Strategyproof and Approximately Maxmin Fair Share Allocation of Chores
aa r X i v : . [ c s . G T ] M a y Strategyproof and Approximately Maxmin Fair Share Allocation of Chores ∗ Haris Aziz , Bo Li and Xiaowei Wu UNSW Sydney and Data61 CSIRO, Australia Department of Computer Science, Stony Brook University, USA Faculty of Computer Science, University of Vienna, [email protected], [email protected], [email protected]
Abstract
We initiate the work on fair and strategyproof allo-cation of indivisible chores. The fairness conceptwe consider in this paper is maxmin share (MMS)fairness. We consider three previously studiedmodels of information elicited from the agents: theordinal model, the cardinal model, and the publicranking model in which the ordinal preferences arepublicly known. We present both positive and neg-ative results on the level of MMS approximationthat can be guaranteed if we require the algorithmto be strategyproof. Our results uncover some inter-esting contrasts between the approximation ratiosachieved for chores versus goods.
Multi-agent resource allocation is one of the major researchtopics in artificial intelligence [Bouveret et al. , 2016]. Weconsider fair allocation algorithms of indivisible chores whenagents have additive utilities. The fairness concept we usein this paper is the intensively studied and well-establishedmaxmin share fairness. The maxmin fair share of an agent isthe best she can guarantee for herself if she is allowed to par-tition the items but then receives the least preferred bundle.In this paper we take a mechanism design perspective tothe problem of fair allocation. We impose the constraint thatthe algorithm should be strategyproof, i.e., no agent shouldhave an incentive for report untruthfully for profile of val-uations. The research question we explore is the followingone.
When allocating indivisible chores, what approximationguarantee of maxmin share fairness can be achieved by strat-egyproof algorithms?
This approach falls under the umbrellaof approximation mechanism design without money that hasbeen popularized by Procaccia and Tennenholtz [2013].Maxmin share fairness was proposed by Budish [2011]as a fairness concept for allocation of indivisible items.The concept coincides with the standard proportionalityfairness concept if the items to be allocated are divisi-ble. There have been several works on algorithms thatfind an approximate MMS allocation [Procaccia and Wang, ∗ The authors are ordered alphabetically. This work is partiallysupported by NSF CAREER Award No. 1553385. et al. , 2015; Barman and Murthy, 2017;Ghodsi et al. , 2018; Aziz et al. , 2017]. None of these workstook a mechanism design perspective to the problem of com-puting approximately MMS allocation. Amanatidis et al. [2016] were the first to embark on a study of strategyproofand approximately MMS fair algorithms. Their work onlyfocussed on the case of goods. There are many settings inwhich agents may have negative utilities such as when choresor tasks are allocated. In this paper, we revisit strategyprooffair allocation by considering the case of chores.
We initiate the study of maxmin share (MMS) allocations of m indivisible chores among n strategic agents. It is assumedthat all agents have underlying cardinal additive utilities overthe chores. We consider three preference models in this work. • Cardinal model : agents are asked to report their cardi-nal utilities over the items. • Ordinal model : agents are only allowed or asked to ex-press their ordinal rankings over the items. • Public ranking model : all agents’ rankings are publicinformation and the agents are asked to report their util-ities that are consistent the known ordinal rankings.First, for cardinal and ordinal models, we design a de-terministic sequential picking algorithm
SequPick , which isstrategyproof and unexpectedly achieves an approximation of O (log mn ) . Roughly speaking, given an order of the agents,a sequential picking algorithm lets each agent i pick a i itemsand leave. Amanatidis et al. [2016] proved that when theitems are goods, the best a sequential picking algorithm canguarantee is an approximation of ⌊ ( m − n + 2) / ⌋ , and suchan approximation can be easily achieved by letting each of thefirst n − agents select one item and allocating all the remain-ing items to the last agent. Compared to their result, we showthat by carefully selecting the a i ’s, when items are chores, weare able to significantly improve the bound to O (log mn ) .Next, we further improve the approximation ratio for car-dinal and ordinal models by randomized algorithms. Partic-ularly, we show that by randomly allocating each item butallowing each agent to recognize a set of “bad” items and to In this paper we use log() to denote log () . oods ChoresLower Upper Lower UpperOrdinal Ω(log n ) O ( m − n ) (D) (R) O (log mn ) (D) O ( √ log n ) (R)Cardinal O ( m − n ) (D)N/A (R) O (log mn ) (D) O ( √ log n ) (R)Public ranking
65 32 for n = 22 for n = 3 O ( n ) for any n (D) for n ≤ for any n Table 1: Lower and upper bounds on approximation of MMS fairness of strategyproof algorithms. The results for goods were provedby Amanatidis et al. [2016], all of which concern deterministic algorithms. The results for chores are proved in this paper, where we use (D)and (R) to indicate deterministic and randomized algorithms, respectively. be able to decline each allocated “bad” item once, the result-ing algorithm is strategyproof and achieves an approximationratio of O ( √ log n ) in expectation.We complement these upper bound results with lowerbound results by showing that for cardinal and ordinal model,no deterministic strategyproof algorithm has a better than approximation. For the ordinal model, we prove that thelower bound of cannot be improved by non-strategyproofalgorithms. For randomized non-strategyproof algorithms,this bound cannot be improved to be better than .Finally, for the public ranking model, we show that thegreedy round-robin algorithm is strategyproof and achieves2-approximation. This is also surprising as when theitems are goods, the best known approximation is O ( n ) byAmanatidis et al. [2016]. When n ≤ , we give a strate-gyproof divide-and-choose algorithm that further improvesthis ratio to . We complement these upper bound resultsby a lower bound of for any deterministic strategyproof al-gorithms.Our results as well as previous results for the case of goodsare summarized in Table 1. MMS fairness is weaker than the proportionality fairness con-cept that requires that each agent gets at least /n of the totalutility she has for the set of all items [Bouveret and Lemaˆıtre,2016]. However for indivisible items, there may not exist anallocation that guarantees any approximation for the propor-tionality requirement.Most of the work on fair allocation of items is forthe case of goods although recently, fair allocation ofchores [Aziz et al. , 2017] or combinations of goods andchores [Aziz et al. , 2018] has received attention as well.Aziz et al. [2017] proved that MMS allocations do not al-ways exist but can be 2-approximated by a simple algorithm.Aziz et al. [2017] also presented a PTAS for relaxation ofMMS called optimal MMS. Barman and Murthy [2017] pre-sented an improved approximation algorithm for MMS allo-cation of chores.Strategyproofness is a challenging property to satisfy forfair division algorithms. Amanatidis et al. [2016] initiated thework on strategyproof goods allocation with respect to MMSfairness. In particular they proved the results covered in the goods part of Table 1. There is also work on the approxima-tion of welfare that can be achieved by strategyproof algo-rithms for allocation of divisible items [Aziz et al. , 2016]. For the fair allocation problem, N is a set of n agents, and M is a set of m indivisible items. The goal of the problemis to fairly distribute all the items to these agents. Differ-ent agents may have different preferences for these items andthese preferences are generally captured by utility or valu-ation functions: each agent i is associated with a function v i : 2 M → R that valuates any set of items. MMS fairness.
Imagine that agent i gets the opportunityto partition all the items into n bundles, but she is the lastto choose a bundle. Then her best strategy is to partition theitems such that the smallest value of a bundle is maximized.Let Π( M ) denote the set of all possible n -partitionings of M .Then the maxmin share (MMS) of agent i is defined as MMS i = max h X ,...,X n i∈ Π( M ) min j ∈ N v i ( X j ) . (1)If agent i finally receives a bundle of items with value at least MMS i , she is happy with the final allocation.In this work, it is assumed that items are chores: v i ( { j } ) ≤ for all i ∈ N and j ∈ M . Then each agent actually wantsto receive as few items as possible. For ease of analysis, weascribe a disutility or cost function c i = − v i for each agent i .In this paper, we assume that the cost function of each agent i is additive. We represent each cost function c i by a vector ( c i , · · · , c im ) where c ij = c i ( { j } ) is the cost of agent i foritem j . Then for any S ⊆ M we have c i ( S ) = P j ∈ S c ij .Agent i ’s maxmin share can be equivalently defined as MMS i = min h X ,...,X n i∈ Π( M ) max j ∈ N c i ( X j ) . (2)Note that the maxmin threshold defined in Equation 2 ispositive which is the opposite number of the threshold definedin Equation 1. Throughout the rest of our paper, we choose touse the second definition. For each agent i , we use a permu-tation over M , σ i : M → [ m ] , to denote agent i ’s ranking onthe items: c iσ i (1) ≥ · · · ≥ c iσ i ( m ) . In other words, item σ i (1) is the least preferred item and σ i ( m ) is the most preferred.et x = ( x i ) i ∈ N be an allocation , where x i = ( x ij ) j ∈ M and x ij ∈ { , } indicates if agent i gets item j under al-location x . A feasible allocation guarantees a partition of M , i.e., P i ∈ N x ij = 1 for any j ∈ M . We somewhatabuse the definition and let X i = { j ∈ M | x ij = 1 } and c i ( x ) = c i ( x i ) = c i ( X i ) . An allocation x is called an MMSallocation if c i ( x i ) ≤ MMS i for every agent i and α -MMSallocation if c i ( x i ) ≤ α MMS i for all agents i .We first state the following simple properties of MMS.Lemma 2.1 implies if an agent receives k items, then its costis at most k · MMS i . Lemma 2.1
For any agent i and any cost function c i , • MMS i ≥ n c i ( M ) ; • MMS i ≥ c ij for any j ∈ M . Proof:
The first inequality is clear as for any partition of theitems, the largest bundle has cost at least n c i ( M ) .For the second inequality, it suffices to show MMS i ≥ c iσ i (1) . This is also clear since in any partitioning of theitems, the largest bundle should have cost at least c iσ i (1) .By Lemma 2.1, it is easy to see that if m ≤ n , any alloca-tion that allocates at most one item to each agent is an MMSallocation. Thus throughout this paper, we assume m > n . Models.
In the cardinal model , the agents are asked to ex-press their cardinal costs over M . A deterministic cardinalalgorithm is denoted by a function M : ( R m ) n → Π( M ) . Ifan algorithm is restricted to only use the resulting rankings ofthe reported cardinal cost functions to allocate the items, wecalled it an ordinal algorithm and the corresponding problemis called the ordinal model . If the algorithm has the informa-tion of all agents’ rankings by default, and every agent hasto report her cost function with respect to the known ranking,the algorithm is called a public ranking algorithm and thecorresponding problem is called the public ranking model . Adeterministic algorithm M is called ( α -approximate) MMSif for any cost functions, it always outputs an ( α -) MMS al-location. A randomized algorithm M returns a distributionover Π( M ) and is called α -approximate MMS if for any costfunctions c , · · · , c n , E x ∼M ( c , ··· ,c n ) [max i ∈ N c i ( x ) MMS i ] ≤ α . In this work, we study the situation when the costs are pri-vate information of the agents. Each agent may withhold hertrue cost function in order to minimize her own cost for the al-location. We call an algorithm strategyproof (SP) if no agentcan unilaterally misreport her cost function to reduce her cost.Formally, a deterministic algorithm M is called SP if forevery agent i , cost function c i and the cost functions c − i of other agents, c i ( M ( c i , c − i )) ≥ c i ( M ( c ′ i , c − i )) holdsfor all c ′ i . We call a randomized algorithm M SP in ex-pectation if for every i , c i and c − i , E x ∼M ( c i ,c − i ) c i ( x ) ≥ E x ∼M ( c ′ i ,c − i ) c i ( x ) holds for all c ′ i . Note that if the α -approximation is defined as for every agent i , E x ∼M ( c , ··· ,c n ) c i ( x ) ≤ α MMS i , the problem becomes trivial asuniform-randomly allocating all items optimizes α to be 1. Example 2.2
Suppose the cost function of an agent on fouritems is c = (1 , , , . In an SP cardinal algorithm, re-porting c minimizes her cost (in expectation, for randomizedalgorithm and the same for the following cases); In an SP or-dinal algorithm, reporting c ≥ c ≥ c ≥ c minimizesher cost; In an SP public ranking algorithm, the algorithmknows c ≥ c ≥ c ≥ c by default, and the agentminimizes her cost by reporting c . By the above definition, we have the following lemma im-mediately, which also appeared in Amanatidis et al. [2016].
Lemma 2.3
An SP α -approximation algorithm for the ordi-nal model is also SP α -approximate for the cardinal model.An SP α -approximation algorithm for the cardinal model isalso SP α -approximate for the public ranking model. We end this section by providing a necessary conditionof all SP algorithms for cardinal and public ranking models,which is mainly used to prove our hardness results.
Definition 2.4
An allocation algorithm M is monotone if forany cost functions c , · · · , c n and x = M ( c , · · · , c n ) , in-creasing c ij for some x ij = 0 , or decreasing c ij for some x ij = 1 does not change x i . First, by perturbing the costs by arbitrarily small differentvalues, we can assume without loss of generality that the cost c i ( S ) of agent i is different for every S ( M . Lemma 2.5
All SP algorithms are monotone.
Proof:
Fix any agent i and let x be the allocation when i reports c i and the others report c − i . We fist consider thecase when x ij = 0 and c ij is increased. Let c ′ be the newcost profile, and x ′ be the new allocation. If x ′ ij = 1 , thenif c ′ i ( x ′ i ) ≤ c i ( x i ) , then agent i has incentive to lie when itstrue cost is c (since c i ( x ′ i ) < c ′ i ( x ′ i ) ); if c ′ i ( x ′ i ) > c i ( x i ) , thenagent i has incentive to lie when its true cost is c ′ . Hence wehave x ′ ij = 0 . For the same reason, we should have c i ( x ′ i ) = c i ( x i ) , which implies x ′ i = x i .Next, we consider the case when x ij = 1 and c ij is de-creased. If x ′ ij = 0 , then if c ′ i ( x ′ i ) = c i ( x ′ i ) < c i ( x i ) ,then agent i has incentive to lie when its true cost is c ; if c ′ i ( x ′ i ) = c i ( x ′ i ) ≥ c i ( x i ) , then agent i has incentive to liewhen its true cost is c ′ (since c ′ i ( x i ) < c i ( x i ) ). Hence wehave x ′ ij = 1 . We further have c i ( x ′ i ) − c ij = c i ( x i ) − c ij asotherwise agent i has incentive to lie when its true cost is theone that results in a higher cost. Hence we have x ′ i = x i . Before we present our algorithm for the ordinal model, wefirst discuss the limitation of deterministic ordinal algorithms.
Lemma 3.1
No deterministic ordinal algorithm (even non-SP) has an approximation ratio smaller than , even for agents and items. roof: Consider the instance with agents, whose rankingon the m = 4 items are identical. Without loss of generality,assume the item with maximum cost is given to the first agent,i.e. x = 1 . If the first agent is allocated only one item, thenfor the case when c = (1 , , , , the approximation ratiois : the second agent has total cost while MMS = 2 .Otherwise for the case when c = (3 , , , , the approxima-tion ratio is at least , as the first agent has total cost at least while MMS = 3 .Next we present a deterministic sequential picking algo-rithm that is O (log mn ) -approximate and SP. Amanatidis et al. [2016] gave a deterministic SP ordinal algorithm which is O ( m − n ) -approximate when the items are goods. In the fol-lowing, we show that if all the items are chores, it is possibleto improve the bound to O (log mn ) . Without loss of gener-ality, we assume that n and mn are at least some sufficientlylarge constant. As otherwise it is trivial to obtain an O (1) -approximation by assigning mn arbitrary items to each agents. Theorem 3.2
There exists a deterministic SP ordinal algo-rithm with approximation ratio O (log mn ) . SequPick . Fix a sequence of integers a , . . . , a n such that P i ≤ n a i = m . Order the agents arbitrarily. For i = n, n − , . . . , , let agent i pick a i items from the remaining items.We note that as long as a i ’s do not depend on the valuationsof agents, the rule discussed above is the serial dictatorshiprule for multi-unit demands. When it is agent i ’s turn to pickitems, it is easy to see that her optimal strategy is to pick thetop- a i items with smallest cost, among the remaining items.Hence immediately we have the following lemma. Lemma 3.3
For any { a i } i ≤ n , SequPick is SP.
It remains to prove the approximation ratio.
Lemma 3.4
There exists a sequence { a i } i ≤ n such that theapproximation ratio of SequPick is O (log mn ) . Proof:
We first establish a lower bound on the approxima-tion ratio in terms of { a i } i ≤ n . Then we show how to fix thenumbers appropriately to get a small ratio. Let r be the ap-proximation ratio of the algorithm.Consider the moment when agent i needs to pick a i items.Recall that at this moment, there are P j ≤ i a j items, and the a i ones with smallest cost will be chosen by agent i . Let c bethe average cost of items agent i picks, i.e., c i ( x ) = c · a i . Onthe other hand, each of the P j ≤ i − a j items left has cost atleast c . Thus we have MMS i ≥ c · l a + ... + a i − n m and r = max i ∈ N (cid:26) c i ( X i ) MMS i (cid:27) ≤ max i ∈ N a i l a + ... + a i − n m . It suffices to compute a sequence of a , . . . , a n that sum to m and minimizes this ratio. Fix K = 2 log mn . Let a i = ( , i ≤ n , min { m − P j n . Note that the first term of the min is to guarantee we leaveenough items for the remaining agents. Moreover, truncating a i is only helpful for minimizing the approximation ratio andthus we only need to consider the case when a i equals thesecond term of the min . In the following, we show that1. all items are picked: P i ∈ N a i = m ;2. for every i > n : a i ≤ K · l a + ... + a i − n m .Note that for i ≤ n , since agent i receives items, theapproximation ratio is trivially guaranteed.The first statement holds because P n i =1 P ni = n +1 (cid:0) K · (1 + Kn ) i − n − (cid:1) = P i ≤ n (cid:0) K · (1 + Kn ) i − (cid:1) + n =(1 + Kn ) n · n − n + n ≥ K · n > m, and a i ’s will be truncated when their sum exceeds m .For i > n , observe that (let l = i − n − ) n ( a + . . . + a i − ) = 1 + n P lj =1 K · (1 + Kn ) j − = 1 + (1 + Kn ) l − Kn ) l . Thus we have a i ≤ (cid:6) K · (1 + Kn ) l (cid:7) ≤ K · (cid:6) (1 + Kn ) l (cid:7) ≤ K · l a + ... + a i − n m , as claimed.We conclude the section by showing that our approxima-tion ratio is asymptotically tight for SequPick . Lemma 3.5 (Limits of
SequPick ) The
SequPick algorithm(with any { a i } i ∈ N ) has approximation ratio Ω(log mn ) . Proof:
Fix K = log mn . Suppose there exists a sequenceof { a i } i ∈ N such that the algorithm is K -approximate.Then the last agent to act must receive at most K items,i.e., a ≤ K . Next we show by induction on i = 2 , , . . . , n that a i ≤ K (1 + Kn ) i − for all i ∈ N .Suppose the statement is true for a , . . . , a i . Then if a i +1 > K (1 + Kn ) i , we have a i +1 a + . . . + a i +1 > K (1 + Kn ) i k · n K ((1 + Kn ) i +1 − ≥ Kn .
Thus we have P ni =1 a i ≤ n · (cid:0) (1 + Kn ) n − (cid:1) ≤ n · (cid:0) e K − (cid:1) < m , which is a contradiction, since not all itemsare allocated. We have shown a logarithmic approximation algorithm
SequPick for the problem. However, the algorithm may stillhave poor performance when the number of items is muchlarger than the number of agents, e.g., m = 2 n . In this sec-tion we present a randomized O ( √ log n ) -approximation or-dinal algorithm, which is SP in expectation.Again, before we show our algorithm, let us first see a lim-itation of the randomized ordinal algorithms. emma 4.1 No randomized ordinal algorithm (even non-SP)has approximation ratio smaller than , even for agents and items. Proof:
Consider the instance with agents, whose rankingon the m = 4 items are identical. Let p be the probability thatthe algorithm assigns items to both agents.If p ≤ , consider the instance with evaluation (1 , , , ,for which MMS = MMS = 2 . Then with probability − p ,the agent receiving at least items has cost at least timesits maximin share, which implies that the expected approxi-mation ratio is at least p + (1 − p ) · = − p ≥ .If p > , then consider the instance with evaluation (3 , , , , for which MMS = MMS = 3 . Then with prob-ability p , the agent receiving the item with cost has cost atleast times its maximin share, which implies an expectedapproximation ratio at least p · + (1 − p ) = 1 + p ≥ .Basically, if we randomly allocate all the items, one isable to show that the algorithm achieves an approximationof O (log n ) . The drawback of this na¨ıve randomized algo-rithm is that it totally ignores the rankings of agents. In thefollowing, we show that if the agents have opportunities todecline some “bad” items, the performance of this random-ized algorithm improves to O ( √ log n ) . Note that since wealready have an O (log mn ) -approximate deterministic algo-rithm for the ordinal model, it suffices to consider the casewhen m ≥ n log n . RandDecl . Let K = ⌊ n √ log n ⌋ . Based on the ordering ofitems submitted by agents, for each agent i , label the K itemswith largest cost as “large”, and the remaining to be “small”.It can be also regarded as each agent reports a set M i of largeitems with | M i | = K . The algorithm operates in two phases. • Phase 1: every item is allocated to a uniformly-at-random chosen agent, independently. After all alloca-tions, gather all the large items assigned to every agentinto set M b . Note that M b is also a random set. • Phase 2: Redistribute the items in M b evenly to allagents: every agent gets | M b | n random items. Theorem 4.2
There exists a randomized SP ordinal algo-rithm with approximation ratio O ( √ log n ) . We prove Theorem 4.2 in the following two lemmas.
Lemma 4.3
In expectation, the approximation ratio of Algo-rithm
RandDecl is O ( √ log n ) . Proof:
We show that with probability at least − n , ev-ery agent i receives a collection of items of cost at most O ( √ log n ) · MMS i . Fix any agent i . Without loss of gen-erality, we order the items according to agent i ’s ranking, i.e., σ i ( j ) = j for any j ∈ M and c i ≥ · · · ≥ c im .For ease of analysis, we rescale the costs such that c i + c i + . . . + c im = n p log n = K. Note that after the scaling, agent i ’s maximin share is MMS i ≥ √ log n . Let x ij denote the random variable in-dicating that the contribution of item j to the cost of agent i . Then for j > K , x ij = c ij with probability n , and x ij = 0 otherwise. For j ≤ K , x ij = 0 with probability . Note that E [ P mi =1 x i ] = n · P mi = K +1 c ij ≤ Kn = √ log n. Moreover, we have c ij ≤ for j > K , as otherwisewe have the contradiction that P Kj =1 c ij > K . Note that { x ij } j ≤ m are independent random variables taking value in [0 , . Hence by Chernoff bound we have Pr[ P mj =1 x ij ≥ √ log n · MMS i ] ≤ Pr[ P mj =1 x ij ≥ n ] ≤ exp (cid:16) − · (cid:16) n E [ P mi =1 x i ] − (cid:17) · E [ P mi =1 x i ] (cid:17) < n . Then by union bound over the n agents, we conclude thatwith probability at least − n , every agent i receives a bundleof items of cost at most O ( √ log n ) · MMS i in phase 1.Now we consider the items received by an agent in thesecond phase. Recall that the items M b will be reallo-cated evenly. By the second argument of Lemma 2.1, toshow that every agent i receives a bundle of items of cost O ( √ log n ) · MMS i in the second phase, it suffices to provethat | M b | = O ( n √ log n ) (with probability at least − n ).Let y j ∈ { , } be the random variable indicating whetheritem j is contained in M b . For every item j , let b j = |{ k : j ∈ M k }| be the number of agents that label item j as “large”.Then we have y j = 1 with probability b j n . Since every agentlabels exactly n √ log n items, we have E [ | M b | ] = E [ P mi =1 y i ] = n P mi =1 b i = n √ log n. Applying Chernoff bound we have
Pr[ P mi =1 y i ≥ n √ log n ] ≤ exp (cid:16) − n √ log n (cid:17) < n . Thus, with probability at least − n , every agent i receivesa bundle of items with cost O ( √ log n · MMS i ) in the twophases combined. Since in the worse case, i receives a totalcost of at most n · MMS i , in expectation, the approximationratio is (1 − n ) · O ( √ log n ) + n · n = O ( √ log n ) . Lemma 4.4
RandDecl is SP in expectation.
Proof:
To prove that the algorithm is SP in expectation, itsuffices to show that for every agent, the expected cost it is as-signed is minimized when being truthful. Let K = n √ log n and fix any agent i . Suppose c i , . . . , c iK are the costs ofitems labelled “large” by the agent; and c i,K +1 , . . . , c im arethe remaining items. Then the expected cost assigned to theagent in the first phase is given by n P mj = K +1 c ij , as everyitem is assigned to the agent with probability n . Now weconsider the cost the agent is assigned in the second phase.Recall that the expected total cost of items to be reallocatedin the second phase is E [ P j ∈ M b c ij ] = P mj =1 c ij · b j n , where b j is the number of agents that label item j “large”. Let E bethis expectation when agent i does not label any item “large”.By labelling c i , . . . , c iK “large”, agent i increases theprobability of each item j ≤ K being included in M b by n .hus it contributes an n P Kj =1 c ij increase to the expectationof total cost of M b . In other words, we have E [ P j ∈ M b c ij ] = E + n P Kj =1 c ij . Since a random subset of | M b | n items from M b will be as-signed to agent i , the expected total value of items assignedto the agent in the two phases is given by n P mj = K +1 c ij + n · (cid:16) E + n P Kj =1 c ij (cid:17) . Obviously, the expression is minimized when c i + . . . + c iK is maximized. Hence every agent minimizes its expectedcost by telling the true ranking over the items. First, we present a lower bound on the approximation ratiofor the all deterministic SP cardinal algorithms.
Lemma 5.1
No deterministic cardinal SP algorithm has ap-proximation ratio smaller than , even for agents and items. Proof:
First, consider c = c = (3 , , , , for which MMS = MMS = 3 . To obtain an approximation smallerthan , the only possible allocation is to assign the first itemto some agent, and the remaining items to the other. Withoutloss of generality, suppose agent receives the first item.By monotonicity of SP algorithms (Lemma 2.5), for thecase when c = (1 , , , and c = (3 , , , . The assign-ment remains unchanged. Now we consider the profile when c = (1 , , , and c = (2 , , , .Note that we also have MMS = MMS = 3 , and thus(to guarantee the approximation ratio) agent cannot receivethe last three items. Moreover, to guarantee SPness, agent cannot receive a proper subset of the last three items, as oth-erwise agent will misreport (2 , , , when its true valueis (3 , , , . Thus the first item must be assigned to agent ,and consequently the second item must be assigned to agent . To guarantee the approximation ratio, agent should notreceive any other item, which means that agent must receivethe last two items, which violates the better-than- approxi-mation ratio.For positive results, by Lemma 2.3, both algorithms in Sec-tions 3 and 4 apply to the cardinal model. Thus we immedi-ately have the following. Corollary 5.1
For the cardinal model, there exists a deter-ministic SP algorithm with approximation ratio O (log mn ) ;and a randomized SP-in-expectation algorithm with approxi-mation ratio O ( √ log n ) . Amanatidis et al. [2016] provided a deterministic SP publicranking algorithm which is O ( n ) -approximate if the items aregoods. In this section, we show that if the items are chores,we can do much better. We first give a simple SP algorithm with an approximationratio of at most 2.
RounRobi . Fix an arbitrary order of the agents, let theagents pick items in the round-robin manner.
Theorem 6.1
For the public ranking model,
RounRobi is SPand has an approximation ratio of − n . Proof:
Aziz et al. [2017] proved that
RounRobi gives an ap-proximation bound of − n if at each round every agent is al-located the item with smallest cost. Note that the algorithm isordinal hence no agent can change the outcome by misreport-ing her cardinal utilities. Furthermore, since the preferencerankings of the agents are public knowledge, agents cannotmisreport by expressing a different ordinal preference. Hencethe algorithm is SP for the public ranking model.By Theorem 6.1, when n = 2 and , the algorithm givesa and approximations, respectively. Indeed, for n = 3 ,we show a divide-and-choose algorithm which is SP and stillguarantees -approximation ratio. Theorem 6.2
For the public ranking model, there exist an SP . -approximation algorithm when n = 3 . Proof:
Without loss of generality, we order the items ac-cording to agent 1’s ranking, i.e., σ ( j ) = j for any j ∈ M and c ≥ · · · ≥ c m . The algorithm runs as follows: • Let S = { } , S = { j ∈ M | j mod 2 = 0 } and S = { j ∈ M | j > and j mod 2 = 1 } . Note that ( S , S , S ) is a partition of all the items. • Let agent 2 select her favourite bundle from S , S , S . • Let agent 3 select her favourite one from the two bundlesleft in Step 2, and assign the last bundle to agent 1.It is easy to see that the above algorithm is SP as the algorithmdoes not use any information reported by agent 1 and both ofagent 2 and agent 3’s best strategy is to report the costs suchthat the bundle with smallest cost is selected. We are left toprove the approximation ratio.For agent 1, note that c ( S ∪ S ) ≥ c ( S ) ≥ c ( S ) .By the first argument of Lemma 2.1, c ( S ) ≤ c ( S ) ≤ c ( M ) ≤ · MMS . By the second argument of Lemma 2.1, c ( S ) ≤ MMS . That is, no matter which bundle is left, thecost of this bundle is at most · MMS to agent 1.As agent 2 gets her best bundle, her cost is at most MMS .For agent 3, since she is still able to select one from twobundles, her cost for the better bundle is at most · c ( M ) which is at most · MMS by Lemma 2.1.In conclusion, the algorithm is a -approximation. Next, we complement the upper bound results with a lowerbound result for the the public ranking model.
Lemma 6.3
For the public ranking model, no deterministicSP algorithm has an approximation ratio smaller than ,even for agents. roof: Let n = 2 and m = 6 . Assume for contradic-tion that there exists some SP algorithm with approxima-tion ratio less than . Suppose c = (1 , , , , , and c = (1 , , , , , , then we have MMS = MMS = 3 .The algorithm must assign every agent exactly items, asotherwise the cost for one of them is ≥ · . Without lossof generality, assume that A = { , , } and A = { , , } .By Lemma 2.5, when c = (1 , , , , , and c =(1 , , , , , , the assignments remain unchanged, i.e., A = { , , } . If we further increase c to , we have MMS = MMS = 5 . If { } ⊆ A , we should have A = { } to guarantee an approximation ratio less than .Then we know that agent has incentive to lie when itstrue cost is c = (1 , , , , , . Hence we have { } ⊆ A , which implies A = { } and A = { , , , , } .Again, by Lemma 2.5, for c = (1 , , , , , and c =(1 , , , , , , the assignment remains unchanged.Applying a similar argument, for c = (1 , , , , , and c = (1 , , , , , , we have A = { , , } . Then for c = (1 , , , , , and c = (1 , , , , , , we have { } ⊆ A , | A | ≤ , and c ( A ) = 4 to guarantee anapproximation ratio less than . Thus by Lemma 2.5, for c = (1 , , , , , and c = (1 , , , , , , the assign-ment does not change, i.e., | A | = 4 , which contradicts theconclusion we draw in the previous paragraph. In this paper, we initiated the study of SP and approximatelymaxmin fair algorithms for chore allocation. Our study leadsto several new questions. The most obvious research ques-tions would be to close the gap between the lower and up-per approximation bounds for SP algorithms and to study thelower bound of randomized SP cardinal algorithms.At present we have two parallel lines of research for goodsand chores. It is interesting to consider similar questions forcombinations of goods and chores [Aziz et al. , 2018]. An-other direction is to study the SP fair allocation algorithmsfor the case of asymmetric agents [Aziz et al. , 2019].
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