Time Horizons, Lattice Structures, and Welfare in
Multi-period Matching Markets∗
Sangram V. Kadam†
Maciej H. Kotowski‡
March 14, 2017
Abstract
We analyze a T -period, two-sided matching economy without monetary transfers.
Under natural restrictions on agents’ preferences, which accommodate switching costs,
status-quo bias, and other forms of inter-temporal complementarity, dynamically stable
matchings exist. We propose an ordering of the set of dynamically stable matchings
ensuring this set forms a lattice. We investigate the robustness of dynamically stable
matchings with respect to the market’s time horizon. We relate our analysis to marketdesign applications, including student-school assignment and labor markets.
Keywords: Two-sided Matching, Dynamic Matching, Stable Matching, Lattice, Marketdesign
JEL: C78, D47, C71
∗
We thank audiences at Match-Up 2015 (University of Glasgow) and the 2015 Canadian Economic Theory
Conference (University of Western Ontario) for helpful comments. Sangram Kadam gratefully acknowledges
the support of the Danielian Travel & Research Grant. Portions of this study were completed while Maciej
H. Kotowski was visiting the Stanford University Economics Department. He thanks Al Roth and the
department for their generous hospitality.
†
Cornerstone Research, 1919 Pennsylvania Avenue NW, Washington DC 20006.
E-mail:
<[email protected]>
‡
(Corresponding Author) John F. Kennedy School of Government, Harvard University, 79 JFK Street,
Cambridge MA 02138. E-mail: <[email protected]>
1
The passage of time is an inescapable dimension of most economic and social interactions. Employees pursue decades-long careers. Students spend years learning at a succession
of institutions. And wedding anniversaries are celebrated—or forgotten. It is therefore surprising that time is largely absent from typical studies of two-sided matching markets, as
originally formulated by Gale and Shapley (1962). In such a market, agents are partitioned
into two groups—men and women, firms and workers, schools and students—and seek to
match together to realize benefits. Each agent has a preference over potential partners and
these preferences need not be aligned. That stable matchings, where no agent or no pair can
independently pursue a mutually-preferable arrangement, are possible is both surprising and
profound.
Accounting for the market’s time horizon forces one to confront several real-world complications that do not arise when interactions are ephemeral. First, with the passage of time,
agents frequently change their partners.1 The possibility of change introduces a degree of
complexity not encountered in the single-period, one-shot case. The order of interactions
matters and inter-temporal tradeoffs assume a prominent role. Second, decision making over
a time horizon is difficult. Complexity and psychology promote habit formation and pathdependence into significant behavioral features. Unlike theory, practice often conflicts with
Samuelson’s (1937) influential discounted-utility framework. Finally, commitment is often
limited and dynamic incentives matter.
This paper examines the positive theory of T -period, two-sided matching markets where
agents’ preferences exhibit non-trivial, inter-temporal dependencies. In our model, agents
are limited in their commitment ability and must be continually incentivized to continue
with a proposed assignment plan. Two questions guide our investigation:
1. What are the welfare characteristics of “stable” market outcomes?
2. How sensitive are stable outcomes to the market’s time horizon?
These questions are natural counterparts, both theoretically and in applications. Their
answers directly inform our understanding of likely outcomes in dynamic markets and can
1
Change occurs in all standard applications of two-sided matching theory, including marriage, employment, and education. Approximately 54 percent of couples married between 1975 and 1979 celebrated their
25th anniversary (Kreider and Ellis, 2011, Table 4) and around a quarter of those born from 1940–49 were
married two or more times by age 55 (Kreider and Ellis, 2011, Table 3). Similarly, longitudinal evidence
from the U.S. Bureau of Labor Statistics (2012) notes that the average person born between 1957 and 1964
held 11.3 jobs between the ages of 18 and 46. Finally, there is considerable “churn” among students enrolled
in schools. Approximately ten percent of students changed schools in Massachusetts in 2008–09 (O’Donnell
and Gazos, 2010, Table 1). During the same time period, the churn rate in Boston public schools was 25.3
percent (O’Donnell and Gazos, 2010, Table 3).
2
help guide the design of matching mechanisms when the market’s time horizon or rematching
frequency are design variables. Our study constitutes a preliminary step in this direction.
To investigate the above questions, we extend the model of Kadam and Kotowski (2016),
who define and analyze “dynamically stable” matchings in a multi-period, one-to-one matching economy. Kadam and Kotowski (2016) focus primarily on a two-period model and investigate conditions that ensure the existence of a stable matching, according to their definition.2 They introduce several restrictions on preferences—“inertia,” “sequential improvement
complementarity,” and others—that ensure an appropriately-generalized version of Gale and
Shapley’s (1962) deferred acceptance algorithm identifies a stable outcome. Among other
conclusions, they show that a successive repetition of the one-period deferred acceptance
procedure often leads to unstable outcomes. They also consider versions of their model with
imperfect information and financial transfers.
We adopt the same solution concept and baseline model as Kadam and Kotowski (2016),
but our analysis differs in terms of thematic focus and technical detail. Foremost, given our
interest in the market’s time horizon, our analysis is set in a T -period setting. In developing this version of the model, we propose a new class of non-time separable preferences
based on the relative persistence of assignments. This class is analytically tractable and
compatible with important behavioral characteristics. Our model allows for status quo bias
and switching costs, two important features of inter-temporal decision making (Samuelson
and Zeckhauser, 1988). Surprisingly, and despite their connotation, these wrinkles need not
reinforce an assignment’s immutability. Stability may necessitate frequent changes among
partners, a period of sacrifice involving a match to an undesirable partner, or even involuntary singlehood. In a labor market application, we often call such outcomes job-hopping,3
internships, and unemployment, respectively. All are everyday phenomena.
To address the question of welfare, we investigate the lattice structure of the set of
dynamically stable matchings. A lattice is a partially-ordered set where any two elements
have a unique supremum and a unique infimum (Birkhoff, 1940). In a static setting, the set of
stable matchings forms a lattice when ordered by the “common preference” of agents on one
side of the market. This ordering, which we define formally below, is an aggregation of agents’
individual preferences. Thus, any emergent systematic structure is naturally tied to welfare
2
In an appendix to their paper, Kadam and Kotowski (2016) explain how some of their proposed algorithms can be extended to a T -period setting. Our analysis does not draw on these generalizations. Given
the class of preferences assumed in our study, a simpler argument is sufficient.
3
“Job hopping” refers to the rapid movement of workers between firms. Fallick et al. (2006) document
this phenomenon among workers in Silicon Valley.
3
and distributional considerations. The lattice of stable matchings has been investigated in
and array of static models (Knuth, 1976; Blair, 1984, 1988; Roth, 1984, 1985b; Sotomayor,
1999; Alkan, 2001, 2002). However, the nature of this structure in a dynamic matching
market (if any) has not been explored.
Our analysis yields subtle conclusions and qualifications. First, we identify cases where
dynamically stable matchings are not Pareto optimal. Moreover, we show that man- or
woman-optimal stable matchings do not always exist.4 Man- and woman-optimal matchings
exist in a wide range of one-period models (Gale and Shapley, 1962; Roth, 1984, 1985b). As a
counterpoint to these intriguing though negative observations, we introduce a mild restriction
on agents’ preferences and we propose a new ordering of the stable set ensuring it forms a
lattice. The new ordering is also derived from agents’ preferences, thereby accommodating
comparative welfare analysis.
As our model emphasizes the multi-period nature of interactions, it is also important to
understand how the stable set changes with the market’s time horizon, the parameter T in
our notation. This is our second focus and it is significant for two reasons. First, the selection
and interpretation of T is often at the analyst’s discretion. Thus, a hint of robustness along
this dimension is desirable. We show that intuitive projections and embeddings of stable
outcomes are possible as T changes. Such conclusions are not immediate since, for instance,
adjusting a market’s time horizon changes the number of blocking opportunities.
Second, in normative applications, the market’s time horizon and rematching frequency
are design variables open to refinement and fine-tuning. For example, the assignment of
students to high schools has emerged as a celebrated application of two-sided matching
theory (Abdulkadiroğlu and Sönmez, 2003). Traditional models of this problem have been
static, one-period affairs. But, students attend high school for (typically) four years and,
therefore, this too is a multi-period assignment problem. A very coarse conflation of the
successive school years yields the typical one-shot assignment model; however, an unbundling
of successive years, semesters, or quarters yields a rich family of multi-period alternatives.
The introduction of coordinated rematching opportunities may improve match quality or
alleviate capacity constraints.
Stepping back from the design of a particular centralized assignment algorithm, as common in the literature on two-sided matching, one can also use a version of our model to evaluate the performance of multi-period markets under different organizing principles. Again,
4
A stable matching is man-optimal if each man prefers his assignment in that matching to his assignment
in all other stable matchings. A woman-optimal stable matching is defined analogously.
4
education offers a germane example. Some school systems require eight years of elementary
school plus four years of high school. Other systems bundle grades in separate primary,
middle, and secondary schools. Clearly, these schemes differ on rematching opportunities
and assignment length. Selecting among them, or transitioning from one scheme to another,
is typically a contentious policy process. More opportunities for change may lead to better matchings (abstracting from dynamic considerations); however, the associated switching
costs may reverse many of these gains.5 A multi-period matching model, like ours, can be
helpful in comparing and evaluating such alternatives more rigorously. While some applications of the theory we develop will be set in more complex settings, perhaps incorporating many-to-many assignments or financial transfers, we believe that the classic one-to-one
matching model investigated here is sufficiently rich to offer valuable insights concerning the
economic tradeoffs salient in multi-period, two-sided markets more generally.
This paper is organized as follows. In Section 1 we provide a brief summary of the
related literature. Section 2 introduces our model. Section 3 examines the stable set’s lattice
structure. Section 4 investigates the relationship between stability and the market’s time
horizon. Section 5 concludes. All proofs are relegated to the appendix.
1
Related Literature
Dynamic models occupy a nook within the expansive matching markets literature. Some
papers, of which Kurino (2014) and Akbarpour et al. (2016) are recent examples, consider
one-sided problems. In contrast, we consider a two-sided market where agents on both sides
of the market, men and women in our nomenclature, have preferences over their partners’
identity. Specific applications have motivated previous studies in this vein. Kennes et al.
(2014a,b) study the assignment of children to Danish daycares. Dur (2012) and Pereyra
(2013) consider school-assignment problems. Though these models include features that are
absent from our analysis, our model is not a special case of any of them. For example,
school-assignment applications naturally introduce an asymmetry between the market’s two
sides. Our model presumes symmetry.
As noted in the introduction, at the center of our analysis is a set of matchings that
are “dynamically stable.” This solution concept is studied by Kadam and Kotowski (2016),
principally in a two-period environment. Kadam and Kotowski (2016) provide an exten5
Recent empirical evidence suggests that student performance is negatively impacted by the transition
from elementary to middle school (Rockoff and Lockwood, 2010; Schwerdt and West, 2013).
5
sive discussion concerning the motivation supporting this solution concept, which we only
briefly touch on in our discussion. They propose generalizations of the deferred acceptance
algorithm that can identify stable outcomes in a wide range of cases, including situations
where preferences feature non-trivial inter-temporal dependencies. They also study several
extensions of their model, including incomplete information, financial transfers, and stronger
definitions of stability. We do not tackle any of these questions below. Rather, the present
paper provides an extension of their analysis focusing on the structural characteristics of the
stable set with respect to the market’s time horizon and on its lattice structure.
Dynamic stability generalizes Gale and Shapley’s (1962) original stability concept to a
multi-period environment. It does so in a manner analogous to Gale’s (1978) “sequential
core,” which was originally proposed to examine a multi-period exchange economy with
limited trust.6 However, it is but one of many plausible generalizations. Beyond the studies
cited above, alternative generalizations of stability are proposed by Damiano and Lam (2005)
and Kurino (2009), who in different ways emphasize the credibility of blocking coalitions.
Unlike Damiano and Lam (2005) and Kurino (2009), our analysis is not restricted to timeseparable preferences. Kotowski (2015) examines an alternative solution concept directly
applicable to our problem. His proposal is neither weaker nor stronger than dynamic stability.
Independently and contemporaneously, Doval (2015) has proposed a solution concept
also called “dynamic stability.” Her definition, which differs from ours, extends and modifies
prior proposals by Corbae et al. (2003) and Kurino (2009) and treats a (dynamic) matching
as a complete contingent plan for the economy as a whole. She studies an economy with
one-time irrevocable assignments that may happen in different periods. Similarly, Baccara
et al. (2016) examine an infinite-horizon, two-sided matching economy where agents arrive
stochastically over time, match one-time, and depart. They focus on the tradeoff between
market thickness (which improves match quality) and waiting costs and they identify an
welfare-maximizing mechanism in their environment. In our model, in contrast, the pool of
agents is fixed and assignments may change over time.7
A conceptual link exists between a dynamic, one-to-one matching market and a static,
many-to-many matching market. Every agent is matched to many partners, though in succession. Our analysis is not a special case of the most general treatments of many-to-many
6
Unlike the sequential core, our definition of dynamic stability is confined to pair-wise blocking coalitions,
like Gale and Shapley’s (1962) original stability notion.
7
Though we do not undertake the exercise here, economies with irrevocable assignments, arrivals, or
departures, can be examined within our framework by introducing a preference restriction where agents
“prefer to be unmatched” in periods when they are absent from the market (see Kadam and Kotowski, 2016).
6
matching markets lodged in matching with contracts framework (Hatfield and Milgrom,
2005). Notably, our model does not satisfy the substitutability condition stressed by Hatfield and Kominers (2012).8 Furthermore, the cross-period linkages embedded within our
definition of blocking lead to a stability concept that differs from its static counterparts in
the many-to-many matching literature. We comment on this distinction again in Section 2.
While dynamics have not received much attention in the literature on two-sided matching
following Gale and Shapley (1962), they are a central pillar of alternative examinations
of two-sided markets. For example, the fact that workers shuffle between jobs has been
formalized in the literature on matching within the search-theoretic paradigm. In the work
of Mortensen and Pissarides (1994), to cite but one well-known example, workers move
between employment and unemployment and jobs are created and destroyed. Change and
churn are features of an economy’s steady state. We identify comparably volatile stable
outcomes in our setting and we hope our analysis serves as a useful step toward better
integrating insights from both “matching” literatures.9 The model of Corbae et al. (2003)
offers another connection and focuses on applications in monetary economics.
2
Model
Let M and W be finite, disjoint sets of agents—labelled men and women, respectively—
who interact over T periods. Most of our results continue to apply if T = ∞, though for
simplicity we assume that T is finite. In every period, each man (woman) can be matched
with one woman (man) or remain single. Following convention, an unmatched agent is
said to be matched to himself/herself. Thus, the set of potential partners for m ∈ M is
Wm := W ∪ {m}. Mw := M ∪ {w} is the set of potential partners for w ∈ W . We use
m’s and w’s to denote specific men and women when a distinction is helpful. Otherwise,
i, j, k, l are generic agents. For notational brevity, we define some concepts only from the
perspective of a typical man. However, our model is symmetric and all definitions apply to
women with obvious changes to notation.
Over a lifetime each agent encounters a sequence of partners, called a partnership plan.
We denote a plan by
x = (x1 , x2 , . . . , xT )
8
Substitutability was introduced into the matching literature by Kelso and Crawford (1982). See Roth
(1984) for its use in a many-to-many matching economy.
9
Crawford and Knoer (1981, p. 437–8) discuss the link between these literatures.
7
where xt is the assigned partner in period t. We will write out a plan as a full T -tuple when
we wish to emphasize its particular components. Otherwise, when confusion is unlikely, we
omit the commas and parentheses from our notation and we write
x = x1 x2 · · · xT
instead. With this more compact notation, we can express truncations or subsets of x as
follows:
x≤t = x1 x2 · · · xt
x≥t = xt xt+1 · · · xT
x[t,t′ ] = xt · · · xt′
x<t = x1 x2 · · · xt−1
x>t = xt+1 xt+2 · · · xT
x(t,t′ ) = xt+1 · · · xt′ −1
We sometimes combine the above expressions when writing out a plan, i.e. x = (x<t , x[t,t′ ] , x>t′ ).
By convention, x<1 = ∅ and x>T = ∅. A constant plan is written as ī = ii · · · i.
Each agent i has a strict, rational preference ≻i defined over partnership plans. Thus,
≻m for m ∈ M ranks all possible plans in WmT . Symmetrically, ≻w ranks all possible plans
in MwT . If agent i prefers plan x to plan y, we write x ≻i y. If x ≻i y or x = y, then x i y.
Adopting standard terminology to describe market outcomes, we call the function µt : M ∪
W → M ∪ W a one-period matching (for period t) if (i) for all m ∈ M, µt (m) ∈ Wm ; (ii)
for all w ∈ W , µt (w) ∈ Mw ; and, (iii) for all i, µt (i) = j =⇒ µt (j) = i. A matching
µ : M ∪ W → (M ∪ W )T is a sequence of one-period matchings, defining a partnership plan
for each agent. Thus, µ(i) = µ1 (i) · · · µT (i) is i’s plan given by the matching µ. The notation
µ≤t , µ>t , etc., follows the conventions introduced above.
2.1
Stable Matchings
When T = 1, the economy described above reduces to Gale and Shapley’s (1962) two-sided
matching market. Gale and Shapley (1962) introduce the idea of “stability” in order to
identify plausible outcomes in their model. A matching is stable in a one-period economy if
each agent’s assignment is individually rational and if no pair of agents prefer to be matched
together in lieu of their assigned partners.
As noted above, there exist multiple generalizations of Gale and Shapley’s (1962) stability
definition to a multi-period setting. In this paper we adopt the definition proposed by Kadam
and Kotowski (2016). According to this definition, a multi-period matching will be stable if
it cannot be “blocked” in each period by either an agent or a pair. As in classic cooperative
models, blocking will involve agents pursuing their best option independently of the others’
8
behavior. Several preliminary definitions are required to formalize this idea in our model.
As a starting point, we first define individual and pairwise blocking. In the one-period
case, an assignment µ is individually rational for agent i if agent i prefers his assignment
µ(i) to his unilateral outside option, which entails remaining unmatched. We can generalize
this intuition to blocking in an arbitrary period t as follows.
Definition 1. Agent i can period-t block the matching µ if (µ<t (i), ī≥t ) ≻i µ(i).
At period t, (µ<t (i), ī≥t ) is the best outcome that agent i can independently guarantee
himself. If t = 1, for example, agent i shuns the market entirely without matching with
anyone. Instead, if t = T , agent i leaves the market only in the final period. If agent i
cannot period-t block µ for all t, then his assignment is dynamically individually rational.
Blocking by a pair extends the preceding intuition by allowing a pairs of agents to pursue
their best alternative together. A pair can block a matching in period t if they can form a
(continuation) partnership plan among themselves for periods t, t+1, t+2, . . ., that they both
prefer given the elapsed history. Such a plan involves a sequence of arrangements among the
blocking pair and is independent of others’ behavior.
{m,w}
Definition 2. The function µt
: {m, w} → {m, w} is a one-period matching (for period
{m,w}
{m,w}
{m,w}
t) among {m, w} if (i) µt
(m) = w and µt
(w) = m; or, (ii) µt
(m) = m and
{m,w}
µt
(w) = w.
Definition 3. Agents m and w can period-t block the matching µ if there exists a (contin{m,w}
{m,w}
{m,w}
{m,w}
uation) matching among {m, w}, µ≥t
= µt
· · · µT
, such that (µ<t (i), µ≥t (i)) ≻i
µ(i) for all i ∈ {m, w}.
A matching is dynamically stable if it cannot be period-t blocked by any agent or by any
pair for all t. Dynamic stability is a succinct multi-period generalization of Gale and Shapley’s original stability definition and reduces accordingly when T = 1. Kadam and Kotowski
(2016) discuss the solution concept at length and provide additional context and motivation
for its formulation. Here, we comment on two of its features. First, our stability definition
is limited to pairwise blocking actions. Other authors studying dynamic matching models,
including Damiano and Lam (2005), Kurino (2009), and Doval (2015),10 among others, have
favored coalition-based definitions. It is simple to extend Definition 3 to arbitrary blocking
coalitions thereby leading to a version of our market’s core. Conditions ensuring the core’s
10
Contemporaneously and independently, Doval (2015) also introduces a stability concept called “dynamic
stability.” Her solution concept differs from ours.
9
non-emptiness are identified by Kadam and Kotowski (2016) for the special case of a twoperiod economy. These conditions are more restrictive than those employed in our analysis
to follow.
Second, dynamic stability imposes chronological restrictions on admissible blocking actions. Specifically, a blocking action initiated in period t carries consequences for all t′ ≥ t.
Thus, period-t blocking is not a mere collection of ephemeral and independent one-period
objections to µ. Hence, it differs from stability definitions employed in static, many-to-many
matching models. To appreciate this difference, note that m and w cannot block a matching
with a one-period partnership in period t and then return to their planned assignment as if
nothing ever happened.11 Contemporaneously with a temporary deviation, for instance, the
wider market might evolve in ways unexpected, or possibly unknown, to m and w given their
period-t action. The agents’ erstwhile partners may identify new partners and, since preferences generally are not time-separable, a cascade of further re-matchings may ensue. These
finer details of the market’s operation are left unspecified by our lean cooperative model.
Via the definition of period-t blocking, dynamic stability posits that agents assess departures
from a plan conjecturing unfavorable developments in the wider market. Acknowledging the
situation’s complexity, particularly when T is large, m and w anticipate that they will be
resigned to matching opportunities only among themselves—as the worst case—should they
as a pair object to a proposed matching.12 Though a conservative benchmark, the robust
decision criterion implicit in the definition of dynamic stability contributes to its adaptability
and portability to many applications.
Several important implications follow from the preceding observations. As mentioned,
agents cannot depart from a planned assignment without (necessarily) incurring consequences in the future. Allowing “one-shot” deviations or unencumbered early period deviations will necessarily lead to a smaller stable set and, in many cases, preclude the existence of a stable matching altogether.13 From a market design point-of-view, this modeling
features is also not particularly demanding. Many centralized markets, with the National
Resident Matching Program (NRMP) being but one example, penalize participants who
pursue arrangements outside of the centralized market. The NRMP market has a particu11
Damiano and Lam (2005) define strict self-sustaining stability, which allows agents to return to an
original plan following a temporary deviation. They acknowledge that it presumes a non-deviating agent
accepts a deviator’s return, which may not always be true in practice.
12
While our model is not specified in characteristic function form, this blocking notion captures the flavor
of a coalition’s α-effectiveness (Aumann and Peleg, 1960; Aumann, 1961).
13
We refer the interested reader to Kadam and Kotowski (2016, Example 2) for a further elaboration of
these and related points concerning fleeting deviations in a multi-period economy.
10
lar multi-period structure due to the existence of primary and specialist programs, which
must be sequenced appropriately by some market participants (Kadam and Kotowski, 2016).
Sanctions may include prevention from participating in future instances of the market. Our
model approximates this benchmark.
2.2
Separability and the Persistent Closure
A dynamically stable matching may not exist if T ≥ 2.14 As common in the literature, we
will narrow our focus to a restricted preference domain that ensures stable matchings exist.
Kadam and Kotowski (2016) identify a family of sufficient conditions ensuring the existence
of a dynamically stable matching. Their analysis is built around a requirement that they
call “sequential improvement complementarity” (SIC).15 While this condition allows for a
general model where assignments may exhibit interesting cross-period complementarities,
it is difficult to draw sharp conclusions concerning the stable set without introducing additional structure. Therefore, in the analysis below we restrict attention to a special class
of preferences balancing analytic tractability and behavioral plausibility. Such preferences
merge a consistent ranking of potential partners and an attitude favoring persistence. This
latter element introduces complementaries across successive assignments and enhances our
analysis’ scope. We describe each component in turn. Thereafter, we explain how to combine
them together by defining the persistent closure of the set of separable preferences.
Separable Preferences
Suppose an agent holds a strict ranking of potential partners, abstracting from all dynamic
considerations. We call such a ranking a spot ranking and we denote it by Pi . If j ranks
above k, we write jPi k. If jPi k or j = k, then we write jRi k. This ranking is distinct from
an agent’s preferences, which in a multi-period economy are defined over entire sequences
of partners, but it will inform preferences in a natural way. We say that the preference ≻i
reflects the spot ranking Pi if for all x and t,
jPi k ⇐⇒ (x1 , . . . , xt−1 , j, xt+1 , . . . , xT ) ≻i (x1 , . . . , xt−1 , k, xt+1 , . . . , xT )
14
Consider a two-person economy where ≻m : wm, ww, mm and ≻w : mm, ww. This economy does not
have a dynamically stable matching (Kadam and Kotowski, 2016).
15
Formally, SIC alone is insufficient to ensure the existence of a stable matching. Kadam and Kotowski
(2016) combine SIC with supplementary technical conditions, singlehood aversion or revealed dominance
of singlehood. These additional conditions address special cases where an agent is unmatched in a single
period.
11
and we let Si be the set of preferences for agent i that reflect a spot ranking. When ≻i ∈ Si ,
agent i prefers the plan with the higher-ranked partner whenever two plans differ only in
their assignment for one period. This idea mirrors Roth’s (1985a) responsiveness condition,
but additionally requires the assignment ordering to be the same. Many distinct preferences
may reflect the same spot ranking.
Persistence
Though Si is an analytically convenient class of preferences, it precludes situations where
successive moments in time exhibit some complementarity. For example, if kPi j and ≻i ∈ Si ,
then agent i must believe that jkj ≻i jjj. However, status-quo bias or switching costs may
tilt i’s preference toward the jjj plan in lieu of plans incorporating many changes (Samuelson
and Zeckhauser, 1988). In practice, plans economizing on switching may be held in favor
even if the assignments are intrinsically less desirable in some time-independent sense. At
times, of course, such attitudes may be (mis-)interpreted as preference reversals.
To model preferences incorporating these features, we first introduce a vocabulary describing partnership plan variability. The plan x = x1 · · · xT is (maximally) persistent if
xt = xt′ for all t and t′ . Otherwise it is volatile. It is maximally volatile if xt 6= xt+1 for all
t. Finally, the following comparison of plans will prove convenient.
Definition 4. The plan x is more persistent than plan y, denoted as x D y, if for all t ≤ t′ ,
yt = · · · = yt′ =⇒ xt = · · · = xt′ .
D is a preorder of partnership plans. It is reflexive (x D x) and transitive (x D y, y D z =⇒
x D z), but not anti-symmetric. If x D y, y D x, but x 6= y, then x is equally persistent to
y and we write x ⊲⊳ y. x ⊲ y when x D y and y 4 x. Finally, x k y if x and y are not
D-comparable. To illustrate, iij ⊲ ijk, iij ⊲⊳ jjk, and iij k ijj.
A natural way to model a bias toward more persistent plans is to allow such outcomes
to rise in an agent’s preference ranking. For example, if agent m’s preference is
≻m : w2 w2 w2 , w2 w1 w1 , w1 w2 w2 , w1 w2 w1 , . . . ,
(1)
then he values persistence more than if his preference was
≻′m : w2 w1 w1 , w1 w2 w1 , w2 w2 w2 , w1 w2 w2 , . . . .
(2)
In both cases, agent m ranks the same four outcomes highly, but ≻m clearly favors w2 w2 w2
12
and w1 w2 w2 , which involve relatively more persistent assignments.
Definition 5. The preference ≻i values persistence more than ≻′i if
(a) x ≻′i y and x D y =⇒ x ≻i y; and,
(b) for all x and y such that x k y, x ≻′i y ⇐⇒ x ≻i y.
With reference to the examples stated in (1) and (2), ≻m values persistence more than ≻′m .
Let Υ(≻′i ) be the set of preferences for agent i that value persistence more than ≻′i . If
≻i ∈ Υ(≻′i ), then ≻i and ≻′i rank non-D-comparable plans in the same way. The same is
true for equally persistent plans.
Lemma 1. If ≻i ∈ Υ(≻′i ) and x ⊲⊳ y, then x ≻′i y ⇐⇒ x ≻i y.
The Persistent Closure
The preference class Si and the mapping Υ(·) are independent objects. However, together
they characterize a tractable and behaviorally interesting family of preferences that is useful
for analysis. We define the persistent closure of Si as
S¯i :=
[
Υ(≻′i ).
≻′i ∈Si
S̄i is the main class of preferences that we employ in our analysis. Intuitively, this class of
preferences is a combination of separable preferences and those that value persistence. Each
of these characteristics is motivated by easy to understand behavioral features, as explained
above. Moreover, every member of S¯i can be constructed with a simple two-step procedure.
First, take any ≻i ∈ Si as a starting point. Next modify this preference by favoring, i.e.
shuffling up in the ranking, relatively more persistent plans. Thus, each preference in S¯i
can be conceptually separated into a time-independent root and an operation capturing the
agent’s proclivity for persistence, susceptibility to status quo bias, or sensitivity to switching
costs. This separation will prove useful in some of the technical arguments to follow.16
When agents’ preferences belong to S¯i , stable outcomes may assume a variety of forms.
We illustrate some possibilities with the following example that we further discuss below.
16
When restricted to two periods, preferences in S̄i satisfy the “inertia” condition of Kadam and Kotowski
(2016). They do not offer a T -period extension of this definition.
13
Example 1. Suppose M = {m1 , m2 }, W = {w1 , w2 }, and T = 3. Table 1 summarizes each
agent’s preferences. Plans are listed in preferred order. It can be verified that ≻i ∈ S̄i for
all i. Table 2 lists all dynamically stable matchings in this economy. To read this table,
µ1 (m1 ) = w1 w1 w1 , and so on.
Table 1: Agent’s preferences in Example 1.
≻m1
w1 w1 w1
w1 m1 w1
w1 w2 w1
m1 m1 w1
w1 m1 m1
m1 w2 w1
m1 m1 m1
≻m2
w2 w2 w2
w2 m2 w2
w2 w1 w2
m2 m2 w2
w2 m2 m2
m2 w1 w2
m2 m2 m2
≻w1
m2 m2 m2
m1 m2 m2
m2 m2 m1
m1 m2 m1
w1 m2 m2
m2 m2 w1
w1 m2 m1
m1 m1 m1
w1 w1 w1
≻w2
m1 m1 m1
m2 m1 m1
m1 m1 m2
m2 m1 m2
w2 m1 m1
m1 m1 w2
w2 m1 m2
m2 m2 m2
w2 w2 w2
Table 2: All dynamically stable matchings in Example 1.
Matching
µ1
µ2
µ3
m1
w1 w1 w1
w1 w2 w1
m1 w2 w1
m2
w2 w2 w2
w2 w1 w2
m2 w1 w2
w1
m1 m1 m1
m1 m2 m1
w1 m2 m1
w2
m2 m2 m2
m2 m1 m2
w2 m1 m2
We make four observations motivated by Example 1. First, maximally-volatile plans can
be dynamically stable, even when ≻i ∈ S̄i for all i. In fact, some agents may prefer volatile
outcomes among all dynamically stable matchings. For instance, w1 and w2 prefer µ2 among
the stable set. Despite the apparent bias toward persistent outcomes, stable matchings are
often volatile.
Second, a stable matching may incorporate a period of sacrifice. In Example 1, µ2 and
µ3 temporarily assign both men to partners who they consider unacceptable absent any
possibilities of change.17 Such an assignment is stable because both men match with a
highly-ranked partner in the final period. Thus, with a long enough time horizon costly
17
That is, m1 m1 m1 ≻m1 w2 w2 w2 and m2 m2 m2 ≻m2 w1 w1 w1 .
14
actions can be adequately incentivized as part of a stable outcome. A myopic, per-period
definition of individual rationality would not capture such inter-temporal tradeoffs.
Third, stable outcomes may include periods of temporary singlehood. The matching µ3
in Example 1 has the surprising property that all agents are unmatched in period 1 but
matched in later periods. This outcome has no analogue when T = 1 and it cannot occur
when T = 2 and ≻i ∈ S¯i for all i (see Lemma A.3 in the Appendix). Thus, if an agent is
unmatched in some period in a dynamically stable matching, s/he may be matched in that
period in some other dynamically stable matching. This outcome would not be possible if
stable matchings were simply a sequence of “stable” one-period assignments.18
Fourth, µ2 (i) ≻i µ3 (i) for all i. Thus, dynamically stable matching need not be Pareto
optimal.19 When T = 1, stable assignments in a one-to-one matching market are Pareto
optimal (Gale and Shapley, 1962).20 The source of this welfare gap is a coordination failure
among agents, which is common in dynamic models.21
2.3
Existence of Dynamically Stable Matchings
In their one-period model, Gale and Shapley (1962) confirm the existence of a stable matching
by proposing an algorithm that finds a stable matching in every one-to-one matching market.
We can summarize this algorithm informally as follows.
• In round 1, all men propose to their most preferred potential partner. Each woman
tentatively accepts her most preferred proposal and rejects the others. If a man’s most
preferred option is to be unmatched, he remains so.
• In subsequent rounds, each man proposes to his most preferred potential partner who
has not rejected him in any previous round. Each woman tentatively accepts only
her most preferred proposal and rejects the others. If an unmatched man finds all
remaining women to be unacceptable, he remains unmatched.
18
The hypothetical situation would contradict Roth’s (1986) rural hospital theorem. One formulation of
this theorem in a one-period, one-to-one matching market is the following: If an agent is unmatched in one
stable matching, s/he is unmatched in all stable matchings (McVitie and Wilson, 1970).
19
The matching µ is Pareto optimal if there does not exist a matching µ′ such that µ′ (i) i µ(i) for all i
and µ′ (i) ≻i µ(i) for some i.
20
Efficiency is not guaranteed in all one-period economies. Stable matchings may not be Pareto optimal
in a one-period, many-to-many matching market (Roth and Sotomayor, 1990, Proposition 5.23).
21
As a simple example, consider an infinitely repeated prisoner’s dilemma. Forever repeating the stagegame Nash equilibrium is an equilibrium despite being Pareto dominated by an equilibrium that sustains
cooperation.
15
The above process continues until no further rejections occur. At this point, tentative assignments are made final and any remaining agents are left unmatched. The algorithm’s womanproposing variant is defined symmetrically with the roles of men and women reversed. Roth
(2008) provides additional details concerning the deferred acceptance algorithm’s properties
and practical applications.
In a multi-period market where each agent’s preference belongs to S̄i , a straightforward
generalization of the deferred acceptance algorithm confirms the following theorem.
Theorem 1. If ≻i ∈ S̄i for all i, there exists a dynamically stable matching.
To prove Theorem 1, we rely on an extension of the Gale and Shapley (1962) deferred acceptance algorithm, which we call the (man-proposing) ex ante deferred acceptance procedure
(EDA). Kadam and Kotowski (2016) study the same procedure in their two-period model
as a special case of a more general matching mechanism. To define the EDA, first define
agent i’s ex ante spot ranking induced by ≻i ,22 denoted P≻i , as jP≻i k ⇐⇒ j̄ ≻i k̄. The
EDA procedure defines the matching µ∗ as follows. For each t, µ∗t is the one-period matching
identified by Gale and Shapley’s (man-proposing) deferred acceptance algorithm when each
agent i makes/accepts proposals according to P≻i .23
In the proof of Theorem 1 we confirm that the EDA assignment is dynamically stable
when ≻i ∈ S¯i for all i. To do so, we first observe that if ≻i ∈ Υ(≻′i ), then P≻i = P≻′i . And
second, we draw on the following technical lemma.
Lemma 2. Suppose the matching µ is dynamically individually rational for m ∈ M and w ∈
W . Suppose this couple can period-t block µ. If ≻m ∈ S¯m and ≻w ∈ S̄w , then (µ<t (m), w̄≥t ) ≻m
µ(m) and (µ<t (w), m̄≥t ) ≻w µ(w).
Lemma 2 has a simple interpretation. It says that if a couple can block a particular matching
µ in period t, then that same couple can block µ in period t with an alternative matching plan
where they are matched together from that period onwards. Thus, when checking whether
a matching is stable or not, it is sufficient to check only a few focal deviations.
22
Kennes et al. (2014a) introduce a similar concept that they call the isolated preference relation. They
use it to define their DA-IP mechanism. This mechanism may generate a dynamically unstable outcome
when ≻i ∈ S̄i for all i (Kadam and Kotowski, 2016).
23
If it is known a priori that ≻i ∈ S̄i for all i, the ex ante deferred acceptance procedure is strategyproof for
the proposing side. This fact is an immediate corollary to the same conclusion concerning Gale and Shapley’s
deferred acceptance algorithm (Dubins and Freedman, 1981). On an unrestricted preference domain, there
does not exist a strategyproof matching procedure that always identifies a dynamically stable matching
(Kadam and Kotowski, 2016).
16
3
Welfare and Conflicting Interests
Typically, agents on different sides of a market have opposing interests. A seller prefers
a price hike while a buyer desires a discount. Similarly, agents on the same side of the
market compete and conflict could emerge concerning desirable outcomes. As cited above,
the mediation of these conflicts in one-period matching markets has been studied extensively.
It is known, for example, that the set of stable matchings enjoys a regularized structure when
stable outcomes are compared with the men’s or women’s common interests in mind. Given
the matchings µ and µ′ , we write µ ≻M µ′ if and only if µ(m) m µ′ (m) for all m ∈ M and
µ(m) ≻m µ′ (m) for some m ∈ M. If µ ≻M µ′ or µ = µ′ , then µ M µ′ . We call ≻M the
men’s common preference. If µ ≻M µ′ then all men agree that µ is the (weakly) superior
market-wide outcome. The women’s common preference, ≻W , is defined analogously. The
following facts, surveyed extensively by Roth and Sotomayor (1990), are known about the
set of stable matchings in a one-period economy.
(C1) There exists a conflict of interest among agents on opposing sides of the market. If
men collectively prefer one stable matching over another, the market’s women hold the
opposite opinion, i.e. if µ and µ′ are stable assignments, then µ M µ′ ⇐⇒ µ′ W µ
(Knuth, 1976; Roth, 1985b).
(C2) Agents on the same side of the market express consensus concerning which stable
outcome is most preferable. If µ is a stable matching, it is M-optimal if for every
other stable matching µ′ , µ M µ′ . When T = 1, the matching identified by the
man-proposing deferred acceptance algorithm is M-optimal (Gale and Shapley, 1962).
(C3) Conclusions (C1) and (C2) stem from the stable set’s lattice structure when ordered
by M or W (attributed to John Conway by Knuth, 1976). A lattice is a partiallyordered set where any two elements have a greatest lower bound and a least upper
bound (Birkhoff, 1940).24 If µ and µ′ are stable matchings, then there exists a stable
matching ν such that ν M µ, ν M µ′ , and µ′′ M ν for all other matchings µ′′ that
are preferred to ν and µ′′ . All men benefit when the market shifts from µ or µ′ to
ν. By (C1), the women are harmed by this transition. This structure ensures that
welfare comparisons among stable matchings are generally straightforward. Roth and
Sotomayor (1990, Chapter 3) survey the importance of this property in a static one-toone matching market. Interestingly, similar relationships among “stable” or equilibrium
Examples of lattices include RN when ordered by the usual coordinate-wise partial order and 2X when
ordered by set inclusion.
24
17
outcomes are observed in many economic models (for example, see Demange and Gale,
1985).
Once T ≥ 2, the conclusions (C1)–(C3) do not apply without further qualifications.
Above, we noted that a dynamically stable matching may not be Pareto optimal. Therefore,
the collective interests of the men and women are no longer necessarily opposed. Moreover, as
illustrated by the following example, a multi-period market may lack an M-optimal matching.
Thus, the set of dynamically stable matchings cannot form a lattice when ordered by M .
Example 2. Consider a two-period economy with three men and three women:
≻m1 : w2 w2 , w2w1 , w2 w3 , w1 w1 , w3 w3 , m1 m1
≻m2 : w3 w3 , w3w2 , w3 w1 , w2 w2 , w1 w1 , m2 m2
≻m3 : w1 w1 , w1w3 , w3 w3 , w1 w2 , w2 w2 , m3 m3
≻w1 : m2 m2 , m1 m1 , m2 m1 , m1 m2 , w1 m2 , m3 m2 , w1 w1
≻w2 : m3 m3 , m2 m2 , m3 m2 , m2 m3 , w2 m3 , m1 m3 , w2 w2
≻w3 : m1 m1 , m1 m3 , m3 m1 , w3 m1 , m2 m1 , m3 m3 , w3 w3
It can be verified that ≻i ∈ S¯i for every agent i ∈ M ∪ W . There are three dynamically
stable matchings in this market. These matchings are boxed, underlined, and boldfaced in
the agents’ preference list. The men disagree as to which stable matching is best. Thus,
there does not exist an M-optimal stable matching.
Persistent Matchings
While it may appear that no systematic conclusions regarding the stable set’s structure are
possible in our multi-period model, this is not entirely true. As a preliminary step, we first
focus on a subset of the dynamically stable set.
Fix an economy and let D be the set of dynamically stable matchings. We sometimes write
(D, M ) to emphasize this set’s ordering by M . The matching µ is maximally persistent
if µ(i) is maximally persistent for each i. Let P ⊂ D be the set of maximally-persistent,
dynamically stable matchings. In light of Theorem 1, when ≻i ∈ S̄i for all i, P 6= ∅. Moreover,
there exists a close connection between P and stable matchings in a one-period economy.
18
Lemma 3. Suppose ≻i ∈ S¯i for all i. Let µ be a maximally-persistent matching. The
matching µ is dynamically stable if and only if, for any t, µt is a stable matching in a
one-period market where each agent i’s preference coincides with P≻i .
The applicability of conclusions (C1)–(C3) on this restricted domain is an immediate corollary.
Corollary 1. Suppose ≻i ∈ S̄i for all i. (i) (P, M ) is a lattice. (ii) The matching identified
by the man-proposing EDA procedure is M-optimal among matchings in P. (iii) For all
µ, µ′ ∈ P, µ M µ′ ⇐⇒ µ′ W µ.
Volatile Matchings
To extend the preceding conclusions beyond P it is necessary to restrict agents’ preferences.
In light of Example 2, we first resolve the non-existence of an M-optimal stable matching.
The restriction we propose is the following.
Definition 6. The preference ≻i is sacrifice averse if for all partnership plans x = x1 · · · xT
and j̄, x i j̄ =⇒ xt i j̄ for all t. Let Ai be the set of preferences for i that are sacrifice
averse.
To paraphrase Definition 6, suppose a worker prefers successive, short-term contracts at
Firm A and then at Firm B in lieu of a stable long-term job at Firm C. If his preferences are
sacrifice averse, then he would also prefer long-term employment at Firm A or Firm B over
long-term employment at Firm C. In Example 2, the preference of m1 violates this condition.
In particular, w2 w3 ≻m1 w1 w1 , but w1 w1 ≻m1 w3 w3 . A similar observation applies to the
preference of m2 as well.
Three brief remarks concerning sacrifice aversion are appropriate. First, sacrifice aversion
does not preclude volatile stable outcome. This fact is illustrated by Examples 3 and 4
below. Consequently, an economy where each agent’s preferences satisfy this condition is
not isomorphic to a static, one-period market. Second, the set of sacrifice averse preferences
is related to the set of preferences that reflect a spot ranking in the following manner. Take
a partnership plan x = x1 · · · xT and suppose xt i j̄ for every t. If ≻i ∈ Si , then it is easy to
show that x i j̄ as well. Third, the restriction to sacrifice averse preferences guarantees the
existence of an M-optimal stable matching. We prove the following theorem as a corollary
to Theorem 3 at the end of this section.
19
Theorem 2. Suppose ≻i ∈ S¯i ∩ Ai for all i. The matching identified by the man-proposing
EDA procedure is M-optimal.
Noting Theorem 2, a tempting conjecture is that when ≻i ∈ S̄i ∩ Ai for all i, (D, M ) is
a lattice. Surprisingly, this need not be the case.
Example 3. Consider an economy with two men and two women. Their preferences are:
≻m1 : w1 w1 w1 , w2 w1 w1 , w1 w1 w2 , w1 w2 w2 , w2 w2 w1 , w2w2 w2 , m1 m1 m1
≻m2 : w2 w2 w2 , w2 w2 w1 , w1 w2 w2 , w1 w1 w2 , w2 w1 w1 , w1w1 w1 , m2 m2 m2
≻w1 : m2 m2 m2 , m1 m2 m2 , m2 m2 m1 , m2 m1 m1 , m1 m1 m2 , m1 m1 m3 , w1w1 w1
≻w2 : m1 m1 m1 , m1 m1 m2 , m2 m1 m1 , m2 m2 m1 , m1 m2 m2 , m2 m2 m2 , w2w2 w2
This market has six dynamically stable matchings (Table 3). (D, M ) is not a lattice since
µ4 and µ5 do not have a unique least upper bound. This conclusion is shown in Figure 1
where we illustrate the transitive reduction of M
Table 3: All dynamically stable matchings in Example 3.
Matching
µ1
µ2
µ3
µ4
µ5
µ6
m1
w1 w1 w1
w2 w1 w1
w1 w1 w2
w1 w2 w2
w2 w2 w1
w2 w2 w2
m2
w2 w2 w2
w1 w2 w2
w2 w2 w1
w2 w1 w1
w1 w1 w2
w1 w1 w1
w1
m1 m1 m1
m2 m1 m1
m1 m1 m2
m1 m2 m2
m2 m2 m1
m2 m2 m2
w2
m2 m2 m2
m1 m2 m2
m2 m2 m1
m2 m1 m1
m1 m1 m2
m1 m1 m1
Further restrictions on the preference domain can preclude cases like Example 3 and
render (D, M ) a lattice. However, to preserve our model’s generality, we will pursue a
different vein. We will arrange the set of dynamically stable matchings with an alternative
partial order. Blair (1988) adopts a similar approach to identify a lattice structure in the set
of stable matchings in a static, many-to-many matching market. The new ordering that we
propose is derived from agents’ preferences and therefore has a behavioral foundation. It is
defined in two steps. First, given a preference ≻i for agent i, we extract the agent’s “decisive
20
µ1
µ3
µ2
µ4
µ5
µ6
Figure 1: Hasse diagram of (D, M ) in Example 3.
preference.” This is a conservative comparison among plans based on their particular perperiod assignments. Second, we define a “decisive common preference” for the men and
women by aggregating each agent’s decisive preference in the usual way. By basing our
ordering on agents’ decisive preferences, we ensure that welfare comparisons among stable
outcomes continue to be possible.
Definition 7. Consider agent i with preference ≻i . Agent i decisively prefers plan x to plan
y, denoted x ≻∗i y, if (i) x ≻i y and (ii) xt i yt′ for all t and t′ . We write x ∗i y when x = y
or x ≻∗i y.
Though the role of ∗i in our analysis is mainly technical, it enjoys at least two further
motivations of independent interest. Both are justifications for the usefulness of incomplete
preference relations, like ∗i , in decision analysis. First, and stepping back from the assumption that ≻i describes i’s true preferences, we can interpret ∗i to be agent i’s best ranking
of available plans after some reflection, though he retains some indecisiveness among similar
options (Aumann, 1962). The relation ∗i completely ranks all maximally-persistent plans,
which is plausible and relatively easy to do in practice. However, the agent’s arbitration
among volatile plans is more clouded. For example, suppose
w1 w1 ≻m w1 w2 ≻m w2 w1 ≻m w2 w2 ≻m w1 w3 ≻m w3 w3 ≻m w2 w3 .
(3)
Figure 2 illustrates ∗m when derived from (3). In this case, m is certain that w1 w1 dominates
w2 w2 , which dominates w3 w3 . However, w1 w2 , w2 w1 , and w1 w3 are more difficult to compare.
21
w1 w1
w1 w2
w2 w1
w1 w3
w2 w2
w2 w3
w3 w3
Figure 2: The ∗m order derived from (3).
Interestingly, ∗m does not rank w2 w3 “in between” w2 w2 and w3 w3 . The matching w2 w3
incorporates an assignment to a better partner but w3 w3 is more persistent. Thus, ∗m
captures some of the tension imparted by switching costs in a dynamic setting. When an
agent is only certain of ∗i , we may interpret ≻i to be a complete ranking of available
outcomes reported by i when pressed for such a response.25
An alternative, and second, interpretation casts ∗i as an outside observer’s best guess
concerning agent i’s true preference (Ok, 2002). The observer is unaware of i’s refined
opinions given by ≻i but may be able to use ∗i as a conservative benchmark for welfare
analysis.
Whether ∗i is motivated by behavioral concerns or analytic limitations, it aggregates in
the usual way. µ ≻∗M µ′ if and only if µ(m) ∗m µ′ (m) for all m ∈ M and µ(m) ≻∗m µ′ (m) for
some m ∈ M. As usual, µ ∗M µ′ if µ ≻∗M µ′ or µ = µ′ . ≻∗W and ∗W are defined analogously.
Lemma 4. Let µ and µ′ be maximally-persistent matchings. For all i, µ(i) i µ′ (i) ⇐⇒
µ(i) ∗i µ′ (i) and µ M µ′ ⇐⇒ µ ∗M µ′ . Therefore, (P, M ) = (P, ∗M ).
The coincidence between M and ∗M does not hold more generally, as illustrated by the
following example.
Example 4. Consider a two-period market with four men and women. The men’s prefer25
In this case, outcomes would be stable with respect to the reported preferences.
22
ences are:
≻m1 : w1 w1 , w4 w4 , w1 w4 , w4 w1 , w2w1 , w2w4 , w1w2 , w2 w2 , w3 w3 , m1 m1
≻m2 : w2 w2 , w3 w3 , w2 w3 , w3 w2 , w1w2 , w1w3 , w2w1 , w1 w1 , w4 w4 , m2 m2
≻m3 : w3 w3 , w2 w2 , w3 w2 , w2 w3 , w4w3 , w4w2 , w3w4 , w4 w4 , w1 w1 , m3 m3
≻m4 : w4 w4 , w1 w1 , w4 w1 , w1 w4 , w3w4 , w3w1 , w4w3 , w3 w3 , w2 w2 , m4 m4
The women’s preferences are:
≻w1 : m2 m2 , m3 m3 , m2 m3 , m3 m2 , m2 m1 , m3 m1 , m1 m2 , m1 m1 , m4 m4 , w1 w1
≻w2 : m1 m1 , m4 m4 , m1 m4 , m4 m1 , m1 m2 , m4 m2 , m2 m1 , m2 m2 , m3 m3 , w2 w2
≻w3 : m4 m4 , m1 m1 , m4 m1 , m1 m4 , m4 m3 , m1 m3 , m3 m4 , m3 m3 , m2 m2 , w3 w3
≻w4 : m3 m3 , m2 m2 , m3 m2 , m2 m3 , m3 m4 , m2 m4 , m4 m3 , m4 m4 , m1 m1 , w4 w4
This market has sixteen dynamically stable matchings (Table 4). Figure 3 summarizes
these matchings when ordered by M and by W . In this case (D, M ) and (D, W ) are
lattices, but W is not the inverse of M . The required conflict of interest is not maintained
throughout the set of dynamically stable outcomes since µ6 Pareto dominates µ11 . Figure 4
presents the same set of stable matchings, but orders the set according to ∗M and ∗W . It
is clear that ∗W is the inverse of ∗M . Of course, (D, ∗M ) and (D, ∗W ) are lattices as well.
The rearrangement of the stable set provided by ∗M is sufficient to ensure that (D, ∗M )
forms a lattice more generally.
Theorem 3. Suppose ≻i ∈ S̄i ∩ Ai for all i. (D, ∗M ) is a lattice. Moreover, if µ, µ′ ∈ D and
µ ∗M µ′ , then µ′ ∗W µ.
A corollary to Theorem 3 is that an M-optimal matching exists (Theorem 2). In particular,
the ≻M -maximal matching in P, which is identified by the EDA procedure, ≻M -dominates
all matchings in D.
4
Stable Matchings and the Market’s Time Horizon
We have thus far taken the market’s time horizon, defined by the parameter T , as given. A
natural question, however, concerns the sensitivity of the set of dynamically stable matchings
23
Table 4: All dynamically stable matchings in Example 4.
Matching
µ1
µ2
µ3
µ4
µ5
µ6
µ7
µ8
µ9
µ10
µ11
µ12
µ13
µ14
µ15
µ16
m1
w1 w1
w1 w1
w1 w1
w1 w1
w2 w1
w2 w1
w2 w1
w2 w1
w1 w2
w1 w2
w1 w2
w1 w2
w2 w2
w2 w2
w2 w2
w2 w2
m2
w2 w2
w2 w2
w2 w2
w2 w2
w1 w2
w1 w2
w1 w2
w1 w2
w2 w1
w2 w1
w2 w1
w2 w1
w1 w1
w1 w1
w1 w1
w1 w1
m3
w3 w3
w4 w3
w3 w4
w4 w4
w3 w3
w4 w3
w3 w4
w4 w4
w3 w3
w4 w3
w3 w4
w4 w4
w3 w3
w4 w3
w3 w4
w4 w4
m4
w4 w4
w3 w4
w4 w3
w3 w3
w4 w4
w3 w4
w4 w3
w3 w3
w4 w4
w3 w4
w4 w3
w3 w3
w4 w4
w3 w4
w4 w3
w3 w3
w1
m1 m1
m1 m1
m1 m1
m1 m1
m2 m1
m2 m1
m2 m1
m2 m1
m1 m2
m1 m2
m1 m2
m1 m2
m2 m2
m2 m2
m2 m2
m2 m2
w2
m2 m2
m2 m2
m2 m2
m2 m2
m1 m2
m1 m2
m1 m2
m1 m2
m2 m1
m2 m1
m2 m1
m2 m1
m1 m1
m1 m1
m1 m1
m1 m1
w3
m3 m3
m4 m3
m3 m4
m4 m4
m3 m3
m4 m3
m3 m4
m4 m4
m3 m3
m4 m3
m3 m4
m4 m4
m3 m3
m4 m3
m3 m4
m4 m4
w4
m4 m4
m3 m4
m4 m3
m3 m3
m4 m4
m3 m4
m4 m3
m3 m3
m4 m4
m3 m4
m4 m3
m3 m3
m4 m4
m3 m4
m4 m3
m3 m3
µ1
µ5
µ9
µ13
µ2
µ6
µ10
µ14
µ3
µ7
µ11
µ15
µ4
µ8
µ12
µ16
Figure 3: The set of dynamically stable matchings in Example 4 ordered by M (solid
arrows) and W (dashed arrows).
24
µ1
µ2
µ3
µ4
µ8
µ5
µ6
µ7
µ10
µ12
µ11
µ9
µ13
µ14
µ15
µ16
Figure 4: The set of dynamically stable matchings in Example 4 ordered by ∗M (solid
arrows) and ∗W (dashed arrows).
to changes in T . While some situations have an “obvious” time horizon and a well-specified
set of rematching opportunities, many do not. Thus, it is important to know whether a
particular dynamically stable matching has an easily-identifiable counterpart as the model’s
time horizon is slightly perturbed or the frequency of revision opportunities is adjusted.
4.1
Abridgment
Comparing markets with different time horizons involves mapping stable matchings from
one case to the other. Thus, the set of agents must be the same and agents’ preferences
need to be coherently related for different values of T . In this regard, abridgments of the
time horizon are simple because matchings and preferences can be truncated. For example,
consider a T -period market where x = x1 · · · xT is a typical partnership plan and ≻i is agent
x
i’s preference. Agent i’s preference conditional on x≤t , denoted ≻i ≤t , is a preference over
plans of length T − t such that
x
(y1 , . . . , yT −t ) ≻i ≤t (z1 , . . . , zT −t ) ⇐⇒ (x≤t , y1 , . . . , yT −t) ≻i (x≤t , z1 , . . . , zT −t ).
25
(4)
By conditioning preferences on the market’s elapsed history, we can relate the final periods
of a stable outcome given a longer time horizon to a stable matching in a shorter market.
Theorem 4. Let µ∗ be a dynamically stable matching in a T -period economy where each
agent’s preference is ≻i . Then µ∗>t = (µ∗t+1 , . . . , µ∗T ) is a dynamically stable matching in a
µ∗ (i)
T − t-period economy where each agent’s preference is ≻i ≤t .
A helpful analogy reinforcing Theorem 4 can be found by considering a sub-game perfect
equilibrium of a multi-stage game. At each moment in time, the continuation of the game
is itself an equilibrium. Similarly, the continuation plan remains dynamically stable in our
model.
A natural follow-up question asks whether stable matchings can be truncated conditioning
on the future, rather than the past? Generally, this is not possible. If µ∗ is dynamically
stable, then the t-period matching µ∗≤t need not be dynamically stable when preferences are
conditioned on µ∗>t in an manner analogous to (4).26 As a counterexample, m1 can block
µ3≤2 in Example 1 above.
4.2
Extension
While every T -period economy can be shortened, introducing additional periods requires
greater finesse. An extension involves embedding the set of dynamically stable matchings
into an economy with a longer time horizon. The key challenge in this exercise concerns
translating the original preferences into the new market. One way of doing so is the following.
ˆ i (for a T̂ = T + 1-period economy) is a one-period extension
Definition 8. The preference ≻
of the preference ≻i (for a T -period economy) if
ˆ i y1 · · · yT yT .
x1 · · · xT ≻i y1 · · · yT ⇐⇒ x1 · · · xT xT ≻
As a complementary motivation for Definition 8, consider an economy that lasts two periods of calendar time. However, institutional constraints do not accommodate re-matching
between periods. In this special case, the comparison outlined in Definition 8 is analogous
to splitting the periods apart and introducing a rematching opportunity.
26
(·,x
)
Agent i’s preference anticipating x>t , denoted ≻i >t , is a preference over plans of length t such that
(·,x )
(y1 , . . . , yt ) ≻i >t (z1 , . . . , zt ) ⇐⇒ (y1 , . . . , yt , x>t ) ≻i (z1 , . . . , zt , x>t ).
26
If ≻i ∈ S¯i in a T -period economy, then it is easy to see that some one-period extensions
of ≻i will not belong to the extended market’s version of S̄i . However, we can always find
preference extensions that belongs to this analytically-convenient set.
Lemma 5. Let Si and Ŝi be the sets of preferences that reflect a spot ranking in a T -period
and a T̂ = T + 1 -period market, respectively.
ˆ i ∈ Ŝi such that ≻
ˆ i is a one-period extension of ≻i .
(i) Suppose ≻i ∈ Si . There exists ≻
ˆ i ∈ Ŝ¯i such that ≻
ˆ i is a one-period extension of ≻i .
(ii) Suppose ≻i ∈ S̄i . There exists ≻
The following theorem embeds all dynamically stable matchings from a T -period market
into the set of stable matchings from a T + 1-period market.
Theorem 5. Let µ∗ = µ∗1 · · · µ∗T be a dynamically stable matching in a T -period market
where ≻i ∈ S̄i for all i. Consider a T̂ = T + 1 -period market with the same set of agents
ˆ i ∈ Ŝ¯i is a one-period extension of ≻i . The matching
and where each agent’s preference ≻
µ̂∗ = µ∗1 · · · µ∗T µ∗T
| {z }
µ∗
is dynamically stable in the T̂ -period market.
The following example, which is adapted from Kadam and Kotowski (2016), illustrates the
relationship between Theorems 4 and 5.
Example 5. Consider a one-period market with two men and two women. Their preferences
are
≻w1 : m2 , m1 , w1
≻w2 : m1 , m2 , w2
≻m1 : w1 , m1 , w2
≻m2 : w2 , m2 , w1
Trivially, ≻i ∈ S¯i for all i. This market’s only stable matching is boxed in the preceding
preference lists. Call this matching µ1 .
27
Consider a one-period extension of this economy where the agents’ preferences are
ˆ m1 : w1 w1 , w1 m1 , w1 w2 , m1 w1 , w2 w1 , m1 m1
≻
µ̂2
µ̂1
ˆ m2 : w2 w2 , w2 m2 , w2 w1 , m2 w2 , w1 w2 , m2 m2
≻
ˆ w1 : m2 m2 , m2 m1 , m1 m2 , m1 m1 , w1 w1
≻
ˆ w2 : m1 m1 , m1 m2 , m2 m1 , m2 m2 , w2 w2
≻
ˆ i ∈ Ŝ¯i for all i in the sense of Lemma 5. There are now two
It can be verified that ≻
dynamically stable matchings. The boxed matching µ̂1 extends µ1 in the sense of Theorem
5. The underlined matching, µ̂2 , is not the extension of any stable matching from the oneperiod economy. However, it can be truncated in the sense of Theorem 4 to recover µ1 .
4.3
Discussion
Temporal manipulations of the kind considered above have practical applications. Recently,
many communities have adopted centralized matching procedures to navigate the conflict
between schools’ capacity constraints and students’ preferences (Abdulkadiroğlu and Sönmez,
2003). Such problems have a multi-period nature because each student is assigned to a
particular school for many years. Most assignment procedures assign students to a school for
a focal duration, such as all four years of high school. But, it is possible to introduce planned
revision opportunities at finer time scales. For instance, a student may like a particular school
only because it offers a specialized upper-year curriculum and may prefer (or be willing) to
enroll elsewhere in early years. Allowing him to do so in a pre-planned manner may ease a
capacity constraint and may better align with his preferences. This possibility is particularly
relevant for tertiary education where it is common for some students to initially attend a
local community college and then transfer to a larger university to complete their degree. For
example, roughly 15,000 California community college students transfer into the University
of California system every year (University of California Office of the President, 2014). Such
a strategy may be associated with lower costs of attendance, among other considerations.
Welfare improvements stemming from the fine-tuning of rematching frequencies are possible too. By allowing for re-matching between periods in Example 5, a novel stable matching,
µ̂2 , emerges. This option is preferred by the women in the market in lieu of the original
assignment’s naive extension, µ̂1 . While µ̂2 involves a welfare loss for the men relative to
28
µ1 or µ̂1 , this tradeoff may be acceptable. In many centralized assignment problems, the
“preferences” of agents on one side of the market are administratively-defined “priorities,”
which are not necessarily reflective of welfare.
5
Concluding Remarks
Dynamics are an integral feature of many two-sided markets. Though our model is stylized,
it accommodates many properties of real-world interactions that unfold over many periods.
While we have maintained Gale and Shapley’s (1962) original terminology by speaking of
a matching between men and women, the theory developed above captures key features of
many labor markets and it can be adapted to tackle important allocation problems, such as
student-school assignment. We hope the preceding analysis provides a stepping stone toward
these applications.
Our analysis points to at least two directions for further research. First, there is considerable scope to refine and extend our analysis’ normative implications. We have thus far
confined our attention to a positive description of multi-period market’s operation (taking
as given our proposed solution concept). We believe, however, that our results can be applied to inform the design and redesign of markets or allocation mechanisms where change is
common. Above, we mentioned the example of tertiary education; however, even elementary
or high-school students change enrollment frequently and for many reasons (O’Donnell and
Gazos, 2010). A multi-period matching model allows one to understand the welfare implications of these dynamics while also accounting for their presence in the design of a matching
mechanism. In the current work we identify broad themes that will color any such application. We also identify new parameters that are valuable additions to a market designer’s
toolkit, such as relationship length and re-matching frequency.
Second, we have focused on small markets, in the economic sense of the term. An analysis of large markets with an eye toward dynamics is necessary to better link our conclusions
with those obtained in parallel dynamic matching literatures, such as those stressing search
behavior.27 In the latter, it is known that equilibria may not be Pareto optimal due to
coordination failures or externalities. The results presented above, including the qualifications surrounding the lattice of stable outcomes, echo those observations. Exploring these
connections further appears to be a promising avenue for future work.
27
In a recent working paper, Kennes et al. (2014b) examine a large dynamic matching market focusing on
agents’ strategic incentives. They identify some desirable properties of the deferred acceptance algorithm in
their setting.
29
A
Proofs
Proof of Lemma 1. (⇒) Suppose x ≻′i y. Since ≻i ∈ Υ(≻′i ) and x D y, Definition 5 implies
that x ≻i y. (⇐) Suppose x ≻i y. Therefore, x 6= y. If y ≻′i x, then y D x and ≻i ∈ Υ(≻′i )
imply that y ≻i x—a contradiction. Therefore, x ≻′i y.
Lemmas A.1 and A.2 are preliminary results used in some of the arguments to follow.
Lemma A.1. Suppose ≻i ∈ S¯i and let x be a volatile partnership plan. xt ≻i x for some t.
Proof. Suppose the contrary. Since ≻i ∈ S̄i , ≻i ∈ Υ(≻′i ) for some ≻′i ∈ Si . Because xt ⊲ x,
x ≻i xt =⇒ x ≻′i xt for all t. Let xt∗ be such that xt∗ R≻′i xt ∀t. x ≻i xt∗ implies that
xt P≻i xt∗ for some t 6= t∗ , which is a contradiction.
Lemma A.2. Suppose ≻i ∈ S̄i . Suppose agent i can period-t block the matching µ for some
t ≥ 2. (i) If µ(i) is maximally persistent, then i can period-1 block µ. (ii) If µ(i) is volatile,
then i can period-t′ block µ at some t′ such that µt′ −1 (i) 6= µt′ (i).
Proof. (i) This fact is shown in the proof of Theorem 1. (ii) Suppose µt−1 (i) = µt (i).
(Otherwise t = t′ satisfies the lemma’s claim.) Without loss of generality, we can write µ(i)
as
µ(i) = (µ<s (i), j̄[s,t) , j , j̄(t,s′ ] , µ>s′ (i)).
↑
t
where µs−1(i) 6= j and µs′ +1 (i) 6= j. By assumption s < t. If we let
µ̃(i) = (µ<s (i), j̄[s,t) , i, ī(t,s′ ] , ī>s′ )
↑
t
then by assumption µ̃(i) ≻i µ(i). There are three cases.
1. If j = i, then t′ = s.
2. If jP≻i i, then (µ<s (i), j̄[s,t) , j, j̄(t,s′ ] , ī>s′ ) ≻′i µ̃(i). By inspection, µ̃(i) k µ(i). Hence,
(µ<s (i), j̄[s,t) , j, j̄(t,s′ ] , ī>s′ ) ≻i µ̃(i), which implies (µ<s (i), j̄[s,t) , j, j̄(t,s′ ] , ī>s′ ) ≻i µ̃(i) ≻i
µ(i). Thus, t′ = s′ + 1.
3. If iP≻i j, then (µ<s (i), ī[s,t) , i, ī(t,s′ ] , ī>s′ ) ≻′i µ̃(i). Since (µ<s (i), ī[s,t) , i, ī(t,s′ ] , ī>s′ ) D µ̃(i),
(µ<s (i), ī[s,t) , i, ī(t,s′ ] , ī>s′ ) ≻i µ̃(i) ≻i µ(i). Thus, t′ = s.
30
{m,w}
Proof of Lemma 2. By assumption, µ̂(i) = (µ<t (i), µ≥t (i)) ≻i µ(i) for i ∈ {m, w}. As µ
cannot be blocked by m or w alone, there exists t′ ≥ t such that µ̂t′ (m) = w and µ̂t′ (w) = m.
Let t′ be the smallest such index. Given t′ define µ̃(m) ≡ (µ<t (m), µ̂[t,t′ ) (m), w̄≥t′ ). By
construction, µ̃(m) D µ̂(m).
Since ≻i ∈ Υ(≻′i ) for some ≻′i ∈ Si , (µ<t (m), m̄≥t ) D µ̂(m) and µ̂(m) ≻m µ(m) ≻m
(µ<t (m), m̄≥t ) imply that µ̂(m) ≻′m (µ<t (m), m̄≥t ). Hence, wP≻m m. Thus, µ̃(m) ′m µ̂(m)
and, therefore, µ̃(m) m µ̂(m). Likewise, we conclude that µ̃(w) w µ̂(w).
If t′ = t, then the proof is complete. Suppose t′ > t. In this case (µ<t (m), w̄≥t ) ≻′m µ̃(m).
There are two cases:
1. If (µ<t (m), w̄≥t ) D µ̃(m), then (µ<t (m), w̄≥t ) ≻m µ̃(m).
2. If (µ<t (m), w̄≥t ) 4 µ̃(m), then µ̃(m) 4 (µ<t (m), w̄≥t ).28 Hence, (µ<t (m), w̄≥t ) ≻m
µ̃(m).
In both cases we see that (µ<t (m), w̄≥t ) ≻m µ̃(m) m µ̂(m) ≻m µ(m). An analogous
argument applies to w.
Proof of Theorem 1. Let µ∗ be the matching identified by the EDA procedure. First, we
verify that µ∗ cannot be period-t blocked by agent i alone. If µ∗ (i) = ī, then clearly µ∗
cannot be period-t blocked by agent i. Instead, suppose µ∗ (i) = j̄ for some j 6= i. Since the
Gale and Shapley deferred acceptance algorithm always identifies an individually rational
assignment, jP≻i i. If µ∗ (i) can be period-t blocked by i, then (j̄<t , ī≥t ) ≻i j̄ ≻i ī where
≻i ∈ Υ(≻′i ) for some ≻′i ∈ Si . But, j̄ ≻′i ī and j̄ D (j̄<t , ī≥t ) imply that j̄ ≻i (j̄<t , ī≥t ), which
is a contradiction. Thus, µ∗ cannot be blocked in period t by agent i.
It remains to verify that no pair of agents can period-t block µ∗ . Again we argue by
contradiction. Suppose m and w can period-t block µ∗ . By Lemma 2, this implies µ̃(m) =
(µ∗ (m), w̄≥t ) ≻m µ∗ (m) and µ̃(w) = (µ∗ (w), m̄≥t ) ≻w µ∗ (w). Since ≻m ∈ S¯m , ≻m ∈ Υ(≻′ )
<t
<t
≻′m ∈
m
∗
∗
≻′m
∗
for some
Sm . As µ (m) D µ̃(m) and µ̃(m) ≻m µ (m), it follows that µ̃(m)
µ (m).
∗
∗
∗
This implies wP≻′m µt (m) and thus, wP≻m µt (m). Likewise, mP≻w µt (w). However, this
implies m and w can block the one-period matching identified by the deferred acceptance
algorithm when agents’ preferences are given by P≻i . This is a contradiction as the deferred
acceptance algorithm identifies a stable one-period matching. As µ∗ cannot be period-t
blocked by any agent or by any pair, it is dynamically stable.
28
This case may occur if µt−1 (m) = m.
31
Lemma A.3. Suppose ≻i ∈ S¯i for all i. Let µ be a dynamically stable matching when T = 2.
Then agent i either has a partner in each period or remains unmatched in both periods.
Proof. Let µ be a dynamically stable matching. Without loss of generality suppose m1 ∈ M
is single in period 1 and matched to w1 ∈ W in period 2. (The argument when m1 is matched
to w1 in period 1 and single in period 2 is identical and we omit it for brevity.)
As µ is dynamically stable, µ(m1 ) = m1 w1 ≻m1 m1 m1 . Since ≻m1 ∈ S̄m1 ,
w1 w1 ≻m1 µ(m1 ) = m1 w1 ≻m1 m1 m1 .
Furthermore, µ(w1) ≻w1 m1 m1 as otherwise m1 and w1 could block µ. As µ(w1 ) ≻w1 w1 w1
and ≻ w1 ∈ S̄w1 , there exists m2 ∈ M, such that
m2 m2 ≻w1 µ(w1 ) = m2 m1 ≻w1 m1 m1 .
As above, µ(m2 ) ≻m2 w1 w1 as otherwise m2 and w1 could block µ. As µ(m2 ) ≻m2 m2 m2
and ≻m2 ∈ S¯m2 , there exists w2 ∈ W such that
w2 w2 ≻m2 µ(m2 ) = w1 w2 ≻m2 w1 w1 .
Continuing by induction, suppose that for k ≥ 2 there exists distinct men m2 , . . . , mk
and distinct women w2 , . . . , wk such that for each k ′ ≤ k
mk′ mk′ ≻wk′ −1 µ(wk′ −1 ) = mk′ mk′ −1 ≻wk′ −1 mk′ −1 mk′ −1 .
(A.1)
wk′ wk′ ≻mk′ µ(mk′ ) = wk′ −1 wk′ ≻mk′ wk′ −1 wk′ −1 .
(A.2)
and
We will show that we can find a new mk+1 and a new wk+1 satisfying (A.1) and (A.2).
Given (A.2) it follows that µ(wk ) ≻wk mk mk and µ2 (wk ) = mk . Thus, there exists
mk+1 ∈ M such that
mk+1 mk+1 ≻wk µ(wk ) = mk+1 mk ≻wk mk mk .
As mk+1 and wk cannot block µ, µ(mk+1 ) ≻mk+1 wk wk . Clearly mk+1 6= m1 , . . . , mk . Otherwise µ(mk+1 ) = wk i = µ(mk′ ) = wk′−1 wk′ , which implies wk′ −1 = wk . But this contradicts
the assumption that w1 , . . . , wk are distinct.
32
It follow, therefore, that there exists wk+1 ∈ W such that
wk+1 wk+1 ≻mk+1 µ(mk+1 ) = wk wk+1 ≻mk+1 wk wk .
Clearly, wk+1 6= wk . If instead wk+1 = wk′ for k ′ < k, then µ(wk+1) = µ(wk′ ) = mk′ +1 mk′
implying that mk+1 = mk′ , which contradicts the preceding analysis. Therefore wk+1 is
distinct from w1 , . . . , wk .
Thus, continuing in this manner, we can construct a sequence of distinct men (m1 , . . .),
and an sequence of distinct women (w1 , . . .), satisfying (A.1) and (A.2). However, this is
impossible since there is a finite number of men and women in the market.
Proof of Lemma 3. (⇒) Suppose µ is dynamically stable. Take any µt and suppose that it
can be blocked by agent i. Then, iP≻i µt (i) =⇒ ī ≻i µ(i), which contradicts the individual
rationality of µ. Likewise, if m and w can block µt , wP≻m µt (m) and mP≻w µt (w) imply that
w̄ ≻m µ(m) and m̄ ≻m µ(w). Thus, µ is not dynamically stable—a contradiction.
(⇐) Suppose µt is a stable one-period matching when each agent’s preference is P≻i .
Suppose ī ≻i µ(i). This implies iP≻i µt (i), which contradicts the individual rationality of µt .
Suppose instead that w̄ ≻m µ(m) and m̄ ≻w µ(w). Hence, wP≻m µt (m) and mP≻w µt (w).
Thus, m and w can block µt , which is a contradiction.
Proof of Corollary 1. By Lemma 3, each maximally-persistent, dynamically stable matching
in a T -period economy corresponds to a stable matching in a one-period economy and vice
versa. The set of stable matchings in a one-period economy is a lattice when ordered by
men’s common preference (Knuth, 1976). Thus, (P, M ) is a lattice as well. Points (ii) and
(iii) follow from Gale and Shapley (1962) and Knuth (1976), as noted in C1–C3 above.
Proof of Lemma 4. Consider agent i. If µ(i) = µ′ (i), the lemma’s first part is clearly true.
Suppose µ(i) ≻i µ′ (i). As both matchings are maximally persistent, µ(i) = µt (i) and
µ′ (i) = µ′t′ (i) for all t and t′ . Thus, µt (i) ≻i µ′t′ (i). Therefore, µ(i) ≻∗i µ′ (i). Conversely, if
µ(i) ≻∗i µ′ (i), then µ(i) ≻i µ′ (i) follows immediately from the definition of ∗i .
Now consider the collective preference. If µ = µ′ , the lemma’s conclusion is again trivial.
If µ ≻M µ′ , then µ(m) m µ′ (m) for all m and µ(m′ ) ≻m µ′ (m′ ) for some m′ . As µ and µ′
are maximally-persistent matchings, µ(m) ∗m µ′ (m) for all m and µ(m′ ) ≻∗m µ′ (m′ ) for some
m′ . Hence, µ ≻∗M µ′ . Conversely, µ ∗M µ′ =⇒ µ(m) ∗m µ′ (m) ∀m ∈ M =⇒ µ(m) m
µ′ (m) ∀m ∈ M =⇒ µ M µ′ .
33
The proof of Theorem 3 relies on several preliminary lemmas that we prove in the appendix. Lemma A.4 is mainly concerned with the relationship between single period assignments and dynamically stable partnership plans when ≻i ∈ S¯i ∩ Ai . Lemma A.5 shows that
every stable matching is ∗M -dominated by a persistent stable matching. Finally, Lemma A.6
confirms that ∗W is the inverse of ∗M (and vice verse). This conclusion is well-illustrated
by Example 4.
Lemma A.4. Suppose ≻i ∈ S¯i ∩ Ai ∀i and let µ ∈ D.
(i) µt = (µt , . . . , µt ) ∈ D.
(ii) If µ(i) is volatile, there exists a period t(i) such that µ(i) ≻i µt(i) (i).
(iii) If µ(i) ≻i µt (i), then for all t′ such that µt′ (i) 6= µt (i), µt′ (i) ≻i µ(i).
Proof.
(i) An implication of Definition 6 is that µt (i)R≻i i for all i. Thus, µt (i) i ī and i cannot
block µ′ alone. Instead, suppose that m and w can block µt . By Lemma 2, m and
w can period-1 block µt . Thus, wP≻m µt (m) and mP≻w µt (w). As µ ∈ D, m and w
cannot period-1 block µ. Thus, µ(m) ≻m w̄ or µ(w) ≻w m̄. Without loss of generality,
suppose µ(m) ≻m w̄. Since ≻m ∈ Am , µt (m)R≻m w. But then µt (m)R≻m wP≻m µt (m),
which is a contradiction. Thus, µt ∈ D.
(ii) We argue by contradiction. Without loss of generality, suppose µ(m1 ) is volatile and
µt (m1 ) ≻m1 µ(m1 ) for all t. As µ ∈ D, µ(m1 ) ≻m1 m1 . Thus, µt (m1 ) = w1 ∈ W for
some t and w1 ≻m1 µ(m1 ). µ(w1 ) ≻w1 m1 ; else, w1 and m1 can block µ.
As m1 is not matched to w1 for all t, µ(w1 ) is volatile. By Lemma A.1, ∃ m2 ∈ M such
that
m2 ≻w1 µ(w1 ) ≻w1 m1
and µt(w1 ) (w1 ) = m2 for some period t(w1 ). Clearly, m1 6= m2 .
As µ ∈ D, µ(m2 ) ≻m2 w1 ; else, m2 and w1 could block µ. Since m2 is not matched to
w1 for all t, µ(m2 ) is volatile. By Lemma A.1, ∃ w2 ∈ W such that
w2 ≻m2 µ(m2 ) ≻m2 w1
and µt(m2 ) (m2 ) = w2 for some period t(m2 ). Clearly w2 6= w1 .
We continue by induction. Suppose that for all 2 ≤ k ′ ≤ k:
34
(a) mk′ ≻wk′ −1 µ(wk′−1 ) ≻wk′ −1 mk′ −1 ;
(b) wk′ ≻mk′ µ(mk′ ) ≻mk′ wk′ −1 ;
(c) µt(wk′ −1 ) (wk′ −1 ) = mk′ ;
(d) µt(mk′ ) (mk′ ) = wk′ ; and,
(e) m1 , . . . , mk and w1 , . . . , wk are all distinct.
We will show that we can find mk+1 and wk+1 distinct from those already identified
satisfying (a)–(e).
First, consider wk . As µ ∈ D, µ(wk ) ≻wk mk ; else, wk and mk can block µ. By Lemma
A.1, ∃ mk+1 ∈ M such that
mk+1 ≻wk µ(wk ) ≻wk mk
and µt(wk ) (wk ) = mk+1 for some period t(wk ).
Clearly mk+1 6= mk . If mk+1 = m1 , then wk and m1 can block µ, which is a contradiction. Finally, if mk+1 = mk′ for 1 < k ′ < k, we know that
wk′ ≻mk′ µ(mk′ ) ≻mk′ wk′ −1
and µt(wk′ −1 ) (wk′ −1 ) = mk′ . As µ ∈ D, µ(mk′ ) ≻mk′ wk . Since ≻mk′ ∈ Amk′ and wk′ −1
and wk are distinct members encountered during the plan µ(mk′ ),
µ(mk′ ) ≻mk′ wk′−1 =⇒ wk ≻mk′ wk′ −1
and
µ(mk′ ) ≻mk′ wk =⇒ wk′ −1 ≻mk′ wk .
Clearly, this is a contradiction. Therefore mk+1 6= mk′ . Hence, mk+1 is distinct from
each m1 , . . . , mk .
Now consider mk+1 . As µ ∈ D, µ(mk+1 ) ≻mk+1 wk ; else mk+1 and wk can block µ. By
Lemma A.1, ∃ wk+1 ∈ W such that
wk+1 ≻mk+1 µ(mk+1 ) ≻mk+1 wk .
and wk+1 = µt(mk+1 ) (mk+1 ) for some period t(mk+1 ).
35
Clearly, wk+1 6= wk . Suppose wk+1 = wk′ for some k ′ < k. We know that
mk′ +1 ≻wk′ µ(wk′ ) ≻wk′ mk′ ,
µt(mk′ ) (wk′ ) = mk′ and µt(mk+1 ) (wk′ ) = mk+1 . As µ ∈ D, µ(wk′ ) ≻wk′ mk+1 ; else, mk+1
and wk′ = wk+1 can block µ. Since ≻wk′ ∈ Awk′ and mk′ and mk+1 are distinct
µ(wk′ ) ≻wk′ mk′ =⇒ mk+1 ≻wk′ mk′
and
µ(wk′ ) ≻wk′ mk+1 =⇒ mk′ ≻wk′ mk+1
Clearly, this is a contradiction. Therefore, wk+1 6= wk′ for all k ′ ≤ k.
Proceeding by induction we can construct an infinite sequence of men m2 , m3 , . . . and
women w2 , w3 , . . . satisfying conditions (a)–(e) above. However, this is not possible as
there is a finite number of agents. Thus, if µ(m1 ) is volatile, there exists a period t
such that µ(m1 ) ≻m1 µt (m1 ).
(iii) It is sufficient to observe that when µt (i) 6= µt′ (i), µ(i) ≻i µt (i) =⇒ µt′ (i) ≻i µt (i)
and µ(i) ≻i µt′ (i) =⇒ µt (i) ≻i µt′ (i), which is a contradiction as ≻i ∈ Ai .
Lemma A.5. Suppose ≻i ∈ S¯i ∩ Ai for all i. If µ ∈ D, ∃ µ′ ∈ P ⊂ D such that µ′ ∗M µ.
Proof. Let µ′ be the M -maximal matching in P. Since (P, M ) is a finite lattice, this
matching exits and is unique. Let µ ∈ D and suppose µ′ 6∗M µ. Hence, ∃m ∈ M such
that µ′ (m) 6∗m µ(m). This implies µ(m) is volatile. Thus, there exists a period t′ such
that µ(m) ≻m µt′ (m) and for all t 6= t′ , µt (m) ≻m µ(m). However, µt ∈ P for all t.
Thus, µ′ (m) m µt (m) ≻m µ(m) ≻m µt′ (m). But this implies µ′ (m) ≻∗m µ(m), which is a
contradiction.
Lemma A.6. Suppose ≻i ∈ S¯i ∩ Ai for all i. Let µ, µ′ ∈ D. µ ∗M µ′ ⇐⇒ µ′ ∗W µ.
Proof. Suppose µ ≻∗M µ′ . Thus, there exists at least one man and at least one woman for
whom µ(i) 6= µ′ (i). If µ′ 6≻∗W µ, then µ′ (w) 6∗w µ(w) for some w ∈ W . Thus, µ′ (w) 6= µ(w)
and µ′ (w) 6≻∗w µ(w). There are two cases.
36
1. There exist t and t′ such that µt (w) ≻w µ′t′ (w). Hence, there exists m ∈ M such that
m = µt (w) and m̄ = µt (w) ≻w µ′t′ (w). Since µ ≻∗M µ′ , it follows that for this agent m,
µ(m) ∗m µ′ (m) and for t and t′ , w̄ = µt (m) m µ′t′ (m). As µt (w) = m 6= µt′ (w), it
must be the case that µt (m) ≻m µ′t′ (m).
By Lemma A.4(i), µ′t′ ∈ D. However, µ′t′ (w) = µ′t′ (w) and µ′t′ (m) = µ′t′ (m). Thus, m
and w can period-1 block µ′t′ , which is a contradiction.
2. Suppose µ(w) ≻w µ′ (w). By Lemma A.1, there exist t and t′ such that µt (w) w µ(w)
and µ′(w) w µ′t′ (w). Thus, the same argument as case (1) applies.
As neither case applies, we arrive at a contradiction. Hence, µ′ ≻∗W µ.
Proof of Theorem 3. By Lemma A.6, ∗W is the inverse of ∗M on D. Thus, to verify that
(D, ∗M ) is a lattice it is sufficient to show that there exists a least upper bound for any two
dynamically stable matchings.
Let µ, µ′ ∈ D. If µ ≻∗M µ′ , then µ is the least upper bound for µ and µ′ . Instead, suppose
µ and µ′ are not ordered by ∗M . From Lemma A.5 we know that µ and µ′ are bounded
above by some µ1 , µ2 ∈ D, i.e.
)
(
µ
µ1
∗
≻M
.
2
µ′
µ
It is sufficient to show that ∃ µ̃ ∈ D such that
µ1
µ2
)
∗M µ̃ ≻∗M
(
µ
.
µ′
Let P(µk ) = {µkt : t = 1, . . . , T }. By Lemma A.4(i), µkt ∈ P. Since (P, ∗M ) is a lattice,
P(µ1 ) ∪ P(µ2 ) ⊂ P has a greatest lower bound in (P, ∗M ). Let λ ∈ P be this greatest lower
bound.
Next, we argue that for each m, µk ∗m λ. There are two properties to check:
1. For each t and t′ , µkt (m) m λt′ (m).
µkt ∈ P(µk ). Hence, µkt ∗M λ. This implies µkt (m) ∗m λ(m) = λt′ (m) for all t′ as
λ ∈ P.
2. µk (m) m λ(m).
Assume the contrary. Then λ(m) ≻m µk (m). From part (1) above we know that for
all t, µkt (m) m µ̃(m) ≻m µk (m). There are two possibilities. If µk (m) ∈ P, then
37
/ P, then by Lemma A.4
µk (m) = µkt (m) and we have a contradiction. If instead µk ∈
µk (m) ≻m µkt (m) for some t. This too is a contradiction. Hence, our assumption to
the contrary was incorrect.
To complete the proof, we verify that λ(m) ∗m µ(m) for all m. (The case of µ′ follows
identically.) As a preliminary point we note that since µk ∗M µ, for all m, t, t′ , and k,
µkt (m) m µt′ (m). Suppose that for some m, λ(m) 6∗m µ(m). There are two possibilities:
1. For some t, µt (m) ≻m λ(m).
Given the observation made immediately above, µt ∈ P is a lower bound for P(µ1 ) ∪
P(µ2 ). But λ is the greatest such lower bound. Therefore, this case cannot be true.
2. µ(m) ≻m λ(m).
If µ(m) ∈ P, then case (1) above applies. If µ(m) ∈
/ P, then by Lemma A.4 µt (m) ≻m
λ(m) for some t. But again, this implies case (1) applies.
As neither case (1) nor case (2) applies, we conclude that in fact λ(m) ∗m µ(m) ∀m.
Proof of Theorem 4. Suppose i can block µ∗>t , i.e.
µ∗ (i)
(µ∗(t,t′ ) (i), ī≥t′ ) ≻i ≤t
µ∗>t (i) ⇐⇒ (µ∗≤t (i), µ∗(t,t′ ) (i), ī≥t′ ) ≻i (µ∗≤t (i), µ∗>t (i)).
Thus, i can block µ∗ , which is a contradiction.
If {m, w} can block µ∗>t , then
{m,w}
(µ∗(t,t′ ) (i), µ≥t′
µ∗ (i)
(i)) ≻i ≤t
{m,w}
µ∗>t (i) ⇐⇒ (µ∗≤t (i), µ∗(t,t′ ) (i), µ≥t′
(i)) ≻i (µ∗≤t (i), µ∗>t (i))
for i ∈ {m, w}. Thus, {m, w} can block µ∗ , which is a contradiction. Hence, µ∗>t is dynamically stable.
Proof of Lemma 5.
(i) We propose a lexicographic extension of ≻i ∈ Si . For all x = x1 · · · xT and y = y1 · · · yT ,
ˆ i as follows:
define ≻
ˆ i y1 · · · yT k ⇐⇒
x1 · · · xT j ≻





x ≻i y
or
x = y & jP≻i k
ˆ i reflects P≻i = P≻ˆ i . Thus, ≻
ˆ i ∈ Sˆi .
It is simple to verify that ≻
38
(A.3)
ˆ ′i ∈ Ŝi be the one-period
(ii) Given ≻i ∈ S̄i , let ≻′i ∈ Si be such that ≻i ∈ Υ(≻′i ). Let ≻
ˆ i as follows:
ˆ ′i reflects P≻ˆ ′ = P≻′i . Define ≻
lexicographic extension, as in (A.3), of ≻′i . ≻
i
′
ˆ i ŷ ⇐⇒ x̂≻
ˆ i ŷ.
(a) If x̂T 6= x̂T +1 or ŷT 6= ŷT +1 , then x̂≻
ˆ i ŷ.
(b) If x̂T = x̂T +1 and ŷT = ŷT +1, then x̂≤T ≻i ŷ≤T ⇐⇒ x̂≻
ˆ i is a one-period extension of ≻i . To confirm that ≻
ˆ i ∈ Υ(≻
ˆ ′i ), we check two condi≻
tions.
′
ˆ i ŷ and x̂ D ŷ. To work toward a contradiction, assume ŷ ≻
ˆ i x̂. Then
(a) Suppose x̂≻
ŷT = ŷT +1 . Since x̂ D ŷ then x̂T = x̂T +1 . Therefore, ŷ≤T ≻i x̂≤T as well. But,
ˆ ′i x̂, which
x̂ D ŷ also implies x̂≤T D ŷ≤T and, therefore, ŷ≤T ≻′i x̂≤T . But then ŷ ≻
ˆ i ŷ.
is a contradiction. Therefore, x̂≻
(b) Suppose x̂ k ŷ. This implies x̂≤T 6= ŷ≤T ; else, the two partnership plans could be
ordered by D.
ˆ ′i ŷ. As x̂≤T 6= ŷ≤T , it follows that x̂≤T ≻′i ŷ≤T . Suppose ŷ ≻
ˆ i x̂.
First, suppose x̂≻
This is possible only if ŷT = ŷT +1 and ŷ≤T ≻i x̂≤T . But if x̂≤T ≻′i ŷ≤T , then it
must be that ŷ≤T D x̂≤T as ≻i ∈ Υ(≻′i ). This implies (ŷ≤T , ŷT +1) D (x̂≤T , x̂T +1 ),
ˆ i ŷ.
which contradicts ŷ 4 x̂. Thus, x̂≻
′
ˆ i ŷ. To derive a contradiction, suppose ŷ ≻
ˆ i x̂. Thus, x̂T = x̂T +1
Conversely, let x̂≻
′
and x̂≤T ≻i ŷ≤T . However, since x̂≤T 6= ŷ≤T , ŷ≤T ≻i x̂≤T . Thus, x̂≤T D ŷ≤T . As
ˆ ′i ŷ.
x̂T = x̂T +1 , then x̂ D ŷ—a contradiction. Therefore, x̂≻
ˆ ′i analogously, i.e. ≻
ˆi ∈
Proof of Theorem 5. Let ≻′i ∈ Si be such that ≻i ∈ Υ(≻′i ). Define ≻
′
ˆ i ). Since µ∗ is dynamically stable, i cannot block it. As ≻
ˆ i extends ≻i ,
Υ(≻
ˆ (µ∗ (i), ī[t,T ] , i) .
µ∗ (i) i (µ∗<t (i), ī[t,T ] ) =⇒ (µ∗1 (i), . . . , µ∗T (i), µ∗T (i)) |
{z
} i | <t {z
}
µ̂∗ (i)
(µ̂∗<t (i),ī[t,T̂ ] )
Hence, i cannot block µ̂∗ in period t ≤ T .
ˆ i µ̂∗ (i). Thus, µ̂∗T (i) 6= i and, therefore,
Suppose i can block µ̂∗ in period T̂ , i.e. (µ̂∗≤T (i), i)≻
ˆ ′i µ̂∗ (i) and, therefore, iP≻ˆ ′ µ̂∗ (i) =⇒ iP≻′i µ̂∗T (i).
µ̂∗ (i) D (µ̂∗≤T (i), i). This implies (µ̂∗≤T (i), i)≻
i
T̂
Let t′ ≤ T be the smallest value such that µ∗t′ (i) = · · · = µ∗T (i). Thus,
µ̃(i) ≡ (µ∗<t′ (i), ī≥t′ ) ≻′i (µ∗<t′ (i), µ∗t′ (i), . . . , µ∗T (i)) = µ∗ (i).
39
But, µ̃(i) D µ∗ (i) and thus, µ̃(i) ≻i µ∗ (i). Hence, µ∗ can be period-t′ blocked by i in the
T -period economy. This is a contradiction. Therefore, i cannot block µ̂∗ .
Consider the pair m and w. They cannot block µ∗ in period t ≤ T . From the definition
of stability, µ∗ (m) m (µ∗<t (m), w̄[t,T ] ) and µ∗ (w) w (µ∗<t (w), m̄[t,T ]). Hence,
ˆ m (µ∗<t (m), w̄[t,T ], w) and
(µ∗≤T (m), µ∗T (m)) {z
}
|
µ̂∗ (m)
Thus, m and w cannot block µ̂∗ in period-t.
ˆ w (µ∗<t (w), m̄[t,T ] , m).
(µ∗≤T (w), m) {z
}
|
µ̂∗ (w)
Suppose m and w can period-T̂ block µ̂∗ :
ˆ m µ̂∗ (m) and µ̃(w) ≡ (µ̂∗≤T (w), m)≻
ˆ w µ̂∗ (w).
µ̃(m) ≡ (µ̂∗≤T (m), w)≻
Of course, this implies µ̂∗T +1 (m) = µ∗T (m) 6= w. By reasoning analogous to the single-agent
ˆ ′m µ̂∗ (m) ⇐⇒ wP≻ˆ ′ µ̂∗ (m). Similarly, mP≻ˆ ′ µ∗T (w).
case above, µ̂∗ (m) D µ̃(m) =⇒ µ̃(m)≻
w
m T̂
Let t′i be the smallest index such that µ∗t′ (i) = · · · = µ∗T (i) and let t′ = max{t′m , t′w }.
i
Then,
ˆ ′m (µ∗<t′ (m), µ∗[t′ ,T ] (m), w) = µ̃(m).
(µ∗<t′ (m), w̄[t′ ,T ] , w)≻
Given the definition of t′ , (µ∗<t′ (m), µ∗[t′ ,T ] (m), w) ⋫ (µ∗<t′ (m), w̄[t′ ,T ] , w). And so,
ˆ m µ̃(m).
(µ∗<t′ (m), w̄[t′ ,T ] , w)≻
ˆ m µ̂∗ (m) and likewise for w, (µ∗<t′ (m), w̄[t′ ,T ] , w)≻
ˆ w µ̂∗ (w). Thus,
But then (µ∗<t′ (m), w̄[t′ ,T ] , w)≻
m and w can block µ̂∗ in period t′ ≤ T . Above, we established that this is not possible.
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