Allocating Indivisible Items in Categorized Domains

04/22/2015 ∙ by Erika Mackin, et al. ∙ Rensselaer Polytechnic Institute 0

We formulate a general class of allocation problems called categorized domain allocation problems (CDAPs), where indivisible items from multiple categories are allocated to agents without monetary transfer and each agent gets at least one item per category. We focus on basic CDAPs, where the number of items in each category is equal to the number of agents. We characterize serial dictatorships for basic CDAPs by a minimal set of three axiomatic properties: strategy-proofness, non-bossiness, and category-wise neutrality. Then, we propose a natural extension of serial dictatorships called categorial sequential allocation mechanisms (CSAMs), which allocate the items in multiple rounds: in each round, the active agent chooses an item from a designated category. We fully characterize the worst-case rank efficiency of CSAMs for optimistic and pessimistic agents, and provide a bound for strategic agents. We also conduct experiments to compare expected rank efficiency of various CSAMs w.r.t. random generated data.

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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 storage111Suppose 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 non-categorized domains have been an active research area at the interface of computer science and economics. In computer science, allocation problems have been studied as multi-agent resource allocation [12]. In economics, allocation problems have been studied as one-sided 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 non-categorized 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 seminar-organization example.

Our technical contributions are two-fold. First, we characterize serial dictatorships for any basic CDAPs with at least two categories by a minimal set of three axiomatic properties: strategy-proofness, non-bossiness, and category-wise neutrality. This helps us understand the possibility of designing strategy-proof 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 worst-case 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 top-ranked bundle that is still available, and pessimistic agents, who always choose the item that gives them best worst-case guarantee. This characterization naturally leads to useful corollaries on worst-case efficiency of various CSAMs. For example, we show that while serial dictatorships with all-optimistic agents have the best worst-case utilitarian rank, they have the worst worst-case egalitarian rank (Proposition 5). On the other hand, balanced CSAMs with all-pessimistic agents have good worst-case egalitarian rank (Proposition 5). For strategic agents, we prove an upper bound on the worst-case 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 all-optimistic agents have good expected utilitarian rank and bad expected egalitarian rank. On the other hand, the balanced CSAMs with all-pessimistic 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 multi-type 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 multi-agent 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 cross-fertilization 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. CP-nets [4]) to represent their preferences, which is otherwise hard [6]. Approaches in combinatorial voting [10] can also be naturally considered in CDAPs.

Technically, one-sided 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 one-sided matching [28, 22, 23, 24, 13, 16]. This is because the category-wise neutrality used in our characterization is weaker than the neutrality used in previous work.

Our analysis of the worst-case 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 worst-case social welfare loss given a social welfare function, as in Proposition 5 through 5.

Finally, there is a literature in social choice on analyzing the outcomes when agents are myopic [9, 19, 8, 21]. These papers focused on voting or cake cutting, while we focus on allocation of indivisible items.

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 non-sharable 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 strategy-proofness 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 non-bossiness 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 category-wise 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, category-wise neutrality degenerates to the traditional neutrality for one-sided matching [28]. When , category-wise 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 top-ranked 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 strategy-proof, non-bossy, and category-wise 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 strategy-proof and non-bossy 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 strategy-proof and non-bossy 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 strategy-proof 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 strategy-proof. Therefore .

By non-bossiness, . 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 strategy-proof and non-bossy 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 strategy-proof and non-bossy 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 strategy-proof 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 strategy-proof. Therefore, or . If , then by non-bossiness . This completes the proof.

We next prove that strategy-proofness, non-bossiness, and category-wise neutrality altogether imply Pareto-optimality, 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 strategy-proof, non-bossy, and category-wise neutral allocation mechanism is Pareto optimal.

Proof

We prove the lemma by contradiction. Let be a strategy-proof, non-bossy, category-wise 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 category-wise neutrality , which is a contradiction.

The next lemma states that for any strategy-proof and non-bossy 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 .

Proof steps for Lemma 3. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

Lemma

Let be a strategy-proof and non-bossy 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 top-ranked 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 top-ranked bundles. This contradicts the Pareto-optimality of (Lemma 3). Hence . By non-bossiness 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 non-bossiness, . This means that is Pareto-dominated by the allocation where gets , gets , and all other agents get their top-ranked bundles. This contradicts the Pareto-optimality 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 Pareto-dominated by the allocation where gets , gets , and all other agents get their top-ranked bundles. This contradicts the Pareto-optimality of . Hence, . By non-bossiness .

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 strategy-proofness of and completes the proof of Lemma 3.

It is easy to check that any serial dictatorship satisfies strategy-proofness, non-bossiness and category-wise 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 strategy-proof and non-bossy. 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 top-ranked bundle in , and for any , let denote agent ’s top-ranked 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 category-wise permutation such that for all , , where we recall that is the item in the -th category in . Let . It follows that for all , . By category-wise 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 strategy-proofness, non-bossiness, and category-wise neutrality are a minimal set of properties that characterize serial dictatorships.

Strategy-proofness 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.222The terms in the utility functions are only used to avoid ties in allocations. In fact, any utility functions where there are no ties satisfy non-bossiness and category-wise 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 non-bossiness. 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 category-wise 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 .

Non-bossiness 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 second-ranked 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 strategy-proofness and category-wise neutrality, and is not a serial dictatorship (where the order must be fixed before seeing the profile).

Category-wise 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 strategy-proofness and non-bossiness, and is not a serial dictatorship.

Remarks. The theorem is somewhat negative, meaning that we have to sacrifice one of strategy-proofness, category-wise neutrality, or non-bossiness. Among the three axiomatic properties, we feel that non-bossiness 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.

Input: An order over .
1 Broadcast to all agents. for  to  do
2      Let be the -th element in . Agent chooses an available item . Broadcast to all agents.
3 end for
Protocol 1 Categorial sequential allocation mechanism (CSAM) .

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 top-ranked 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 top-ranked available bundle for agent (optimistic), so she chooses from . When , the available bundles are . If agent chooses from , then the worst-case available bundle is , and if agent chooses from , then the worst-case 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.333This 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