfies the notion Q .

Original:  Fair Mechanisms
Revised:  Fair Mechanisms

Original: In this section we study whether non obvious manipulability is compatible with EF1.
Revised: In this section we study whether non obvious manipulability is compatible with EF1.

Original: For deterministic mechanisms, it is known that no truthful mechanism can always outputs an EF1 allocation, even for the case of two agents [ABCM17].
Revised: For deterministic mechanisms, it is known that no truthful mechanism can always outputs an EF1 allocation, even for the case of two agents [ABCM17].

Original: In sharp contrast, we show that Round-Robin, arguably the simplest EF1 algorithm, is not obviously manipulable.
Revised: In sharp contrast, we show that Round-Robin, arguably the simplest EF1 algorithm, is not obviously manipulable.

Original: Recall that the Round-Robin algorithm allocates items to agents over a sequence of rounds.
Revised: Recall that the Round-Robin algorithm allocates items to agents over a sequence of rounds.

Original: In each round, the agents choose one item each (a highest-value remaining item, as per their valuation 3 ) following a fixed, arbitrary order.
Revised: In each round, the agents choose one item each (a highest-value remaining item, as per their valuation 3 ) following a fixed, arbitrary order.

Original: Theorem 1.
Revised: Theorem 1.

Original: Round-Robin is not obviously manipulable .
Revised: Round-Robin is not obviously manipulable .

Original: Proof.
Revised: Proof.

Original: We prove that the two inequalities in Definition 1 hold for Round-Robin.
Revised: We prove that the two inequalities in Definition 1 hold for Round-Robin.

Original: Let i ∈ N be the i -th agent in the Round-Robin order, and let v i be her valuation vector.
Revised: Let i ∈ N be the i -th agent in the Round-Robin order, and let v i be her valuation vector.

Original: Assume without loss of generality that v i,j ≥ v i,j +1 for all j = 0 , . . .
Revised: Assume without loss of generality that v i,j ≥ v i,j +1 for all j = 0 , . . .

Original: , m − 1 .
Revised: , m − 1 .

Original: Let ℓ be the number of items agent i receives (either ⌊ nm ⌋ or ⌊ nm ⌋ + 1 , depending on i , n and m ), and notice that ℓ does not depend on her report.
Revised: Let ℓ be the number of items agent i receives (either ⌊ nm ⌋ or ⌊ nm ⌋ + 1 , depending on i , n and m ), and notice that ℓ does not depend on her report.

Original: We first prove inequality (1) from the definition of NOM, i.e. the worst-case guarantee.
Revised: We first prove inequality (1) from the definition of NOM, i.e. the worst-case guarantee.

Original: The k -th time agent i gets to pick an item, there is an un-allocated item that she values at least v i,i +( k − 1) n , since only ( k − 1) n + i − 1 items have been allocated at that point.
Revised: The k -th time agent i gets to pick an item, there is an un-allocated item that she values at least v i,i +( k − 1) n , since only ( k − 1) n + i − 1 items have been allocated at that point.

Original: Therefore, the worst-case for agent i under honest reporting is realized when all other agents rank the items in the same order as her, and her utility is exactly (cid:80) ℓk
Revised: Therefore, the worst-case for agent i under honest reporting is realized when all other agents rank the items in the same order as her, and her utility is exactly (cid:80) ℓk

Original: =1 v i,i +( k − 1) n .
Revised: =1 v i,i +( k − 1) n .

Original: Now consider the case when all agents other than i rank items in the same order as i (i.e. v i ′ ,j ≥ v i ′ ,j +1 for all i ′ ∈ N ), and i reports some vector b i .
Revised: Now consider the case when all agents other than i rank items in the same order as i (i.e. v i ′ ,j ≥ v i ′ ,j +1 for all i ′ ∈ N ), and i reports some vector b i .

Original: Let j 1 , . . .
Revised: Let j 1 , . . .

Original: , j ℓ be the items she receives, in the order she picked them.
Revised: , j ℓ be the items she receives, in the order she picked them.

Original: First, without loss of generality, i picks these items in decreasing order with respect to her true valuation, i.e. v i,j k ≥ v i,j k +1 : if she picks an item k at round t , and at a later round t ′
Revised: First, without loss of generality, i picks these items in decreasing order with respect to her true valuation, i.e. v i,j k ≥ v i,j k +1 : if she picks an item k at round t , and at a later round t ′

Original: she could pick an item k ′ with v i,k ′ > v i,k , then by the choice of valuation for agents other than i , picking k ′ in round t and k in round t ′ is also possible.
Revised: she could pick an item k ′ with v i,k ′ > v i,k , then by the choice of valuation for agents other than i , picking k ′ in round t and k in round t ′ is also possible.

Original: Second, v i,j k ≤ v i,i +( k − 1) n : when i picks an item for the k -th time, ( i − 1) + ( k − 1)( n − 1) of her top i +( k − 1) n items have been picked by other agents, and k − 1 of her top i +( k − 1) n items have been picked by herself (since items are picked in decreasing order with respect to the true valuation).
Revised: Second, v i,j k ≤ v i,i +( k − 1) n : when i picks an item for the k -th time, ( i − 1) + ( k − 1)( n − 1) of her top i +( k − 1) n items have been picked by other agents, and k − 1 of her top i +( k − 1) n items have been picked by herself (since items are picked in decreasing order with respect to the true valuation).

Original: Therefore v i,i +( k − 1) n is the largest value v i,j k can have; i ’s utility is at most (cid:80) ℓk
Revised: Therefore v i,i +( k − 1) n is the largest value v i,j k can have; i ’s utility is at most (cid:80) ℓk

Original: =1 v i,i +( k − 1) n .
Revised: =1 v i,i +( k − 1) n .

Original: Thus, the worst-case utility is maximized under truthful reporting.
Revised: Thus, the worst-case utility is maximized under truthful reporting.

Original: Next, we prove inequality (2) (the best-case guarantee).
Revised: Next, we prove inequality (2) (the best-case guarantee).

Original: Since agent i gets exactly ℓ items, her utility is at most (cid:80) ℓk
Revised: Since agent i gets exactly ℓ items, her utility is at most (cid:80) ℓk

Original: =1 v i,k .
Revised: =1 v i,k .

Original: This utility is realized when she reports v i , and everyone else ranks the items in the opposite order, i.e. v i ′ ,
Revised: This utility is realized when she reports v i , and everyone else ranks the items in the opposite order, i.e. v i ′ ,

Original: j < v i ′ ,j +1 for all i ′ , and it cannot be improved upon.
Revised: j < v i ′ ,j +1 for all i ′ , and it cannot be improved upon.

Original: In fact, we can strengthen this result to give a “best-of-both-worlds” NOM mechanism.
Revised: In fact, we can strengthen this result to give a “best-of-both-worlds” NOM mechanism.

Original: Specifically, we prove that the PS-Lottery algorithm of [Azi20b], which outputs randomized allocations that are ex-ante EF and ex-post EF1 , is NOM in expectation; proof has been deferred to Appendix B.
Revised: Specifically, we prove that the PS-Lottery algorithm of [Azi20b], which outputs randomized allocations that are ex-ante EF and ex-post EF1 , is NOM in expectation; proof has been deferred to Appendix B.

Original:  Efficient Mechanisms
Revised:  Efficient Mechanisms

Original: In this section we study whether natural efficiency notions are compatible with NOM.
Revised: In this section we study whether natural efficiency notions are compatible with NOM.

Original: We consider the three most popular notions of efficiency: utilitarian social welfare, Nash social welfare and egalitarian social welfare (see Appendix C).
Revised: We consider the three most popular notions of efficiency: utilitarian social welfare, Nash social welfare and egalitarian social welfare (see Appendix C).

Original: Under truthfulness, the only Pareto efficient and truthful algorithm is a dictatorship, which immediately implies that one cannot truthfully achieve any non-trivial approximations with respect to any of these notions.
Revised: Under truthfulness, the only Pareto efficient and truthful algorithm is a dictatorship, which immediately implies that one cannot truthfully achieve any non-trivial approximations with respect to any of these notions.

Original: Omitted proofs can be found in Appendix C.
Revised: Omitted proofs can be found in Appendix C.

Original: Utilitarian Social Welfare.
Revised: Utilitarian Social Welfare.

Original: We start by considering the utilitarian social welfare maximizing algorithm, i.e. the algorithm which allocates each item to the agent that values it the most.
Revised: We start by considering the utilitarian social welfare maximizing algorithm, i.e. the algorithm which allocates each item to the agent that values it the most.

Original: When the winner for an item is not unique the algorithm needs to break ties; the choice of the tie-breaking rule will be crucial for our positive result in this section.
Revised: When the winner for an item is not unique the algorithm needs to break ties; the choice of the tie-breaking rule will be crucial for our positive result in this section.

Original: In the context of fair division it is standard to assume that agents’ values are normalized when analyzing utilitarian social welfare.
Revised: In the context of fair division it is standard to assume that agents’ values are normalized when analyzing utilitarian social welfare.

Original: Specifically, the most common assumption is that the agents’ values add up to 1 ; see [Azi20a] for a number of justifications for this assumption.
Revised: Specifically, the most common assumption is that the agents’ values add up to 1 ; see [Azi20a] for a number of justifications for this assumption.

Original: For the remainder of this section we also assume that (cid:80) j ∈ M v i,j =for all agents i .
Revised: For the remainder of this section we also assume that (cid:80) j ∈ M v i,j =for all agents i .

Original: We note that without this assumption, the utilitarian social welfare maximizing algorithm is NOM, since no matter what an agent reports, the best case for her is that she wins all items she values positively, and the worst case is that she loses all items, so no misreport can help increase either the worst-case or best-case utility.
Revised: We note that without this assumption, the utilitarian social welfare maximizing algorithm is NOM, since no matter what an agent reports, the best case for her is that she wins all items she values positively, and the worst case is that she loses all items, so no misreport can help increase either the worst-case or best-case utility.

Original: Theorem 2.
Revised: Theorem 2.

Original: Every (randomized or deterministic) mechanism for n = 2 agents that always outputs utilitarian social welfare maximizing allocations is obviously manipulable.
Revised: Every (randomized or deterministic) mechanism for n = 2 agents that always outputs utilitarian social welfare maximizing allocations is obviously manipulable.

Original: Surprisingly, this impossibility result does not hold for more than two agents.
Revised: Surprisingly, this impossibility result does not hold for more than two agents.

Original: Theorem 3.
Revised: Theorem 3.

Original: The utilitarian social welfare maximizing algorithm, coupled with an appropriate tie-breaking rule, is not obviously manipulable for n ≥ 3 agents.
Revised: The utilitarian social welfare maximizing algorithm, coupled with an appropriate tie-breaking rule, is not obviously manipulable for n ≥ 3 agents.

Original: Proof.
Revised: Proof.

Original: First, we describe our algorithm.
Revised: First, we describe our algorithm.

Original: The algorithm, given reports b 1 , . . .
Revised: The algorithm, given reports b 1 , . . .

Original: , b n , allocates each item to an agent with the largest reported value.
Revised: , b n , allocates each item to an agent with the largest reported value.

Original: In case of a tie, the item is allocated to the agent with the smallest index (i.e. 1 ≻ 2 ≻ 3 . . . ), except if the tie is exactly between agents 1 and n , in which case the item goes to agent n .
Revised: In case of a tie, the item is allocated to the agent with the smallest index (i.e. 1 ≻ 2 ≻ 3 . . . ), except if the tie is exactly between agents 1 and n , in which case the item goes to agent n .

Original: Consider an arbitrary agent i with true valuation v i .
Revised: Consider an arbitrary agent i with true valuation v i .

Original: Towards proving inequality (1), we have that if b i,j < 1 for all j ∈ M , there exists a choice for v − i such that agent i does not have the highest value for any item (e.g., some agent k could have a value of 1 for i ’s favorite item, and a different agent k ′ can out-bid i in all remaining items).
Revised: Towards proving inequality (1), we have that if b i,j < 1 for all j ∈ M , there exists a choice for v − i such that agent i does not have the highest value for any item (e.g., some agent k could have a value of 1 for i ’s favorite item, and a different agent k ′ can out-bid i in all remaining items).

Original: If b i,j = 1 for some j ∈ M , then, again, i can again end up with no items, since by the choice of our tie-breaking rule, all agents can lose in the tie-breaking (agents other than 1 lose to some smaller index agent, while agent 1 loses to agent n ).
Revised: If b i,j = 1 for some j ∈ M , then, again, i can again end up with no items, since by the choice of our tie-breaking rule, all agents can lose in the tie-breaking (agents other than 1 lose to some smaller index agent, while agent 1 loses to agent n ).

Original: Therefore, no matter what agent i reports, in the worst-case she gets a utility of zero.
Revised: Therefore, no matter what agent i reports, in the worst-case she gets a utility of zero.

Original: Next we prove inequality (2), the best-case guarantee.
Revised: Next we prove inequality (2), the best-case guarantee.

Original: By the normalization assumption and tie-breaking rule, any agent i (except agent 1 ) cannot out-bid every other agent on every single item, no matter what b i and v − i are.
Revised: By the normalization assumption and tie-breaking rule, any agent i (except agent 1 ) cannot out-bid every other agent on every single item, no matter what b i and v − i are.

Original: Therefore, the best-case outcome for agent i is that she is allocated all items except her least favorite one.
Revised: Therefore, the best-case outcome for agent i is that she is allocated all items except her least favorite one.

Original: Additionally, note that this outcome can be realized when agent i reports honestly and v − i is such that v i ′ ,j = 1 , for all i ′
Revised: Additionally, note that this outcome can be realized when agent i reports honestly and v − i is such that v i ′ ,j = 1 , for all i ′

Original: ̸
Revised: ̸

Original: = i , where j is the item that i values the least (i.e. j ∈ argmin k ∈ M { v i,k } ) .
Revised: = i , where j is the item that i values the least (i.e. j ∈ argmin k ∈ M { v i,k } ) .

Original: Finally, if agent 1 reports her true valuation, the best-case happens when all other agents also report agent 1’s true valuation — in which case, as per the tie-breaking rule, she gets all the items; this outcome also cannot be improved upon.
Revised: Finally, if agent 1 reports her true valuation, the best-case happens when all other agents also report agent 1’s true valuation — in which case, as per the tie-breaking rule, she gets all the items; this outcome also cannot be improved upon.

Original: Nash Social Welfare.
Revised: Nash Social Welfare.

Original: The next natural maximization objective, and arguably the most popular one in the context of fair division, is the Nash social welfare.
Revised: The next natural maximization objective, and arguably the most popular one in the context of fair division, is the Nash social welfare.

Original: Here, we show that — unlike utilitarian social welfare maximization — there does not exist a NOM mechanism that always outputs allocations that maximize the Nash social welfare, for any number of agents.
Revised: Here, we show that — unlike utilitarian social welfare maximization — there does not exist a NOM mechanism that always outputs allocations that maximize the Nash social welfare, for any number of agents.

Original: The case of n = 2 agents follows from
Revised: The case of n = 2 agents follows from

Original: Theorem 10 (Appendix E) that details our “best-of-both-worlds” results.
Revised: Theorem 10 (Appendix E) that details our “best-of-both-worlds” results.

Original: In the following theorem (proof deferred to Appendix C), we show that the impossibility extends to n ≥ 3 agents.
Revised: In the following theorem (proof deferred to Appendix C), we show that the impossibility extends to n ≥ 3 agents.

Original: Theorem 4.
Revised: Theorem 4.

Original: Every (randomized or deterministic) mechanism that always outputs an allocation that maximizes the Nash social welfare is obviously manipulable, even for 3 additive agents and 4 items.
Revised: Every (randomized or deterministic) mechanism that always outputs an allocation that maximizes the Nash social welfare is obviously manipulable, even for 3 additive agents and 4 items.

Original:  Fair and Efficient Mechanisms
Revised:  Fair and Efficient Mechanisms

Original: Maximizing the Nash social welfare results in allocations that are fair in addition to being economically efficient [CKM + 19].
Revised: Maximizing the Nash social welfare results in allocations that are fair in addition to being economically efficient [CKM + 19].

Original: However, as we established in Theorem 4, mechanisms that output MNW allocations are obviously manipulable for any number of agents.
Revised: However, as we established in Theorem 4, mechanisms that output MNW allocations are obviously manipulable for any number of agents.

Original: In this section we state our main result, where we show that there is a deterministic NOM mechanism that is fair and economically efficient for additive valuations.
Revised: In this section we state our main result, where we show that there is a deterministic NOM mechanism that is fair and economically efficient for additive valuations.

Original: This result is established by showing a black-box reduction from the problem of designing NOM mechanisms that output EF1 allocations to the problem of designing algorithms that output EF1 allocations.
Revised: This result is established by showing a black-box reduction from the problem of designing NOM mechanisms that output EF1 allocations to the problem of designing algorithms that output EF1 allocations.

Original: Additionally, this black-box reduction preserves Pareto efficiency guarantees.
Revised: Additionally, this black-box reduction preserves Pareto efficiency guarantees.

Original: Missing proofs can be found in Appendix D.
Revised: Missing proofs can be found in Appendix D.

Original: The following theorem formally states our main result.
Revised: The following theorem formally states our main result.

Original: Theorem 5.
Revised: Theorem 5.

Original: For additive valuations, there exists a black-box reduction, which preserves Pareto efficiency guarantees, from the problem of designing a NOM and EF1 mechanism, to designing an algorithm that computes clean, non-wasteful and EF1 allocations.
Revised: For additive valuations, there exists a black-box reduction, which preserves Pareto efficiency guarantees, from the problem of designing a NOM and EF1 mechanism, to designing an algorithm that computes clean, non-wasteful and EF1 allocations.

Original: By combining Theorem 5 with known algorithmic results we can get NOM mechanisms, with the same fairness and efficiency guarantees.
Revised: By combining Theorem 5 with known algorithmic results we can get NOM mechanisms, with the same fairness and efficiency guarantees.

Original: Specifically, [GM21] give a pseudo-polynomial time algorithm that computes fractionally Pareto efficient and EF1 allocations.
Revised: Specifically, [GM21] give a pseudo-polynomial time algorithm that computes fractionally Pareto efficient and EF1 allocations.

Original: Fractional Pareto efficiency implies non-wastefulness, and without loss of generality we can assume that this algorithm outputs clean allocations.
Revised: Fractional Pareto efficiency implies non-wastefulness, and without loss of generality we can assume that this algorithm outputs clean allocations.

Original: Hence, by applying Theorem 5 we get the following application.
Revised: Hence, by applying Theorem 5 we get the following application.

Original: Corollary 1.
Revised: Corollary 1.

Original: (via [GM21]).
Revised: (via [GM21]).

Original: For agents with additive valuations, there exists a fractionally Pareto efficient, EF1 , and NOM mechanism, that runs in pseudo-polynomial time.
Revised: For agents with additive valuations, there exists a fractionally Pareto efficient, EF1 , and NOM mechanism, that runs in pseudo-polynomial time.

Original: The Reduction.
Revised: The Reduction.

Original: Our reduction, Mechanism 1, takes as input reported
Revised: Our reduction, Mechanism 1, takes as input reported

Original: valuations b = ( b 1 , . . .
Revised: valuations b = ( b 1 , . . .

Original: , b n ) , and black-box access to a deterministic algorithm M ∗ .
Revised: , b n ) , and black-box access to a deterministic algorithm M ∗ .

Original: 6 Our reduction requires the algorithm M ∗ to always output non-wasteful, clean EF1 allocations for every possible input valuation functions.
Revised: 6 Our reduction requires the algorithm M ∗ to always output non-wasteful, clean EF1 allocations for every possible input valuation functions.

Original: The reduction is based on two key ideas, first, through a careful construction of cases we ensure that if an agent i ∈ N reports valuation b i , then the set of allocations that can be returned by our reduction,
Revised: The reduction is based on two key ideas, first, through a careful construction of cases we ensure that if an agent i ∈ N reports valuation b i , then the set of allocations that can be returned by our reduction,

Original: Mechanism 1, for every possible b − i can be precisely characterized (Lemma 3).
Revised: Mechanism 1, for every possible b − i can be precisely characterized (Lemma 3).

Original: Second, we prove a structural result (Lemma 4) concerning this set of possible output allocation.
Revised: Second, we prove a structural result (Lemma 4) concerning this set of possible output allocation.

Original: This structural result plays a central role in establishing that Mechanism 1 is not obviously manipulable.
Revised: This structural result plays a central role in establishing that Mechanism 1 is not obviously manipulable.

Original: Further, we show that, by construction, our reduction always outputs fPO + EF1 allocations.
Revised: Further, we show that, by construction, our reduction always outputs fPO + EF1 allocations.

Original: We begin by defining some notation.
Revised: We begin by defining some notation.

Original: For each agent i ∈ N , let D i be the set of goods that have strictly positive value for i i.e. D
Revised: For each agent i ∈ N , let D i be the set of goods that have strictly positive value for i i.e. D

Original: i := { g ∈ M | b i,g > 0 } .
Revised: i := { g ∈ M | b i,g > 0 } .

Original: Let (cid:98) D i :=
Revised: Let (cid:98) D i :=

Original: M \ ∪ j ̸ = i
Revised: M \ ∪ j ̸ = i

Original: D j be the goods remaining after removing all goods desired by agents other than i .
Revised: D j be the goods remaining after removing all goods desired by agents other than i .

Original: Let R i , for each agent i ∈ N , be the indicator for the event that the subsets { D j } j ∈ N \ { D i } are pairwise disjoint, i.e. R i
Revised: Let R i , for each agent i ∈ N , be the indicator for the event that the subsets { D j } j ∈ N \ { D i } are pairwise disjoint, i.e. R i

Original: = 1 iff ( { D j } j ∈ N ) \
Revised: = 1 iff ( { D j } j ∈ N ) \

Original: { D i } are pairwise disjoint, and R i = 0 otherwise.
Revised: { D i } are pairwise disjoint, and R i = 0 otherwise.

Original: Mechanism 1 sequentially considers four cases based on the sets { D j } nj =1 and the values { R j } nj =1 , to find a temporary allocation A ∗ .
Revised: Mechanism 1 sequentially considers four cases based on the sets { D j } nj =1 and the values { R j } nj =1 , to find a temporary allocation A ∗ .

Original: See Appendix D.2 for the pseudo-code.
Revised: See Appendix D.2 for the pseudo-code.

Original: Case I: The sets { D j } nj =1 are pairwise disjoint (equivalently, R i = 1 for all
Revised: Case I: The sets { D j } nj =1 are pairwise disjoint (equivalently, R i = 1 for all

Original: i ∈ N ).
Revised: i ∈ N ).

Original: In this case,
Revised: In this case,

Original: A ∗ allocates the bundle D j to agent j for each agent j ∈ N .
Revised: A ∗ allocates the bundle D j to agent j for each agent j ∈ N .

Original: Case II: R i = 1 for exactly one agent i ∈ N .
Revised: Case II: R i = 1 for exactly one agent i ∈ N .

Original: This can occur if D i intersects two or more D j s (and these are the only intersections among pairs of subsets in { D k } nk =1 ).
Revised: This can occur if D i intersects two or more D j s (and these are the only intersections among pairs of subsets in { D k } nk =1 ).

Original: If allocating the bundle (cid:98) D i to agent i , and the bundle D j to each agent j ∈ N , for each j ̸
Revised: If allocating the bundle (cid:98) D i to agent i , and the bundle D j to each agent j ∈ N , for each j ̸

Original: = i results in an EF1 allocation, then A ∗ is set to this allocation.
Revised: = i results in an EF1 allocation, then A ∗ is set to this allocation.

Original: Otherwise, if this allocation is not EF1 , A ∗ is the allocation returned by M ∗ .
Revised: Otherwise, if this allocation is not EF1 , A ∗ is the allocation returned by M ∗ .

Original: Case III:
Revised: Case III:

Original: There are exactly two agents i, j ∈ N such that R i =
Revised: There are exactly two agents i, j ∈ N such that R i =

Original: R j = 1 .
Revised: R j = 1 .

Original: The only way this is possible is if D i , D j intersect each other and any other pair of subsets D k , D l where { k, l } ̸
Revised: The only way this is possible is if D i , D j intersect each other and any other pair of subsets D k , D l where { k, l } ̸

Original: = { i, j } , are disjoint.
Revised: = { i, j } , are disjoint.

Original: In this case, Mechanism 1 considers whether the set of goods D i ∩ D j are valued more by agent i or agent j ; each of these two subcases are similar to Case II (see Lines 12-21).
Revised: In this case, Mechanism 1 considers whether the set of goods D i ∩ D j are valued more by agent i or agent j ; each of these two subcases are similar to Case II (see Lines 12-21).

Original: Case IV:
Revised: Case IV:

Original: None of the previous cases holds (equivalently, R i = 

Revised: None of the previous cases holds (equivalently, R i =

Original: for all i ∈ N ).
Revised: for all i ∈ N ).

Original: In this case, A ∗ is simply the allocation returned by M ∗ .
Revised: In this case, A ∗ is simply the allocation returned by M ∗ .

Original: The last step of Mechanism 1 (Line 24) is to make the bundle allocated to each agent i in the temporary allocation clean , i.e. remove items from her bundle that she does not value.
Revised: The last step of Mechanism 1 (Line 24) is to make the bundle allocated to each agent i in the temporary allocation clean , i.e. remove items from her bundle that she does not value.

Original: This is necessary for one of our technical lemmas (specifically, for characterizing the set of all allocations that are possible outputs of our mechanism).
Revised: This is necessary for one of our technical lemmas (specifically, for characterizing the set of all allocations that are possible outputs of our mechanism).

Original: Note that this step does not affect efficiency or envy-freeness up to one item (i.e. if the allocation satisfied any of these notions before this step, it continues to do so).
Revised: Note that this step does not affect efficiency or envy-freeness up to one item (i.e. if the allocation satisfied any of these notions before this step, it continues to do so).

Original: For the case of two agents, our reduction is especially simple: give each agent i all items she has a positive value and the other agent has a zero value, and give all items that both agents want (if any) to the agent with the largest value for them; if this is not EF1, run the black-box mechanism.
Revised: For the case of two agents, our reduction is especially simple: give each agent i all items she has a positive value and the other agent has a zero value, and give all items that both agents want (if any) to the agent with the largest value for them; if this is not EF1, run the black-box mechanism.

Original: The case of strictly positive valuations.
Revised: The case of strictly positive valuations.

Original: It is interesting to notice that when valuations are strictly positive, i.e. v i,j > 0 for all i, j , Mechanism 1 simply outputs the allocation of M ∗ (Case IV).
Revised: It is interesting to notice that when valuations are strictly positive, i.e. v i,j > 0 for all i, j , Mechanism 1 simply outputs the allocation of M ∗ (Case IV).

Original: However, this does not imply that every PO and EF1 algorithm is NOM: in calculating worst-case outcomes, agents consider the possibility that others have zero valuations.
Revised: However, this does not imply that every PO and EF1 algorithm is NOM: in calculating worst-case outcomes, agents consider the possibility that others have zero valuations.

Original: This fact is critical in our proof of correctness (see the proof of Lemma 3).
Revised: This fact is critical in our proof of correctness (see the proof of Lemma 3).

Original: Establishing the main result.
Revised: Establishing the main result.

Original: Towards establishing Theorem 5, we begin by proving the following supporting lemmas.
Revised: Towards establishing Theorem 5, we begin by proving the following supporting lemmas.

Original: We show that Mechanism 1 preserves the efficiency and EF1 guarantees of M ∗ .
Revised: We show that Mechanism 1 preserves the efficiency and EF1 guarantees of M ∗ .

Original: Lemma 1.
Revised: Lemma 1.

Original: If M ∗ is α -Pareto efficient (resp.
Revised: If M ∗ is α -Pareto efficient (resp.

Original: α -fractionally Pareto efficient) then Mechanismalways outputs an α -Pareto efficient (resp.
Revised: α -fractionally Pareto efficient) then Mechanismalways outputs an α -Pareto efficient (resp.

Original: α -fractionally Pareto efficient) partial allocation.
Revised: α -fractionally Pareto efficient) partial allocation.

Original: Lemma 2.
Revised: Lemma 2.

Original: If M ∗ outputs non-wasteful, clean, and EF1 allocations then Mechanism 1 always outputs non-wasteful, clean, and EF1 partial allocations.
Revised: If M ∗ outputs non-wasteful, clean, and EF1 allocations then Mechanism 1 always outputs non-wasteful, clean, and EF1 partial allocations.

Original: The following lemma characterizes the set of allocations that Mechanism 1 could possibly return given the reported valuation of a particular agent.
Revised: The following lemma characterizes the set of allocations that Mechanism 1 could possibly return given the reported valuation of a particular agent.

Original: Specifically, we will show that every clean, nonwasteful and EF1 allocation, that is consistent with the reported valuation, is a possible output of
Revised: Specifically, we will show that every clean, nonwasteful and EF1 allocation, that is consistent with the reported valuation, is a possible output of

Original: Mechanism 1.
Revised: Mechanism 1.

Original: Before stating the lemma, we define some useful notations.
Revised: Before stating the lemma, we define some useful notations.

Original: For any agent i ∈ N and valuation function v i , let EF1( i, v i ) be the set of (partial) allocations A
Revised: For any agent i ∈ N and valuation function v i , let EF1( i, v i ) be the set of (partial) allocations A

Original: = ( A 1 , . . .
Revised: = ( A 1 , . . .

Original: , A n ) that are clean, non-wasteful and envy-free up to one item with respect to agent i when her valuation function is v i , i.e., EF1( i, v i )
Revised: , A n ) that are clean, non-wasteful and envy-free up to one item with respect to agent i when her valuation function is v i , i.e., EF1( i, v i )

Original: =
Revised: =

Original: { A ∈ Π n (M) | ∀
Revised: { A ∈ Π n (M) | ∀

Original: g ∈
Revised: g ∈

Original: A i v i,g > 0 , and ∀ j ∈ N with A j ̸
Revised: A i v i,g > 0 , and ∀ j ∈ N with A j ̸

Original: = ∅ , v i ( A i ) ≥ v i ( A j \ { g } ) for some g ∈ A j , and v i (M \ ∪ k ∈ N A k ) = 0 } .
Revised: = ∅ , v i ( A i ) ≥ v i ( A j \ { g } ) for some g ∈ A j , and v i (M \ ∪ k ∈ N A k ) = 0 } .

Original: Lemma 3.
Revised: Lemma 3.

Original: Given any agent i ∈ N and valuation function v i , for every allocation A ∈ EF1( i, v i ) there exists a set of valuations v − i such that Mechanism 1 on input ( v i , v − i )
Revised: Given any agent i ∈ N and valuation function v i , for every allocation A ∈ EF1( i, v i ) there exists a set of valuations v − i such that Mechanism 1 on input ( v i , v − i )

Original: outputs allocation A .
Revised: outputs allocation A .

Original: Lemma 3 establishes that all partial allocations in EF1( i, v i ) can possibly be returned by the mechanism if agent i reports v i .
Revised: Lemma 3 establishes that all partial allocations in EF1( i, v i ) can possibly be returned by the mechanism if agent i reports v i .

Original: Therefore, the problem of whether Mechanism 1 is obviously manipulable (partially) reduces to whether some set EF1( i, v ′ i ) has a better worst-case outcome than EF1( i, v i ) , with respect to the valuation vector v i .
Revised: Therefore, the problem of whether Mechanism 1 is obviously manipulable (partially) reduces to whether some set EF1( i, v ′ i ) has a better worst-case outcome than EF1( i, v i ) , with respect to the valuation vector v i .

Original: Before describing the subsequent lemma which develops this idea, we define some relevant notations.
Revised: Before describing the subsequent lemma which develops this idea, we define some relevant notations.

Original: Given any set of (partial) allocations
Revised: Given any set of (partial) allocations

Original: S ⊆
Revised: S ⊆

Original: Π n (M) , an agent i ∈ N and a valuation vector v define ℓ ( i, v , S ) =
Revised: Π n (M) , an agent i ∈ N and a valuation vector v define ℓ ( i, v , S ) =

Original: min
Revised: min

Original: A ∈ S v ( A i ) .
Revised: A ∈ S v ( A i ) .

Original: Lemma 4.
Revised: Lemma 4.

Original: For any valuations v , v ′ and any agent i ∈ N , ℓ ( i, v , EF1( i, v )) ≥ ℓ ( i, v , EF1( i, v ′ )) .
Revised: For any valuations v , v ′ and any agent i ∈ N , ℓ ( i, v , EF1( i, v )) ≥ ℓ ( i, v , EF1( i, v ′ )) .

Original: Proof.
Revised: Proof.

Original: Let A ∗ ∈ EF1( i, v ) be an allocation such v ( A ∗ i ) =
Revised: Let A ∗ ∈ EF1( i, v ) be an allocation such v ( A ∗ i ) =

Original: ℓ ( i, v , EF1( i, v )) , i.e. A ∗ is a worst-case outcome for agent i when her valuation is v .
Revised: ℓ ( i, v , EF1( i, v )) , i.e. A ∗ is a worst-case outcome for agent i when her valuation is v .

Original: We can assume, without loss of generality, that A ∗ is a complete allocation because if we assign the unallocated items, M \ ∪ k ∈ N
Revised: We can assume, without loss of generality, that A ∗ is a complete allocation because if we assign the unallocated items, M \ ∪ k ∈ N

Original: A ∗ k , to agents other than agent i arbitrarily then still the resultant allocation would continue to belong in EF1( i, v ) .
Revised: A ∗ k , to agents other than agent i arbitrarily then still the resultant allocation would continue to belong in EF1( i, v ) .

Original: This follows from the fact that A is non-wasteful, v (M \ ∪ k ∈ N A ∗ k )
Revised: This follows from the fact that A is non-wasteful, v (M \ ∪ k ∈ N A ∗ k )

Original: = 0 .
Revised: = 0 .

Original: If A ∗ ∈ EF1( i, v ′ ) , the lemma immediately follows, since the minimum v ( B i ) , over all allocations
Revised: If A ∗ ∈ EF1( i, v ′ ) , the lemma immediately follows, since the minimum v ( B i ) , over all allocations

Original: B ∈ EF1( i, v ′ ) , is at most v ( A i ) .
Revised: B ∈ EF1( i, v ′ ) , is at most v ( A i ) .

Original: Otherwise, A ∗ / ∈ EF1( i, v ′ ) .
Revised: Otherwise, A ∗ / ∈ EF1( i, v ′ ) .

Original: Since A ∗ is complete, and thereby non-wasteful, this is possible if either bundle A i is not clean wrt valuation v ′ or agent i envies some other agent even after removal of one good.
Revised: Since A ∗ is complete, and thereby non-wasteful, this is possible if either bundle A i is not clean wrt valuation v ′ or agent i envies some other agent even after removal of one good.

Original: Denote by A the allocation obtained by cleaning the bundle A ∗ i wrt valuation v ′ , all other bundles remain as it is, i.e., A j =
Revised: Denote by A the allocation obtained by cleaning the bundle A ∗ i wrt valuation v ′ , all other bundles remain as it is, i.e., A j =

Original: A ∗ j for all j ̸
Revised: A ∗ j for all j ̸

Original: = i .
Revised: = i .

Original: Now if allocation A ∈ EF1( i, v ′ ) , the lemma follows, because v ( A ∗ i ) ≥ v ( A i ) and ℓ ( i, v , EF1( i, v ′ )) is at most v ( A i ) .
Revised: Now if allocation A ∈ EF1( i, v ′ ) , the lemma follows, because v ( A ∗ i ) ≥ v ( A i ) and ℓ ( i, v , EF1( i, v ′ )) is at most v ( A i ) .

Original: In the rest of the proof we will handle the case when A / ∈ EF1( i, v ′ ) .
Revised: In the rest of the proof we will handle the case when A / ∈ EF1( i, v ′ ) .

Original: Towards this we will construct an allocation A ′ with the following properties: ( i ) v ( A ′ i ) ≤ v ( A i ) , and ( ii ) A ′ ∈ EF1( i, v ′ ) .
Revised: Towards this we will construct an allocation A ′ with the following properties: ( i ) v ( A ′ i ) ≤ v ( A i ) , and ( ii ) A ′ ∈ EF1( i, v ′ ) .

Original: These two properties together establish the desired inequality, since ℓ ( i, v , EF1( i, v ))
Revised: These two properties together establish the desired inequality, since ℓ ( i, v , EF1( i, v ))

Original: =
Revised: =

Original: v ( A ∗ i ) ≥ v ( A i ) ≥ ( i ) v ( A ′ i ) ≥ ( ii ) ℓ
Revised: v ( A ∗ i ) ≥ v ( A i ) ≥ ( i ) v ( A ′ i ) ≥ ( ii ) ℓ

Original: ( i, v , EF1( i, v ′ )) .
Revised: ( i, v , EF1( i, v ′ )) .

Original: Let S ∗ := { j ∈ N \{ i } | A j ̸ = ∅} be the subset of agents with non-empty bundles in A .
Revised: Let S ∗ := { j ∈ N \{ i } | A j ̸ = ∅} be the subset of agents with non-empty bundles in A .

Original: Define agent j ∗ := arg max k ∈ S ∗ min
Revised: Define agent j ∗ := arg max k ∈ S ∗ min

Original: g ∈
Revised: g ∈

Original: A k v ′ ( A k \ { g } ) , to be the agent in S ∗ whose allocation “up to one item” has maximum value with respect to v ′ , and let g ∗ =
Revised: A k v ′ ( A k \ { g } ) , to be the agent in S ∗ whose allocation “up to one item” has maximum value with respect to v ′ , and let g ∗ =

Original: arg min g ∈
Revised: arg min g ∈

Original: A j ∗ v ′
Revised: A j ∗ v ′

Original: ( A j ∗ \ { g } ) be the favourite good with respect to v ′ that j ∗ has.
Revised: ( A j ∗ \ { g } ) be the favourite good with respect to v ′ that j ∗ has.

Original: A ′ is defined as follows.
Revised: A ′ is defined as follows.

Original: A ′ i
Revised: A ′ i

Original: =
Revised: =

Original: A j ∗ \ { g ∗ } , A ′ j ∗ =
Revised: A j ∗ \ { g ∗ } , A ′ j ∗ =

Original: A i ∪ { g ∗ } , and for all agents k ∈ N \ { i, j ∗ } we have A ′ k
Revised: A i ∪ { g ∗ } , and for all agents k ∈ N \ { i, j ∗ } we have A ′ k

Original: =
Revised: =

Original: A k .
Revised: A k .

Original: It remains to show that A ′ satisfies ( i ) and ( ii ) .
Revised: It remains to show that A ′ satisfies ( i ) and ( ii ) .

Original: Towards proving ( i ) we have v ( A ′ i )
Revised: Towards proving ( i ) we have v ( A ′ i )

Original: = v ( A j ∗ \ { g ∗ } ) ≤ v ( A i ) , where the last inequality follows from the fact that A ∈ EF1( i, v ) .
Revised: = v ( A j ∗ \ { g ∗ } ) ≤ v ( A i ) , where the last inequality follows from the fact that A ∈ EF1( i, v ) .

Original: For ( ii ) , note that N \ { i } can be written as the union of three disjoint sets S 1 , S 2 and S 3 , where S 1 = N \
Revised: For ( ii ) , note that N \ { i } can be written as the union of three disjoint sets S 1 , S 2 and S 3 , where S 1 = N \

Original: ( S ∗ ∪ { i } ) , S 2 = ( S ∗ \ { j ∗ } ) , and S 

Revised: ( S ∗ ∪ { i } ) , S 2 = ( S ∗ \ { j ∗ } ) , and S

Original: = { j ∗ } .
Revised: = { j ∗ } .

Original: We will show that in allocation A ′ agent i with valuation vector v ′ , does not envy, up to one item, any of the agents in S 1 , S 2 , or S 3 ; this establishes that A ′ ∈ EF1( i, v ′ ) .
Revised: We will show that in allocation A ′ agent i with valuation vector v ′ , does not envy, up to one item, any of the agents in S 1 , S 2 , or S 3 ; this establishes that A ′ ∈ EF1( i, v ′ ) .

Original: First, for every agent k ∈ S 1 we have that A ′ k
Revised: First, for every agent k ∈ S 1 we have that A ′ k

Original: =
Revised: =

Original: A k = ∅ , by the definition of S ∗ , so agent i cannot envy any such agent.
Revised: A k = ∅ , by the definition of S ∗ , so agent i cannot envy any such agent.

Original: Second, given the definition of agent j ∗ , we know that for each agent k in S 2 = S ∗ \ { j ∗ } , we have
Revised: Second, given the definition of agent j ∗ , we know that for each agent k in S 2 = S ∗ \ { j ∗ } , we have

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

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: k for all agents except i and j ∗ )
Revised: k for all agents except i and j ∗ )

Original: [Equation]
Revised: [Equation]

Original: (by the definition of j ∗ )
Revised: (by the definition of j ∗ )

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

Original: Therefore, agent i with valuation v ′ does not envy up to one item (and, in fact, up to any item) any agent in S 2 .
Revised: Therefore, agent i with valuation v ′ does not envy up to one item (and, in fact, up to any item) any agent in S 2 .

Original: Third, recall that A / ∈ EF1( i, v ′ ) .
Revised: Third, recall that A / ∈ EF1( i, v ′ ) .

Original: By definition, agent j ∗ has the bundle that v ′ likes the most “up to one item”, and g ∗ is that most valuable (in terms of marginal value) item in j ∗ ’s bundle, so v ′
Revised: By definition, agent j ∗ has the bundle that v ′ likes the most “up to one item”, and g ∗ is that most valuable (in terms of marginal value) item in j ∗ ’s bundle, so v ′

Original: ( A i ) <
Revised: ( A i ) <

Original: v ′ ( A j ∗ \ { g ∗ } ) .
Revised: v ′ ( A j ∗ \ { g ∗ } ) .

Original: Therefore, we have
Revised: Therefore, we have

Original: [Equation]
Revised: [Equation]

Original: g
Revised: g

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

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

Original: That is, agent i with valuation v ′ does not envy, up to one item, any agent in S 

Revised: That is, agent i with valuation v ′ does not envy, up to one item, any agent in S

Original: = { j ∗ } .
Revised: = { j ∗ } .

Original: Note that, allocation A ′ is non-wasteful wrt agent i .
Revised: Note that, allocation A ′ is non-wasteful wrt agent i .

Original: Furthermore, the bundle A ′ i can be cleaned maintaining the fact that A ′ is EF1 wrt agent i .
Revised: Furthermore, the bundle A ′ i can be cleaned maintaining the fact that A ′ is EF1 wrt agent i .

Original: This establishes that A ′ ∈ EF1( i, v ′ ) and concludes the proof.
Revised: This establishes that A ′ ∈ EF1( i, v ′ ) and concludes the proof.

Original: Given the previous lemmas, we can show our main theorem with respect to NOM.
Revised: Given the previous lemmas, we can show our main theorem with respect to NOM.

Original: Theorem 6.
Revised: Theorem 6.

Original: If agents’ valuation functions are additive and M ∗ outputs EF1 , non-wasteful and clean allocations, then Mechanism 1 is not obviously manipulable.
Revised: If agents’ valuation functions are additive and M ∗ outputs EF1 , non-wasteful and clean allocations, then Mechanism 1 is not obviously manipulable.

Original: Proof.
Revised: Proof.

Original: We first show that agents cannot improve their worst-case utilities by misreporting.
Revised: We first show that agents cannot improve their worst-case utilities by misreporting.

Original: Let i ∈ N be an agent whose true valuation is v i and let her misreport be b i .
Revised: Let i ∈ N be an agent whose true valuation is v i and let her misreport be b i .

Original: By Lemma 3, for every reported valuation vector b of agent i , and for every allocation A ∈ EF1( i, b ) , there exists a valuation profile v − i such that A is the output of Mechanism 1.
Revised: By Lemma 3, for every reported valuation vector b of agent i , and for every allocation A ∈ EF1( i, b ) , there exists a valuation profile v − i such that A is the output of Mechanism 1.

Original: On the other hand, by
Revised: On the other hand, by

Original: Lemma 2, Mechanism 1 always outputs allocations
Revised: Lemma 2, Mechanism 1 always outputs allocations

Original: A that are EF1, clean, and non-wasteful.
Revised: A that are EF1, clean, and non-wasteful.

Original: Therefore, every output allocation of Mechanism 1 is in EF1( i, b ) .
Revised: Therefore, every output allocation of Mechanism 1 is in EF1( i, b ) .

Original: Therefore, if agent i reports v i (respectively b i ), then the set of all possible allocations of Mechanism 1 is exactly EF1( i, v i ) (respectively EF1( i, b i ) ).
Revised: Therefore, if agent i reports v i (respectively b i ), then the set of all possible allocations of Mechanism 1 is exactly EF1( i, v i ) (respectively EF1( i, b i ) ).

Original: Overloading notation, let A ∗
Revised: Overloading notation, let A ∗

Original: i ( v i , v − i ) be the allocation of agent
Revised: i ( v i , v − i ) be the allocation of agent

Original: i in Mechanism 1, on input ( v i , v − i ) .
Revised: i in Mechanism 1, on input ( v i , v − i ) .

Original: For the worst-case utility of agent i we have min v − i v i
Revised: For the worst-case utility of agent i we have min v − i v i

Original: ( A ∗ i ( v i , v − i ))
Revised: ( A ∗ i ( v i , v − i ))

Original: =
Revised: =

Original: min
Revised: min

Original: A ∈ EF1( i, v i ) v i ( A i ) =
Revised: A ∈ EF1( i, v i ) v i ( A i ) =

Original: ℓ ( i, v i , EF1( i, v i )) ≥
Revised: ℓ ( i, v i , EF1( i, v i )) ≥

Original: Lem.
Revised: Lem.

Original: 4 ℓ ( i, v i , EF1( i, b i ))
Revised: 4 ℓ ( i, v i , EF1( i, b i ))

Original: =
Revised: =

Original: min v − i v i
Revised: min v − i v i

Original: ( A ∗ i ( b i , v − i )) .
Revised: ( A ∗ i ( b i , v − i )) .

Original: This establishes Inequality (1) from the definition of NOM.
Revised: This establishes Inequality (1) from the definition of NOM.

Original: To conclude the proof, it remains to show that the best-case utility of an agent is also not improved by misreporting.
Revised: To conclude the proof, it remains to show that the best-case utility of an agent is also not improved by misreporting.

Original: Note that the best case for agent i ∈ N occurs when all other agents have no value for the items, i.e. D j = ∅ for all j ∈ N \ { i } .
Revised: Note that the best case for agent i ∈ N occurs when all other agents have no value for the items, i.e. D j = ∅ for all j ∈ N \ { i } .

Original: Hence the sets { D k } k ∈ [ n ] are pairwise-disjoint, and Mechanism 1 (via Case I) allocates the entirety of D i to agent i , which is impossible to improve upon.
Revised: Hence the sets { D k } k ∈ [ n ] are pairwise-disjoint, and Mechanism 1 (via Case I) allocates the entirety of D i to agent i , which is impossible to improve upon.

Original: The proof of the above theorem along with the supporting lemmas establishes Theorem 5.
Revised: The proof of the above theorem along with the supporting lemmas establishes Theorem 5.

Original: We note that that it is not possible to improve upon this result by adding ex-ante fairness guarantees; see
Revised: We note that that it is not possible to improve upon this result by adding ex-ante fairness guarantees; see

Original: Appendix D.3 for details.
Revised: Appendix D.3 for details.

Original: Acknowledgements
Revised: Acknowledgements

Original: The authors are supported in part by an NSF CAREER award CCF-2144208, a Google Research
Revised: The authors are supported in part by an NSF CAREER award CCF-2144208, a Google Research

Original: Scholar Award and by the Algorand Centres of Excellence program managed by Algorand Foundation.
Revised: Scholar Award and by the Algorand Centres of Excellence program managed by Algorand Foundation.

Original: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Algorand Foundation.
Revised: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Algorand Foundation.

Original: Checklist
Revised: Checklist

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

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

Original: [Yes] See “Our contribution” in intro.
Revised: [Yes] See “Our contribution” in intro.

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: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 2.
Revised: [Yes] 2.

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

Original: [Yes] See Section 2.
Revised: [Yes] See Section 2.

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

Original: [Yes] All our theorems and lemmas have proofs, either in the first 9 pages, or in the supplementary material.
Revised: [Yes] All our theorems and lemmas have proofs, either in the first 9 pages, or in the supplementary material.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

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

Original: [N/A] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
Revised: [N/A] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

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

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

Original: [N/A] 4.
Revised: [N/A] 4.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [N/A] (b) Did you mention the license of the assets?
Revised: [N/A] (b) Did you mention the license of the assets?

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

Original: [N/A] (d) Did you discuss whether
Revised: [N/A] (d) Did you discuss whether

Original: and how consent was obtained from people whose data you’re using/curating?
Revised: and how consent was obtained from people whose data you’re using/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?

Original: [N/A] 5.
Revised: [N/A] 5.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

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

Original: A Missing preliminaries
Revised: A Missing preliminaries

Original: Allocations.
Revised: Allocations.

Original: A randomized allocation R = { ( p z , A z ) } kz =1 is a probability distribution (or a lottery) over a set of integral allocations, i.e., for every z ∈ [ k ] , A z is an integral allocation that occurs with a probability p z .
Revised: A randomized allocation R = { ( p z , A z ) } kz =1 is a probability distribution (or a lottery) over a set of integral allocations, i.e., for every z ∈ [ k ] , A z is an integral allocation that occurs with a probability p z .

Original: The sum of the probabilities is equal to one, (cid:80) kz
Revised: The sum of the probabilities is equal to one, (cid:80) kz

Original: =1 p z = 1 .
Revised: =1 p z = 1 .

Original: The integral allocations
Revised: The integral allocations

Original: A 1 , A 2 , . . .
Revised: A 1 , A 2 , . . .

Original: , A k constitute the support of R .
Revised: , A k constitute the support of R .

Original: For each randomized allocation R = { ( p z , A z ) } kz
Revised: For each randomized allocation R = { ( p z , A z ) } kz

Original: = there exists a corresponding expected fractional allocation X
Revised: = there exists a corresponding expected fractional allocation X

Original: = (cid:80) kz
Revised: = (cid:80) kz

Original: =1 p z · A z , where for each agent i ∈ N , the fraction of items given to agent
Revised: =1 p z · A z , where for each agent i ∈ N , the fraction of items given to agent

Original: i is X i
Revised: i is X i

Original: = (cid:80) kz
Revised: = (cid:80) kz

Original: =1 p z · A zi .
Revised: =1 p z · A zi .

Original: Here X i,j can be thought of as the probability with which agent i is allocated item j in an integral allocation that is sampled from R .
Revised: Here X i,j can be thought of as the probability with which agent i is allocated item j in an integral allocation that is sampled from R .

Original: For notational clarity, we will use the letters X, Y to denote allocations that are fractional or integral,
Revised: For notational clarity, we will use the letters X, Y to denote allocations that are fractional or integral,

Original: A, B to denote allocations that are exclusively integral and R for randomized allocations.
Revised: A, B to denote allocations that are exclusively integral and R for randomized allocations.

Original: Mechanisms.
Revised: Mechanisms.

Original: A mechanism R is randomized if for every reported valuations b it outputs a randomized allocation, i.e. it returns integral allocations that are drawn from a probability distribution corresponding to a randomized allocation.
Revised: A mechanism R is randomized if for every reported valuations b it outputs a randomized allocation, i.e. it returns integral allocations that are drawn from a probability distribution corresponding to a randomized allocation.

Original: Since every randomized allocation has an associated expected fractional allocation, the output of a randomized mechanism for reported valuations b can also be interpreted as representing a fractional allocation.
Revised: Since every randomized allocation has an associated expected fractional allocation, the output of a randomized mechanism for reported valuations b can also be interpreted as representing a fractional allocation.

Original: We use R ( b ) =
Revised: We use R ( b ) =

Original: ( R 1 ( b ) , . . .
Revised: ( R 1 ( b ) , . . .

Original: , R n ( b )) to denote the expected fractional allocation that a randomized mechanism outputs given bids b ; the vector R i ( b ) ∈ [0 , 1] m represents the probabilities with which agent i receives each item in the sampled integral allocation.
Revised: , R n ( b )) to denote the expected fractional allocation that a randomized mechanism outputs given bids b ; the vector R i ( b ) ∈ [0 , 1] m represents the probabilities with which agent i receives each item in the sampled integral allocation.

Original: For a randomized mechanism, v i ( R i ( b )) denotes the expected utility of agent i in the sampled integral allocation when the input to the mechanism R is b .
Revised: For a randomized mechanism, v i ( R i ( b )) denotes the expected utility of agent i in the sampled integral allocation when the input to the mechanism R is b .

Original: To define NOM for a randomized mechanism R , we compare the expected utilities v i ( R i ( · , · )) in inequality (1) and (2), instead of v i ( M i ( · , · )) as in the case of deterministic mechanisms.
Revised: To define NOM for a randomized mechanism R , we compare the expected utilities v i ( R i ( · , · )) in inequality (1) and (2), instead of v i ( M i ( · , · )) as in the case of deterministic mechanisms.

Original: Notice that for randomized mechanisms, the definition of NOM takes an expectation over the randomness of the mechanism, and minimum/maximum are over the reports of other agents; we sometimes write “NOM in expectation” when referring specifically to a randomized mechanism.
Revised: Notice that for randomized mechanisms, the definition of NOM takes an expectation over the randomness of the mechanism, and minimum/maximum are over the reports of other agents; we sometimes write “NOM in expectation” when referring specifically to a randomized mechanism.

Original: B Missing from Section
Revised: B Missing from Section

Original: The PS-Lottery algorithm is based on the well-known probabilistic serial algorithm, which outputs fractional allocations that are envy-free.
Revised: The PS-Lottery algorithm is based on the well-known probabilistic serial algorithm, which outputs fractional allocations that are envy-free.

Original: On a high level, the PS-Lottery algorithm uses Birkhoff’s algorithm 8 to implement the fractional allocation output by probabilistic serial as a randomized allocation (a lottery) over a set of EF1 allocations.
Revised: On a high level, the PS-Lottery algorithm uses Birkhoff’s algorithm 8 to implement the fractional allocation output by probabilistic serial as a randomized allocation (a lottery) over a set of EF1 allocations.

Original: For the sake of completeness, the PS-Lottery algorithm is formally described in Appendix B.1.
Revised: For the sake of completeness, the PS-Lottery algorithm is formally described in Appendix B.1.

Original: We begin by proving a lemma that highlights a connection between not obvious manipulability and randomized mechanisms that output ex-ante proportional allocations.
Revised: We begin by proving a lemma that highlights a connection between not obvious manipulability and randomized mechanisms that output ex-ante proportional allocations.

Original: We note that a similar observation is made in [OSH19] in the context of cake cutting.
Revised: We note that a similar observation is made in [OSH19] in the context of cake cutting.

Original: Lemma 5.
Revised: Lemma 5.

Original: Inequality (1) (the worst-case guarantee) is satisfied for every randomized mechanism R that outputs ex-ante proportional allocations.
Revised: Inequality (1) (the worst-case guarantee) is satisfied for every randomized mechanism R that outputs ex-ante proportional allocations.

Original: Proof.
Revised: Proof.

Original: Let R be a randomized mechanism that outputs ex-ante proportional allocations.
Revised: Let R be a randomized mechanism that outputs ex-ante proportional allocations.

Original: Consider an agent i ∈ N with true valuation v i .
Revised: Consider an agent i ∈ N with true valuation v i .

Original: Suppose agent i reports her true valuation v i .
Revised: Suppose agent i reports her true valuation v i .

Original: Since the mechanism outputs ex-ante proportional allocations, for every possible reports of other agents, b − i , the expected fractional allocation output by R , Y =
Revised: Since the mechanism outputs ex-ante proportional allocations, for every possible reports of other agents, b − i , the expected fractional allocation output by R , Y =

Original: ( Y 1 , Y 2 , . . .
Revised: ( Y 1 , Y 2 , . . .

Original: , Y n ) will be such that v i ( Y i ) ≥ 1 n v i (M) .
Revised: , Y n ) will be such that v i ( Y i ) ≥ 1 n v i (M) .

Original: As a consequence, when agent i reports v i , her worst-case expected utility is at least 1 n v i (M) .
Revised: As a consequence, when agent i reports v i , her worst-case expected utility is at least 1 n v i (M) .

Original: Next, we show that when every agent j ∈ N \ { i } reports her valuation to be v i (the true valuation of agent i ), then the worst-case expected utility of agent i , as per her true valuation and irrespective of her report, will be at most 1 n v i (M) , which implies the lemma.
Revised: Next, we show that when every agent j ∈ N \ { i } reports her valuation to be v i (the true valuation of agent i ), then the worst-case expected utility of agent i , as per her true valuation and irrespective of her report, will be at most 1 n v i (M) , which implies the lemma.

Original: Consider the case when every agent j ∈ N \ { i } reports her valuation to be b j = v i and agent i reports valuation b
Revised: Consider the case when every agent j ∈ N \ { i } reports her valuation to be b j = v i and agent i reports valuation b

Original: i .
Revised: i .

Original: We know that the expected fractional allocation Y ′
Revised: We know that the expected fractional allocation Y ′

Original: =
Revised: =

Original: ( Y ′ 1 , Y ′ 2 , . . .
Revised: ( Y ′ 1 , Y ′ 2 , . . .

Original: , Y ′ n ) returned by R will be proportional, i.e., for each agent j ∈ N \ { i } we have b
Revised: , Y ′ n ) returned by R will be proportional, i.e., for each agent j ∈ N \ { i } we have b

Original: j ( Y ′ j )
Revised: j ( Y ′ j )

Original: =
Revised: =

Original: v i ( Y ′ j ) ≥ 1 n v i (M) .
Revised: v i ( Y ′ j ) ≥ 1 n v i (M) .

Original: Adding up over all j ∈ N \ { i } we have that (cid:80) j ∈ N \{ i } v i
Revised: Adding up over all j ∈ N \ { i } we have that (cid:80) j ∈ N \{ i } v i

Original: ( Y ′ j ) = v i (M) − v i ( Y ′ i ) ≥ n − 1 n v i (M) .
Revised: ( Y ′ j ) = v i (M) − v i ( Y ′ i ) ≥ n − 1 n v i (M) .

Original: On rearranging we get the required inequality v i ( Y ′ i ) ≤ 1 n v i (M) .
Revised: On rearranging we get the required inequality v i ( Y ′ i ) ≤ 1 n v i (M) .

Original: Using Lemma 5 we can prove the following theorem.
Revised: Using Lemma 5 we can prove the following theorem.

Original: Theorem 7.
Revised: Theorem 7.

Original: The PS-Lottery algorithm of [Azi20b], which is ex-ante envy-free and ex-post EF1 , is not obviously manipulable in expectation.
Revised: The PS-Lottery algorithm of [Azi20b], which is ex-ante envy-free and ex-post EF1 , is not obviously manipulable in expectation.

Original: Proof.
Revised: Proof.

Original: Inequality (1) is implied by Lemma 5, since the PS-Lottery algorithm is ex-ante envy-free (see Appendix B.1), and therefore ex-ante proportional.
Revised: Inequality (1) is implied by Lemma 5, since the PS-Lottery algorithm is ex-ante envy-free (see Appendix B.1), and therefore ex-ante proportional.

Original: It remains to prove Inequality (2).
Revised: It remains to prove Inequality (2).

Original: First, it holds that the expected fractional allocation returned by the PS-Lottery algorithm, irrespective of the agents’ reports, is such that each agent receives mn (fractional) items in total; see Propertyin Appendix B.1.
Revised: First, it holds that the expected fractional allocation returned by the PS-Lottery algorithm, irrespective of the agents’ reports, is such that each agent receives mn (fractional) items in total; see Propertyin Appendix B.1.

Original: Second, the best-case for an agent i ∈ N who, without loss of generality, values items in the order v i, 1 ≥ v i, 2 . . .
Revised: Second, the best-case for an agent i ∈ N who, without loss of generality, values items in the order v i, 1 ≥ v i, 2 . . .

Original: ≥ v i,m , and reports honestly, occurs when the reported valuation of other agents induce an opposite preference order on items, i.e., for each agent j ∈ N \ { i } we have b j,m ≥ b j,m − 1 . . .
Revised: ≥ v i,m , and reports honestly, occurs when the reported valuation of other agents induce an opposite preference order on items, i.e., for each agent j ∈ N \ { i } we have b j,m ≥ b j,m − 1 . . .

Original: ≥ b j, 1 .
Revised: ≥ b j, 1 .

Original: In this case, agent i would receive items 1 , 2 , . . .
Revised: In this case, agent i would receive items 1 , 2 , . . .

Original: , ⌊ mn ⌋ in their entirety and a mn − ⌊ mn ⌋ fraction of item ⌊ mn ⌋ + 1 .
Revised: , ⌊ mn ⌋ in their entirety and a mn − ⌊ mn ⌋ fraction of item ⌊ mn ⌋ + 1 .

Original: This allocation results in the maximum possible expected utility that agent i can get subject to the constraint that she gets mn fraction of items, and therefore it cannot be improved upon, no matter what her report is.
Revised: This allocation results in the maximum possible expected utility that agent i can get subject to the constraint that she gets mn fraction of items, and therefore it cannot be improved upon, no matter what her report is.

Original: B.1 The PS-Lottery Algorithm of [Aziz, 2020b]
Revised: B.1 The PS-Lottery Algorithm of [Aziz, 2020b]

Original: A square matrix M ∈ [0 , 1] k · k is bistochastic iff the sum of entries in each of its rows and columns is equal to one, i.e., for each i ∈
Revised: A square matrix M ∈ [0 , 1] k · k is bistochastic iff the sum of entries in each of its rows and columns is equal to one, i.e., for each i ∈

Original: [ k ] , we have (cid:80) kj =1 M i,j = (cid:80) kj =1 M j,i = 1 .
Revised: [ k ] , we have (cid:80) kj =1 M i,j = (cid:80) kj =1 M j,i = 1 .

Original: Additionally, a bistochastic matrix N ∈ { 0 , 1 } k · k is a permutation matrix — each row and column of a permutation matrix contains exactly one entry having value one and all other entries are zero.
Revised: Additionally, a bistochastic matrix N ∈ { 0 , 1 } k · k is a permutation matrix — each row and column of a permutation matrix contains exactly one entry having value one and all other entries are zero.

Original: In essence, the PS-Lottery algorithm of [Azi20b] is based on the following two well-known algorithms:
Revised: In essence, the PS-Lottery algorithm of [Azi20b] is based on the following two well-known algorithms:

Original: Birkhoff’s algorithm.
Revised: Birkhoff’s algorithm.

Original: Given a bistochastic matrix M ∈ [0 , 1] k · k as input, Birkhoff’s algorithm can be used to decompose, in polynomial time, the matrix M into a convex combination of permutation matrices.
Revised: Given a bistochastic matrix M ∈ [0 , 1] k · k as input, Birkhoff’s algorithm can be used to decompose, in polynomial time, the matrix M into a convex combination of permutation matrices.

Original: That is, Birkhoff’s algorithm outputs permutation matrices { M i } ti =1 and positive real numbers { p i } ti =1 such that M = (cid:80)
Revised: That is, Birkhoff’s algorithm outputs permutation matrices { M i } ti =1 and positive real numbers { p i } ti =1 such that M = (cid:80)

Original: ki =1 p i M i and (cid:80) i p i = 1 ; here t =
Revised: ki =1 p i M i and (cid:80) i p i = 1 ; here t =

Original: O ( k 2 ) is a positive integer.
Revised: O ( k 2 ) is a positive integer.

Original: Probabilistic serial algorithm.
Revised: Probabilistic serial algorithm.

Original: Given a fair division instance wherein agents have additive valuations, the probabilistic algorithm outputs a fractional allocation that is envy-free (and hence proportional).
Revised: Given a fair division instance wherein agents have additive valuations, the probabilistic algorithm outputs a fractional allocation that is envy-free (and hence proportional).

Original: The fractional allocation output by probabilistic serial can be interpreted as the output of the following continuous procedure: starting from time t = 0 , simultaneously, each agent start consuming items in the order of their preference (i.e., if v i,j 1 ≥ v i,j 2 . .
Revised: The fractional allocation output by probabilistic serial can be interpreted as the output of the following continuous procedure: starting from time t = 0 , simultaneously, each agent start consuming items in the order of their preference (i.e., if v i,j 1 ≥ v i,j 2 . .

Original: , ≥ v i,j m then the items are consumed in the order j 1 , j 2 , . . .
Revised: , ≥ v i,j m then the items are consumed in the order j 1 , j 2 , . . .

Original: , j m 9 ) and at a rate of one item per unit time.
Revised: , j m 9 ) and at a rate of one item per unit time.

Original: If an item is fully consumed, then agents start consuming the next item as per their preference order.
Revised: If an item is fully consumed, then agents start consuming the next item as per their preference order.

Original: The algorithm terminates when all items have been consumed, which happens at time t =
Revised: The algorithm terminates when all items have been consumed, which happens at time t =

Original: mn .
Revised: mn .

Original: As a direct consequence of this, ( i ) each agent gets exactly mn fraction of items at the end, and ( ii ) the resultant fractional allocation is envy-free, since at every point agents are consuming their most-valued remaining item.
Revised: As a direct consequence of this, ( i ) each agent gets exactly mn fraction of items at the end, and ( ii ) the resultant fractional allocation is envy-free, since at every point agents are consuming their most-valued remaining item.

Original: The PS-Lottery algorithm uses Birkhoff’s algorithm to decompose the fractional allocation output by probabilistic serial into a convex combination over integral allocations, i.e., a randomized allocation.
Revised: The PS-Lottery algorithm uses Birkhoff’s algorithm to decompose the fractional allocation output by probabilistic serial into a convex combination over integral allocations, i.e., a randomized allocation.

Original: In addition, each integral allocation in the support of the randomized allocation is EF1.
Revised: In addition, each integral allocation in the support of the randomized allocation is EF1.

Original: In Section 3, towards showing that the PS-Lottery algorithm is NOM in expectation, we use the following properties of the PS-Lottery algorithm.
Revised: In Section 3, towards showing that the PS-Lottery algorithm is NOM in expectation, we use the following properties of the PS-Lottery algorithm.

Original: Property 1.
Revised: Property 1.

Original: The randomized allocation output by the PS-Lottery algorithm is ex-ante EF and ex-post EF1.
Revised: The randomized allocation output by the PS-Lottery algorithm is ex-ante EF and ex-post EF1.

Original: Property 2.
Revised: Property 2.

Original: The total fraction of items that each agent gets in
Revised: The total fraction of items that each agent gets in

Original: the expected fractional allocation
Revised: the expected fractional allocation

Original: returned by the PS-Lottery algorithm is mn
Revised: returned by the PS-Lottery algorithm is mn

Original: Additionally, suppose that agent i ∈ N prefers the items in the order v i, 1 ≥ v i,j 2 . . .
Revised: Additionally, suppose that agent i ∈ N prefers the items in the order v i, 1 ≥ v i,j 2 . . .

Original: ≥ v i,j m , and the preference order to all the other agents is opposite, i.e., for all agents k ∈ [ n ] \ { i } , we have v k,m ≥ v k,m − 1 . . .
Revised: ≥ v i,j m , and the preference order to all the other agents is opposite, i.e., for all agents k ∈ [ n ] \ { i } , we have v k,m ≥ v k,m − 1 . . .

Original: ≥ v k, 1 , then in the fractional allocation output by probabilistic serial, agent i gets the items 1 , 2 , . . .
Revised: ≥ v k, 1 , then in the fractional allocation output by probabilistic serial, agent i gets the items 1 , 2 , . . .

Original: , ⌊ mn ⌋ entirely, and a fraction mn − ⌊ mn ⌋ of the item ⌊ mn ⌋ + 1 .
Revised: , ⌊ mn ⌋ entirely, and a fraction mn − ⌊ mn ⌋ of the item ⌊ mn ⌋ + 1 .

Original: C Missing from Section
Revised: C Missing from Section

Original: Proof of Theorem 2.
Revised: Proof of Theorem 2.

Original: We prove the theorem for the case of two items; the proof can be easily generalized to hold for any number of items.
Revised: We prove the theorem for the case of two items; the proof can be easily generalized to hold for any number of items.

Original: Let M be the utilitarian social welfare maximizing algorithm, coupled with any tie-breaking rule.
Revised: Let M be the utilitarian social welfare maximizing algorithm, coupled with any tie-breaking rule.

Original: Assume that both agents report a value of 1 for the first item, and zero for the other item, i.e. b i, 1 = 1 and b i, 2 = 0 for i ∈ { 1 , 2 } .
Revised: Assume that both agents report a value of 1 for the first item, and zero for the other item, i.e. b i, 1 = 1 and b i, 2 = 0 for i ∈ { 1 , 2 } .

Original: Given these reports, there must be an agent who gets item 1 with probability at least 12 (if M is deterministic then this probability will be exactly 1 ); assume that this is agent 1 , without loss of generality.
Revised: Given these reports, there must be an agent who gets item 1 with probability at least 12 (if M is deterministic then this probability will be exactly 1 ); assume that this is agent 1 , without loss of generality.

Original: Now, consider the case when the true valuation of agent 1 is
Revised: Now, consider the case when the true valuation of agent 1 is

Original: v 1 = ( 23 + ϵ, 13 − ϵ ) , for some small ϵ > 0 .
Revised: v 1 = ( 23 + ϵ, 13 − ϵ ) , for some small ϵ > 0 .

Original: Additionally, suppose that agent 1 reports her true valuation (i.e., b 1 = v 1 ) and agent 2 ’s reported value for item 1 , b 2 , 1 > 23 + ϵ .
Revised: Additionally, suppose that agent 1 reports her true valuation (i.e., b 1 = v 1 ) and agent 2 ’s reported value for item 1 , b 2 , 1 > 23 + ϵ .

Original: In this case, the utilitarian social welfare maximizing allocation gives the first item to agent 2 .
Revised: In this case, the utilitarian social welfare maximizing allocation gives the first item to agent 2 .

Original: Therefore, the worst-case utility of agent 1 when she reports her true valuation is at most 13 − ϵ .
Revised: Therefore, the worst-case utility of agent 1 when she reports her true valuation is at most 13 − ϵ .

Original: Next, consider the dishonest report b 1 = ( b 1 , 1 , b 1 , 2 )
Revised: Next, consider the dishonest report b 1 = ( b 1 , 1 , b 1 , 2 )

Original: =
Revised: =

Original: Given this, if b 2 , 1 < 1 , then agent 1 gets the first item for a utility of 23 + ϵ .
Revised: Given this, if b 2 , 1 < 1 , then agent 1 gets the first item for a utility of 23 + ϵ .

Original: Otherwise if b 2 , 1 = 1 , then agent 1 gets the first item with probability at least 1 / 2 (this follows from our choice of agent 1 ), thus, her expected utility is at least 12 ( 23 + ϵ )
Revised: Otherwise if b 2 , 1 = 1 , then agent 1 gets the first item with probability at least 1 / 2 (this follows from our choice of agent 1 ), thus, her expected utility is at least 12 ( 23 + ϵ )

Original: = 

Revised: =

Original: + ϵ 2 .
Revised: + ϵ 2 .

Original: In either case, her utility is strictly larger than 13 − ϵ , her worst-case utility under honest reporting.
Revised: In either case, her utility is strictly larger than 13 − ϵ , her worst-case utility under honest reporting.

Original: Therefore, M is obviously manipulable.
Revised: Therefore, M is obviously manipulable.

Original: Proof of Theorem 4.
Revised: Proof of Theorem 4.

Original: We prove the theorem for the case of n = 3 agents and m = 4 items; we describe how our arguments can be adjusted to work for n > 3 agents and any number of items at the end of this proof.
Revised: We prove the theorem for the case of n = 3 agents and m = 4 items; we describe how our arguments can be adjusted to work for n > 3 agents and any number of items at the end of this proof.

Original: Let M be a (possibly randomized) mechanism that always outputs a Nash social welfare maximizing allocation.
Revised: Let M be a (possibly randomized) mechanism that always outputs a Nash social welfare maximizing allocation.

Original: Let the true valuation of agent 1 be v 1 = (3 . 9 , 3 , 2 , 0 . 9) .
Revised: Let the true valuation of agent 1 be v 1 = (3 . 9 , 3 , 2 , 0 . 9) .

Original: The subsequent proof has two parts: first, we will show that if agent 1 reports her true valuation, then the worst-case utility is exactly 2 , and second, if agent 1 reports b 1 = (2 , 2 , 1 , 1) , then the worst-case would be strictly more than 2 ; the theorem follows.
Revised: The subsequent proof has two parts: first, we will show that if agent 1 reports her true valuation, then the worst-case utility is exactly 2 , and second, if agent 1 reports b 1 = (2 , 2 , 1 , 1) , then the worst-case would be strictly more than 2 ; the theorem follows.

Original: Worst-case utility when reporting honestly: Consider the case where agent 1 reports her true valuation, i.e., the report b 1 = v 1 .
Revised: Worst-case utility when reporting honestly: Consider the case where agent 1 reports her true valuation, i.e., the report b 1 = v 1 .

Original: We know that mechanism M must output Nash social welfare maximizing allocations, and such allocations are necessarily EF1 [CKM + 19]; randomized M will output ex-post
Revised: We know that mechanism M must output Nash social welfare maximizing allocations, and such allocations are necessarily EF1 [CKM + 19]; randomized M will output ex-post

Original: EF1 allocations.
Revised: EF1 allocations.

Original: Consequently, agent 1 must be allocated at least one item, since otherwise some other agent will receive at least two items, and agent 1 will envy that agent even upon the removal of any one item.
Revised: Consequently, agent 1 must be allocated at least one item, since otherwise some other agent will receive at least two items, and agent 1 will envy that agent even upon the removal of any one item.

Original: Furthermore, if agent 1 is allocated only item 4 the overall allocation can’t be EF1, since agent 1 will envy (even upon the removal of any item) the agent that gets two of the first three items ( v 1 , 4 is smaller than all other values).
Revised: Furthermore, if agent 1 is allocated only item 4 the overall allocation can’t be EF1, since agent 1 will envy (even upon the removal of any item) the agent that gets two of the first three items ( v 1 , 4 is smaller than all other values).

Original: Thus, in the worst-case, agent 1 will get a bundle whose value is at least 2 , her value for item 3 .
Revised: Thus, in the worst-case, agent 1 will get a bundle whose value is at least 2 , her value for item 3 .

Original: Next, we show that her worst-case utility is exactly equal to 2 .
Revised: Next, we show that her worst-case utility is exactly equal to 2 .

Original: Let the reported valuations of agent 2 be b 2 = (0 , 1 , 0 , 0) and of agent 3 be b 3 = (2 , 0 , 0 , 1) .
Revised: Let the reported valuations of agent 2 be b 2 = (0 , 1 , 0 , 0) and of agent 3 be b 3 = (2 , 0 , 0 , 1) .

Original: Given this, the Nash welfare maximizing allocation is unique, and this allocation is such that agent 1 gets only item 3 , for a total utility of 2 .
Revised: Given this, the Nash welfare maximizing allocation is unique, and this allocation is such that agent 1 gets only item 3 , for a total utility of 2 .

Original: To see that this allocation is unique, first notice that item 2 must go to agent 2 (otherwise agent 2 ’s utility, and therefore Nash social welfare, will be zero).
Revised: To see that this allocation is unique, first notice that item 2 must go to agent 2 (otherwise agent 2 ’s utility, and therefore Nash social welfare, will be zero).

Original: Additionally, item 3 must go to agent 1 since she is the only agent with a non-zero value for it.
Revised: Additionally, item 3 must go to agent 1 since she is the only agent with a non-zero value for it.

Original: The remaining items ( 1 and 4 ), which are valued only by agents 1 and 3 , must be allocated in a way that the Nash social welfare of the resultant allocation is maximized.
Revised: The remaining items ( 1 and 4 ), which are valued only by agents 1 and 3 , must be allocated in a way that the Nash social welfare of the resultant allocation is maximized.

Original: A simple case analysis shows that the unique allocation that maximizes Nash social welfare gives both items to agent 3 .
Revised: A simple case analysis shows that the unique allocation that maximizes Nash social welfare gives both items to agent 3 .

Original: A consequence of having a unique Nash social welfare maximizing allocation is that mechanism M must output it, irrespective of whether M is randomized or deterministic.
Revised: A consequence of having a unique Nash social welfare maximizing allocation is that mechanism M must output it, irrespective of whether M is randomized or deterministic.

Original: Worst-case utility when misreporting: Consider the case where agent 1 misreports her valuation as b 1 = (2 , 2 , 1 , 1) .
Revised: Worst-case utility when misreporting: Consider the case where agent 1 misreports her valuation as b 1 = (2 , 2 , 1 , 1) .

Original: Given this report, the mechanism M must allocate at least one item to agent 1 ; otherwise its allocation would not be EF1 (ex-post EF1 for if M is randomized).
Revised: Given this report, the mechanism M must allocate at least one item to agent 1 ; otherwise its allocation would not be EF1 (ex-post EF1 for if M is randomized).

Original: We will show that agent 1 ’s allocation cannot be only item 3 nor only item 4 (i.e., a bundle of value 1 with respect to the reported values), irrespective of the reports of agents 2 and agent 3 .
Revised: We will show that agent 1 ’s allocation cannot be only item 3 nor only item 4 (i.e., a bundle of value 1 with respect to the reported values), irrespective of the reports of agents 2 and agent 3 .

Original: Specifically, agent 1 must be allocated either ( i ) at least two items, ( ii ) only item 1 , or ( iii ) only item 2 .
Revised: Specifically, agent 1 must be allocated either ( i ) at least two items, ( ii ) only item 1 , or ( iii ) only item 2 .

Original: In each of these cases, the value of the items allocated to agent 1 (as per her true valuation) will be strictly more than 2 , which completes the proof.
Revised: In each of these cases, the value of the items allocated to agent 1 (as per her true valuation) will be strictly more than 2 , which completes the proof.

Original: We proceed to show that agent 1 cannot be allocated just item 3 or just item 4 in any Nash social welfare maximizing allocation.
Revised: We proceed to show that agent 1 cannot be allocated just item 3 or just item 4 in any Nash social welfare maximizing allocation.

Original: Towards a contradiction, assume that in a Nash social welfare maximizing allocation A , agent 1 is allocated either only item 3 or only item 4 .
Revised: Towards a contradiction, assume that in a Nash social welfare maximizing allocation A , agent 1 is allocated either only item 3 or only item 4 .

Original: Since agent 1 is allocated only one item, there must be some other agent who is allocated at least two items.
Revised: Since agent 1 is allocated only one item, there must be some other agent who is allocated at least two items.

Original: We consider the following exhaustive cases based on the items allocated to this other agent.
Revised: We consider the following exhaustive cases based on the items allocated to this other agent.

Original: Case I: There is an agent j ∈ { 2 , 3 } having both items 1 and 2 .
Revised: Case I: There is an agent j ∈ { 2 , 3 } having both items 1 and 2 .

Original: In this case, the allocation A is not
Revised: In this case, the allocation A is not

Original: EF1 — and hence not maximizing Nash social welfare — since agent 1 envies agent j (with respect to the reported valuation b 1 ), even upon the removal of one item.
Revised: EF1 — and hence not maximizing Nash social welfare — since agent 1 envies agent j (with respect to the reported valuation b 1 ), even upon the removal of one item.

Original: Case II: There is an agent j ∈ { 2 , 3 } who gets one item from the set of items { 1 , 2 } and one item from { 3 , 4 } .
Revised: Case II: There is an agent j ∈ { 2 , 3 } who gets one item from the set of items { 1 , 2 } and one item from { 3 , 4 } .

Original: Without loss of generality assume that agent j is allocated items 2 and 3 , and agent 1 gets item 4 .
Revised: Without loss of generality assume that agent j is allocated items 2 and 3 , and agent 1 gets item 4 .

Original: We will show that in this case A can never be a Nash social welfare maximizing allocation.
Revised: We will show that in this case A can never be a Nash social welfare maximizing allocation.

Original: Since allocation A maximizes Nash social welfare, it must be that transferring item 2 from agent j to agent 1 does not increase the Nash social welfare, i.e., the following inequalities must hold:
Revised: Since allocation A maximizes Nash social welfare, it must be that transferring item 2 from agent j to agent 1 does not increase the Nash social welfare, i.e., the following inequalities must hold:

Original: [Equation]
Revised: [Equation]

Original: (substituting b 1 , 3 = b 1 , 4 = 1 )
Revised: (substituting b 1 , 3 = b 1 , 4 = 1 )

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

Original: Similarly, transferring item 3 from agent j to agent 1 must also not increase the Nash social welfare:
Revised: Similarly, transferring item 3 from agent j to agent 1 must also not increase the Nash social welfare:

Original: [Equation]
Revised: [Equation]

Original: substituting b 1 , 2 = 2 and b 1 , 4 = 1 )
Revised: substituting b 1 , 2 = 2 and b 1 , 4 = 1 )

Original: [Equation]
Revised: [Equation]

Original: Combining inequalities (1) and (2) we have b j, 3 ≥ 2 b j, 3 , which can be true only if b j, 3 = 0 .
Revised: Combining inequalities (1) and (2) we have b j, 3 ≥ 2 b j, 3 , which can be true only if b j, 3 = 0 .

Original: However, since A allocates item 3 to agent j , and b j, 3 = 0 , but b 1 , 3 > 0 , A is not Nash social welfare maximizing (in fact, not even Pareto efficient).
Revised: However, since A allocates item 3 to agent j , and b j, 3 = 0 , but b 1 , 3 > 0 , A is not Nash social welfare maximizing (in fact, not even Pareto efficient).

Original: The same argument can be extended to instances having n > 3 agents and m = n + 1 items by considering the case wherein agent 1 ’s true valuation, v 1 = (3 . 9 , 3 , 3 , . .
Revised: The same argument can be extended to instances having n > 3 agents and m = n + 1 items by considering the case wherein agent 1 ’s true valuation, v 1 = (3 . 9 , 3 , 3 , . .

Original: Here, the misreported valuation which leads to an improvement in her (expected) worst-case utility is b 1 = (2 , 2 , 2 , . . .
Revised: Here, the misreported valuation which leads to an improvement in her (expected) worst-case utility is b 1 = (2 , 2 , 2 , . . .

Original: C.1 Egalitarian Social Welfare
Revised: C.1 Egalitarian Social Welfare

Original: In this section we will show that any mechanism — randomized or deterministic — that maximizes egalitarian social welfare is obviously manipulable.
Revised: In this section we will show that any mechanism — randomized or deterministic — that maximizes egalitarian social welfare is obviously manipulable.

Original: The egalitarian social welfare of an (integral or fractional) allocation X , denoted as ESW ( X ) , is defined as the minimum utility that any agent derives from allocation X , i.e., ESW ( X ) =
Revised: The egalitarian social welfare of an (integral or fractional) allocation X , denoted as ESW ( X ) , is defined as the minimum utility that any agent derives from allocation X , i.e., ESW ( X ) =

Original: min i ∈ N v i ( x i ) .
Revised: min i ∈ N v i ( x i ) .

Original: An integral allocation is egalitarian social welfare maximizing iff ( i ) it maximizes, among the set of all integral allocations, the number of agents having positive utility and ( ii ) for any such maximal set of agents S , it maximizes the egalitarian social welfare, i.e., the minimum utility of agents in S .
Revised: An integral allocation is egalitarian social welfare maximizing iff ( i ) it maximizes, among the set of all integral allocations, the number of agents having positive utility and ( ii ) for any such maximal set of agents S , it maximizes the egalitarian social welfare, i.e., the minimum utility of agents in S .

Original: Our result rules out the existence of NOM mechanisms that output leximin allocations, 10 since leximin allocations are, by definition, utilitarian social welfare maximizing.
Revised: Our result rules out the existence of NOM mechanisms that output leximin allocations, 10 since leximin allocations are, by definition, utilitarian social welfare maximizing.

Original: Similar to utilitarian social welfare, when discussing egalitarian social welfare, we will assume that the valuations of agents are normalized: for every agent, the combined value for the set of all items is 1 ; see [AFRC + 16] for a thorough discussion.
Revised: Similar to utilitarian social welfare, when discussing egalitarian social welfare, we will assume that the valuations of agents are normalized: for every agent, the combined value for the set of all items is 1 ; see [AFRC + 16] for a thorough discussion.

Original: For the case of n = 2 agents, there are no NOM mechanisms that maximize egalitarian social welfare for that case; this follows from Theorem 10 in Appendix
Revised: For the case of n = 2 agents, there are no NOM mechanisms that maximize egalitarian social welfare for that case; this follows from Theorem 10 in Appendix

Original: E.
Revised: E.

Original: The following theorem establishes that this impossibility continues to hold for the case of n ≥ 3 agents.
Revised: The following theorem establishes that this impossibility continues to hold for the case of n ≥ 3 agents.

Original: Theorem 8.
Revised: Theorem 8.

Original: Every (randomized or deterministic) mechanism that always outputs an allocation that maximizes the egalitarian social welfare is obviously manipulable, even for n = 3 agents and m =items.
Revised: Every (randomized or deterministic) mechanism that always outputs an allocation that maximizes the egalitarian social welfare is obviously manipulable, even for n = 3 agents and m =items.

Original: Proof of Theorem 8.
Revised: Proof of Theorem 8.

Original: Consider any (randomized or deterministic) mechanism M that maximizes the minimum utility among agents.
Revised: Consider any (randomized or deterministic) mechanism M that maximizes the minimum utility among agents.

Original: To show that M is obviously manipulable, we consider an instance with n = 3 agents and m = 4 items.
Revised: To show that M is obviously manipulable, we consider an instance with n = 3 agents and m = 4 items.

Original: Let the true (normalized) valuation of agent 1 be v 1 = (0 . 3 , 0 . 3 , 0 . 3 , 0 . 1) .
Revised: Let the true (normalized) valuation of agent 1 be v 1 = (0 . 3 , 0 . 3 , 0 . 3 , 0 . 1) .

Original: First, we will show that if agent 1 reports honestly, then the worst-case case outcome is that she is allocated only item 4 , i.e., her utility in the worst-case is 0 .
Revised: First, we will show that if agent 1 reports honestly, then the worst-case case outcome is that she is allocated only item 4 , i.e., her utility in the worst-case is 0 .

Original: Note that since M maximizes the number of agents with positive utility, and agent 1 has a positive value for all 4 items, it must allocate at least one item to agent 1 ; otherwise either agent 2 or 3 has more than one item, one of which could be transferred to agent 1 to make her utility positive.
Revised: Note that since M maximizes the number of agents with positive utility, and agent 1 has a positive value for all 4 items, it must allocate at least one item to agent 1 ; otherwise either agent 2 or 3 has more than one item, one of which could be transferred to agent 1 to make her utility positive.

Original: Now, consider the case where the reports of other agents are b 2 = (0 , 0 , 1 , 0) and b 3 = (0 . 05 , 0 . 05 , 0 . 9 , 0) .
Revised: Now, consider the case where the reports of other agents are b 2 = (0 , 0 , 1 , 0) and b 3 = (0 . 05 , 0 . 05 , 0 . 9 , 0) .

Original: Here, item 3 must be allocated to agent 2 because it is the only item that she desires.
Revised: Here, item 3 must be allocated to agent 2 because it is the only item that she desires.

Original: By a straightforward case analysis, the unique way in which we can allocate the remaining items (items 1 , 2 and 4 ) to maximize the minimum utility is by allocating item 4 to agent 1 and items 1 and 2 to agent 3 .
Revised: By a straightforward case analysis, the unique way in which we can allocate the remaining items (items 1 , 2 and 4 ) to maximize the minimum utility is by allocating item 4 to agent 1 and items 1 and 2 to agent 3 .

Original: Since this allocation is unique, it must be the output of M , irrespective of whether it is deterministic or randomized.
Revised: Since this allocation is unique, it must be the output of M , irrespective of whether it is deterministic or randomized.

Original: Therefore, the utility of agent 1 is 0 .
Revised: Therefore, the utility of agent 1 is 0 .

Original: 1 under honest reporting, in the worst case.
Revised: 1 under honest reporting, in the worst case.

Original: Now, consider the case where agent 1 reports b 1 = ( 13 , 13 , 13 , 0) .
Revised: Now, consider the case where agent 1 reports b 1 = ( 13 , 13 , 13 , 0) .

Original: Since M maximizes the number of agents having non-zero utility, and agent 1 positively values the first three items, she must get at least one of them: otherwise some other agent will have at least two of them, and transferring one of them to agent 1 would strictly increase the number of agents with non-zero utility.
Revised: Since M maximizes the number of agents having non-zero utility, and agent 1 positively values the first three items, she must get at least one of them: otherwise some other agent will have at least two of them, and transferring one of them to agent 1 would strictly increase the number of agents with non-zero utility.

Original: Hence, the worst-case utility of agent 1 when reporting b 1 but her true values are v 1 is 0 .
Revised: Hence, the worst-case utility of agent 1 when reporting b 1 but her true values are v 1 is 0 .

Original: 3 , her true value for each of the first three items.
Revised: 3 , her true value for each of the first three items.

Original: Since the worst-case utility improved, M is obviously manipulable.
Revised: Since the worst-case utility improved, M is obviously manipulable.

Original: The same construction can be extended to n > 3 agents and m = n + 1 items by considering the case wherein agent 1 ’s true valuation v 1 = ( 1 n − ϵ, 1 n − ϵ, 1 n − ϵ, . .
Revised: The same construction can be extended to n > 3 agents and m = n + 1 items by considering the case wherein agent 1 ’s true valuation v 1 = ( 1 n − ϵ, 1 n − ϵ, 1 n − ϵ, . .

Original: , 1 n − ϵ, n.ϵ ) for any positive constant ϵ < 1 n ( n +1) , and she misreports to b 1 = ( 1 n , 1 n , 1 n , . . .
Revised: , 1 n − ϵ, n.ϵ ) for any positive constant ϵ < 1 n ( n +1) , and she misreports to b 1 = ( 1 n , 1 n , 1 n , . . .

Original: , 1 n , 0) leading to an improvement in her worst-case utility.
Revised: , 1 n , 0) leading to an improvement in her worst-case utility.

Original: D Proofs missing from SectionD.1 Missing proofs
Revised: D Proofs missing from SectionD.1 Missing proofs

Original: Proof of Lemma 1.
Revised: Proof of Lemma 1.

Original: First, note that Line 24 (making the temporary assignment clean) does not affect the utility of any agent i for her own bundle.
Revised: First, note that Line 24 (making the temporary assignment clean) does not affect the utility of any agent i for her own bundle.

Original: Therefore if the allocation before this step was α -PO (respectively α -fPO) then after Line 24 these efficiency guarantees will continue to hold.
Revised: Therefore if the allocation before this step was α -PO (respectively α -fPO) then after Line 24 these efficiency guarantees will continue to hold.

Original: Considering all possible cases for Mechanism 1, if all agents j ∈ N receive the bundle D j (as in case I), the allocation is fractionally Pareto efficient.
Revised: Considering all possible cases for Mechanism 1, if all agents j ∈ N receive the bundle D j (as in case I), the allocation is fractionally Pareto efficient.

Original: If all agents j receive the bundle D j , with the exception of a single agent i that receives (cid:98) D i , then all agents except i have the maximum possible utility, i.e. their utility for all items M .
Revised: If all agents j receive the bundle D j , with the exception of a single agent i that receives (cid:98) D i , then all agents except i have the maximum possible utility, i.e. their utility for all items M .

Original: Furthermore, it is impossible to improve the utility of i without decreasing the utility of some other agent j ; this follows from the definition of (cid:98) D i and D j .
Revised: Furthermore, it is impossible to improve the utility of i without decreasing the utility of some other agent j ; this follows from the definition of (cid:98) D i and D j .

Original: Therefore, this allocation is fractionally Pareto efficient as well.
Revised: Therefore, this allocation is fractionally Pareto efficient as well.

Original: This allocation is considered in cases II and III (if the black-box algorithm M ∗ is not called), and it is the output of Mechanism 1 if it is also EF1 .
Revised: This allocation is considered in cases II and III (if the black-box algorithm M ∗ is not called), and it is the output of Mechanism 1 if it is also EF1 .

Original: In the remaining cases (if the aforementioned allocation is not EF1 , or we are in case IV)
Revised: In the remaining cases (if the aforementioned allocation is not EF1 , or we are in case IV)

Original: Mechanismreturns the allocation computed by the black-box algorithm M ∗ (modulo Line 24), which is an α -PO (respectively α -fPO) partial allocation by definition.
Revised: Mechanismreturns the allocation computed by the black-box algorithm M ∗ (modulo Line 24), which is an α -PO (respectively α -fPO) partial allocation by definition.

Original: Proof of Lemma 2.
Revised: Proof of Lemma 2.

Original: Line 24 does not affect the utility of any agent for her own bundle, and since valuations are additive, removal of items could only decrease her utility for the bundle of a different agent.
Revised: Line 24 does not affect the utility of any agent for her own bundle, and since valuations are additive, removal of items could only decrease her utility for the bundle of a different agent.

Original: Therefore if the allocation before this step was EF1 , it remains EF1 afterwards.
Revised: Therefore if the allocation before this step was EF1 , it remains EF1 afterwards.

Original: This step also ensures that the allocation is clean.
Revised: This step also ensures that the allocation is clean.

Original: Next, considering all possible cases, we first have that if every agent j ∈ N receives the bundle D j (as in case I), the allocation is envy-free, and therefore EF1 .
Revised: Next, considering all possible cases, we first have that if every agent j ∈ N receives the bundle D j (as in case I), the allocation is envy-free, and therefore EF1 .

Original: In cases II and III, Mechanism 1 either checks whether an allocation is EF1 before outputting it, or calls M ∗ .
Revised: In cases II and III, Mechanism 1 either checks whether an allocation is EF1 before outputting it, or calls M ∗ .

Original: In case IV M ∗ is called.
Revised: In case IV M ∗ is called.

Original: Whenever M ∗ is called, then the output allocation is EF1 by definition.
Revised: Whenever M ∗ is called, then the output allocation is EF1 by definition.

Original: Finally, note that Mechanism 1 returns non-wasteful allocations: it is easy to confirm that whenever
Revised: Finally, note that Mechanism 1 returns non-wasteful allocations: it is easy to confirm that whenever

Original: Mechanism 1 doesn’t call M ∗ its allocation is non-wasteful, while if M ∗ is called, the allocation of
Revised: Mechanism 1 doesn’t call M ∗ its allocation is non-wasteful, while if M ∗ is called, the allocation of

Original: M ∗ the returned allocation is non-wasteful and clean, so it remains non-wasteful after Line 24.
Revised: M ∗ the returned allocation is non-wasteful and clean, so it remains non-wasteful after Line 24.

Original: Proof of Lemma 3.
Revised: Proof of Lemma 3.

Original: Given the valuation vector v i and an allocation A =
Revised: Given the valuation vector v i and an allocation A =

Original: ( A 1 , A 2 , . . .
Revised: ( A 1 , A 2 , . . .

Original: , A n ) ∈
Revised: , A n ) ∈

Original: EF1( i, v i ) we will construct valuations for the other agents, v − i , such that Mechanism 1 outputs A on input ( v i , v − i ) .
Revised: EF1( i, v i ) we will construct valuations for the other agents, v − i , such that Mechanism 1 outputs A on input ( v i , v − i ) .

Original: The valuation of each agent j ∈ (N \{ i } ) will be such that v j,g = 0 for all g / ∈
Revised: The valuation of each agent j ∈ (N \{ i } ) will be such that v j,g = 0 for all g / ∈

Original: A j , and v j,g > 0 for each good g ∈
Revised: A j , and v j,g > 0 for each good g ∈

Original: A j (the precise value of v j,g will depend on how D i intersects with A j ).
Revised: A j (the precise value of v j,g will depend on how D i intersects with A j ).

Original: That is, for each agent j ∈ (N \ { i } ) , D j = A j .
Revised: That is, for each agent j ∈ (N \ { i } ) , D j = A j .

Original: Therefore, by construction, the subsets { D j } nj =1 \ { D i } are pairwise disjoint, and therefore R i = 1 (so, we are never in case IV of Mechanism 1).
Revised: Therefore, by construction, the subsets { D j } nj =1 \ { D i } are pairwise disjoint, and therefore R i = 1 (so, we are never in case IV of Mechanism 1).

Original: Also, by construction, Line 24 will not affect any bundle.
Revised: Also, by construction, Line 24 will not affect any bundle.

Original: In the rest of the proof, we consider the three exhaustive cases, based on how many sets from A − i =
Revised: In the rest of the proof, we consider the three exhaustive cases, based on how many sets from A − i =

Original: ( A 1 , . . . ,
Revised: ( A 1 , . . . ,

Original: A i − 1 , A i +1 , . . .
Revised: A i − 1 , A i +1 , . . .

Original: , A n ) the set D i intersects.
Revised: , A n ) the set D i intersects.

Original: The first case is when D i does not intersect any bundle from A − i .
Revised: The first case is when D i does not intersect any bundle from A − i .

Original: Since v i,g = 0 for every unallocated good g (by the definition of EF1( i, v i ) ), D i ⊆
Revised: Since v i,g = 0 for every unallocated good g (by the definition of EF1( i, v i ) ), D i ⊆

Original: A i , and therefore R j = 1 for all j ∈ N .
Revised: A i , and therefore R j = 1 for all j ∈ N .

Original: In this case we can set v j,g to be an arbitrary value for all g ∈
Revised: In this case we can set v j,g to be an arbitrary value for all g ∈

Original: A j , e.g. v j,g = 1 , for all agents j ̸ = i .
Revised: A j , e.g. v j,g = 1 , for all agents j ̸ = i .

Original: Therefore,
Revised: Therefore,

Original: Mechanism 1 considers Case I, and it allocates the set D j to every agent j .
Revised: Mechanism 1 considers Case I, and it allocates the set D j to every agent j .

Original: By construction, for all j ̸
Revised: By construction, for all j ̸

Original: = i , D j
Revised: = i , D j

Original: =
Revised: =

Original: A j .
Revised: A j .

Original: Additionally, for agent i , it must be that the bundle
Revised: Additionally, for agent i , it must be that the bundle

Original: A i =
Revised: A i =

Original: D i , since A i is also clean (by Line 24).
Revised: D i , since A i is also clean (by Line 24).

Original: Therefore, the output of Mechanism 1 is precisely A .
Revised: Therefore, the output of Mechanism 1 is precisely A .

Original: The second case is that D i intersects exactly one bundle from A − i .
Revised: The second case is that D i intersects exactly one bundle from A − i .

Original: Let j ∗ ∈
Revised: Let j ∗ ∈

Original: (N \{ i } ) be the (unique) agent such that A j ∗ ∩ D i ̸ = ∅ .
Revised: (N \{ i } ) be the (unique) agent such that A j ∗ ∩ D i ̸ = ∅ .

Original: Additionally, assume that i < j ∗ ; an almost identical argument works if i > j ∗ .
Revised: Additionally, assume that i < j ∗ ; an almost identical argument works if i > j ∗ .

Original: For each agent k ∈ N \ { i, j ∗ } , we define v k,g = 1 if good g ∈
Revised: For each agent k ∈ N \ { i, j ∗ } , we define v k,g = 1 if good g ∈

Original: A k and v k,g = 0 otherwise.
Revised: A k and v k,g = 0 otherwise.

Original: The valuations of agent j ∗ are as follows
Revised: The valuations of agent j ∗ are as follows

Original: [Equation]
Revised: [Equation]

Original: Note that, the set of desired goods D k
Revised: Note that, the set of desired goods D k

Original: = A k for all agents k ∈ (N \ { i } ) , and (cid:98) D i ⊇ A i .
Revised: = A k for all agents k ∈ (N \ { i } ) , and (cid:98) D i ⊇ A i .

Original: Furthermore, since A is non-wasteful (by the definition of EF1( i, v i ) ), goods in (cid:98) D i \
Revised: Furthermore, since A is non-wasteful (by the definition of EF1( i, v i ) ), goods in (cid:98) D i \

Original: A i must have zero value for i .
Revised: A i must have zero value for i .

Original: Therefore, v i ( (cid:98) D i )
Revised: Therefore, v i ( (cid:98) D i )

Original: =
Revised: =

Original: v i ( A i ) .
Revised: v i ( A i ) .

Original: The construction satisfies that the bundles { D k } k ∈ N \{ i } are pairwise disjoint, and the bundle D i only intersects D j ∗ =
Revised: The construction satisfies that the bundles { D k } k ∈ N \{ i } are pairwise disjoint, and the bundle D i only intersects D j ∗ =

Original: A j ∗ .
Revised: A j ∗ .

Original: Therefore, R i =
Revised: Therefore, R i =

Original: R j ∗ = 1 and
Revised: R j ∗ = 1 and

Original: R k = 0 for every other agent k .
Revised: R k = 0 for every other agent k .

Original: Therefore, Mechanism 1, given valuations ( v i , v − i ) , considers
Revised: Therefore, Mechanism 1, given valuations ( v i , v − i ) , considers

Original: Case III (Lines 11-21) to compute the final allocation.
Revised: Case III (Lines 11-21) to compute the final allocation.

Original: In Case III, the mechanism first checks whether v i ( D i ∩ D j ∗ ) <
Revised: In Case III, the mechanism first checks whether v i ( D i ∩ D j ∗ ) <

Original: v j
Revised: v j

Original: ( D i ∩ D j ∗ ) holds, which is true for our construction since v j ∗ ,g = 

Revised: ( D i ∩ D j ∗ ) holds, which is true for our construction since v j ∗ ,g =

Original: v i
Revised: v i

Original: ( D i ∩ D j ∗ ) for all g in A j ∗ ∩ D i =
Revised: ( D i ∩ D j ∗ ) for all g in A j ∗ ∩ D i =

Original: D j ∗ ∩ D i .
Revised: D j ∗ ∩ D i .

Original: Afterwards, the mechanism checks if the allocation ( D 1 , . . .
Revised: Afterwards, the mechanism checks if the allocation ( D 1 , . . .

Original: , (cid:98) D i , . .
Revised: , (cid:98) D i , . .

Original: , D j ∗ , . . .
Revised: , D j ∗ , . . .

Original: , D n ) is EF1 (Line 13).
Revised: , D n ) is EF1 (Line 13).

Original: Since v i ( (cid:98) D i )
Revised: Since v i ( (cid:98) D i )

Original: = v i ( A i ) , and A ∈ EF1( i, v i ) , this allocation is indeed EF1 for agent i , and since all other agents are envy-free, Mechanism 1 sets it as the temporary assignment.
Revised: = v i ( A i ) , and A ∈ EF1( i, v i ) , this allocation is indeed EF1 for agent i , and since all other agents are envy-free, Mechanism 1 sets it as the temporary assignment.

Original: After removing all zero valued items from (cid:98) D i (Line 24) we are left with the bundle
Revised: After removing all zero valued items from (cid:98) D i (Line 24) we are left with the bundle

Original: A i , hence, the final allocation is exactly A .
Revised: A i , hence, the final allocation is exactly A .

Original: The third and final case is that D i intersects more than one bundle from A − i .
Revised: The third and final case is that D i intersects more than one bundle from A − i .

Original: Here, the valuation vector of every agent j ∈ N \ { i } is such that v j,g = 1 for g ∈
Revised: Here, the valuation vector of every agent j ∈ N \ { i } is such that v j,g = 1 for g ∈

Original: A j and v j,g = 0 otherwise.
Revised: A j and v j,g = 0 otherwise.

Original: By construction, the set of desired goods D j =
Revised: By construction, the set of desired goods D j =

Original: A j for every agent j ∈ N \ { i } , (cid:98) D i ⊇ A i , and since
Revised: A j for every agent j ∈ N \ { i } , (cid:98) D i ⊇ A i , and since

Original: A is non-wasteful, we have v i ( (cid:98) D i )
Revised: A is non-wasteful, we have v i ( (cid:98) D i )

Original: = v i ( A i ) .
Revised: = v i ( A i ) .

Original: D i intersecting more than one bundle from A − i , is equivalent to D i intersecting more than one subsets from { D j } j ̸
Revised: D i intersecting more than one bundle from A − i , is equivalent to D i intersecting more than one subsets from { D j } j ̸

Original: = i , and therefore R j = 0 for all j ∈ N \{ i } .
Revised: = i , and therefore R j = 0 for all j ∈ N \{ i } .

Original: Thus, given the valuation profile ( v i , v − i ) , Mechanism 1 considers Case II (Lines 6-10), where it first checks whether the allocation ( D 1 , D 2 , . . .
Revised: Thus, given the valuation profile ( v i , v − i ) , Mechanism 1 considers Case II (Lines 6-10), where it first checks whether the allocation ( D 1 , D 2 , . . .

Original: , (cid:98) D i , . .
Revised: , (cid:98) D i , . .

Original: , D n ) is EF1 .
Revised: , D n ) is EF1 .

Original: This allocation has the same value as A for agent i (and all other agents have exactly the same allocation), so agent i has an envy of at most one item by definition, and all other agents have no envy at all.
Revised: This allocation has the same value as A for agent i (and all other agents have exactly the same allocation), so agent i has an envy of at most one item by definition, and all other agents have no envy at all.

Original: Therefore,
Revised: Therefore,

Original: Mechanism 1 sets this as the temporary assignment.
Revised: Mechanism 1 sets this as the temporary assignment.

Original: After removing all zero valued items from (cid:98) D i (Line 24) the final allocation is exactly A .
Revised: After removing all zero valued items from (cid:98) D i (Line 24) the final allocation is exactly A .

Original: This concludes the proof of Lemma 3.
Revised: This concludes the proof of Lemma 3.

Original: D.2 Pseudo-code for MechanismREDUCTION
Revised: D.2 Pseudo-code for MechanismREDUCTION

Original: : Black-box reduction
Revised: : Black-box reduction

Original: Input: Reported valuation agents algorithm M
Revised: Input: Reported valuation agents algorithm M

Original: Output: A partial integral allocation
Revised: Output: A partial integral allocation

Original: A
Revised: A

Original: ( A
Revised: ( A

Original: . . , A n ) 1: Set D i ← { g ∈ M | b i,g > 0 } .
Revised: . . , A n ) 1: Set D i ← { g ∈ M | b i,g > 0 } .

Original: 2: Set (cid:98) D i ← M \ ∪ j ̸ = i
Revised: 2: Set (cid:98) D i ← M \ ∪ j ̸ = i

Original: D j for each i ∈ N .
Revised: D j for each i ∈ N .

Original: 3: Set R i ← 1 if subsets { D j } nj =1 \ { D i } are pairwise disjoint; 0 otherwise.
Revised: 3: Set R i ← 1 if subsets { D j } nj =1 \ { D i } are pairwise disjoint; 0 otherwise.

Original: 4: if the subsets { D j } nj =1 are pairwise disjoint then 5:
Revised: 4: if the subsets { D j } nj =1 are pairwise disjoint then 5:

Original: A ∗ ← ( D 1 , . . .
Revised: A ∗ ← ( D 1 , . . .

Original: , D n ) 6: else if ∃ i ∈ N such that R i
Revised: , D n ) 6: else if ∃ i ∈ N such that R i

Original: = 1 and R j = 0 for every j ∈ N \ { i } then 7: if the allocation ( D 1 , D 2 , . .
Revised: = 1 and R j = 0 for every j ∈ N \ { i } then 7: if the allocation ( D 1 , D 2 , . .

Original: , (cid:98) D i , . .
Revised: , (cid:98) D i , . .

Original: , D n ) is EF1 then 8:
Revised: , D n ) is EF1 then 8:

Original: A ∗ ← ( D 1 , D 2 , . . .
Revised: A ∗ ← ( D 1 , D 2 , . . .

Original: , (cid:98) D i , . .
Revised: , (cid:98) D i , . .

Original: , D n ) 9: else 10:
Revised: , D n ) 9: else 10:

Original: A ∗ ← M ∗ ( b 1 , b 2 , . . .
Revised: A ∗ ← M ∗ ( b 1 , b 2 , . . .

Original: , b n ) 11: else if ∃ i, j ∈ N such that i <
Revised: , b n ) 11: else if ∃ i, j ∈ N such that i <

Original: j and R i
Revised: j and R i

Original: =
Revised: =

Original: R j = 1 , and R k = 0 for k ∈ [ n ] \ { i, j } then 12: if b i ( D
Revised: R j = 1 , and R k = 0 for k ∈ [ n ] \ { i, j } then 12: if b i ( D

Original: i ∩ D j ) <
Revised: i ∩ D j ) <

Original: b j ( D i ∩ D j ) then 13: if ( D 1 , . . .
Revised: b j ( D i ∩ D j ) then 13: if ( D 1 , . . .

Original: , (cid:98) D i , . .
Revised: , (cid:98) D i , . .

Original: , D j , . . .
Revised: , D j , . . .

Original: , D n ) is EF1 then 14:
Revised: , D n ) is EF1 then 14:

Original: A ∗ ← ( D 1 , . . .
Revised: A ∗ ← ( D 1 , . . .

Original: , (cid:98) D i , . .
Revised: , (cid:98) D i , . .

Original: , D j , . . .
Revised: , D j , . . .

Original: , D n ) 15: else 16:
Revised: , D n ) 15: else 16:

Original: A ∗ ← M ∗ ( b 1 , b 2 , . . . , b n ) 17: else 18: if ( D 1 , . . .
Revised: A ∗ ← M ∗ ( b 1 , b 2 , . . . , b n ) 17: else 18: if ( D 1 , . . .

Original: , D i , . . .
Revised: , D i , . . .

Original: , (cid:98) D j , . .
Revised: , (cid:98) D j , . .

Original: , D n ) is EF1 then 19:
Revised: , D n ) is EF1 then 19:

Original: A ∗ ← ( D 1 , . . .
Revised: A ∗ ← ( D 1 , . . .

Original: , D i , . . .
Revised: , D i , . . .

Original: , (cid:98) D j , . .
Revised: , (cid:98) D j , . .

Original: , D n ) 20: else 21:
Revised: , D n ) 20: else 21:

Original: A ∗ ← M ∗ ( b 1 , b 2 , . . .
Revised: A ∗ ← M ∗ ( b 1 , b 2 , . . .

Original: , b n ) 22: else 23:
Revised: , b n ) 22: else 23:

Original: A ∗ ← M ∗ ( b 1 , b 2 , . . .
Revised: A ∗ ← M ∗ ( b 1 , b 2 , . . .

Original: , b n ) 24:
Revised: , b n ) 24:

Original: For all i ∈ N , iteratively remove goods g ∈
Revised: For all i ∈ N , iteratively remove goods g ∈

Original: A ∗ i such that b i ( A ∗ i \ { g } ) =
Revised: A ∗ i such that b i ( A ∗ i \ { g } ) =

Original: b i ( A ∗ i ) .
Revised: b i ( A ∗ i ) .

Original: 25: return A ∗
Revised: 25: return A ∗

Original: Case I
Revised: Case I

Original: Case II
Revised: Case II

Original: Case III
Revised: Case III

Original: Case IV
Revised: Case IV

Original: D.3 Ex-ante EF, Ex-post fPO, Ex-post EF1 and NOM
Revised: D.3 Ex-ante EF, Ex-post fPO, Ex-post EF1 and NOM

Original: In the previous section we established (Application 1) the existence of a deterministic NOM mechanism that outputs ex-post fPO and EF1 allocations for additive agents.
Revised: In the previous section we established (Application 1) the existence of a deterministic NOM mechanism that outputs ex-post fPO and EF1 allocations for additive agents.

Original: Here we make a short remark that it is not possible to improve upon this result by adding ex-ante fairness guarantees.
Revised: Here we make a short remark that it is not possible to improve upon this result by adding ex-ante fairness guarantees.

Original: Specifically, there does not exist a randomized mechanism that is NOM in expectation and outputs ex-post fPO and EF1 allocations, that is additionally ex-ante envy-free.
Revised: Specifically, there does not exist a randomized mechanism that is NOM in expectation and outputs ex-post fPO and EF1 allocations, that is additionally ex-ante envy-free.

Original: This follows directly from the following impossibility result of [FSV20].
Revised: This follows directly from the following impossibility result of [FSV20].

Original: Theorem 9 ([FSV20]) .
Revised: Theorem 9 ([FSV20]) .

Original: There exists instances with additive valuation where there is no randomized allocation that is simultaneously ex-post fPO and EF1 , and ex-ante envy-free.
Revised: There exists instances with additive valuation where there is no randomized allocation that is simultaneously ex-post fPO and EF1 , and ex-ante envy-free.

Original: E Best of Both Worlds
Revised: E Best of Both Worlds

Original: In the previous sections we saw how fairness guarantees can be turned into NOM guarantees.
Revised: In the previous sections we saw how fairness guarantees can be turned into NOM guarantees.

Original: For example, Lemma 5 says that the worst-case guarantee of NOM is satisfied for ex-ante proportional algorithms, while Theorem 5 turns any EF1 algorithm into a NOM (plus EF1 ) mechanism.
Revised: For example, Lemma 5 says that the worst-case guarantee of NOM is satisfied for ex-ante proportional algorithms, while Theorem 5 turns any EF1 algorithm into a NOM (plus EF1 ) mechanism.

Original: In this section we show that this connection can be exploited in the other direction to prove impossibility results for fair algorithms.
Revised: In this section we show that this connection can be exploited in the other direction to prove impossibility results for fair algorithms.

Original: Specifically, we show that certain “best-of-both-worlds” fairness guarantees, that is, randomized allocations that satisfy an ex-ante and ex-post guarantees simultaneously, are not guaranteed to exist.
Revised: Specifically, we show that certain “best-of-both-worlds” fairness guarantees, that is, randomized allocations that satisfy an ex-ante and ex-post guarantees simultaneously, are not guaranteed to exist.

Original: Our impossibility result uses the following theorem.
Revised: Our impossibility result uses the following theorem.

Original: Theorem 10.
Revised: Theorem 10.

Original: Every deterministic (or randomized) mechanism for n = 2 agents with additive and normalized utilities, that always outputs allocations that (ex-post) maximize the number of agents having positive utility, is obviously manipulable.
Revised: Every deterministic (or randomized) mechanism for n = 2 agents with additive and normalized utilities, that always outputs allocations that (ex-post) maximize the number of agents having positive utility, is obviously manipulable.

Original: Proof of Theorem 10.
Revised: Proof of Theorem 10.

Original: Let M be such a deterministic (or randomized) mechanism.
Revised: Let M be such a deterministic (or randomized) mechanism.

Original: We will prove the theorem (i.e., show that M is obviously manipulable) for the case of m = 2 items; the proof easily generalizes to more than 2 items.
Revised: We will prove the theorem (i.e., show that M is obviously manipulable) for the case of m = 2 items; the proof easily generalizes to more than 2 items.

Original: Consider the case when both the agents report a value of 1 for the first item and zero for the other item, i.e. b i = (1 , 0) for i ∈ { 1 , 2 } .
Revised: Consider the case when both the agents report a value of 1 for the first item and zero for the other item, i.e. b i = (1 , 0) for i ∈ { 1 , 2 } .

Original: Given these reports, there must be an agent who gets item 1 with probability at least 12 (if M is deterministic then this probability will be exactly 1 ); assume that this is agent 1 , without loss of generality.
Revised: Given these reports, there must be an agent who gets item 1 with probability at least 12 (if M is deterministic then this probability will be exactly 1 ); assume that this is agent 1 , without loss of generality.

Original: Consider the case that the true valuation of agent 1 is v 1 = ( 23 + ϵ, 13 − ϵ ) , for some small constant ϵ > 0 .
Revised: Consider the case that the true valuation of agent 1 is v 1 = ( 23 + ϵ, 13 − ϵ ) , for some small constant ϵ > 0 .

Original: The worst case for agent 1 under honest reporting happens when the reported valuation of agent 2 b 2 = (1 , 0) , in which case item 1 must be allocated to agent 2 and item 2 to agent 1 – irrespective of whether the mechanism M is deterministic or randomized since M maximizes the number of agents having positive utility ex-post , and allocating item 1 to agent 2 and item 2 to agent 1 is the unique allocation that gives both agents non-zero utility.
Revised: The worst case for agent 1 under honest reporting happens when the reported valuation of agent 2 b 2 = (1 , 0) , in which case item 1 must be allocated to agent 2 and item 2 to agent 1 – irrespective of whether the mechanism M is deterministic or randomized since M maximizes the number of agents having positive utility ex-post , and allocating item 1 to agent 2 and item 2 to agent 1 is the unique allocation that gives both agents non-zero utility.

Original: Therefore, the worst-case utility of agent 1 is at most 13 − ϵ .
Revised: Therefore, the worst-case utility of agent 1 is at most 13 − ϵ .

Original: Now, consider the dishonest report b 1 = ( b 1 , 1 , b 1 , 2 )
Revised: Now, consider the dishonest report b 1 = ( b 1 , 1 , b 1 , 2 )

Original: =
Revised: =

Original: If b 2 , 2 > 0 , then mechanism M , in order to maximize the number of agents having positive utility, will allocate item 2 to agent 2 and item 1 to agent 1 , i.e., agent 1 ’s utility in this case (per v 1 ) would be 23 + ϵ .
Revised: If b 2 , 2 > 0 , then mechanism M , in order to maximize the number of agents having positive utility, will allocate item 2 to agent 2 and item 1 to agent 1 , i.e., agent 1 ’s utility in this case (per v 1 ) would be 23 + ϵ .

Original: On the other hand, if b 2 , 2 = 0 (and therefore b 2 , 1 = 1 ) then as per our choice of mechanism M , agent 1 will receive itemwith probability at least 12 .
Revised: On the other hand, if b 2 , 2 = 0 (and therefore b 2 , 1 = 1 ) then as per our choice of mechanism M , agent 1 will receive itemwith probability at least 12 .

Original: Hence, her utility in this case would be at least 12 v 1 , 1 = 13 + ϵ 2 .
Revised: Hence, her utility in this case would be at least 12 v 1 , 1 = 13 + ϵ 2 .

Original: The above case analysis shows that if agent 1 reports valuation b 1 then her worst-case utility will be 13 + ϵ 2 , which is strictly larger then her worst-case utility when reporting true valuation.
Revised: The above case analysis shows that if agent 1 reports valuation b 1 then her worst-case utility will be 13 + ϵ 2 , which is strictly larger then her worst-case utility when reporting true valuation.

Original: Therefore, M is obviously manipulable.
Revised: Therefore, M is obviously manipulable.

Original: We note that we can apply Theorem 10 to show that for n = 2 there is no NOM mechanism
Revised: We note that we can apply Theorem 10 to show that for n = 2 there is no NOM mechanism

Original: that outputs (ex-post) and MNW solution, since MNW is scale free (and therefore normalization comes without loss of generality).
Revised: that outputs (ex-post) and MNW solution, since MNW is scale free (and therefore normalization comes without loss of generality).

Original: We are now ready to prove our main result for this section.
Revised: We are now ready to prove our main result for this section.

Original: Theorem 11.
Revised: Theorem 11.

Original: Under additive valuations, randomized allocations that are ex-ante proportional, ex-post Pareto efficient and ex-post maximize the number of agents having positive utility do not exist, for any number of agents and any number of items.
Revised: Under additive valuations, randomized allocations that are ex-ante proportional, ex-post Pareto efficient and ex-post maximize the number of agents having positive utility do not exist, for any number of agents and any number of items.

Original: Proof.
Revised: Proof.

Original: We begin by establishing the result for n = 2 agents.
Revised: We begin by establishing the result for n = 2 agents.

Original: Towards a contradiction, assume that for additive valuations, randomized allocations that are ex-ante proportional, ex-post Pareto efficient and ex-post maximize the number of agents having positive utility always exits, and let R be a randomized mechanism that, given the reported valuations of agents, outputs such a allocation.
Revised: Towards a contradiction, assume that for additive valuations, randomized allocations that are ex-ante proportional, ex-post Pareto efficient and ex-post maximize the number of agents having positive utility always exits, and let R be a randomized mechanism that, given the reported valuations of agents, outputs such a allocation.

Original: Note that this mechanism need not be computationally efficient.
Revised: Note that this mechanism need not be computationally efficient.

Original: We will show that R is NOM in expectation, in direct contradiction to Theorem 10.
Revised: We will show that R is NOM in expectation, in direct contradiction to Theorem 10.

Original: Since, R outputs ex-ante proportional allocations, the worst-case guarantee of NOM (inequality (1)) holds, via Lemma 5.
Revised: Since, R outputs ex-ante proportional allocations, the worst-case guarantee of NOM (inequality (1)) holds, via Lemma 5.

Original: Additionally, the best-case guarantee (inequality (2)) is also satisfied: if an agent i ∈ M reports her true valuation, then in the best case, all other agents only desire her least-valued item; in this case, agent i receives all items except her least-valued item.
Revised: Additionally, the best-case guarantee (inequality (2)) is also satisfied: if an agent i ∈ M reports her true valuation, then in the best case, all other agents only desire her least-valued item; in this case, agent i receives all items except her least-valued item.

Original: This outcome cannot be (strictly) improved as agent i must lose at least one item: the mechanism maximizes the number of agents having positive utility and each agent reports having a positive utility of at least one item, therefore, all items cannot be allocated to the same agent.
Revised: This outcome cannot be (strictly) improved as agent i must lose at least one item: the mechanism maximizes the number of agents having positive utility and each agent reports having a positive utility of at least one item, therefore, all items cannot be allocated to the same agent.

Original: Therefore, R is NOM is expectation, a contradiction.
Revised: Therefore, R is NOM is expectation, a contradiction.

Original: Thus, for n = 2 agents randomized allocation that are ex-ante proportional, ex-post Pareto efficient, and ex-post maximize the number of agents having positive utility do not always exist.
Revised: Thus, for n = 2 agents randomized allocation that are ex-ante proportional, ex-post Pareto efficient, and ex-post maximize the number of agents having positive utility do not always exist.

Original: In particular, there is a 2 agent instance where such randomized allocations do not exist.
Revised: In particular, there is a 2 agent instance where such randomized allocations do not exist.

Original: This instance can be easily converted into a n > 2 agent instance by adding dummy agents a 1 , a 2 , . .
Revised: This instance can be easily converted into a n > 2 agent instance by adding dummy agents a 1 , a 2 , . .

Original: . , a k and dummy items j 1 , j 2 , . .
Revised: . , a k and dummy items j 1 , j 2 , . .

Original: , j k such that ( i ) for each i ∈ [ k ] , agent a
Revised: , j k such that ( i ) for each i ∈ [ k ] , agent a

Original: i ’s value for item
Revised: i ’s value for item

Original: j 1 is 1 and her value for every other item is 0 , ( ii ) the existing agents have zero value for each dummy item.
Revised: j 1 is 1 and her value for every other item is 0 , ( ii ) the existing agents have zero value for each dummy item.

Original: In this new instance, every Pareto efficient allocation must be such that agent a i is allocated item j i and no other item that existing agents value positively.
Revised: In this new instance, every Pareto efficient allocation must be such that agent a i is allocated item j i and no other item that existing agents value positively.

Original: Thus, a randomized allocation that is ex-ante proportional, ex-post Pareto efficient, and ex-post maximizes the number of agents having positive utility would continue to satisfy these properties after removing the dummy agents and dummy items.
Revised: Thus, a randomized allocation that is ex-ante proportional, ex-post Pareto efficient, and ex-post maximizes the number of agents having positive utility would continue to satisfy these properties after removing the dummy agents and dummy items.

Original: However, as already established, such allocations do not exist for n = 2 agents.
Revised: However, as already established, such allocations do not exist for n = 2 agents.

Original: This concludes the proof of
Revised: This concludes the proof of

Original: Theorem 11.
Revised: Theorem 11.

Original: The existence of ex-ante proportional, ex-post Pareto efficient and ex-post EF1 allocations remains an elusive open problem.
Revised: The existence of ex-ante proportional, ex-post Pareto efficient and ex-post EF1 allocations remains an elusive open problem.

Original: Our result shows that replacing ex-post EF1 with a different, mild fairness guarantee, namely ex-post maximizing the number of agents with positive utility, is impossible.
Revised: Our result shows that replacing ex-post EF1 with a different, mild fairness guarantee, namely ex-post maximizing the number of agents with positive utility, is impossible.

Original: The following corollary is immediate (and we note that point #3 in the following corollary is shown by [FSV20]).
Revised: The following corollary is immediate (and we note that point #3 in the following corollary is shown by [FSV20]).

Original: Corollary 2.
Revised: Corollary 2.

Original: Under additive valuations, randomized allocations that satisfy the following properties do not always exist.
Revised: Under additive valuations, randomized allocations that satisfy the following properties do not always exist.

Original: ex-ante proportional, ex-post Pareto efficient and ex-post egalitarian welfare maximizing 2.
Revised: ex-ante proportional, ex-post Pareto efficient and ex-post egalitarian welfare maximizing 2.

Original: ex-ante proportional and ex-post leximin allocations 3.
Revised: ex-ante proportional and ex-post leximin allocations 3.

Original: ex-ante proportional and ex-post MNW.
Revised: ex-ante proportional and ex-post MNW.