1 Introduction
Suppose we are organizing a seminar and must allocate discussion topics and dates to students. Students have heterogeneous and combinatorial preferences over (topic, date) bundles: a student’s preferences over the topics may depend on the date and vice versa, because she may prefer an early date if she gets an easy topic and may prefer a late date if she gets a hard topic.
This example illustrates a common setting for allocating multiple indivisible items, which we formulate as a categorized domain. A categorized domain contains multiple indivisible items, each of which belongs to one of the categories. In categorized domain allocation problems (CDAPs), we want to design a mechanism to allocate the items to agents without monetary transfer, such that each agent gets at least one item per category. In the above example, there are two categories: topics and dates, and each agent (student) must get a topic and a date.
Many other allocation problems are CDAPs. For example, in cloud computing, agents have heterogeneous preferences over multiple types of items including CPU, memory, and storage^{1}^{1}1Suppose each type contains discrete units of resources that are essentially indivisible for operational convenience. [15, 14, 1]; patients must be allocated multiple types of resources including surgeons, nurses, rooms, and equipment [17]; college students need to choose courses from multiple categories per semester, e.g. computer science courses, math courses, social science courses, etc.
The design and analysis of allocation mechanisms for noncategorized domains have been an active research area at the interface of computer science and economics. In computer science, allocation problems have been studied as multiagent resource allocation [12]. In economics, allocation problems have been studied as onesided matching, also known as assignment problems [27]. Previous research faces three main barriers.
Preference bottleneck: When the number of items is not too small, it is impractical for the agents to express their preferences over all (exponential) bundles of items.
Computational bottleneck:
Even if the agents can express their preferences compactly using some preference language, computing an “optimal” allocation is often a hard combinatorial optimization problem.
Threats of agents’ strategic behavior: An agent may have incentive to report untruthfully to obtain a more preferred bundle. This may lead to a socially inefficient allocation.
Our Contributions. We initiate the study of mechanism design under the novel framework of CDAPs towards breaking the three aforementioned barriers. CDAPs naturally generalize classical noncategorized allocation problems, which are CDAPs with one category. CDAPs are our main conceptual contribution.
As a first step, we focus on basic categorized domain allocation problems (basic CDAPs), where the number of items in each category is exactly the same as the number of agents, so that each agent gets exactly one item from each category. See e.g. the seminarorganization example.
Our technical contributions are twofold. First, we characterize serial dictatorships for any basic CDAPs with at least two categories by a minimal set of three axiomatic properties: strategyproofness, nonbossiness, and categorywise neutrality. This helps us understand the possibility of designing strategyproof mechanisms to overcome the third barrier, i.e. threats of agents’ strategic behavior.
Second, to overcome the preference bottleneck and the computational bottleneck, and to go beyond serial dictatorships, we propose categorial sequential allocation mechanisms (CSAMs), which are a large class of indirect mechanisms that naturally extend serial dictatorships [28], sequential allocation protocols [7], and the draft mechanism [11]. For agents and categories, a CSAM is defined by an ordering over all (agent, category) pairs: in each round, the active agent picks an item that has not been chosen yet from the designated category. CSAMs have low communication complexity and low computational complexity.
We completely characterize the worstcase rank efficiency of CSAMs, measured by agents’ ranks of the bundles they receive, for any combination of two types of myopic agents: optimistic agents, who always choose the item in their topranked bundle that is still available, and pessimistic agents, who always choose the item that gives them best worstcase guarantee. This characterization naturally leads to useful corollaries on worstcase efficiency of various CSAMs. For example, we show that while serial dictatorships with alloptimistic agents have the best worstcase utilitarian rank, they have the worst worstcase egalitarian rank (Proposition 5). On the other hand, balanced CSAMs with allpessimistic agents have good worstcase egalitarian rank (Proposition 5). For strategic agents, we prove an upper bound on the worstcase rank efficiency for all CSAMs, and a matching lower bound for two agents.
We also use computer simulation to further compare expected utilitarian rank and expected egalitarian rank of some natural CSAMs where agents’ preferences are generated from the Mallows model [20]. We observe that serial dictatorships with alloptimistic agents have good expected utilitarian rank and bad expected egalitarian rank. On the other hand, the balanced CSAMs with allpessimistic agents have good expected egalitarian rank.
Related Work and Discussions. We are not aware of previous work that explicitly formulates the CDAP framework. Previous work on multitype resource allocation assumes that items of the same type are interchangeable, and agents have specific preferences, e.g. Leontief preferences [15] and threshold preferences [17]. CDAPs are more general as agents’ preferences are only required to be rankings but not otherwise restricted.
From the modeling perspective, ignoring the categorial information, CDAPs become standard centralized multiagent resource allocation problems. However, the categorial information opens more possibilities for designing natural allocation mechanisms such as CSAMs. More importantly, we believe that CDAPs provide a natural framework for crossfertilization of ideas and techniques from other fields of preference representation and aggregation. For example, the combinatorial structure of categorized domains naturally allows agents to use graphical languages (e.g. CPnets [4]) to represent their preferences, which is otherwise hard [6]. Approaches in combinatorial voting [10] can also be naturally considered in CDAPs.
Technically, onesided matching problems are basic CDAPs with one category. Our characterization of serial dictatorships for basic CDAPs are stronger than characterizations of serial dictatorships and similar mechanisms for onesided matching [28, 22, 23, 24, 13, 16]. This is because the categorywise neutrality used in our characterization is weaker than the neutrality used in previous work.
Our analysis of the worstcase rank efficiency of categorial sequential allocation mechanisms resembles the price of anarchy [18], which is defined for strategic agents together with a social welfare function that numerically evaluates the quality of outcomes. Our theorem is also related to distortion in the voting setting [26, 5], which concerns the social welfare loss caused by agents reporting a ranking instead of a utility function. Nevertheless, our approach is significantly different because we focus on allocation problems for myopic and strategic agents, and we do not assume the existence of agents’ cardinal preferences nor a social welfare function, even though our theorem can be easily extended to study worstcase social welfare loss given a social welfare function, as in Proposition 5 through 5.
2 Categorized Domain Allocation Problems
Definition
A categorized domain is composed of categories of indivisible items, denoted by . In a categorized domain allocation problem (CDAP), we want to allocate the items to agents without monetary transfer, such that each agent gets at least one item from each category.
In a basic categorized domain for agents, for each , , , and each agent’s preferences are represented by a linear order over . In a basic categorized domain allocation problem (basic CDAP), we want to allocate the items to agents without monetary tranfer, such that every agent gets exactly one item from each category.
In this paper, we focus on basic categorized domains and basic CDAPs for nonsharable items [12], that is, each item can only be allocated to one agent. Therefore, for all , we write . Each element in is called a bundle. For any , let denote a linear order over and let denote the agents’ (preference) profile. An allocation is a mapping from to , such that , where for any and , is the bundle allocated to agent and is the item in category allocated to agent . An allocation mechanism is a mapping that takes a profile as input, and outputs an allocation. We use to denote the bundle allocated to agent by for profile .
We now define three axiomatic properties for allocation mechanisms. The first two properties are common in the literature [28], and the third is new.
A direct mechanism satisfies strategyproofness if no agent benefits from misreporting her preferences. That is, for any profile , any agent , and any linear order over , , where is composed of preferences of all agents except agent .
satisfies nonbossiness if no agent is bossy. An agent is bossy if she can report differently to change the bundles allocated to some other agents without changing her own allocation. That is, for any profile , any agent , and any linear order over , .
satisfies categorywise neutrality if after applying a permutation over the items in a given category the allocation is also permuted in the same way. That is, for any profile , any category , and any permutation over , we have , where for any bundle , .
When there is only one category, categorywise neutrality degenerates to the traditional neutrality for onesided matching [28]. When , categorywise neutrality is much weaker than the traditional neutrality.
A serial dictatorship is defined by a linear order over such that agents choose items in turns according to . A truthful agent chooses her topranked bundle that is still available in each step.
Example
Let and . . Agents’ preferences are as follows.
Suppose the agents are truthful. Let . In the first round of the serial dictatorship, agent chooses ; in the second round, agent cannot choose or because item in is unavailable, so she chooses ; in the final round, agent chooses .
3 An Axiomatic Characterization
Theorem
For any and , an allocation mechanism for a basic categorized domain is strategyproof, nonbossy, and categorywise neutral if and only if it is a serial dictatorship. Moreover, the three axioms are minimal for characterizing serial dictatorships.
Proof
We first prove four lemmas. The first three lemmas are standard in proving characterizations for serial dictatorships. The last one (Lemma 3) is new, whose proof is the most involved and heavily relies on the categorial information.
The first lemma (roughly) says that for all strategyproof and nonbossy mechanisms and all profiles , if every agent reports a different ranking without enlarging the set of bundles ranked above (and she can shuffle the bundles ranked above and she can shuffle the bundles ranked below ), then the allocation to all agents does not change in the new profile. This resembles (strong) monotonicity in social choice.
Lemma
Let be a strategyproof and nonbossy allocation mechanism over a basic categorized domain with . For any pair of profiles and such that for all , , we have .
Proof
We first prove the lemma for the special case where and only differ on one agent’s preferences. Let be an agent with and . We will prove that .
Suppose for the sake of contradiction . If then it means that is not strategyproof since has incentive to report when her true preferences are . If then , which means that when agent ’s preferences are she has incentive to report ,. This again contradicts the assumption that is strategyproof. Therefore .
By nonbossiness, . The lemma is proved by recursively applying this argument to .
For any linear order over and any bundle , we say a linear order is a pushup of from , if can be obtained from by raising the position of while keeping the relative positions of other bundles unchanged. The next lemma states that for any strategyproof and nonbossy mechanism , if an agent reports her preferences differently by only pushing up a bundle , then either the allocation to all agents does not change, or she gets .
Lemma
Let be a strategyproof and nonbossy allocation mechanism over a basic categorized domain with . For any profile , any , any bundle , and any that is a pushup of from , either (1) or (2) .
Proof
We first prove that or . Suppose on the contrary that is neither nor . If , then is not strategyproof since when agent ’s true preferences are and other agents’ preferences are , she has incentive to report to make her allocation better. If , then since , we have . In this case when agent ’s true preferences are and other agents’ preferences are , she has incentive to report to make her allocation better, which means that is not strategyproof. Therefore, or . If , then by nonbossiness . This completes the proof.
We next prove that strategyproofness, nonbossiness, and categorywise neutrality altogether imply Paretooptimality, which states that for any profile , there does not exist an allocation such that all agents prefer their bundles in to their bundles in , and some of them strictly prefer their bundles in .
Lemma
For any basic categorized domains with , any strategyproof, nonbossy, and categorywise neutral allocation mechanism is Pareto optimal.
Proof
We prove the lemma by contradiction. Let be a strategyproof, nonbossy, categorywise neutral, but non(Pareto optimal) allocation mechanism. Let denote a profile such that is Pareto dominated by an allocation . For any , let denote the permutation over so that for every , is permuted to . Let . It follows that for all , .
Let denote an arbitrary ranking where is ranked at the top place, and is ranked at the second place if it is different from . Let denote an arbitrary ranking where is ranked at the top place, and is ranked at the second place if it is different from . Let and . and are illustrated as follows.
Since Pareto dominates , by Lemma 3 we have , because for any , in the only bundle ranked ahead of is , if it is different from , and is also ranked ahead of in . By Lemma 3 again we have . Comparing and , we observe that the only differences are the orderings among . Applying Lemma 3 to and , we have that . However, by categorywise neutrality , which is a contradiction.
The next lemma states that for any strategyproof and nonbossy allocation mechanism , any profile , and any pair of agents , there is no bundle that only contains items allocated to agent and by , such that both and prefer to their bundles allocated by .
Lemma
Let be a strategyproof and nonbossy allocation mechanism over a basic categorized domain with . For any profile and any , let and , there does not exist such that and , where is the th component of .
Proof
Suppose for the sake of contradiction that such a bundle exists. Let denote the bundle such that . More precisely, for all , . For example, if , , and , then .
The rest of the proof derives a contradiction by proving the series of observations illustrated in Table 3. In each step, we prove that the boxed bundles are allocated to agent and agent respectively, and all other agents get their topranked bundles.
Step 1. Let , , where “others” represents an arbitrary linear order over the remaining bundles, and for any , let ]. By Lemma 3, .
Step 2. Let be a pushup of from . We will prove that . Since is a pushup of from , by Lemma 3, is either or . We now show that the former case is impossible. Suppose for the sake of contradiction , then cannot be , , or since otherwise some item will be allocated twice. This means that is Pareto dominated by the allocation where gets , gets , and all other agents get their topranked bundles. This contradicts the Paretooptimality of (Lemma 3). Hence . By nonbossiness we have .
Step 3. Let be a pushup of from . We will prove that in , gets , gets , and all other agents get the same items as in . Since is a pushup of from , by Lemma 3, is either or . We now show that the former case is impossible. Suppose for the sake of contradiction that . By nonbossiness, . This means that is Paretodominated by the allocation where gets , gets , and all other agents get their topranked bundles. This contradicts the Paretooptimality of (Lemma 3).
Step 4. Let be a pushup of from . By Lemma 3, .
Step 5. Let be a pushup of from . We will prove that . Since is a pushup of from , by Lemma 3, is either or . We now show that the former case is impossible. Suppose for the sake of contradiction that . Then in , agent cannot get , , or , which means that is Paretodominated by the allocation where gets , gets , and all other agents get their topranked bundles. This contradicts the Paretooptimality of . Hence, . By nonbossiness .
Step 6. We note that is a pushup of from (and is still below ). By Lemma 3, . We note that the right hand side is the profile in Step 2.
Contradiction. Finally, the observations in Step 5 and Step 6 imply that when agents’ preferences are as in Step 6, agent has incentive to report in Step 5 to improve the bundle allocated to her (from to ). This contradicts the strategyproofness of and completes the proof of Lemma 3.
It is easy to check that any serial dictatorship satisfies strategyproofness, nonbossiness and categorywise neutrality. We now prove that any mechanism satisfying the three axioms must be a serial dictatorship. Let be a linear order over that satisfies the following conditions:
.
For any , the bundles ranked between and are those satisfying the following two conditions: 1) at least one component is , and 2) all components are in . Let denote these bundles. That is, and .
For any and any , if the number of ’s in is strictly larger than the number of ’s in , then .
The next claim states that agrees with a serial dictatorship on a specific profile.
Claim
Let . For any , there exists such that .
Proof
The claim is proved by induction on . When , for the sake of contradiction suppose there is no with . Then there exist a pair of agents and such that both and contain in at least one category.
Let be the bundle obtained from by replacing items in categories where takes to . More precisely, we let , where
It follows that in , and since the number of ’s in is strictly larger than the number of ’s in or . By Lemma 3, this contradicts the assumption that is strategyproof and nonbossy. Hence there exists with .
Suppose the claim is true for . We next prove that there exists such that . This follows after a similar reasoning to the case. Formally, suppose for the sake of contradiction there does not exist such a . Then, there exist two agents who get and in such that both and contain in at least one category. By the induction hypothesis, items in all categories have been allocated, which means that all components of and are at least as large as . Let be the bundle obtained from by replacing items in all categories where takes to . We have and , leading to a contradiction by Lemma 3. Therefore, the claim holds for . This completes the proof of Claim 3.
W.l.o.g. we let , , , denote the agents in Claim 3. For any profile , we define bundles as follows. Let denote the topranked bundle in , and for any , let denote agent ’s topranked available bundle given that items in have already been allocated. That is, is the most preferred bundle in according to . In other words, are the bundles allocated to agents through by the serial dictatorship . We next prove that this is exactly the allocation by .
For any , we define a categorywise permutation such that for all , , where we recall that is the item in the th category in . Let . It follows that for all , . By categorywise neutrality and Claim 3, in agent gets .
Comparing to , we notice that for all and all bundles , if then . This is because if there exists such that but , then is still available after have been allocated, and is ranked higher than in . This contradicts the selection of . By Lemma 3, , which proves that is the serial dictatorship w.r.t. the order .
Next, we show that strategyproofness, nonbossiness, and categorywise neutrality are a minimal set of properties that characterize serial dictatorships.
Strategyproofness is necessary
Consider the allocation mechanism that maximizes the social welfare w.r.t. the following utility functions. For any and , the bundle ranked at the th position in agent ’s preferences gets points.^{2}^{2}2The terms in the utility functions are only used to avoid ties in allocations. In fact, any utility functions where there are no ties satisfy nonbossiness and categorywise neutrality, but some of them are equivalent to serial dictatorships, which are the cases we want to avoid in our proof. It is not hard to check that for any pair of different allocations, the social welfares are different. It follows that this allocation mechanism satisfies nonbossiness. This is because if agent ’s allocation is the same when only she reports differently, then the set of items left to the other agents is the same, which means that the allocation to the other agents by the mechanism is the same. Since the utility of a bundle only depends on its position in the agents’ preferences rather than the name of the bundle, the allocation mechanism satisfies categorywise neutrality. This mechanism is not a serial dictatorship. To see this, consider the basic categorized domain with , , and . A serial dictatorship will either give to agent and give to agent , or give to agent and give to agent , but the allocation that maximizes social welfare w.r.t. the utility function described above is to give to agent and give to agent .
Nonbossiness is necessary
Consider the following “conditional serial dictatorship”: agent chooses her favorite bundle in the first round, and the order over the remaining agents depends on agent ’s preferences in the following way: if the first component of agent ’s secondranked bundle is the same as the first component of her top choice, then the order over the rest of agents is ; otherwise it is . It is not hard to verify that this mechanism satisfies strategyproofness and categorywise neutrality, and is not a serial dictatorship (where the order must be fixed before seeing the profile).
Categorywise neutrality is necessary
Consider the following “conditional serial dictatorship”: agent chooses her favorite bundle in the first round, and the order over agents depends on the allocation to agent in the following way: if agent gets , then the order over the rest of agents is ; otherwise it is . It is not hard to verify that this mechanism satisfies strategyproofness and nonbossiness, and is not a serial dictatorship.
Remarks. The theorem is somewhat negative, meaning that we have to sacrifice one of strategyproofness, categorywise neutrality, or nonbossiness. Among the three axiomatic properties, we feel that nonbossiness is the least natural one.
4 Categorial Sequential Allocation Mechanisms
Given a linear order over , the categorial sequential allocation mechanism (CSAM) allocates the items in steps as illustrated in Protocol 1. In each step , suppose the th element in is , (equivalently, ). Agent is called the active agent in step and she chooses an item that is still available from . Then, is broadcast to all agents and we move on to the next step.
In CSAMs, in each step the active agent must choose an item from the designated category. Hence, CSAMs are different from sequential allocation protocols [7] and the draft mechanism [11], where in each step the active agent can choose any available item from any category.
Example
The serial dictatorship w.r.t. is a CSAM w.r.t. .
For any even number , given any linear order over the agents, we define the balanced CSAM to be the mechanism where agents choose items in phases, such that for each , in phase all agents choose from w.r.t. if
is odd, and w.r.t. inverse
if is even.For example, when , , and , the balanced CSAM uses the order .
Similar to sequential allocations [7], CSAMs can be implemented in a distributed manner. Communication cost for CSAMs is much lower than for direct mechanisms, where agents report their preferences in full to the center, which requires bits per agent, and thus the total communication cost is . For CSAMs, the total communication cost of Protocol 1 is , which has multiplicative saving. In light of this, CSAMs preserve more privacy as well.
We will consider two types of myopic agents. For any , we let denote the set of available items in at the beginning of round .

Optimistic agents. An optimistic agent chooses the item in her topranked bundle that is still available, given the items she chose in previous steps.

Pessimistic agents. A pessimistic agent in round chooses an item from , such that for all with , agent prefers the worst available bundle whose th component is to the worst available bundle whose th component is .
In this paper, we assume that whether an agent is optimistic or pessimistic is fixed before applying a CSAM.
Example
Let , . Consider the same profile as in Example 2, which can be simplified as follows.
Let . Suppose agent 1 and agent 2 are optimistic and agent 3 is pessimistic. When , agent (optimistic) chooses item from . When , item is the topranked available bundle for agent (optimistic), so she chooses from . When , the available bundles are . If agent chooses from , then the worstcase available bundle is , and if agent chooses from , then the worstcase available bundle is . Since agent prefers to , she chooses from . When , agent chooses from . When , agent choses from and when , agent choses from . Finally, agent gets , agent gets , and agent gets .
5 Rank Efficiency of CSAMs for Myopic Agents
In this section, we focus on characterizing the rank efficiency of CSAMs measured by agents’ ranks of the bundles they receive.^{3}^{3}3This is different from the ordinal efficiency for randomized allocation mechanisms [2]. For any linear order over and any bundle , we let denote the rank of in , such that the highest position has rank and the lowest position has rank . Given a CSAM , we introduce the following notation for any