A Discussion of the Two-Phase Approach

# Strategyproof Pareto-Stable Mechanisms for Two-Sided Matching with Indifferences

We study variants of the stable marriage and college admissions models in which the agents are allowed to express weak preferences over the set of agents on the other side of the market and the option of remaining unmatched. For the problems that we address, previous authors have presented polynomial-time algorithms for computing a “Pareto-stable” matching. In the case of college admissions, these algorithms require the preferences of the colleges over groups of students to satisfy a technical condition related to responsiveness. We design new polynomial-time Pareto-stable algorithms for stable marriage and college admissions that correspond to strategyproof mechanisms. For stable marriage, it is known that no Pareto-stable mechanism is strategyproof for all of the agents; our algorithm provides a mechanism that is strategyproof for the agents on one side of the market. For college admissions, it is known that no Pareto-stable mechanism can be strategyproof for the colleges; our algorithm provides a mechanism that is strategyproof for the students.

## 1 Introduction

Gale and Shapley [7] introduced the stable marriage model and its generalization to the college admissions model. Their work spawned a vast literature on two-sided matching; see Manlove [11] for a recent survey. The present paper is primarily concerned with variants of the stable marriage and college admissions models where the agents have weak preferences, i.e., where indifferences are allowed.

In the most basic stable marriage model, we are given an equal number of men and women, where each man (resp., woman) has complete, strict preferences over the set of women (resp., men); we refer to this model as SMCS. For SMCS, an outcome is a matching that pairs up all of the men and women into disjoint man-woman pairs. A man-woman pair is said to form a blocking pair for a matching if prefers to his partner in and prefers to her partner in . A matching is stable if it does not have a blocking pair. It is straightforward to prove that any stable matching is also Pareto-optimal. Gale and Shapley presented the deferred acceptance (DA) algorithm for the SMCS problem and proved that the man-proposing version of the DA algorithm produces the unique man-optimal (and woman-pessimal) stable matching. Roth [12] showed that the associated mechanism, which we refer to as the man-proposing DA mechanism, is strategyproof for the men, i.e., it is a weakly dominant strategy for each man to declare his true preferences. Unfortunately, the man-proposing DA mechanism is not strategyproof for the women. In fact, Roth [12] showed that no stable mechanism for SMCS is strategyproof for all of the agents.

The SMCW model is the generalization of the SMCS model in which each man (resp., woman) has weak preferences over the set of women (resp., men). When indifferences are allowed, we need to refine our notion of a blocking pair. A man-woman pair is said to form a strongly blocking pair for a matching if prefers to his partner in and prefers to her partner in . A matching is weakly stable if it is individually rational and does not have a strongly blocking pair. Two other natural notions of stability, namely strong stability and super-stability, have been investigated in the literature (see Manlove [11, Chapter 3] for a survey of these results). We focus on weak stability because every SMCW instance admits a weakly stable matching (this follows from the existence of stable matchings for SMCS, coupled with arbitrary tie-breaking), but not every SMCW instance admits a strongly stable or super-stable matching. It is straightforward to exhibit SMCW instances (with as few as two men and two women) for which some weakly stable matching is not Pareto-optimal. Sotomayor [16] argues that Pareto-stability (i.e., Pareto-optimality plus weak stability) is an appropriate solution concept for SMCW and certain other matching models with weak preferences, and proves that every SMCW instance admits a Pareto-stable matching.

Erdil and Ergin [4] and Chen and Ghosh [2] present polynomial-time algorithms for computing a Pareto-stable matching of a given SMCW instance; in fact, these algorithms are applicable to certain more general models to be discussed shortly. Given the existence of a stable mechanism for SMCS that is strategyproof for the men (or, symmetrically, for the women), it is natural to ask whether there is a Pareto-stable mechanism for SMCW that is strategyproof for the men. We cannot hope to find a Pareto-stable mechanism for SMCW that is strategyproof for all agents, since that would imply a stable mechanism for SMCS that is strategyproof for all agents. A similar statement holds for the SMIW model, the generalization of the SMCW model in which the agents are allowed to express incomplete preferences. See Section 4 for a formal definition of the SMIW model and the associated notions of weak stability and Pareto-stability. Throughout the remainder of the paper, when we say that a mechanism for a stable marriage model is strategyproof, we mean that it is strategyproof for the agents on one side of the market; moreover, unless otherwise specified, it is to be understood that the mechanism is strategyproof for the men. The Pareto-stable algorithms of Erdil and Ergin, and of Chen and Ghosh, are based on a two-phase approach where the first phase runs the Gale-Shapley DA algorithm after breaking all ties arbitrarily. In Appendix A we show that this approach does not provide a strategyproof mechanism.

This paper provides the first Pareto-stable mechanism for SMIW (and also SMCW) that is shown to be strategyproof. We present a nondeterministic algorithm for SMIW that generalizes Gale and Shapley’s DA algorithm as follows: in each iteration, a nondeterministically chosen unmatched man “proposes” simultaneously to all of the women in his next-highest tier of preference (i.e., the highest tier to which he has not already proposed); the women respond to this proposal by solving a certain maximum-weight matching problem to determine which man becomes unmatched (i.e., the man making the proposal or one of the tentatively matched men). Our generalization of the DA mechanism admits a polynomial-time implementation.

The college admissions model with weak preferences, which we denote CAW, is a further generalization of the SMIW model. In the CAW model, students and colleges are being matched rather than men and women, and each college has a positive integer capacity representing the number of students that it can accommodate. See Section 5 for a formal definition of the CAW model and the associated notions of weak stability and Pareto-stability.

A key difference between CAW and SMIW is that in addition to expressing preferences over individual students, the colleges have preferences over groups of students. This characteristic is shared by the CAS model, which is the restriction of the CAW model to strict preferences. It is known that no stable mechanism for CAS is strategyproof for the colleges [13]; the proof makes use of the fact that the colleges do not (in general) have unit demand. It follows that no Pareto-stable mechanism for CAW is strategyproof for the colleges. Throughout the remainder of the paper, when we say that a mechanism for a college admissions model is strategyproof, we mean that it is strategyproof for the students.

Gale and Shapley’s DA algorithm generalizes easily to the CAS model. Roth [13] has shown that the student-proposing DA algorithm provides a strategyproof stable mechanism for CAS when the preferences of the colleges are responsive. When the colleges have responsive preferences, the student-proposing DA mechanism is also known to be student-optimal for CAS [13].

Erdil and Ergin [4] consider the special case of the CAW model where the following restrictions hold for all students and colleges : is not indifferent between being assigned to and being left unassigned; is not indifferent between having one of its slots assigned to and having that slot left unfilled. We remark that this special case of CAW corresponds to the HRT problem discussed in Manlove [11, Chapter 3].1 For this special case, Erdil and Ergin present a polynomial-time algorithm for computing a Pareto-stable matching when the preferences of the colleges satisfy a technical restriction related to responsiveness. We consider the same class of preferences, which we refer to as minimally responsive; see Section 5 for a formal definition. The algorithm of Erdil and Ergin does not provide a strategyproof mechanism. Chen and Ghosh [2] build on the results of Erdil and Ergin by considering the many-to-many generalization of HRT in which the agents on both sides of the market have capacities (and the agent preferences are minimally responsive). For this generalization, Chen and Ghosh provide a strongly polynomial-time algorithm. No strategyproof mechanism (even for the agents on one side of the market) is possible in the many-to-many setting, since it is a generalization of CAS. We provide the first Pareto-stable mechanism for CAW that is shown to be strategyproof. As in the work of Erdil-Ergin and Chen-Ghosh, we assume that the preferences of the colleges are minimally responsive. We can also handle the class of college preferences “induced by additive utility” that is defined in Section 5.2.

In the many-to-many matching setting addressed by Chen and Ghosh [2], a pair of agents (on opposite sides of the market) can be matched with arbitrary multiplicity, as long as the capacity constraints are respected. Chen [1] presents a polynomial-time algorithm for the variation of many-to-many matching in which a pair of agents can only be matched with multiplicity one. Kamiyama [8] addresses the same problem using a different algorithmic approach. (The algorithms of Chen and Kamiyama are strongly polynomial, since we can assume without loss of generality that the capacity of any agent is at most the number of agents on the other side of the market.) Since this variation of the many-to-many setting also generalizes CAS, it does not admit a strategyproof mechanism, even for the agents on one side of the market.

Erdil and Ergin [5, 4] and Kesten [9] consider a second natural solution concept in addition to Pareto-stability. In the context of SMIW (or its special case SMCW), this second solution concept seeks a weakly stable matching that is “man optimal” in the following sense: for all weakly stable matchings , either all of the men are indifferent between and , or at least one man prefers to . Erdil and Ergin [4] present a polynomial-time algorithm to compute such a man optimal weakly stable matching for SMIW; in fact, their algorithm is presented for the generalization of SMIW to CAW. Erdil and Ergin [5] and Kesten [9] prove that no strategyproof man optimal weakly stable mechanism exists for SMCW. Prior to our work, it was unclear whether such an impossibility result might hold for strategyproof Pareto-stable mechanisms for SMCW (or its generalizations to SMIW and CAW).

The assignment game of Shapley and Shubik [15] can be viewed as an auction with multiple distinct items where each bidder is seeking to acquire at most one item. This class of unit-demand auctions has been heavily studied in the literature (see, e.g., Roth and Sotomayor [14, Chapter 8]). In Section 2, we define the notion of a “unit-demand auction with priorities” (UAP) and establish a number of useful properties of UAPs; these are straightforward generalizations of corresponding properties of unit-demand auctions. Section 3 builds on the UAP notion to define the notion of an “iterated UAP” (IUAP), and establishes a number of important properties of IUAPs; these results are nontrivial to prove and provide the technical foundation for our main results. Section 4 presents our first main result, a polynomial-time algorithm for SMIW that provides a strategyproof Pareto-stable mechanism. Section 5 presents our second main result, a polynomial-time algorithm for CAW that provides a strategyproof Pareto-stable mechanism assuming that the preferences of the colleges are minimally responsive.

## 2 Unit-Demand Auctions with Priorities

In this section, we formally define the notion of a unit-demand auction with priorities (UAP). In Section 2.1, we describe an associated matroid for a given UAP and we use this matroid to define the notion of a “greedy MWM”. In Section 2.2, we establish a result related to extending a given UAP by introducing additional bidders. In Section 2.3, we discuss how to efficiently compute a greedy MWM in a UAP. In Section 2.4, we introduce a key definition that is helpful for establishing our strategyproofness results. We start with some useful definitions.

A (unit-demand) bid for a set of items is a subset of such that no two pairs in share the same first component. (So may be viewed as a partial function from to .)

A bidder for a set of items is a triple where is an integer ID, is a bid for , and is a real priority. For any bidder , we define as , as , as , and as the union, over all in , of .

A unit-demand auction with priorities (UAP) is a pair satisfying the following conditions: is a set of items; is a set of bidders for ; each bidder in has a distinct ID.

### 2.1 An Associated Matroid

A UAP may be viewed as an edge-weighted bipartite graph, where the set of edges incident on bidder correspond to : for each pair in , there is an edge of weight . We refer to a matching (resp., maximum-weight matching (MWM), maximum-cardinality MWM (MCMWM)) in the associated edge-weighted bipartite graph as a matching (resp., MWM, MCMWM) of . For any edge in a given UAP, the associated weight is denoted or . For any set of edges , we define as . For any UAP , we let denote the weight of an MWM of .

###### Lemma 1.

Let be a UAP, and let denote the set of all subsets of such that there exists an MWM of that matches every bidder in . Then is a matroid.

###### Proof.

The only nontrivial property to show is the exchange property. Assume that and belong to and that . Let be an MWM of that matches every bidder in , and let be an MWM of that matches every bidder in . If matches some bidder in , then belongs to , as required. Thus, in what follows, we assume that does not match any bidder in . The symmetric difference of and , denoted , corresponds to a collection of vertex-disjoint paths and cycles. Since does not match any bidder in , we deduce that each bidder in is an endpoint of one of the paths in this collection. Since , , and , we have . It follows that there is at least one path in this collection, call it , such that one endpoint of is a bidder in and the other endpoint of is a vertex that does not belong to . Moreover, does not belong to : if the length of is odd, then is an item and hence does not belong to ; if the length of is even, then is not matched in and hence does not belong to . Since does not belong to and does not belong to , we conclude that does not belong to . The edges of alternate between and . Let denote the edges of that belong to , and let denote the edges of that belong to . Since is an MWM of and is a matching of , we deduce that . Since is an MWM of and is a matching of , we deduce that . Hence and and are MWMs of . The MWM matches all of the vertices on except for . Since does not belong to , we conclude that matches all of the vertices in , and so the exchange property holds. ∎

For any UAP , we define as the matroid of Lemma 1.

For any UAP and any independent set of , we define the priority of as the sum, over all bidders in , of . For any UAP , the matroid greedy algorithm can be used to compute a maximum-priority maximal independent set of .

For any matching of a UAP , we define as the set of all bidders in that are matched in . We say that an MWM of a UAP is greedy if is a maximum-priority maximal independent set of . For any UAP , we define the predicate to hold if for all greedy MWMs and of .

For any matching of a UAP, we define the priority of , denoted , as the sum, over all bidders in , of . Thus an MWM is greedy if and only if it is a maximum-priority MCMWM.

###### Lemma 2.

All greedy MWMs of a given UAP have the same distribution of priorities.

###### Proof.

This is a standard matroid result that follows easily from the exchange property and the correctness of the matroid greedy algorithm. ∎

For any UAP and any real priority , we define as the (uniquely defined, by Lemma 2) number of matched bidders with priority in any greedy MWM of .

###### Lemma 3.

Let be a UAP. Let be a bidder in such that belongs to , , and is not matched in any greedy MWM of . Let be a bidder in such that belongs to , , and is matched to in some greedy MWM of . Then .2

###### Proof.

Let be a greedy MWM in which is matched to . Thus is not matched in . Let denote the matching . Since is an MCMWM of and , we conclude that . If , the claim of the lemma follows. Assume that . In this case, is an MCMWM of since and . Since is a greedy MWM of and , we conclude that . If then is a greedy MWM of that matches , a contradiction. Hence , as required. ∎

### 2.2 Extending a UAP

Let be a UAP and let be a bidder such that is not equal to the ID of any bidder in . Then we define as the UAP . For any UAPs and , we say that extends if and .

###### Lemma 4.

Let be a UAP, let be a bidder in that is not matched in any greedy MWM of , and let be a UAP that extends . Then is not matched in any greedy MWM of .

###### Proof.

In what follows, we derive a contradiction by proving that is matched in a greedy MWM of . We need to prove that is matched in some greedy MWM of . Let denote a greedy MWM of . If is matched in , we are done, so assume that is not matched in . Thus contains a unique path with as an endpoint. The edges of alternate between and . Let denote the edges of that belong to , and let denote the edges of that belong to .

Since is matched in and not in , the other endpoint of is either an item, or it is a bidder that is matched in and not in . Either way, we deduce that all of the vertices on belong to . Thus is a matching in . Since is an MWM of and is a matching of , we deduce that and hence that . Since all of the vertices on belong to , we conclude that is a matching in . Since is an MWM of and is a matching of , we deduce that and hence that . Thus , and we conclude that is an MWM of .

Since is matched in and not in , we deduce that and hence that . Since is a greedy MWM of , we know that is an MCMWM of , and hence that . Thus and hence and is an MCMWM of . Since , the other endpoint of is a bidder that is matched in and not in . Since is a greedy MWM of and is an MCMWM of , we deduce that and hence that .

Since and , we deduce that is an MCMWM of . Since is a greedy MWM of and is an MCMWM of , we deduce that and hence that . Since we argued above that , we conclude that , and hence that is a greedy MWM of . This completes the proof, since is matched in . ∎

### 2.3 Finding a Greedy MWM

In this section, we briefly discuss how to efficiently compute a greedy MWM of a UAP via a slight modification of the classic Hungarian method for the assignment problem [10]. In the (maximization version of the) assignment problem, we are given a set of agents, a set of tasks, and a weight for each agent-task pair, and our objective is to find a perfect matching (i.e., every agent and task is required to be matched) of maximum total weight. The Hungarian method for the assignment problem proceeds as follows: a set of dual variables, namely a “price” for each task, and a possibly incomplete matching are maintained; an arbitrary unmatched agent is chosen and a shortest augmenting path from to an unmatched task is computed using “residual costs” as the edge weights; an augmentation is performed along the path to update the matching, and the dual variables are adjusted in order to maintain complementary slackness; the process repeats until a perfect matching is found.

Within our UAP setting, the set of bidders can be larger than the set of items, and some bidder-item pairs may not be matchable, i.e., the associated bipartite graph is not necessarily complete. In this setting, we can use an “incremental” version of the Hungarian method to find an (not necessarily greedy) MWM of a given UAP as follows. For the purpose of simplifying the presentation of our method, we enlarge the set of items by adding a dummy item such that is connected to each bidder with an edge of weight and we always maintain in the residual graph with a price of . We start with the empty matching . Then, for each bidder in (in arbitrary order), we process via an “incremental Hungarian step” as follows: let denote the set of bidders that are matched by ; let denote the set of items that are not matched by ; find the shortest paths from to each item in in the residual graph; let denote the minimum path weight among these shortest paths; choose a path that is either (1) a shortest path of weight from to an item in , or (2) a shortest path from to a bidder in such that extending with the edge yields a shortest path of weight from to ; augment along ; adjust the prices in order to maintain complementary slackness; update the residual graph. The algorithm terminates when every non-reserve bidder has been processed. The algorithm performs incremental Hungarian steps and each incremental Hungarian step can be implemented in time by utilizing Fibonacci heaps [6], where denotes the number of edges in the residual graph, which is .

In order to find a greedy MWM, we slightly modify the implementation described in the previous paragraph. Lemmas 7 and 8 established below imply that choosing the path in the following way results in a greedy MWM: if a path of type (1) exists, we arbitrarily choose such a path; if no path of type (1) exists, then we identify the nonempty set of all bidders such that a path of type (2) exists, and we choose a shortest path that terminates at a minimum priority bidder in . It is easy to see that the described modification does not increase the asymptotic time complexity of the algorithm. In the remainder of this section, we establish Lemmas 7, 8, and 9; Lemma 9 is used in Section 3.2.1 to prove Lemma 19. We start with some useful definitions.

Let and be UAPs, and let be an MWM of . We define as the edge-weighted digraph that may be obtained by modifying the subgraph of induced by the set of vertices as follows: for each edge that belongs to , we direct the edge from item to bidder and leave the weight unchanged; for each edge that does not belong to , we direct the edge from bidder to item and negate the weight.

###### Lemma 5.

Let and be UAPs, let be an MWM of , and let denote . Then does not contain any negative-weight cycles.

###### Proof.

Such a cycle could not involve (since only has outgoing edges) so it has to be a negative-weight cycle that already existed before was added, a contradiction since is an MWM of . ∎

Let and be UAPs, let be an MWM of , and let denote . We define a set of items , and a set of bidders , as follows. By Lemma 5, the shortest path distance in from bidder to any vertex reachable from is well-defined. We define as the set of all items in such that is unmatched in and the weight of a shortest path in from to is . We define as the set of all bidders such that the weight of a shortest path in from to is equal to .

Let and be UAPs, let be an MWM of , and let be a directed path in that starts at , has weight , and terminates at either an item in or a bidder in . (Note that could be a path of length zero from to .) Let denote the edges in that correspond to item-to-bidder edges in , and let denote the edges in that correspond to bidder-to-item edges in . It is easy to see that the set of edges is an MWM of . We define this MWM of as .

###### Lemma 6.

Let be a UAP, let be a greedy MWM of , let be a bidder that does not belong to , let denote the UAP , and let denote a greedy MWM of minimizing . Then contains a directed path satisfying the following conditions: has weight ; starts at ; the bidder-to-item edges in correspond to the edges in ; the item-to-bidder edges in correspond to the edges in ; if is nonempty, then terminates at an item in ; if is empty, then terminates at a minimum-priority bidder in .

###### Proof.

The edges of form a collection of disjoint cycles and paths of positive length.

We begin by arguing that does not contain any cycles. Suppose there is a cycle in . Let denote the edges of that belong to , and let denote the edges of that belong to . Let denote , which is a matching in since is unmatched in and hence does not belong to . Since is an MWM of and , we conclude that . Let denote , which is a matching in . Since is an MWM of and , we conclude that . Thus and hence , implying that is an MWM of . Moreover, since matches the same set of bidders as , we find that is a greedy MWM of . This contradicts the definition of since .

Next we argue that if is a path in , then is an endpoint of . Suppose there is a path in such that is not an endpoint of . Thus does not appear on since is unmatched in . Let denote the edges of that belong to , and let denote the edges of that belong to . Let denote , which is a matching in since does not belong to . Since is an MWM of and , we conclude that . Let denote , which is a matching in . Since is an MWM of and , we conclude that . Thus and hence and , implying that is an MWM of and is an MWM of . Since is a greedy MWM and hence an MCMWM of , the set of bidders matched by is not properly contained in the set of bidders matched by ; we conclude that . Since is a greedy MWM and hence an MCMWM of , the set of bidders matched by is not properly contained in the set of bidders matched by ; we conclude that . Thus , so the length of path is even. We consider two cases.

Case 1: The endpoints of are items. In this case, and match the same set of bidders, and hence is a greedy MWM of . This contradicts the definition of , since has positive length and hence .

Case 2: The endpoints of are bidders. Since has positive length, one endpoint, call it , is matched in but not in , and the other endpoint, call it , is matched in but not in . Since is a greedy MWM of and is an MWM of , we deduce that . Since is a greedy MWM of and is an MWM of , we deduce that . Thus . It follows that . Hence is a greedy MWM of . This contradicts the definition of since .

From the preceding arguments, we deduce that either or corresponds to a positive-length path with as an endpoint. Equivalently, is the edge set of a path that has as an endpoint and may have length zero (i.e., the path may begin and end at ). We claim if the edges of this path are directed away from endpoint , we obtain a directed path satisfying the six conditions stated in the lemma. It is easy to see that satisfies the first four of these conditions. It remains to establish that satisfies the fifth and sixth conditions.

For the fifth condition, assume that is nonempty. We need to prove that terminates at an item in . Since is nonempty, we deduce that , and hence that terminates at some item . Since has weight , we deduce that belongs to , as required.

For the sixth condition, assume that is empty. We need to prove that terminates at a minimum-priority bidder in . Suppose terminates at some item . Since has weight , we deduce that belongs to , a contradiction. Thus terminates at some bidder . Since has weight , we deduce that belongs to . If is not a minimum-priority bidder in , it is easy to argue that is not a greedy MWM of , a contradiction. Thus terminates at a minimum-priority bidder in . ∎

###### Lemma 7.

Let be a UAP, let be a greedy MWM of , let be a bidder that does not belong to , let denote the UAP , let be a directed path in of weight from to an item in , and let denote . Then is a greedy MWM of .

###### Proof.

The definition of implies that is an MWM of . Let denote a greedy MWM of minimizing . Let denote the set of bidders in matched by . Since is nonempty, Lemma 6 implies that the set of bidders in matched by is . Since is an MWM of that also matches the set of bidders , we deduce that is a greedy MWM of . ∎

###### Lemma 8.

Let be a UAP, let be a greedy MWM of , let be a bidder that does not belong to , and let denote the UAP . Assume that is empty. Let denote a minimum-priority bidder in (which is nonempty by Lemma 6), let be a directed path in of weight from to , and let denote . Then is a greedy MWM of .

###### Proof.

The definition of implies that is an MWM of . Let denote a greedy MWM of minimizing . Let denote the set of bidders in matched by . Since is empty, Lemma 6 implies that the set of bidders in matched by is , where is some minimum-priority bidder in . It is straightforward to check that has the same weight, cardinality, and priority as . Thus is a greedy MWM of , as required. ∎

###### Lemma 9.

Let and be two UAPs such that extends , let be a greedy MWM of , and let be a greedy MWM of . Then .

###### Proof.

Immediate from Lemmas 7 and 8. ∎

### 2.4 Threshold of an Item

In this section, we define the notion of a “threshold” of an item in a UAP. This lays the groundwork for a corresponding IUAP definition in Section 3.2. Item thresholds play an important role in our strategyproofness results.

###### Lemma 10.

Let be a UAP and let be an item in . Let be the set of bidders such that is a UAP and is of the form . Then there is a unique pair of reals such that for any bidder in , the following conditions hold, where denotes , denotes , and denotes : (1) if then is matched in every greedy MWM of ; (2) if then is not matched in any greedy MWM of ; (3) if then is matched in some but not all greedy MWMs of .

###### Proof.

Let be a greedy MWM of , let denote , and let denote . Let denote the set of matchings of that do not match , let denote the maximum-weight elements of , let denote the maximum-cardinality elements of , let denote the maximum-priority elements of , and observe that there is a unique pair of reals such that any matching in has weight and priority . It is straightforward to verify that the unique choice of satisfying the conditions stated in the lemma is . ∎

For any UAP and any item in , we define the unique pair of Lemma 10 as .

## 3 Iterated Unit-Demand Auctions with Priorities

In this section, we formally define the notion of an iterated unit-demand auction with priorities (IUAP). An IUAP allows the bidders, called “multibidders” in this context, to have a sequence of unit-demand bids instead of a single unit-demand bid. In Section 3.1, we define a mapping from an IUAP to a UAP by describing an algorithm that generalizes the DA algorithm, and we establish Lemma 15 that is useful for analyzing the matching produced by Algorithm 4.1 of Section 4. Lemma 15 is used to establish weak stability (Lemmas 27, 28, and 29) and Pareto-optimality (Lemma 30). In Section 3.2, we define the threshold of an item in an IUAP and we establish Lemma 18, which plays a key role in establishing our strategyproofness results. We start with some useful definitions.

A multibidder for a set of items is a pair where is a real priority and is a sequence of bidders for such that all the bidders in have distinct IDs and a common priority . We define as . For any integer such that , we define as the bidder . For any integer such that , we define as . We define as .

An iterated UAP (IUAP) is a pair where is a set of items and is a set of multibidders for . In addition, for any distinct multibidders and in , the following conditions hold: ; if belongs to and belongs to , then . For any IUAP , we define as the union, over all in , of .

### 3.1 Mapping an IUAP to a UAP

Having defined the notion of an IUAP, we now describe an algorithm ToUap that maps a given IUAP to a UAP. Algorithm ToUap generalizes the DA algorithm. In each iteration of the DA algorithm, a single man is nondeterministically chosen, and this man reveals his next choice. In each iteration of ToUap, a single multibidder is nondeterministically chosen, and this multibidder reveals its next bid. We prove in Lemma 14 that, like the DA algorithm, algorithm ToUap is confluent: the output does not depend on the nondeterministic choices made during an execution. We conclude this section by establishing Lemma 15, which is useful for analyzing the matching produced by Algorithm 4.1 in Section 4. Lemma 15 is used to establish weak stability (Lemmas 27, 28, and 29) and Pareto-optimality (Lemma 30). We start with some useful definitions.

Let be a UAP and let be an IUAP . The predicate is said to hold if and for any multibidder in , for some .

A configuration is a pair where is a UAP, is an IUAP, and holds.

Let be a configuration, where and , and let be a bidder in . Then we define as the unique multibidder in such that belongs to .

Let be a configuration where and . For any in , we define as .

Let be a configuration where . We define as the set of all bidders in such that and where .

{algorithm}

ToUap {algorithmic}[1] \RequireAn IUAP \State \State \While is nonempty \State an arbitrary bidder in \State \EndWhile\State\Return

Our algorithm for mapping an IUAP to a UAP is Algorithm 3.1. The input is an IUAP and the output is a UAP such that holds. The algorithm starts with the UAP consisting of all the items in but no bidders. At this point, no bidder of any multibidder is “revealed”. Then, the algorithm iteratively and nondeterministically chooses a “ready” bidder and “reveals” it by adding it to the UAP that is maintained in the program variable . A bidder associated with some multibidder is ready if is not revealed and for each bidder that precedes in , is revealed and is not matched in any greedy MWM of . It is easy to verify that the predicate is an invariant of the algorithm loop: if a bidder belonging to a multibidder is to be revealed at an iteration, and for some integer at the beginning of this iteration, then after revealing , where is the UAP that is maintained by the program variable at the beginning of the iteration. No bidder can be revealed more than once since a bidder cannot be ready after it has been revealed; it follows that the algorithm terminates. We now argue that the output of the algorithm is uniquely determined (Lemma 14), even though the bidder that is revealed in each iteration is chosen nondeterministically.

For any configuration , we define the predicate to hold if for any bidder that is matched in some greedy MWM of , we have where denotes .

###### Lemma 11.

Let be a configuration where and assume that holds. Then for each in .

###### Proof.

The claim of the lemma easily follows from the definition of . ∎

###### Lemma 12.

The predicate is an invariant of the Algorithm 3.1 loop.

###### Proof.

It is easy to see that holds when the loop is first encountered. Now consider an iteration of the loop that takes us from configuration where to configuration where . We need to show that holds. Let be a bidder that is matched in some greedy MWM of . Let denote the bidder that is added to in line 3.1, and consider the following three cases.

Case 1: . Let denote