Multi-period Matching

Transcription

1 Multi-period Matching February 10, 2018 Sangram V. Kadam and Maciej H. Kotowski 1 Manuscript Received January 27, 2017; revised July 13, 2017 and August 28, We thank seminar audiences at Harvard/MIT, Santa Clara University, Stanford, the University of Melbourne, the University of Western Ontario, and Yale, and conference participants at the 2014 Meeting of the Society for Social Choice and Welfare (Boston), the 2014 International Conference on Game Theory (Stony Brook), and the 2014 International Workshop on Game Theory and Economic Applications honoring Marilda Sotomayor (Sao Paulo) for helpful suggestions that greatly improved this study. We thank Fanqi Shi, Neil Thakral, Norov Tumennasan and, particularly, Chris Avery for their comments on earlier versions of this manuscript. We are grateful to Mei Liang and Dr. Laurie Curtin of the National Resident Matching Program for comments on a draft of this manuscript. Sangram V. Kadam gratefully acknowledges the support of the Danielian Travel & Research Grant. Portions of this study were completed when Maciej H. Kotowski was visiting the Stanford University Economics Department. He thanks Al Roth and the department for their generous hospitality. 1

2 Abstract We examine a dynamic, two-sided, one-to-one matching market, where agents on both sides interact over a period of time. We define and identify sufficient conditions for the existence of a dynamically stable matching, which may require revisions to initial assignments. A generalization of the deferred acceptance algorithm can identify dynamically stable outcomes in a large class of economies, including cases with intertemporal preference complementarities. We relate our analysis to market unraveling and to common market design applications, including the medical residency match. Keywords: Dynamic matching, two-sided market, stability, market design JEL: C78, D47 Running Head: Multi-period Matching Author Affiliations: Kadam: Charles River Associates, 1201 F Street NW, Washington DC <sangram.kadam@gmail.com> Kotowski: John F. Kennedy School of Government, Harvard University, 79 JFK Street, Cambridge MA <maciej_kotowski@hks.harvard.edu> 2

3 1 Introduction Gale and Shapley s (1962) concept of stability is the leading theoretical organizing principle for two-sided matching markets. The term s connotation, however, suggests an immutability in a market s outcome. But, this is not what we often observe. Consider a few consequences of seemingly, or aspirationally, stable pairings in two-sided markets. 1. After freshman year, a student transfers to another college. 2. A married couple divorces. Each then finds a new spouse. 3. A business graduate initially works in management consulting. After a year, she quits to start her own business. In the preceding cases, commitment is limited and intended long-term (or permanent) pairings often see revisions after a few periods. Gale and Shapley s (1962) model of ephemeral one-time assignments cannot capture these changes, which unfold with time. In this paper we examine an extension of Gale and Shapley s model to a multi-period setting, where agents match and possibly change their assignments over time. As part of our analysis, we propose a new stability definition dynamic stability that extends Gale and Shapley s solution in an intuitive and tractable way. Drawing on familiar intuitions, a matching is dynamically stable if at each moment in time it is individually rational and no pair of agents can arrange a mutually-preferable relationship plan conjecturing that the wider market evolves in an unfavorable manner. We consider dynamic stability to be the simplest and most natural multi-period generalization of Gale and Shapley s original idea. 3

4 We identify sufficient conditions for the existence of a dynamically stable matching and we propose a new generalization of Gale and Shapley s (1962) deferred acceptance algorithm to find such assignments. Our model captures the periodic re-matching observed in many two-sided markets. Though we adopt Gale and Shapley s terminology of a matching between men and women, our model s applications extend beyond the study of interpersonal relations. Labor markets offer a germane application. According to the U.S. Bureau of Labor Statistics (2012), individuals born from 1957 to 1964 held an average of 11.3 jobs from ages 18 to 46. Our model and solution accommodate such dynamics. 2 For simplicity, we suppress the role of financial transfers in our main exposition, but they can be incorporated into our model, as we show in the online appendix. Our analysis is primarily a positive description of a multi-period economy and we make no presumption concerning the existence of a centralized authority determining agents assignments. Nevertheless, the existence of algorithms that can identify stable outcomes has spurred the establishment of specialized clearinghouses to coordinate participants interaction in some markets (Roth, 2002). Our results can inform these applications as well since they often involve a multi-period component. As an example, consider the National Resident Matching Program R (NRMP R ) medical residency match (Roth and Peranson, 1999). 3 The 2 Labor market dynamics are a central focus of the search-and-matching literature (Rogerson et al., 2005). Our model offers a complementary perspective on this phenomenon building more directly on Gale and Shapley s (1962) model. 3 National Resident Matching Program and NRMP are registered trademarks of the National Resident Matching Program. 4

5 NRMP is a clearinghouse that matches graduating medical students to hospital residency programs in the United States using an algorithm. While the initial match of a trainee-doctor to a residency program draws the most attention, a deeper look at this market reveals a rich multi-period structure. Some residency programs, for example, only provide introductory instruction and students must also match with an advanced specialty. Other programs bundle all years of training together. Attrition (McAlister et al., 2008; Yaghoubian et al., 2012) and transfers between residency programs occur as well. 4 These features suggest that an assessment of the algorithm s success and of participants welfare requires a longer-term perspective, going beyond the initial Match Day headlines. Multi-period models, like ours, provide a framework for such an analysis. This paper is organized as follows. Section 2 briefly reviews the related literature. Section 3 introduces our model and defines our main solution concept, dynamic stability. For simplicity, we focus on a two-period setting, but our model generalizes. 5 Section 4 identifies sufficient conditions for the existence of a dynamically stable matching and presents a new multi-period generalization of Gale and Shapley s (1962) deferred acceptance algorithm. Sections 5 and 6 explore two applications of our model. First, Section 5 examines several multi-period matching mechanisms, including the aforementioned NRMP algorithm, in greater detail. We contrast their operation with the algorithms that we propose. And second, Section 6 considers the role of learning in a multi-period matching market. We ap- 4 Losada (2010) describes the process of switching programs and specialities, including anecdotal accounts of students experiences. 5 We offer a T-period version of our model in the online appendix. See also Kadam and Kotowski (2018). 5

6 ply our definition of dynamic stability to offer a new explanation for market unraveling in situations where agents interact over multiple periods. Section 7 concludes by summarizing our contributions and by outlining extensions of our analysis, which we investigate in the online appendix. All omitted proofs are in the appendix. 2 Literature We study an economy where agents interact over multiple periods, forming and revising their assignments with time. Similar two-sided, 6 one-to-one matching markets are studied by Damiano and Lam (2005), Kurino (2009), and Kotowski (2015). These studies propose alternative definitions of stability emphasizing different characteristics of dynamic markets. Among these papers, Kotowski (2015) is the closest to our setting and motivation. He proposes an alternative solution concept motivated by the literature on matching with externalities (Sasaki and Toda, 1996; Pycia and Yenmez, 2017). His proposal is neither weaker nor stronger than our definition of dynamic stability. Self-sustaining stability, proposed by Damiano and Lam (2005), incorporates the idea of coalition proofness (Bernheim et al., 1987). Kurino s (2009) proposal, credible group stability, draws on the intuition provided by the bargaining set (Zhou, 1994; Klijn and Massó, 2003). An important precursor to Kurino s (2009) analysis is Corbae et al. (2003). They develop a dynamic matching model to investigate questions in monetary economics, which is not the focus of our study. 6 Abdulkadiroğlu and Loertscher (2007), Bloch and Cantala (2013), and Kurino (2014), among others, study dynamic one-sided markets. We do not study this case, though our analysis is complementary. 6

7 Several authors have also considered multi-period, many-to-one or many-to-many matching models. Dur (2012), Bando (2012), Pereyra (2013), and Kennes et al. (2014) propose models in this vein. Though these papers span many cases, our analysis is not a special case of any of them. Notably, our model allows for richer forms of inter-temporal complementarity in preferences, our stability concepts are distinct, and our algorithms for constructing stable assignments are new. In our model, agents interact over multiple periods and revise their assignments over time. A complementary class dynamic matching models examines one-time matchings that arise over multiple periods. Here focus has ranged from questions of preference formation (Kadam, 2015) and unraveling (Roth and Xing, 1994) to managing the (stochastic) arrival and departure of agents or objects (Ünver, 2010; Akbarpour et al., 2017; Baccara et al., 2016; Thakral, 2016; Leshno, 2017). Doval (2017) examines an economy of this latter type and independently proposes a definition of dynamic stability, which is distinct from our proposal. Incorporating the intuition of sub-game perfection, her definition relies on an expanded definition of a matching, which also differs from our approach. 7 An interesting parallel exists between multi-period, one-to-one matching markets and one-period, many-to-many matching markets. Over a lifetime, each agent can have many partners. Recently, Hatfield and Kominers (2012b) have examined such markets in the matching with contracts framework (Hatfield and Milgrom, 2005). Dynamic matching problems can be analyzed within this paradigm by allowing agents to encode the date(s) of their 7 Corbae et al. (2003), Kurino (2009), and Doval (2017) propose stability definitions that assume agents coordinate on a contingent plan for the economy conditional on each counterfactual history. 7

8 relationship(s). 8 However, our model is not subsumed by their analysis. Two differences are paramount. First, we do not presume substitutable preferences, which are key requirement for the existence of stable outcomes in static many-to-many matching markets. And second, as we explain below, our definition of blocking links successive assignments through agents conjectures regarding the market s future development. Thus, our definition of stability differs from the definition employed in a static setting. The possible absence of a man-optimal stable matching 9 further contrasts our model with classic many-to-many matching models (Roth, 1984). Our definition of dynamic stability is tailored to the analysis of multi-period, two-sided matching markets, which are a particular class of dynamic cooperative games. Other solutions, including dynamic or sequential versions of the core, have been proposed in related models. As explained below, our definition of dynamic stability is a natural analogue to Gale s (1978) sequential core, which he proposed to study an exchange economy with limited trust. Kranich et al. (2005) examine generalized versions of this idea the strong and weak sequential cores in the context of dynamic games with transferable utility. 10 Lehrer and Scarsini (2012) propose the fair, stable, and credible cores as alternative solutions for this class of games. All of the above concepts emphasize different aspects of multi-period 8 For example, see Dimakopoulos and Heller (2014). 9 See Example A.1 in the online appendix. See also Kadam and Kotowski (2018). 10 Of these concepts, the strong sequential core is closest to our definition of dynamic stability. It too allows a coalition to block at each moment in time. Predtetchinski et al. (2002) apply the strong sequential core to study two-period exchange economies. See Habis and Herings (2010) for further discussion of the weak sequential core. 8

9 interaction, such as credibility, commitment, and foresight. Different stances on these points also underpin the collection of solutions proposed for multi-period matching markets, as noted above. A follow-up companion paper to this study, Kadam and Kotowski (2018), examines a T-period version of the model developed below, albeit on a more restricted preference domain. It investigates the (non-)lattice structure of the set of dynamically matchings and the robustness of stable outcomes to changes in the market s time horizon. 3 The Model Mindful of the noted applications, for expositional ease we present our model using Gale and Shapley s terminology of a matching between men and women. Alternative applicationspecific labels would be students and schools, doctors and hospitals, or workers and firms. For brevity, we define some concepts only from the perspective of a typical man. Our model is symmetric and all definitions apply to women with obvious changes in notation. 3.1 The One-Period Market To introduce notation and to provide a benchmark, we briefly review Gale and Shapley s (1962) one-period matching market. There are finite, disjoint sets of men, M = {m 1,...,m n }, and women, W = {w 1,...,w n }. Each man (woman) can be matched to one woman (man) or not matched at all. By convention, a man (woman) who is not matched to a woman (man) is treated as matched to himself (herself). Thus, W m := W {m} is the set 9

10 of potential partners for m M. The set M w := M {w} is defined analogously for w W. Each agent has a strict preference ranking of potential partners. A matching is a function that assigns each agent to a partner. More formally, the function µ: M W M W is a one-period matching if µ(m) W m for all m M, µ(w) M w for all w W, and µ(i) = j = µ(j) = i for all i. A matching is stable if it cannot be blocked by any agent or pair. That is, (i) each agent weakly prefers his/her assigned partner to being not matched; and, (ii) no pair prefers to be together in lieu of their assigned partners. Gale and Shapley (1962) introduce the (man-proposing) deferred acceptance algorithm to construct a stable matching in every one-period market. It proceeds as follows: 1. In round 1, each man proposes to his most preferred partner. (If a man finds all women unacceptable, he remains single.) Given the received proposals, each woman tentatively engages her favorite suitor and rejects the others. 2. In roundτ 2, each rejected man in roundτ 1 proposes to his most preferred partner who has not yet rejected him. (If a man finds all remaining women unacceptable, he remains single.) Each woman evaluates any received proposals and her engaged partner (if any) and tentatively engages her favorite suitor and rejects the others. The above process continues until no rejections occur. At this point, engaged pairs are matched and all others remain unmatched. The resulting assignment is stable. 10

11 3.2 A Multi-period Market Extending the model, suppose agents interact over two periods. In every period, each man (woman) can be matched with one woman (man) or not matched at all. An agent s partners in periods t and t may differ. Thus, (j,k) is a partnership plan for i where he is matched with j in period 1 and with k in period 2. When confusion is unlikely, we write jk for (j,k). The plan jk is persistent if j = k. Else, it is volatile. Each agent has a strict and rational preference over partnership plans. If i prefers plan jk to plan j k, we write jk i j k. As usual, jk i j k if jk i j k or jk = j k. The function µ: M W (M W) 2 is a multi-period matching if for all i, µ(i) = (µ 1 (i),µ 2 (i)) and both µ 1 and µ 2 are one-period matchings. Henceforth, we refer to a multi-period matching simply as a matching. In a one-period market, stability combines an individual-rationality requirement and a pairwise no-blocking condition. The conditions assert that an agent or a pair cannot benefit by pursuing his/her/their best option outside of the market independently of others actions. We extend this idea to a multi-period setting on a period-by-period basis. At the market s beginning, each agent s unilateral outside option is to always remain unmatched. Thus, agent i can period-1 block the matching µ if ii i µ(i). Similar logic guides blocking by a pair, though the set of available outside options is exponentially richer due to scheduling opportunities. The pair {m,w} can period-1 block the matching µ if 1. ww m µ(m) and mm w µ(w); 2. wm m µ(m) and mw w µ(w); 3. mw m µ(m) and wm w µ(w); or, 11

12 4. mm m µ(m) and ww w µ(w). 11 Continuing to period 2, we say that agent i can period-2 block the matching µ if (µ 1 (i),i) i µ(i). Similarly, the pair {m,w} can period-2 block the matching µ if 1. (µ 1 (m),w) m µ(m) and (µ 1 (w),m) w µ(w); or, 2. (µ 1 (m),m) m µ(m) and (µ 1 (w),w) w µ(w). A matching is ex ante individually rational if it cannot be period-1 blocked by any agent. An ex ante stable matching cannot be period-1 blocked by any agent or pair. A matching is dynamically individually rational if for all t it cannot be period-t blocked by any agent. A matching is dynamically stable if for all t it cannot be period-t blocked by any agent or pair. 12 Dynamic stability, the focus of our study, refines its ex ante counterpart by additionally allowing blocking conditional on the market s elapsed history. The following example illustrates this refinement s implications. Example 1. Consider a market with two men and two women. Their preferences are: m1 : w 1 w 1,w 1 m 1,w 1 w 2,m 1 w 1, w 2 w 1,m 1 m 1 w1 : m 2 m 2, m 2 m 1,m 1 m 2,m 1 m 1,w 1 w 1 m2 : w 2 w 2,w 2 m 2,w 2 w 1,m 2 w 2, w 1 w 2,m 2 m 2 w2 : m 1 m 1, m 1 m 2,m 2 m 1,m 2 m 2,w 2 w 2 Here, w 1 w 1 is m 1 s most preferred plan, w 1 m 1 is second best, and so on. Unlisted plans are not individually rational. This market has three ex ante stable matchings. These are 11 Condition 4 may seem redundant. We include it since single-agent and pairwise definitions of period-t blocking are special cases of more general coalition-based definitions. See Section 7 and the online appendix. 12 Doval (2017) has contemporaneously and independently proposed an alternative solution concept also called dynamic stability. 12

13 boldfaced, underlined, and boxed in the agents preference lists. 13 Focusing on the underlined matching, we note that m 1 is paired with w 2 in period 2. However, w 1 m 1 m1 w 1 w 2. Thus, it seems unlikely m 1 would agree to a continuation of w 1 w 2 after his period 1 assignment to w 1. He prefers to renege on his commitment and the matching is not dynamically stable. The remaining matchings are both dynamically stable. Before addressing the existence of a dynamically stable matching, we provide further context for our solution, which we regard as the simplest multi-period generalization of Gale and Shapley s original idea. Defining stability in a multi-period economy involves resolving many conceptual issues that are absent from the static, one-period case (Damiano and Lam, 2005). Our definition pursues a tractable compromise along three dimensions. (1) Commitment Commitment or its absence plays an important role when interaction occurs over multiple periods. 14 Most cooperative solutions, such as stability or the core, are formulated from an ex ante point of view and they cannot capture the absence of trust or agents changing incentives when interaction is sequential. Gale (1978) recognized this phenomenon in a multi-period exchange economy. Dynamic stability is the natural (pairwise) counterpart in our model to Gale s (1978) definition of the sequential core, which allows blocking ex ante and conditional on the passage of time. It captures some of the intuition 13 Formally, the boldface plans correspond to the matching µ(m 1 ) = w 1 w 1 ;µ(m 2 ) = w 2 w 2 ;µ(w 1 ) = m 1 m 1 ;µ(w 2 ) = m 2 m 2. The other matchings are read similarly. 14 Diamantoudi et al. (2015) study the role of commitment in a dynamic matching market. Assuming additively-separable, time-invariant, and history-independent preferences, they show that repetitions of a single-period stable matching is a stationary sequential equilibrium when commitment is not possible. 13

14 of sub-game perfection, thought within a cooperative context (Gale, 1978, p. 461). (2) Size of Blocking Coalitions We define dynamic stability with pairwise blocking. Others have considered blocking actions by larger groups in similar models (Damiano and Lam, 2005; Kurino, 2009; Doval, 2017). We prefer the pairwise specification for three reasons. First, we wish to maintain a minimal departure from classic studies of two-sided markets. Second, the practical difficulties of large-scale coordination or collusion, commonly acknowledged in static models, continue to apply in our environment. Third, coalition-based definitions of blocking necessitate the introduction of more restrictive assumptions to ensure positive results. In the online appendix, we identify sufficient conditions ensuring that dynamically stable matchings are immune to blocking by larger coalitions. (3) Conjectures About the Future In a multi-period economy agents must subscribe to a model of the economy s subsequent evolution following a departure from a proposed plan. Favoring the spareness of a cooperative model, dynamic stability embeds a robustness criterion in the definition of blocking. When agents block, our definition assumes that each agent behave as if the market will evolve in the most unfavorable manner (to them) in response to their deviation. In practice, of course, this implies complete exclusion from the wider market. 15 Excluded agents anticipate implementing the best continuation plan 15 In a market design application, market exclusion may be real rather than conjectured since the return of non-cooperative agents is controllable by the designer. For example, in the NRMP matching process, applicants/residency programs that fail to honor the prescribed outcome, without securing a waiver, may be barred from accepting alternative positions/applicants or participating in future NRMP Matches (NRMP, 14

15 given their conjectured restricted circumstances. Though dynamic stability differs in some key ways, a similar intuition is sometimes used to motivate a cooperative game s α-core (Aumann and Peleg, 1960; Aumann, 1961). The result is a robust and straightforward model of behavior. 4 Existence of Dynamically Stable Matchings Dynamic stability provides a natural extension of Gale and Shapley s (1962) stability concept to a multi-period economy. Unlike the one-period case, however, the existence of a dynamically stable matching is not guaranteed. Example 2. Consider a market with one man and one woman. 16 Their preferences are: m : wm,ww,mm w : mm,ww There are only two candidate stable matchings. The matching where µ(m) = mm is not ex ante stable as the couple can period-1 block it. The matching where µ (m) = ww is ex ante stable. However, it is not dynamically stable since m will renege after period 1. To ensure the existence of a dynamically stable matching, we will impose additional structure on the admissible preference domain. We introduce this structure by expanding its scope, thus accommodating progressively richer behavioral features. 2014b). Other centrally-coordinated markets, such as ride-sharing applications, can enforce similar sanctions. 16 Hatfield and Kominers (2012a) consider a similar example of a doctor and a hospital contracting morning and afternoon shifts. They suggest the doctor and the hospital should sign a unified contract covering both shifts. In our model, that suggestion corresponds to the ex ante stable outcome. 15

16 The simplest way to define a multi-period preference involves aggregating a time-independent ranking of potential partners. Pursuing this benchmark, let P i be an ordinal ranking of agent i s potential partners abstracting from temporal considerations. We call P i a spot ranking. As usual, we write jp i k if j is superior to k and jr i k if jp i k or j = k. We say that the (multi-period) preference i reflects P i if (S) [jr i j & kp i k ] or [jp i j & kr i k ] = jk i j k. Condition (S) resembles responsiveness (Roth, 1985), but additionally accounts for assignment timing. Concordant with intuition, (S) implies plans with unambiguously superior partners are always preferable. Though analytically appealing, preferences that reflect a spot ranking fail to capture many critical characteristics of inter-temporal decision-making, such as status-quo bias or switching costs (Samuelson and Zeckhauser, 1988). Thus, we propose two further conditions accommodating a wider range of preferences. The first is inertia, which allows successive assignments to the same partner to be complementary. The preference i exhibits inertia if (I) [jk i jj = kk i jk] & [jk i kk = jj i jk]. If i reflects a spot ranking, it also exhibits inertia; however, inertia is a much weaker requirements. Preferences with inertia have been invoked in some student-school assignment applications. 17 They are particularly applicable in cases with non-trivial switching costs or when long-term pairing is comparatively valued, as in labor markets with specialized skills. 17 Dur (2012) argues that families with multiple children typically want their younger child (the period 2 match) to attend the same school as his older sibling (the period 1 match). In their model of daycare 16

17 A further weakening of (I) allows us to capture some forms of chronological complementarity among different assignments. The preference i satisfies sequential improvement complementarity if (SIC) jk i jj i ii = kk i jj. In words, if agent i is willing to switch from an initial match with j to a match with k, in spite of the switching costs, then it must be that he prefers kk over jj. Thus, if an agent prefers to switch assignments after period 1, then the change ought to be toward an ex ante better (long-term) option. To highlight the behavioral differences between (S), (I), and (SIC), consider the preference i : kk,kj,jk,jj,... It reflects the spot ranking kp i j. Given a preferences satisfying (S), inertia allows, but does not require, persistent plans to rise in rank. For instance, the preference i : kk,kj,jj,jk,... biases agent i toward prolonging a period-1 assignment to j. However, (SIC) additionally allows an agent to (particularly) appreciate directional improvement in his assignments. For example, the preference satisfies (SIC) but not (I). Neither i i : jk,kk,kj,jj,... nor i reflect any spot ranking. assignment, Kennes et al. (2014) assume that children s preferences satisfy a condition called rankability. Condition (I) is weaker. 17

18 Regrettably, (SIC) alone is too weak to guarantee the existence of a dynamically stable matching. 18 Particular complications arise (only) when agents are unmatched. Thus, in our analysis we will invoke two further technical restrictions. The first is singlehood aversion: (SA) [ij i ii = jj i ij] & [ji i ii = jj i ji]. The second is revealed dominance of singlehood: (RDS) [jk i ji i ii = kk i ji] & [ik i ij i ii = kk i ij]. (SA) posits a distaste for being unmatched in any period. It is commonly satisfied in applications workers loathe unemployment and firms dislike vacancies. Preferences with inertia satisfy both (SIC) and (SA). 19 (RDS) is more technical and is best interpreted as a mild strengthening of (SIC), applicable only when an agent is unmatched in one period. 20 Behaviorally, the condition allows our model to address cases where an agent believes a particular partner is an acceptable match only in a specific period, a situation that (SA) precludes (see Example 3 and footnote 25). Lemma 1. In every economy, there exists an ex ante stable matching. Theorem 1. There exists a dynamically stable matching if (a) i satisfies (SIC) and (SA) for all i; or, (b) i satisfies (SIC) and (RDS) for all i See Example A.2 in the online appendix. 19 See the proof of Corollary Preferences with inertia do not satisfy (RDS). We are grateful to Fanqi Shi for alerting us to this fact. 21 Parts (a) and (b) cannot be merged. If some preferences satisfy only (SIC)+(SA) and others only (SIC)+(RDS), a dynamically stable matching may not exist. See Example A.2 in the online appendix. 18

19 Corollary 1. (a) If each agent s preference i reflects a spot ranking, there exists a dynamically stable matching. (b) If each agent s preference i exhibits inertia, there exists a dynamically stable matching. As with all formal claims, we prove the above results in the appendix. Our proofs are constructive, relying on an algorithm to identify a stable outcome. Below, we present our algorithm in two parts. First, we define an algorithm that identifies an ex ante stable matching in every market, thus proving Lemma 1. Thereafter, we augment this algorithm with an adjustment step, which ensures that the resulting assignment is dynamically stable when the conditions of Theorem 1 are satisfied. A simple generalization of the deferred acceptance procedure identifies an ex ante stable matching. In the plan deferred acceptance procedure, each man proposes to one woman at a time and specifies their exclusive relationship s timing. A man and woman may be together for both periods or together in one period and single otherwise. Such proposals are made just like in the usual deferred acceptance algorithm with the woman tentatively accepting her best available option. Once no proposals are rejected, the final matching is set and is ex ante stable. Algorithm 1 (PDA). The (man-proposing) plan deferred acceptance procedure identifies a matching µ as follows. For each m M let X 0 m = w W{ww,wm,mw}. At τ = 0, no plans in X 0 m have been rejected. In round τ 1: 1. Let X τ m X0 m be the subset of plans that have not been rejected in some round τ < τ. If Xm τ = or mm m x for all x Xm τ, then m does not make any proposals. 19

20 Otherwise, m proposes to the woman identified in his most preferred plan in Xm. τ If ww is his most preferred plan, he proposes a two-period relationship to w. If wm (mw) is his most preferred plan, he proposes a one-period partnership with w for period 1 (2). In period 2 (1), both m and w are to be unmatched. 2. Let X τ w be the set of plans made available to w. If ww w x for all x X τ w, w rejects all proposals. Otherwise, w (tentatively) accepts her most preferred plan in X τ w and rejects the others. A woman may accept at most one plan at a time. 22 The above process continues until no rejections occur. If w accepts m s proposal in the final round, define µ (m) and µ (w) accordingly. If i does not make or receive any proposals in the final round, set µ (i) = ii. Remark 1. The PDA procedure can be interpreted as men proposing from a restricted set of contracts, as in the setting of Hatfield and Milgrom (2005). The existence of an ex ante stable matching then follows as a corollary to their analysis. The analysis of Hatfield and Milgrom (2005) does not imply the existence of a dynamically stable matching. Most notably, the definitions of blocking and the preference domains differ. Next, we extend the PDA procedure by introducing an adjustment phase. Algorithm 2 (PDAA). The (man-proposing) plan deferred acceptance procedure with adjustment identifies a matching µ as follows: 22 In this algorithm, a woman may accept at most one plan at a time. Thus, she cannot accept the plans mw and wm although one could seemingly combine them as mm without affecting any other matches simultaneously. This technical feature both simplifies the algorithm and the analysis of its output. 20

21 Step 1. Implement the PDA procedure and call the resulting matching µ 1 = ( µ 1 1, µ 1 2). For each agent i who is assigned a partner in period 2 (i.e. µ 1 2 (i) i), set µ (i) = µ 1 (i) and exclude the agent from further consideration. Step 2. For each remaining agent i, define a conditional preference given the period- 1 assignment as follows: j µ1 1 (i) i j ( µ 1 1(i),j) i ( µ 1 1(i),j ). 23 Next, implement Gale and Shapley s (1962) (man-proposing) deferred acceptance algorithm where each agent makes/accepts proposals according to his/her conditional preference, µ1 1 (i) i. If µ 2 2 ( ) is the resulting one-period matching, for all agents involved set µ (i) = ( µ 1 1 (i), µ2 2 (i)) as the final outcome. The PDAA operates in two steps, and both are required to ensure stability under the conditions of Theorem 1. Step 1 secures the matching s ex ante stability while step 2 ensures that the final outcome cannot be period-2 blocked. 24 The following example illustrates the PDAA s operation and the importance of the adjustment step. Example 3. Consider an economy with three men and three women. m1 : w 1 w 2,w 2 w 2,w 1 m 1,m 1 m 1 w1 : m 1 m 1,m 1 m 2,m 1 w 1,w 1 w 1 m2 : w 2 w 1,w 1 w 1,w 2 m 2,m 2 m 2 w2 : m 2 m 2,m 2 m 1,m 2 w 2,w 2 w 2 m3 : w 1 w 1,w 2 w 2,w 3 w 3,m 3 m 3 w3 : m 3 m 3,w 3 w 3 23 Kennes et al. (2014) present a similar definition when introducing their isolated preference relation. 24 The adjustment step in the PDAA adjusts the assignments of agents who are unmatched in period 2. Allowing all agents to adjust their period-2 assignments may lead to an unstable matching (see Example 4). Allowing agents who are unmatched in period 1 to adjust their period-1 assignment, holding their period-2 assignment fixed, may also result in an unstable matching (see Example A.3 in the online appendix). 21

22 Each agent s preference satisfies (SIC) and (RDS). 25 [Table 1 about here.] Table 1 summarizes the PDAA s operation. The first step, coinciding with the PDA, terminates in three rounds. In round 1, m 1 proposes a two-period partnership to w 2 and is rejected. The proposals of m 2 and m 3 are similarly rejected. By the third round, each man s proposal is accepted. The resulting interim matching is ex ante stable. Since m 3 and w 3 are matched in period 2, their final matching is set. The others go on to the PDAA s second step. Their conditional preferences at µ 1 1 ( ) are: w 1 m 1 : w 2,m 1 w 2 m 2 : w 1,m 2 m 1 w 1 : m 2,w 1 m 2 w 2 : m 1,w 2 Conditional on µ 1 1 ( ), m 1 and w 2 wish to match together for period 2. Similarly, m 2 and w 1 wish to match together. Of course, the deferred acceptance algorithm leads to this outcome. The final assignment is this economy s only dynamically stable matching. Remark 2. The PDAA algorithm and Theorem 1 generalize to a T-period setting. We present this extension in the online appendix. 5 Multi-period Matching Mechanisms Many market-design applications have a multi-period structure and several dynamic generalizations of the deferred acceptance algorithm have been proposed. Two types of mechanisms 25 The preferences of m 1 and m 2 do not satisfy (SA). In this example, m 1 does not consider w 1 to be an acceptable period-2 match. In fact, he prefers to remain unmatched in period 2. 22

23 are common. The first involves implementing a sequence of stable one-period assignments; the second works backwards starting with the final period assignment. Below we examine these alternatives with dynamic stability and the PDAA as benchmarks for comparison. We conclude this section by considering some strategic aspects of multi-period, two-sided matching mechanisms. 5.1 Repeating Stable One-Period Assignments The simplest multi-period version of the deferred acceptance algorithm implements successive, one-period assignments derived with the original procedure. Employing such a mechanism in our setting requires care. Since preferences are defined over partnership plans, there need not exist a stand-alone, single-period preference that can be used to identify a stable one-period matching. One-period preferences need to be inferred by the mechanism. A natural approach involves identifying an agent s ranking of one-period assignments with his preference for persistent plans. Given (any) i, define agent i s ex ante spot ranking induced by i as jp i k jj i kk. 26 P i is a time-independent ranking of agent i s potential partners. It can be used to define a matching using Gale and Shapley s (1962) algorithm. Algorithm 3 (EDA). The (man-proposing) ex ante deferred acceptance procedure assigns in each period the one-period matching identified by the (man-proposing) deferred acceptance algorithm where each agent makes/accepts proposals according to P i. 26 Kennes et al. (2014) present a similar definition when introducing their isolated preference relation. If i already reflects a spot ranking P i, then P i = P i. 23

24 Lemma 2. If all preferences satisfy (SA), then the EDA and PDAA assignments coincide. Thus, by Theorem 1, when preferences satisfy (SIC) and (SA), the EDA assignment is dynamically stable. A practical advantage of the EDA is that it draws on less information than the PDAA. Beyond P i, an agent need not know or communicate i in its entirety. A natural extension of the EDA procedure captures the intuition of successive spot markets by repeating Gale and Shapley s deferred acceptance procedure conditional on the period-1 assignment to determine (or adjust) the period-2 assignment. Several authors investigate variants of this approach (Damiano and Lam, 2005; Kurino, 2009; Dur, 2012; Pereyra, 2013). The following operationalization of this idea is a specialization of a mechanism proposed by Kennes et al. (2014, 2018) for the assignment of children to daycares in Denmark. Algorithm 4 (SDA). The (man-proposing) spot-market deferred acceptance procedure defines the matching µ = ( µ 1, µ 2 ) on a period-by-period basis as follows: Period 1. Define µ 1 as the one-period matching identified by the (man-proposing) deferred acceptance algorithm where each agent i makes/accepts proposals according to his/her ex ante spot ranking, P i. Period 2. Define agent i s conditional spot ranking at j as kp j i l (j,k) i (j,l). Set µ 2 to be the one-period matching identified by the (man-proposing) deferred acceptance algorithm where each agent i makes/accepts proposals according to his/her conditional spot ranking at µ 1 (i), P µ 1(i) i. When preferences reflect a spot ranking, the SDA and EDA matchings coincide. More 24

25 generally, the SDA procedure ignores any perceived complementarities between successive assignments. Thus, it may fail to arrive at a dynamically stable matching, as illustrated by the following example. Example 4. Consider the following market with three men and three women. Each agent s preference exhibits inertia: m1 : w 2 w 2,w 1 w 2,w 1 w 1,... m2 : w 1 w 1,w 3 w 3,w 3 w 1,... m3 : w 1 w 1,w 2 w 1,w 2 w 2,... w1 : m 1 m 1,m 2 m 2,m 3 m 3,m 1 m 2,m 1 m 3,... w2 : m 3 m 3,m 1 m 1,m 3 m 1,... w3 : m 2 m 2,... The SDA matching is boldfaced in the agents preference lists. This matching is neither ex ante nor dynamically stable. 27 For example, it can be period-1 blocked by m 1 and w 2. This economy s only dynamically stable assignment, which is identified by the PDAA, is underlined in the agents preference list. This case also illustrates why the PDAA procedure does not allow all agents to re-match conditional on their period-1 assignment. In this example, that would lead to the unstable SDA matching. 27 The SDA procedure is a specialization of the DA-IP mechanism proposed by Kennes et al. (2014) to assign children to daycares. To nest Example 4 in their framework, call men children and women daycares with unit capacity. All preferences satisfy their assumptions. If {P w1,p w2,p w3 } is the initial priority structure, the SDA matching is stable according to their definition (Kennes et al., 2014, Definition 8). Hence, our definitions of ex ante and dynamic stability are distinct from, and not weaker than, their proposal. 25

26 5.2 Backward Induction and the NRMP All of the mechanisms considered above define matchings chronologically. First, assignments for period 1, and possibly period 2, are specified. And then, period 2 adjustments are made. This operation contrasts with the backward induction reasoning familiar to dynamic problems and encountered in some real-world matching applications. As an example, consider the NRMP s Main Residency Match R. 28 Four program types participate in the matching process, leading to a specific multi-period structure (Table 2). Students can enter Categorical and Primary programs immediately after medical school. These programs lead to certification in their specialty, usually after 3 6 years of training. Preliminary programs provide one or two years of training, are open to students immediately after medical school, but do not lead to certification. Instead, they are prerequisites for Advanced programs, which students enter subsequently for an additional 3 5 years of training. Physician positions are advanced positions that start in the current year but are available only to students who have completed graduate medical education. [Table 2 about here.] There exists a sequential complementarity between Preliminary and Advanced training. The NRMP algorithm addresses this complementarity anti-chronologically. For each Advanced program that a student ranks, he may also submit a supplemental ranking of Preliminary programs. (Submitting supplemental lists is not required.) If the algorithm matches an applicant to an Advanced program, it then tries to match him to a Preliminary 28 Main Residency Match is a registered trademark of the National Resident Matching Program. 26

27 program from the associated supplemental ranking. Each of these steps is carried out using a variation of the (one-period) deferred acceptance algorithm (Roth, 1996; Roth and Peranson, 1999). Successfully matching to Preliminary and/or Advanced program is not assured. This procedure may pose problems when some particular programs are sequential complements. For example, consider a student applying to advanced programs a 2 and b 2 in cities a and b. Each city also hosts a Preliminary program, say a 1 and b 1. For concreteness, suppose the student s (true) preference is a 1 a 2 i b 1 b 2 i b 1 a 2 i a 1 b 2 where the period 1 assignment corresponds to preliminary training and the period 2 assignment corresponds to advanced training. Within the NRMP, the student can submit supplemental rankings accompanying each advanced program. For example, a plausible submission given his preferences may be (1) a 2 [a 1 ˆ i b 1 ] ˆ i b 2 [b 1 ˆ i a 1 ]. Since a 1 a 2 i b 1 a 2, [a 1 ˆ i b 1 ] is the supplemental list accompanying advanced program a 2. Suppose the algorithm assigns the student to his most preferred advanced program, a 2, and then to his second-choice preliminary program, b 1. The b 1 a 2 outcome may not be dynamic stable. The student has an incentive to transfer at the preliminary program s conclusion since b 1 b 2 i b 1 a Though a transfer benefits student i, his initial assignment to a 2 may have excluded another student from matching with that program, thus masking a potential welfare loss in aggregate. The sequencing among Preliminary and Advanced programs is but one of many practical 29 Students are the proposers in the NRMP algorithm. Therefore, program b 2 may find student i acceptable since it did not reject him during the initial NRMP match if (1) was the student s report. 27

28 challenges that the NRMP matching process must address. Others include couples wishing to match together, programs requiring even or odd numbers of residents, and the transfer of unfilled positions among programs within the same hospital. Theoretically, any of these features can compromise a matching s stability (Roth, 1996). Thus far, however, the algorithm appears to have successfully navigated the portfolio of complications. 5.3 Strategic Issues The incentive properties of matching mechanisms are of great practical interest. A matching procedure is called strategyproof if it is a dominant strategy for each agent to truthfully reveal his preferences to the mechanism. Otherwise, the mechanism is manipulable. In the one-period case, there does not exist a strategyproof matching mechanism that always yields a stable outcome (Roth, 1982). It is a dominant strategy for each man to be truthful in the man-proposing deferred acceptance algorithm, but the women often have a profitable manipulation (Dubins and Freedman, 1981; Gale and Sotomayor, 1985). The possibilities for manipulation increase in a multi-period market. Consider again the SDA procedure and Example 4. Had w 1 shunned the initial proposal of m 1, her favorite partner, the SDA procedure would have matched her with m 2 in both periods, which she prefers. Curiously, in some cases a proposing agent may also wish to strategize. In the NRMP example above, if program b 2 considered student i acceptable, he could have improved his match by claiming b 2 [b 1 ˆ i a 1 ] ˆ i a 2 [a 1 ˆ i b 1 ] instead of (1). 30 More positive conclusions 30 Arguably, the student must be strategic since the algorithm s preference elicitation method does not allow him to communicate his full (true) preference. 28

29 apply to the mechanisms we have proposed. Theorem 2. (a) If all preferences satisfy (SIC) and (SA), then the EDA procedure is strategyproof for the proposing side. (b) If all preferences satisfy (SIC) and (RDS), then the PDAA procedure is strategyproof for the proposing side Limited Information and Learning Most agents enter into multi-period relationships with limited knowledge about future preferences. Learning justifies assignment revisions and we can investigate this theme in our framework. Consider the following amendment to our model. Each agent i has a preference i over partnership plans, but in period 1 only knows the following: (L1) His preferences have inertia. (L2) His ex ante spot ranking is P i. Thus, if jp i kp i l then agent i knows that jj i kk i ll, but the relative ranking of kj is unknown. As time passes, each agent i learns more. 31 The PDAA requires a minor modification in this theorem s particular context. If an agent s preference report does not satisfy (SIC) and (RDS), the mechanism leaves him/her unassigned in both periods. Otherwise, the PDAA operates as defined. This modification ensures that each agent s preference announcement remains in the appropriate domain. 29

30 (L3) If in period 1 agent i is assigned to k, in period 2 he learns his preferences for plans of the form kl, for all l. For instance, agent i may discover that jj i kj i kk i ll i kl. Together, (L1) (L3) outline a basic model of path-dependent learning. For simplicity, we do not introduce further beliefs or priors. To study this market, we assume that its final outcome is the result of a matching mechanism meeting certain properties. Though the term matching mechanism has the connotation of a centralized process, here we mean any regularized interaction leading to a matching in an economy. An economy is a tuple e = (M,W, ) encompassing the agents and their preferences. The function A( ) is a (deterministic) matching mechanism if it assigns a matching to each economy. Thus, A(e) = (A 1 (e),a 2 (e)) is a matching for economy e. The algorithms defined above are all examples of matching mechanisms. Our restricted information structure draws attention to mechanisms with two pertinent properties. First, the mechanism should not leverage information that agents themselves do not know. In period 1, agent i knows only P i. A matching mechanism A( ) is nonprophetic if for all economies e = (M,W, ) and e = (M,W, ) such that P i = P i for each i, A 1 (e) = A 1 (e ). The EDA and SDA procedures are both non-prophetic mechanisms. Second, the mechanism should lead to a dynamically stable matching. Dynamic stability is an appealing benchmark in markets with limited information since it subsumes an appealing no-regret property when (L1) (L3) are satisfied. Whereas an agent cannot turn back the clock to period-1 block once on period 2 s threshold, by (L3) he can assess an assignment s 30

31 continuation relative to persistent alternatives. If m and w discover that ww m µ(m) and mm w µ(w), both will regret not pairing at an earlier opportunity. A dynamically stable matching insulates agents from such regret. If I is the domain of preferences that exhibit inertia, then we call a matching mechanism dynamically stable on (preference domain) I if it identifies a dynamically stable matching whenever all preferences exhibit inertia. Many mechanisms are both non-prophetic and dynamically stable on I. 32 Though spartan, the preceding model lets us answer two related questions. First, what properties do dynamically stable matchings enjoy in this market? And second, when is rematching after period 1, and drawing on new information, welfare-enhancing? Two theorems, which we interpret below, together answer these questions. Theorem 3. Let A be a non-prophetic matching mechanism that is dynamically stable on I. Suppose that in economy e agents preferences exhibit inertia and A(e) = (µ 1,µ 2 ). Then µ = (µ 1,µ 1 ) is (also) a dynamically stable matching in economy e. Theorem 4. Assume each agent s preference exhibits inertia and µ = (µ 1,µ 2 ) is dynamically stable. Let µ = (µ 1,µ 1 ). If µ(i) i µ(i), the µ(j) j µ(j) for j = µ 2 (i). Theorem 3 begins with the dynamically stable matching generated by a non-prophetic mechanism and it shows that the matching s first-period assignment can be prolonged with- 32 The EDA is an example. As another example, consider a mechanism that selects the volatile underlined, but dynamically stable, matching in the following economy: m1 : w 1 w 1,w 1 w 2,w 2 w 2,w 2 w 1,m 1 m 1 m2 : w 2 w 2,w 2 w 1,w 1 w 1,w 1 w 2,m 2 m 2 w1 : m 2 m 2,m 2 m 1,m 1 m 2,m 1 m 1,w 1 w 1 w2 : m 1 m 1,m 1 m 2,m 2 m 1,m 2 m 2,w 2 w 2 In all other economies, the mechanism assigns the EDA matching. 31