aa r X i v : . [ ec on . T H ] M a r Dynamically Stable Matching ∗ Laura Doval † March 6, 2020
Abstract
I introduce a stability notion, dynamic stability , for two-sided dynamicmatching markets where (i) matching opportunities arrive over time, (ii) match-ing is one-to-one, and (iii) matching is irreversible. The definition addressestwo conceptual issues. First, since not all agents are available to match atthe same time, one must establish which agents are allowed to form block-ing pairs. Second, dynamic matching markets exhibit a form of externalitythat is not present in static markets: an agent’s payoff from remaining un-matched cannot be defined independently of what other contemporaneousagents’ outcomes are. Dynamically stable matchings always exist. Dynamicstability is a necessary condition to ensure timely participation in the econ-omy by ensuring that agents do not strategically delay the time at which theyare available to match. K EYWORDS : dynamic stability, dynamic matching, stable matching, non-transferable utility, externalities,credibility, market design, dynamic arrivals, aftermarkets JEL
CLASSIFICATION : D47, C78 ∗ This paper supercedes my previous paper, “A theory of stability in dynamic matching mar-kets.” It is a revised version of the first chapter of my dissertation at Northwestern University.I am indebted to Eddie Dekel, Jeff Ely, and Alessandro Pavan for many fruitful conversationsand for their continuing guidance and support. I also wish to thank H´ector Chade, FedericoEchenique, Jan Eeckhout, Guillaume Haeringer, John Hatfield, George Mailath, Pablo Schenone,James Schummer, Vasiliki Skreta, Alex Teytelboym, Asher Wolinsky, Leeat Yariv, Vijay Vazirani,Charles Zheng, and especially, Erik Eyster and Jacob Leshno for useful discussions. All errors are,of course, my own. † California Institute of Technology, Pasadena, CA 91125. E-mail: [email protected] Introduction
I formulate a notion of stability, denoted dynamic stability , for two-sided dynamicmatching markets where (i) matching opportunities arrive over time, (ii) match-ing is one-to-one, and (iii) matching is irreversible. Stability notions provide ananalyst with a set of predictions for the self-enforcing outcomes of decentralizedmatching markets that depend only on the primitive payoff structure. While sta-bility notions are extensively used in the study of static matching markets, theyhave not been systematically studied for dynamic matching markets, even thoughthe latter are ubiquitous and cover many important applications, such as labormarkets and child adoption.Defining stability in a dynamic matching market brings forth two new chal-lenges that arise when taking into account agents’ intertemporal incentives. First,since not all agents are available to match at the same time, it is natural to askwhich pairs of agents can object to a proposed matching. Dynamic stability as-sumes that only agents who are available to match at the same time can form ablocking pair. Second, whether an agent finds their matching partner acceptabledepends on what their value of remaining unmatched is. In turn, this value de-pends on what matching the agent conjectures would ensue upon their decision toremain unmatched. Given a conjectured continuation matching, one could definean agent’s acceptable partners to be those who are preferred to the continuationmatching. This, together with a specification of the set of blocking pairs, is enoughto determine whether a matching is stable in the dynamic economy: it should haveno blocking pairs and agents should always be matched to acceptable partners.The missing step is then to determine what matching the agent conjectureswould result following their decision to remain unmatched. The first difficultyis that the set of agents available to match from tomorrow onward depends bothon the arrivals into the economy and on who remains unmatched from previ-ous periods. In other words, today’s matching together with tomorrow’s arrivalsdefine the set of feasible continuation matchings. When contemplating remain-ing unmatched, the agent then needs to conjecture both who else remains un-matched today and tomorrow’s continuation matching. Thus, as in the litera-ture on the core with externalities (see, for instance, Shapley and Shubik (1969);Rosenthal (1971); Richter (1974); Sasaki and Toda (1996); Pycia and Yenmez (2017);Rostek and Yoder (2017)), an agent’s payoff from remaining unmatched cannotbe defined independently of what other contemporaneous agents’ matching out-comes are. This externality sets apart dynamic matching markets from their static2ounterparts.Given a conjecture about who else remains unmatched today, not all continua-tion matchings are equally reasonable . Indeed, the agent should correctly anticipatethat the continuation matching should be itself self-enforcing. Thus, for a givenconjecture about today’s matching outcome, the agent rules out those continua-tion matchings that are not self-enforcing. This is still not enough to pin down aunique continuation matching and, thus, the value of remaining unmatched. Fora given conjecture about who else remains unmatched today, there can be manyself-enforcing matchings. Moreover, there can be many conjectures about who elseremains unmatched today. Thus, the last step in determining the value of remain-ing unmatched is an assumption on how the agent selects amongst the reasonableconjectures. Like Sasaki and Toda (1996), I assume the agent’s value of remain-ing unmatched coincides with the agent’s payoff at the most pessimistic conjec-ture. Unlike Sasaki and Toda (1996), the agent does not entertain all continuationmatchings but only those that are self-enforcing in the continuation economy.Dynamic stability (Definition 5) builds on the elements described above. Amatching for the dynamic economy is dynamically stable if (i) there is no pair ofagents who are available to match at the same time who prefer to match togetherand (ii) there is no agent who is matched to someone who is unacceptable. Con-trary to static notions of stability, the set of acceptable partners today is definedusing the set of dynamically stable matchings from tomorrow onward. Dynami-cally stable matchings always exist in any finite horizon economy (Theorem 1); Idiscuss their properties in Section 5.The study of dynamic matching markets opens up new questions of interest tomarket design, two of which are analyzed in Section 6. First, in a dynamic market,agents can manipulate a matching outcome by delaying the time at which they areavailable to match. Proposition 4 in Section 6.1 shows that dynamic stability is anecessary condition for timely participation in the market: whenever a matchingfails to satisfy dynamic stability, then market participants have an incentive todelay the time at which they are available to match. This echoes the result fromstatic matching markets that stability is a necessary condition for participation(Roth (1984a)). Second, the time at which a matching is finalized and, therefore,when an agent can object to a matching, also becomes part of the design. This isparticularly important for sequential assignment problems, and in particular, af-termarkets in school choice (Pathak (2016)). Section 6.2 analyzes the implicationsof the theory for this application and highlights the assumptions regarding thetiming of matching and blocking implicit in the definition of dynamic stability.3 elated literature
The paper contributes mainly to three strands of the litera-ture. The first strand is the literature on market design, which studies dynamicmatching markets such as those in this paper, but from the point of view of op-timality instead of stability ( ¨Unver (2010), Leshno (2017), Akbarpour et al. (2017),Anderson et al. (2015), Schummer (2015), Bloch and Cantala (2017),Ashlagi et al.(2018), Arnosti and Shi (2018),Baccara et al. (2019),Thakral (2019)). The presentstudy of stability is important because stability is considered a key property forthe success of algorithms (Roth (1991)) and because it highlights the potential is-sues in applying the static notions of stability to dynamic matching markets.The second strand is the literature on matching with frictions, which studiesdynamic matching markets such as those in this paper in a non-cooperative frame-work (see Burdett and Coles (1997), Eeckhout (1999), Adachi (2003), and Lauermann and N ¨oldeke(2014) for non-transferable utility, and Shimer and Smith (2000) for transferableutility). Like in this strand of the literature, an agent’s value of remaining un-matched (their continuation value) is determined endogenously by the remainingagents in the market and the future matching opportunities.The third strand studies stability notions for markets where matching oppor-tunities are fixed and pairings can be revised over time (e.g., Damiano and Lam(2005); Kurino (2009); Kadam and Kotowski (2018); Liu (2018); Kotowski (2019)).The contribution relative to this strand is to provide stability notions for marketswhere matching opportunities arrive (stochastically) over time and matching isirreversible. As discussed in the introduction, that matching opportunities arriveover time and matching is irreversible introduces a primitive externality that isabsent from these papers and must be addressed when defining what stabilitymeans. In particular, while this paper shares with Liu (2018) and Kotowski (2019)the perfection requirement (see also, Doval (2015, 2018)) and with Kotowski (2019)the use of pessimistic conjectures as in Sasaki and Toda (1996), the motivation forusing pessimistic conjectures is different. While the externality in my paper is anintrinsic feature of the environment that must be addressed by the stability notion,the externality in Kotowski (2019) is a consequence of the perfection requirementin his stability notion, together with the assumption of non-separable payoffs.
Organization
The rest of the paper is organized as follows. The next sectionprovides an informal overview of the solution concept through two examples.Section 3 describes the model and Section 4 defines dynamic stability. Section 5shows that dynamically stable matchings exist and discusses their properties.4ection 6 discusses two practical implications of dynamic stability. Section 6.1shows that dynamic stability ensures timely participation in the market. Section 6.2contrasts dynamic stability with the solution concept in Doval (2015, 2018), dis-cussing the implications for sequential assignment problems, in particular, after-markets in school choice. All proofs are in the appendices.
I now illustrate dynamic stability and the main issues at hand by means of a two-period example. The reader interested in the model can skip to Section 3 and toSection 4 for the solution concepts, with little loss of continuity.The purpose of the example is three-fold. First, I illustrate how one can ex-tend the stability notion for static matching markets to the dynamic economy; Idenote this solution concept, the core . Second, I illustrate the conceptual issuespresented in the introduction and how dynamic stability deals with these. Finally,while in the example the core is a subset of the set of dynamically stable match-ings, a slight modification of the example shows that this is not always the case.Section 5 provides a further example where the core and dynamic stability makenon-overlapping predictions.The economy lasts for two periods, t ∈ {
1, 2 } . There are two sides, A and B . Theagents on side A are Erd˝os, Kuhn, and Gale; they arrive at t =
1. Agents on side B are R´enyi and Shapley, who arrive at t =
1, and Tucker and Nash, who arrive at t = One way to approach the problem is to ignore the dynamics and treat the model asa static matching model where an agent on side A prefers to match with Shapleyover matching with Tucker (resp., Nash) only if in the original economy, the utilityof matching with Shapley in t = discounted utility of match-ing with Tucker (resp., Nash) in t =
2. This is analogous to Debreu’s dated-goodsapproach in general equilibrium: matching with Tucker is defined by Tucker’s5haracteristics and the time at which he becomes available to match. Table 1 be-low lists the agents’ preferences. If ( Tucker, 1 ) (resp., ( Nash, 1 ) ) appears before ( Shapley, 0 ) in an agent’s ranking, then they prefer to wait 1 period to match withTucker (resp., Nash) over matching immediately with Shapley; that is, the 0s and1s are the exponents of the discount factor, and the list provides the ranking of thediscounted utilities. For side B , the lists represent the ranking of the utilities. A :Erd ˝os : ( Tucker, 1 ) (
R´enyi, 0 ) (
Nash, 0 ) Kuhn : ( Tucker, 1 ) (
Nash, 0 ) (
Shapley, 0 ) (
Nash, 1 ) Gale: ( Shapley, 0 ) (
Tucker, 1 ) B :R´enyi : Erd ˝osShapley : Kuhn GaleTucker: Gale Erd ˝os KuhnNash: Kuhn Erd ˝os Table 1: Preference listsIn a static economy, the core captures the set of self-enforcing outcomes. Recallthat a matching is in the core if there is no pair of agents who prefer each otherto the partners assigned by the matching and no agent who prefers the matchingin which they are single to the proposed matching. That is, the matching must bepairwise stable and individually rational. There is a unique core matching in thisexample, denoted by m C and represented in the left panel of Figure 1 below. Inwhat follows, a horizontal line separates matchings that occur in different periods. m C = Erd˝os R´enyiKuhn ShapleyGale Tucker ∅ Nash m N C = Erd˝os R´enyiGale ShapleyKuhn Tucker ∅ Nash
Figure 1: Core matching (left) and non-core matching (right) This approach is also analogous to matching with contracts : the contract specifies the agent onematches with and the time at which the matching occurs, which must be no earlier than the dateon which the agent becomes available. For an application, see Dimakopoulos and Heller (2019). Alternatively, I could have written the preferences over the dated goods, i.e., over ( Shapley, 1 ) , ( Tucker, 2 ) , etc. However, this would not contain all the information needed to evaluate the match-ings both from the perspective of t = t = .2 Back to the dynamic matching market The matching m C in Figure 1 is obtained by assuming that all pairs of agents canform a blocking pair and the set of acceptable matching partners corresponds tothe set of agents who are preferred to matching with oneself. Importantly, the setof acceptable partners can be defined from the primitives. I now follow a differentapproach and consider defining the set of blocking pairs and the set of acceptablepartners directly in the dynamic economy. The main lesson is that both matchingsin Figure 1 are consistent with dynamic stability.Dynamic stability assumes that only agents who are available to match at thesame time may form blocking pairs. This precludes agents from engaging in a di-rect agreement before both of them are present to carry it out. To see the differencebetween this assumption and the one in the core, consider why the matching m N C in Figure 1 is not in the core, even though it has no blocks involving a pair that isavailable to match at the same time. Note that Erd˝os and Tucker prefer to matchwith each other over m N C . Since the core allows all pairs to form a block, it is asif Erd˝os and Tucker can agree in advance to match with each other, even if Tuckeris not present to carry this plan out. However, since Erd˝os matches in t = m N C , he is not available to match when Tucker arrives; therefore, they are notallowed to form a blocking pair. Thus, for Erd˝os to match with Tucker, he mustprefer to wait for Tucker to arrive over matching with R´enyi. The next step is thento define an agent’s payoff from remaining unmatched; this, in turn, allows us todetermine whether Erd˝os prefers to wait for Tucker when the matching is m N C .Defining an agent’s payoff from remaining unmatched is not immediate: whatan agent can achieve by remaining single depends on the period the agent is sup-posed to match under the proposed matching. To see this, consider the matching m N C in Figure 1. In t =
2, Kuhn should be able to guarantee that his outcome is atleast as good as that of remaining single. This, together with the observation thatthere should not be any pairwise blocks amongst agents who match in period 2,implies that the period-2 matching must be in the core of the economy formed bythe remaining agents from t = t = t =
1. Erd˝os’ payoff fromremaining single in t = t =
1. Since the t = together with the arrivals defines the set of feasible matchingsin t =
2, Erd˝os needs to conjecture who else remains unmatched today and what7atching will result tomorrow. Thus, as in the literature of the core with externali-ties (see, for instance, Shapley and Shubik (1969); Rosenthal (1971); Richter (1974);Sasaki and Toda (1996)), one needs to specify everyone else’s outcomes in t = t = t =
1. This maynot be enough, however, to pin down Erd˝os’ value from remaining unmatched.Indeed, Figure 2 illustrates two matchings that are consistent with (i) Erd˝os re-maining single in t =
1, (ii) the period-2 matching being a (period-2) core match-ing for the remaining and newly arriving agents, and (iii) the period-1 matchingbeing individually rational and having no pairwise blocks amongst the agentswho match in t = m E = Kuhn ShapleyErd˝os R´enyiGale Tucker ∅ Nash m E = Gale ShapleyErd˝os TuckerKuhn Nash ∅ R´enyi
Figure 2: Two conjectures Erd˝os may have about the matching that ensues whenhe waits to be matched in period 2Note how Erd˝os’ matching outcome depends on the set of agents who match in t =
1. In the example, for each set of agents in t =
2, there is a unique (period-2) core matching. However, different conjectures about who matches in t = t =
2, as illustrated in Figure 2.The final step to determine Erd˝os’ payoff from remaining unmatched is to for-mulate an assumption about how Erd˝os evaluates the non-excluded conjectures.Dynamic stability assumes that Erd˝os forms a pessimistic conjecture: Erd˝os prefersto remain single over matching with R´enyi in t = all of thenon-excluded matchings to matching with R´enyi in t = reasonable matchings, the analyst would fail to rule outmatchings where such certainty does not exist. Another way is to acknowledgethe uncertainty faced by Erd˝os, both in terms of the stochasticity of tomorrow’sarrivals and the need to predict other period-1 agents’ outcomes. It is then natu-ral to assume that Erd˝os chooses to remain single in t = t = t =
1. Since m E is a reasonableconjecture for Erd˝os, it follows that by waiting to be matched, he can guarantee,at most, his payoff from matching with R´enyi in t =
2. Therefore, Erd˝os prefers tomatch with R´enyi in t = m E , in t = t =
2, there are instances in which Tucker is notwilling to match with Erd˝os, once Erd˝os waits for him to arrive.The above discussion implies that the matching m N C satisfies the conditions ofdynamic stability (Definition 5). The core matching m C is also dynamically sta-ble. Indeed, the matching m E in Figure 2 is a valid conjecture for Kuhn when heconsiders remaining single in t =
1. Since Kuhn prefers to match with Shapley in t = t =
2, he cannot object to the core matching bywaiting to be matched. In this example, the set of dynamically stable matchingsis a superset of the core. Hence, an outside observer may mistakenly rule out thematching m N C in Figure 1 by using the core as a solution concept. This is similar in spirit to the approaches of Ambrus (2006) and Liu et al. (2014). This has been assumed away in the example. .3 Not all core matchings are dynamically stable I now modify the above example to illustrate one final point: not all core match-ings are dynamically stable. The example is modified as follows. Kuhn and Galecontinue to arrive at t =
1, while Erd˝os now arrives at t =
2. Arrivals on side B are as before, except that now R´enyi no longer arrives. Preferences are given by Erd ˝os : NashKuhn : ( Tucker, 1 ) (
Nash, 0 ) (
Shapley, 0 ) (
Nash, 1 ) Gale: ( Shapley, 0 ) (
Tucker, 1 ) Shapley: Kuhn GaleTucker: Gale KuhnNash: Kuhn Erd ˝os
There are two matchings in the core of this economy, illustrated in Figure 3: m L = Kuhn ShapleyGale TuckerErd˝os Nash m R = Gale ShapleyKuhn TuckerErd˝os Nash Figure 3: Two core matchings; the one on the left is not dynamically stableHowever, matching m L is not dynamically stable: Kuhn can guarantee to bematched with Tucker by remaining unmatched in t =
1. To see this, note that Galeneeds to also match in t = t = t = m L by waiting to bematched. Since the core assumes that Kuhn’s value from remaining unmatched ishis payoff from remaining single, it fails to capture that in the dynamic economyShapley is not acceptable to Kuhn. The economy lasts for T < ∞ periods. There are two sides A and B . Agents onside A are labeled a ∈ A , while agents on side B are labeled b ∈ B , where A , B are finite sets. Appendix E shows an economy where the two core matchings that correspond to the out-comes of the deferred acceptance algorithm are not dynamically stable. t ≥
1, let E t denote the subset of ( A × B ) t that satisfies the followingproperty. A tuple E t = ( A , B , . . . , A t , B t ) ∈ E t only if A i ∩ A j = ∅ and B i ∩ B j = ∅ , whenever i = j . An economy of length T is a distribution G T on E T . In whatfollows, I refer to an element E t of E t as a realization, to a tuple ( A , . . . , A t ) as aside-A arrival, and to a tuple ( B , . . . , B t ) as a side-B arrival.Given a realization E t = ( A , B , . . . , A t , B t ) , let A s ( E t ) = ∪ st ′ = A t ′ denote theimplied arrivals on side A through period s ≤ t ; similarly, let B s ( E t ) = ∪ st ′ = B t ′ denote the implied arrivals on side B through period s ≤ t . Finally, if s ≤ t and E t = ( E s , E t − s ) , then I say that E s is a truncation of E t and that E t follows E s .Fix an economy G T . Definitions 1 and 2 below define the set of feasible alloca-tions for G T . Definition 1.
A period- t matching for realization E t is a mapping m t : A t ( E t ) ∪ B t ( E t ) A t ( E t ) ∪ B t ( E t ) such that1. For all a ∈ A t ( E t ) , m t ( a ) ∈ { a } ∪ B t ( E t ) ,2. For all b ∈ B t ( E t ) , m t ( b ) ∈ A t ( E t ) ∪ { b } ,3. For all k ∈ A t ( E t ) ∪ B t ( E t ) , m t ( m t ( k )) = k . Definition 2.
A matching m for G T is a map on ∪ Tt = E t such that1. For all t ∈ {
1, . . . , T } , for all E t ∈ E t , m ( E t ) is a period-t matching,2. For all t ∈ {
1, . . . , T } , for all E t ∈ E t , for all a ∈ A t ( E t ) , if m ( E t )( a ) = a , then m ( E s )( a ) = m ( E t )( a ) for all s ≥ t and E s that follow E t .Part 2 of Definition 2 incorporates the idea that matchings in the economy areirreversible. Let M T denote the set of matchings for an economy of length T . Finally, given a matching m ∈ M T and a realization E t , let M T ( m , E t ) denote thesubset of M T that coincide with m at realizations E s that do not follow E t . To economize on notation, I assume that no two agents with the same characteristic arrivewithin or across periods. Doval (2020) extends all results to the case where multiple agents withthe same characteristics can arrive within a period and across periods. The particular economy of length T , i.e., the distribution G T on E T , defines which arrivalsequences on the domain of m have positive probability. m and a realization of the arrivals through period t , E t = ( E t − , ( A t , B t )) .This determines the agents who are available to match at E t : A ( m t − , E t ) = { a ∈ A t − ( E t ) : m ( E t − )( a ) = a } ∪ A t B ( m t − , E t ) = { b ∈ A t − ( E t ) : m ( E t − )( b ) = b } ∪ B t .In the above definition, m t − is defined as ( m ( E ) , . . . , m ( E t − )) where for each s ≤ t − E s is a truncation of E t .A matching m and a realization E t also define a continuation economy of length T − t , where the distribution of arrivals G T − t ( m t , E t ) assigns probability G T (( A t + , B t + , . . . , A T , B T ) | E t ) to the length T − t arrival ( A ( m t , ( E t , A t + , B t + )) , B ( m t , ( E t , A t + , B t + )) , . . . , A T , B T ) ,and 0 to all others.I close the model by defining agents’ preferences. Each a ∈ A defines a discountfactor δ a ∈ [
0, 1 ) and a Bernoulli utility, u ( a , · ) : B ∪ { a } 7→ R . Similarly, each b ∈ B defines a discount factor δ b ∈ [
0, 1 ) and a Bernoulli utility, v ( · , b ) : A ∪ { b } 7→ R . I assume that for all a ∈ A and all b ∈ B , u ( a , a ) = v ( b , b ) = m and a realization E t . Let a ∈ A ( m t − , E t ) . For each E T =( E t , · ) , let t m ( a , E T ) denote the smallest t ≤ s such that m ( E s )( a ) = a and E s is atruncation of E T ; otherwise, let t m ( a , E T ) = T . Then, let U ( a , m , E t ) = E G ( ·| E t ) [ δ t m ( a , E t , · ) − ta u ( a , m ( E t , · )( a ))] ,denote a ’s payoff from matching m at date t when the realization is E t . Similarly,for b ∈ B ( m t − , E t ) , let V ( b , m , E t ) = E G ( ·| E t ) [ δ t m ( b , E t , · ) − tb v ( m ( E t , · )( b ) , b )] ,denote b ’s payoff from matching m at date t when the realization is E t .12 Stability and the core
Section 4 formally introduces the two solution concepts, dynamic stability and thecore. To define dynamic stability, I must introduce three concepts. Definition 3extends the notion of pairwise stability from static markets to dynamic markets,where only agents who are available to match at the same time can form blockingpairs. Definition 4 introduces a minimal requirement that a matching must satisfywithin a given period. I use this requirement to define recursively the set of con-tinuation matchings an agent conjectures when the agent contemplates waitingto be matched. With these three objects in hand, Definition 5 contains the formalstatement of dynamic stability.In what follows, I denote a matching m as individually rational if for all real-izations E T , for all a ∈ A T ( E T ) , and all b ∈ B T ( E T ) , u ( a , m ( E T )( a )) ≥ v ( m ( E T )( b ) , b ) ≥ Definition 3.
A matching m is pairwise stable if there is no realization E T and pair ( a , b ) ∈ A T ( E T ) × B T ( E T ) and t ≤ T such that1. ( a , b ) ∈ A ( m t − , E t ) × B ( m t − , E t ) ,2. u ( a , b ) > U ( a , m , E t ) ,3. v ( a , b ) > V ( b , m , E t ) .When T =
1, Definition 3 is exactly the definition of pairwise stability for staticmatching markets; hence, the name for the condition. Definition 3 states that thereshould be no pair, ( a , b ) such that (i) they are both available to match at the samerealization, and (ii) they prefer to match together rather than to match accordingto the proposed matching, m . Condition (i) does not imply that the agents in thepair ( a , b ) arrive in the same period. Instead, it reflects the notion that in a dy-namic matching market, agents who arrive in different periods can make directagreements only if they are available to match in the same period.While Definition 3 rules out the presence of blocking pairs for a given matching m , Definition 4 rules out the presence of blocking pairs in a given period: Definition 4.
Fix a matching m , a period t ≤ T , and a realization E t . m ( E t ) is stableamongst those who match at E t if the following holds:1. For all a ∈ A ( m t − , E t ) , u ( a , m ( E t )( a )) ≥ b ∈ B ( m t − , E t ) , v ( m ( E t )( b ) , b ) ≥ ( a , b ) ∈ A ( m t − , E t ) × B ( m t − , E t ) such that m ( E t )( a ) = a , m ( E t )( b ) = b , u ( a , b ) > u ( a , m ( E t )( a )) , and v ( a , b ) > v ( m ( E t )( b ) , b ) .There are three differences between Definitions 3 and 4. First, Definition 3 doesnot require that the matching be individually rational. That is, a matching m cansatisfy Definition 3 but fail Definition 4 because at realization E t , a ∈ A ( m t − , E t ) is matched to some b ∈ B ( m t − , E t ) such that u ( a , b ) <
0. Second, while Definition 4checks for the presence of blocking pairs within a period, Definition 3 also con-trols for blocking pairs across periods. As an example of this, recall Figure 2. Thematching m E satisfies Definition 4 for every period t ∈ {
1, 2 } . However, it is notpairwise stable because Kuhn and Shapley, who are available to match in t = t =
1. Finally, Definition 4 only checks for thepresence of blocking pairs amongst agents who match within a period. That is,a matching m can satisfy Definition 4 but fail Definition 3 because at realization E t there is a pair a , b ∈ A ( m t − , E t ) ∪ B ( m t − , E t ) such that m ( E t )( a ) = a and m ( E t )( b ) = b , who prefers to match together than to their eventual matchingpartners under m .Dynamic stability is defined recursively: to determine if m is dynamically stablefor G T , one needs to know which matchings are dynamically stable for the contin-uation economies that are consistent with G T , G T − t ( m t , E t ) . As in the example inSection 2, the dynamically stable continuation matchings define an agent’s conjec-tures. The last piece of notation before Definition 5 accomplishes this recursion.Given an economy of length t , G t , let D t ( G t ) denote the dynamically stablematchings for G t . Fix an economy of length T , G T . Fix a matching m ∈ M T , arealization E t , and an agent k ∈ A ( m t − , E t ) ∪ B ( m t − , E t ) . Suppose one is giventhe correspondences D , . . . , D T − t . This is enough to define the set of continuationmatchings that k conjectures may ensue if k decides to match later than period t .The set of matchings that k conjectures at E t , denoted by M D ( k , m , E t ) , is definedas follows. Since one cannot alter the matchings through period t − M D ( k , m , E t ) is a subset of M T ( m , E t ) . Moreover, any matching m in M D ( k , m , E t ) satisfies thefollowing properties: (i) k is unmatched at E t , (ii) the continuation matching isdynamically stable for the continuation economy G T − t ( m t , E t ) , and (iii) m ( E t ) isstable amongst those who match at E t . Condition (ii) implies that k conjecturesthat the continuation economy abides by the same rules when determining whichoutcomes are self-enforcing. Condition (iii) implies that k rules out that the agentswho match at E t , when k remains unmatched, would agree on a matching that14hey could have improved upon. Formally, M D ( k , m , E t ) = m ∈ M T ( m , E t ) : ( i ) m ( E t )( k ) = k ( ii ) ( m ( E t + s )) s ∈{ T − t } ∈ D T − t ( G T − t ( m t , E t ))( iii ) m ( E t ) satisfies Definition 4 .(1) There is a slight abuse of notation in item (ii) of Equation 1. While D T − t ( G T − t ( m t , E t )) defines a matching only for the agents that are unmatched from t + m also specifies the outcome for those who have matched through realization E t .Item (ii) should then be read as “ ( m ( E t + s )) s ∈{ T − t } coincides with an elementof D T − t ( · ) for the agents who are yet to be matched at the end of period t .”We are now ready to define dynamic stability: Definition 5.
Let G T denote an economy of length T . Suppose one has defined thecorrespondences ( D t ) t ≤ T − . A matching m is dynamically stable for G T if1. m is pairwise stable,2. For all t ∈ {
1, . . . , T } , all realizations E t on the support of G T , and all a ∈A ( m t − , E t ) such that m ( E t )( a ) = a , there exists m ∈ M D ( a , m , E t ) such that U ( a , m , E t ) ≥ U ( a , m , E t ) .3. For all t ∈ {
1, . . . , T } , all realizations E t on the support of G T , and all b ∈B ( m t − , E t ) such that m ( E t )( b ) = b , there exists m ∈ M D ( b , m , E t ) such that V ( b , m , E t ) ≥ V ( b , m , E t ) .Let D T ( G T ) denote the matchings in M T that are dynamically stable for G T .I now unpack the definition of dynamic stability by considering different val-ues of T . Consider first T = m for the one-period economy.Condition 1 in Definition 5 states that for all realizations E , the matching m ( E ) Condition (iii) allows k to conjecture that no one matches at realization E t as a consequenceof their decision to wait to be matched. However, as evidenced by the proof of Theorem 1, thisconjecture never plays a role, except in the case where all the agents in A ( m t − , E t ) ∪ B ( m t − , E t ) do not find each other acceptable, in which case any other matching would violate the requirementof individual rationality in Definition 4. E = ( A , B ) and consider now Condi-tion 2 for a ∈ A . Note that all matchings m in the set M D ( a , m , E ) satisfy that m ( a ) = a (part (ii) in the definition of M D is vacuous). Thus, Condition 2 sim-ply states that a prefers m to remaining single. Thus, when T =
1, Definition 5reduces, for each realization E , to the definition of stability for static economies.It follows that the correspondence D maps one-period economies to the set of ex-post stable matchings.Suppose now that T > m denote an element of M T . Again, Condition 1implies that m has no pairwise blocks amongst agents who are available to matchat the same realization. The discussion in the previous paragraph implies that m ( E T ) is individually rational. Thus, m ( E T ) is in the core of the economy definedby ( A ( m T − , E T ) , B ( m T − , E T )) .Consider now the period T − m ( E T − ) . Conditions 2 and 3 implythe following. Consider a ∈ A ( m T − , E T − ) who is supposed to match at E T − .If instead a remains single at E T − , a anticipates that any continuation matching m satisfies that m ( E T ) is a core matching for the remaining unmatched agents inperiod T , i.e., m ( E T − , · ) ∈ D ( · ) . The elements of M D ( a , m , E T − ) differ in twoaspects. First, if m , m ′ ∈ M D ( a , m , E T − ) match the same set of agents at E T − , thenthey can differ in the core matching they select for the continuation economies inperiod T . Second, m , m ′ may correspond to different configurations of agents whomatch at E T − , which in turn induces different configurations of agents who areavailable to match in period T . Condition 2 states that a waits until period T only if all such matchings are strictly preferred to matching at E T − . Workingbackward, a matching m is dynamically stable if there is no period t , realization E t , and an agent who is supposed to match at E t such that the agent prefers allmatchings starting from E t that specify (i) dynamically stable continuations and(ii) a matching that is stable for the agents who match at E t .Note that when considering whether to match at E t or wait to be matched,the agents use the correct distribution over continuation arrivals when calculat-ing their payoffs. Thus, while Definition 5 has a flavor of pessimistic beliefs , thispessimism only manifests through the matching the agent conjectures would en-sue in the event that they remain unmatched.While Definition 5 never explicitly requires that the matching be individuallyrational, it follows from it that a dynamically stable matching is individually ra-tional. As the previous discussion shows, if m is dynamically stable, then for all E T , m ( E T ) is individually rational for the agents in A ( m T − , E T ) ∪ B ( m T − , E T ) .16his, in turn, implies that all agents who match at a realization E T − accordingto m must be matched to partners whom they prefer to remaining single. Oth-erwise, Conditions 2 and 3 of Definition 5 would be violated: all matchings in M D ( k , m , E T − ) involve individually rational period- T matchings. Working back-ward through T − m is individually rational.The above observation formalizes the idea that in the dynamic economy the de-cision to wait to be matched replaces the decision to remain single in the staticeconomy. That is, Conditions 2 and 3 extend the static notion of individual ratio-nality to the dynamic economy. Moreover, it highlights that in a dynamic econ-omy an agent’s “continuation value” is determined endogenously by what he canachieve by remaining unmatched in one period and possibly matching in the fu-ture. I conclude the section by introducing the definition of the core for the dynamiceconomy. To simplify the exposition, I only do so for the case in which there is nouncertainty over the arrivals, i.e., G T assigns probability 1 to one realization E T .Appendix C presents the statement for the case in which there is uncertainty overthe arrivals. When G T is not degenerate, the core has a many-to-many structure,which complicates its definition. After all, when there is uncertainty over the ar-rivals, one must take into account that, given a realization E t , an agent a ∈ A t ( E t ) may be matched with different side- B agents depending on the realization in pe-riod t +
1, as long as a does not match at E t . Thus, when comparing matchings, a is comparing sets of matching partners. Definition 6.
Fix an economy, E T . A matching m is in the core if1. For all t ∈ {
1, . . . , T } , for all a ∈ A t , b ∈ B t U ( a , m , E t ) ≥ V ( b , m , E t ) ≥
02. There is no pair ( a , b ) ∈ A T × B T such that δ max { s − t ,0 } a u ( a , b ) > U ( a , m , E t ) and δ max { t − s ,0 } b v ( a , b ) > V ( b , m , E s ) ,whenever a ∈ A t , b ∈ B s . Let C T ( E T ) denote the set of core matchings of E T . This resembles the logic in the literature of search and matching. Instead, when there is no uncertainty over the arrivals, the dated-goods economy reduces to astatic, one-to-one, two-sided matching market as in Gale and Shapley (1962). Thus, to determinewhether a matching is in the core it is without loss of generality to check that it is individuallyrational and has no blocking pairs. a and b can form a blocking coalition even if m t ( a ) = a and b arrives at a period later than t (this is similar to Erd˝os and Tucker inSection 2). Implicit in this is that b can promise to match with a when a waits for b to arrive. Second, in the core, blocking coalitions compare the payoffs they obtainby blocking with the payoffs from m . Whereas in dynamic stability, an agent a ,who blocks by waiting, compares his payoff from m with the payoffs he obtainsat the continuation matching originated by the block. Theorem 1 presents the main existence result: the set of dynamically stable match-ings is non-empty.
Theorem 1.
For all T ∈ N , the correspondence D T is non-empty valued on { G T : G T ∈ ∆ ( E T ) } . The proof of Theorem 1 is in Appendix A. The proof shows how to construct amatching (labeled m ⋆ in the proof) that is dynamically stable. The proof constructsthe matching forward from t = t = T . This is necessary because the setof conjectures of an agent that matches in period t depends on the set of agentsavailable to match from that period onwards, which, in turn, depends on howagents have matched before period t .To construct the matching, I adapt the proof technique in Sasaki and Toda (1996)to the dynamic matching market. Consider the case in which T =
2. Fix a real-ization E . For each agent k ∈ A ∪ B , I calculate k ’s continuation value to bethe payoff from the worst matching in M D ( k , · , E ) . When T =
2, the latter set isnon-empty because the set of static stable matchings in period 2 is non-empty re-gardless of the realization of the arrivals. With these continuation values at hand,I truncate k ’s preferences so that k only deems acceptable period-1 agents that arepreferred to this continuation value. The matching at E , m ⋆ ( E ) , is constructedby running deferred acceptance with agents in A proposing to agents in B withthe truncated preferences. This, in turn, determines the set of unmatched agentsat the end of period 1. For each E , one then chooses a stable matching amongstthe newly arriving agents and the remaining ones from period 1. The proof thenchecks that the matching constructed this way is pairwise stable.18espite the similarities with Sasaki and Toda (1996), there are also differences.First, the set of available conjectures in a given period is endogenous to the match-ing being constructed. Second, the elements in the set of conjectures must them-selves satisfy the conditions of dynamic stability in the continuation economy. Thelatter means that the proof must proceed by induction on T : to show that dynamicstability is well defined for T , one must show that it is well defined for T ′ < T . Fi-nally, one should note that the conjectures in Sasaki and Toda (1996) describe whateach agent k expects others will do in the event that k matches with someone onthe other side, but not what would happen in case k remains unmatched. Indeed,Sasaki and Toda (1996) define stability as the absence of pairwise blocks. Con-trast this with the definition of dynamic stability, where the conjectures describewhat an agent k expects others will do in the event that k remains unmatched ina given period, but not what others will do if k forms a pairwise block. This is areflection of the nature of the externality in the dynamic environment under con-sideration. Since there are no payoff externalities, a pair can evaluate the payofffrom matching together independently of what other agents in the economy maydo. On the contrary, when contemplating whether to wait an additional period tobe matched, an agent needs to take into account who else may remain unmatchedsince this determines the set of feasible matchings in the continuation economy. Sasaki and Toda (1996) show that a necessary condition for existence is that theset of conjectures contains the set of all matchings. This, however, does not contra-dict the result in Theorem 1. The reason is two-fold. First, while Sasaki and Toda(1996) allow for general payoff externalities, the externalities present in the dy-namic environment are different in nature. Second, conjectures are endogenouslydetermined under dynamic stability, while they are exogenously given in Sasaki and Toda(1996). Indeed, Hafalir (2008) argues that it is the exogenous nature of the conjec-tures in Sasaki and Toda (1996) which implies their (negative) result.While dynamically stable matchings are guaranteed to exist, as Proposition 1below shows the core may be empty: This final difference between the two models manifests also in the proof of Theorem 1.Sasaki and Toda (1996) use their exogenous conjectures to define for each agent k a ranking overthe agents on the other side. They then show that the matchings that are pairwise stable relativeto these artificial rankings are stable also under the original rankings. Similarly, I use the conjec-tures to define an agent’s value of remaining unmatched, which in turn determines their set ofacceptable partners within a period. Similar logic to that in Sasaki and Toda (1996) implies thatthe matching, m ⋆ , I construct using these values of remaining unmatched satisfies that no agenthas an incentive to delay the time at which they are matched. This does not immediately implythat m ⋆ has no pairwise blocks, so the last step of the proof needs to check this is indeed the case. roposition 1. When there is no uncertainty about arrivals, C T is non-empty valued on { E T : E T ∈ E T } . Moreover, there exist economies G T such that C T ( G T ) = { ∅ } . The first part of Proposition 1 follows from applying the algorithm in Gale and Shapley(1962) to the “dated-goods” economy as in the example in Section 2. The secondpart, which is illustrated in Example 2 in Appendix C, follows from the observa-tion that when there is uncertainty over the arrivals, the core has a many-to-manystructure. In a companion working paper Doval (2016), I show that the dynamiceconomy can be embedded in a many-to-many matching model so that the core inDefinition 8 corresponds to the core of the static many-to-many model. Moreover,when there is uncertainty over the arrivals, the value of matching with an agenton side B at a particular realization depends on who else is available to match atother realizations. This implies that the preferences in the corresponding many-to-many model exhibit complementarities. Consider the following example. In t = B = { b } , and, in t =
2, with probability p , b arrives, and with the re-maining probability, no one arrives on side B . Assume u ( a , b ) > u ( a , b ) and pu ( a , b ) + ( − p ) u ( a , b ) > u ( a , b ) > pu ( a , b ) . Then, a ’s willingness to matchwith b depends on whether b is willing to wait to match in t =
2. It is well-known in static many-to-(one)many matching that complementarities precludeexistence (Kelso and Crawford (1982)); however, the observation that these com-plementarities can be brought about by the stochasticity of the arrivals is uniqueto the dynamic setting considered in this paper.The examples in Section 2 suggest that when the core is non-empty, there isalways a core matching that is dynamically stable. However, as Proposition 2records, this is an artifice of setting T = Proposition 2.
Let T = and G T be such that there is no uncertainty over the arrivals.Then, C ( G ) ∩ D ( G ) = ∅ . When T ≥ , there exist economies G T with no uncer-tainty over the arrivals and such that C T ( G T ) ∩ D T ( G T ) = ∅ . The proof of the first part of Proposition 2 is in Appendix C. An implication isthat, when the core is unique, then it is dynamically stable. One may be temptedto infer from this that the matchings obtained by deferred acceptance are dynam-ically stable. Example 3 in Appendix E shows, however, that this is not the case.In the example, the matchings obtained by using deferred acceptance with eitherside proposing are not dynamically stable; however, the median matching is dy-namically stable.To understand the second part of Proposition 2, consider the following versionof the example in Section 2: 20 xample 1.
Let T =
3. Assume there is no uncertainty in the arrivals and they areas follows. In t =
1, Erd˝os, R´enyi, and Shapley arrive. In t =
2, Kuhn and Galearrive. Finally, in t =
3, Tucker and Nash arrive. Preferences are as follows:
Erd ˝os : ( Tucker, 2 ) (
Shapley, 0 ) (
R´enyi, 0 ) Kuhn : ( Tucker, 1 ) (
Nash, 0 ) (
Shapley, 0 ) (
Nash, 1 ) Gale: ( Shapley, 0 ) (
Tucker, 1 ) R´enyi : Erd ˝osShapley : ( Kuhn, 1 ) (
Gale, 0 ) (
Erd ˝os, 0 ) (
Gale, 1 ) Tucker: Gale Erd ˝os KuhnNash: Kuhn
It is easy to show that there is a unique core matching and that matching partnersare as in the matching m C in Figure 1: Erd˝os matches with R´enyi in t =
1, Kuhnmatches with Shapley in t =
2, and Gale matches with Tucker in t = { Kuhn, Gale, Shapley } , Kuhncan guarantee that he matches with Tucker by remaining single in t =
2. Indeed,the unique dynamically stable matching for that initial period 2 population hasGale and Shapley match in t = t = t =
1, can guaran-tee, at most, the payoff of matching with Gale in t =
2. Thus, anticipating this,he is willing to match with Erd˝os in t =
1. Indeed, the following is a dynamicallystable matching: m D = Erd˝os Shapley ∅ Kuhn NashGale Tucker ∅ R´enyi
The example highlights once again how feasible payoffs depend on the set ofagents available to match. Once Shapley matches in t =
1, Kuhn is no longer ableto match with Tucker. Kuhn and Shapley would both benefit from being able tomatch together in t =
2. However, this is not credible: if Erd˝os matches with R´enyiwhen Shapley waits for Kuhn to arrive, the only matching that is self-enforcingmatches Shapley with Gale.It is interesting to understand the difference between T = T ≥ M D ( · ) . Since the core is unique in Example 1, it seems one should be ableto replicate the construction in Proposition 2 for the case T = ∅ Gale ShapleyKuhn NashErd˝os Tucker ∅ R´enyi
Note that in this matching Erd˝os matches in t = M D ( Kuhn, m C , E ) , where m C is the core matching. After all, the set M D ( Kuhn, m C , E ) is constructed under the notion that bygones are bygones: onceone gets to t =
2, it is not possible to undo Erd˝os’ match with R´enyi. I believe thisis a natural restriction in a dynamic matching market. Notwithstanding this, itis easy to see from the proof of the first part of Proposition 2 that, without thisrestriction, there is always a core matching that satisfies a version of dynamic sta-bility that does not involve this constraint. The issue is that if this core matchingis not truly dynamically stable, then once time starts running, the matching is notself-enforcing unless one is allowed to undo past matchings.One of the most useful properties of stability in static, two-sided, one-to-onematching markets is the lattice property. Naturally, when there is no uncertaintyover the arrivals, the core inherits this property. As Example 5 in Appendix Eillustrates, dynamic stability does not have the lattice property.
The study of stability in dynamic matching markets opens up new questionsfor market design, two of which are explored in this section. Section 6.1 showsthat dynamic stability is a necessary condition to ensure timely participation inthe market: the agents can “game” any individually rational and pairwise sta-ble matching that is not dynamically stable by delaying the time at which theymake themselves available to match. This echoes the observation from staticmatching markets that stability ensures participation in the market (Roth (1984a)).Section 6.2 discusses the assumptions about the timing at which agents match and22orm blocking coalitions implicit in dynamic stability. I connect this discussion tothe literature on sequential assignments and aftermarkets in school choice.
The starting point of this section is the following simple observation. Suppose T = G assigns probability 1to E = ( A , B , A , B ) . Let m A − DA denote the matching obtained by running thedeferred acceptance algorithm with agents in A ∪ A making proposals (usingtheir intertemporal preferences) to agents in B ∪ B . The following holds: Proposition 3.
Let T = and let G be as above. Then, for all a ∈ A , there existsm ∈ M D ( a , m A − DA , E ) such thatU ( a , m A − DA , E ) ≥ U ( a , m , E ) .Proposition 3 shows that, in a two-period economy with no uncertainty in thearrivals, agents in A cannot improve on the outcome of side-A deferred accep-tance by waiting to be matched. The most interesting part of this observation ishow it is proved (see Section B.2). I show that if an agent in A could improveon deferred acceptance by waiting to be matched, then they would be able to ma-nipulate the outcome of deferred acceptance by pretending to be an agent whoarrives in period 2. That is, they would be able to obtain an improvement by lyingabout the time at which they are available to match. This observation motivatesthe analysis in this section.While in a static matching market agents can manipulate a matching outcomeby misrepresenting their preferences, a new form of manipulation is feasible in adynamic matching market: agents may misrepresent the time at which they be-come available to match. There are two ways in which they can do so. First, theycould arrive in the economy and not make themselves available to match imme-diately. Second, they could exit the economy before receiving a match (possiblyto rejoin the economy at a later point in time). These are similar to the ideas of balking and reneging in queueing, so I use this terminology below.Dynamic stability already embodies a notion of no reneging. After all, it stat-ess that there is no realization E t and agent k who matches at E t , who wouldinstead benefit from remaining unmatched at E t , possibly matching at some later23eriod, whenever the continuation matching is itself a dynamically stable match-ing. Proposition 4 below shows that dynamic stability also embodies no balking.Indeed, it states that if a pairwise stable and individually rational matching failsto satisfy dynamic stability for an agent who arrives in period t , then this agentmay benefit from balking. These two observations together imply that dynamicstability is a necessary condition to ensure timely participation in the market.In what follows, I explain the assumptions needed to state Proposition 4. Fix aneconomy of length T with stochastic arrivals, G T . Let m denote a matching for G T .Consider the following game. In each period, agents on both sides arrive accord-ing to G T . Upon arrival, each agent observes the remaining unmatched agentsfrom the previous period and the set of agents that have arrived within a period(and their characteristics). Upon arrival, each agent chooses whether to reveal thatthey have arrived. If they reveal this, however, their characteristics are immedi-ately observed. Given a sequence of reported arrivals through period t , ˆ E t , agentsare matched according to m ( ˆ E t ) .The result requires that the distribution over E T satisfies an independence con-dition. Say G T satisfies exchangeability if G T ( E , . . . , E T ) = G T ( ˜ E , . . . , ˜ E T ) when-ever A T ( E T ) = A T ( ˜ E T ) and B T ( E T ) = B T ( ˜ E T ) . That is, the probability of tworealizations E T and ˜ E T only depends on the arrivals they induce through period T (and not on how they came to be.) Finally, say that a matching m for economy G T is not dynamically stable for arriving agents if whenever there exists a realization E t and an agent k ∈ A ( m t − , E t ) ∪ B ( m t − , E t ) who would benefit from waitingto be matched, then there exists an agent k ′ who would benefit from waiting to bematched and arrived in period t . We are now ready to state Proposition 4: Proposition 4.
Suppose G T has full support and satisfies exchangeability. Let m ∈ M T be pairwise stable and individually rational and suppose that m is not dynamically stablefor arriving agents. Then, there exists a period t, a realization E t = ( E t − , A t , B t ) , andk ∈ A t ∪ B t such that1. m ( E t )( k ) = k, and2. k prefers m ( E t \ { k } , · ) to m ( E t ) . Formally, if k = a, then E t \ { a } = ( E t − , A t \{ a } , B t ) and u ( a , m ( E t )( a )) < E G ( ·| E t ) [ δ t m ( E t \{ a } , E t + ∪{ a } , · ) a u ( a , m T ( E t \ { a } , E t + ∪ { a } , · )( a ))] , (2) and similarly if k = b, then E t \ { b } = ( E t − , A t , B t \ { b } ) and v ( m ( E t )( b ) , b ) < E G ( ·| E t ) [ δ t m ( E t \{ b } , E t + ∪{ b } , · ) b v ( m T ( E t \ { b } , E t + ∪ { b } , · )( b ) , b )] . (3)
24o fix ideas, suppose k = a . Then, Equation 2 shows that a would improveon his outcome by waiting until period t + That is, when m is not dynamically stable, a has an incentive to balk , i.e., leavewithout joining the economy. When everyone else reports their arrivals truthfully, a induces in period t the matching m ( E t \ { a } ) by balking; next period, when a joins the economy, the matching is then m ( E t \ { a } , E t + ∪ { a } , · ) (note that a onlyassigns positive probability to realizations E t + such that a / ∈ E t + .)When T =
2, Proposition 4 follows straight from the definition of dynamic sta-bility. When T ≥
3, however, this is not the case. The issue is that m may failto be dynamically stable for arriving agents at E t , and yet a reveals his arrivaltruthfully because m ( E t \ { a } , · ) does not entail a continuation matching that isdynamically stable. In this case, m ( E t \ { a } , · ) could be worse than any reason-able conjecture a may have under the assumption of dynamic stability. The keyis then to find the longest realization E t for which dynamic stability fails. Thisensures that m ( E t \ { a } , · ) does pick continuation matchings that satisfy dynamicstability for G T ( ·| E t \ { a } , E t + ∪ { a } ) from period t + G T ensures that these continuations are also dynamically stable when using G T ( ·| E t , E t + ) .The above discussion suggests that, when offering pairwise stable and individ-ually rational matchings, there are two ways to incentivize timely participationin the market: either the matching is dynamically stable or one is allowed to offer(and enforce) non-credible continuation matchings. Otherwise, it is not possible toprevent agents from either balking or reneging in the dynamic matching market. In this section, I discuss an alternative definition of stability for dynamic match-ing markets, which I denote passive dynamic stability . The purpose of this istwo-fold. First, by contrasting both stability notions, I can highlight the role theassumptions about the timing at which matching, blocking, and arrivals occurplay in Theorem 1 and Proposition 4. As I explain below, these assumptions are This does not imply, however, that delaying by one period is the best that a can do. Since m is pairwise stable and individually rational, m ( E t \ { a } ) is stable amongst those whomatch in period t . Passive dynamic stability corresponds to the solution concept introduced in Doval (2015,2018). The reason for the term “passive” will become clear with the definition. T =
1, which highlights another novel feature of the dynamiceconomy. Second, the timing assumptions implicit in passive dynamic stabilitycoincide with those of the algorithms used in sequential assignment and schoolchoice with aftermarkets (see Pathak (2016)). Since the date at which matchingsare finalized is part of the design, understanding the difference between the pre-dictions of passive dynamic stability and dynamic stability is important for thedesign of the algorithms used in sequential assignment and school choice.The only difference between dynamic stability and passive dynamic stability isthe set of matchings that an agent conjectures would ensue when they decide toremain unmatched in a given period. Let P t ( G t ) denote the set of passive dy-namically stable matchings for G t . Fix a matching m , a period t and an agent k ∈ A ( m t − , E t ) ∪ B ( m t − , E t ) . The set M P ( k , m , E t ) is defined as follows: M P ( k , m , E t ) = m ∈ M T ( m , E t ) : ( i ) m t ( k ) = k ( ii ) ( m ( E t + s )) s ∈{ T − t } ∈ P T − t ( G T − t ( m t , E t ))( iii ) m ( E t )( k ′ ) = m ( E t )( k ′ ) if k ′ / ∈ { k , m ( E t )( k ) } ,(4) Definition 7.
A matching m for G T is passive dynamically stable if it satisfies the con-ditions of Definition 5 substituting the correspondences ( D s ) s ≤ T − for ( P s ) s ≤ T − .To understand the difference between dynamic stability and passive dynamicstability, contrast the sets defined in equations (1) and (4). These sets only differin condition (iii). While dynamic stability allows agent k to conjecture any period- t matching consistent with individual rationality and pairwise stability amongstthe agents who match in period t , in passive dynamic stability agent k believesthat the agents in period t (except k ’s matching partner) still match according to m ( E t ) . This is the reason for the prefix passive : agent k assumes that no one elsereacts to their decision to wait to match at some future period. As an illustration of this difference, consider the example in Section 2. Whenconsidering whether Erd˝os could block the matching m N C in Figure 1, both match-ings in Figure 2 are possible conjectures for Erd˝os under dynamic stability. How-ever, only the matching m E in Figure 2 is a valid conjecture under passive dynamicstability, since Gale’s assignment is the same as in m N C . Then, under passive dy-namic stability, Erd˝os blocks the matching m N C in Figure 1. This is related to, but different from, the discussion of what a coalition can enforce, which isrelevant for defining farsighted stability ; see, for instance, Mauleon et al. (2011) and Ray and Vohra(2015). In my model, when k waits to be matched, he does not force others to change their matching;however, he understands that his decision to wait may trigger others to change how they match. t , the timing of thedecisions in the economy is as follows: t .1 The unmatched agents through period t − t propose a period- t matching. t .2 Assuming that other period- t agents have no objections to the matching, eachagent who is supposed to exit that period checks whether they would benefitfrom staying or whether there is an agent with whom they could form a pair-wise block. t .2.1 If there are no objections, then agents match according to the period- t matching and exit. The remaining unmatched agents are joined by thenew arrivals and we proceed to step ( t + ) .1. t .2.2 If there are any objections, then return to step t .1.In contrast, passive dynamic stability presumes that in each period t , the timing isas follows: t .1 The unmatched agents through period t − t match according to a period- t matching. t .2 Assuming that other period- t agents have no objections to the matching, eachagent who is supposed to exit that period checks whether they would benefitfrom staying or whether there is an agent with whom they could form a pair-wise block. t .3 Objections are carried out. The remaining agents are joined by the new entrantsand we proceed to step ( t + ) .1.The set of stable matchings for economies of length 1 does not depend on the posi-tion the analyst takes with respect to when matches and blocks are formed withineach period. Indeed, both timings described above are used to describe the staticnotion of stability by different authors: Aumann (1961) and Roth and Sotomayor(1992) use the first timing, while Gale and Shapley (1962) and Liu et al. (2014) usethe second one. This is a reflection of a deeper observation. For any economy However, when discussing what matchings a coalition can enforce, Roth and Sotomayor(1992) adopt the perspective in passive dynamic stability: a coalition can only alter the match-ing amongst its own members and (potentially) the matching of their previous matching partners,who are now unmatched.
27f length 1, it is easy to show that D = P : they both coincide with the set ofpairwise stable and individually rational matchings at each realization E .However, the set of stable matchings in the dynamic economy does depend onwhen matches and blocks are formed within a period. To understand the differ-ence between both timings, let T = m denote a dynamically stable match-ing. Thus, there would be no objections to this matching under the first timing.Moreover, under the second timing, since m is pairwise stable, there would beno objections involving pairs of agents who would rather match together. How-ever, there could be an agent k who is supposed to match at some realization E and once everyone other than k is matched according to m ( E ) , k would benefitfrom remaining unmatched at E . Formally, since M P ( k , m , E ) ⊆ M D ( k , m , E ) ,it could be that the conjecture m ∈ M D ( k , m , E ) that prevents k from blockingunder dynamic stability requires other agents in t = m ( E ). Thus, when k assumes that everyone else is matching according to m ( E ), k may improve on m by remaining unmatched. If m is not passive dynamicallystable, then there are gains to be made by blocking that are not exhausted whenthe match forms in t = Theorem 2.
Let T ≥ . Then, the following hold:1. For all economies of length T, P T ( G T ) ⊆ D T ( G T ) , and there exist economies suchthat the inclusion is strict (see Example 4).2. There exist economies for which P T ( G T ) = { ∅ } . The example in Section 2 is an instance in which P ( G ) = { ∅ } . To see this, notethat the two matchings in Figure 1 are the only two pairwise stable matchings (inthe sense of Definition 3). As explained above, under passive dynamic stability,Erd˝os can improve on his outcome under m N C by remaining unmatched in t =
1. The core matching m C is also not passive dynamically stable, since Kuhn canimprove on his outcome under m C by remaining unmatched in t =
1. When Erd˝os Du and Livne (2016) model the National Resident Matching Program (NRMP) as a two-periodeconomy, where hospitals and residents arrive over time, preferences are supermodular, and in pe-riod 2, an algorithm matches all unmatched agents according to a stable matching. They considera set of matchings satisfying weaker conditions than passive dynamic stability. They show that, asthe number of agents in the economy grows large, this set may be empty. While their paper neitherprovides a stability notion for dynamic matching markets nor studies their properties, their resultsuggests that the non-existence of dynamically stable matchings persists even in a large market. t =
1, there is a unique stable matching in t =
2, where Kuhnmatches with Tucker.That dynamic and passive dynamic stability may differ when T ≥ sequential assignment .
18, 19
The latter are problems where assign-ments are performed in multiple stages because both market participants andtheir matching opportunities become available over time. Between stages, par-ticipants decide either to keep their assignment and exit or to stay for the nextround, in which case both they and their assignment are available to match in thenext round. Note the similarity with the timing implicit in passive dynamic stabil-ity: market participants take as given others’ assignments when they make theirdecisions to exit or stay. While sequential assignment is usually a many-to-one problem, the discussion that followsfocuses on the incentives of the one side. For instance, in the case of school choice, I focus on thebehavior of the students and regard the schools as non-strategic. Because my analysis is about aone-to-one matching market, the analysis in this paper has nothing in principle to say about thebehavior of schools. See, for instance, Westkamp (2013); Dur and Kesten (2014); Feigenbaum et al. (2017);Dogan and Yenmez (2018); Haeringer and Iehl´e (2019); Andersson et al. (2018); Mai and Vazirani(2019). Except for the last two papers, this literature studies models that are in a sense static: theagents and their matching opportunities are fixed. Even in a static model, running deferred accep-tance sequentially may cause problems since deferred acceptance is not consistent . That is, if oneremoves some agents and their assignments under deferred acceptance, m A − DA , and calculatesthe matching obtained by deferred acceptance for the remaining agents and objects, one does notobtains the restriction of m A − DA to the remaining set of agents and objects. This, in turn, may giveagents incentives to wait to be matched in a second round.
29s an example, consider aftermarkets in school choice (see Pathak (2016)). Pub-lic school districts run their matching algorithms several times to accommodatenewly incoming students, newly available seats, and also the timing of decisionsof private and charter schools. While private and charter schools do not partic-ipate of the centralized matching procedure, they are relevant matching oppor-tunities for students who apply to both public and private /charter schools. Inbetween rounds, families must decide whether to take their assigned seat in pub-lic school and exit, take their seat at the private/charter school (if they have beenadmitted to one) and exit, or stay for the next round, in which case both they andtheir assigned seat (if any) are available to match in the next round. Families maydecide to wait either because they may find out the result of their application toprivate school or because new seats may become available at a more desirablepublic school as a result, for instance, of other families exiting.There are three reasons why the non-existence result for passive dynamic stabil-ity is relevant for sequential assignment problems. First, it implies that runningat each stage a matching algorithm that produces stable matchings amongst theagents who match in that period may not be enough to prevent agents from tryingto improve on their outcomes by waiting to be matched. Indeed, as described inAndersson et al. (2018), some school districts run their algorithms multiple times,even very close to the start of classes. This is a reflection that all such match-ings can be improved upon by the agents under passive dynamic stability. Sec-ond, it suggests that even if school choice algorithms are strategy-proof underthe assumptions that (i) the market is static and (ii) all schools participate of thealgorithm, this property may be compromised once we take into account that nei-ther (i) and (ii) hold. As argued in Section 6.1, dynamic stability of the matchingimplemented by the algorithm is a necessary condition for timely participation.Even then, when passive dynamically stable matchings fail to exist, the agentsmay give up on assignments that they declared acceptable. Finally, given thatdynamically stable matchings do exist, this suggests that an important aspect ofthe design is the time at which the matching is finalized. In the current designs,agents have the option of giving up their current assignment and proceeding tothe next stage. This is in contrast to the National Resident Matching Program,where there is a clause that restricts undoing and performing matches outside thealgorithm once it has finalized. Thus, like the NRMP, one could envision a de- Narita (2018) suggests that learning might be another reason for strategy-proofness to fail in adynamic matching market. This would make the timing of decisions similar to that in thedefinition of dynamic stability, as opposed to the one in passive dynamic stability.Given the above, the final result in the paper lists sufficient conditions underwhich passive dynamically stable matchings exist. All the conditions are formallydefined in Appendix D:
Theorem 3.
Suppose that either of the following hold:1. Market participants are patient (Definition 9),2. There are no cycles in preferences (Definition 10),3. Agents on side B are not allowed to block by waiting and there is no uncertainty inarrivals and { v ( · , b ) } b ∈ B satisfies Ergin-acyclicity ,then P T ( · ) = ∅ . The proof is in Appendix D. When the conditions in Theorem 3 hold, one doesnot need to worry about which is the correct timing at which blocks and matchesare formed. Moreover, since passive dynamically stable matchings are also dy-namically stable, they inherit the property of ensuring timely participation in themarket.It is worth noting that, albeit for different reasons (see footnote 19), Ergin-acyclicityhas been identified to be an important property in the literature on sequential as-signments. Theorem 3 provides yet another reason: when considering side B asobjects to be assigned in a dynamic matching market, Ergin-acyclicity guaranteesthat there is a matching the agents on side A cannot improve on by waiting to bematched. As the example in Doval (2020) shows, Ergin-acyclicity is not enoughwhen there is uncertainty over the arrivals. Note that this does not contradict theexisting results in the literature since in most models all objects arrive in period 1. Thus, like in Section 6.1, agents would have a choice of when to join the algorithm, but oncethey receive an assignment they must take it. The result in that section suggests that if the agentsexpect the outcome to be a dynamically stable matching, then they would have an incentive to joinwhen they are available to match Conclusions
I formulate a notion of stability, dynamic stability , for two-sided dynamic matchingmarkets where matching opportunities arrive over time and matching is one-to-one and irreversible. I also compare dynamic stability to other definitions, like thecore and passive dynamic stability. The comparison highlights the assumptionsabout which agreements are feasible and the timing at which they occur, implicitin each definition.Several avenues are worth exploring and are left for future work. First, dy-namic stability assumes that only pairs of agents or a single agent can block amatching. This assumption is irrelevant in static, two-sided, one-to-one match-ing markets. Extending dynamic stability to allow groups of agents blocking towait to be matched requires making assumptions on what conjectures the groupshould hold (and whether their members should hold a common conjecture).Second, while dynamic stability (and passive dynamic stability) are necessaryfor timely participation in the market, the question remains as to whether thejoint manipulation of preferences and arrival times can be avoided. One reasonthis may be important is to understand the data coming out of school choice algo-rithms that have aftermarkets: even if the centralized algorithm is strategy-proof,this property may be lost once we take into account the interaction with the after-market.Finally, it is common to use stability as an identifying assumption for empiricalwork. The framework developed in this paper has the potential to allow empiri-cal researchers to carry out similar exercises in dynamic matching markets. Thereare a few papers that incorporate the dynamics of matching within their empir-ical frameworks (see, for instance, Choo (2015); Narita (2018); Grenet et al. (2019)).Notwithstanding the caveats in the previous paragraph regarding strategy-proofness,dynamic stability would be another way to approach preference identification inthese settings. 32 eferences A DACHI , H. (2003): “A search model of two-sided matching under nontransfer-able utility,”
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A Proof of Theorem 1
The proof proceeds by induction on T ≥
2. As argued in the main text D coin-cides with the set of pairwise stable matchings for the static economy. Let P ( T ) denote the following inductive statement: P ( T ) : The correspondence D T is non-empty valued.I first prove P ( ) =
1. Let G denote an economy of length T =
2. For each E = ( A , B ) on the support of G , I construct a matching m ⋆ ( E ) as follows. Foreach k ∈ A ∪ B , let m kE denote the payoff minimizing matching m such that1. m ( E )( k ) = k ,2. For each E = ( E , A , B ) in the support of G , m ( E , A , B ) is stable for hA ( m , E ) , B ( m , E ) i ,3. m ( E ) is stable amongst those who match at E .Note that the set of matchings m that satisfy these three requirements is finiteand it is non-empty. To see that it is non-empty, note first that for any set of un-matched agents, the set of stable matchings in t = t = A ∪ B \ { k } is non-empty. Pick any such t = A ∪ B by leaving agent k unmatched. Note that it is stable amongst thosewho match in t =
1. Then, for each E , pick a t = hA ( m , E ) , B ( m , E ) i . The matching m formed this way satisfies the three re-quirements. Also note that item 2 implies that any such matching m satisfies therequirement that m ( E , · ) ∈ D ( G ( m , E )) .For each a ∈ A , consider the preference list on B defined as follows: b ∈ B isacceptable to a only if u ( a , b ) ≥ U ( a , m aE , E ) and is not acceptable to a otherwise.Moreover, if b , b ′ ∈ B are acceptable to a , then they are ordered according to u ( a , · ) in a ’s preference list. 37imilarly, for b ∈ B , consider the preference list on A defined as follows: a ∈ A is acceptable to b only if v ( a , b ) ≥ V ( b , m bE , E ) and is not acceptableto b otherwise. Moreover, if a , a ′ ∈ A are acceptable to b , then they are orderedaccording to v ( · , b ) in b ’s preference list.Define m ⋆ ( E ) so that it coincides with the outcome of running the deferred ac-ceptance algorithm with agents in A proposing to agents in B using the truncated preference lists.(If there are ties, fix a tie-breaking procedure and run deferred ac-ceptance.)For each period-2 economy E = ( E , A , B ) , let m ⋆ ( E ) coincide with m ⋆ ( E ) for those agents who matched at E and have it coincide with the outcome ofrunning deferred acceptance on A ( m ⋆ , E ) , B ( m ⋆ , E ) , elsewhere. Note that m ⋆ defined in this way is an element of M .I note the following properties of m ⋆ . First, for all E , m ⋆ ( E ) is stable for A ( m ⋆ , E ) , B ( m ⋆ , E ) . Second, for each E , m ⋆ satisfies the following: (i) thereis no pair ( a , b ) such that m ⋆ ( E )( a ) = a , m ⋆ ( E )( b ) = b that prefer matching witheach other than to match according to m ⋆ ( E ) , i.e., m ⋆ ( E ) is stable amongst thosewho match in period 1; (ii) there is no k ∈ A ∪ B such that m ⋆ ( E )( k ) = k and forall m ∈ M D ( k , m ⋆ , E ) , k prefers m to m ⋆ . To see that (ii) holds, note the following.By construction m k can always be taken to be an element of M D ( k , m ⋆ , E ) by hav-ing it coincide with m ⋆ at all realizations E t , t ∈ {
1, 2 } that do not (weakly) follow E . Moreover, if k is matched at E , then it must be that k ’s matching partner is(weakly) preferred to m k .Thus, to show that m ⋆ ∈ D ( G ) , it remains to be argued that it is pairwisestable. To do so, it is only necessary to check that there is no realization E =( A , B ) and pair ( a , b ) ∈ A × B who prefer to match together at E rather thanmatch according to m ⋆ . Toward a contradiction, suppose there is such a pair.Since m ⋆ is stable amongst those who match in period 1, it cannot be the case thatboth a and b are matched at E . Then, either a or b (or both) declared the otherunacceptable at E and is unmatched at E . Without loss of generality, assumethat it is b , then v ( a , b ) < V ( b , m bE , E ) . (5)Moreover, it must be that v ( a , b ) > V ( b , m ⋆ , E ) since ( a , b ) form a pairwise block Again, if there are indifferences, run deferred acceptance after having broken ties. m ⋆ . It then follows that V ( b , m ⋆ , E ) < V ( b , m bE , E ) . (6)Note, however, that m ⋆ satisfies the following: (i) m ⋆ ( E )( b ) = b , (ii) ( m ⋆ ( E , · )) ∈D ( G ( m ⋆ , E )) , and (iii) m ⋆ ( E ) is stable amongst those who match in period1. Then, m ⋆ ∈ M D ( b , m ⋆ , E ) . This, together with Equation 6, contradicts thedefinition of m bE . By assuming instead that a is unmatched, a similar contradictionfollows. I conclude that m ⋆ is pairwise stable.This concludes the proof that P ( ) = P ( T ′ ) = T ′ < T . Ishow that P ( T ) = E T = ( A , B , . . . , A T , B T ) in the support of G T , consider thefollowing procedure to construct m ⋆ ( E t ) for each truncation E t of E T . For each 1 ≤ t ≤ T −
1, having defined ( m ⋆ ( E ) , . . . , m ⋆ ( E t − )) , let hA ( m ⋆ t − , E t ) , B ( m ⋆ t − , E t ) i denote the agents available to match in period t (if t =
1, then A ( ∅ , E ) = A , B ( ∅ , E ) = B .) For each k ∈ A ( m ⋆ t − , E t ) ∪ B ( m ⋆ t − , E t ) , let m kE t denote thematching m ∈ M T ( m ⋆ t − , E t ) that minimizes k ’s payoff at E t and satisfies that1. m ( E t )( k ) = k ( m ( E t + s )) s ∈{ T − t } ∈ D T − t ( G T − t ( m t , E t )) .3. m ( E t ) is stable amongst those who match in period t .By the inductive hypothesis, D s is non-empty valued for s < T . Since t ≥
1, then D T − t ( · ) is non-empty. Thus, for all k ∈ A ( m ⋆ t − , E t ) ∪ B ( m ⋆ t − , E t ) , m kE t is welldefined since the above set of matchings is non-empty and finite. For each a ∈ A ( m ⋆ t − , E t ) , consider the preference list on B ( m ⋆ t − , E t ) defined asfollows: b ∈ B ( m ⋆ t − , E t ) is acceptable to a only if u ( a , b ) ≥ U ( a , m aE t , E t ) and isnot acceptable to a otherwise. Moreover, if b , b ′ ∈ B ( m ⋆ t − , E t ) are acceptable to a ,then they are ordered according to u ( a , · ) in a ’s preference list. That is, if k = a , I am minimizing U ( a , · , E t ) , while if k = b , I am minimizing V ( b , · , E t ) . Recall that any period-t matching that is (statically) stable for the economy formed by A ( m ⋆ t − , E t ) ∪ B ( m ⋆ t − , E t ) \ { k } can be extended to a period-t matching for the agents in A ( m ⋆ t − , E t ) ∪ B ( m ⋆ t − , E t ) so that agent k is unmatched and the period-t matching is stableamongst those who match at E t . b ∈ B ( m ⋆ t − , E t ) , consider the preference list on A ( m ⋆ t − , E t ) de-fined as follows: a ∈ A ( m ⋆ t − , E t ) is acceptable to b only if v ( a , b ) ≥ V ( b , m bE t , E t ) and is not acceptable to b otherwise. Moreover, if a , a ′ ∈ A ( m ⋆ t − , E t ) are accept-able to b , then they are ordered according to v ( · , b ) in b ’s preference list.Define m ⋆ ( E t ) to be the outcome of deferred acceptance on hA ( m ⋆ t − , E t ) , B ( m ⋆ t − , E t ) i using the above preference lists.For t = T , let m ⋆ ( E T ) coincide with m ⋆ ( E T − ) for all those who have matchedat E T − and with the outcome of running deferred acceptance with agents on A ( m ⋆ T − , E T ) proposing to agents on B ( m ⋆ T − , E T ) , elsewhere.Note that this procedure on all E t and all 1 ≤ t ≤ T implies that m ⋆ is an elementof M T . I now make some observations about m ⋆ .First, for all periods t ∈ {
1, . . . , T } , and all realizations E t , m ⋆ ( E t ) is stableamongst those who match at E t . Second, for all periods t ∈ {
1, . . . , T − } , and allrealizations E t , there is no k ∈ A ( m ⋆ t − , E t ) ∪ B ( m ⋆ t − , E t ) such that m ⋆ ( E t )( k ) = k and for all m ∈ M D ( k , m ⋆ , E t ) , k prefers m to m ⋆ . To see that this holds note the fol-lowing. By construction m kE t can always be taken to be an element of M D ( k , m ⋆ , E t ) by having it coincide with m ⋆ at all realizations E s that do not follow E t . Moreover,if k is matched at E t , it must be that their matching partner is (weakly) preferredto m kE t .Fix t = T and note that for all E T , m ⋆ ( E T ) is pairwise stable and individually ra-tional. Let t ≤ T − E t denote a realization. Suppose that one has shownthat ( m ⋆ ( E t + s )) s ∈{ T − t } is pairwise stable starting from period t + ( m ⋆ ( E t + s )) s ∈{ T − t } ∈ D T − t ( G T − t ( m ⋆ t , E t )) .I now show that there are no pairwise blocks involving agents that have arrivedby E t . Toward a contradiction, suppose there exists a pair ( a , b ) ∈ A ( m ⋆ t − , E t ) ×B ( m ⋆ t − , E t ) such that u ( a , b ) > U ( a , m ⋆ , E t ) , v ( a , b ) > V ( b , m ⋆ , E t ) .Since m ⋆ ( E t ) is stable amongst those who match in period t , then it must be thateither a or b are unmatched at m ⋆ ( E t ) . Without loss of generality, suppose that itis b . Thus, b declared a as unacceptable in the truncated preference lists so that v ( a , b ) < V ( b , m bE t , E t ) . (7)40ombining the above inequalities, it follows that V ( b , m ⋆ , E t ) < V ( b , m bE t , E t ) . (8)Note, however, that m ⋆ ∈ M D ( b , m ⋆ , E t ) . This, together with Equation 8, con-tradicts the definition of m bE t . Thus, m ⋆ is pairwise stable from period t onward.Proceeding this way, it follows that m ⋆ is pairwise stable. This proves P ( T ) = B Proofs of Section 6.1
B.1 Proof of Proposition 3
Let G and m A − DA be as in the statement of Proposition 3. Let a ∈ A ( E ) besuch that m A − DA ( E )( a ) = a . Construct a matching m as follows. Run deferredacceptance as if the economy was given by ( A \ { a } , B , A ∪ { a } , B ) , that is:(a) a makes proposals as if a arrived in t = a proposes to b ∈ B ( E ) before b ′ ∈ B ( E ) if, and only if, u ( a , b ) > u ( a , , b ′ ) (b) b ∈ B ( E ) accepts a ’s offer over a ′ ∈ A only if δ b v ( a , b ) > v ( a ′ , b ) .Note that m ∈ M D ( a , m A − DA , E ) . Toward a contradiction, suppose that U ( a , m , E ) > U ( a , m A − DA , E ) . Let A + denote the set of agents on side A who prefer m to m A − DA . According to a lemma by J.S. Hwang (see Gale and Sotomayor (1985) fora proof), there exists a ′ / ∈ A + and b ∈ m ( E )( A + ) that block m as in Definition 6.This is a contradiction since m is in the core of the economy ( A \ { a } , B , A ∪{ a } , B ) . Thus, it cannot be that a strictly prefers m to m A − DA . B.2 Proof of Proposition 4
Let G T and m satisfy the assumptions of Proposition 4. Let t ≤ T − s ≤ T − E s such that either Conditions2 or 3 of Definition 5 fail. Letting E t = ( E t − , A t , B t ) denote the realization forwhich either Conditions 2 or 3 of Definition 5 fail, it follows that there exists k ∈ This, of course, uses that we have shown that m ⋆ ( E t , · ) is pairwise stable and has no blocks bywaiting. ( m t − , E t ) ∪ B ( m t − , E t ) such that m ( E t )( k ) = k that can improve by remainingunmatched at E t . Moreover, since m is not dynamically stable for arriving agentsit follows that k ∈ A t ∪ B t . Without loss of generality, assume that k = a ∈ A t .In a slight abuse of notation, let E t \ { a } denote ( E t − , A t \ { a } , B t ) . Note that A ( m t − , E t ) \ { a } = A ( m t − , E t \ { a } ) and B ( m t − , E t ) = B ( m t − , E t \ { a } ) I now define a matching, m , and argue that m ∈ M D ( a , m t − , E t ) . First, let m ∈M T ( m t − , E t ) . Second, let m ( E t )( k ′ ) = (cid:26) m ( E t \ { a } )( k ′ ) if k ′ = aa otherwiseFinally, for each 1 ≤ s ≤ T − t and realizations E t + s = ( E t , E t + , . . . , E t + s ) , let m ( E s )( k ) = m ( E t \ { a } , E t + ∪ { a } , . . . , E t + s )( k ) for each k ∈ A ( m t + s − , E t + s ) ∪ B ( m t + s − , E t + s ) . Note that m ( E t ) is stable amongstthose who match at E t by assumption. Moreover, a is unmatched at E t . It re-mains to be argued that ( m ( E t + s )) s ∈{ T − t } ∈ D T − t ( G T − t ( m , E t )) . Note thatby definition ( m ( E t \ { a } , E t + ∪ { a } , . . . , E t + s )) s ∈{ T − t } satisfies the definitionof dynamic stability under distribution G T ( ·| E t \ { a } , E t + ∪ { a } , . . . , E t + s ) . Ex-changeability implies that this is the same as G T ( ·| E t , E t + , . . . , E t + s )) . Thus, m ∈ M D ( a , m , E t ) .Since m is not dynamically stable at E t , it follows that for all matchings in M D ( a , m , E t ) , U ( a , · , E t ) > U ( a , m , E t ) . In particular, U ( a , m , E t ) > U ( a , m , E t ) .The result then follows. C The core of a dynamic matching market
In this section, I extend the definition of the core to any economy G T . When G T is not degenerate, the core has a many-to-many structure: an agent k may bematched to different agents on the other side for different realizations E t . In partic-ular, this implies that, given a matching m , k may want to improve on k ’s matchingoutcome at certain realizations, while keeping their matching outcome the same42t some others. Definition 8 below allows these improvements as long as the restof the agents who match with k are not hurt by k ’s block.Let G T denote an economy. For each realization E T on the support of G T andeach truncation E t = ( A , B , . . . , A t , B t ) , a sequence C E t = ( A C , B C , . . . , A Ct , B Ct ) isa coalition at E t if for all s ≤ t , A Cs ∪ B Cs ⊆ A s ∪ B s .Given an economy G T , the set C = { C E t : t ∈ {
1, . . . , T } , ∑ ( E t + ,..., E T ) G T ( E t , · ) > } is a feasible coalition for G T if, for all t ∈ {
1, . . . , T } and all realizations E t , if E s follows E t , then C E s follows C E t .A matching for a feasible coalition structure C is a mapping on C such that1. m ( C E t ) : A t ( C E t ) ∪ B t ( C E t ) A t ( C E t ) ∪ B t ( C E t ) is a period-t matching, and2. For all t ∈ {
1, . . . , T } , for all realizations E t , for all a ∈ A t ( C E t ) , if m ( C E t )( a ) = a , then m ( C E s )( a ) = m ( C E t )( a ) for all t ≤ s and E s that follow E t .Finally, fix an economy G T and a feasible coalition structure, C . Let m denotea matching for G T and m a feasible matching for C . Let C E t ∈ C be such that C E t = ( A C , B C , . . . , A Ct , B Ct ) . For k ∈ A Ct ∪ B Ct , let M ( k , m , m ) = { m ′ ∈ M T : ( ∀ t ≤ s ≤ T )( ∀ E s that follows E t ) m ′ ( E s )( k ) ∈ { k , m ( E s )( k ) , m ( E s )( k ) }} ,denote the set of all matchings such that, for each realization E s that follows E t , k is either single or matched to those agents to whom k is matched under m or m . Definition 8.
A matching m is in the core of economy G T if there is no feasiblecoalition structure C and matching m feasible for coalition structure C such that1. For all t ∈ {
1, . . . , T } , for all realizations E t on the support of G T , if C E t =( A C , B C , . . . , A Ct , B Ct ) , then U ( a , m , E t ) ≥ U ( a , m , E t ) for all a ∈ A Ct V ( b , m , E t ) ≥ V ( b , m , E t ) for all b ∈ B Ct
2. There exists t ∈ {
1, . . . , T } and E t in the support of G T such that the aboveinequalities are strict for some agent k ∈ A Ct ∪ B Ct ,43. For all t ∈ {
1, . . . , T } , for all realizations E t on the support of G T , if C E t =( A C , B C , . . . , A Ct , B Ct ) , thenfor all a ∈ A Ct , U ( a , m , E t ) ≥ U ( a , m ′ , E t ) for all m ′ ∈ M ( a , m , m ) for all b ∈ B Ct , V ( b , m , E t ) ≥ V ( b , m , E t ) for all m ′ ∈ M ( b , m , m ) When G T assigns probability 1 to one realization E T , then Definition 8 coincideswith Definition 6. Condition 3 imposes a form of “dynamic consistency” on theblocks: no blocking agent k can do better by being able to keep one of k ’s formermatching partners under m by possibly dropping k ’s new matching partner under m . As explained in the main body of the paper, the core may be empty when G T isnot degenerate. This is illustrated in the following example: Example 2.
The example is based on the following economy. Arrivals are givenby: A = { a , a , a , a } , B = { b , b , b } B = { b , b } , B ′ = { b ′ } . G is such that G ( A , B , ∅ , B ) = p = − G ( A , B , ∅ , B ′ ) . In what follows, let E = ( A , B , ∅ , B ) and E ′ denote the other realization. Assume that agents onside B do not discount the future and their preferences are given by Echenique and Oviedo (2006) call this the Blair core (see Blair (1988), and Roth (1984b)). ( a , b ) > > max a = a v ( a , b ) min a = a v ( a , b ) > > v ( a , b ) min { p , 1 − p } v ( a , b ) > max { p , 1 − p } v ( a , b ) min { p , 1 − p } v ( a , b ) > max { p , 1 − p } v ( a , b ) v ( a , b ) , v ( a , b ) > > v ( a , b ) , v ( a , b ) min { p , 1 − p } v ( a , b ) > max { p , 1 − p } v ( a , b ) v ( a , b ) > v ( a , b ) > > v ( a , b ) , v ( a , b ) v ( a , b ) > > max a = a v ( a , b ) v ( a , b ′ ) > v ( a , b ′ ) > > max a = a , a v ( a , b ′ ) Preferences on side A are given by u ( a , b ) > u ( a , b ) > u ( a , b ) > δ a [ pu ( a , b ) + ( − p ) u ( a , b )] u ( a , b ) > u ( a , b ) > u ( a , b ) > u ( a , b ) > δ a [ pu ( a , b ) + ( − p ) u ( a , b )] > δ a [ pu ( a , b ) + ( − p ) u ( a , b )] > u ( a , b ) u ( a , b ) > u ( a , b ′ ) > u ( a , b ) > δ a min { p , 1 − p } u ( a , b ) > δ a max { p , 1 − p } u ( a , b ′ ) > u ( a , b ) min { u ( a , b ) , u ( a , b ′ ) } > u ( a , b ) > δ a [ pu ( a , b ) + ( − p ) u ( a , b ′ )] > u ( a , b ) > δ a [ pu ( a , b ) + ( − p ) u ( a , b ′ )] I now show that there is no matching in the core. The proof is by contradiction.Assume there is such a matching, denote it by m .45 tep 1: If m is in the core, then a has to be matched.Otherwise, a and b block. Step 2:
There is no matching in the core such that a matches with b at t = a has to be matched -note that a can always matchwith b -, and so does a - a can always match with b -. Suppose first that a matches in t = b . Then, a matches with b in t = a matches with b - a only waits for b ′ if a canget b -. However, this is blocked by a matching with b ′ . Suppose, then, that m ( E )( a ) = a , m ( E )( a ) = b , m ( E ′ )( a ) = b . The same steps as before leadto a matching m that is blocked by a and b matching at E . Step 3:
There is no m in the core in which a matches with b at t = ( a , b ) . Step 4:
There is no m in the core such that m ( E )( a ) = a , m ( E )( a ) = b , m ( E ′ )( a ) = b ′ .Note that a is always matched; a can always block with b . However, if a matches with b at t =
1, this can be blocked by coalition C E = ( { a , a } ∅ , ∅ , { b } ) , C E ′ =( { a , a } , ∅ , ∅ , { b ′ } ) and matching m ( a ) = m ( a ) , m ( E )( a ) = b , m ( E ′ )( a ) = b . Hence, suppose that a matches with b at E , and with b at E ′ . Now, it hasto be that a matches with b - a has to be matched because otherwise a blockswith b , and this match is blocked by ( a , b ) . Then, a blocks by matching with b at E , and b ′ at E ′ . Step 5:
There is no m in the core such that m ( E )( a ) = b , m ( E ′ )( a ) = b ′ .Then, a has to be matched with b at t =
1. Hence, a blocks with b . Step 6:
There is no m in the core such that m ( E )( a ) = b , m ( E ′ )( a ) = b .As before, it has to be that a , a are matched. Note that the only possibilityfor a is to match with b at E and with b at E ′ . Thus, a matches with b at E , and a with b . However, C E = ( { a , a } , { b } , ∅ , { b } ) , C E ′ =( { a , a } , { b } , ∅ , { b ′ } ) and m such that m ( E )( a ) = b , m ( E ′ )( a ) = b , m ( E )( a ) = b , m ( E ′ )( a ) = b ′ is a block of m . Step 7:
There is no m in the core such that m ( E )( a ) = b , m ( E ′ )( a ) = b .46n this case, a has to match with b at E , and with b at E ′ . This is blocked by ( a , b ) . C.1 Proof of Proposition 2
Assume in what follows that preferences over matchings are strict. For each b in B , let m b denote the matching m ∈ M that minimizes b ’s payoff amongstthose that satisfy (i) m ( E )( b ) = b , (ii) m ( E ) ∈ D ( G ( m , E )) , and (iii) m ( E ) isstable amongst those who match at E . Eliminate from b ’s preferences all agents a ∈ A ( E ) such that δ [ b ∈ A ] b v ( a , b ) < V ( b , m , E ) .Let m ⋆ denote the outcome of deferred acceptance with side A proposing andagents on side B using the “truncated” preferences. I claim that m ⋆ ∈ C ( E ) ∩D ( E ) .Note first that for all b ∈ B , Proposition 3 implies that V ( b , m B − DA , E ) ≥ V ( b , m b , E ) , where m B − DA denotes the outcome of deferred acceptance with side A proposing using the true preferences. Therefore, m B − DA is in the core of E under the “truncated” preferences as well. Therefore, if b ∈ B is matched un-der m B − DA , it follows from Theorem 2.22 in Roth and Sotomayor (1992) that b ismatched under m ⋆ . Thus, m ⋆ is in the core of E under the original preferences.I claim now that m ⋆ is dynamically stable. Clearly, m ⋆ is pairwise stable. More-over, by definition, for each b ∈ B such that m ⋆ ( E )( b ) = b , it follows that V ( b , m ⋆ , E ) ≥ V ( b , m b , E ) , so that b cannot improve on m ⋆ by remaining un-matched in t = a ∈ A such that m ⋆ ( E )( a ) = a , let M ⋆ D ( a , m ⋆ , E ) denote the set of match-ings that a deems possible when remaining unmatched in t = B are given by the “truncated” preferences. Proposition 3 implies thatthere exists m ⋆ ∈ M ⋆ D ( a , m ⋆ , E ) such that U ( a , m ⋆ , E ) ≥ U ( a , m ⋆ , E ) . I claim that m ⋆ ∈ M D ( a , m ⋆ , E ) , so that m ⋆ is dynamically stable.Toward a contradiction, suppose that for all m ∈ M D ( a , m ⋆ , E ) , one has that U ( a , m , E ) > U ( a , m ⋆ , E ) . It then follows that m ⋆ / ∈ M D ( a , m ⋆ , E ) . Therefore,it must be that there exists b ∈ B such that m ⋆ ( E )( b ) = b and under the truepreferences, b has a pairwise block in t = Let ˆ B denote the set of all such b . Let To see this, suppose that for all b ∈ B , either m ⋆ ( E )( b ) = b or m ⋆ ( E ) = b = m ⋆ ( E )( b ) .Then, since m ⋆ ( E ) is stable amongst those who match in t = m ⋆ ( E ) is individually rational m ( E ) denote a period 2 matching such that ˜ m ( E )( k ) = m ⋆ ( E )( k ) if k matchesin t = m ⋆ ; otherwise, let ˜ m ( E ) form an individually rational and stablematching for hA ( m ⋆ , E ) , B ( m ⋆ , E ) i under the original preferences. Note that itcannot be the case that all b ∈ ˆ B are matched to someone they deemed acceptableunder the truncated preferences; otherwise, ˜ m is also individually rational andpairwise stable under the truncated preferences. The latter is a contradiction sincethe set of agents who are single is the same across all stable matchings. Then, thereexists b ∈ ˆ B who is matched to someone they do not find acceptable under thetruncated preferences. Note then that since ( m ⋆ ( E ) , ˜ m ( E )) ∈ M D ( b , m ⋆ , E ) , itfollows that m b is not the payoff minimizing matching for b in that set. This is acontradiction. It then follows that m ⋆ ∈ M D ( a , m ⋆ , E ) .Thus, m ⋆ ∈ D ( E ) , completing the proof. D Proofs of Section 6.2
I omit the proof of part 1 of Theorem 2 since it follows by applying the definitions.Trivially, the inclusion holds when P T ( G T ) = ∅ . When P T ( G T ) = ∅ and letting m ∈ P T ( G T ) , then for each realization E t the set of conjectures M P ( k , m , E t ) = ∅ for all k ∈ A ( m t − , E t ) ∪ B ( m t − , E t ) . Applying the definitions backward fromthe final period to the first it is possible to establish that the payoff minimizingelements of M P ( k , m , E t ) are in M D ( k , m , E t ) .The rest of the section contains the omitted definitions and proofs for Section 6.2for economies G T such that there is no uncertainty about the arrivals. Doval (2020)includes the definitions and proofs for the case in which arrivals are uncertain.All definitions and results are stated for the case T =
2. It is immediate, thoughtedious, to extend the conditions so that they apply for an economy of length T ≥ E = ( A , B , A , B ) denote the realization such that G ( E ) = Definition 9.
Say that E is patient if1. For all a ∈ A , for all b , b ∈ B ( E ) , whenever u ( a , b ) > u ( a , b ) , then δ a u ( a , b ) > u ( a , b ) , and pairwise stable, it follows that m ⋆ ∈ M D ( a , m ⋆ , E ) .
48. For all b ∈ B , for all a , a ∈ A ( E ) , whenever v ( a , b ) > v ( a , b ) , then δ b v ( a , b ) > v ( a , b ) .For the next definition, for k ∈ A ( E ) ∪ B ( E ) , let t k denote the date on whichagent k arrives. Definition 10. A simultaneous preference cycle of length N + a , b , . . . , a N , b N such that:1. For all n ∈ {
0, . . . , N } , δ t an − a n u ( a n , b n + ) ≥ δ a n u ( a n , b n ) ≥ n ∈ {
0, . . . , N } , δ t bn − b n v ( a n , b n ) ≥ δ a n u ( a n − , b n ) ≥ N + E has no simultaneous preference cycles if for all N ≥
1, there is no simultane-ous preference cycle of length N + Definition 11.
Suppose that for all b ∈ B , δ b =
1. Say that v ( · , b ) satisfies Ergin-acyclicity if there is no b , b ′ ∈ B ( E ) and no a , a ′ , a ′′ ∈ A ( E ) such that v ( a , b ) > v ( a ′ , b ) > v ( a ′′ , b ) , and v ( a ′′ , b ′ ) > v ( a , b ′ ) .I am now ready to prove Theorem 3: Proof of part 1
Let m denote an element of C ( E ) . Let a ∈ A be such that m ( E )( a ) = a . I claim that the matching m such that m ( E ) = m ( E ) and m ( E )( k ) = (cid:26) m ( E )( k ) if k / ∈ { a , m ( E )( k ) } k otherwiseis an element of M P ( a , m , E ) . To see this, note that m ( E ) satisfies conditions(i) and (iii) in the definition of M P ( · ) . To see that m ( E ) is a stable matchingamongst those who match in t =
2, note that if it is not, then there must be apairwise block involving either a or a ’ s matching partner. Without loss of gen-erality, suppose that the block involves a . Let b ∈ B ( m , E ) denote the agentwho blocks m ( E ) with a . Then, u ( a , b ) > u ( a , m ( E )( a )) . If b ∈ B , this con-tradicts that m is a core matching. Then, b ∈ B . Since E is patient, then u ( a , b ) > u ( a , m ( E )( a )) implies that δ a u ( a , b ) > u ( a , m ( E )( a )) , also a contra-diction to m being a core matching. Thus, m ∈ M P ( a , m , E ) . Note that by con-struction, U ( a , m , E ) ≥ U ( a , m , E ) .Since one can apply a similar construction to each b ∈ B as well, it follows that m ∈ P ( E ) . 49 roof of part 2. Let m ⋆ denote the unique matching in C ( E ) . Fix a ∈ A suchthat m ⋆ ( E )( a ) = a . Suppose there exists m ∈ M P ( a , m ⋆ , E ) such that U ( a , m , E ) > U ( a , m ⋆ , E ) .Let A + = { a ′ ∈ A ∪ A : U ( a ′ , m , E ) > U ( a ′ , m ⋆ , E ) } B − = { b ′ ∈ B ∪ B : V ( b ′ , m , E ) < V ( b ′ , m ⋆ , E ) } .Similar arguments to those in the Decomposition Lemma (Corollary 2.21 in Roth and Sotomayor(1992)) imply that m ( E )( A + ) ⊆ B − and m ⋆ ( E )( B − ) ⊆ A + .It then follows that m ( E )( A + ) = m ⋆ ( E )( A + ) = B − . Let G denote the followingdirected graph. Nodes are A + ∪ B − . There is an edge from a to b if b = m ( E )( a ) ;there is an edge from b to a if b = m ⋆ ( E )( a ) . Fix a ′ ∈ A + and follow the pathstarting from a . I claim that the path cycles. Clearly, it cannot end at an unreached a or b since, by definition, these agents are unmatched both at m ⋆ and m . Thus,the path cycles, and let ( a , b , . . . , a N , b N ) denote the cycle. The definitions of A + and B − imply that ( a , b , . . . , a N , b N ) form a simultaneous preference cycle,a contradiction.Thus, no such m exists. A similar argument shows that it cannot be that anyagent on side B can improve on m ⋆ by remaining unmatched in t =
1. Thus, m ⋆ is passive dynamically stable. Proof of part 3.
The proof follows from three steps. First, Proposition 5 belowstates a property of cycles when priorities satisfy acyclicity. Second, let m A − DA denote the outcome of deferred acceptance with side A proposing. The proof ofthe previous part implies that if there is an a in A who can improve on m A − DA byremaining unmatched in t =
1, then there must be a simultaneous preference cy-cle. The first step determines the cycle’s length. Third, the workings of deferredacceptance imply that the cycle is a violation of Ergin-acyclicity, a contradiction.
Definition 12.
A sequence ( a , b , ..., a N , b N ) , N ≥ improvement cycle if forall i ∈ {
1, . . . , N } , v ( a i , b i ) > v ( a i + , b i ) where the indices are taken modulo N . Proposition 5. If { v ( · , b ) : b ∈ B } satisfies Ergin-acyclicity, and ( a , b , ..., a N , b N ) isan improvement cycle, then N = . roof. Let ( a , b , ..., a N , b N ) be an improvement cycle, and suppose N >
2. Then,for all i ∈ {
1, ..., N − } v ( a i , b i ) > v ( a i + , b i ) , and v ( a N , b N ) > v ( a , b N ) .I claim that Ergin-acyclicity implies ( ∀ i )( ∀ j = i , i + ) v ( a j , b i ) > v ( a i + , b j ) . First,note that Ergin-acyclicity implies that ( ∀ i ) v ( a i − , b i ) > v ( a i + , b i ) , where i = ⇒ i − = N ; otherwise, v ( a i , b i ) > v ( a i + , b i ) > v ( a i − , b i ) and v ( a i − , b i − ) > v ( a i , b i − ) , a contradiction. Hence, v ( a i , b i ) > v ( a i + , b i ) , and v ( a i − , b i ) > v ( a i + , b i ) .Similarly, it must be the case that v ( a i − , b i ) > v ( a i + , b i ) ; otherwise v ( a i − , b i ) > v ( a i + , b i ) > v ( a i − , b i ) and v ( a i − , b i − ) > v ( a i − , b i − ) holds, violating Ergin-acyclicity. Thus, v ( a i , b i ) > v ( a i + , b i ) , v ( a i − , b i ) > v ( a i + , b i ) , and v ( a i − , b i ) > v ( a i + , b i ) , and we can proceed inductively and complete this for all j / ∈ { i , i + } .Now, take j / ∈ { i , i + } . If v ( a j , b i ) > v ( a i , b i ) > v ( a i + , b i ) and v ( a i + , b j − ) > v ( a j , b j − ) - note that j − = i -, then it has to be that v ( a i , b i ) > v ( a j , b i ) > v ( a i + , b i ) and v ( a i + , b i − ) > v ( a i , b i − ) , a contradiction. Thus, N = m A − DA denote the outcome of deferred acceptance with side A propos-ing. Suppose there exists a in A such that m A − DA ( E )( a ) = a and there exists m ∈ M P ( a , m A − DA , E ) such that U ( a , m , E ) > U ( a , m A − DA , E ) . It follows fromthe proof of the previous part, that there exists a simultaneous preference cycle, ( b , a , . . . , b N , a N ) . By Proposition 5, it follows that N =
2. Hence, write the cy-cle as ( b , a , b , a ) . Moreover, it follows from the proof of the previous step that b = m A − DA ( E )( a ) , b = m A − DA ( E )( a ) .Let r be the last step of the deferred acceptance algorithm in which an agent in { a , a } makes an offer (and, is accepted by) the agent to whom they are matchedunder m A − DA . Without loss of generality, say that a proposes to b at step r .Let l ∈ {
1, 2 } . Since u ( a l , m A − DA ( E )( a l + )) > u ( a l , m A − DA ( E )( a l )) , a l was re-jected by m A − DA ( E )( a l + ) before step r . Hence, at step r − m A − DA ( E )( a l + ) was matched to some agent, and at r − b had an upstanding offer from someagent and it is not a ; otherwise a is rejected in step r when a makes an of-fer to b . Hence, ( ∃ ˆ a / ∈ { a , a } ) such that v ( a , b ) > v ( ˆ a , b ) > v ( a , b ) . Since v ( a , b ) > v ( a , b ) , this contradicts Ergin-acyclicity. Therefore, m A − DA ∈ P ( E ) . Here is where N > j = i − Examples
Example 3. [An economy where the extremal matchings are not dynamically sta-ble] Consider the following two-period economy with deterministic arrivals:1. Arrivals are given by A = { a , a ′ , a ′ } , A = { a , a , a ′ } B = { b , b ′ , b } , B = { b , b ′ , b ′ }
2. Preferences are given by a : ( b , 0 ) ( b ′ , 0 ) ( b , 0 ) a ′ : ( b ′ , 1 ) ( b ′ , 0 ) ( b ′ , 0 ) ( b ′ , 1 ) a ′ : ( b ′ , 0 ) ( b , 0 ) ( b ′ , 0 ) a : ( b , 0 ) ( b , 0 ) a ′ : ( b ′ , 0 ) a : ( b , 0 ) ( b , 0 ) b : ( a , 1 ) ( a , 0 ) ( a , 0 ) ( a , 1 ) b ′ : ( a ′ , 0 ) ( a , 0 ) ( a , 0 ) b : ( a , 0 ) ( a ′ , 0 ) ( a , 0 ) b : ( a , 0 ) b ′ : ( a ′ , 0 ) ( a ′ , 0 ) b ′ : ( a ′ , 0 ) ( a ′ , 0 ) The following two core matchings are the side-A optimal matching (left) and theside-B optimal matching (right): m A − DA = a b a ′ b ′ a ′ b ′ a b a ′ b ′ a b m B − DA = a b a ′ b ′ a ′ b ′ a b a ′ b ′ a b I now show that V ( b , m , E ) > V ( b , m A − DA , E ) for all matchings m ∈ M D ( b , m A − DA , E ) .Toward a contradiction, suppose there exists one such matching m such that V ( b , m A − DA , E ) ≥ V ( b , m , E ) . Then, it must be the case that b does not match with a under m . Toguarantee this, note that b must also match in t =
2. This implies that, at most,one pair can be matched in t =
1. Below, for each such pair (including the possibil-ity that no one matches in period 1), the period-2 matching is obtained by running As usual, the horizontal line divides the matchings that happen in t = t = A proposing deferred acceptance on the remaining agents: a b ′ a b a b a ′ b a ′ b ′ a ′ b ′ a ′ b ′ a b a b a b a ′ b ′ a ′ b ′ a ′ b ′ a b a b a b a ′ b ′ a ′ b ∅ a b a b a b a ′ b ′ a ′ b ′ a ′ b ′ Note that in each case b ends up being matched with a . Since this is her mostpreferred matching, she blocks m A − DA by waiting.Similarly, it is possible to show that a ′ blocks m B − DA by waiting. In order for a ′ not to block m B − DA by waiting, it must be that he never matches with b ′ in t = a ′ must also match in t =
2. As before, this implies that, at most, one paircan form in t =
1. Below, for each such pair (including the possibility that no onematches in t = t = a b a b a b a ′ b ′ a ′ b ′ a ′ b ′ a b a b a b a ′ b ′ a ′ b ′ a ′ b ′ a b ′ a b a b a ′ b ′ a ′ b ′ a ′ b ∅ a b a b a b a ′ b ′ a ′ b ′ a ′ b ′ Note that in each case a ′ matches with b ′ who is a ′ ’s preferred match. Thus, a ′ blocks m B − DA by waiting.Nevertheless, there is a non-extremal core matching that is dynamically stableand given by a b ′ a ′ b a b a b a ′ b ′ a ′ b ′ In this matching, note that b is matched to her partner under m B − DA , a , and a ′ is matched to his partner under m A − DA , b ′ .53 xample 4. [An economy where P T ( D T .] Consider the following economy:1. Arrivals are given by A = { a , a } , A = { a } B = { b , b , b } , B = { b }