Multi-Type Resource Allocation with Partial Preferences

Multi-Type Resource Allocation with Partial Preferences Haibin Wang,1 Sujoy Sikdar,2 Xiaoxi Guo,1 Lirong Xia,3 Yongzhi Cao,1* Hanpin Wang4,1 1Key Laboratory of High Conﬁdence Software Technologies (MOE), Department of Computer Science and Technology, Peking University, China 2Computer Science & Engineering, Washington University in St. Louis 3Department of Computer Science, Rensselaer Polytechnic Institute 4School of Computer Science and Cyber Engineering, Guangzhou University, China [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], *Corresponding Author

Abstract first class: agents consume their favorite remaining item one- We propose multi-type probabilistic serial (MPS) and multi- by-one according to an ordering drawn from the uniform type random priority (MRP) as extensions of the well-known distribution. Each agent is allocated a fraction of each item PS and RP mechanisms to the multi-type resource allocation equal to the probability that they consume the item. It is easy problems (MTRAs) with partial preferences. In our setting, to see that RP satisfies ex-post-efficiency and equal treat- there are multiple types of divisible items, and a group of ment of equals. Additionally, RP satisfies notions of envy- agents who have partial order preferences over bundles con- freeness and strategyproofness through the idea of stochas- sisting of one item of each type. We show that for the unre- tic dominance (sd). Given strict preferences, a fractional al- stricted domain of partial order preferences, no mechanism location p dominates another q, if at every item o, the total satisfies both sd-efficiency and sd-envy-freeness. Notwith- share of o and items preferred to o under p is at least the standing this impossibility result, our main message is posi- total share under q. RP satisfies weak-sd-envy-freeness and tive: When agents’ preferences are represented by acyclic CP- nets, MPS satisfies sd-efficiency, sd-envy-freeness, ordinal sd-strategyproofness (Bogomolnaia and Moulin 2001). fairness, and upper invariance, while MRP satisfies ex-post- The probabilistic serial (PS) mechanism belongs to the efficiency, sd-strategyproofness, and upper invariance, recov- class of “simultaneous eating” mechanisms (Bogomolnaia ering the properties of PS and RP. Besides, we propose a and Moulin 2001). PS proceeds in multiple rounds. In each hybrid mechanism, multi-type general dictatorship (MGD), round, all agents simultaneously “eat” their favorite re- combining the ideas of MPS and MRP, which satisfies maining item at a constant, uniform rate, until one of the sd-efficiency, equal treatment of equals and decomposability items being consumed is exhausted. This terminates when under the unrestricted domain of partial order preferences. all items are fully consumed, and the output allocates each agent with a fraction of each item that they would con- Introduction sume by this procedure. PS satisfies sd-efficiency, sd-envy- Consider the example of rationing (Elster 1992) two types freeness, and weak-sd-strategyproofness (Bogomolnaia and of divisible resources: food (F) and beverage (B) among two Moulin 2001). Besides, PS is the only mechanism that si- families who have heterogeneous preferences over combi- multaneously satisfies sd-efficiency, sd-envy-freeness, and nations of food and beverage they wish to consume. For ex- bounded invariance (Bogomolnaia and Heo 2012; Bogomol- ample, a family may prefer water with rice, and milk with naia 2015). wheat. How should we distribute available resources to the Our work is the first to consider the design of fair and effi- families fairly and efficiently? cient mechanisms for MTRAs with partial preferences, and In this paper, we consider the problems of divisible multi- the first to extend fractional mechanisms to MTRAs with type resource allocation problems (MTRAs) (Mackin and partial preferences, to the best of our knowledge. Katta and Xia 2016) with partial preferences. Here, there are p ≥ 1

arXiv:1906.06836v3 [cs.AI] 29 Oct 2020 Sethuraman (2006) mention that PS can be extended to par- types of n divisible items per type, with one unit of supply of tial orders but we are not aware of a (formal or informal) each item, and a group of n agents with partial preferences work that explicitly defines such an extension and studies over receiving bundles consisting of one unit of each type. its properties. Monte and Tumennasan (2015) and Mackin Our goal is to design mechanisms to fairly and efficiently and Xia (2016) consider the problem of MTRAs under lin- allocate one unit of items of each type to every agent given ear preferences, but do not fully address the issue of fair- their partial preferences over bundles. Such mechanisms are ness. Ghodsi et al. (2011) consider the problem of allocating called fractional mechanisms, because an agent may receive multiple types of resources, when the resources of each type fractions of items. are indistinguishable, and agents have different demands for When there is one type (p = 1), fractional mechanisms each type of resources. However, the problem of finding fair broadly fall in two classes. The random priority (RP) mech- and efficient assignments for MTRAs with partial prefer- anism (Abdulkadiroglu˘ and Sonmez¨ 1998) exemplifies the ences remains open. Copyright © 2020, Association for the Advancement of Artificial Our mechanisms output fractional assignments, where Intelligence (www.aaai.org). All rights reserved. each agent receives a fractional share of bundles consisting Table 1: Properties of MRP, MPS and MGD under different domain restrictions on partial preferences. A “Y” indicates that the row mechanism satisfies the column property, and an “N” indicates that it does not. Results annotated with † are from (Bogo- molnaia and Moulin 2001), ‡ are from (Hashimoto et al. 2014). Other results are proved in this paper. Mechanism and Preference Domain SE EPE OF SEF WSEF ETE UI SS WSS DC General partial preferences N† Y N‡ N† Y Y N N Y Y MRP CP-nets N† Y N‡ N† Y Y Y Y Y Y Independent CP-nets N† Y N‡ N† Y Y Y Y Y Y General partial preferences Y N N N Y Y N N† N N MPS CP-nets Y N Y Y Y Y Y N† N N Independent CP-nets Y N Y Y Y Y Y N† Y N General partial preferences Y Y N‡ N N Y N N N Y MGD CP-nets Y Y N‡ N N Y N N N Y Independent CP-nets Y Y N‡ N N Y N N N Y of an item of each type, which together amount to one unit (SE) and sd-envy-freeness (SEF) as we prove in Proposi- per type. The fractional assignments output by our mecha- tion 3. Despite this impossibility result, MRP, MPS and nisms also specify for each agent how to form bundles for MGD satisfy several desirable properties: We show in consumption from the assigned fractional shares of items. - Theorem 1 that MRP satisfies ex-post-efficiency (EPE), Our setting may be interpreted as a special case of cake weak-sd-envy-freeness (WSEF), equal treatment of equals cutting (Brams and Taylor 1996; Brams, Jones, and Klam- (ETE), weak-sd-strategyproofness (WSS), and decomposi- ler 2006; Procaccia 2013), where the cake is divided into bility (DC); parts of unit size of p types, and n parts per type, and - Theorem 2 that MPS satisfies sd-efficiency (SE), equal agents have complex combinatorial preferences over being treatment of equals (ETE), weak-sd-envy-freeness (WSEF); assigned combinations of parts of the cake which amount to - Theorem 3 that MGD satisfies sd-efficiency (SE), ex-post- one unit of each type. efficiency (EPE), equal treatment of equals (ETE), and de- Our Contributions. Our work is the first to provide fair and composibility (DC). efficient mechanisms for MTRAs, and the first to extend PS Remarkably, we recover the fairness properties of MPS, and RP both to MTRAs and to partial preferences, to the and the truthfulness and invariance properties for MRP and best of our knowledge. We propose multi-type probabilis- MPS under the well-known and natural domain restriction tic serial (MPS) and multi-type random priority (MRP) as of acyclic CP-net preferences (Boutilier et al. 2004a). We the extensions of PS and RP to MTRAs, respectively. Our show in: main message is positive: Under the well-known and natu- - Theorem 4, that MRP is sd-strategyproof (SS); ral domain restriction of CP-net preferences (Boutilier et al. - Theorem 5, that MPS is sd-envy-free (SEF) and ordinally 2004a), MRP and MPS satisfy all of the fairness and effi- fair (OF); ciency properties of their counterparts for single types and - Proposition 5 that MPS is upper invariant (UI); and complete preferences. - Proposition 6 that MPS is weak-sd-strategyproof (WSS) Unlike single-type resources allocations, in MTRAs, not under the special case where agents’ CP-nets are indepen- all fractional assignments are decomposable, where assign- dent between each type. ments can be represented as a probability distribution over Discussions. MGD may be viewed as a hybrid between assignments where each agent receives a bundle consist- MRP and MPS: on the one hand as a modification of MRP ing of whole items. Unfortunately, the output of MPS may where the priorities depend on the preference profile, in- be indecomposable. In response to this, we propose a new stead of being drawn from the uniform distribution; and mechanism, multi-type general dictatorship (MGD), which on the other hand, as a modification of MPS where only a is decomposable and matches the efficiency of MPS, and subset of the agents are allowed to eat simultaneously in satisfies equal treatment of equals. each round. The design of MGD is motivated by the natu- Our technical results are summarized in Table 1. We ex- ral desire for decomposability (not satisfied by MPS), while tend stochastic dominance to compare two fractional alloca- maintaining the stronger efficiency notion of sd-efficiency tions under partial preferences. Here, a fractional allocation (not satisfied by MRP), and the basic fairness property of p is said to stochastically dominate another allocation q w.r.t. equal treatment of equals. Decomposable assignments are an agent’s partial preference, if at any bundle, the fractional desired when sharing of items imposes overhead costs, as share of weakly dominating bundles in p is larger than or noted by (Sandomirskiy and Segal-Halevi 2019). However, equal to the fractional shares of the bundles in q according to (i) for MTRAs, fractional assignments are not guaranteed to her preference. Formal definitions of stochastic dominance be decomposable as we show in assignment (3) of Exam- and properties in Table 1 can be found in Preliminaries. ple 2, and indeed, MPS does not guarantee decomposable For the unrestricted domain of general partial preferences, assignments; (ii) fractional mechanisms are inevitable when unfortunately, no mechanism satisfies both sd-efficiency certain fairness guarantees are desired. Even when there is one type (p = 1), no mechanism which assigns each item literature assume CP-net preferences (Rossi, Venable, and fully to a single agent, can satisfy the basic fairness property Walsh 2004; Lang 2007), and that agents’ CP-nets have a of equal treatment of equals, whereby, everything else being common dependence structure (see (Lang and Xia 2016) for equal, agents with the same preferences should receive the a recent survey). same share of the resources (e.g. two agents having identical strict preferences). On the flip side, MGD does not satisfy Preliminaries weak-sd-envy-freeness, while MRP and MPS both do. A multi-type resource allocation problem (MTRA) (Mackin and Xia 2016), is given by a tuple (N,M,R). Here, Related Work (1) N = {1, . . . , n} is a set of agents. (2) M = D1∪· · ·∪Dp MTRAs belong to a long line of research on mechanism de- is a set of items of p types, where for each i ≤ p, Di is a set sign for multi-agent resource allocation (see (Chevaleyre of n items of type i, and there is one unit of supply of each et al. 2006) for a survey), where the literature focuses on item in M. We use D = D1 × · · · × Dp to denote the set the settings with a single type of items. Mackin and Xia of bundles. (3) R = ( j)j≤n is a preference profile, where (2016) characterize serial dictatorships for MTRAs by strat- for each j ≤ n, j represents the preference of agent j, and egyproofness, neutrality, and non-bossiness. The exchange R−j represents the preferences of agents in N \{j}. We use economy of multi-type housing markets (Moulin 1995) is R to denote the set of all possible preference profiles. considered in (Sikdar, Adali, and Xia 2017; Sikdar, Adali, Bundles. For any type i ≤ p, we use ki or kt to refer to the k- and Xia 2018) under lexicographic preferences, while Fujita th item of type i where t represents the name of type i. Each et al. (2015) consider the exchange economy where agents bundle x ∈ D is a p-tuple, and we use o ∈ x to indicate that may consume multiple units of a single type of items under bundle x contains item o. We define T = {D1,...,Dp}, lexicographic preferences. and for any S ⊆ T , we define ΠS = ×D∈S D, and −S = Our work is the first to extend RP and PS under partial ˆ T \S. For any S ⊆ T , S ⊆ T \S, and any x ∈ ΠS, y ∈ ΠSˆ, preferences, to the best of our knowledge despite the vast (x, y) denotes the bundles consisting of all items in x and literature on fractional assignments. Hosseini and Larson y. For any S ⊆ T , D ∈ T \ S, and any x ∈ S, o ∈ D, (o, x) (2019) consider RP under lexicographic preferences. The denotes the bundles consisting of o and the items in x. remarkable properties of PS has encouraged extensions to Partial Preferences and Profiles. A partial preference several settings. Hashimoto et al. (2014) provide two char- is a partial order over D, which is an irreflexive, anti- acterizations of PS: (1) by sd-efficiency, sd-envy-freeness, symmetric, and transitive binary relation. Given a partial and upper invariance, and (2) by ordinal fairness and non- preference over D, we define the corresponding prefer- wastefulness. In (Heo 2014; Hatfield 2009), there is a single ence graph, denoted by G , to be the directed graph whose type of items, and agents have multi-unit demands. In (Sa- nodes are the bundles in D, and for every x, y ∈ D, there is ban and Sethuraman 2014), the supply of items may be dif- an directed edge (x, y) if and only if x y and there exists ferent, while agents have unit demand and are assumed to no z ∈ D such that x z and z y. Given a partial order have lexicographic preferences. Other works extend PS to over D, we define the upper contour set of at a bundle settings where indifference relationships are allowed (Katta x ∈ D as U( , x) = {xˆ : xˆ x or xˆ = x}. and Sethuraman 2006; Heo and Yılmaz 2015). Aziz et al. Acyclic CP-nets. A CP-net (Boutilier et al. 2004a) over (2015) consider fair assignments when indifference allowed the set of variables D has two parts: (i) a directed graph in preferences (but not incomparabilities). Yilmaz (2009), G = (T,E) called the dependency graph, and (ii) for each Athanassoglou and Sethuraman (2011) extend PS to the i ≤ p, there is a conditional preference table CPT (D ) that housing markets problem (Shapley and Scarf 1974). Bou- i contains a linear order x over D for each x ∈ Π , veret, Endriss, and Lang (2010) study the complexity of i P a(Di) P a(D ) computing fair and efficient allocations under partial prefer- where i is the set of types corresponding to the par- D G G ences represented by SCI-nets for allocation problems with ents of i in . When is (a)cyclic we say that is a a single type of indivisible items. (a)cyclic CP-net. The partial order induced by an acyclic CP-net over D is the transitive closure of {(o, x, z) (ô, x, z): i ≤ p; o, oˆ ∈ D ; o x oˆ; x ∈ Π ; z ∈ CP-net Preferences i P a(Di) Π }. A CP-profile is a profile of agents’ pref- Compact preference representations are a common approach −(P a(Di)∪{Di}) to deal with the preference formation and elicitation bottle- erences, each of which is represented by an acyclic CP-net. neck faced in MTRAs, where the number of bundles grows We say the acyclic CP-net with a trivial dependency graph exponentially with the number of types. CP-nets (Boutilier such that there is no edge in the dependency graph as inde- et al. 2004b) are perhaps the most well-studied and natural pendent CP-net. An independent CP-profile is a profile of compact preference representation language allowing agents agents’ preferences, each of which is represented by an in- to express conditional (in)dependence of their preferences dependent acyclic CP-net. over combinations of different types (see Example 1). CP- Example 1. Consider MTRA (N,M,R) with p = 2 types, nets are an important restriction on the domain of partial food (F) and beverage (B), where N = {1, 2},M = preferences, and induce a partial ordering on the set of all {1F , 2F , 1B, 2B}, where 1B is item 1 of type B and so on. bundles. Sikdar, Adali, and Xia (2017) design mechanisms Let agent 1’s preference 1 be represented by the acyclic for multi-type housing markets under lexicographic exten- CP-net in Figure 1, where the dependency graph (Figure 1 sions of CP-nets. Several works in the combinatorial voting (a)) shows that her preference on type B depends on her as- Example 2. Consider the situation in Example 1 with the three fractional assignments (1), (2), and (3) shown below. The four upper contour sets for agent 1 are {1F 1B}, {1F 1B, 1F 2B}, {1F 1B, 1F 2B, 2F 2B}, and {1F 1B, 1F 2B, 2F 1B, 2F 2B}. The allocations of the four upper contour sets for agent 1 in assignment (2) are 0.5, 0.5, 1, 1, respectively and in assignment (3) are 0, 0.5, 0.5, 1. The former is greater than or equal to the latter respectively, and the same can be concluded for agent 2 by considering the upper contour sets {1F 1B}, {2F 2B}, {2 1 }, and {1 1 , 1 2 , 2 1 , 2 2 }. Hence assign- Figure 1: Agent 1’s preferences are represented by an acyclic F B F B F B F B F B ment (2) stochastically dominates (3). We can check that (2) CP-net with dependence graph (a), and CP-tables (b), whose does not dominate (1), and that the reverse is true by con- preference graph is in (c). Agent 2’s preferences are repre- sidering the sets of {1 1 } and {2 1 } for agent 2. sented by the preference graph in (d). F B F B

1F 1B 1F 2B 2F 1B 2F 2B signment in type F. The corresponding conditional prefer- Agent 1 0.5 0.5 0 0 (1) ence tables (Figure 1 (b)) show that agent 1 prefers 1B with Agent 2 0 0 0.5 0.5 1F , and 2B with 2F . This induces the preference graph in Figure 1 (c) which happens to be a linear order. Let agent 1F 1B 1F 2B 2F 1B 2F 2B 2’s preference 2 be represented by the preference graph in Figure 1 (d) which represents a partial order, where 1F 2B Agent 1 0.5 0 0 0.5 (2) is the least preferred bundle, and providing no information Agent 2 0.5 0 0 0.5 on the relative ordering of other bundles.

Assignments. A discrete assignment A : N → D is a one 1F 1B 1F 2B 2F 1B 2F 2B to one mapping from agents to bundles such that no item Agent 1 0 0.5 0.5 0 (3) is assigned to more than one agent. A fractional allocation shows the fractional shares an agent acquires over D, repre- Agent 2 0.5 0 0 0.5 1×|D| sented by a vector p = [px]x∈D, p ∈ [0, 1] such that P x∈D px = 1. We use Π to denote the set of all possible fractional allocations on an agent. A fractional assignment Desirable Properties. A fractional assignment P satisfies: (i) sd-efficiency, if there is no fractional assignment P is a combination of all agents’ fractional allocations, and sd can be represented by a matrix P = [p ] , P ∈ Q 6= P such that Q P , (ii) ex-post-efficiency, j,x j≤n,x∈D if P can be represented as a probability distribution over [0, 1]|N|×|D|, such that (i) for every j ≤ n, P p = 1, x∈D j,x sd-efficient discrete assignments, (iii) sd-envy-freeness, if (ii) for every o ∈ M, So = {x : x ∈ D and o ∈ x}, ˆ sd ˆ P for every pair of agents j, j ≤ n, P (j) j P (j), pj,x = 1. The j-th row of P represents agent j≤n,x∈So (iv) weak-sd-envy-freeness, if for every pair of agents j’s fractional allocation under P , denoted P (j). We use P j, ˆj ≤ n, P (ˆj) sd P (j) =⇒ P (j) = P (ˆj), to denote the set of all possible fractional assignments. j (v) equal treatment of equals, if for every pair of agents Mechanisms. A mechanism f : R → P is a mapping from j, ˆj ≤ n j ˆj profiles to fractional assignments. For any profile R ∈ R, such that agents and have the same pref- ˆ we use f(R) to refer to the fractional assignment output by erence, P (j) = P (j), (vi) ordinal fairness, if for ev- f, and for any agent j ≤ n and any bundle x ∈ D, we use ery bundle x ∈ D and every pair of agents j, ˆj ≤ n P P f(R) with Pj,x > 0, Pj,xˆ ≤ Pˆ , and j,x to refer to the value of the element of the matrix xˆ∈U( j ,x) xˆ∈U( ˆj ,x) j,xˆ f(R) indexed by j and x. (vii) decomposability, if P can be represented as a proba- Stochastic Dominance. In this paper we will use the nat- bility distribution over discrete assignments. ural extension of stochastic dominance (Bogomolnaia and A mechanism f satisfies X ∈ {sd-efficiency, ex-post- Moulin 2001) to compare fractional allocations. efficiency, sd-envy-freeness, weak-sd-envy-freeness, equal Definition 1. (stochastic dominance for fractional alloca- treatment of equals, ordinal fairness, decomposability}, if tions) Given a partial preference over D, the stochastic for every R ∈ R, f(R) satisfies X. A mechanism f sat- sd isfies: (i) sd-strategyproofness if for every profile R ∈ R, dominance relation associated with , denoted is 0 0 0 a weak ordering over Π such that for any pair of fractional every agent j ≤ n, every R ∈ R such that R = ( j, −j), sd 0 allocations p, q ∈ Π, p stochastically dominates q, denoted it holds that f(R) j f(R ), and (ii) weak-sd-strate- sd P p q, if and only if for every x ∈ D, pxˆ ≥ gyproofness if for every profile R ∈ R, every agent j ≤ n, xˆ∈U( ,x) 0 0 0 P every R ∈ R such that R = ( , −j), it holds that qxˆ. j xˆ∈U( ,x) 0 sd 0 f(R ) j f(R) =⇒ f(R )(j) = f(R)(j). sd sd We use P j Q to denote P (j) j Q(j). We write Given any partial preferences , we denote |B by the sd sd P Q if for every j ≤ n, we have P j Q. restriction of to B ⊆ D, i.e., |B is a preference relation over B such that for all x, y ∈ B, x |B y ⇔ x y. Then Algorithm 2 MPS 0 for any j ≤ n, j is an upper invariant transformation of Input: An MTRA (N,M,R) j at x ∈ D under a fractional assignment P if for some Output: Assignment P Z ⊆ {y ∈ D | P = 0} U( 0 , x) = U( , x) \Z j,y , j j and 1: j ≤ n 0 0 For each , compute a linear ordering j corre- j|U( 0 ,x)= j|U( 0 ,x). A mechanism f satisfies upper j j sponding to a deterministic topological sort of G j . 0 |N|×|D| 0 invariance if it holds that f(R)ˆj,x = f(R )ˆj,x for ev- 2: P ← 0 and M ← M. For every o ∈ M, ery ˆj ≤ n, j ≤ n, R ∈ R, R0 ∈ R, and x ∈ D, supply(o) ← 1, B ← ∅, progress ← 0. 0 0 0 0 such that R = ( j, −j) and j is an upper invariant 3: while M 6= ∅ do 0 0 transformation of j at x under f(R). 4: top(j) ← Ext( j,M ) for every agent j ≤ n. 5: Consume. Mechanisms for MTRAs with Partial 5.1: For each o ∈ M 0, consumers(o) ← |{j ∈ N : Preferences o is in top(j)}|. supply(o) In this section, we propose MRP (Algorithm 1 as extension 5.2: progress ← mino∈M 0 consumers(o) . of RP), MPS (Algorithm 2, as extension of PS), and MGD 5.3: For each j ≤ n, Pj,top(j) ← Pj,top(j) + progress (Algorithm 3), which can be seen as not only an eating algo- o ∈ M 0 (o) ← (o) − rithm but a special random priority algorithm. 5.4: For each , supply supply × (o) The three mechanisms operate on a modified preference progress consumers . profile of strict preferences, where for every agent with par- supply(o) 0 0 tial preference , an arbitrary deterministic topological sort- 6: B ← arg mino∈M 0 consumers(o) , M ← M \ B 0 ing is applied to obtain a strict ordering over D, such that 7: return P for any pair of bundles x, y ∈ D, x y =⇒ x 0 y. Given a strict order 0 obtained in this way, and remaining M 0, we use Ext( 0,M 0) to denote the first available bun- Algorithm 3 MGD dle in 0, which we refer to as the agents’ favorite bundle. Input: An MTRA (N,M,R) It is easy to see that no available bundle is preferred over Output: Assignment P 0 0 0 Ext( ,M ) according to . 1: For each j ≤ n, compute a linear ordering j corre- Our results apply to arbitrary deterministic topological sponding to a deterministic topological sort of G j . sortings (induced by a fixed ordering over items), even 2: P ← 0|N|×|D| and M 0 ← M. though different topological sortings may lead to different 3: for j = 1 to n do outputs. 0 0 4: top(j) ← Ext( j,M ). 5: For each j0 ∈ Group(j, 0), let Algorithm 1 MRP Input: An MTRA (N,M,R) 1 P 0 ← Output: Assignment P j ,top(j) |Group(j, 0)| 1: For each j ≤ n, compute a linear ordering 0 corre- j 6: M 0 ← M 0 \{o ∈ M : o is in top(j)}. sponding to a deterministic topological sort of G . j 7: return P 2: P ← 0|N|×|D| and M 0 ← M. 3: Pick a random priority order B over agents. 4: Successively pick a highest priority agent j∗ according ∗ 0 0 Given an instance of MTRA with agents’ partial prefer- to B. x ← Ext( j∗ ,M ) and set Pj∗,x∗ ← 1. Re- ∗ ∗ 0 ences, MGD proceeds by operating on an arbitrary deter- move j , and remove all items contained by x in M . ministic topological sorting 0. Let Group(j, 0) denote the 5: return P set of all agents who have the same order with agent j in 0. MGD proceeds in n rounds as follows. In each round Given an instance of MTRA with agents’ partial prefer- j ≤ n, agent j comes and invites other agents with the same ences, MRP fixes an arbitrary deterministic topological sort- topological sort 0 to consume her favorite available bundle ing 0 of agents’ preferences, and sorts agents uniformly at j Ext( 0 ,M 0) with 1 unit of time and one unit of random. Then agents get one unit of their favorite available j |Group(j, 0)| bundle from the remaining M 0 in turns as in RP. eating rate. We present MGD as its eating algorithm ver- Given an instance of MTRA with agents’ partial prefer- sion here and show that MGD(R) is the expected result ences, MPS involves applying the PS mechanism to a modi- of a special random priority algorithm when we prove its fied profile 0 over D using an arbitrary deterministic topo- decomposability. logical sorting in multiple rounds as follows. In each round, Example 3. Consider the situation in Example 1, the topo- 0 0 each agent consumes their favorite available bundle by con- logical sort of agent 1’s preference is 1F 1B 1F 2B 0 0 0 0 suming each item in the bundle at an uniform rate of one 2F 2B 2F 1B but both 2F 1B 1F 1B 2F 2B 0 0 0 unit of an item per type per unit of time, until one of the 1F 2B and 1F 1B 2F 2B 2F 1B 1F 2B can be bundles being consumed becomes unavailable because the the topological sort of agent 2’s preference. If the topologi- supply for one of the items in it is exhausted. cal sort of agent 2 is the former, in MPS, agent 1 consumes 1F 1B and agent 2 consumes 2F 1B at the beginning. When Given an assignment P and a partial preference profile they both consume 0.5 fraction, 1B is exhausted. 1F 1B and R, for any x, xˆ ∈ D, (x, xˆ) is an improvable tuple, de- 2F 1B become unavailable. They turn to identify the next noted by Imp(P,R), if there exists an agent j < n such bundles in line 4. For example, the first two bundles 2F 1B that x j xˆ and pj,xˆ > 0. We use Imp(P ) for short and 1F 1B are unavailable for agent 2, so she turns to the when the preferences are clear from the context. Bogo- third bundle 2F 2B. Agent 1 turns to 1F 2B. Then MPS goes molnaia and Moulin (2001) show that an assignment P to next round of consumption until all left items are ex- is sd-efficient if and only if the binary relation Imp(P ) hausted at the same time. The result is shown in assignment has no cycle in single-type resources allocations, but the (1) in Example 2. However, if the topological sort of agent sufficient condition fails to hold for MTRAs. As Exam- 2 is the latter, agent 1 and agent 2 both get 0.5 of 1F 1B ple 2 shows, assignment (3) is sd-inefficient, but the set and 0.5 of 2F 2B as assignment (2) shows. It is easy to check of the improvable tuples, {(1F 1B, 1F 2B), (1F 1B, 2F 1B), that MRP has the same conclusion. Suppose agent 1 has the (1F 2B, 2F 1B), (2F 2B, 2F 1B)}, has no cycle. same partial preference with agent 2 in Example 1 in MGD. Theorem 2. Under general partial preferences, MPS If their topological sorts are both the former, agent 1 invites satisfies sd-efficiency, weak-sd-envy-freeness, and equal agent 2 to consume 2 1 in round 1 and agent 2 invites F B treatment of equals. agent 1 to consume 1F 2B in round 2. But if their topological sorts are both the latter, agent 1 invites agent 2 to consume To prove the sd-efficiency of MPS, we relax the cycle 1F 1B in round 1 and agent 2 invites agent 1 to consume from bundles to items and find a sufficient condition for 2F 2B in round 2. sd-efficiency in MTRAs under general partial preferences. For example in assignment (3), agent 1 can extract 1 from There is an unique best available bundle w.r.t. any acyclic B 2F 1B to match 1F and extract 2B from 1F 2B to match 2F CP-net preference and remaining supply of items, which to improve her result. We show MPS satisfies sd-efficiency can be computed in polynomial time by induction on the by proving MPS satisfies the sufficient condition. We find types according to the dependency graph as we show in that one agent does not envy others at the first bundle of her Proposition 1. This is an extension of the well-known result topological sort since she consumes the bundle from the be- of Boutilier et al. (2004a) that there is a unique best bundle ginning to the end when the bundle is exhausted. Based on w.r.t. any acyclic CP-net. that, We can prove she does not envy others at any upper 0 0 0 0 contour set under MPS by induction on bundles under the Proposition 1. Let T = {D1 ⊆ D1,...,Dp ⊆ Dp}, D = ΠT 0 , and let be any acyclic CP-net over D. Then, there topological sorting. 0 0 exists unique x ∈ D such that for every y 6= x ∈ D , Remark 2. MPS is not ex-post-efficient since its output may x y. not be decomposable when coming to multi-type resources. All missing proofs can be found in Appendix. MPS is not ordinally fair, sd-envy-free and upper invariant We use T op( ,M 0) to denote the best available bundle in under general partial preferences. remaining M 0 given an acyclic CP-net . Under the domain PS is both sd-efficient and decomposable in single-type restriction of acyclic CP-net preferences, for any topologi- resources allocations (Bogomolnaia and Moulin 2001), but 0 0 0 cal sorting algorithms, Ext( j,M ) is exact T op( j,M ) not decomposable in MTRAs. Serial dictatorship (Mackin for any M 0 and j ≤ n by Proposition 1. Therefore, we can and Xia 2016; Hosseini and Larson 2019) maintains both sd- 0 0 0 remove line 1 and instead Ext( j,M ) with T op( j,M ) efficiency and decomposability even in MTRAs but perform in the three mechanisms to save the time and space. badly in fairness. Thankfully, MGD not only satisfies sd- efficiency and decomposability but the common requirement MRP, MPS and MGD run in O(np+1) time. Proposition 2. for fairness, equal treatment of equals. We note that the size of the preference representation is p+1 Theorem 3. Under general partial preferences, MGD O(n ), and forms a part of the input. satisfies sd-efficiency, ex-post-efficiency, equal treatment of equals, and decomposability. Properties under General Partial Preferences When agents have different topological sorts with each Theorem 1. Under general partial preferences, MRP other, MGD comes to serial dictatorship. Since the dictator satisfies ex-post-efficiency, weak-sd-envy-freeness, equal only invite agents who have the same topological sort with treatment of equals, weak-sd-strategyproofness, and decom- her to share her bundle, the sd-efficiency of serial dictator- posability. ship maintains in MGD. Since agents with the same preference have the same topological sorts, MGD satisfies equal The proof of weak-sd-envy-freeness involves showing treatment of equals. MGD is decomposable since its output that for any two agents j and ˆj, agents j and ˆj receive equal can be seen as an expected result of a special random prior- shares of every bundle in expectation, due to a bijective map- ity algorithm. Decomposability and sd-efficiency induce ex- ping from the set of orders where j picks before ˆj and vice- post-efficiency (Bogomolnaia and Moulin 2001). versa. Remark 3. MGD is not ordinally fair, weak-sd-envy-free, Remark 1. Under general partial preferences, MRP is not upper invariant and weak-sd-strategyproof even in single- upper invariant and sd-strategyproof. type resources allocations under linear preferences. PS is both sd-efficient and sd-envy-free under linear pref- P Pi Pî x∈U( ,x ) P1,x = k=1 P1,xk ≤ k=1 P2,xˆk ≤ erences (Bogomolnaia and Moulin 2001) but this is no P 1 i x∈U( ,x ) P2,x. Therefore, for any y ∈ D with P1,y > 0, longer true under general partial preferences, as the impos- P2 i P we have P1,x ≤ P2,x. sibility result in Proposition 3 shows. x∈U( 1,y) x∈U( 2,y) Proposition 3. No mechanism can satisfy both sd-efficiency Hashimoto et al. (2014) shows that ordinal fairness char- and sd-envy-freeness under general partial preferences. acterizes PS when we come to single-type resources. How- ever, this is not the case in MTRAs: for example, if two Properties under Acyclic CP-net Preferences agents’ preferences are both as (c) in Figure 1 shows, the Theorem 4. Given any CP-profile R, MRP (R) is sd- result that two agents both get 0.5 1F 2B and 0.5 2F 1B is strategyproof. ordinally fair but not the output of MPS.

0 Proposition 5. Given any CP-profile R, MPS(R) is upper Proof. Let P and P be the expected assignments of MRP 0 0 0 0 invariant for any other CP-profile R . given the CP-profile R and R ∈ R such that R = ( j , −j) for some j ≤ n. For an arbitrary fixed priority order, Given any upper invariant transformation at some y ∈ D, in any agent j’s turn, the set of available bundles is the same we show the consumption processes are identical until y is under R and R0. By Proposition 1, the best available bundle exhausted whether the misreport agent is truthful or lies, by is also the same and unique. If j lies, she may get a smaller induction on the bundles consumed by the misreport agent share of her best available bundle. Therefore, the result of under R. Details are in the Appendix. a lie is stochastic dominated by the result of truthfulness. 0 Remark 4. MPS is not weak-sd-strategyproof under acyclic Since this is true for any fixed priority order, P is stochastic CP-net profiles even when all agents have a shared depen- dominated by P . dency graph, and agents are only allowed to misreport their CPTs. Proposition 4. Given any CP-profile R, MRP (R) is upper invariant for any other CP-profile R0. Proposition 6. Given any independent CP-profile R, 0 Thanks to Proposition 1, given any upper invariant trans- MPS(R) is weak-sd-strategyproof for any profile R . formation at some y ∈ D, if y is available in the misreport Independent CP-nets are a special case of acyclic CP-nets agent’s turn, she gets the same bundle despite her lie. Oth- where the dependency graph has no edges, and the CPT erwise if y is unavailable, the lie does not affect the assign- for each type represents preferences over items of that type ment of y. without any conditional dependencies on the allocation of Theorem 5. Given any CP-profile R, MPS(R) is sd-envy- items of other types. Strict preferences over a single type of free and ordinally fair. resources are a special case where p = 1. Moreover, when p = 1, MPS is equivalent to PS, and at any point during Proof. W.l.o.g we only prove the case between agent 1 and the execution of MPS agents consume their favorite remain- agent 2. Let P = MPS(R). Let D1 = {x ∈ D : P1,x > 0} ing item until it is exhausted. The proof for Proposition 6 is and n1 = |D1|. By Proposition 1, we have an order over D1 along similar lines to the proof of weak-sd-strategyproofness such that x1 1 x2 1 · · · 1 xn1 , xi ∈ D1 for any i ≤ n1. of PS due to (Bogomolnaia and Moulin 2001). For agent 2, we can define D2 and n2 and have an order over

D2, xˆ1 2 xˆ2 2 · · · 2 xˆn2 , similarly. sd Conclusion and Future Work (1) sd-envy-freeness. We need to prove P (1) 1 P (2). For any y ∈ D, let xi be the least favorable bundle of We proposed and studied MRP and MPS as extensions of RP agent 1 in U( , y) ∩ D . If i = n , P P = and PS to MTRAs. We also proposed MGD that is both sd- 1 1 1 x∈U( 1,y) 1,x Pn1 P efficient and decomposable. For future work, we are inter- P1,x = 1 ≥ P2,x. If i < n1, xi+1 ∈/ k=1 k x∈U( 1,y) ested in axiomatic characterization of MRP, MPS and MGD. U( 1, y). When agent 1 starts to consume xi+1, yˆ is un- More generally, designing desirable mechanism for MTRAs available for any yˆ ∈ U( 1, y). Otherwise by Proposi- is still an interesting and challenging open question. tion 1, we have xi+1 1 yˆ or xi+1 = yˆ both indicating x ∈ U( , y) t(x ) i+1 1 , a contradiction. Suppose i be the Acknowledgments time when xi is exhausted and agent 1 starts to consume P Pi We are grateful to the anonymous reviewers for their xi+1. Then x∈U( ,y) P1,x = k=1 P1,xk = t(xi) ≥ P 1 helpful comments. LX acknowledges NSF #1453542 and P2,x for any y ∈ D. Hence P (1) 1 P (2). x∈U( 1,y) #1716333 for support. YC acknowledges NSFC under (2) ordinal fairness. For any y ∈ D1, we need to prove P P Grants 61772035, 61751210, and 61932001, and the Na- P1,x ≤ P2,x. For any xi ∈ D1, x∈U( 1,y) x∈U( 2,y) tional Science and Technology Major Project for IND (in- suppose that when xi is exhausted, agent 2 is consum- vestigational new drug) under Grant 2018ZX09201-014 for ing xˆˆ ∈ D2 or xˆˆ is exhausted at the same time. If i P i P support. HW acknowledges NSFC under Grants 61572003 xˆˆ = xi, we have P1,x = P2,x. i x∈U( 1,xi) x∈U( 2,xi) and 61972005, and the National Key R&D Program under If xˆî 6= xi, since xi is available when agent 2 starts to Grants 2018YFB1003904 and 2018YFC1314200 for sup- ˆ consume xˆî, we have xˆk 2 xi for any k ≤ i. Hence, port. References [Ghodsi et al. 2011] Ghodsi, A.; Zaharia, M.; Hindman, B.; [Abdulkadiroglu˘ and Sonmez¨ 1998] Abdulkadiroglu,˘ A., Konwinski, A.; Shenker, S.; and Stoica, I. 2011. Dominant and Sonmez,¨ T. 1998. Random serial dictatorship and resource fairness: Fair allocation of multiple resource types. the core from random endowments in house allocation In Proceedings of the 8th USENIX Conference on Networked problems. Econometrica 66(3):689–702. Systems Design and Implementation, 323–336. [Hashimoto et al. 2014] Hashimoto, T.; Hirata, D.; Kesten, [Athanassoglou and Sethuraman 2011] Athanassoglou, S., ¨ and Sethuraman, J. 2011. 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Mechanism design for multi-type housing markets. rounds at most to consume. In each round of consumption, Proceedings of the 31st AAAI Conference on Artificial In MPS cost O(np) in line 5.1, 5.2, 5.4 and O(n) in line 5.3. Intelligence . Hence, MPS cost O(n2p2) to consume totally. Since p is a [Sikdar, Adali, and Xia 2018] Sikdar, S.; Adali, S.; and Xia, small number, the cost of MPS is still O(np+1). L. 2018. Top-trading-cycles mechanisms with acceptable (3) MGD. Line 1 cost O(np+1) in MGD. Line 4 cost bundles. In Proceedings of the Thirty-Second AAAI Confer- O(np) in each round and O(np+1) totally. There are some ence on Artificial Intelligence, AAAI, volume 18. previous work for getting Group(j, 0) in line 5. We should [Yilmaz 2009] Yilmaz, O.¨ 2009. Random assignment un- compute the segmentation of N such that agents with the der weak preferences. Games and Economic Behavior same topological sort is in and only in the same set. To do 66(1):546–558. so, we let all agents be in a same set with np rounds following. In each round k ≤ np, for each set, visit the k-th bundles Appendix in the topological sorts of the agents in the set, divide agents in the set into several new sets such that the visited bundles Proof of Proposition 1. of agents in the same new set is same. That can be done in 0 0 0 0 p Proposition 1. Let T = {D1 ⊆ D1,...,Dp ⊆ Dp}, D = linear time by using a map with n size in each round. The ΠT 0 , and let be any acyclic CP-net over D. Then, there final sets make up the segmentation. The previous work cost exists unique x ∈ D0 such that for every y 6= x ∈ D0, O(np+1) totally. After the previous work, line 5 can be done x y. in O(n). The total cost in line 6 is O(np) since there are only np items totally. Hence, the running time of MGD is Proof. Given any acyclic CP-net , suppose the depen- O(np+1). dency graph in the CP-net is G. As G has no circle, there

is Di1 ∈ T such that Di1 has no parent and there is o1 in D0 such that the agent prefers o than any other item in i1 1 Proof of Theorem 1. 0 D . Then there is Di ∈ T such that P a(Di ) ⊆ {Di }. i1 2 2 1 Theorem 1. Under general partial preferences, MRP By CPT (Di2 ), we know the agent has a linear preference 0 satisfies ex-post-efficiency, weak-sd-envy-freeness, equal over Di given o1. So there is o2 ∈ D such that the agent 2 i2 treatment of equals, weak-sd-strategyproofness, and decom- prefers o than any other item in D0 when she is holding 2 i2 posability. o1. The same procedure can be easily adapted to obtain o3 D0 o D0 o D0 Proof. (1) ex-post-efficiency. Given any priority order of in i3 , 4 in i4 ,..., p in ip . We claim that o1o2...op oˆ1oˆ2...oˆp for any oˆ1oˆ2...oˆp ∈ agents in MRP, suppose P is the output. Assume for the sake 0 D and oˆ1oˆ2...oˆp 6= o1o2...op. If o1 6=o ˆ1, the agent of contradiction that there exists an assignment Q 6= P ∈ P sd prefers o1 than oˆ1 and we have o1oˆ2...oˆp oˆ1oˆ2...oˆp, such that Q P . Let j be the first agent in the set of agents who get different bundles between P and Q. Then, else we have o1oˆ2...oˆp =o ˆ1oˆ2...oˆp. Since P a(Dik+1 ) ⊆ {Di ,Di , ..., Di } for any k < p, the agent prefers ok+1 any bundle she gets in Q is available in her turn in MRP. Let 1 2 k sd than oˆk+1 given o1, o2, ..., ok or ok+1 =o ˆk+1. There- x be the bundle agent j gets in P . Since Q P , there is a for, o1o2...okok+1oˆk+2...oˆp o1o2...okoˆk+1oˆk+2...oˆp or bundle y ∈ D such that y j x, Qj,y > 0 and y is available o1o2...okok+1oˆk+2...oˆp = o1o2...okoˆk+1oˆk+2...oˆp. Since in agent j’s turn. That is contradicted with that agent j gets is transitive and o1o2...op 6=o ˆ1oˆ2...oˆp, we have o1o2...op x in her turn. Therefore, the result of MRP is sd-efficient for oˆ1oˆ2...oˆp. Since the selection of ok (k ≤ p) only depends on any fixed priority order. The expected result of MRP is the the selections of its parents, o1o2...op is unique. result by randomizing the priority orders, so it is the linear combination of some sd-efficient discrete assignments. (2) weak-sd-envy-freeness. W.l.o.g we only consider agent 1 and agent 2. Let assignment P be the expected result of Proof of Proposition 2. sd MRP. Suppose P (2) 1 P (1). We need to prove that Proposition 2. MRP, MPS and MGD run in O(np+1) time. P (2) = P (1). Suppose the topological order of agent 1’s preference in p 0 0 0 Proof. (1) MRP. For a preference graph with n nodes, we MRP is x1 x2 ... xnp . Let O be the set of any pos- p can get its topological sort in O(n ) via a queue and a vec- sible priority order, O1 be the set of orders in which agent 1 tor to mark. Hence, the running time of line 1 in MRP is is ranked in the front of agent 2 and O2 be the set of orders O(np+1). Line 1 run in O(n). In line 4, MRP cost O(np) to in which agent 2 is ranked in the front of agent 1. It is obvi- p+1 get Ext for each agent. Hence, line 4 cost O(n ) totally. ous that {O1,O2} is one division of O. Let f : O1 7→ O2 o ∈ O f(o ) 0 P 0 = P such that for any 1 1, 1 is the order that exchange , since 1,x1 1,x, there is a priority order B2 such 0 0 the positions of agent 1 and agent 2 in o1. f is a one-to-one that agent 1 gets x1 under R and gets x2 where x2 x1 mapping and a surjection from O1 to O2 obviously. under R. Continue the process until we find the first bundle 0 We define the proposition F (x) as follow. For any o1 ∈ xi in and there is no such priority order for xi. Hence, 0 0 sd O1, o2 ∈ O2 and f(o1) = o2, either case (i) or case (ii) P > P P P 1,xi 1,xi , a contradiction with 1 . Therefore, happens: (i) agent 1 gets x under o1 or o2 while agent 2 gets P 0(1) = P (1). x under the other order; (ii) both agent 1 and agent 2 do not (4) equal treatment of equals and decomposability. That get x under o1 and o2. is obvious. Then, we prove that F (x1) is true. We consider two cases.

1. For any o2 ∈ O2, let o1 ∈ O1 such that f(o1) = o2. If agent 2 gets x1 under o2, x1 is available for agent 1 under Counterexample for Remark 1. o1 and agent 1 gets x1 under o1. Remark 1. Under general partial preferences, MRP is not 2. For any o1 ∈ O1, agent 1 is front of agent 2 and x1 is the upper invariant and sd-strategyproof. first bundle in topological order of agent 1’s preference, so agent 2 has no chance to get x1 under o1. Proof. Consider the case where N = {1, 2}, M = D1 = {1F , 2F }, and R = { 1, 2} where 1= ∅, 2= {1F 2 Based on the two cases above, we can conclude that p ≤ 0 2,x1 2F }. In MRP, the fixed topological orders are 1F 2F for sd 0 p1,x1 . But P (2) 1 P (1) leads to p2,x1 ≥ p1,x1 . So 1 and 1F 2F for 2. It is obvious that agents both get p = p that induce that F (x ) is true in addition to 0 0 2,x1 1,x1 1 0.5 1F and 0.5 2F under this situation. Let R = { 1, 2} the two points above. 0 0 0 where 1= {2F 1 1F }. Under R , agent 1 gets 2F and p 0 Next, supposing for any i < k (k ∈ {2, 3, ..., n }), agent 2 gets 1F . Let P and P be the expected output of p = p and F (x ) is true, we prove that p = p sd 0 2,xi 1,xi i 2,xk 1,xk MRP. We don’t have P 1 P . Hence, MRP is not sd- and F (x ) is true. Consider the two cases again. 0 k strategyproof under general partial preferences. 1 is an upper invariant transformation of 1 at 2F under P , but 1. For any o2 ∈ O2, there is o1 ∈ O1 such that f(o1) = o2. P 0 6= P 1,2F 1,2F . Hence, MRP is not upper invariant under If agent 2 gets xk under o2, xk is available for agent 1 general partial preferences. under o1. Besides, it means that agent 2 does not get xi for any i < k under o2. So the case (ii) in F (xi) happens, agent 1 does not get xi under o1. Agent 1 choose xk in her turn under o1. Proof of Theorem 2. 2. For any o1 ∈ O1, there is o2 ∈ O2 such that f(o1) = o2. Theorem 2. Under general partial preferences, MPS If agent 2 dose not get xk under o1, it dose not influence satisfies sd-efficiency, weak-sd-envy-freeness, and equal the result that the expectation of xk agent 1 gets is bigger treatment of equals. than what agent 2 gets. If agent 2 gets xk under o1, xk is available for agent 1 under o2. Agent 2 does not get xi for Proof. (1) sd-efficiency. Given any partial preference R and any i < k under o1 and the case (ii) in F (xi) happens, any assignment P for an MTRA, a set C ⊆ Imp(P ) is a that indicates Agent 1 does not get xi for any i < k under generalized cycle if it holds for every o ∈ M that: if there o2. Hence, Agent 1 gets xk under o2. exists an improvable tuple (x1, xˆ1) ∈ C such that o ∈ x1, then there exists a tuple (x2, xˆ2) ∈ C, such that o ∈ xˆ2. The two cases above imply that p2,xk ≤ p1,xk .Since sd Claim 1. Given any partial preference profile R and assign- P (2) 1 P (1) and p2,xi = p1,xi for any i < k and ment P , P is sd-efficient if P admits no generalized cycle at U( 1, xk) ⊆ {xi : i ≤ k}, we have p2,xk ≥ p1,xk . Hence R. p2,xk = p1,xk that induce that F (xk) is true in addition to the two points above. We have P (2) = P (1) by induction. Proof: By inversions, we assume that the assignment P admits no generalized cycle at R and there exists another (3) weak-sd-strategyproofness. W.l.o.g Suppose agent 1 sd is the misreport agent. Given any partial preference R, let assignment Q 6= P such that Q P . We try to find a 0 0 0 generalized cycle in P . R = ( 1, −1) and be the fixed topological order over 0 ˆ ˆ 1 in MRP. Let assignment P and P be the expected re- Let N = {j ∈ N : P (k) 6= Q(k)} and N 6= ∅ because 0 0 sd Q 6= P . Let C be the set of all tuples (x, xˆ) such that for sults of MRP under R and R . Suppose P 1 P . As- 0 ˆ sume for the sake of contradiction that P (1) 6= P (1). Let some j ∈ N, x j xˆ and qj,x > pj,x, qj,xˆ < pj,xˆ. Because x 0 P 0 6= P p > q ≥ 0, we have C ⊆ Imp(P ). 0 be the first bundle in such that 1,x0 1,x0 . We j,xˆ j,xˆ P 0 > P P 0 sd P For any j ∈ Nˆ, there exists x ∈ D such that q 6= p . have 1,x0 1,x0 since 1 . Under a priority or- j,x j,x 0 der of agent, if agent 1 gets y ∈ D such that x0 y If we have qj,x > pj,x for all qj,x 6= pj,x, we would have 0 P P under R, x0 is unavailable under R and R . Hence, there 1 = x∈D qj,x > x∈D pj,x = 1 which is a contradic- 0 tion. Hence there exists x ∈ D such that q < p . is a priority order B1 such that agent 1 gets x0 under R 0 j,x0 j,x0 0 We have P q ≥ P p because and gets x1 where x1 x0 under R, otherwise we have x∈U( j ,x0) j,x x∈U( j ,x0) j,x P 0 ≤ P x Q P x ∈ D 1,x1 1,x, a contradiction. If 1 isn’t the first bundle in . Hence, there exists some 1 such that x1 j x0 and qj,x1 > pj,x1 , otherwise we induce a contra- Suppose for any i < k, p1,xi = p2,xi and agent 1 and diction that P q < P p . There- agent 2 are both consuming the same bundle at any time x∈U( k,x0) j,x x∈U( k,x0) j,x fore, (x1, x0) ∈ C and C 6= ∅. in [ti−1, ti). We want to prove agent 1 and agent 2 are Suppose for the sake of contradiction that C is not a gen- both consuming the same bundle at any time in [tk−1, tk). eralized cycle. Then there exists an item o such that for any If tk = tk−1, that is true for [tk−1, tk). If tk > tk−1, x ∈ D containing o and there is x in some tuple in C, x agent 1 is consuming xk in the whole interval of [tk−1, tk) is always the left component of any improvable tuple in- but never in [t0, tk−1). Hence, agent 2 never consumes volving x. Obviously, we have q = p for any j∈ / Nˆ. xk in [t0, tk−1). Since xk is exhausted at tk, we have j,x j,x p = t − t ≥ p P sd P ˆ 1,xk k k−1 2,xk . Since 2 1 1, we Suppose there exists j ∈ N such that qj,x < pj,x. Since P P have p2,x ≥ p1,x. Since U( 1 P q ≥ P p , there exists some x∈U( 1,xk) x∈U( 1,xk) xˆ∈U( j ,x) j,xˆ xˆ∈U( j ,x) j,xˆ , xk) ⊆ {x1, x2, ..., xk−1, xk} and p1,xi = p2,xi for any xˆ ∈ D such that xˆ j x and qj,xˆ > pj,xˆ. Hence (xˆ, x) ∈ C i < k, we have p ≥ p . Therefore, p = p and x is the right component which is a contradiction to x 2,xk 1,xk 1,xk 2,xk and agent 1 and agent 2 are both consuming xk in the whole is always the left component. Therefore, for any j ∈ Nˆ, we interval of [tk−1, tk). have qj,x ≥ pj,x. Since x is the left component of some tuple p By induction, we have p1,xk = p2,xk for any k ≤ n . in C, there exists j1 ∈ N, such that qj ,x > pj ,x. Therefore P P 1 1 Hence P (1) = P (2). 1 = j∈N,o∈x qj,x > j∈N,o∈x pj,x = 1, a contradiction. (3) equal treatment of equals. That is obvious. Therefore for any tuple (x, xˆ) ∈ C and o ∈ x, there is a tuple (yˆ, y) ∈ C such that o ∈ y. Hence C is a generalized cycle. Counterexamples for Remark 2. Suppose P is the output of MPS given any partial preference profile R. We next prove P admits no generalized Remark 2. MPS is not ex-post-efficient since its output may cycle, and therefore is sd-efficient by Lemma 1. not be decomposable when coming to multi-type resources. Suppose for the sake of contradiction that P admits a gen- MPS is not ordinally fair, sd-envy-free and upper invariant eralized cycle C. Let (x, xˆ) be an arbitrary improvable tu- under general partial preferences. ple in C. Let Seq be the partial order on M which reflects Proof. (1) MPS is not ex-post-efficient coming to multi-type the time when items are exhausted. Formally, for any pair of resources. We consider a special case for partial prefer- ˆ items o and oˆ exhausted at times t and t during the execution ences, CP-net preferences with the shared dependency graph ˆ of ESP respectively, if t ≤ t, then o Seq oˆ. as shown in Figure 2. We can induce that the preferences ˆ Let M be the set of items that are in some left compo- are 1F 2B 1 1F 1B 1 2F 1B 1 2F 2B and 1F 1B 2 ˆ nents involved in C, and let oˆ ∈ M be one of top items 1F 2B 2 2F 2B 2 2F 1B. MPS(R) is shown in assign- according to Seq. By definition of generalized cycles, there ment 4 which cannot be realized obviously. is an improvable tuple (x, xˆ) ∈ C such that oˆ ∈ xˆ. Since ˆ (x, x) ∈ Imp(P ), there exists an agent j ∈ N such that 1 1 1 2 2 1 2 2 x xˆ and p > 0. Hence when agent j starts to consume F B F B F B F B j j,xˆ Agent 1 0 0.5 0.5 0 (4) xˆ, x is unavailable and there is an item o ∈ x such that o is Agent 2 0.5 0 0 0.5 unavailable when j starts to consume xˆ. If we use t(o) and t(ô) to stand for the times when o and oˆ exhausted in MPS respectively, we have t(o) ≤ t(ô) − pj,xˆ < t(ô). Besides, since (x, xˆ) ∈ C, we have o ∈ Mˆ , a contradiction to that oˆ is one of top items according to Seq in Mˆ . (2) weak-sd-envy-freeness. Suppose P is the output of MPS given any partial preference profile R. W.l.o.g we only sd prove the case for agent 1 and agent 2, that is if P (2) 1 P (1), then P (1) = P (2). sd Suppose P (2) 1 P (1). Suppose the topological order 0 0 0 of agent 1’s preference in MPS is x1 x2 ... xnp . There exists 0 = t0 ≤ t1 ≤ · · · ≤ tnp = 1 such that for any i ≤ np, if p > 0, t > t and t is the time when x is 1,xi i i−1 i i Figure 2: CP-net preferences with the shared dependency exhausted, else ti = ti−1 and xi is unavailable at ti. graph: (a) the dependency graph for agents; (b) CPTs for At the beginning, any bundle is available. Hence, p > 1,x1 agent 1; (c) CPTs for agent 2. 0 and agent 1 starts to consume x1 at t0 = 0 till x1 is ex- p ≥ p P (2) sd hausted. That indicates 1,x1 2,x1 . Since 1 (2) MPS is not ordinally fair and sd-envy-free under gen- P (1) and x1 is the first bundle in the topological or- P eral partial preferences. Consider the case where N = der over 1, we have p2,x = p2,x ≥ 1 x∈U( 1,x1) {1, 2}, M = D1 = {1F , 2F }, and R = { 1, 2} where P p = p . Therefore, p = p and x∈U( 1,x1) 1,x 1,x1 1,x1 2,x1 1= ∅, 2= {1F 2 2F }. In MPS, the fixed topologi- 0 0 agent 1 and agent 2 are both consuming x1 in the whole in- cal orders are 2F 1F for 1 and 1F 2F for 2. It terval of [t0, t1). is obvious agent 1 gets 2F and agent 2 gets 1F . Let P = P MPS(R). We have P2,1F > 0 and x∈U( ,1 ) P2,x > agents in the same group have the same partial preference, P 2 F P1,x. Hence, MPS is not ordinally fair. We it is obvious the expected result of the mechanism is that x∈U( 1,1F ) 0 0 1 sd for each j ∈ Group(j, ), Pj0,top(j) ← |Group(j, 0)| . don’t have P (1) 1 P (2). Hence, MPS is not sd-envy- free. That is same as MGD(R). Since the expected result of (3) MPS is not upper invariant under general partial pref- random priority mechanism is decomposable, MGD(R) is erences. The counterexample of upper invariance for MRP decomposable. can be also an counterexample for MPS. (4) ex-post-efficiency. Decomposability and sd-efficiency induce ex-post-efficiency (Bogomolnaia and Moulin 2001).

Proof of Theorem 3. Counterexamples for Remark 3. Theorem 3. Under general partial preferences, MGD satisfies sd-efficiency, ex-post-efficiency, equal treatment of Remark 3. MGD is not ordinally fair, weak-sd-envy-free, equals, and decomposability. upper invariant and weak-sd-strategyproof even in single- type resources allocations under linear preferences. Proof. Given any partial preference R, there are n eating rounds in MGD. For each round j ≤ n, let top(j) Proof. Consider the case where N = {1, 2, 3}, M = D1 = be the first available bundle in agent j’s topological sort {1F , 2F , 3F }, and R = { 1, 2, 3} where 1= {1F 1 0 and Group(j, ) be the set of agents who have the same 2F , 1F 1 3F , 2F 1 3F }, 2= {1F 2 3F , 1F 2 0 0 0 topological sort j. Then for each j ∈ Group(j, ), 2F , 3F 2 2F }, and 3= {3F 3 1F , 3F 3 2F , 1F 3 1 2 }. It is obvious that agent 1 gets 1 , agent 2 gets 3 Pj0,top(j) ← 0 . F F F |Group(j, )| 3 2 P = MGD(R) (1) sd-efficiency. Given any partial preference profile R, and agent gets F in MGD. Let . Since P = 1 > P and 1 is better than 3 for agent 2, let P = MGD(R). Assume for the sake of contradiction 1,1F 2,1F F F that there exists an assignment Q 6= P ∈ P such that Q sd MGD is not ordinally fair and weak-sd-envy-free. Suppose 3 1 P . agent misreport her preference as same as agent , which is an upper invariant transformation of at 2 under P . In round j = 1, for each j0 ∈ Group(j, 0), 3 F Agent 1 and agent 3 share their bundles. Agent 3 gets 0.5 P 0 P 0 since x∈U( 0 ,top(1)) Pj ,x ≤ x∈U( 0 ,top(1)) Qj ,x j j 1F and 0.5 2F which is better than only 2F for him. Hence, and U( j0 , top(1)) = {top(1)}, we have Pj0,top(1) ≤ MGD is not upper invariant and weak-sd-strategyproof. P Qj0,top(1). Since j0∈Group(1, 0) Pj0,top(1) = 1 ≥ P j0∈Group(1, 0) Qj0,top(1), we have Pˆj,top(1) = Qˆj,top(1) ˆ for each j ≤ n. Suppose for j < k, Pˆj,top(j) = Proof of Proposition 3. ˆ 0 Qˆj,top(j) for each j ≤ n. When j = k, for any j ∈ Proposition 3. No mechanism can satisfy both sd-efficiency 0 Group(k, ) and any x 6= top(k) ∈ U( j0 , top(k)), and sd-envy-freeness under general partial preferences. x is unavailable. For any x 6= top(k) ∈ U( j0 , top(k) ), Qj0,x ≤ Pj0,x since Qj0,x > Pj0,x implies x is Proof. We just consider the case of the allocation of single- available in round k. Hence, Qj0,x = Pj0,x, otherwise, type items and it is easy to example in similar manner when sd Qj0,x < Pj0,x would cause a contradiction with Q p ≥. Consider the case where N = {1, 2, 3}, M = D = P P 1 P . By Pj0,x ≤ Qj0,x, {1F , 2F , 3F }, and R = ( j)j≤3 where 1= {1F 1 x∈U( j0 ,top(k)) x∈U( j0 ,top(k)) 0 2 , 1 3 , 2 3 }, = {3 2 , 3 we have Pj0,top(k) ≤ Qj0,top(k) for each j ∈ F F 1 F F 1 F 2 F 2 F F 2 0 P 1F , 2F 2 1F }, and 3= ∅. It is easy to conclude that Group(k, ). Since j0∈Group(k, 0) Pj0,top(k) = 1 ≥ P assignment (5) is the only assignment that agent 3 does not Q 0 P = Q j0∈Group(k, 0) j ,top(k), we have ˆj,top(k) ˆj,top(k) envy others under the concept of sd-envy-freeness. However, ˆ for each j ≤ n. By induction on k, we have Pˆj,top(j) = it is dominated by assignment (6). Qˆ for each ˆj, j ≤ n. Hence, P = Q, a contradiction. j,top(j) 1 2 3 Therefore, P is sd-efficient. F F F (2) equal treatment of equals. That is obvious. Agent 1 1 1 1 3 3 3 (5) (3) decomposability. Let S = {Group(j, 0): j ≤ n} 1 1 1 Agent 2 3 3 3 which is an segmentation of N. For l ≤ |S|, sl ∈ S and 1 1 1 Agent 3 3 3 3 m ≤ |sl|, sort agents in sl in ascending order and let sl,m be the m-th agent. Let k be the least common multiple of 1F 2F 3F {|s| : s ∈ S}. We forge a set of k priority orders denoted O. Agent 1 2 1 0 For u ≤ k and v ≤ n, let Ou,v be the v-th agent in the u-th 3 3 1 2 (6) priority order in O. For each u ≤ k, l ≤ |S|, and m ≤ |sl|, Agent 2 0 3 3 let w = ((m + u − 2) mod |s |) + 1 and O = s . 1 1 1 l u,(sl,m) l,w Agent 3 Consider the special random priority mechanism that ran- 3 3 3 1 dom in O with k possibility for each priority order. Since 0 Proof of Proposition 4. tk. If agent 1 turns to consume xk+1 under R at tk, the consumption processes are same between tk and tk+1 because Proposition 4. Given any CP-profile R, MRP (R) is upper 0 invariant for any other CP-profile R0. the preferences of the other agents are same under R and R . Next, we prove that xk+1 is the exact bundle agent 1 turns 0 to at tk. Suppose D ⊆ D is the set of all available bundle Proof. W.l.o.g we only prove the cases when agent 1 lies. 0 at tk under R and R . By Proposition 1, xk+1 1 xˆ for any Given any CP-profile R = ( j)j≤n, let P be the expected 0 0 0 0 xˆ 6= xk+1 ∈ D . As the same reason, there is x ∈ D such output of MRP under R. For any CP-profile R = ( 1 0 0 0 that x 1 xˆ for any xˆ 6= x ∈ D . Since k < i0, y is avail- , −1) where 1 is an upper invariant transformation of 1 0 0 0 able at tk. Hence, xk+1 ∈ U( 1, y) and x ∈ U( 1, y). at y ∈ D, let P be the expected output of MRP under R . 0 0 Since P1,xk+1 > 0, we have xk+1 ∈ U( 1, y) by the Next, we prove that Pj,y = Pj,y for any j ≤ n by showing definition of upper invariant transformation. If x 6= xk+1, that given any priority order B over agents, any agent gets y 0 under R0 if and only if she gets y under R. we have x 1 xk+1 and xk+1 1 x which is a contradiction to the definition of upper invariant transformation Since the preferences of other agents keep same, given 0 where | 0 = | 0 . Hence, x = x , agent 1 1 U( 1,y) 1 U( 1,y) k+1 B, the sets of available bundles are same in agent 1’s turn 0 0 0 turns to xk+1 at tk under R and the consumption processes under R and R . Suppose D ⊆ D is the set of all available 0 bundles in agent 1’s turn in MRP under R and R0. Suppose are same between tk and tk+1 under R and R . 0 0 0 0 By induction, the consumption processes of all agents are agent 1 gets x ∈ D under R and x ∈ D under R . By 0 0 0 0 0 same before ti0 under R and R . Hence, the consumption Proposition 1, x 1 z for any z 6= x ∈ D and x 1 z for 0 0 0 0 processes are same until y is exhausted. Hence, P = P any z 6= x ∈ D . Given B, we divide all situations to two j,y j,y cases, y ∈ D0 and y ∈/ D0. for any j ≤ n. 0 0 Case 1, y ∈ D . We have x ∈ U( 1, y) and x ∈ 0 0 U( 1, y). Since P1,x > 0, we have x ∈ U( 1, y) by the definition of upper invariant transformation. If x 6= x0, 0 0 0 Counterexample for Remark 4. we have x 1 x and x 1 x which is a contradiction to the definition of upper invariant transformation where Remark 4. MPS is not weak-sd-strategyproof under acyclic 0 0 | 0 = | 0 . Hence, x = x and agent 1 gets CP-net profiles even when all agents have a shared depen- 1 U( 1,y) 1 U( 1,y) x under R and R0. Therefore, all agents get the same bun- dency graph, and agents are only allowed to misreport their dles under R and R0 in this case. CPTs. y ∈/ D0 1 1 Case 2, . Agents after agent in B and agent get Proof. Consider the MTRA where N = {1, 2, 3}, M = y R R0 1 no both under and . Besides, agents before agent in {1 , 2 , 3 , 1 , 2 , 3 }, and let agents’ preferences be the R R0 F F F B B B B get the same bundles under and . Hence, one agent same as in Figure 3, where agents have a shared dependency y R0 y R gets under if and only if she gets under in this case. graph shown in Figure 3(a). The CPTs of agent 1 are shown To sum up the two cases, given any priority order B over as Figure 3(b) while agent 2 and agent 3 have the same agents, any agent gets y under R0 if and only if she gets y 0 CPTs as Figure 3(c) shows. Suppose that agent 1 misreports under R. Therefore, Pj,y = Pj,y for any j ≤ n. her CPTs as Figure 3(d). Then, Figures 3(e), 3(f), and 3(g) are the preference graphs corresponding to the CPTs in Fig- ures 3(b), 3(c), and 3(d) respectively with the shared depen- Proof of Proposition 5. dency graph in Figure 3(a). When agent 1 reports her CPTs as Figure 3(b), she gets 1 1 2 , 1 2 1 and 1 2 3 under Proposition 5. Given any CP-profile R, MPS(R) is upper 3 F B 3 F B 3 F B 0 MPS. But if agent 1 misreports her CPTs as Figure 3(d), she invariant for any other CP-profile R . 1 1 2 1 gets 3 1F 2B, 3 2F 1B, 9 2F 2B and 9 2F 3B which is a better result for her under the notion of stochastic dominance. Proof. W.l.o.g we only prove the cases when agent 1 lies. Hence, MPS is not weak-sd-strategyproof even under this Given any CP-profile R = ( j)j≤n and any CP-profile 0 0 0 restriction. R = ( 1, −1) where 1 is an upper invariant transfor- 0 mation of 1 at y ∈ D, let P = MPS(R) and P = MPS(R0). Proof of Proposition 6. Let Dˆ = {x ∈ D : P1,x > 0} and nˆ = |D|ˆ . By Propo- Proposition 6. Given any independent CP-profile R, sition 1, we have an order over Dˆ such that x1 1 x2 1 ˆ MPS(R) is weak-sd-strategyproof for any profile R0. · · · 1 xnˆ , xi ∈ D for any i ≤ nˆ. There is xi0 such that R 1 x y under , when agent starts to consume i0 , is available Proof. Before the main body of the proof, we list three x y t and after i0 is exhausted, is unavailable. Suppose i is the claims about independent CP-nets for convenience. The time when xi is exhausted for i ≤ nˆ under R. Let t0 = 0. proofs of Claim 3 and 4 are based on Claim 2. Next, we prove that the consumption processes of all agents 0 Claim 2. Given any independent CP-net , for any D ∈ T , are same before ti0 under R and R by induction. i ˆ Suppose before tk (k < i0), the consumption processes of o ∈ Di and oˆ ∈ Di, if there are some y ∈ Π−{Di}, y ∈ 0 all agents are same under R and R (That is true for k = 0). Π−{Di} such that (o, y) (ô, yˆ), then we have o ∈ U( Hence, the remaining resources are same under R and R0 at , oˆ). Figure 3: CP-net preferences with the shared dependency graph: (a) the shared dependency graph for agents; (b) CPTs for agent 1; (c) CPTs for agent 2 and 3; (d) The misreported CPTs; (e) The corresponding preference graph of (b); (f) The corresponding preference graph of (c); (g) The corresponding preference graph of (d).

Given any independent CP-net , for any Di ∈ T , o ∈ y ∈ Π−{Di}, we have (o, y) j x0. Hence, D , oˆ ∈ D , suppose there are some y ∈ Π , yˆ ∈ i i −{Di} X X X X Q = Q ≤ Q . Π−{Di} such that (o, y) (ô, yˆ). Then there exists one path j,o j,(o,y) j,x y∈Π from (o, y) to (ô, yˆ) in the preference graph of . Suppose o∈U( j ,oˆ) o∈U( j ,oˆ) −{Di} x∈U( j ,x0) the path is (o1, y1) (o2, y2) · · · (ot, yt) such that For any (o, y) ∈ D such that (o, y) j x0, by Claim 2, we o1 = o, y1 = y, ot =o ˆ, yt = yˆ, and only one pair of have o ∈ U( j, oˆ). Hence, items is different between (ok, yk) and (ok+1, yk+1) for any k < t. For any z ∈ Π , we try to find a path from (o, z) X X X X −{Di} Q = Q ≥ Q . to (ô, z) by referring the path from (o, y) to (ô, yˆ). Initialize j,o j,(o,y) j,x o∈U( ,oˆ) o∈U( ,oˆ) y∈Π x∈U( ,x ) E as ∅. For any k < t, suppose the different items between j j −{Di} j 0 P P (ok, yk) and (ok+1, yk+1) belong to Dik ∈ T . If Dik = Di, Therefore, Q = Q . In like o∈U( j ,oˆ) j,o x∈U( j ,x0) j,x we have yk = yk+1 which implies (ok, z) (ok+1, z). Add P P manner, we have o∈U( ,oˆ) Pj,o = x∈U( ,x ) Pj,x. the edge between (ok, z) and (ok+1, z) to E. It is easy to see j j 0 Since Q sd P , P Q ≥ P P . that the edges in E together make up a path from (o, z) to j x∈U( j ,x0) j,x x∈U( j ,x0) j,x (ô, z). Therefore (o, z) (ô, z) for any z ∈ Π . Hence, Hence we have P Q ≥ P P . −{Di} o∈U( j ,oˆ) j,o o∈U( j ,oˆ) j,o o ∈ U( , oˆ). Claim 4. Under MPS, given any independent CP-profile R, for any Di ∈ T and j ≤ n, there exists o1 j o2 j · · · j Claim 3. Given any independent CP-profile R and two as- ol and 0 = t0 < t1 < ··· < tl = 1 such that ok ∈ Di, tk signments Q and P , if Q sd P , for any D ∈ T , oˆ ∈ D , j i i is the time when ok is exhausted and agent j consumes ok in we have the time interval [t , t ] for k ≤ l. X X k−1 k Qj,o ≥ Pj,o. Given Di ∈ T and an independent CP-net j which is o∈U( j ,oˆ) o∈U( j ,oˆ) the preference of agent j ∈ N. By Proposition 1, at the beginning of MPS, agent j starts to consume her unique fa- Given any independent CP-profile R and two assignments vorite bundle y1 ∈ D. Let o1 ∈ Di such that o1 ∈ y1. sd Q and P , suppose Q j P . For any x ∈ D and D ∈ T , Suppose agent j turns to consume yˆ such that o1 ∈/ yˆ before we define function fD as fD(x) = o such that o ∈ x and o1 is exhausted. Let yˆ|o1 be the bundle which changes the o ∈ D. For any Di ∈ T and oˆ ∈ Di, there is x0 ∈ D such item of Di in yˆ to o1. Then yˆ|o1 is available when agent j that fDi (x0) =o ˆ and for any D 6= Di ∈ T , o ∈ D, we turns to yˆ. By y1 j yˆ and Claim 2, we have yˆ|o1 j yˆ, a have fD(x0) = o or o j fD(x0). For any o ∈ U( j, oˆ), contradiction. Hence, agent j consumes o1 in [0, t1) where o1 is exhausted at t1. Suppose agent j turns to consume y2 we have: 0 at t1. Let o2 ∈ Di such that o2 ∈ y2. In the same man- j ∈ N(ok, t) ⇒ j ∈ N (ok, t). (7) ner, we have agent j consumes o2 in [t1, t2) where o2 is ex- Suppose for the sake of contradiction that there is an agent hausted at t2. Continue the process. Finally, we find orders, j, j 6= 1, and a time t, 0 ≤ t < τ(ok) such that j ∈ N(ok, t) o1 j o2 j · · · j ol and 0 = t0 < t1 < ··· < tl = 1, 0 and j ∈ N (ô1, t), for some item oˆ1 6= ok ∈ Di. As such that ok ∈ Di, tk is the time when ok is exhausted and 0 0 t < τ(ok) ≤ τ (ok), ok is available at time t under R . agent j consumes ok in the time interval [tk−1, tk] for k ≤ l. Then oˆ1 j ok because the preferences of agent j are same 0 After the proofs of the three claims, we begin the main under R and R . So oˆ1 is unavailable at t under R. This 0 body of the proof. W.l.o.g we prove that it is not benefi- says τ(ô1) ≤ t < τ (ô1). Let A = {o ∈ Di : o 6= 0 cial for agent 1 to misreport. Let R be an arbitrary inde- ok ∧ τ(o) < τ (o)}, we have oˆ1 ∈ A and A 6= ∅. We pendent CP-profile. Suppose for the sake of contradiction, pick oˆ2 in A such that τ(ô2) = min{τ(o): o ∈ A}. Note 0 0 that R = ( 1, −1) is a profile where agent 1 misre- that τ(ô2) < τ(ok) because τ(ô1) < τ(ok) and oˆ1 ∈ A. 0 ports her preference as any partial preference relation 1, ˆ 0 sd We claim that at some t < τ(ô2), there is an agent j such such that MPS(R ) 1 MPS(R). Let P = MPS(R) and 0 0 0 0 sd that ˆj ∈ N(ô2, t) and ˆj∈ / N (ô2, t). Otherwise we have P = MPS(R ). We prove that P 1 P implies that 0 0 N(ô2, t) ⊆ N (ô2, t) for t ∈ [0, τ(ô2)). And there is some P (1) = P (1). 0 0 sub interval of [τ(ô2), τ (ô2)) such that n (ô2, t) ≥ 1 for any For the sake of convenience, we define the following 0 0 t in the sub interval since τ (ô2) = sup{t : n (ô2, t) ≥ 1}. quantities to track the execution of MPS(R). At any time t Hence during the execution of MPS(R), let S(j, t, D ) be the item i 0 Z τ(ô2) Z τ (ô2) o ∈ Di that agent j is consuming at t, and for any item 0 o ∈ M, let N(o, t) be the set of agents who are consuming n(ô2, t)dt < n (ô2, t)dt, (8) 0 0 an item o at t and n(o, t) = |N(o, t)|. For any o ∈ M, let τ(o) be the time when o is exhausted during the execution of a contradiction with 0 MPS(R). Then, we have τ(o) = sup{t : n(o, t) ≥ 1}. For Z τ(ô2) Z τ (ô2) 0 0 0 0 the execution of MPS(R ), we define S (j, t, Di), N (a, t), n(ô2, t)dt = 1 = n (ô2, t)dt. (9) n0(a, t) and τ 0(o) in a similar way. 0 0 Our proof involves showing that S(j, t, Di) = ˆj 6= 1 1 o 0 Note that because agent is consuming k dur- S (j, t, Di) for any j ≤ n, t ∈ [1, 0) and Di ∈ T , which ˆ ˆ 0 ing [tk−1, τ(ok)) under R. Suppose j ∈ N(ô2, t) and j ∈ is a stronger condition than P1 = P . 0 1 N (ô3, t), for some t < τ(ô2) and oˆ3 ∈ Di. As oˆ2 is avail- For any D ∈ T , let O = {o , ··· , o } be the set of items 0 0 i i 1 l able at t under R (because t < τ(ô2) < τ (ô2)), we have that agent 1 gets a positive fractional share of in MPS(R), oˆ oˆ . Hence oˆ is unavailable at t under R (ˆj 6= 1 i.e. for each item o ∈ O , P > 0, and for each oˆ ∈ D \O , 3 ˆj 2 3 i 1,o i i and her preferences are same under R and R0). Therefore, P1.oˆ = 0. W.l.o.g. let o1 1 · · · 1 ol. Then, by Claim 4, 0 τ(ô3) < t < τ (ô3) and τ(ô3) < t < τ(ô2), a contradiction there are time steps 0 = t0 < t1 < ··· < tl = 1, such that with τ(ô2) = min{τ(o): o ∈ A}. for each k ≤ l, agent 1 only consumes ok from Di during the whole interval [t , t ). Thus we have proved claim (7) is true. Suppose that agent k−1 k 1 consumes o such that o 6= o in the sub interval of i ≤ p k Consider any type . We prove by induction that [t , t ] under R0. Then o is available after t . If t = 1, S(j, t, D ) = S0(j, t, D ) j ≤ n t ∈ [1, 0) k−1 k k k k i i for any , . Since we have a contradiction. If t < 1, we have P < 1 and S(j, t, D ) = S0(j, t, D ) k 1,ok the proof for the base case where i i o is allocated to at least two agents. Since agents continue [t , t ) k for the interval 0 1 is similar with the inductive steps, consuming an item until the item is exhausted once they k ≤ l we only prove the induction hypothesis that for any , start to consume the item in MPS under independent CP- S(j, t, D ) = S0(j, t, D ) = o [t , t ) i i k for the interval k−1 k , nets by Claim 4. There is at least one agent except agent 1 for every j ≤ n. 0 consumes ok just before tk under R and R and the agent Assume that the induction hypothesis is true for each 0 continues consuming ok after tk under R . In that case, m < k. We claim that there exists no o ∈ D such 0 Pn 0 Pn i we have P = 1 − P < 1 − Pj,o = that o o , o∈ / O , and P 0 > 0. Otherwise, since 1,ok j=2 j,ok j=2 k 1 k i 1,o P , a contradiction with P 0 ≥ P . Therefore, we 0 1,ok 1,ok 1,ok S(j, t, Di) = S (j, t, Di) for any j ≤ n, t ∈ [0, tk−1), 0 have shown N(ok, t) ⊆ N (ok, t) for t ∈ [0, τ(ok)). If o is available at tk−1, and o 1 ok. However, agent 1 0 τ(ok) < τ (ok), we would have a contradiction like (8) and starts to consume ok at tk−1, a contradiction to our assump- 0 0 (9). Hence τ(ok) = τ (ok) = tk. Since P ≥ P1,o and tion that P = MPS(R). Therefore, by P 0 sd P and 1,ok k Pk 0 P 0 P1,ok = τ(ok) − tk−1, agent 1 is consuming ok during the Claim 3, we have P = P ≥ 0 0 0 m=1 1,om o∈U( 1,ok) 1,o whole interval [tk−1, τ (ok)) under R and P1,o = P1,ok . P Pk k o∈U( ,o ) P1,o = m=1 P1,om . By the induction hy- Since the preferences of all agents except agent 1 keep same 1 k 0 0 P = P 0 m < k under R and R , S(j, t, Di) = S (j, t, Di) for any j ∈ N pothesis, we have 1,om 1,om for any . Hence, P 0 ≥ P 1 o and t ∈ [tk−1, tk). 1,ok 1,ok . Agent is consuming k during the whole 0 By induction, we have S(j, t, Di) = S (j, t, Di) for any interval [tk−1, τ(ok)) under R but during the sub interval of 0 0 0 0 j ≤ N, t ∈ [0, 1) and Di ∈ T . Hence, P (1) = P (1). [tk−1, τ (ok)) under R . Therefore, τ(ok) ≤ τ (ok). We claim that for all t ∈ [0, τ(ok)) and all agents j, j 6= 1,