Best-of-Both-Worlds Fair-Share Allocations
aa r X i v : . [ c s . G T ] M a r Best-of-Both-Worlds Fair-Share Allocations
Moshe Babaioff ∗ , Tomer Ezra † , Uriel Feige ‡ March 4, 2021
Abstract
We consider the problem of fair allocation of indivisible items among n agents with additive valuations,when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds(BoBW) fairness mechanisms aim to give all agents both an ex-ante guarantee (such as getting theproportional share in expectation) and an ex-post guarantee. Prior BoBW results have focused on ex-post guarantees that are based on the “up to one item” paradigm, such as envy-free up to one item(EF1). In this work we attempt to give every agent a high value ex-post, and specifically, a constantfraction of her maximin share (MMS). The up to one item paradigm fails to give such a guarantee, andit is not difficult to present examples in which previous BoBW mechanisms give some agent only a n fraction of her MMS.Our main result is a deterministic polynomial-time algorithm that computes a distribution overallocations that is ex-ante proportional, and ex-post, every allocation gives every agent at least herproportional share up to one item, and more importantly, at least half of her MMS. Moreover, thislast ex-post guarantee holds even with respect to a more demanding notion of a share, introduced inthis paper, that we refer to as the truncated proportional share (TPS). Our guarantees are nearly bestpossible, in the sense that one cannot guarantee agents more than their proportional share ex-ante, andone cannot guarantee agents more than a n n − fraction of their TPS ex-post. In this paper we consider fair allocation of indivisible items to agents with additive valuations. An instance I = ( v, M , N ) of the fair allocation problem consists of a set M of m indivisible items, a set N of n agents,and vector v = ( v , v , . . . , v n ) of non-negative additive valuations, with the valuation of agent i ∈ N for set S ⊆ M being v i ( S ) = P j ∈ S v i ( j ), where v i ( j ) denotes the value of agent i for item j ∈ M . We assumethat the valuation functions of the agents are known to the social planer, and that there are no transfers(no money involved). We further assume that all agents have equal entitlement to the items. An allocation A is a collection of n disjoint bundles A , . . . , A n (some of which might be empty), where A i ⊆ M forevery i ∈ N . A randomized allocation is a distribution over deterministic allocations. We wish to designrandomized allocations that enjoy certain fairness properties.Before discussing some standard fairness properties, we briefly motivate the best of both worlds (BoBW)framework, that considers both ex-ante and ex-post properties of randomized allocations. Consider a simpleallocation instance I with two agents and two equally valued items. Intuitively, any fair allocation in thiscase is an allocation that gives each agent one of the items. Giving both items to one of the agents andno item to the other agent is not considered fair. Consider now an instance I with two agents and justone item. As we want to allocate the item (to achieve Pareto efficiency) but the item is indivisible, we giveit to one of the agents, and then the other agent gets no item. The fact that some agent receives no item ∗ Microsoft Research — E-mail: [email protected] . † Tel Aviv University — E-mail: [email protected] . ‡ Weizmann Institute — E-mail: [email protected] . Part of the work was conducted at Microsoft Research,Herzeliya.
1s unavoidable, and in this respect the allocation is fair. Yet, the agent not getting the item might arguethat this deterministic allocation is unfair as she has the same right to the item as the other agent. Indeed,we can improve the situation at least ex-ante: We can invoke a lottery to decide at random which of thetwo agents gets the item. While for any realization inevitably one agent is left with nothing, the allocationmechanism is ex-ante fair (each agent has a fair chance to win the lottery). Going back to instance I , wecould also have a lottery for I , and have the winner receive both items. This too would be ex-ante fair,but ex-post (with respect to the final allocation) it would not be fair (as we did have the option to choosean allocation that gives every agent one item). Examples such as those above illustrate why we want ourallocation mechanism to concurrently enjoy both ex-ante and ex-post fairness guarantees, as each guaranteeby itself seems not to be sufficiently fair.For the purpose of defining ex-ante fairness properties of randomized allocations, we assume that agentsare risk neutral. That is, the ex-ante value that an agent derives from a distribution over bundles is the sameas the expected value of a bundle selected at random from this distribution. Consequently, when consideringa distribution D over allocations (of M to N ), we also consider the expectation of this distribution, whichcan be interpreted as a fractional allocation . In this fraction allocation, the fraction of item i given to agent j exactly equals the probability with which agent i receives item j under D . We naturally extend the additivevaluation functions of agents to fractional allocations, by considering the expected valuation, that is, anadditive valuation where the value of a fraction q j of item j to agent i is q j · v i ( j ). We briefly review some properties of allocations from the literature, properties that are most relevant to thecurrent work and to prior related work.We start with standard share definitions. The proportional share of agent i is P S i = v i ( M ) n . We say thatan allocation A = ( A , . . . , A n ) is proportional if every agent i gets value at least P S i (that is, v i ( A i ) ≥ v i ( M ) n = P S i ), and a fractional (randomized) allocation is ex-ante proportional if she gets her proportionalshare in expectation. We say that an allocation is proportional up to one item (Prop1) if every agent i getsvalue at least P S i − max j ∈M v i ( j ). The maximin share M M S i of agent i is the maximum value that i couldsecure if she was to partition M into n bundles, and receive the bundle with the lowest value under v i .We next discuss envy. An allocation is envy free (EF) if every agent (weakly) prefers her own bundle overthat of any other agent, and a fractional (randomized) allocation is ex-ante envy free if for every agent, theexpected value of her own allocation is at least as high as the expected value of the allocation of any otheragent. Note that an allocation that is ex-ante envy free is ex-ante proportional. An allocation is envy-freeup to one good (EF1) ( envy-free up to any good (EFX) , respectively) if every agent weakly prefers her ownbundle over that of any other agent, up to the most (least, respectively) valuable item in the other agent’sbundle. Note that EF1 implies Prop1. Finally, an allocation is envy-free up to one good more-and-less ( EF ) if no agent i envies another agent j after removing one item from the set j gets, and adding one item (notnecessarily the same item) to i . Note that EF is weaker than EF Pareto dominates another (fractional)allocation if it is weakly preferred by all agents, and strictly so by at least one. An integral allocation is
Pareto optimal (PO) if no integral allocation Pareto dominates it. An allocation (integral or fractional)is fractionally Pareto optimal (fPO) if it is Pareto optimal, and moreover, no fractional allocation Paretodominates it. Another notion of efficiency is that of
Nash Social Welfare maximization . The
Nash SocialWelfare (NSW) of allocation A = ( A , . . . , A n ) is (cid:0)Q i ∈N v i ( A i ) (cid:1) n . In case of fractional allocations, we usethe notation fNSW. The state of the art BoBW results for additive valuations are presented in the two recent papers ofFreeman et al. [2020], Aziz [2020]. Both of these works are based on the well known paradigm that wecall here “faithful implementation of a fractional allocation”: a distribution over deterministic allocations is2 faithful implementation of the fractional allocation if the ex-ante (expected) value of every agent under thedistribution is the same as it is in the fractional allocation, and ex-post (for any realization) it is the same asthe expectation, up to the value of one item. Both papers use versions of the result of Budish et al. [2013]showing that any fractional allocation can be faithfully implemented. Various versions of these results werepresented in the past, and in Appendix B we survey those results. In Section 2 we formally present a versionof “faithful implementation” that summarizes the prior results, stated as Lemma 9.By “faithful implementing” the fractional allocation that is the outcome of multiple executions of theprobabilistic serial mechanism (a.k.a. eating mechanism) of Bogomolnaia and Moulin [2001] till there areno more items, the following BoBW result was proved in [Aziz, 2020]. (The same theorem was establishedearlier in [Freeman et al., 2020], but with a somewhat more complicated proof.)
Theorem 1 ([Freeman et al., 2020, Aziz, 2020])
There is a deterministic polynomial-time faithful im-plementation of a fractional allocation that is ex-ante envy free (and thus ex-ante proportional), and theimplementation is supported on allocations that are (ex-post) EF1.
By “faithful implementing” the fractional allocation that maximizes the fractional Nash Social Welfare,the following BoBW result was proved in [Freeman et al., 2020].
Theorem 2 ([Freeman et al., 2020])
There is a deterministic polynomial-time faithful implementationof a fractional allocation that is ex-ante fPO and ex-ante proportional, and the implementation is supportedon allocations that are (ex-post) fPO, Prop1, and EF . The “up to one item” paradigm used in the ex-post guarantees of Theorems 1 and 2 is most useful whena difference of one item does not make a big difference in value. However, when items do have large values,it does not guarantee agents a high ex-post value. In contrast, we aim to give each agent “high enoughvalue” ex-post, where value is measure compared to “what the agent deserves”, captured by her fair share.Specifically, we aim to give every agent a large fraction (“an approximation”) of her “fair share”, e.g. halfthe agent’s MMS share. The following allocation instance shows that neither Theorem 1 nor Theorem 2provide a constant approximation for the MMS ex-post, and both are supported only on allocations that areintuitively very unfair. Moreover, in this instance the MMS nearly equals the proportional share, and henceone cannot dismiss this example as one in which the MMS is too small for the agents to care about.Consider an instance with n identical items, each value of n . In this case it is clear each agent shouldget one item ex-post. Now, assume the first of those big items is split into n small items, each of value 1.In this case we want one agent to get all of these small items, and each other agent to get one of the bigitems. Our next example shows that once these small items are not completely identical, but rather eachagent slightly prefers a different one of them, both prior BoBW results that use “faithful implementation”will end up with one of the agents only getting one of the small items ex-post, ending with a value of about1, instead of every agent getting about the same value of n (the MMS is n , and the proportional share ofeach agent is about n ). Specifically: Example 3
The instance has n − items { s , s , . . . , s n } ∪ { b , b , . . . , b n − } . For some small ǫ > , forevery agent i ∈ N , the additive valuation function v i is as follows: • v i ( s i ) = 1 + ǫ . • v i ( s j ) = 1 for every ≤ j ≤ n such that j = i . • v i ( b j ) = n for every ≤ j ≤ n − .The MMS of every agent is n : one bundle contains all small items { s , . . . , s n } , and the remainingbundles each contain one of the remaining, big, items. The proportional share of every agent is n + ǫn .The algorithms of Theorem 1 and Theorem 2 both start with a fractional allocation. In both cases (theallocation maximizing fractional Nash Social Welfare; the outcome of the eating mechanism), the fractionalallocation allocates the small item s i integrally to agent i , for every i ≤ n . Consequently, also ex-post every gent i gets the respective item s i . By the pigeon-hole principle, in an ex-post allocation there is an agentthat receives no item among the big items { b , b , . . . , b n − } . This agent i receives only the small item s i ,and hence only a ǫn fraction of her MMS. In this paper we aim for a Best-of-Both-Worlds fairness result: a randomized allocation that gives everyagent at least her proportional share ex-ante, and some guaranteed value ex-post. The ex-post guarantee wegive is at least half the MMS, and in fact, stronger. We introduce a new notion of share that we refer to asthe truncated proportional share (TPS) , which we believe might be of independent interest. We show thatthe TPS is at least as large as the MMS, and our BoBW result guarantees half of the TPS ex-post (and thushalf the MMS ex post), while also giving each agent her proportional share ex-ante.
We next define the Truncated Proportional Share of an agent with an additive valuation. As we will seelater, this share has two advantages over MMS: it is at least as high as the MMS, and while the MMS isNP-hard to compute, the TPS is easy to compute. We alert the reader that in this paper we define TPS onlywith respect to additive valuation functions (while the definition of MMS extends without change beyondadditive valuations).
Definition 4
For a setting with n agents and a set of items M , the truncated proportional share T P S i = T P S i ( n, M , v i ) of agent i with additive valuation function v i is the largest value t such that n P j ∈M min[ v i ( j ) , t ] = t . From the definition of TPS it is immediate to see that
T P S i ≤ P S i , but T P S i is smaller than P S i whenthere is at least one over-proportional item , which is an item that by itself gives agent i value larger than P S i . Observe that if the value of every over-proportional items is reduced to T P S i , then the TPS is theproportional share of the resulting valuation function after these reductions. In absence of over-proportionalitems, clearly T P S i = P S i . Yet, when there are over-proportional items, T P S i might be much smallerthan P S i . For example, whenever there are less items than agents (e.g. a single item and two agents thatdesire it) then for every agent T P S i = M M S i = 0 while P S i >
0. In any such case, it is clearly impossibleto concurrently give all agents a positive fraction of their proportional share ex post (while the truncatedproportional shares and the maximin share are small enough to make their approximation plausible).Moreover, regardless of the presence of over-proportional items,
T P S i ≥ M M S i . This is because taking t = M M S i satisfies n P j ∈M min[ v j , t ] ≥ t (as every one of the n bundles in the partition that determines M M S i contributes at least t to the sum), which implies that t in Definition 4 is at least as large as M M S i .Hence M M S i ≤ T P S i ≤ P S i . In particular, guarantees with respect to the TPS imply at least the sameguarantees with respect to MMS, and sometimes better.The TPS is a more tractable object than the MMS. It is not difficult to see that the TPS can berecursively defined as follows: when n = 1 then T P S i = T P S i (1 , M , v i ) = v i ( M ), and when n ≥ T P S i is the minimum among v i ( M ) n , the proportional share of agent i , and her TPS in a reduced instance in which an item j of highest value is removed as well as one of the agents, that is, T P S i in this case is T P S i ( n − , M \ { j } , v i ). This procedure provides a simple polynomial time algorithm for computing theTPS: if the proportional share of the reduced instance is smaller than that of the original instance, compute T P S i for the reduced instance. If not, then T P S i is the proportional share of the original instance. (Incontrast, computing the MMS is NP-hard.)Moreover, consider ρ T P S , the highest fraction such that in every instance, there is an allocation givingevery agent a ρ T P S fraction of her TPS. It is easy to determine the exact value of ρ T P S , which turns outto be n n − . (In contrast, the exact value of the corresponding ρ MMS is unknown [Kurokawa et al., 2018,Garg and Taki, 2020].) To see that ρ T P S ≥ n n − , we observe that a polynomial time allocation algorithmof Lipton et al. [2004] gives every agent a n n − fraction of her TPS. (For more details, see Appendix A.) To4ee that ρ T P S ≤ n n − , consider an instance with 2 n − n − n , but in every allocation, at least one of the agents gets at most one item, and hence value at most 1.The example above also shows that the TPS of an agent can be factor n − n larger than her MMS. Thisratio is tight, because M M S i ≥ n n − T P S i for every agent i . This follows by considering n agents with thesame valuation function v i , and recalling that there is an allocation that gives every agent at least a n n − fraction of her TPS. The n bundles of this allocation each have a value of at least n n − T P S i , and hencethey form a partition of M that shows that M M S i ≥ n n − T P S i . We summarize the above discussion inthe following proposition: Proposition 5
For any setting with n agents and any additive valuation v i it holds that P S i ≥ T P S i ≥ M M S i ≥ n n − T P S i Moreover, each of the above inequalities is strict for some instance, and holds as equality for some otherinstance.
We now return to present our main result. Due to the difficulties alluded to in Example 3 and Proposition 7,we do not follow the paradigm of starting with a simple to describe fractional allocation, and then faithfullyimplementing it (using Lemma 9). Instead, we design an algorithm that generates a distribution overallocations that each gives every agent at least half of her TPS, with the additional property that everyagent gets at least her proportional share in expectation. Along the way, we do use Lemma 9, but weapply it on fractional allocations that involve only carefully selected subsets of M , rather than a fractionalallocation that involves all of M . Our main result is the following. Theorem 6
For every allocation instance with additive valuations, there is a randomized allocation that isex-ante proportional, and gives each agent at least half of her TPS ex-post (and hence also at least half ofher MMS), as well as being Prop1 ex-post. Moreover, there is a deterministic polynomial time algorithmthat, given the valuation functions of the agents, computes a faithful implementation of such a randomizedallocation, supported on at most n allocations. Theorem 6 is nearly the best possible in multiple senses. It is not possible to guarantee every agentstrictly more than her proportional share ex-ante (e.g., if all agents have the same valuation function). Theex-post Prop1 guarantee is as close as possible to being proportional, if one measures the distance fromproportionality by the number of missing items. The highest possible fraction of the truncated proportionalshare that can be guaranteed ex-post is n n − = + n − (see Example 3 above), which tends to half as n grows large, and the theorem indeed ensures a fraction of half. We also remark that for the instance inExample 3, while in the BoBW results from prior work [Freeman et al., 2020, Aziz, 2020] there is always anagent that gets just about n fraction of her MMS, the algorithm of Theorem 6 gives every agent more thana n − n fraction of her TPS (and her MMS).Another aspect in which Theorem 6 cannot be improved is with respect to its Pareto properties. While theprior result of Freeman et al. [2020] present a BoBW result (Theorem 2) with a distribution over allocationsthat is ex-ante fPO, our result does not give ex-ante fPO. We next show that if we want every agent toreceive ex-post at least a constant fraction of her maximin share, getting the guarantee of ex-ante fPO isimpossible. Moreover, this conflict between ex-ante fPO and half the MMS concerns every ex-post allocationthat might potentially be in the support, not just one of them. Proposition 7
For every n ≥ and every ǫ > there are allocation instances with additive valuations,with the following property: for every ex-ante Pareto optimal (fPO) randomized allocation (whether ex-anteproportional or not), every allocation in its support does not give some agent more than a ǫn fraction ofher maximin share. Corollary 8
For every allocation instance with additive valuations, there is a randomized allocation that issupported on at most n allocations, is ex-ante proportional, and ex-post it gives every agent at least half ofher TPS (and hence also at least half her MMS), as well as being ex-post PO and Prop1. The maximin share was introduced by Budish [2011] as a relaxation of the proportional share. Kurokawa et al.[2018] showed that for agents with additive valuations, an allocation that gives each agent her MMS maynot exist. A series of papers [Kurokawa et al., 2018, Amanatidis et al., 2017b, Barman and Krishnamurthy,2020, Ghodsi et al., 2018, Garg et al., 2019 ,Garg and Taki, 2020] considered the best fraction of the MMSthat can be concurrently guaranteed to all agents, and the current state of the art (for additive valuations)is a + Ω( n ) fraction of the MMS.The fairness notion of Prop1 was introduced by Conitzer et al. [2017]. The fairness notion of EF1 wasimplicitly used in [Lipton et al., 2004], and was formally defined in [Budish, 2011]. EFX (envy-free up toany good) was introduced in [Caragiannis et al., 2019] The notion of envy-free up to one good more-and-less( EF ) was defined in [Barman and Krishnamurthy, 2019], relaxing EF1.For a subclass of additive valuations, that of additive dichotomous valuations, very strong BoBW resultsare known [Halpern et al., 2020, Aziz, 2020, Babaioff et al., 2020], which among other properties, are EF ex-ante, EFX ex-post, maximize welfare, and the underlying allocation mechanism is universally truthful. Sucha strong combination of results is impossible to achieve for general additive valuations. In particular, theresults of Amanatidis et al. [2017a] imply that every universally truthful randomized allocation mechanismfor two agents that allocates all items must sometimes not give an agent more than a m fraction of her MMSex-post. See Section 4.3 for a more extensive discussion on truthfulness.In Section 1.2 we already discussed some previous BoBW results. We further remark that in [Freeman et al.,2020] they present an instance for which there is no randomized allocation that is ex-ante proportional,ex-post EF1 and ex-post fPO. For the same instance, there is no randomized allocation that is ex-anteproportional, is ex-post fPO, and gives every agent a positive fraction of her MMS. We consider fair allocation of indivisible items to agents with additive valuations. An instance I = ( v, M , N )of the fair allocation problem consists of a set M of m indivisible items, a set N of n agents, and vector v = ( v , v , . . . , v n ) of non-negative additive valuations, with the valuation of agent i ∈ N for set S ⊆ M being v i ( S ) = P j ∈ S v i ( j ), where v i ( j ) denotes the value of agent i for item j ∈ M . We assume that thevaluation functions of the agents are known to the social planer, and that there are no transfers (no moneyinvolved). We further assume that all agents have equal entitlement to the items. An allocation A is acollection of n disjoint bundles A , . . . , A n (some of which might be empty), where A i ⊆ M for every i ∈ N .As we shall be dealing with randomized allocations, let us introduce terminology that we shall use in thiscontext. A random allocation is a distribution D over integral allocations A , A , . . . . It induces an expected allocation A ∗ , where A ∗ ij specifies for agent i and item j the probability that agent i receives item j , when anallocation is chosen at random from the underlying distribution D . These probabilities can be interpreted6s fractions of the item that an agent receives ex-ante. Hence the expected allocation A ∗ can be viewedas a fractional allocation , in which items are divisible. Conversely, we say that the distribution D (namely,the random allocation) implements the fractional allocation A ∗ when the expectation of D is A ∗ . Finally,we note that an additive valuation function can be extended in a natural way from allocations to fractionalallocations, by considering the expected valuation. That is, the value of a p j fraction of item j to agent i tois p j · v i ( j ), and the value of a fractional allocation A ∗ to agent i is P j ∈M A ∗ ij · v i ( j ).For the issue of computing randomized allocations there are two different notions of polynomial time com-putation. In a random polynomial time implementation , there is a randomized polynomial time algorithmthat samples an allocation from the distribution D . In a polynomial time implementation , there is a deter-ministic polynomial time algorithm that lists all allocations in the support of D (implying in particular thatthe support contains at most polynomially many allocations), together with their associated probabilities. For general additive valuations, there is a very useful lemma that greatly simplifies the design of BoBW alloca-tions. We refer to it here as the faithful implementation lemma . The lemma (sometimes with slight variations)was previously stated and used in BoBW results [Budish et al., 2013, Freeman et al., 2020, Halpern et al.,2020, Aziz, 2020], and was used even earlier in approximation algorithms for maximizing welfare [Srinivasan,2008]. Restricted variants of it were introduced for scheduling problems [Lenstra et al., 1990], and werelater used for allocation problems [Bez´akov´a and Dani, 2005]. For an extensive discussion of the faithfulimplementation lemma, as well as its proof (presented for completeness), see Appendix B.
Lemma 9
Let A ∗ be a fractional allocation of m items to n agents with additive valuations, and let f denotethe number of strictly fractional variables in A ∗ (number of pairs ( i, j ) such that in A ∗ , the fraction of item j allocated to agent i is strictly between 0 and 1). Then there is a deterministic polynomial time implementationof A ∗ , supported only on allocations in which every agent gets value (ex-post) equal her ex-ante value (inthe fractional allocation A ∗ ), up to the value of one item. (For agent i , the corresponding one item is theitem most valuable to i , among those items that are assigned to i under A ∗ in a strictly fractional fashion.Moreover, the values that the agent gets in any two allocations differ by at most the value of this single item.)The distribution of the implementation is supported over at most f + 1 allocations. Using Lemma 9, one trivially gets the following BoBW result (implicit in previous work), which is abaseline against which other BoBW results can be compared.
Proposition 10
There is a deterministic polynomial time implementation of a fractional allocation that isex-ante envy free, and the implementation is supported on allocations that are (ex-post) Prop1.
Proof.
Consider the uniform fractional allocation , that assigns a fraction of n of every item to every agent.It is ex-ante envy free, as all agents get the same fractional allocation. Applying Lemma 9, it is implementedin deterministic polynomial time by allocations that are Prop1. (cid:4) We now restate and prove our main result, Theorem 6.
Theorem 6
For every allocation instance with additive valuations, there is a randomized allocation that isex-ante proportional, and gives each agent at least half of her TPS ex-post (and hence also at least half ofher MMS), as well as being Prop1 ex-post. Moreover, there is a deterministic polynomial time algorithmthat, given the valuation functions of the agents, computes a faithful implementation of such a randomizedallocation, supported on at most n allocations. In Section 4 we discuss possible extensions of the theorem with regard to three aspects: fairness, efficiencyand truthfulness. We present impossibilities of some natural extensions, as well as some open problems.7 .1 Proof Overview
Let I = ( v, M , N ) be an input instance. For any instance I we denote the proportional share and thetruncated proportional share of agent i by P S i ( I ) and T P S i ( I ) respectively. For the original instance, weomit the instance and denote the proportional share and the truncated proportional share of agent i by P S i and T P S i , respectively.To prove the theorem we present a deterministic polynomial time algorithm that, given the input instance I = ( v, M , N ), computes an implementation of a randomized allocation that gives every agent at least herproportional share ex-ante, and at least half of her truncated proportional share ex-post, and is supportedon at most n allocations. Items that each by itself gives an agent her TPS will play a central rule in ouralgorithm. We say that item j is exceptional for agent i if v i ( j ) ≥ T P S i . Our algorithm has several phases:1. Find a distribution over 4 n matchings. Each of these matchings partitions the agents to two disjoint sets N and N , and the items to three disjoint sets M ( N ) , M ( N ) and M ( M = M ( N ) ∪M ( N ) ∪M , |N | = |M ( N ) | and |N | = |M ( N ) | ). Each agent in N is matched with an item in M ( N ), and eachagent in N is matched with an item in M ( N ). The distribution over these matchings is computedin two steps:(a) Compute (using LP1) a distribution in which in every matching, every agent in N is matched toan item that is exceptional item for him in M ( N ) and such that:i. No unallocated items (items in M \ M ( N )) is exceptional to any agent in N = N \ N .ii. The distribution over these 4 n matchings gives each agent her PS conditioned on every agentin N eventually getting her TPS in expectation (as indeed is guaranteed by 1(b)ii below).(b) Complete each partial matching to a complete matching by matching each agent in N to an itemin M ( N ), such that:i. Each agent prefers the item matched to him over any unmatched item (item in M ).ii. For the unmatched items, there still is a fractional allocation of M such that for each agent in N , her expected value for the combination of her matched item and her fractional allocationis at least her TPS.2. For each matching above, find (by LP3) a distribution over m +1 deterministic allocations that allocate M , the unmatched items, with the following properties:(a) in each allocation in the support, every agent in N gets (in total, over the matched item and theremaining allocation) at least half her TPS.(b) In expectation, every agent in N gets (in total) her TPS.3. From the distribution over 4 n ( m + 1) allocations defined above, find a distribution over at most n ofthese allocations that is ex-ante proportional (all ex-post properties are preserved).Before elaborating on these steps we make the following remark. Remark 11
Below we present an allocation algorithm with properties as in the theorem. Some componentsin this algorithm are flexible. In particular, this applies to the objective functions of LP1 and LP3. Makinguse of this flexibility, we set the objective functions of these LPs so that they maximize welfare (subject to theconstraints of the respective LPs). We find it natural to measure welfare in units of “proportional share”.Hence we assume (without loss of generality) that in I the valuation function of every agent i is scaled sothat v i ( M ) = n . Consequently, P S i ( I ) = 1 (and T P S i ( I ) ≤ ). This assumption is not used in the proof ofTheorem 6, and is presented in this remark only so as to explain our particular choice of objective functionsfor LP1 and LP3. Note that
P S i and T P S i depend on v i , but not on the valuations of the other agents. .2 The Algorithm and the Proof We next move to formally describe all the steps of the algorithm and prove the theorem.
Phase 1a:
Maximal allocation of exceptional items.We start by transforming the input instance I into a new instance I . In I , we add n auxiliary itemsto M , and denote them by a , . . . , a n , thus obtaining a set M = M ∪ { a , . . . , a n } . For every i ∈ N , wemodify the original valuation function v i to the following unit demand valuation function u i . • For every item j ∈ M , if v i ( j ) ≥ T P S i ( I ) then u i ( j ) = v i ( j ). • For every item j ∈ M , if v i ( j ) < T P S i ( I ) then u i ( j ) = 0. • u i ( a i ) = T P S i ( I ). • u i ( a j ) = 0 for j = i . • u i is unit demand . Namely, for u i ( S ) = max j ∈ S u i ( j ) for every S ⊂ M .We now set up a linear program that finds a fractional allocation that maximizes welfare in I , subjectto the constraint that the fractional value received by every agent i is at least P S i ( I ) (hence at least 1, dueto our scaling). Variable x ij denotes the fraction of item j received by agent i . Variable s i denotes the valuethat agent i derives from the fractional allocation. We refer to the following linear program as LP1. Maximize P i ∈N s i subject to: P i ∈N x ij ≤ j ∈ M . (Every item is fractionally allocated at most once.)2. P j ∈M x ij = 1 for every agent i ∈ N . (Agent i gets item fractions that sum to one item).3. s i = P j ∈M u i ( j ) x ij for every agent i ∈ N . (Agent’s i value is the sum of fraction of values that shereceives from the fractional allocation.)4. s i ≥ P S i ( I ) for every agent i ∈ N . (Agent’s i value is at least as high as P S i ( I ).)5. x ij ≥ i ∈ N and item j ∈ M . Proposition 12
LP1 is feasible.
Proof.
Let E i denote the set of items that are exceptional for agent i in the original instance I . Necessarily, | E i | ≤ n . Consider a solution for LP1 with x ij = n for every i ∈ N and j ∈ E i , and x ia i = 1 − P j ∈ E i x ij =1 − | E i | n . It clearly satisfies constraints 1,2,3 and 5. In remains to establish that this solution satisfiesconstraint 4, that is, the constraint s i ≥ P S i ( I ).We shall use the facts that P S i ( I ) = n v i ( E i ) + n v i ( M \ E i ) and T P S i ( I ) = v i ( M\ E i ) n −| E i | . We can see thatconstraint 4 is satisfied by this solution: s i = 1 n u i ( E i ) + (cid:18) − | E i | n (cid:19) u i ( a i ) = 1 n v i ( E i ) + n − | E i | n T P S i = 1 n ( v i ( E i ) + v i ( M \ E i )) = P S i ( I ) (cid:4) We solve LP1. Let A ∗ denote the optimal fractional solution that is found. We assume without loss ofgenerality that ties are broken toward real items: In A ∗ there is no agent i that is fractionally allocatedher auxiliary item a i ( x ia i >
0) such that there is some real item j ∈ M with value v i ( j ) = T P S i ( I ) thatis not fully allocated (it is easy to make sure that is the case by shifting mass from the auxiliary item tosuch a real item if not.) Moreover, without loss of generality, in A ∗ there are at most 2 n − n . Hence either there are n items that are fully allocated, or there are fewer than n fully allocated items. In the latter case, and only9n the latter case, in addition there are items that are partly allocated. The number of partly allocated itemsneed not exceed n , because no agent needs to receive fractions from two different partly allocated items.The agent can instead increase her share in the more valuable of the two, and reduce her share in the other.)Consequently, constraint 1 involves at most 2 n − n additionalconstraints. As all n variables s i are positive, the number of positive x ij variables is at most 4 n − A ∗ . The following proposition follows immediately fromthe properties of A ∗ and Lemma 9, and hence its proof is omitted. Proposition 13
The faithful randomized rounding of A ∗ produces a distribution over allocations with thefollowing properties:1. In every allocation, every agent i gets exactly one item from M . This item is either one of herexceptional items, or her auxiliary item a i . In either case the u i value of that item is least T P S i ( I ) .2. The distribution is supported on at most n allocations.3. In expectation, every agent i gets value s i with respect to u i . Recall that s i ≥ P S i ( I ) . Consider now an arbitrary allocation A ′ in the support of the faithful randomized rounding of A ∗ . Withrespect to A ′ , let N = N ( A ′ ) denote the set of agents that receive an item that was exceptional for them,and let N denote the set of agents that receive their auxiliary item. (Note that N ∪ N = N .)The first phase ends by giving each agent of N the item that she receives under A ′ , and not givingagents of N any item (as the auxiliary items do not really exist). Thus we have that M ( N ) is the set ofitems matched to agents in N . Observe that every agent i ∈ N gets at least T P S i ex-post. The remainingphases will ensure that agents in N get at least half their TPS ex-post. They will also ensure that ex-ante,every agent gets at least her proportional share (this will make use of item 3 of Proposition 13). Phase 1b:
Completing the matching.If N is empty, we go directly to Phase 2. Hence here we assume that N is non-empty.Let M ⊂ M denote the subset of original items (not including the auxiliary items) that remain unal-located in A ′ (those items not allocated to N ). Let I denote the allocation instance that has M as itsset of items, N as its set of agents, and the valuation function of every agent i ∈ N remains v i (restrictedto the items in M .) As I is obtained from I by removing |N | agents and |N | items, it holds that T P S i ( I ) ≥ T P S i ( I ) for every agent i ∈ N .Importantly, recall that we may assume without loss of generality that M has no item that accordingto instance I was exceptional for an agent of N . (If M contains an item j that is exceptional for i ∈ N ,then in A ′ , give j instead of a i to agent i , by this moving agent i out of N and into N .) The fact that T P S i ( I ) ≥ T P S i ( I ) (for i ∈ N ) implies that also in I , M has no item that is exceptional for an agentof N . Consequently, we infer that for every agent i ∈ N : • T P S i ( I ) = v i ( M ) |N | , and consequently also v i ( M ) |N | ≥ T P S i ( I ). • There are strictly more than |N | items j ∈ M with v i ( j ) > B i ⊂ M denote the set of |N | items of highest value to agent i ∈ N , breaking ties arbitrarily. Let W i = v i ( B i ). As M has more than |N | items of positive value for i , it follows that W i < v i ( M ).We now transform the instance I into a new instance I ′ . The set of items in I ′ is M , and the set ofagents is N . Every agent i ∈ N has a unit demand valuation function w i , defined as follows. For j ∈ B i we have w i ( j ) = v i ( j ) v i ( M ) − W i (observe that the denominator is positive), and for j B i we have w i ( j ) = 0.In the matching completion phase, we find a welfare maximizing allocation B ∗ in I ′ . Observe that thiscan be done in polynomial time, because agents are unit demand, and hence finding B ∗ amounts to solvingan instance of maximum weight matching in a bipartite graph G , with N as the set of left side vertices, M as the set of right side vertices, and weight w i ( j ) for edge ( i, j ). In B ∗ , every agent i ∈ N receives an10tem from her respective set B i (this follows because | B i | ≥ |N | ). We have that M ( N ) is the set of itemsmatched to agents in N .By the end of the matching phase, every agent holds one item. Agents in N received their item inPhase 1a (under A ′ ), whereas agents in N received their item in Phase 1b (under B ∗ ). Let e i denote theitem that has been allocated to agent i , and let M = M \ { e , . . . , e n } denote the set of items that are notyet allocated. A key property established by the first two phases is summarized in the following proposition. Proposition 14
For every agent i ∈ N it holds that v i ( e i ) ≥ max j ∈M v i ( j ) . Proof.
For an agent i ∈ N , the proposition follows from the optimality of the fractional allocation A ∗ .If there is an item i ∈ M with v i ( j ) > v i ( e i ), then in A ′ item j could replace item e i for agent i , thusincreasing the welfare of A ′ . This would imply that LP1 has a fractional solution of value higher than thatof A ∗ , contradicting the optimality of A ∗ .For an agent i ∈ N , the proposition follows from the optimality of the integral allocation B ∗ . (cid:4) We are now ready to move to the next phase of our algorithm.
Phase 2:
Allocating unmatched items.In this phase, for each matching computed before, we allocate the items of M , the items not in thematching. Every agent i ∈ N has her original valuation function v i (with v i ( M ) = n ).We first compute a fractional allocation for the items of M . This is done by solving a linear programthat we refer to as LP3. In LP3, variable x ij denotes the fraction of item j ∈ M allocated to agent i , and s i denotes the value that agent i derives from the fraction allocation (under valuation function v i ). Theparameters f i are treated as constants in LP3. Their values are computed based on Phase 1b. Specifically, f i = vi ( M |N | − v i ( e i ) v i ( M ) − W i for i ∈ N (where e i is the item allocated to agent i in Phase 1b, and W i = v i ( B i ), asdefined in Phase 1b). We now present LP3. Maximize P i ∈N s i subject to: P i ∈N x ij ≤ j ∈ M . (Every item is fractionally allocated at most once.)2. s i = P j ∈M v i ( j ) x ij for every agent i ∈ N . (Agent’s i value is the sum of the fractions of values thatshe receives from the fractional allocation.)3. s i ≥ f i v i ( M ) for every agent i ∈ N . (This is the key constraint that ties LP3 with the allocation B ∗ of Phase 1b. It applies only to agents in N .)4. x ij ≥ i ∈ N and item j ∈ M .Note that LP3 may fractionally allocate items from M to agents in N , but only after each agent in N receives items of sufficiently high value as dictated by Constraint 3. Lemma 15
LP3 is feasible.
Proof.
Constraints 2 and 4 are satisfied by every solution in which x i,j ≥ i and j ). It remains toshow that constraints 1 and 3 can be satisfied simultaneously.Recall the bipartite graph G from Phase 1b. In G , consider a fractional matching F = { y ij } , where y ij = |N | for every agent i ∈ N and item j ∈ B i , and y ij = 0 if j B i . Observe that for every agent i ∈ N we have P j ∈ B y ij = 1 and for every item j ∈ M we have P i ∈N y ij ≤ |N | |N | = 1. Henceindeed F defines a fractional matching. In I the fractional matching F gives agent i ∈ N fractional value P j ∈ B i y ij v i ( j ) = |N | v i ( B i ) = W i |N | .Being a fractional matching, F can be represented as a distribution D over integral matchings. Inevery one of these integral matchings, every agent i ∈ N is matched, because i is fully matched in F .Select a matching at random from the distribution D . Then in expectation, agent i gets an item of value11 j ∈ B i y ij v i ( j ) = W i |N | . Using E D to denote expectation over choice from distribution D , and denoting by e i the item received by i , we have that E D [ v i ( e i )] = W i |N | . Hence the expectation of f i is E D [ vi ( M |N | − v i ( e i ) v i ( M ) − W i ] = vi ( M |N | − Wi |N | v i ( M ) − W i = |N | . By linearity of expectation, E D [ P i ∈N f i ] = 1. This implies that there is a matchingin G under which the sum of the respective f i satisfies P i ∈N f i ≤
1. The matching that maximizes P i ∈ [ n ] v i ( e i ) v i ( M ) − W i (which is B ∗ that we use in the matching step, because we defined w i ( j ) to be v i ( j ) v i ( M ) − W i )also minimizes P i ∈N f i , and hence has P i ∈N f i ≤
1. This implies that the solution with x ij = f i for every i ∈ N and j ∈ M , and x ij = 0 for every i ∈ N , is feasible for LP3. (cid:4) Let C ∗ be a fractional allocation of M that is an optimal solution to LP3. Phase 2 ends by performingfaithful randomized rounding of C ∗ . The following proposition follows immediately from the properties of C ∗ and Lemma 9, and hence its proof is omitted. Proposition 16
The faithful randomized rounding of C ∗ produces a distribution over allocations of theitems of M , with the following properties:1. The distribution is supported on at most m + 1 allocations. (The number of constraints in LP3 is |M | + n + |N | . In a basic feasible solution, at least |N | of the s i variables are positive, and so atmost |M | + n = m of the x ij variables are positive.)2. Every agent i ∈ N gets ex-ante value s i ≥ f i · v i ( M ) .3. Every agent i ∈ N gets ex-post value at least s i , up to one item. That is, at least s i − max j ∈M [ v i ( j )] . The allocation algorithm above computes a distribution over 4 n matchings in Phases 1a and 1b, and foreach such matching, in Phase 2 it computes a distribution over m + 1 allocations of M . We thus have adistribution over 4 n ( m + 1) allocations and we next prove that it satisfies the requirements of Theorem 6(except the support reduction to n allocations, that will be handled in Phase 3 below). Every agent gets her proportional share ex-ante.
By item 3 of Proposition 13, with respect to A ∗ , every agent i gets value at least P S i ( I ) ex-ante. However, this value might have been attained by beingallocated the respective auxiliary item a i , of value T P S i ( I ). In this case, agent i does not actually get a i ,but is instead included in N . Hence we need to show that for every agent i ∈ N , her combined ex-antevalue from Phases 1b and 2 is at least T P S i ( I ). This ex-ante value is at least v i ( e i ) + f i · v i ( M ). We claimthat indeed v i ( e i ) + f i · v i ( M ) ≥ T P S i ( I ).Recall that f i = vi ( M |N | − v i ( e i ) v i ( M ) − W i . Observe also that v i ( M ) − W i ≤ v i ( M ), because the total value for i of the |N | items allocated under B ∗ cannot be larger than W i = v i ( B i ) (as B i contains the |N | items ofhighest value). Combining these observations we have that: f i = v i ( M ) |N | − v i ( e i ) v i ( M ) − W i ≥ v i ( M ) |N | − v i ( e i ) v i ( M ) = v i ( M ) − |N | · v i ( e i ) |N | · v i ( M )We can now establish the claim. v i ( e i ) + f i v i ( M ) ≥ v i ( e i ) + v i ( M ) − |N | · v i ( e i ) |N | · v i ( M ) v i ( M ) = v i ( M ) |N | ≥ T P S i ( I )(for the last equality, see discussion in Phase 1b). Every agent gets at least half her TPS ex-post.
For agents in N , this holds by definition. Foragents i ∈ N , we have already shown that ex-ante they get at least T P S i ( I ). Item 3 of Proposition 16implies that ex-post agent i gets a value of at least T P S i ( I ) − max j ∈M [ v i ( j )]. If max j ∈M [ v i ( j )] ≤ T P S i ( I )2 ,then at least a value of T P S i ( I )2 remains. If max j ∈M [ v i ( j )] > T P S i ( I )2 , then also v i ( e i ) ≥ T P S i ( I )2 (byProposition 14), and hence i gets half her TPS already after Phase 1b.12 he allocation is prop1 ex-post. If there is an item that is exceptional for agent i , then an item that i values most, denoted as item j , necessarily satisfies v i ( j ) ≥ P S i (if v i ( j ) < P S i then T P S i = P S i , andthen j is not exceptional for i ). In this case, every allocation gives i her proportional share, up to the item j . If there is no item that is exceptional for agent i , then T P S i ( I ) = P S i ( I ), and also, i ends up in N .Item 3 of Proposition 16 ensures that she gets T P S i ( I ) up to one item, which in this case is equivalent to P S i ( I ) up to one item. The randomized allocation is computed in polynomial time.
The TPS of every agent can becomputed in polynomial time. In various steps, the algorithm involves scaling of the valuation functionsby constant factors, which too can be done in polynomial time. The heavier computational aspects of thealgorithm are the following. Phases 1a and 2 each involve solving an LP, and then performing faithfulrandomized rounding. Phase 1b involves finding a maximum weight matching. Also, these heavier steps canbe done in polynomial time, using standard algorithms.
The randomized allocation is supported on n allocations. The combination of item 2 of Propo-sition 13 and item 1 of Proposition 16 implies that the randomized allocation is supported over at most4 n ( m + 1) allocations. In Phase 3 (to be described next) of our algorithm, we reduce this number to n . Phase 3:
Reducing the support size to be at most n .So far we established that there is a distribution D over allocations, giving every agent at least herproportional share ex-ante, and supported on at most 4 n ( m + 1) “good” allocations: allocations that giveevery agent at least half her TPS, and are Prop1. We now explain how to reduce the support size to at most n . Set up the following linear program. For every allocation in the support of D (index these allocationsas G k ) there is a variable z k (representing the probability that G k is selected in our new distribution), andfor every agent i there is a variable y i (representing her ex-ante value). The coefficients a ik denote the valuethat agent i derives from the items allocated to him under allocation G k . The linear program LP4 is asfollows: Minimize z subject to: P k z k = z .2. y i = P k a ik z k for every agent i ∈ N . ( y i represents the ex-ante value of the randomized allocation toagent i .)3. y i ≥ P S i for every i ∈ N . (Every agent gets at least her proportional share ex-ante.)4. z k ≥ G k ( z k is proportional to the probability of G k ).The distribution D shows that the optimal value z ∗ of LP4 satisfies z ∗ ≤
1. LP4 has 2 n + 1 constraints(excluding non-negativity constraints), and hence an optimal basic feasible solution is supported on at most2 n + 1 positive variables. As z and the variables y i are all positive, there are at most n variables z k that arepositive. Scale the z k variables of the optimal solution by z ∗ , so that they form a probability distribution.Likewise, scale the y i variables by z ∗ so that constraint 2 remains satisfied. Constraint 3 also remainssatisfied, as z ∗ ≥
1. The scaled values of the z k variables represent a randomized allocation that provesTheorem 6. We have presented a best-of-both-worlds result, showing that for every allocation instance with additivevaluations, there is a randomized allocation that gives every agent at least her proportional share ex-ante,and at least half of her TPS (and MMS) ex-post. Moreover, we have shown that there is a deterministicpolynomial time algorithm that, given the valuation functions of the agents, computes a faithful implemen-tation of such a randomized allocation, supported on at most n allocations. We next discuss directions in13hich our results can possibly be improved upon, presenting impossibilities of some natural extensions, aswell as some open problems. Theorem 6 guarantees every agent at least half her TPS ex-post. This is nearly the best possible, in thesense that there are allocation instances for which no allocation gives every agent more than a n n − fractionof her TPS. Still, it might be interesting to see if the ex-post BoBW guarantee can be improved to n n − ofthe TPS.For the maximin share, MMS, it might be possible to offer every agent a ρ -fraction of her MMS ex-post,for some < ρ < . Even if so, it is not clear if such a guarantee will be betterthan half the TPS, because the gap between MMS and TPS may be a factor of 2 − n .An alternative BoBW result that one might consider is a result in which we replace the ex-post guaranteeto be EFX. As every EFX allocation gives every agent a n n − fraction of her TPS (see proof in Appendix A),such a result will, in particular, strengthen our result and obtain the best possible TPS fraction of n n − . Amajor hurdle in obtaining such a result is that EFX allocations are not known to always exist, so achievingEFX in the BoBW setting seems to be currently beyond reach. Moreover, we observe that even if EFXallocations do always exist, ex-post EFX conflicts with Pareto optimality, another important property thatwe wish to have. This follows from the next simple example: There are two items ( a and b ), and two agentswith valuations v ( a ) = 1, v ( b ) = 0, v ( a ) = 2 and v ( b ) = 1. This instance has only one allocation that isboth EFX and PO, namely, agent 1 gets a and agent 2 gets b . Hence requiring EFX ex-post does not allowus to obtain BoBW result in which agent 2 gets her proportional share ex-ante, and the allocation is POex-post. As noted in Corollary 8, our BoBW result can be supported on allocations that are Pareto optimal (thoughour polynomial time randomized allocation of Theorem 6 does not guarantee ex-post PO). Some strongereconomic efficiency properties cannot be achieved, as they contradict our ex-post fairness properties. Forexample, ex-ante Pareto optimal (fPO) cannot be achieved, as shown in Proposition 7. Likewise, the next ex-ample shows that one should not attempt to approximately maximize welfare, not even if valuation functionsare normalized and scaled so that every agent has the same value for the set of all items M .Consider the following example with m = n , with n being a perfect square. For every agent i ≤ √ n , everyitem j with j = i modulo √ n has value √ n (and the rest of the items have value 0). For the remaining agents,every item has value 1. Observe that v i ( M ) = n for every agent i , so valuations are indeed normalized. TheTPS of each of the remaining agents is 1, and hence if we want every agent to get at least a constant fractionof her TPS, in every ex-post allocation each of them must receive at least one item. The welfare of everysuch allocation is at most n − √ n + √ n · √ n < n , whereas the maximum welfare allocation gives each ofthe first √ n agents value of n ( √ n items, each of value √ n ), resulting in optimal welfare of n . Thus, anyallocation that gives every agent a constant fraction of her TPS does not approximate the maximum welfare(even when valuations are normalized) to any factor better than Ω( √ n ).This leads us to consider Nash Social Welfare (NSW). We cannot hope to exactly maximize the fractionalNSW ex-ante, as this allocation is fractionally PO, and Proposition 7 shows an impossibility result for thiscase. However, we can hope to get a constant approximation for the maximum NSW, either in an ex-ante sense or in an ex-post sense (or both). It would be interesting to understand whether our allocationalgorithm from the proof of Theorem 6, possibly with a change to the objective of LP3, provides a constantapproximation to the maximum NSW. We discuss here truthfulness aspects for individual agents, and do not address in our discussion more de-manding aspects of group strategyproofness. 14oBW allocation mechanisms are randomized. As such, one may consider either ex-post or ex-antetruthfulness notions. The most straightforward notion is that of universal truthfulness – reporting the truevaluation function is a dominant strategy, with respect to both ex-post and ex-ante values simultaneously, forevery realization of the coin tosses of the randomized allocation. A BoBW result with universal truthfulnesswas achieved in [Babaioff et al., 2020] in the special case of additive dichotomous valuations (and also forsubmodular dichotomous valuations). However, for general additive valuations, there are impossibility resultsfor truthful mechanisms, and they carry over to universally truthful mechanisms. In particular, it is provedin [Conitzer et al., 2017] that every truthful allocation mechanism for two agents that allocates all items must,in some instances, give an agent not more than a m fraction of her MMS. Consequently, every universallytruthful randomized allocation mechanism for two agents that allocates all items must sometimes not givean agent more than a m fraction of her MMS ex-post. Moreover, this implies the next proposition regarding n agents. The proposition is a direct corollary from the result of Amanatidis et al. [2017a]. Proposition 17
Every universally truthful randomized allocation mechanism for n agents and m items thatis ex-post PO must sometimes not give an agent more than an O ( nm ) fraction of her MMS ex-post. This proposition holds by adapting the two agents impossibility result proof, having large enough m andadding one auxiliary item and n − truthful in expectation (TIE), which is an ex-ante notion,and postulates that agents attempt to maximize their expected utility. In the BoBW setting, it is indeedreasonable to assume that agents are expectation maximizers, as these are the type of guarantees that theyare given ex-ante. However, TIE tacitly assumes that agents are not strategic concerning their ex-postguarantees, an aspect that is somewhat problematic in BoBW settings. A TIE BoBW result for additivedichotomous valuations is achieved in [Halpern et al., 2020].None of the BoBW results (the previous Theorems 2 and 1, and our Theorem 6) provides a truthfulmechanism, not even TIE. On the other hand, the weaker and rather trivial Proposition 10 does give a TIEmechanism, but with rather weak BoBW guarantees.In the mechanism of Proposition 10 an agent that maximizes expected utility has no incentive to lie, butalso no incentive to be truthful. Using a trick of Mossel and Tamuz [2010], we can modify that mechanism soas to make truthfulness the unique dominant strategy. After receiving the valuation functions of all agents,first generate a fractional solution A fR at random. If A fR Pareto dominates the uniform fractional allocation,then faithfully implement A fR . If not, then faithfully implement the uniform fractional allocation. Thismechanism is ex-ante proportional and ex-post Prop1.We do not know if there is a TIE mechanism that offers every agent at least a constant fraction of herMMS ex-post (even without requiring any ex-ante guarantee). References
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Approximation, Randomization andCombinatorial Optimization. Algorithms and Techniques , pages 247–253. Springer, 2008.16
Missing Proofs
We first show that there is a polynomial time algorithm that gives every agent (with an additive valuation)at least a n n − fraction of her TPS.We say that an allocation A = ( A , . . . , A n ) is half-fair if for every two agents i and j , if | A j | > v i ( A i ) ≥ v i ( A j ). In other words, if an agent i prefers bundle A j over her own bundle A i , then either A j contains only one item, or A j is at most twice as valuable to i than A i .Proposition 18 below shows that in every half-fair allocation, every agent gets at least a n n − fraction ofher TPS. We note that every EFX allocation is half-fair, and thus gives every agent at least a n n − fractionof her TPS. Unfortunately, EFX allocations are not known to always exist. However, EF1 allocations doalways exist, as shown in [Lipton et al., 2004]. Though EF1 allocations are not necessarily half-fair (recallExample 3), the EF1 allocations generated by the algorithm of Lipton et al. [2004] are half-fair. Hence thealgorithm of Lipton et al. [2004] produces an allocation that gives every agent at least a n n − fraction of herTPS. Proposition 18
Every half-fair allocation A = ( A , . . . , A n ) gives agent i at least a n n − fraction of herTPS. Proof.
Let A = ( A , . . . , A n ) be a half-fair allocation. Recall that T P S i has the property that for everysingle item e , T P S i ( n − , M \ { e } , v i ) ≥ T P S i ( n, M , v i ). Let K denote the set of items that are in bundles(excluding A i ) that contain only a single item, and let M ′ = M \ K . Then T P S i ( n − | K | , M ′ , v i ) ≥ T P S i ( n, M , v i ). As allocation A is half-fair, we have that v i ( M ′ ) ≤ (2( n − | K | ) − v i ( A i ), since the itemsin M ′ \ A i are divided between n − | K | − v i ) ofat most 2 v i ( A i ). Hence: T P S i ≤ T P S i ( n − | K | , M ′ , v i ) ≤ P S i ( n − | K | , M ′ , v i ) ≤ n − | K | ) − n − | K | v i ( A i ) ≤ n − n v i ( A i ) , which concludes the proof. (cid:4) We now restate and prove Proposition 7, showing that ex-ante fPO is in conflict with the ex-post fairnessproperties that we desire.
Proposition 7
For every n ≥ and every ǫ > there are allocation instances with additive valuations,with the following property: for every ex-ante Pareto optimal (fPO) randomized allocation (whether ex-anteproportional or not), every allocation in its support does not give some agent more than a ǫn fraction ofher maximin share. Proof.
Recall the instance of Example 3, where the maximin share of every agent is n . Consider an arbitraryfractional fPO allocation A ∗ for this instance. We may assume that every agent i receives fractions eitherfrom at least one of the items { b , b , . . . , b n − } or from at least two of the items in { s , . . . , s n } , as otherwiseagent i cannot get ex-post a value larger than 1 + ǫ . In either of these cases, agent i holds some fraction ofan item (say, item e j ) different from s i . Fractional Pareto optimality of A ∗ then implies that A ∗ allocates s i in full to agent i . (Otherwise A ∗ can be Pareto improved. Agent i , who values s i more than other agentsdo, and values e j not more than other agents do, can trade a fraction of e j with a fraction of s i , benefitinghimself, and without hurting the agent who originally holds the fraction of item s i .) Consequently, everyagent i receives the corresponding item s i in every ex-post allocation. By the pigeon-hole principle, in anex-post allocation there is an agent that receives no item among { b , b , . . . , b n − } . This agent i receives only s i , and hence only a ǫn fraction of her MMS. (cid:4) Faithful Implementations of Fractional Allocations
In this section we present a self contained explanation of a usage of randomized rounding to obtain BoBWfairness, the concept we refer to as faithful implementation. We provide some historical context as to thedevelopment of various components of it, and present the proof of Lemma 9, which summarizes the resultregarding faithful implementations.Consider a fractional allocation A ∗ of m items to n agents with additive valuations. Denote the fractionalallocation to agent i by A ∗ i , with A ∗ ij denoting the fraction of item j given to agent i in A ∗ . Let M fi = { j | < A ∗ ij < } denote the set of items for which some positive proper fraction (neither 0 nor 1) is allocatedto i , and let f = P i ∈N | M fi | denote the number of variables that are strictly fractional.We consider generating a distribution over integral allocations from the fractional allocation A ∗ (a “round-ing procedure”). We distinguish between three kinds of rounding: • Deterministic rounding.
Produces a single integral allocation. • Randomized rounding.
Produces a distribution over integral allocations. • Implementation.
Randomized rounding, where the expectation of the associated distribution is exactly A ∗ .We consider two notions of polynomial-time algorithms for performing randomized rounding. • Randomized polynomial time.
There is a randomized polynomial time algorithm that samples an integerallocation from the associated distribution. • Deterministic polynomial time.
There is a deterministic polynomial time algorithm that lists all integralallocations in the support of the distribution, together with the associated probability of each allocation.In particular, this implies that the size of the support is upper bounded by some polynomial in n and m .We list several faithfulness properties that may be associated with the rounding.1. Ex-post faithfulness , which satisfy both of the following properties:(a)
Faithfulness from above.
In the rounded integral allocation A , every agent i gets a bundle of valueat most her fractional value, up to the value of one of her fractionally allocated items. That is, v i ( A i ) ≤ v i ( A ∗ i ) + max j ∈M fi v i ( j ).(b) Faithfulness from below.
In the rounded integral allocation A , every agent i gets a bundle of valueat least her fractional value, up to the value of one of her fractionally allocated items. That is, v i ( A i ) ≥ v i ( A ∗ i ) − max j ∈M fi v i ( j ).For an implementation of a fractional allocation, Ex-post faithfulness follows from the following singleproperty: • Small spread.
For every agent i , the difference in values that i receives in any two rounded integralallocations is at most max j ∈M fi v i ( j ).We refer to a distribution over allocations as a faithful implementation of A ∗ if it is an implementationthat satisfies small spread.2. Ex-ante faithfulness.
In the randomized rounding, every agent i gets in expectation value at leastequal to her fractional value. E [ v i ( A i )] ≥ v i ( A ∗ i ). Observe that by definition, an implementation ofthe fractional allocation is ex-ante faithful. 18aithful rounding of fractional solutions has a long history, where in different times researchers addedadditional ingredients (from those mentioned above) that they wished to satisfy. We briefly mention a fewpast relevant works.Independent randomized rounding has numerous applications for approximation algorithms. The round-ing allocates each item to at most one agent, independently of the allocation of other items. That is, eachitem j is independently (from other items) allocated to at most a single agent, with each agent i gettingitem j with probability equal to A ∗ ij . This procedure provides a randomized polynomial time implementa-tion for the fractional allocation (and hence is ex-ante faithful), but it does not provide ex-post faithfulnessguarantees.Deterministic (polynomial time) rounding that is faithful from above was developed in [Lenstra et al.,1990] in the context of scheduling problems. For allocation problems, faithfulness from below is a morenatural requirement, and this version was presented in [Bez´akov´a and Dani, 2005]. A randomized polynomialtime faithful implementation (showing that the small spread property holds and making explicit use it) waspresented in [Srinivasan, 2008]. A randomized polynomial time faithful implementation for a more generalsetting (referred to as a bi-hierarchy) was presented in [Budish et al., 2013]. Later work was concernedwith deterministic (rather than randomized) polynomial time faithful implementations, with one approachdescribed in [Freeman et al., 2020], and a somewhat simpler approach presented in [Aziz, 2020]. Summarizingthe above discussion, and marginally improving over it (in terms of the upper bound on the support of thedistribution), we have the following lemma. Lemma 9
Let A ∗ be a fractional allocation of m items to n agents with additive valuations, and let f denotethe number of strictly fractional variables in A ∗ (number of pairs ( i, j ) such that in A ∗ , the fraction of item j allocated to agent i is strictly between 0 and 1). Then there is a deterministic polynomial time implementationof A ∗ , supported only on allocations in which every agent gets value (ex-post) equal her ex-ante value (inthe fractional allocation A ∗ ), up to the value of one item. (For agent i , the corresponding one item is theitem most valuable to i , among those items that are assigned to i under A ∗ in a strictly fractional fashion.Moreover, the values that the agent gets in any two allocations differ by at most the value of this single item.)The distribution of the implementation is supported over at most f + 1 allocations. Proof.
The proof of the lemma has two parts, neither one of them is new. The first (and main) part provesthe lemma but without the upper bound of f + 1, and the second part observes that standard techniquesreduce the support to size f + 1.For the first part, we sketch for completeness the proof approach of Aziz [2020]. Recall the Birkhoff –von Neumann theorem that says that every doubly stochastic matrix can be decomposed into a weightedsum of permutation matrices. Equivalently, every perfect fractional matching in a bipartite graph can bedecomposed into a weighted sum of perfect (integral) matchings. Moreover, this can be done in polynomialtime, via repeatedly finding and peeling off a bipartite perfect matchings.We reduce the setting of Lemma 9 to that of the Birkhoff – von Neumann theorem, showing how we cantake A ∗ and generate a distribution over matchings of “clones” of each agent, that can be use to generate adistribution over allocations that is a faithful implementation of A ∗ . For every agent i we do the following.Let f i = P j A ∗ ij denote the total sum of fractions of items (not their values) received by i under A ∗ . Wereplace i by ⌈ f i ⌉ clones c i , . . . , c ⌈ f i ⌉ i as follows. Sort all items in order of decreasing v i value. This gives apriority order for the following sequential “eating” process. The clones of i “eat” the fractional allocation of i , where each clone in its turn consumes one unit of the fractional allocation (starting consuming only afterthe prior clone completed consuming), where the unit is chosen according to the priority order. The lastclone might have less than a single unit to consume.Having done the above for all agents, we now have a fractional matching between clones and items. Thisis not a perfect fractional matching (the last clone of an agent may consume less than one item), but theBirkhoff – von Neumann theorem still applies (e.g., one can add dummy clones and items as needed soas to complete the instance to a perfect fractional matching on a larger bipartite graph). Hence we candecompose the fractional matching into integral matchings. In every integral matching, every agent gets theitems received by her clones. 19x-post faithfulness follows from the fact that for every agent i , in every integral allocation, each of i ’s clones (except for perhaps the last one) receives one item. Let S i, max ( S i, min , respectively) be the setof items obtained by taking for each of i ’s clones the highest priority (lowest priority, respectively) itemthat the clone may possibly receive. Then every allocation that agent i may receive has value in the range[ v i ( S i, min ) , v i ( S i, max )]. Observe that v i ( S i, min ) ≥ v i ( S i, max ) − max j ∈M fi v i ( j ). This last statement can beverified by removing the most valuable item (that of clone 1) from S i, max , and then using the fact that forevery j ≤
1, the item of clone j in S i, min is at least as valuable as the item of clone j + 1 in S i, max . Thisestablished the small spread property, which implies ex-post faithfulness.The first part of the proof provided a deterministic polynomial time implementation of A ∗ as a distribution D over polynomially many allocations A , A , . . . A ℓ , where every allocation in the support is ex-post faithful.In the second part we reduce the size of the support to f + 1. For this we set up a linear program. Thevariable x k specifies the extent to which we include allocation A k in the new implementation of A ∗ . The set F contains those pairs ( i, j ) for which in A ∗ , agent i is allocated a strictly fractional part of item j , and A ∗ i,j denotes this fraction. Observe that | F | = f . For every allocation A k in the support of D , we use A ki,j as anindicator of whether item j is allocated to agent i in A k . The A ∗ i,j and A ki,j values serve as coefficients inour LP. The constraints of the LP are (the objective function can be set to 0):1. P ≤ k ≤ ℓ x k = 1.2. P i ∈N , ≤ k ≤ ℓ A ki,j x k = A ∗ i,j for every ( i, j ) ∈ F .3. x k ≥ ≤ k ≤ ℓ .The above LP is feasible, as the probabilities that D assigns to each A k serve as a feasible solution.In polynomial time, one can find a basic feasible solution to the LP. The number of non-zero variables inthis solution is no larger than the number of constraints (excluding the non-negativity constraints), whichis f + 1, as desired. (cid:4)(cid:4)