Dominant Resource Fairness with Meta-Types
DDominant Resource Fairness with Meta-Types
Steven Yin, Shatian Wang, Lingyi Zhang, Christian Kroer IEOR Department, Columbia [email protected], [email protected], [email protected], [email protected]
Abstract
Inspired by the recent COVID-19 pandemic, we study a gen-eralization of the multi-resource allocation problem with het-erogeneous demands and Leontief utilities. Unlike existingsettings, we allow each agent to specify a constraint to onlyaccept allocations from a subset of the total supply for eachresource. Such constraints often arise from location con-straints (e.g. among all of the volunteer nurses, only thosewho live nearby can work at hospital A due to commute con-straints. So hospital A can only receive allocations of volun-teers from a subset of the total supply). This can also modela type of substitute effect where some agents need 1 unitof resource A or B, but some other agents specifically wantA, and some specifically want B. We propose a new mech-anism called Group Dominant Resource Fairness which de-termines the allocations by solving a small number of linearprograms. The proposed method satisfies Pareto optimality,envy-freeness, strategy-proofness, and a notion of sharing in-centive for our setting. To the best of our knowledge this is thefirst mechanism to achieve all four properties in our setting.Furthermore, we show numerically that our method scalesbetter to large problems than alternative approaches. Finally,although motivated by the problem of medical resource al-location in a pandemic, our mechanism can be applied morebroadly to resource allocation under Leontief utilities withaccessibility constraints.
The recent COVID-19 pandemic has brought forward anumber of questions that are particularly relevant to theoperations research community. While infectious diseasespread modeling and resource demand forecasting provideguidance for the policy making process, an equally impor-tant and often overlooked problem is the effective and fairallocation of resources, such as volunteer medical workers,ventilators, emergency field hospital beds, personal protec-tive equipment, etc.There are several challenges unique to the medical re-source allocation problem in the face of an infectious diseaseoutbreak.
First , utilities from different types of resources arenot additive nor linear. For example, when there are enoughnurses but not enough doctors, the marginal utility of hav-ing one additional nurse on staff is very low.
Second , notall resources are accessible to all hospitals / organizations( agents ). For instance, the home location of each volunteer medical worker largely affects where she can commute towork; thus, she can only be assigned to agents within hercommutable radius.
Third , agents have different capacities(big medical centers versus small hospitals) and are in differ-ent stress levels (hospitals in an epicenter versus the ones inrural areas with few cases), so they naturally have different priorities over the resources.Another setting that has the above characteristics is thecompute resource sharing problem with sub-types. For ex-ample, suppose a compute server has several compute nodes,and there are different types of GPU/CPU on the variousnodes (e.g. NVIDIA vs. AMD GPU, size of RAM on theGPU card, Intel vs AMD CPU, etc). Some users might belooking for a specific hardware configuration while othersmight be less selective.In this paper, we propose a new market mechanism thattackles the three challenges outlined above and achieves de-sirable fairness properties including Pareto optimality, envy-freeness, strategy-proofness, and sharing incentive. In ournumerical experiments, we demonstrate that compared tothe Maximum Nash Welfare (MNW) approach, our mech-anism is cheaper to compute (sometimes significantly) andenjoys theoretical properties that MNW does not have.
There has been a flurry of recent papers coming out of theoperations research, statistics, and computer science com-munities addressing various aspects of the pandemic. Wefocus on literature that deals with emergency resource al-locations in the aftermath of a pandemic, as well as generalfair division techniques that are most relevant in our setting.
Much work has been done to address the various sup-ply shortages caused by COVID-19. Many studied medi-cal resource allocations from a qualitative perspective, ad-dressing ethical and medical questions in a pandemic (Barret al. 2008; Emanuel et al. 2020; Truog, Mitchell, and Da-ley 2020). From a mechanism design view point, (Jalota,Pavone, and Ye 2020) proposed a mechanism for allocationof public goods that are capacity constrained due to socialdistancing protocols, focusing on achieving a market clear-ing outcome. (Mehrotra et al. 2020) studied the allocation of a r X i v : . [ ec on . T H ] S e p entilators under a stochastic optimization framework, mini-mizing the expected number of shortages in ventilators whilealso considering the cost of transporting ventilators. (Kanter2015; Powell, Christ, and Birkhead 2008) provided guide-lines for deciding whether a patient should be allocateda ventilator. (Zenteno 2013) combined influenza modelingtechniques with robust optimization to handle workforceshortfall in a pandemic. (Xiang and Zhuang 2016) mod-eled the deteriorating health of victims using a stochas-tic model and optimized for the expected number of sur-vivals. (Arora, Raghu, and Vinze 2010) studied the trade-offs between building redundant capacity and using mutualaid. (Boreskie, Boreskie, and Melady 2020) proposed guide-lines for physicians when deciding if a patient should be al-located an ICU unit. All of these papers focus on optimizingsome objective function but do not address fairness proper-ties. Our work serves as a complement to the existing work. Under a fairly general class of utility functions includingthe Leontief utility, computing market equilibrium underthe fisher market setting (divisible goods) can be done us-ing an Eisenberg-Gale (EG) convex program (Eisenberg andGale 1959). Market equilibrium solutions satisfy Pareto op-timality, proportionality, and envy-freeness. It is also knownthat EG convex program implicitly maximizes Nash welfare,which is the product of all agents’ utilities. However, MNWis generally not strategyproof, and can be computationallyexpensive for large problems.For Leontief utilities, (Ghodsi et al. 2011a) introduced theDominant Resource Fairness allocation mechanism (DRF)which in addition to the three properties satisfied by marketequilibrium solutions, is also strategyproof. Later (Parkes,Procaccia, and Shah 2015) extended the setting to allowagents to have priority weights as well as zero demand oversome resources while maintaining all four desiderata. How-ever they do not handle the accessibility constraints in oursetting, and as we discuss later in the paper, a naive adapta-tion of this generalized DRF mechanism to our setting doesnot yield Pareto optimal allocations in general.For indivisible resources, (Caragiannis et al. 2019)showed that maximizing Nash welfare under the indivisiblesetting satisfies envy-freeness up to one resource unit andhas nice guarantees on the Max-Min Share ratio. Althoughexact market equilibrium might not exist in indivisible set-tings, (Budish 2011) showed that a close approximation ofit exists in the unweighted, binary allocation case. This waslater put into practice for course allocation in (Budish et al.2017). However the theory does not provide useful approx-imation bounds when assignments are not binary (e.g., eachstudent only needs one seat from each class, but each hospi-tal may require hundreds of doctors). Furthermore, the exist-ing computational approaches do not scale to the size of theproblems encountered in pandemic situations, where tens ofthousands of volunteers need to be allocated to hundreds ofhospitals.
For the remainder of the paper we use local medical per-sonnel allocation as a running example, even though otherresource allocation problems can be formulated in a similarfashion. We group resources into meta-types : volunteer doc-tors, nurses, ventilators, emergency field hospital beds, etc.Within each meta-type (e.g., doctors), we have types (e.g.,doctors from the Bronx, doctors from Brooklyn, doctorsfrom Manhattan, etc. ). We assume that demands are givenover meta-types (e.g., a hospital is indifferent to where doc-tors assigned to it come from). However, each agent some-times can only receive allocation from a subset of the re-sources in a meta-type because of constraints such as loca-tion (e.g., a hospital in the Bronx might only accept vol-unteer doctors from the Bronx and Manhattan because thecommute would be too long otherwise). We refer to suchsubsets of resource types in each meta-type as the agents’demand groups .We use Ω , Ω , · · · , Ω L to denote the meta-types. Eachmeta-type Ω l is a set that contains resource types which be-long to it. We assume that Ω i ∩ Ω j = ∅ for any two differentmeta-types i, j and use R to denote the set of all resourcetypes: R = ∪ l ∈ [ L ] Ω l , and m = | R | to denote the total num-ber of resource types. Note that each resource type belongsto only one meta-type. We let N be the set of agents, and n = | N | be the total number of agents.We use S r to denote the supply of resource type r . Weassume that the supplies are normalized within each meta-type: (cid:88) r ∈ Ω l S r = 1 ∀ l ∈ [ L ] . Each agent i ∈ N submits a similarly normalized demandvector [ d i , . . . , d iL ] where d il denotes the fraction of avail-able units among meta-type l that agent i needs in order toget one unit of utility (one can think of this as each agenttrying to complete as many units of work as possible, whereeach unit of work requires d il units of meta-type l ). Addi-tionally, each agent also submits a group structure constraintin the form of a set of demand groups.Let G i = { g il ⊆ Ω l : l ∈ [ L ] , d il > } , be the set of de-mand groups for agent i , where g il ⊆ Ω l is agent i ’s demandgroup for l , specifying the subset of resource types belong-ing to meta-type l that agent i accepts/can access. Note thatwe only include in G i demand groups for meta-types thatagent i has non-zero demand of. This is to simplify notationin the later analysis. Intuitively, the introduction of meta-types models the substitution effects, and the introduction ofdemand groups models the accessibility constraints. When i is clear from the context, we sometimes use g l instead of g il to simplify the notation.We also allow each agent i to have a different prior-ity weight w il for each meta-type l . Note that if we let w i = . . . = w iL , then this reduces to having a single pri-ority weight for each agent. We assume weights are normal-ized within each meta-type: (cid:80) i ∈ N w il = 1 for l ∈ [ L ] . Manhattan, the Bronx, and Brooklyn are three boroughs ofNew York City. igure 1: All three hospitals can accept both types of doc-tors. However, hospitals I and II can only accept Nurse typeC, while hospital III accepts only Nurse type D.Using these notations, we further define l ∗ i := arg min l ∈ [ L ] w il d il d i ∗ := d il ∗ i w i ∗ := w il ∗ i Namely, l ∗ i is the meta-type from which agent i demands thebiggest proportional share, adjusted by the priority weights.We refer to l ∗ i as the dominant resource meta-type for agent i . d i ∗ is the proportional share demanded by agent i from itsdominant resource meta-type to finish one unit of work.For each meta-type l , (cid:80) r ∈ g il x ir is the fraction of the totalsupply of meta-type l that is assigned to agent i , where x ir is the assignment of each individual resource type. We use x i to denote the allocations vector of agent i and x to denotethe assignment matrix that encodes the allocations vectorsof all agents. We define the utility as such: u i ( x i ) := min g l ∈ G i (cid:40) d il (cid:88) r ∈ g l x ir (cid:41) . (1)Since agent i needs d il proportion of the supply units fromeach meta-type l to finish a unit of work, u i ( x i ) is the totalunits of work that agent i can finish given allocation vector x i . This form of utility measure is called a Leontief utility .We now give a concrete example, which is also illustratedin Figure 1. To keep the example simple we assume thateach agent has the same weight over all meta-types: onlyone weight w i is defined for each agent i . Example 1.
Consider a case of three agents (hospitals) { , , } and two resource meta-types. The first meta-typeconsists of two types of doctors (resource A, B ), and thesecond consists of two types of nurses(resource
C, D ): Ω = { A, B } , Ω = { C, D } . The normalized weights for the threehospitals are: w = w = , w = . The supply foreach type of doctor and nurse is 500. Thus, the total avail-able units of each meta-type is
500 + 500 = 1000 , and S r = = ∀ r ∈ { A, B, C, D } . All three hospitalscan accept both types of doctors but hospitals , can onlyaccept nurse type C while the third hospital only acceptsnurse type D : G = { g = { A, B } , g = { C }} , G = { g = { A, B } , g = { C }} , G = { g = { A, B } , g = { D }} . Hospital 1 demands 4 doctors and 1 nurse for ev-ery unit of work. Hospital 2 demands 1 doctor and 4 nurses for every unit of work. The third hospital demands 1 doc-tor and 1 nurse for every unit of work. Since the total unitsof supply for each meta-type is , d = [ , ] , d = [ , ] , d = [ , ] . We now formally define the fairness properties we discussin this paper.
Pareto optimality
An allocation mechanism is Pareto op-timal if compared to the output allocation x , there does notexist another allocation x (cid:48) where some agent is strictly bet-ter off without some other agent being strictly worse off: ∃ i s.t. u i ( x (cid:48) i ) > u i ( x i ) = ⇒ ∃ j s.t. u ( x (cid:48) j ) < u ( x j ) . Weighted envy-freeness u i (cid:16) x jr w il w jl ∀ r ∈ g il , l ∈ [ L ] (cid:17) − u i ( x i ) is how much i envies j . An allocation is weightedenvy free if this quantity is non-positive for any i, j ∈ N : u i (cid:18) x jr w il w jl ∀ r ∈ g il , l ∈ [ L ] (cid:19) ≤ u i ( x i ) Intuitively, this means an agent prefers her allocation overthe allocation of any other agent scaled by the weight ratiosof the two agents. Note that since there is a separate weightfor every meta-type l , the allocations for each resource type r is scaled according the corresponding weight for the meta-type that it belongs. Strategy-proofness
In the existing literature, agents canonly be strategic by misreporting their demand vector. In oursetting however, the agents have the additional possibility ofmisreporting their accessibility constraints/demand groupsfor the meta-types (e.g. One can report that she accepts bothIntel and AMD CPUs but in fact my program only runson Intel CPU). Our definition of strategy-proofness guardsagainst both types of misreporting.Let x be the allocation returned by the mechanism undertruthful reporting from all agents. Let x (cid:48) be an allocationreturned by the mechanism when all agents report truthfullyexcept agent i reports an alternative demand vector and/oralternative demand groups. The mechanism is strategy-proofif u i ( x i ) ≥ u i ( x (cid:48) i ) . Sharing Incentive
In settings where the supplies for eachresource comes from the participants’ contribution, sharingincentive is satisfied when the resulting allocation gives eachparticipant at least as much utility as she could have gottenwithout participating in the pool. More specifically, for each i ∈ N and l ∈ [ L ] , let s il be the proportion of meta-type l contributed by agent i that she can also access . We canalso think of s il as the amount of “useful” resource agent i originally possessed of meta-type l (she might contributemore than s il to the pool, but we only count the part thatshe can access herself). Without being part of the resourcesharing, agent i ’s utility would be u oi := min g l ∈ G i (cid:26) s il d il (cid:27) . haring incentives says that u i ( x i ) ≥ u oi ∀ i ∈ N , i.e.,all agents have incentives to share (pool) their individual re-sources for reallocation. We also define a related conceptcalled proportionality: Proportionality
An allocation x satisfies proportionalityif u i ( x i ) ≥ u i ( x (cid:48) i ) for all i , where x (cid:48) ir = w il S r for each r ∈ g il and l ∈ [ L ] . u i ( x (cid:48) ) can be explicitly written out as u i ( x (cid:48) ) := min g l ∈ G i (cid:40) w il d il (cid:88) r ∈ g l S r (cid:41) . Remark . In the existing resource allocation literature,sharing incentive and proportionality are often used inter-changeably. Indeed, when the priority weights are set ac-cording to the agents’ proportional accessible contributionsto the resource pool ( w il = s il ), the two notions are equiv-alent in settings where there are no accessibility constraints.With accessibility constraints, however, they are not equiv-alent. Under our definition of proportionality, the amountof accessible resource meta-type l that agent i receives is s il · (cid:80) r ∈ g il S r . Since (cid:80) r ∈ g il S r < if agent i cannot ac-cess the entire supply of meta-type l , u i ( x (cid:48) ) can be smallerthan u oi . Therefore when priority weights are set accordingto agents’ contributions, proportionality is a weaker notionthan sharing incentive. However, since proportionality canbe defined for arbitrary weights, regardless of whether or notwe are in a setting where agents bring their own supplies, itis a more flexible concept. We focus on Sharing Incentive inthe main paper and provide a detailed discussion of propor-tionality in the Appendix. We now present our fair allocation mechanism, whichwe call
Group Dominant Resource Fairness (GDRF). Themechanism proceeds in rounds and agents are gradually“eliminated”. In each round t , we use the linear programdescribed in Equation 2 to maximize a fractional value y t sothat each remaining agent i can receive at least y t × w i ∗ frac-tion of the total supply from its’ dominant resource meta-type l ∗ i , and more generally y t × w i ∗ × d il /d i ∗ fractionof each demanded meta-type l . Based on this solution, we An alternative is to make each remaining agent receive a y t × w i ∗ × d il /d i ∗ fraction of the total supply from each of its resourcegroup g l ∗ i , as agent i can only derive utilities from resources in g l ∗ i ⊆ Ω l ∗ i . To do so, we can multiply the left hand side of theallocation constraints in Equation 2 by (cid:80) r ∈ g il S r . This alternativesetup, however, does not lead to a mechanism with envy-freenessand strategy-proofness.As a simple example, assume that there are five agents , , , , of equal weights, one meta-type, and two resource types A, B thatfall under this meta-type, with equal supply. Agent , accept onlytype A ; agent , , accept only type B . With simple calculation,we have that the largest y we can get is 1/3: everyone receives / of their accepted supply. The only possible allocation to achievethat is by assigning 1/3 of A each to agents 1,2, and 1/3 of B each toagents 3, 4, 5. However, if agent strategically stated that he couldtake both A and B , the resulting allocation would be assigning 1/3 eliminate at least one resource and one agent using Defini-tion 1 and 2 (although the algorithm only needs to explicitlymaintain a list of active/eliminated agents, not resources).For each agent i eliminated in round t , we set γ i = y t . Wefix the fraction of dominant meta-type l ∗ i assigned to agent i to γ i × w i ∗ , without fixing the specific allocations of theresources . It is not too hard to show the following (proof isin the Appendix): Fact 1.
For any round t , the allocation constraints in Equa-tion 2 for i / ∈ N t have to be tight for optimal solutions. This fact effectively says that when an agent is eliminated,her utility in the final allocation is decided, even though theexact allocation is not. Not fixing the allocation is a deliber-ate choice because agents who are flexible with their demandgroups should accommodate agents who are more restrictive(ex: if agent 1 accepts both type A and B, and agent 2 onlyaccepts type A, then we should allocate agent 1 mostly typeB resource, and leave type A resource for agent 2). Howeverwhen the number of agents and resource types is large, it isdifficult to characterize such dynamics explicitly. So it’s cru-cial to not fix the allocation to the agents until the very lastiteration.We will show that there is at least one new resource andone agent being eliminated in each round. Thus our algo-rithm requires at most min( m, n ) rounds (in practice it of-ten terminates in 2-3 rounds even with a large number ofresources types and agents). Since each round requires solv-ing a polynomial-sized linear program, the overall procedurecan be run in polynomial time.Let N t , R t be the set of active agents and resources atthe beginning of round t . The LP for round t is defined asfollows. max y t s.t. (active agents allocation constraints) y t × w i ∗ × d il d i ∗ ≤ (cid:88) r ∈ g l x ir ∀ i ∈ N t , g l ∈ G i (eliminated agents allocation constraints) γ i × w i ∗ × d il d i ∗ ≤ (cid:88) r ∈ g l x ir ∀ i (cid:54)∈ N t , g l ∈ G i (2)(supply constraints) (cid:88) i ∈ N x ir ≤ S r ∀ r ∈ R (non-negativity constraints) x ir ≥ ∀ i ∈ N, r ∈ R The allocation constraints are saying that remaining agentsneed to receive at least y t × w i ∗ fraction of the total supplyof their dominant resource meta-type. The eliminated agentsonly need γ i × w i ∗ where γ i was determined in a previousround. Note that the ratio d il d i ∗ is simply making sure that of A to agent 1, 2/3 of A to agent 2, and 1/3 of B each to agents3, 4, 5. In this new allocation, the largest y is still / , but sincethe total accepted supply for agent 2 is larger, he receives more.Furthermore, agent 1 would now envy agent 2. here is no waste in the allocation. For an agent who hasbeen eliminated, γ i w i ∗ d i ∗ is her final utility. If agent i is not yeteliminated after round t , then y t w i ∗ d i ∗ represents how muchutility she is currently guaranteed to receive (it will neverdecrease in later rounds, see Fact 2). Fact 2.
The optimal value for Equation 2 increases over thenumber of rounds: y ∗ ≤ y ∗ ≤ ... , where y ∗ t is the optimalobjective function value of the LP in round t . This follows because the constraints on eliminated agentsare less restrictive than the constraints on active agents, andthe set of active agents is decreasing over time.
Definition 1.
Resource r is eliminated when (cid:80) i ∈ N x ir = S r for every optimal x .By Fact 2 it is also easy to see that the set of remainingresources R t decreases over time. Definition 2.
We give two equivalent definitions for elimi-nating agents: • Agent i is eliminated in round t when there exists g l ∈ G i such that g l ∩ R t +1 = ∅ . • Agent i is eliminated in round t when there exists g l ∈ G i such that y t × w i ∗ × d il d i ∗ = (cid:80) r ∈ g l x ir for every optimal x, y t .It is not hard to see that they are equivalent: both defini-tions are saying that the agent can not improve her utilityfurther in (2). We provide a proof in the Appendix. We nowpresent the full algorithm: Algorithm 1:
Group Dominant Resource Fairness(GDRF) Input: Agents N , resources R , supplies S r ∀ r ∈ R ,demand groups G i ∀ i ∈ N , normalized demands d il ∀ i ∈ N, g l ∈ G i , priority weights w il ∀ i ∈ N, l ∈ [ L ] Initialize N = N for t ← , , , ... do y ∗ t ← Solve Equation 2 Update the remaining active agents N t +1 (usingClaim 2) for agent i eliminated in this round do γ i ← y ∗ t end if N t +1 = ∅ then Solve Equation 2 and assign resourcesaccording to x ir with rounding break end end First we address the question of whether the GDRF canbe efficiently implemented. We defer most of the proofs tothe Appendix, but include the proof of Claim 1 here becauseit is a good representation of the flavor of arguments that weuse to prove many of the other results
Claim 1.
In each round t , at least one remaining resource r ∈ R t and one remaining agent i ∈ N t is eliminated. Proof. Suppose no resource is eliminated, then for each r ∈ R t , there exists an optimal solution such that (cid:80) i ∈ N x ir , thereexists r (cid:48) ∈ g il such that r (cid:48) is not eliminated. By the sameconvex combination argument above, we know that there isan optimal solution such that (cid:80) i x ir (cid:48) < S r (cid:48) for every such r (cid:48) . Then for every such agent we can remove (cid:15) allocationof r from her and replace it with (cid:15) allocation of the corre-sponding r (cid:48) . This gives us an allocation that has the sameobjective as before without using up the entire supply of r ,contradicting r being eliminated.This result shows that our algorithm can be imple-mented efficiently by solving at most min( m, n ) number ofpolynomial-size linear programs. However, it does not tellus how to find the eliminated agents. The following theoremsays that we can do so by looking at the dual variables of theLP. Note that the algorithm does not need to explicitly main-tain a list of active resources (Equation 2 does not depend on R t ). Claim 2.
This claim has two parts: • If the shadow price of an active allocation constraint inthe LP for round t is positive, then its corresponding agentneeds to be eliminated in round t . • In each round t , at least one allocation constraint corre-sponding to an agent in N t has a positive shadow price. Now we state our main results.
Lemma 1.
GDRF is Pareto optimal.
Lemma 2.
GDRF is weighted envy-free.
Lemma 3.
GDRF is strategy-proof.
The proofs for these three lemmas all involve a case anal-ysis of different scenarios and showing that the undesirableoutcomes violate either the optimality of the LP or the def-inition of eliminated resources/agents, similar to the argu-ments presented in the proof Claim 1.
Lemma 4.
Assume that demands, weights and supplies areall rational numbers. If priority weights of the algorithmare set according to the each agent’s contribution to the re-source pool (for each meta-type), then GDRF satisfies Shar-ing Incentive.
The proof constructs a bipartite graph of supplies anddemands of the resources, then uses Hall’s Theorem (Hall1935) to show that there exists a feasible solution to the firstround’s LP that already gives every agent at least as muchutility as they could get without participating in the pool.ince agents’ utilities only increase in later rounds, the finalallocation must also satisfy sharing incentive.
The core difference between our problem setup and the ex-isting DRF settings (Ghodsi et al. 2011b; Parkes, Procac-cia, and Shah 2015) is the added accessibility constraints.Namely, there are resource meta-types Ω , Ω , · · · , Ω L , andeach agent i can only accept a subset g il ⊂ Ω l of each meta-type l . When no such allocation constraint is present, as in(Ghodsi et al. 2011b; Parkes, Procaccia, and Shah 2015), itis equivalent to set | Ω l | = 1 for each l ∈ [ L ] in our setting.Then the meta-types notion collapses to the normal resourcetypes notion used in the DRF settings, and our GDRF mech-anism is equivalent to the DRF mechanisms introduced in(Ghodsi et al. 2011b; Parkes, Procaccia, and Shah 2015).Note that in these simplified settings one can write out theclosed form solution to the Linear Programs, so no actualoptimization needs to be performed. As discussed in Section 2, the other most suitable ap-proach in our setting is the Maximum Nash Welfare ap-proach. When the weights are equal, it is also commonlyreferred to as the Competitive Equilibrium with Equal In-come (CEEI) approach. It can be shown that MNW indeedis still a market equilibrium, even with our new accessibilityconstraints, which means that it satisfies Pareto Optimality,envy-freeness, and sharing incentive. However it is knownthat MNW is not strategyproof (see Section 6 of (Ghodsiet al. 2011b) for an example). We show in Section 5 that ourGDRF mechanism also works better in practice in terms ofcomputational cost.
We currently assume that resources and demands follow ameta-type/group/type structure. One might be interested ina general group structure where a demand group can containany subset of all possible resources (not necessarily froma single meta-type). The problem with this kind of flexiblegroup structure is that it opens up possibilities for people tocheat the system by misreporting their true demand structure(e.g.: instead of reporting that I am indifferent to resource Aand B, and that I only need one unit of either one to finisha unit of work, I claim that I need one unit each from bothA and B to finish one unit of work). In particular, DominantResource Fairness based approaches will likely not work,since it is unclear how one would even define the dominantresource under such a general setting. We leave this as anopen question for future work.
So far we have implicitly assumed that the resources are di-visible, and all fairness results are stated with respect to the fractional assignment output of Algorithm 1. In practice weround down the output to obtain the final assignment, sinceresources such as ventilators are indivisible, Each agent loses at most 1 unit of assignment of each typeof resource through rounding. Because we focus on prob-lems where each agent receives hundreds of units of each re-source, the performance loss due to rounding is often small.For example, starting with an envy-free fractional allocation,one agent can envy another by at most m items after round-ing. In Example 1 m = 4 , while the total allocations eachagent receives are in the hundreds. So an envy of m itemsis not significant. Note that such divisibility assumption isalso standard in existing DRF literature, which often focuson the compute resource sharing problem: even though CPUcores are discrete, it’s common to treat the problem as a con-tinuous problem since there is a large quantity of cores in acompute cluster.There are many existing works that focus on fair alloca-tion of indivisible goods (Caragiannis et al. 2019). Indivis-ible resource allocation is particularly important when thequantities of the resources are small (for instance, fairly as-signing a car, a house, and a ring to two people). However, asis the case with most discrete optimization problems, thesealgorithms don’t scale well to the sizes that we deal with in apandemic with hundreds of hospitals and tens of thousandsof volunteers / ventilators, or in the cloud compute settingwhere each data center contains millions of CPU cores. Insettings where each agent receives hundreds of units of eachresource, the performance loss due to rounding is often smallcompared to the dramatic increase in computational cost forsolving Mixed Integer Programs. In this section we compare GDRF with the MNW approachusing three metrics: social welfare, max envy, and compu-tation time. Social welfare is defined to be the sum of theagents’ utilities. Max envy is the maximum amount of envybetween any pair of agents (see Subsection 3.1). Note thatthese metrics are computed on the rounded allocations, sinceboth approaches output fractional allocations. Additionally,both approaches are envy-free so the non-zero envy pre-sented in this section is the result of rounding (which as wewill show later, is very small).Solving for MNW is an exponential cone program, whichuntil recently did not have a reliable commercial solver. Thischanged with the introduction of MOSEK version 9, whichadded support for such cones (ApS 2020). We implementedthe MNW using the MOSEK solver and our GDRF approachusing GUROBI. We made little effort to optimize either ap-proach beyond the off-the-shelf implementations.The first experiment compares the social welfare achievedby GDRF and MNW. We use the following meta-type struc-ture: Ω = { } , Ω = { , } , Ω = { , , } , Ω = { , , , } . The group structure/accessibility constraints aregenerated randomly for each agent in each trial. The de-mands and weights (before normalization) are sampled uni-formly from [1 , , and the number of agents is n = 5 .The supply for each resource is uniformly sampled from [ n × , n × . We ran for 300 trials. As shown inFigure 2, the bar graph shows that the mean social welfareachieved by the two algorithms over all trials are very close.igure 2b shows the normalized difference in social wel-fare between MNW and GDRF, where the normalized dif-ference is defined by the difference between the social wel-fare achieved by the two algorithms, divided by the socialwelfare achieved by the MNW approach. 95% of the trialshave a normalized difference of less than 13%, which meansthat in 95% of the times, our algorithm obtained at least of the social welfare achieved by the MNW approach. (a)(b) Figure 2: Mean social welfare over all trials (a) and normal-ized difference between social welfare achieved by the twoalgorithms (b).Figure 3 shows that rounding introduces very little envyin terms of utility. The error bar shows the th and th percentile.Figure 3: Average Max Envy over all trials.For the next experiment we keep the same meta-typestructure but scale up the number of agents. We plot the runtime for the two algorithms in Figure 4a. The error regionrepresents one standard deviation from the mean, calculatedfrom 16 iterations. GDRF has noticeably larger variance in its’ run time. This is because the number of LP’s we are solv-ing in each instance may vary, while the MNW approachalways solve only one optimization problem. The overheadof formulating a new optimization problem in each iterationof GDRF using GUROBI already significantly increases theoverall compute time. Despite the extra overhead, our ap-proach is still significantly faster.We then fix the number of agents to n = 5 , the number oftypes of resources in each meta-type to , and then scale upthe number of meta-types. The supply for each resource isstill sampled from [ n × , n × as before. As shownin Figure 4b, our algorithm is orders of magnitude faster inthis setting. The error region is again computed from 16 it-erations. (a)(b) Figure 4: Run time comparison of GDRF and MNW.
We proposed a novel resource allocation mechanism for ageneralized demand structure under Leontief utilities. Our meta-type/type/group demand structure models the substitu-tion and accessibility constraints that are common in appli-cations.Our linear programming-based mechanism is Pareto op-timal, envy free, strategy-proof, and satisfies sharing incen-tive. We drew an interesting connection to Hall’s theorem inour proof of sharing incentive. To the best of our knowledge,we are the first to propose a fair resource allocation mecha-nism with these properties for this general demand structure.Finally, we leave as future work to design a mechanism thatworks with more flexible group demand structures. eferences
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Missing Proofs of Results
A.1 Proof of Equivalence in Definition 2
Proof.
Suppose the second definition holds but the first onedoes not. Then by the definition of eliminated resource, thereexists an optimal solution such that for every g l ∈ G i , thereexists r ∈ g l such that (cid:80) i ∈ N x ir < S r . Then for every g l ∈ G i we can assign i a little more of the resource type above,and have y t × w i ∗ × d il d i ∗ < (cid:80) r ∈ g l x ir . This contradicts thesecond definition.Now suppose the first definition holds but the second defi-nition does not. This means that there exists an optimal solu-tion such that y t × w i ∗ × d il d i ∗ < (cid:80) r ∈ g l x ir for every g l ∈ G i .Consider the g l such that g l ∩ R t +1 = ∅ by the first defini-tion (every r ∈ g l is eliminated by the end of round t ). Wecan reduce the allocation of resources in that demand groupto i by a little bit without sacrificing optimality because theallocation constraints were satisfied strictly. But this meanswe have an optimal solution that does not use up the supplyof r ∈ g l : this contradicts the elimination of these resourcesin the first definition. A.2 Proof of Claim 2
Proof.
The first part is straightforward. If q ig > is thedual variable for the allocation constraint for some agent i ∈ N t , g ∈ G i , then by complementary slackness every optimalsolution needs to satisfy y t × w i ∗ × d il d i ∗ = (cid:80) r ∈ g l x ir , whichmeans agent i needs to be eliminated by Definition 2.Let’s now rewrite the linear program solved at time t : max y t (3)s.t. − (cid:88) r ∈ g l x ir + d il d i ∗ w i y t ≤ ∀ i ∈ N t , g l ∈ G i (4) − (cid:88) r ∈ g l x ir ≤ − d il d i ∗ γ i w i ∀ i (cid:54)∈ N t , g l ∈ G i (5) (cid:88) i ∈ N x ir ≤ S r ∀ r ∈ R (6) x ir ≥ ∀ i ∈ N, r ∈ R This LP is in canonical form, where the objective coeffi-cient vector is c T = [0 , ..., , . Let q ig be the dual variablesthat correspond to the allocation constraints (constraint (4)and (5)), and q r the dual variables corresponding to the sup-ply constraints (constraint (6)). Let y ∗ t be the value of y t inan optimal solution to the linear program and let ¯ q be the op-timal solution to the corresponding dual program. By com-plementary slackness we know that ¯ q (cid:62) A y = c y = 1 , where A y is the last column of the primal constraint matrix. Notethat the entries in A y are either positive or zero. Therefore, ¯ q ig must be positive for some i ∈ N t , g ∈ G i . This finishesthe proof of the second part. A.3 Proof of Lemma 1
Proof.
Suppose x is the output of Algorithm 1 and there ex-ists allocation x (cid:48) such that agent i is strictly better off whileother agents have just as much utility. Let y (cid:48) × w i ∗ be thefraction of the dominant resource meta-type that i receiveswith allocation x (cid:48) .Let t be the round in which i was eliminated in Algorithm1. Since i is strictly better off with allocation x (cid:48) , y (cid:48) > y ∗ t .Now we construct a new allocation by scaling down agent i ’s allocation from x (cid:48) i to x (cid:48) i y ∗ t y (cid:48) . Since we know other agentshave at least as much utility as with allocation x , this newsolution has an LP objective value at least as high as before,satisfies all the allocation/supply constraints, and does not use up all the resources that i cares about. This contradicts i being eliminated in round t .Note that by the same argument as above we know that theallocation constraint in Equation 2 for the eliminated agentshas to be satisfied with equality (otherwise we can scale thisallocation down to make the constraint tight, and that agentwould not have been eliminated in an earlier round). Thuswe have also shown Fact 1 A.4 Proof of Lemma 2
Proof.
For any pair of agents i, j ∈ N , we will show that i does not envy j . Let x be the allocation returned by Algo-rithm 1. Starting from the LHS of the definition of weightedenvy-freeness: u i (cid:18) x jr w il w jl ∀ r ∈ g il , l ∈ [ L ] (cid:19) = min g l ∈ G i (cid:80) r ∈ g il x jr w il w jl d il = min g l ∈ G i (cid:80) r ∈ g jl ∩ g il x jr w il w jl d il ≤ min g l ∈ G i (cid:80) r ∈ g jl x jr w il w jl d il = min g l ∈ G i w il w jl (cid:80) r ∈ g jl x jr d il . The first equality is the definition of Leontief utility in(1). The second equality holds because x jr = 0 for r ∈ Ω l but r / ∈ g jl (If the output allocation does contain inaccessi-ble resources then we can simply remove them without af-fecting the utilities of agents). The inequality follows fromnon-negativity of x jr .Now let t i , t j be the rounds in which agent i and j areeliminated respectively. Note that from the LP in Equation, we know (cid:80) r ∈ g jl x jr = y ∗ t j w j ∗ d jl /d j ∗ . So min g l ∈ G i w il w jl (cid:80) r ∈ g jl x jr d il = min g l ∈ G i w il w jl y ∗ t j w j ∗ d jl /d j ∗ d il = min g l ∈ G i w j ∗ d j ∗ d jl w jl y ∗ t j w il d il ≤ min g l ∈ G i y ∗ t j w il d il = y ∗ t j w i ∗ d i ∗ where the inequality follows from the definition of domi-nant resource meta-type ( w j ∗ d j ∗ = min l w jl d jl ).If y ∗ t j ≤ y ∗ t i (which means t j ≤ t i , by Fact 2 and Fact 1),we have y ∗ t j w i ∗ d i ∗ ≤ y ∗ t i w i ∗ d i ∗ = u i ( x i ) . Now suppose y ∗ t j > y ∗ t i (which means t j > t i ), and that i envies j . Note that this implies that for every group g il ∈ G i ,there exists at least one r ∈ g il such that x jr > .Consider an alternative allocation x (cid:48) that scales the al-location to agent j to y ∗ ti y ∗ tj x j while keeping the allocationsto other agents the same as in x , namely, x (cid:48) j = x j y ∗ ti y ∗ tj and x (cid:48) k = x k ∀ k (cid:54) = j . This alternative allocation gives everyagent as much utility as they had before in round t i whilemaintaining slack in at least one resource from each demandgroup of G i . This contradicts the definition of t i becauseagent i was eliminated in round t i (see Definition 2). A.5 Proof of Lemma 3
Our proof approach is adapted from (Parkes, Procaccia, andShah 2015) with important modifications. We first introducesome new notations and prove two helpful results. Let i bethe only agent who reports her demands untruthfully. Let d be the true demand vector for all agents and d (cid:48) be an alter-native demand where only the elements belonging to agent i might be different. Let t ∗ be the first round in which agent i is eliminated in Algorithm 1, either with truthful or un-truthful reporting (minimum of the two). Let N t , N (cid:48) t , and y ∗ t , y ∗(cid:48) t represent the remaining active agents at the begin-ning of round t , and the optimal LP objective in round t ,under d and d (cid:48) respectively, Claim 3.
If agent i is not eliminated in round t , then if weremove the allocation constraint for agent i and omit thevariables related to agent i from the supply constraints inEquation 2, the optimal value as well as agents eliminatedin that round do not change.Proof. First we show that x ir = 0 if r is one of the elimi-nated resources in that round. Suppose x ir > . Since i isnot eliminated, there must exist another resource r (cid:48) in thesame demand group of r for agent i that is not eliminated.This means that we could replace some of the allocation of r with a little more allocation of r (cid:48) . But this would then con-tradict r being an eliminated resource. Note that by the samelogic x ir = 0 holds in all future rounds too. This allows us to remove x ir from the supply constraints.Now the allocation constraint can be written as y t × w i ∗ × d il d i ∗ ≤ (cid:88) r ∈ g l ∩ R t +1 x ir ∀ g l ∈ G i Since the remaining resources are not constrained by sup-ply, this inequality can always hold without posing limitson other variables. So we can remove this constraint com-pletely.
Claim 4.
For all t ≤ t ∗ , N t = N (cid:48) t . For all t < t ∗ , y ∗ t = y ∗(cid:48) t .Proof. We use proof by induction. t = 0 holds trivially.We assume the claim holds for t . Suppose t + 1 < t ∗ .Then by Claim 3, we can remove the constraints related toagent i from the optimization problem. But the only differ-ences between these two optimization problems are thoserelated to agent i , so they have the same solutions and weare eliminating the same agents.Now we prove Lemma 3. Proof.
Let x and x (cid:48) be the allocations returned by Algorithm1 given demand d (truthful reporting) and d (cid:48) (agent i mis-reports) respectively. We consider the following four casesseparately. • y ∗ t ∗ ≤ y ∗(cid:48) t ∗ and agent i is eliminated in t ∗ reporting d .By Claim 4, we know N t ∗ = N (cid:48) t ∗ . Although we do notknow the exact round in which agents in N (cid:48) t ∗ are even-tually eliminated under d (cid:48) , we know that their dominantresource shares are all at least y ∗(cid:48) t ∗ ≥ y ∗ t ∗ , because theoptimal objective value of the optimization problem canonly increase over time by Fact 2.Suppose u i ( x (cid:48) i ) > u i ( x i ) . Now consider x (cid:48) as a candi-date solution for the optimization problem in round t ∗ of truthful reporting. Every agent j in N t ∗ receives atleast y ∗(cid:48) t ∗ w j ∗ ≥ y ∗ t ∗ w j ∗ fraction of their dominant resourcemeta-type, while agent i receives strictly more. This con-tradicts either the optimality of y ∗ t or the fact that agent i was eliminated in round t ∗ reporting d (see Definition 2). • y ∗ t ∗ ≥ y ∗(cid:48) t ∗ and the agent is eliminated in t ∗ reporting d (cid:48) .Suppose the dominant resource meta-type is the same un-der d (cid:48) . Since y ∗(cid:48) t ∗ × w i ∗ is the fraction of the total supplyof dominant resource that i receives, agent i must be re-ceiving less of that under d (cid:48) .Now suppose the reported dominant resource meta-typeis different under d (cid:48) . Let • denote the new dominant re-source meta-type. Let w i • , d i (cid:48)• be the new dominant re-source weight and demand. Let d i (cid:48)∗ be the new demand forthe original dominant resource meta-type. The amount oforiginal dominant resource meta-type i receives is y ∗(cid:48) t ∗ w i • d i (cid:48)• d i (cid:48)∗ ≤ y ∗(cid:48) t ∗ w i ∗ ≤ y ∗ t ∗ w i ∗ The first inequality follows from the definition dominantresource meta-type: w i • d i (cid:48)• := min l ∈ [ L ] w il d i (cid:48) l . The final expressions the amount of original dominant resource that agent i receives under truthful reporting. • y ∗(cid:48) t ∗ > y ∗ t ∗ and the agent is not eliminated reporting d butis eliminated reporting d (cid:48) . We argue that this case cannothappen. By Claim 4, we can remove the allocation con-straints related to i in round t ∗ under truthful reporting.But then we are left with two optimization problems withthe same constraints, except that with untruthful reportingthe optimization problem has extra allocation constraint(for agent i ), and an extra non-negative term in the supplyconstraints. Extra constraints and extra terms in the sup-ply constraints can only make the optimization problemharder. • y ∗(cid:48) t ∗ < y ∗ t ∗ and the agent is eliminated reporting d but noteliminated reporting d (cid:48) . This is the symmetric case as theprevious one and so cannot happen either.Finally, a closer inspection of the above shows we did notneed the group structure of agent i to stay the same, so theresult holds for misreporting group structures as well. A.6 Proof of Lemma 4
Proof.
Recall that we use s il denote the proportion that isboth accessible to and contributed by agent i . Each agentmight have access to other people’s contributions as well.We set w il = s il . Since an agent might not have access to allof the supplies that she brings, (cid:80) i ∈ N w il might be strictly lessthan one. In that case we can pretend that there is a phan-tom agent with weight − (cid:80) i ∈ N w il for each meta-type l , andthat his demand vector is zero. Note that we do not need toimplement this phantom agent when running the algorithm,because GDRF is invariant to the scale of weights. We areonly adding this weight to make our definition of sharingincentive consistent with the assumption (cid:80) i w il = 1 Now note that (cid:80) r ∈∪ i ∈ N (cid:48) g il S r ≥ (cid:80) i ∈ N (cid:48) s il because each agenthas access to her own accessible supply. By the definition ofdominant resource w i ∗ d i ∗ d il ≤ w il . So for any N (cid:48) ⊂ N, l ∈ [ L ] (cid:88) r ∈∪ i ∈ N (cid:48) g il S r / (cid:32) (cid:88) i ∈ N (cid:48) w i ∗ d il d i ∗ (cid:33) ≥ (cid:80) i ∈ N (cid:48) s il (cid:80) i ∈ N (cid:48) w il = 1 . After rearranging the terms, we have (cid:88) r ∈∪ i ∈ N (cid:48) g il S r ≥ (cid:32) (cid:88) i ∈ N (cid:48) w i ∗ d il d i ∗ (cid:33) ∀ N (cid:48) ⊆ N, l ∈ { . . . L } . (7)For every agent i and every meta-type l , consider w i ∗ d il d i ∗ as the “total demand” for resource meta-type l from agent i .Then, we construct a bipartite graph as follows: for theleft-hand nodes, we create a node for every (cid:15) unit of to-tal demand from each agent for each resource meta-type.Thus, each node is associated with some specific agent i and resource meta-type l . For the right-hand nodes, we cre-ate a node for every (cid:15) unit of supply of each resource type( r ∈ R ). Note that since there is a finite number of agentsand resource types, there exists an (cid:15) small enough that it canperfectly divide up all the demands and supplies, assumingthat all the weights are rational.Next, we create an edge between each pair of left andright-hand side nodes if and only if the supply side node be-longs to the demand group of that agent for that meta-type: r ∈ g il .Eq.(7) now tells us that for every subset of the demandside nodes, the number of neighbors of that subset is greaterthan or equal to the size of the subset. This is precisely thecondition in Hall’s Theorem , which states that if this condi-tion holds, then there exists a matching in the bipartite graphsuch that the demand side nodes are covered.Consider such a matching obtained via Hall’s Theorem.We construct a solution x by setting x ir equal to (cid:15) times thenumber of matched edges corresponding to ir . This yieldsan assignment that gives each agent w i ∗ d il d i ∗ of each meta-type. By the construction of the matching this is a legal allo-cation. Then, we can set y t = 1 to obtain a feasible solutionto the optimization problem in (2).This means that after the first round of GDRF, agent i ’sutility is at least u i ( x i ) = min g l ∈ G i w i ∗ d il d i ∗ d il = w i ∗ d i ∗ = min l : d il (cid:54) =0 s il d il The final expression is exactly the utility agent i gets fromher own supplies. B Beyond Sharing Incentive: Proportionality
As discussed in Subsection 3.1, proportionality is a weakernotion than sharing incentive, when weights are set accord-ing to contribution. However, when priority weights and ac-cessibility constraints are set independently, proportional-ity is a more flexible concept. Unfortunately proportionalitydoes not hold generally. We prove proportionality under thefollowing assumption:
Assumption 1. min N (cid:48) ⊆ N,l ∈{ ...L } (cid:88) r ∈∪ i ∈ N (cid:48) g il S r / (cid:32) (cid:88) i ∈ N (cid:48) w i ∗ d il /d i ∗ (cid:33) ≥ max i ∈ N min l : d il (cid:54) =0 w il (cid:80) r ∈ g il S r w i ∗ d il /d i ∗ Lemma 5.
Assume that demands, weights and supplies areall rational numbers. Then under Assumption 1, GDRF sat-isfies proportionality.
The proof is very similar to that of Lemma 4, but we firstgive some intuition for Assumption 1.Since (cid:80) r ∈ g il S r ≤ for every i, l , and min l w il d il = w i ∗ d i ∗ for every i , the right hand side of Assumption 1 is up- DRFAllocations
Hospital 1( w = 1 / ) Hospital 2( w = 1 / ) Hospital 3( w = 1 / )Doctor A 100 400Doctor B 400 100Nurse C 100 400Nurse D 500 Utilities
GDRF 100 100 500Proportional 62.5 31.25 250Table 1: Allocations from GDRF in Example 1 and the com-parison of the resulting utilities with utilities of proportionalallocation.per bounded by 1. (cid:80) r ∈∪ i ∈ N (cid:48) g il S r is the union of the accept-able supply of resource meta-type l from every agent in N (cid:48) . (cid:80) i ∈ N (cid:48) w i d il /d i ∗ is the total weighted demand from agents inset N (cid:48) . Note that d il ≤ d i ∗ .So, what the condition says intuitively is that wheneverthere is a group of agents who have a large combinedweighted demand on meta-type l , they need to also collec-tively have access to/accept a large fraction of the total sup-ply of l .To provide more intuition for this assumption, we look attwo examples. First we check that Example 1 satisfies As-sumption 1. The RHS of the assumption evaluates to (withhospital and resource meta-type ). One can check that theminimum on the LHS is achieved by picking N (cid:48) = N and l = 1 which gives us > . Thus Assumption 1 is sat-isfied. The resulting allocation and utilities using GDRF isgiven in Table 1. Clearly our allocation is better for everyonethan the proportional allocation.However, by adjusting the weights of the hospitals we canalso construct an example that does not satisfy Assumption1. Take the same parameters of Example 1 with the fol-lowing modification on weights: w = 0 . , w = 0 . , w = 0 . . The RHS value of Assumption 1 does notchange. However, because the weights of hospitals , nowdominate the market, the minimum of LHS is achieved with N (cid:48) = { , } and l = 2 , which gives us / . × . × / < . So the assumption is violated. Intuitively, the problemwith this setup is that even though hospitals 1 and 2 accountfor vast majority of the weighted demand for the nurse meta-type, they are both severely constrained to the same half ofthe total supply of nurses.Under this setup, the GDRF assignments/utilities do notchange. With proportional allocation however, the utilitiesfor the three agents are [122 . , . , . . So Agent 1 re-ceived more utility under proportional allocation than the al-location given by GDRF, at the expense of significantly hurt-ing the social welfare: the sum of the utilities is less than , compared to generated by the GDRF allocation.Now we prove Lemma 5 Proof.
Let ˆ y denote the RHS of Assumption 1. After rear- ranging we have for all N (cid:48) ⊆ N and l ∈ { . . . L } : (cid:88) r ∈∪ i ∈ N (cid:48) g il S r ≥ ˆ y (cid:32) (cid:88) i ∈ N (cid:48) w i ∗ d il /d i ∗ (cid:33) For every agent i and every meta-type l , consider ˆ yw i ∗ d il /d i ∗ as the “total demand” for resource meta-type l from agent i .Then we construct a bipartite graph and apply Hall’s the-orem the same way as in the proof of Lemma 4. This yieldsan assignment that gives each agent at least ˆ yw i ∗ d il /d i ∗ ofeach meta-type. By the definition of ˆ y , it follows that theutility of each agent after the first round is at least: d il d i ∗ ˆ yw i ∗ d il = w i ∗ d i ∗ ˆ y ≥ w i ∗ d i ∗ min l : d il (cid:54) =0 w il (cid:80) r ∈ g il S r w i ∗ d il /d i ∗ = min l : d il (cid:54) =0 w il (cid:80) r ∈ g il S r d ilil