stability in repeated matching...

Stability in Repeated Matching Markets*

Ce Liu†

Job Market Paper

First Uploaded: September 1, 2018

Current Version: December 10, 2018

Abstract

I develop a framework for studying repeated matching markets, where in every pe-

riod, a new generation of short-lived agents on one side of the market is matched to a

fixed set of long-lived institutions on the other. Within this framework, I characterize

self-enforcing arrangements for two types of environments. When wages are rigid, as in

the matching market for hospitals and medical residents, players can be partitioned into

two sets: regardless of patience level, some players can be assigned only according to

a static stable matching; when institutions are patient, the other players can be assigned

in ways that are unstable in one-shot interactions. I discuss these results’ implications

for allocating residents to rural hospitals. When wages can be flexibly adjusted, insti-

tutions can be divided into a hierarchy of congested and uncongested segments. I use

this hierarchical structure to characterize equilibrium payoffs. As an application, I show

that with flexible wages, repeated interaction resolves well-known non-existence issues:

while static stable matchings may fail to exist with complementarities and/or peer effects,

self-enforcing matching processes always exist if institutions are sufficiently patient.

*I am indebted to Nageeb Ali for many fruitful conversations and his continuing guidance and support. Iam grateful to the Economics Department at Penn State for their hospitality during a productive visit in 2018. Ithank Joel Sobel for his detailed comments and financial support during my visit to Penn State. I have also ben-efited from comments and suggestions by Mohammad Akbarpour, Christopher Chambers, Vincent Crawford,Henrique de Oliviera, Miaomiao Dong, Simone Galperti, Ed Green, Nima Haghpanah, Fuhito Kojima, ScottKominers, Maciej Kotowski, Mark Machina, Ran Shorrer, Joel Watson, Alexander Wolitzky, Leeat Yariv, aswell as seminar participants at UCSD. All errors are my own.

†Contact: Department of Economics, University of California, San Diego. E-mail: [email protected]. Web-site: http://econweb.ucsd.edu/~cel013/.

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http://econweb.ucsd.edu/%7Ecel013/pdfs/wp/celiu_jmp.pdf

mailto: [email protected]

http://econweb.ucsd.edu/%7Ecel013/

1 Introduction

Summary: College admission, hospital-resident matching, and entry-level hiring are allongoing matching processes that take place every year. One side of these markets—theinstitutions—are long-lived players, whereas the other side—namely the students, residents,or workers—participate in the matching process on only a few occasions (sometimes onlyonce). Yet, much of our theoretical analysis of matching environments treats both sides ofthe market as being short-lived, ignoring the possibility for dynamic incentives that could beused to motivate long-lived players. To understand the scope for such dynamic interactions,this paper develops a framework to study ongoing matching processes in repeated matchingmarkets, where one side of the market is long-lived while the other is short-lived. In eachperiod, every long-lived institution is matched to multiple short-lived agents on the otherside of the market.

In this environment, I define and study a solution concept called self-enforcing matchingprocess. A matching process is a complete contingent plan specifying a current matchingoutcome as a function of past histories. A matching process is self-enforcing if it is immuneto not only unilateral deviations by institutions or agents, but also blocking coalitions thatsimultaneously involve an institution and groups of agents over a possibly infinite horizon. Aself-enforcing matching process therefore represents self-fulfilling market expectations thatare immune to both individual and coalitional deviations. A central theme of my analysisis that in repeated matching markets, long-lived coalition members—the institutions— canbe disciplined through continuation play. In other words, dynamic incentives can enforcematching outcomes that are not stable in one-shot interactions. The goal of my paper is tocharacterize what can be supported, both with and without flexible wages between institutionsand their employees.

The first contribution of this paper is to provide a framework and solution concept thattakes into account the possibility of institutions’ dynamic interactions. In two-sided matchingmarkets, any appropriate stability notion must allow for joint deviations that are carried outby coalitions of agents from opposite sides of the market. Modeling these deviations througha non-cooperative game-theoretic approach, however, would inevitably anchor the analysisto specific extensive-form assumptions on the coalition formation process. Instead, I adopt acooperative approach consistent with the analysis of static matching environments. I assumethat any deviation involving coalitions of players must be profitable for all its members, andstudy matching processes that admit no such deviations. By meshing coalitional reasoningdirectly into a repeated matching environment, the resulting framework explicitly accounts

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for the institutions’ intertemporal tradeoffs, while relying only on assumptions on the players’primitive payoff structure.

In Section 3, I analyze markets where wages are fixed, as in Gale and Shapley (1962). Thisis the canonical assumption for the labor market of medical residents. I show that there is adichotomy of players in this environment: in every self-enforcing matching process, playersin the top coalition sequence, identified through an algorithm, must be matched accordingto a static stable matching, regardless of institutions’ patience level; by contrast, when in-stitutions are patient, it is possible to assign players outside of the top coalition sequence inways that are unstable in one-shot interactions—I characterize the set of self-enforcing ar-rangements with patient institutions. In an extreme case, when all agents share a commonordinal ranking over institutions, every player is in the top coalition sequence. As a result,no self-enforcing matching process can enforce any outcome beyond the unique static stableoutcome—dynamic enforcement is completely powerless. In the motivation section, I discussthese results’ implication for the matching market of medical residents.

Section 4 turns to the environment where institutions and their employees can redistributematch surplus through wages—this is the case considered in Kelso and Crawford (1982). Ishow that the institutions can be partitioned into a congestion hierarchy, which consists of acongested segment as well as a sequence of uncongested ones: institutions in the congestedsegment can sanction each other through costly bidding wars for employees; the uncongestedsegments, by contrast, can always adapt their hiring decisions in response to the congestedsegment and obtain maximum surplus. This market structure determines the set of stableoutcomes when institutions are patient. As an application, I show that—in contrast to thefindings from the static matching literature—self-enforcing arrangements always exist wheninstitutions are patient, despite complementarities and peer effects in preferences. In particu-lar, treating groups of agents as objects, a Random Serial Dictatorship among the congestedsegment is always self-enforcing when institutions are sufficiently patient.

Motivation: The interest in repeated matching markets and self-enforcing matching pro-cesses is motivated by both normative and positive considerations.

From a normative standpoint, in the matching market for medical residents, dynamicincentives can be harnessed as an instrument for enforcing matching outcomes that favorrural hospitals. In the U.S., rural communities typically feature more elderly patients, higherrate of accidental injuries, and lower average income; the closure of rural hospitals has alsobeen associated to significant changes in medical utilization and the health status of localpopulation. See, for example, Bindman, Keane and Lurie (1990), Hadley and Nair (1991)

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and Rosenbach and Dayhoff (1995). These have motivated the interests in policy intervention(U.S. Government Accountability Office 1991); nevertheless, the lack of complete medicalfacilities, excessive workloads, and cultural and geographic isolation make it difficult for ruralhospitals to attract healthcare providers.

According to the Rural Hospital Theorem (Roth 1986), static market incentives are com-pletely powerless for the purpose of allocating more residents to rural hospitals: regardless ofwhat a (static) matching program may propose, a rural hospital failing to fill its vacancies willend up with the exact same matches across all stable market outcomes. A history-dependentmatching program, however, can tap into hospitals’ dynamic incentives, and propose match-ing processes that sustain more favorable outcomes for rural hospitals. One particular wayof exploiting history dependence is to excludes hospitals that go against the program’s pro-posed matchings from future participation, effectively recommending empty matches to thedeviating hospitals in subsequent periods—this is the method used by the Japanese medicalresident matching program (JRMP) (Kamada and Kojima 2015). However, in both the U.S.and Japanese markets, matchings proposals are not externally enforced: hospitals can cir-cumvent the program and hire through direct negotiation (see also, for example, Roth 1991).The proposed matching process must then be self-enforcing in order not to create incentivesfor defection.

The results in Section 3 are valuable for understanding what can be sustained by a history-dependent matching program. In particular, Theorem 2 suggests that even in the completeabsence of coercive power, a history-dependent matching program can alleviate the staffingshortages faced by rural hospitals—this should be implemented by targeting hospitals out-side of the top coalition sequences. Theorem 1, on the other hand, indicates that residentsin the top coalition sequence can only be allocated to rural hospitals through alternativemeans of intervention: existing measures to help rural hospitals, such as student loan re-payment/forgiveness programs, should prioritize residents in the top coalition sequence. Infact, as indicated in Corollary 2, if medical residents have highly homogenous preferencesover hospitals, no changes in the assignment of residents can be achieved through history-dependence alone.

For descriptive purpose, static stability notions (see, for example, Gale and Shapley 1962,Kelso and Crawford 1982 and Hatfield and Milgrom 2005) overlook important intertemporaltradeoffs that form the basis of firms’ collusive behaviors in matching markets.

For instance, in a 2010 antitrust litigation, the Department of Justice alleged that Adobe,Apple, Google, Intel, Intuit and Pixar had reached unwritten agreements that “eliminated asignificant form of competition” in order to suppress the wages paid to their matched em-

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ployees. In particular, according to the DOJ, these agreements were “not ancillary to anylegitimate collaboration”, and “much broader than reasonably necessary for the formationor implementation of any collaborative effort“, but instead aimed at “disrupting the normalprice-setting mechanisms that apply in the labor setting” (Department of Justice 2010).

Collusive agreements like this cannot be explained through static stability notions: in theabsence of future repercussions, competing firms would form blocking pairs with workerspaid at excessively low wages. In a self-enforcing matching process, however, current poach-ing behaviors can be met with future sanctions. As a result, firms refrain from poaching inthe current period in order to maintain on-going surplus extraction from their employees.

Dynamic incentives also provide an explanation for what happens in matching marketswhere static stable outcomes fail to exist—such non-existence issues are prevalent when pref-erences exhibit complementarities and/or peer effects, both of which are realistic and impor-tant features in labor markets. In those cases, static solution concepts fail to produce anymeaningful descriptions of market outcomes. The results in Section 4 point to a novel channelthat sustains stability in markets with non-canonical preferences: institutions in the congestedsegment have interdependent blocking opportunities—this is the source of (static) instability,but also gives rise to intertemporal tradeoffs that stabilize the market. The uncongested seg-ments, by contrast, can never be motivated dynamically, but can always be matched in waysthat create no myopic incentives to deviate.

In the next section, I illustrate the framework and results of this paper through a stylizedexample based on the NRMP.

1.1 Motivating Example:

Three hospitals I1, I2 and Ir each has 2 positions to fill every year. Each year, five medicalstudents a1, a2, a3, a4, a5 enter the market looking for internship. In the NRMP, wages arenon-negotiable, so players’ payoffs are determined by the identity of their match partners.

Students are short-lived players, so only their ordinal preferences are relevant. The leftpanel of Table 1 lists every student’s preference ranking over hospitals. Observe that hospitalIr is the least favorite for every student. In the matching literature, Ir is usually interpreted asa hospital located in the rural area that faces difficulties in filling its vacancies.

Hospitals’ Bernoulli utility from matched students are shown in the right panel of Table 1.Assume each hospital has additively separable utility from matched students, and derives zeroutility from unfilled positions. Observe that a5 is every hospital’s least preferred student.

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a1 I2 I1 Ir

a2 I2 I1 Ir

a3 I1 I2 Ir

a4 I1 I2 Ir

a5 I1 I2 Ir

uI(a) 5 4 3 2 1

I1 a1 a2 a3 a4 a5

I2 a3 a2 a4 a1 a5

Ir a2 a4 a3 a1 a5

TABLE 1. Preferences of Students and Hospitals

Static Stability: There are only two static stable matchings in this market: as is shownin Fig. 1, the matchings mI and mA correspond, respectively, to the hospital-optimal andstudent-optimal matchings. Both these matchings leave the rural hospital Ir with an unfilledposition, while the other position is filled by the worst student a5.

I1

I2

Ir

a1

a2

a3

a4

a5

mI

I1

I2

Ir

a1

a2

a3

a4

a5

mA

FIGURE 1. Static Stable Matchings

For the welfare of rural population, the matching m0, shown in Fig. 2, is a more desirableoutcome: Ir still has one unfilled position, but its situation is improved compared to mI ormA, as it gets to hire the more desirable resident a2. However, m0 is unstable: I1 and a2

will form a blocking pair. This phenomenon is known more generally as the Rural HospitalTheorem:

Rural Hospital Theorem. (Roth 1986). When preferences over individuals arestrict, any hospital that does not fill its quota at some (static) stable matching isassigned precisely the same set of students at every (static) stable matching.

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I1

I2

Ir

a1

a2

a3

a4

a5

FIGURE 2. m0: An Unstable Matching

According to the Rural Hospital Theorem, reallocating physicians to rural hospitals will in-evitably create unstable matches.

A Self-Enforcing Matching Process: Now suppose the matching program can vary pro-posed matches based on past history. Consider the matching process µ0 depicted in Fig. 3:hospitals and students start off by matching according to m0. So long as no blocking involv-ing a hospital has been observed in the past, players are repeatedly matched at m0; if anyblocking involving a hospital is observed, players will be matched according to mA forever.Hospitals’ per-period cardinal payoffs frommA andm0 are also listed in the diagram for easyreference.

If the market evolves according to µ0, then in every period the static matching m0 isrealized. It remains to check that µ0 is self-enforcing.

FIGURE 3. A Self-Enforcing Matching Process Implementing m0

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Students are short-lived players. The only deviation a student can do unilaterally is tosever the match with his hospital.

Hospitals, on the other hand, are long-lived players. Hospital I1 can contemplate a devia-tion plan d1, which is a complete contingent plan specifying a group of students, d1(h), thatI1 intends to be matched with at every possible future history h. Recall that the matching pro-cess µ0 is also a complete contingent plan. At history h, µ0 also specifies a group of students,denoted µ0(I1|h), for I1. Whenever d1(h) and µ0(I1|h) are in conflict, I1 can unilaterally

fire everyone in µ0(I1|h)\d1(h). Students in d1(h)\µ0(I1|h), however, need to be poachedfrom other hospitals. Since poaching involves joint deviations by both hospital and student,we say d1(h) is feasible when everyone in d1(h)\µ0(I1|h) finds I1 more desirable than theirmatched hospital recommended by µ0.

For the matching process µ0 to be self-enforcing, it must be immune to both the unilat-eral and joint deviations outlined above. A one-shot deviation principle, established later inLemma 1, implies that in markets with fixed wages like the NRMP, a matching process isself-enforcing if and only if at every possible history of the market:

1. no student wishes to unilaterally sever her match, and

2. no hospital finds it profitable to deviate once with a feasible group of student, andpermanently revert to following µ0.

Suppose no deviation involving hospitals has occurred. Every student is getting a strictlypositive payoff, so no students would want to unilaterally leave their match. By following µ0,I1 receives a payoff of 6 every period from matching with a1, a5. If I1 were to fire a5 andblock with a2, it would enjoys a one-period payoff of 9, but subsequently receive per-periodpayoff of 5. As long as I1’s discount factor δ satisfies

(1− δ)9 + δ5 < 6,

or δ > 34, such blocking will not be profitable. A similar argument rules out any profitable

joint deviations involving I2.Now suppose a hospital-deviation has occurred. According to µ0, they should expect to

be matched at mA forever. Again, no students are willing to individually break off from theirmatches. For hospitals, current period blocking no longer carries any future consequences—the market is permanently stuck at mA regardless of what transpires in the matching market.Since mA is statically stable, by definition, a hospital’s matched partners in mA are moredesirable than any other students that are feasible, so no hospital has any profitable one-shotblocking either.

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In light of Lemma 1, µ0 is a self-enforcing matching process. Note that in a static world,pairwise blocking between I1 and a2 would have rendered the matching m0 unstable. In thematching process µ0, however, one-shot blockings like this will permanently throw the marketinto a less desirable matching, mA, for I1. Such blocking is no longer jointly profitable.

The Problem with Common Ranking over Hospitals: The analysis so far has demon-strated the possibility of using history dependence to enforce outcomes that are not stablein one-shot interactions. Now I illustrate the limitation of dynamic enforcement when allstudents share a common preference over hospitals.

a1 I1 I2 Ir

a2 I1 I2 Ir

a3 I1 I2 Ir

a4 I1 I2 Ir

a5 I1 I2 Ir

uI(a) 5 4 3 2 1

I1 a1 a2 a3 a4 a5

I2 a3 a2 a4 a1 a5

Ir a2 a4 a3 a1 a5

TABLE 2. A Market with Aligned PreferencesConsider the market in Table 2, where everything is identical to the one in Table 1, except

now all students share a common ranking I1 I2 Ir over hospitals.

I1

I2

Ir

a1

a2

a3

a4

a5

FIGURE 4. m∗: The Unique Stable Matching

There is a unique static stable matching m∗ in this market, as depicted in Fig. 4. I willshow that regardless of how patient the hospitals are, no self-enforcing matching process canimplement any matching other than m∗.

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To see why, first observe that as the top choice for all the students, I1 always finds its mostpreferred students a1, a2 feasible: in any matching process µ and at any history h, a1, a2are either already matched to I1, or matched to a hospital that they find less desirable thanI1. If I1 is matched with any other students at a history h, then I1 has a feasible deviationplan: match with a1, a2 at every future history, including h. This is profitable because itoffers I1 an immediate improvement at h, as well as the highest possible continuation value inany matching process. To rule out profitable deviations like this, all self-enforcing matchingprocesses must match I1 with a1, a2 regardless of the history of the market.

Since a1 and a2 are always matched to I1 in a self-enforcing matching process, there isno way to reward I2 with a better match than a3, a4. On the other hand, the only way topunish I2 with a match worse than a3, a4 is to match a3 or a4 with I1—otherwise I2 cansimply poach them back. This is, again, impossible because the hiring quota for I1 is alwaysfilled by a1 and a2 at every possible history of the market. It follows that in any self-enforcingmatching process, I2 is always matched precisely with a3, a4, on and off equilibrium path.

A similar “peeling” argument along students’ shared preference list ensures Ir is matchedwith a5 in any self-enforcing matching process. The only self-enforcing matching processtherefore always has players matched according to m∗.

In the above example, depending on players’ preference configuration, history depen-dence may or may not expand the set of stable outcomes. It is also worth pointing out that theuniqueness of static stable outcome is not responsible for the collapse of dynamic enforce-ment: in fact, if the hospitals share a common (cardinal or ordinal) preference over residents,the market will also have a unique static stable matching; nevertheless, it is still possibleto expand the set of stable outcomes through history dependence. The results in Section 3characterizes how the scope of this expansion relates to players’ preferences.

1.2 Related Literature

There is now a growing interest in dynamic interactions in matching markets. The exist-ing literature can largely be divided into two strands, based on the nature of the matchingenvironment being investigated.

The first strand focuses on matching markets where players leave the market permanentlyonce matched: this is the natural assumption for organ transplant or child adoption. In suchcontext, players optimally trade off the cost of waiting against the arrival of better matchingopportunities. Unver (2010), Anderson, Ashlagi, Gamarnik and Kanoria (2015), Baccara, Leeand Yariv (2015), Leshno (2017), and Akbarpour, Li and Oveis Gharan (2017) investigate the

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welfare implications of various matching algorithms; Du and Livne (2016) and Doval (2018)consider the existence of self-enforcing matching arrangements.

Another strand of the literature investigates matching between long-lived players, wherematching arrangements can be revised over time. Depending on the application, Corbae,Temzelides and Wright (2003), Damiano and Lam (2005), Kurino (2009), Newton and Sawa(2015), Kotowski (2015), Kadam and Kotowski (2018a) and Kadam and Kotowski (2018b)propose and analyze different solution concepts in this environment. The most closely re-lated paper to mine is Corbae et al. (2003), which allows full history dependence in directedmatching, and considers applications in monetary theory.

My paper differs from the above papers in that I consider a different kind of matchingenvironment: a fixed set of long-lived players on one side of the market repeatedly matchwith new generations of short-lived players from the other. This is the environment that arisesnaturally from college admission and various entry-level labor markets. Dynamic incentivesalso play a different role in such environments: they can be used as carrots and sticks fordisciplining long-lived players.

My analysis builds on the techniques from the repeated games literature. In particular,the environment I consider bears many similarities to repeated games with perfect monitor-ing; I also adapt the techniques for equilibrium construction in Fudenberg and Maskin (1986)and Abreu, Dutta and Smith (1994). The main difference of my paper is that I consider atwo-sided cooperative environment, where the basic unit of analysis is not actions for indi-vidual players, but moves by coalitions. This introduces novel features that are not presentin standard repeated games: regardless of patience level, the matching environment can im-pose severe restrictions on the payoffs of long-lived players. This is in sharp contrast to the“anything goes” result from folk theorems.

The top coalition sequence, featured prominently in the analysis in Section 3, is closelyrelated to, but different from the top coalition property. The top coalition property is firstintroduced in Banerjee, Konishi and Sonmez (2001) in the context of coalition formation, andsubsequently used in a variety of cooperative game settings. See, for example, Niederle andYariv (2009) and Doval (2018) for the use in matching settings. The top coalition property isan assumption which requires that every subset of players to have a top coalition; top coalitionsequence, by contrast, is not an assumption about the underlying game, but is instead thecollection of top coalitions that can be found through iterative eliminations.

The current paper is not the first attempt to mesh coalitional reasoning with dynamic con-sideration. Bernheim and Slavov (2009) considers the dynamic extension of Condorcet Win-ner in majoritarian voting settings; in Ali and Liu (2018), we consider coalition moves in a

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general repeated cooperative environments. Both these papers allow full history dependence—the most important difference of my approach from these papers is in the form of effectivecoalitions: in both these papers, the effective coalitions consist of subsets of long-lived play-ers; in the matching environment I consider, however, the effective coalitions are those thatconsist of a single long-lived player and multiple generations of short-lived players. There isalso an extensive literature on farsighted coalition formation: Chwe (1994), Ray and Vohra(1997), Ray and Vohra (1999), Konishi and Ray (2003), Gomes and Jehiel (2005), and Ray(2007), among others, consider the (implicit or explicit) coalition formation process amongforward-looking players. My approach is different from these papers in that I allow full his-tory dependence in a specialized matching environment, where there is no persistent statevariable.

Finally, the dynamic interactions analyzed in this paper speak to two strands literature onstatic matching markets. The first strand is the literature on matching with complementar-ities or externalities. See, for example, Sasaki and Toda (1996), Dutta and Masso (1997),Echenique and Yenmez (2005), Kominers (2010), Pycia (2012), Flanagan (2015), and Pyciaand Yenmez (2017). In particular, Che, Kim and Kojima (2017) and Azevedo and Hatfield(2018) consider existence of stable matching in large markets, where institution preferencesmay exhibit complementarities. My results complement the existing literature by pointingout a novel channel for maintaining stability, when market size is not necessarily large. Thesecond strand is the literature on matching with constraints. See, for example, Kamada andKojima (2015), Fragiadakis and Troyan (2017), Goto, Kojima, Kurata, Tamura and Yokoo(2017), Kamada and Kojima (2017), and Kamada and Kojima (2018). This literature focuseson understanding and improving the welfare properties of a variety of real-life matchingsmarkets, which are subject to restrictions other than those covered by standard capacity con-straints. My paper complements this literature by providing a framework for understandingthe enforceability of these restrictions, when the matching authority can exploit institutions’dynamic incentives.

1.3 Structure of the Paper

Section 2 introduce the model and discuss the solution concept, making minimal assumptionson players’ preferences and allowing for flexible wages between the institutions and theirmatched agents. In Section 3, I focus on the canonical model for the matching market ofmedical residents: I assume that wages are held at an exogenously fixed level, and imposestandard restrictions on preferences. Section 4 analyzes the flexible-wage model with general

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preferences. Finally, Section 5 concludes. All omitted proofs can be found in Appendix A.

2 Model and Solution Concept

2.1 Repeated Matching Market

Time is discrete. In every period t = 0, 1, 2 . . ., a new generation of agentsA ≡ a1, . . . , aJenter a matching market to match with the institutions I ≡ I1, . . . , IK. Institutions arelong-lived players that persist through time. Agents are short-lived, and remain in the marketfor only one period.

Each institution Ik has a per-period hiring quota qk > 0, and Bernoulli utility functionuk : 2A → R over the sets of agents it matches with in every period, where 2A is the set ofall subsets of A. The utility of staying unmatched, uk(∅), is normalized to 0. Institutions’preferences are not assumed to satisfy the gross substitutes condition (see, for example, Kelsoand Crawford 1982). Institutions share a common discount factor δ, and evaluate a sequenceof flow utilities (u1, u2, . . .) through discounting:

(1− δ)∞∑

t=0

δtut

Agents are allowed to care about not only the institution they are matched with, but alsotheir colleagues. Each agent aj has utility function vj :

(I × 2Aaj

)∪ (∅, ∅) → R over

institutions and colleagues, where 2Aaj is the set of subsets of A that contain aj , while (∅, ∅)represents staying unmatched. For every agent aj , vj(∅, ∅) is normalized to 0.

Finally, I assume players have quasilinear utilities over money.

2.2 Stage-Game Matchings:

In this section, I introduce the static matching game that is played repeatedly in every period.As a point of comparison for the dynamic solution concept, I then review what it means for amatching to be stable in one-shot interactions.

In each period, a single institution can recruit multiple agents who are currently in themarket—institutions and agents are matched according to a static one-to-many matching. Astage-game matching is described by a static assignment of players along with wage vectorsfrom institutions.

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Definition 2.1. A static assignment φ is a mapping defined on the set I ∪ A which satisfiesfor all Ik ∈ I and aj ∈ A:

1. agents are either unmatched, or matched to institution and peers: φ(aj) ∈ (I × 2Aaj) ∪(∅, ∅);

2. institutions are either unmatched, or matched to agents: φ(Ik) ∈ 2A and |φ(Ik)| ≤ qk;and

3. consistency requirements for the mapping φ: if aj ∈ φ(Ik) then φ(aj) = (Ik, φ(Ik)); ifφ(aj) = (Ik, A) then φ(Ik) = A.

Let Φ denote the set of all static assignments.

A wage vector ζk ∈ RJ from institution Ik describes the wages paid by institution to eachagent. Any non-zero wage payments can be made from an institution only to its employees.

Definition 2.2. A static matching is a pairm = (φ, ζkKk=1), where φ is a static assignment,and each ζk = [ζkj]

Jj=1 is a vector of wages from institution Ik to agents. Together φ and

ζkKk=1 satisfy ζkj 6= 0 only if aj ∈ φ(Ik).

I use M to denote the set of all static matchings.

Players’ preferences over static matchings are induced by their preferences over theirassigned partners and the payments. For m = (φ, ζkKk=1), I write

uk(m) = uk(φ(Ik))−∑

aj∈φ(Ik)

ζkj,

andvj(m) = vj(φ(aj)) + ζkj

for all aj ∈ φ(Ik).

In canonical models of static matching markets, stability is defined in a symmetric way forinstitutions and agents. For example, as in Gale and Shapley (1962) and Kelso and Crawford(1982), the set of possible deviations consist of those that can be carried out by an institution,an agent, or a coalition involving players from both sides of the market.

In my model, institutions are long-lived while agents are short-lived. Due to this inherentasymmetry between two sides of the market, it is convenient to define stability in a way thatemphasizes the role played by institutions, which is still consistent with the canonical stabilitynotions.

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Definition 2.3. A feasible deviation by institution Ik from static matching m = (φ, ζlKl=1)

is a vector (A′k, ζ′k) where A′k is a set of agents that satisfies |A′k| ≤ qk, and ζ ′k ∈ RJ a vector

of wages from institution Ik. Together A′k and ζ ′k satisfy:

1. institutions can make non-zero wage payments only to their employees: ζ ′kj = 0 for allaj /∈ A′k; and

2. a deviation is feasible only if all involved agents find it profitable: vj(Ik, A′k) + ζ ′kj >

vj(m) for all aj ∈ A′k.

I use Dk(m) to denote the set of all feasible deviations from m by institution Ik.

A feasible deviation thus encodes both an institution’s ability to fire any subset of itsemployees, as well as its ability to organize a blocking coalition with agents from the otherside of the market: such coalitions are available to Ik as long as every member of A′k finds itprofitable. A feasible deviation (A′k, ζ

′k) is profitable for Ik if

uk(A′k)−

∑

aj∈A′k

ζ ′kj > uk(m).

A player is said to be autarkic in the stage game if it is not matched to any other player.The following definition of static stability extends that of Kelso and Crawford (1982) to thecurrent environment.

Definition 2.4. A static matching m = (φ, ζkKk=1) is stable if

1. no agent prefers autarky: vj(m) ≥ 0 for every aj ∈ A; and

2. no institution has any profitable feasible deviation: if Ik has a feasible deviation (A′k, ζ′k)

from m, then (A′k, ζ′k) is not profitable for Ik.

Notice that this definition allows for peer effects. I use M∗ to denote the set of all staticstable matchings.

2.3 Matching Processes

I now turn to repeated matching markets, and introduce the dynamic counterparts for theconcepts reviewed in the previous section. In particular, a matching process is the dynamicextension of a static matching; self-enforcing matching process replaces stable matching out-come as the stability notion in this dynamic environment.

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In a repeated matching market, a matching process specifies a proposed or anticipated

stage-game matching today, based on the history of the market and the realization of thepublic randomization device.

At any time t = 0, 1, . . ., a period t ex ante history records past realizations of the publicrandomization device and the ensuing matching outcomes up to the beginning of period t.Formally, letH0 ≡ ∅; for t > 0, the set of period t ex ante histories isHt ≡ (Ω×M)t, whereΩ is the state space of the arbitrary public randomization device. I useH ≡ ∪∞t=0Ht to denotethe set of all possible ex ante histories. Ht ≡ Ht ×Ω is the set of period t ex post histories,andH ≡ H×Ω the set of all ex post histories. LetH∞ = (Ω×M)∞ be the set of outcomepaths of the game. For each h ∈ H ∪ H ∪ H∞, let ωt(h) and mt(h) denote the period trealization of public randomization device and stage-game matching in h, respectively. Foreach ω ∈ Ω, I use H(ω) ≡ h ∈ H : ω0(h) = ω to denote the set of ex post histories withω as the initial draw from the public randomization device.

Definition 2.5. A matching process µ is a mapping µ : H → ∆(M).

Equivalently, making explicit the realization from the public randomization device, a match-ing process is also a mapping µ : H →M .

In what follows, I will often write µ = (ψ, ξkKk=1). ψ : H → Φ is an assignmentprocess, which maps an ex post history h to the static assignment in the stage-game matchingµ(h); for each institution Ik ∈ I, ξk is a wage plan ξk : H → RJ such that ξk(h) is the wagevector from institution Ik in µ(h).

For an assignment process ψ, ex post history h, and institution Ik, I use ψ(Ik|h) to denotethe (possibly empty) set of agents matched to Ik in the static assignment ψ(h), and ψ(aj|h)

for the institution and peers matched to agent aj .A matching process can capture various ways in which players coordinate their expecta-

tions. In markets like the NRMP, one can interpret a matching process as a matching protocolthat proposes a (lottery over) stage-game matchings in every period based on the market’shistory. In labor markets without an explicit coordination mechanism, a matching processrepresents a shared understanding among players on how hiring outcomes in the past impactthe market’s future employment decisions.

A matching process µ, together with the public randomization device, induces a proba-bility measure over H∞. Institutions’ preferences over matching processes are induced fromthe distribution over the sequences of stage-game matchings. That is, institutions evaluate

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matching processes by

uk(µ) ≡ EµH∞[(1− δ)

∞∑

t=0

δtuk(mt(h∞))].

For every t ≥ 0, a matching process µ also induce a probability measure over the periodt ex post historiesHt. Let utk(µ) ≡ EµHt

[uk(µ(h)

)]denote institution Ik’s expected payoff in

period t from the matching process µ. We have

uk(µ) =∞∑

t=0

δtutk(µ).

As long-lived players, in a repeated matching market, institutions can conceive a feasibledeviation plan from a matching process. A feasible deviation plan for an institution is acomplete contingent plan, which specifies a static feasible deviation for the same institutionafter every possible history of the market.

Definition 2.6. A feasible deviation plan from the matching process µ = (ψ, ξlKl=1) byinstitution Ik is a vector (d′k, ξ

′k). d′k : H → 2A is a mapping from ex post histories to sets of

agents, and ξ′k : H → RJ is a wage plan. Together d′k and ξ′k satisfy that (d′k(h), ξ′k(h)) is afeasible deviation from µ(h) for Ik, at every ex post history h.

A feasible deviation plan (d′k, ξ′k) from matching process µ = (ψ, ξlKl=1) is a one-shot

deviation if there is a unique ex post history h in the domain of (d′k, ξ′k), such that

d′k(h) = ψ(Ik|h) and ξ′k(h) = ξk(h) for all h ∈ H\h.

I will refer to a one-shot deviation (d′k, ξ′k) by (d′k(h), ξ′k(h)).

Similar to Definition 2.3, the feasibility of a deviation plan encodes the constraint thatparticipation in a coalitional deviation must be voluntary: for a coalitional deviation to beavailable to institution Ik at history h, every participating agent must find the deviation prof-itable.

A feasible deviation plan can be thought of as an institution’s plan for how to manipulatea matching process, in every possible contingency of the market. If the plan is carried out,it generates, together with the matching process, a path of realized stage-game matchings.Institutions evaluate a feasible deviation plan through the discounted sum of utilities collectedfrom this path of manipulated stage-game matchings.

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A conceptual question needs to be resolved in order to determine an institution’s pay-off from a feasible deviation plan: if institution Ik were to commit a stage-game deviation(A′k, ζ

′k) when players anticipate the stage-game outcome to be matching m, what is the re-

sulting stage-game matching? Since the matching process adjusts future expectations basedon past realized matchings, without an answer to the question above, it will be impossible todetermine how a matching process responds to a deviation as history unfolds.

To this end, I assume that in a stage-game, if institution Ik were to commit a deviation,other players do not further deviate, and remain matched according to m whenever possi-ble. Formally, given a static matching m = (φ, ζlKl=1) and a feasible deviation (A′k, ζ

′k)

from m by institution Ik, let [m, (Ik, A′k, ζ′k)] denote the manipulated matching. I make the

following assumption.

Assumption 1. Supposem = (φ, ζlKl=1). The static matching [m, (Ik, A′k, ζ′k)] = (φ, ζ lKl=1)

satisfies

1. φ(Ik) = A′k; φ(Il) = φ(Il)\A′k for all l 6= k; and φ(a) = (∅, ∅) for all a ∈ φ(Ik)\A′k.

2. ζk = ζ ′k; ζ lj = ζlj for l 6= k, aj ∈ φ(Il); and ζ lj = 0 for l 6= k, aj /∈ φ(Il).

Most static matching models are silent about what happens in the wake of a deviationbecause in one-shot interactions, the profitability of a deviation does not depend on howother players react.1

The only role Assumption 1 plays in my model is in establishing Lemma 2 below: anydeviation involving a single institution can be unequivocally attributed to the said institution.One can imagine alternative assumptions on how other players may further deviate in thestage-game upon seeing an initial deviation. As long as players understand which institutionwas involved in the initial deviation, all results in this paper remain unaffected.

I now define the manipulated matching process resulting from a feasible deviation plan,as well as the profitability of a feasible deviation plan.

Definition 2.7. Given a matching process µ and a feasible deviation plan (d′k, ξ′k) by institu-

tion Ik, the manipulated matching process [µ, (Ik, d′k, ξ′k)] : H → M is a matching process

defined by[µ, (Ik, d

′k, ξ′k)](h) = [µ(h), (Ik, d

′k(h), ξ′k(h))] ∀h ∈ H

1There are a few exceptions in the literature on matching with externailities, where the utility from a devi-ation depends on how other players are matched. In these static models, the reactions of other player are oftenbuilt implicitly into the players’ choice functions. See, for example, Sasaki and Toda (1996), Bando (2012),Pycia and Yenmez (2017).

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A feasible deviation plan (d′k, ξ′k) is profitable for institution Ik if uk([µ, (Ik, d′k, ξ

′k)]) >

uk(µ).

To state what it means for a matching process to be self-enforcing, it is convenient todefine the continuation of a matching process and a deviation plan, at both ex ante and ex

post histories.

Definition 2.8. Fix a matching process µ. At ex ante history h ∈ H, the continuationmatching process µ|h : H →M is defined by

µ|h(h) = µ(h, h) for all h ∈ H;

At ex post history h = h × ω ∈ H, the continuation matching process µ|h : H(ω) → M isdefined by

µ|h(h) = µ(h, h) for all h ∈ H(ω);

The continuation of a feasible deviation plan is defined analogously.

Definition 2.9. Fix a feasible deviation plan (d′k, ξ′k). At ex ante history h ∈ H, the continu-

ation deviation plan (d′k|h, ξ′k|h) : H → 2A × RJ is defined by

d′k|h(h) = d′k(h, h) and ξ′k|h(h) = ξ′k(h, h) for all h ∈ H.

At ex post history h = h × ω ∈ H, the continuation deviation plan (d′k|h, ξ′k|h) : H(ω) →2A × RJ is defined by

d′k|h(h) = d′k(h, h) and ξ′k|h(h) = ξ′k(h, h) for all h ∈ H(ω).

At an ex post history, the continuations of a matching process and a deviation plan haveto follow the most recent realization from the public randomization device, which imposesthe restriction on their domains. To reduce the notational burden, however, I will suppressthis dependence on the realization of ω whenever it causes no confusion.

As the dynamic counterpart to static stable matching, a self-enforcing matching pro-cess represents players’ self-fulfilling expectations over how the repeated matching marketevolves. Such expectations must be robust against both individuals and coalitional devia-tions.

Definition 2.10. A matching process µ : H →M is self-enforcing if at every ex post historyh,

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1. no agent prefers autarky: vj(µ(h)) ≥ 0 for every aj ∈ A; and

2. no institution has any profitable feasible deviation plan from the continuation matchingprocess µ|h.

The lack of profitable deviation is imposed on the continuation matching processes af-ter every possible ex post history: this essentially imposes the same sequential rationalityrequirement that subgame perfection does in a non-cooperative environment.

2.4 Some Intermediate Results

Before getting to the main results of the paper, I first establish some lemmas that will beuseful for characterizing self-enforcing matching processes. Readers who are familiar withthe repeated games literature will immediately recognize them as translations of classicalresults. In particular, these results make it possible to import the language of carrot and stickinto the current cooperative environment.

Lemma 1 (One-shot Deviation Principle). A matching process µ is self-enforcing if and only

if

1. vj(µ(h)) ≥ 0 for all aj ∈ A and every h ∈ H, and

2. no institution has any profitable one-shot deviation from the continuation matching

process µ|h at any h ∈ H, and

3. the set uk(µ(h)) : h ∈ H is bounded for all Ik ∈ I.

In order to verify that a matching process is self-enforcing, it is sufficient to verify thatthere is no one-shot individual or coalitional deviation after any ex post history. The bound-edness condition is needed to rule out Ponzi schemes.

Lemma 2 (Identifiability of Deviating Institution). Let m be a static matching. If (A′k1 , ζ′k1

)

is a feasible deviation by institution Ik1 and (A′k2 , ζ′k2

) is a feasible deviation by institution

Ik2 , then [m, (Ik1 , A′k1, ζ ′k1)] = [m, (Ik2 , A

′k2, ζ ′k2)] 6= m only if Ik1 = Ik2 .

This is an identification result: if stage-game matching m′ is different from m, and is theresult of a deviation from m triggered by a single institution, then the deviating institutioncan be uniquely identified by comparing m′ and m. Note that this does not imply that allm′ 6= m can be attributed to a single institution. In fact, a deviation that is attributed to

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institution Ik is not even necessarily caused by Ik: if Ik’s employee in m chooses to beautarkic, Ik will be identified as the deviator. However, since in a self-enforcing matchingprocess agents are always individually rational, employees will never choose to leave theirmatched institutions. When constructing self-enforcing matching processes, this result thenallows deviating institutions to be punished for their disobedience.

The next result is a direct consequence of the above two lemmas.

Lemma 3. Let m be a static stable matching. For any discount factor 0 ≤ δ < 1, the

matching process µ : H →M defined by

µ(h) = m ∀h ∈ H

is a self-enforcing matching process.

3 Markets with Fixed Wages

In this section, I first specialize the general environment introduced in Section 2 to marketswhere wages are fixed. This case is considered in Gale and Shapley (1962) and Roth andSotomayor (1992), and is the canonical environment to study the NRMP. I characterize theset of self-enforcing assignment processes in this environment, and discuss how these resultsprovide insights and caveats on using dynamic enforcement to redistribute medical residents.

3.1 Model

In the fixed-wage model, players’ utilities are determined by the identity of their match: thereare no flexible transfers that adjust matching payoffs. Institutions’s preferences over agentssatisfy the gross substitutes condition. In addition, agents are indifferent about which otheragents are matched to the same institution. Finally, all players’ preferences in the stage-gameare strict.

Market: Formally, in a fixed-wage environment, a stage-game matching m = (φ, ζkKk=1)

must satisfy ζk = 0 for all 1 ≤ k ≤ K, where 0 ∈ RJ is the zero wage vector. This is withoutloss of generality, as the fixed wages’s impacts can be incorporated into the functions uk(.)and vj(.). I will suppress the redundant wage vectors—instead of stage-game matchings, Iwill focus on stage-game assignments in this section.

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Instead of matching processes, it is without loss to focus on assignment processes ψ :

HF → ∆(Φ), where HF ≡ ∪∞t=0(Ω × Φ)t is the set of ex ante histories under fixed-wageenvironment. The set of ex post histories, HF , and outcome paths, HF

∞, are also modifiedaccordingly to reflect the restriction on wages.

In the matching market for medical residents, an assignment process represents a match-ing program that can suggest different future assignments based on past history. For example,the matching program can exclude certain hospitals from the matching program in responseto a past violation, which is equivalent to suggesting an assignment that leaves these hospitalsunmatched.

The wage offer in any feasible deviation (A′l, ζ′l) must also be held at ζ ′l = 0. I will also

suppress the wage offer here and use DAk (φ) to denote the subsets of agents that, combinedwith a zero wage offer, constitute a feasible deviation from the assignment φ:

DAk (φ) ≡ B ⊆ A : (B,0k) ∈ Dk(φ, 0lKl=1)

where 0l ∈ RJ is the zero wage vector for all 1 ≤ l ≤ K. A feasible deviation plan (d′k, ζ′k)

reduces to the mapping d′k : HF → 2A.

Preferences: For each institution Ik, the utility from matched agents, uk : 2A → R is strict:uk(.) satisfies uk(A) 6= uk(A

′) for different sets of agents A 6= A′. The gross substitutescondition, when specialized to the fixed-wage environment, requires that for all A ⊆ A andall aj, aj′ ∈ A,

aj ∈ arg maxB⊆A∪aj ,aj′,|B|≤qk

uk(B) ⇒ aj ∈ arg maxB⊆A∪aj,|B|≤qk

uk(B)

For each agent aj , the utility function vj : I → R is only a function of his matchedinstitution, and satisfies vj(I) 6= vj(I

′) for all I 6= I ′ ∈ I. Without concerns for peer effects,there is no need to keep track of agents’ peers when describing an assignment: instead ofwriting φ(aj) ∈ I × 2Aaj , I use φ(aj) ∈ I to denote agent aj’s matched institution.

3.2 The Scope of Dynamic Enforcement

In this section, I explore the possibility of using assignment processes to redistribute medicalresidents to rural hospitals. Before stating the main results, it is well known that static stableassignment exists when institutions’ preferences satisfy the gross substitutes condition and

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agents’ preferences are absent of peer effects (see, for example, Roth and Sotomayor (1992),Theorem 6.5). As a corollary to Lemma 3, the following result guarantees the existence ofself-enforcing assignment process for any patience level.

Corollary 1. When institutions’ preferences satisfy the gross substitutes condition and agents

do not care about peer effects, self-enforcing assignment processes exist for every discount

factor 0 ≤ δ < 1.

I first define the “top coalition sequence” in the matching market. Fix an arbitrary subsetof institutions and agents I ′∪A′ ⊆ I∪A. An institution Ik ∈ I ′ and a set of agents Ak ⊆ A′form a top coalition of I ′ ∪ A′ if

uk(Ak) ≥ uk(B) for all B ⊆ A′ and vj(Ik) ≥ vj(I) for all I ∈ I ′

Players in a top coalition find each other to be the most desirable match, given that I ′∪A′are the potential pool of players to be matched with. A top coalition sequence generalizesthis notion by iteratively finding and eliminating top coalitions in the remaining players, untilno new top coalition can be found.

Definition 3.1. 2 The top coalition sequence is the ordered set T = (Ikg , Akg)Gg=1 producedby the following algorithm:

1. Set g = 0 and T0 = ∅;

2. For g ≥ 1:

• If (I ∪ A)\Tg−1 has a top coalition (Ikg , Akg): set Tg = Tg−1 ∪ (Ikg , Akg) andg = g+ 1. If there are multiple top coalitions, break ties in favor of the institutionwith the smallest index;

• If (I ∪ A)\Tg−1 has no top coalition: set T = Tg−1 and stop;

For each top coalition (I, A) in T , I will refer to I and A as each other’s top coalitionpartners.

I use ΦT to denote the set of static assignments that assign players in T to their topcoalition partners, and are individually rational for all the agents:

ΦT = φ ∈ Φ : φ(Ik) = Ak ∀Ik ∈ T , and vj(φ) ≥ 0 ∀aj ∈ A2Whenever it causes no confusion, I will use T and Tg to denote the (I, A) pairs or the set of players that

are contained therein without distinction.

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All static stable assignments must be contained in ΦT . In particular, in a static stableassignment, players in the top coalition sequence must be matched to their top coalition part-ners.

Lemma 4. If φ is a static stable assignment, then φ ∈ ΦT .

Let κ(I\T ) ≡ k : Ik ∈ I\T denote the indices for the institutions not in the topcoalition sequence. For every k ∈ κ(I\T ), the fixed-wage minmax payoff, uTk , captures theinstitution’s participation constraint, and is defined as

uTk = minφ∈ΦT

maxB∈DAk (φ)

uk(B).

Theorem 1. For every 0 ≤ δ < 1 and in every self-enforcing assignment process ψ:

1. regardless of history, players in the top coalition sequence must be matched with their

top coalition partners; all agents must obtain at least their autarkic payoffs:

ψ(h) ∈ ΦT ∀h ∈ HF ;

2. all institutions must obtain at least their fixed-wage minmax payoffs:

uk(ψ) ≥ uTk ∀Ik ∈ I\T .

Together with Lemma 4, Theorem 1 implies that in the matching market for medical res-idents, a history-dependent matching program can never alter the assignments for T beyondwhat is static stable. In addition, the worst possible continuation value for each hospitalcannot be lower than uTk .

The reasoning behind Theorem 1 is that in a top coalition sequence T = (Ikg , Akg)Gg=1,Ik1’s favorite agents Ak1 is always a feasible deviation, so there is no credible way to vary Ik1’scontinuation value. As the result, Ik1 must always behave in the myopically optimal way—being matched to Ak1 . By induction, the only credible way to vary Ikg ’s continuation valueis to match agents in Akg to institutions in Ikii≤g−1. This is impossible because Ikii≤g−1

must always be matched to their own top coalition partners. Accordingly, Ikg must alwaysbe matched to Akg . For the second half of the theorem, since players have to be matchedaccording to some assignment in ΦT , every Ik ∈ I\T can ensure a long-run payoff of uTk byblocking with a coalition of feasible agents in every period.

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As the next example illustrates, if the matching market for medical residents can be parti-tioned into specialties, then within every specialty, the residents of the “top programs” cannotbe redistributed.

Example 1. The institutions (hospital programs) and agents (residents) in the matching mar-

ket can be partitioned into specialties. Specifically, let S be the set of specialties. The set of

institutions can be partitioned into I = ∪s∈SI(s), where I(s) = Is,k : 1 ≤ k ≤ K(s),K(s) being the number of institutions in specialty s; the set of agents can be partitioned into

A = ∪s∈SA(s).

Suppose that within every specialty s ∈ S, Is,1 is the top institution for agents in the same

specialty: vj(Is,1) > vj(I) for all aj ∈ A(s), I ∈ I, and I 6= Is,1. Suppose that institutions

prefer to hire within their own specialties: us,k(A) > us,k(A′) for every institution Is,k,

A ⊆ A(s), |A| ≤ qs,k, and A′ * A(s), |A′| ≤ qs,k. Then the top institution in all specialties,

along with their favorite agents, are in the top coalition sequence.

When all agents share a common ranking over institutions, every player is in the top coali-tion sequence. In particular, if medical residents’ preferences over hospitals are highly ho-mogenous, it is impossible to enforce any assignment than the (unique) static stable assignment—history dependent matching programs offer no room for improvement compared to staticones.

Corollary 2. When agents share a common ranking over institutions, there is a unique self-

enforcing assignment process, where players match according to the unique static stable

assignment at every ex post history of the market.

Theorem 2 complements Theorem 1, and shows that with patient hospitals, if a random-ization over assignments in ΦT gives hospitals in I\T payoffs higher than uTk , then it can besupported as the time-invariant outcome of a self-enforcing assignment process.

To simplify the statement of Theorem 2, I make the following “non-triviality” assump-tions.

Assumption 2. There are at least two institutions not in the top coalition sequence, and for

every institution Ik ∈ I\T , there exists aj ∈ A\T such that vj(Ik) > 0.

Assumption 2 simplifies the statement of Theorem 2 by ruling out two simple but notationallycumbersome cases: if an institution Ik ∈ I\T satisfies vj(Ik) < 0 for all aj ∈ A\T , it cannever be matched to any agent, and can simply be removed from the game. If there is only

25

one institution Ik ∈ I\T , then

uTk = minφ∈ΦT

maxB∈DAk (φ)

uk(B) = maxB⊆A\T :vj(Ik)>0∀aj∈B

uk(B) = maxφ∈ΦT

uk(φ).

According to Theorem 1, for every self enforcing assignment process ψ,

uTk ≤ uk(ψ) ≤ maxφ∈ΦT

uk(φ) = uTk .

The second inequality above follows because all stage-game assignments in a self-enforcingmatching process are in ΦT . Due to the strictness of Ik’s preference, this would imply thatplayers can only be matched according to arg maxφ∈ΦT

uk(φ), regardless of patience or his-tory.

Define Λ∗ ≡ λ ∈ ∆(ΦT ) : Eλ[uk(φ)] > uTk ∀k ∈ κ(I\T ).

Theorem 2. For every λ ∈ Λ∗, there is a δ such that for every δ ∈ (δ, 1), there exists a

self-enforcing assignment process that randomizes over stage-game assignments according

to λ in every period.

The proof for Theorem 2 adapts the “Folk Theorem” equilibrium construction from Fu-denberg and Maskin (1986) and Abreu et al. (1994). To see the connection, note that thecondition Eλ[uk(φ)] > uTk ∀k ∈ κ(I\T ) in the definition of Λ∗ ensures that the equilibriumpayoffs for all players in I\T are above their respective minmax levels. I then constructa matching process by varying continuation play within ΦT , and using the analogues for“player-specific punishments” and “minmax punishment” in the matching environment.

As patience converges to one, this punishment scheme is able to ward off any deviationsinvolving I\T ; since all stage-game assignments are drawn from ΦT , players in I ∩ T aregiven the same match on and off equilibrium path and have no profitable stage-game devi-ations; for the same reason, agents are always individually rational and have no profitabledeviations. The constructed matching process is therefore self-enforcing.

To wrap up, the findings in this section suggest that in repeated matching markets, historydependence can substantially expand the set of feasible allocations. However, the scope ofdynamic enforcement is tempered by the presence of top coalition sequence. As a result, cau-tion should be exercised when a dynamic matching program attempts to enforce outcomesbeyond those that are static stable—this is especially relevant when there are reasons to sus-pect homogeneity in agents preferences.

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4 Markets with Flexible Wages

In this section, I return to the model introduced in Section 2: wages can be flexibly adjusted;in addition, I allow institutions to have complementary preferences over agents, and agentsto have non-trivial preferences over colleagues.

The analysis in the current section is arranged in the following order. First, I introducea congestion hierarchy over institutions, which can be identified through an algorithm; usingthis congestion hierarchy, I characterize self-enforcing matching processes when institutionsare patient. Second, I discuss the lack of stable outcomes in one-shot matching marketswith complementarities and/or peer effects, and show that self-enforcing matching processesalways exist in repeated matching markets, when institutions are sufficiently patient.

4.1 Congestion Hierarchy

In the current environment, employers can flexibly divide match surpluses with their employ-ees. The implication of this mechanism is two-fold:

1. Institutions can engage in a costly bidding war, where they compete for employeesthrough wage compensation;

2. Under the protection of a no-poaching agreement, by contrast, institutions can set lowwages and extract match surplus from their employees, without losing workers to com-petitors;

In a self-enforcing matching process, institutions can use bidding wars as promised pun-ishments for deviant behaviors; on the other hand, surplus extraction, protected by no-poachingagreements, can be used as future rewards for cooperation. Both bidding wars and no-poaching agreements must themselves be self-enforcing.

To motivate an institution dynamically, its punishment payoff when targeted in a biddingwar must be strictly lower than the maximum reward from surplus extraction. As I illustratein Example 2, the congestion hierarchy is instrumental for disentangling institutions that canbe motivated dynamically from those that cannot. As a result, it will also determine the set ofstage-game outcomes that can be credibly promised.

Example 2. There are four firms and five workers. Each firm can hire at most one worker.

Firms’ gross profits uk(aj) are as listed in Table 3. Workers derive 0 utility from unemploy-

27

ment or working for any firm.

a1 a2 a3 a4 a5

I1 1 0 1 2 2

I2 4 1 2 4 3

I3 2 1 2 2 2

I4 3 3 3 3 3

TABLE 3. Gross Profit: uk(aj)What is the worst possible payoff that can be imposed upon I2 in a bidding war? One

might be tempted to use the stage-game matchingm′2 in Fig. 5: all firms other than I2 conspire

to bid away its favorite workers. This removes a1, a4, a5 from I2’s choice set. In addition,

a3 is employed by I2 at wage 1, which leaves I2 a payoff of 1. I2 will be forced to accept

this arrangement, since otherwise its best option is to block with a2 (illustrated as the dashed

arrow)—this will at most earn it a payoff of 1, which is no more profitable compared to m′2.

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I3<latexit sha1_base64="EUzxi7f1hln3oc4G7mdgaWtn9hQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi94q2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzdRvPaHSPJaPZpygH9GB5CFn1Fjp4a533itX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhFd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXcqnd/Uald53EU4QiO4RQ8uIQa3EIdGsBgAM/wCm+OcF6cd+dj3lpw8plD+APn8wfHD41z</latexit><latexit sha1_base64="EUzxi7f1hln3oc4G7mdgaWtn9hQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi94q2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzdRvPaHSPJaPZpygH9GB5CFn1Fjp4a533itX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhFd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXcqnd/Uald53EU4QiO4RQ8uIQa3EIdGsBgAM/wCm+OcF6cd+dj3lpw8plD+APn8wfHD41z</latexit><latexit sha1_base64="EUzxi7f1hln3oc4G7mdgaWtn9hQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi94q2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzdRvPaHSPJaPZpygH9GB5CFn1Fjp4a533itX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhFd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXcqnd/Uald53EU4QiO4RQ8uIQa3EIdGsBgAM/wCm+OcF6cd+dj3lpw8plD+APn8wfHD41z</latexit><latexit sha1_base64="EUzxi7f1hln3oc4G7mdgaWtn9hQ=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi94q2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzdRvPaHSPJaPZpygH9GB5CFn1Fjp4a533itX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhFd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXcqnd/Uald53EU4QiO4RQ8uIQa3EIdGsBgAM/wCm+OcF6cd+dj3lpw8plD+APn8wfHD41z</latexit>


45 = 2<latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit><latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit><latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit><latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit>

45 = 2<latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit><latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit><latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit><latexit sha1_base64="csxmUe50ClGtb+jui16sZ1kvE1w=">AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkpSKvYiFLx4rGA/oA1hs520SzebsLsR2tBf4sWDIl79Kd78N27bHLT1wcDjvRlm5gUJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqaTQpjGPZS8gCjgT0NZMc+glEkgUcOgGk7uF330CqVgsHvU0AS8iI8FCRok2km+XBzPQxM/q13N8i2vYtytO1VkCbxI3JxWUo+XbX4NhTNMIhKacKNV3nUR7GZGaUQ7z0iBVkBA6ISPoGypIBMrLlofP8aVRhjiMpSmh8VL9PZGRSKlpFJjOiOixWvcW4n9eP9Vhw8uYSFINgq4WhSnHOsaLFPCQSaCaTw0hVDJzK6ZjIgnVJquSCcFdf3mTdGpV16m6D/VKs5HHUUTn6AJdIRfdoCa6Ry3URhSl6Bm9ojdrZr1Y79bHqrVg5TNn6A+szx/SbJHW</latexit>

34 = 3<latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit><latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit><latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit><latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit>

34 = 3<latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit><latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit><latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit><latexit sha1_base64="iCDI4OrxYZ9HZY1kY8w+rcXG9zM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKeyagLkIAS8eI5gHJGuYnfQmQ2YfzPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3ebEUGm3728qtrW9sbuW3Czu7e/sHxcOjlo4SxaHJIxmpjsc0SBFCEwVK6MQKWOBJaHvj65nffgClRRTe4SQGN2DDUPiCMzTSfe8JkPXTSnVKr2ilXyzZZXsOukqcjJRIhka/+NUbRDwJIEQumdZdx47RTZlCwSVMC71EQ8z4mA2ha2jIAtBuOr96Ss+MMqB+pEyFSOfq74mUBVpPAs90BgxHetmbif953QT9mpuKME4QQr5Y5CeSYkRnEdCBUMBRTgxhXAlzK+UjphhHE1TBhOAsv7xKWhdlxy47t9VSvZbFkScn5JScE4dckjq5IQ3SJJwo8kxeyZv1aL1Y79bHojVnZTPH5A+szx//HJF6</latexit>

11 = 3<latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit><latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit><latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit><latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit>

11 = 3<latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit><latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit><latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit><latexit sha1_base64="P2tS0t3gQtipOJJrg1tVmtbxFGI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyqYC5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1PMm5Iqcd4slt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7UapWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf3bZF1</latexit>

23 = 1<latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit><latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit><latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit><latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit>

23 = 1<latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit><latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit><latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit><latexit sha1_base64="Hj7AQSlYHDXOf0ugts4XpEQxDVg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZSScZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmkOdRzLSrYAZkEJBHQVKaMUaWBhIaAaj66nffABtRKTucByDH7KBEn3BGVrpvvMEyLpp+XxCr6jXLRTdkjsDXSZeRookQ61b+Or0Ip6EoJBLZkzbc2P0U6ZRcAmTfCcxEDM+YgNoW6pYCMZPZ1dP6KlVerQfaVsK6Uz9PZGy0JhxGNjOkOHQLHpT8T+vnWC/4qdCxQmC4vNF/URSjOg0AtoTGjjKsSWMa2FvpXzINONog8rbELzFl5dJo1zy3JJ3e1GsVrI4cuSYnJAz4pFLUiU3pEbqhBNNnskreXMenRfn3fmYt6442cwR+QPn8wf5AZF2</latexit>

22 = 0<latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit><latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit><latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit><latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit>

22 = 0<latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit><latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit><latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit><latexit sha1_base64="I1fV3yYrC/pZp9whXySRljnHwsk=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKewGwVyEgBePEcwDkhhmJ73JkNnZZaZXiUv+w4sHRbz6L978GyfJHjSxoKGo6qa7y4+lMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2UaA4NHslIt31mQAoFDRQooR1rYKEvoeWPr2d+6wG0EZG6w0kMvZANlQgEZ2il++4TIOunlcqUXlG3Xyy5ZXcOukq8jJRIhnq/+NUdRDwJQSGXzJiO58bYS5lGwSVMC93EQMz4mA2hY6liIZheOr96Ss+sMqBBpG0ppHP190TKQmMmoW87Q4Yjs+zNxP+8ToJBtZcKFScIii8WBYmkGNFZBHQgNHCUE0sY18LeSvmIacbRBlWwIXjLL6+SZqXsuWXv9qJUq2Zx5MkJOSXnxCOXpEZuSJ00CCeaPJNX8uY8Oi/Ou/OxaM052cwx+QPn8wf19JF0</latexit>

m′2 : u2(m′2) = 1

FIGURE 5. Not Credible

1. Uncongested Segment(s)—Lack of Dynamic Motivation:

However, a closer examination of the matchingm′2 will cast its credibility in doubt: inm′2,

in order to remove a5 from I2, I4 is behaving sub-optimally in the stage-game: I4 is paying

a5 a wage of 2, leaving itself a profit of 1, while it could have blocked with the unemployed

a2 at 0 wage, which will earn it a higher profit of 3.

28

In fact, I4 faces an uncongested market: no matter what other players are doing, I4 can

always block with an unemployed worker and earn a profit of 3—this is no less than the

highest possible payoff I4 can obtain in a stage-game. Therefore, I4 cannot be motivated

dynamically through the threat of being targeted in a bidding war. As a result, I4 has no

reason to accept the suboptimal stage-game arrangement m′2. In fact, in any credible stage-

game matching, I4’s employee must be paid a wage of 0. Without the participation of I4,

bidding wars against I2 will not be as effective: m′2 is no longer a credible punishment for

I2. When constructing the congestion hierarchy, the first step is to identify institutions like I4,

so they can be discarded from potential participants in bidding wars.

The argument above goes one step further—if it’s understood that I4 will not participate

in any bidding wars, I3 will reason that it also faces an uncongested market: in any credible

stage-game matching, there are at least two workers, either employed by I4 or unemployed,

who are earning 0 wage; one of these workers can earn I3 a payoff of 2. This is, again, no

less than I3’s highest possible stage-game payoff. An argument similar to that for I4 ensures

that I3 will not participate in any bidding wars. The second step in the construction of the

congestion hierarchy is to identify and discard institutions like I3.



















15 = 1<latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit><latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit><latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit><latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit>

15 = 1<latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit><latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit><latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit><latexit sha1_base64="xnZ77lSqRl1R/NtYZo+mipu7S2Q=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidYslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf6iZF3</latexit>

24 = 1<latexit sha1_base64="x5Do6x1sv0AyT9r4xAUQOknVMbg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2melV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mMpDLrut7O2vrG5tZ3bye/u7R8cFo6OmyZKNIcGj2Sk2z4zIIWCBgqU0I41sNCX0PLH1zO/9QDaiEjd4SSGXsiGSgSCM7TSffcJkPXTcmVKr6jXLxTdkjsHXSVeRookQ71f+OoOIp6EoJBLZkzHc2PspUyj4BKm+W5iIGZ8zIbQsVSxEEwvnV89pedWGdAg0rYU0rn6eyJloTGT0LedIcORWfZm4n9eJ8Gg2kuFihMExReLgkRSjOgsAjoQGjjKiSWMa2FvpXzENONog8rbELzll1dJs1zy3JJ3WynWqlkcOXJKzsgF8cglqZEbUicNwokmz+SVvDmPzovz7nwsWtecbOaE/IHz+QP6ipF3</latexit><latexit sha1_base64="x5Do6x1sv0AyT9r4xAUQOknVMbg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2melV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mMpDLrut7O2vrG5tZ3bye/u7R8cFo6OmyZKNIcGj2Sk2z4zIIWCBgqU0I41sNCX0PLH1zO/9QDaiEjd4SSGXsiGSgSCM7TSffcJkPXTcmVKr6jXLxTdkjsHXSVeRookQ71f+OoOIp6EoJBLZkzHc2PspUyj4BKm+W5iIGZ8zIbQsVSxEEwvnV89pedWGdAg0rYU0rn6eyJloTGT0LedIcORWfZm4n9eJ8Gg2kuFihMExReLgkRSjOgsAjoQGjjKiSWMa2FvpXzENONog8rbELzll1dJs1zy3JJ3WynWqlkcOXJKzsgF8cglqZEbUicNwokmz+SVvDmPzovz7nwsWtecbOaE/IHz+QP6ipF3</latexit><latexit sha1_base64="x5Do6x1sv0AyT9r4xAUQOknVMbg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2melV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mMpDLrut7O2vrG5tZ3bye/u7R8cFo6OmyZKNIcGj2Sk2z4zIIWCBgqU0I41sNCX0PLH1zO/9QDaiEjd4SSGXsiGSgSCM7TSffcJkPXTcmVKr6jXLxTdkjsHXSVeRookQ71f+OoOIp6EoJBLZkzHc2PspUyj4BKm+W5iIGZ8zIbQsVSxEEwvnV89pedWGdAg0rYU0rn6eyJloTGT0LedIcORWfZm4n9eJ8Gg2kuFihMExReLgkRSjOgsAjoQGjjKiSWMa2FvpXzENONog8rbELzll1dJs1zy3JJ3WynWqlkcOXJKzsgF8cglqZEbUicNwokmz+SVvDmPzovz7nwsWtecbOaE/IHz+QP6ipF3</latexit><latexit sha1_base64="x5Do6x1sv0AyT9r4xAUQOknVMbg=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2melV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mMpDLrut7O2vrG5tZ3bye/u7R8cFo6OmyZKNIcGj2Sk2z4zIIWCBgqU0I41sNCX0PLH1zO/9QDaiEjd4SSGXsiGSgSCM7TSffcJkPXTcmVKr6jXLxTdkjsHXSVeRookQ71f+OoOIp6EoJBLZkzHc2PspUyj4BKm+W5iIGZ8zIbQsVSxEEwvnV89pedWGdAg0rYU0rn6eyJloTGT0LedIcORWfZm4n9eJ8Gg2kuFihMExReLgkRSjOgsAjoQGjjKiSWMa2FvpXzENONog8rbELzll1dJs1zy3JJ3WynWqlkcOXJKzsgF8cglqZEbUicNwokmz+SVvDmPzovz7nwsWtecbOaE/IHz+QP6ipF3</latexit>


32 = 0<latexit sha1_base64="oALgFFtcHMKnp+USLK4aeg/I6es=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/PyhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybu9KFYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf3fpF1</latexit><latexit sha1_base64="oALgFFtcHMKnp+USLK4aeg/I6es=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/PyhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybu9KFYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf3fpF1</latexit><latexit sha1_base64="oALgFFtcHMKnp+USLK4aeg/I6es=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/PyhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybu9KFYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf3fpF1</latexit><latexit sha1_base64="oALgFFtcHMKnp+USLK4aeg/I6es=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0omIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/PyhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybu9KFYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf3fpF1</latexit>


43 = 0<latexit sha1_base64="POuzWmq3d+9phZu0gayKO3HW5Sg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hV0NmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RxXvLckndbLlYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6kZF3</latexit><latexit sha1_base64="POuzWmq3d+9phZu0gayKO3HW5Sg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hV0NmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RxXvLckndbLlYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6kZF3</latexit><latexit sha1_base64="POuzWmq3d+9phZu0gayKO3HW5Sg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hV0NmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RxXvLckndbLlYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6kZF3</latexit><latexit sha1_base64="POuzWmq3d+9phZu0gayKO3HW5Sg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hV0NmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RxXvLckndbLlYrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6kZF3</latexit>


11 = 0<latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit><latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit><latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit><latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit>

11 = 0<latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit><latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit><latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit><latexit sha1_base64="D8uaAZIvmUbURHW/rAqu+ZH6vag=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgpleJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT5Q06Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVAhoiVrFu+9yAkhE0UKKCdqKBh76Clj+6nvqtB9BGxtEdjhPwQj6IZCAFRyvdd58AeS9jbEKvqNsrld2KOwNdJiwnZZKj3it9dfuxSEOIUChuTIe5CXoZ1yiFgkmxmxpIuBjxAXQsjXgIxstmV0/oqVX6NIi1rQjpTP09kfHQmHHo286Q49AselPxP6+TYlD1MhklKUIk5ouCVFGM6TQC2pcaBKqxJVxoaW+lYsg1F2iDKtoQ2OLLy6R5XmFuhd1elGvVPI4COSYn5Iwwcklq5IbUSYMIoskzeSVvzqPz4rw7H/PWFSefOSJ/4Hz+APLhkXI=</latexit>

R

8>><>>:

<latexit sha1_base64="r03J0bE/Bz8/6Wm1Ju20kC0eysY=">AAACFXicbVDLSgMxFM34rPU16tJNsAgupMyIoMuiG5dV7AM6pWTSO21o5kFyRyhDf8KNv+LGhSJuBXf+jZl2Ftp6IeHknHu5OcdPpNDoON/W0vLK6tp6aaO8ubW9s2vv7Td1nCoODR7LWLV9pkGKCBooUEI7UcBCX0LLH13neusBlBZxdI/jBLohG0QiEJyhoXr2qRcyHHIms7sJ9SQESL2MeiqVkDk8nGSuucxbDIZY7dkVp+pMiy4CtwAVUlS9Z395/ZinIUTIJdO64zoJdjOmUHAJk7KXakgYH7EBdAyMWAi6m01dTeixYfo0iJU5EdIp+3siY6HW49A3nbkHPa/l5H9aJ8XgspuJKEkRIj5bFKSSYkzziGhfKOAoxwYwroT5K+VDphhHE2TZhODOW14EzbOq61Td2/NK7aqIo0QOyRE5IS65IDVyQ+qkQTh5JM/klbxZT9aL9W59zFqXrGLmgPwp6/MHYP+e7g==</latexit><latexit sha1_base64="r03J0bE/Bz8/6Wm1Ju20kC0eysY=">AAACFXicbVDLSgMxFM34rPU16tJNsAgupMyIoMuiG5dV7AM6pWTSO21o5kFyRyhDf8KNv+LGhSJuBXf+jZl2Ftp6IeHknHu5OcdPpNDoON/W0vLK6tp6aaO8ubW9s2vv7Td1nCoODR7LWLV9pkGKCBooUEI7UcBCX0LLH13neusBlBZxdI/jBLohG0QiEJyhoXr2qRcyHHIms7sJ9SQESL2MeiqVkDk8nGSuucxbDIZY7dkVp+pMiy4CtwAVUlS9Z395/ZinIUTIJdO64zoJdjOmUHAJk7KXakgYH7EBdAyMWAi6m01dTeixYfo0iJU5EdIp+3siY6HW49A3nbkHPa/l5H9aJ8XgspuJKEkRIj5bFKSSYkzziGhfKOAoxwYwroT5K+VDphhHE2TZhODOW14EzbOq61Td2/NK7aqIo0QOyRE5IS65IDVyQ+qkQTh5JM/klbxZT9aL9W59zFqXrGLmgPwp6/MHYP+e7g==</latexit><latexit sha1_base64="r03J0bE/Bz8/6Wm1Ju20kC0eysY=">AAACFXicbVDLSgMxFM34rPU16tJNsAgupMyIoMuiG5dV7AM6pWTSO21o5kFyRyhDf8KNv+LGhSJuBXf+jZl2Ftp6IeHknHu5OcdPpNDoON/W0vLK6tp6aaO8ubW9s2vv7Td1nCoODR7LWLV9pkGKCBooUEI7UcBCX0LLH13neusBlBZxdI/jBLohG0QiEJyhoXr2qRcyHHIms7sJ9SQESL2MeiqVkDk8nGSuucxbDIZY7dkVp+pMiy4CtwAVUlS9Z395/ZinIUTIJdO64zoJdjOmUHAJk7KXakgYH7EBdAyMWAi6m01dTeixYfo0iJU5EdIp+3siY6HW49A3nbkHPa/l5H9aJ8XgspuJKEkRIj5bFKSSYkzziGhfKOAoxwYwroT5K+VDphhHE2TZhODOW14EzbOq61Td2/NK7aqIo0QOyRE5IS65IDVyQ+qkQTh5JM/klbxZT9aL9W59zFqXrGLmgPwp6/MHYP+e7g==</latexit><latexit sha1_base64="r03J0bE/Bz8/6Wm1Ju20kC0eysY=">AAACFXicbVDLSgMxFM34rPU16tJNsAgupMyIoMuiG5dV7AM6pWTSO21o5kFyRyhDf8KNv+LGhSJuBXf+jZl2Ftp6IeHknHu5OcdPpNDoON/W0vLK6tp6aaO8ubW9s2vv7Td1nCoODR7LWLV9pkGKCBooUEI7UcBCX0LLH13neusBlBZxdI/jBLohG0QiEJyhoXr2qRcyHHIms7sJ9SQESL2MeiqVkDk8nGSuucxbDIZY7dkVp+pMiy4CtwAVUlS9Z395/ZinIUTIJdO64zoJdjOmUHAJk7KXakgYH7EBdAyMWAi6m01dTeixYfo0iJU5EdIp+3siY6HW49A3nbkHPa/l5H9aJ8XgspuJKEkRIj5bFKSSYkzziGhfKOAoxwYwroT5K+VDphhHE2TZhODOW14EzbOq61Td2/NK7aqIo0QOyRE5IS65IDVyQ+qkQTh5JM/klbxZT9aL9W59zFqXrGLmgPwp6/MHYP+e7g==</latexit>

P2

n<latexit sha1_base64="O4U8xMJ682nJERKvp8QcZdGwFmM=">AAACGnicbVDLSgMxFM3UV62vUZdugkVwNczUgi6LblxWsA/olJJJ77ShmQfJHaEM/Q43/oobF4q4Ezf+jeljoa0HQg7n3EtyTpBKodF1v63C2vrG5lZxu7Szu7d/YB8eNXWSKQ4NnshEtQOmQYoYGihQQjtVwKJAQisY3Uz91gMoLZL4HscpdCM2iEUoOEMj9WzPjxgOOZN5fdKrUF9CiNTPqa8yCbnLo0nuOtULcxtJDIbo9Oyy67gz0FXiLUiZLFDv2Z9+P+FZBDFyybTueG6K3ZwpFFzCpORnGlLGR2wAHUNjFoHu5rNoE3pmlD4NE2VOjHSm/t7IWaT1OArM5DSIXvam4n9eJ8PwqpuLOM0QYj5/KMwkxYROe6J9oYCjHBvCuBLmr5QPmWIcTZslU4K3HHmVNCuO5zreXbVcu17UUSQn5JScE49ckhq5JXXSIJw8kmfySt6sJ+vFerc+5qMFa7FzTP7A+voBHsmgQw==</latexit><latexit sha1_base64="O4U8xMJ682nJERKvp8QcZdGwFmM=">AAACGnicbVDLSgMxFM3UV62vUZdugkVwNczUgi6LblxWsA/olJJJ77ShmQfJHaEM/Q43/oobF4q4Ezf+jeljoa0HQg7n3EtyTpBKodF1v63C2vrG5lZxu7Szu7d/YB8eNXWSKQ4NnshEtQOmQYoYGihQQjtVwKJAQisY3Uz91gMoLZL4HscpdCM2iEUoOEMj9WzPjxgOOZN5fdKrUF9CiNTPqa8yCbnLo0nuOtULcxtJDIbo9Oyy67gz0FXiLUiZLFDv2Z9+P+FZBDFyybTueG6K3ZwpFFzCpORnGlLGR2wAHUNjFoHu5rNoE3pmlD4NE2VOjHSm/t7IWaT1OArM5DSIXvam4n9eJ8PwqpuLOM0QYj5/KMwkxYROe6J9oYCjHBvCuBLmr5QPmWIcTZslU4K3HHmVNCuO5zreXbVcu17UUSQn5JScE49ckhq5JXXSIJw8kmfySt6sJ+vFerc+5qMFa7FzTP7A+voBHsmgQw==</latexit><latexit sha1_base64="O4U8xMJ682nJERKvp8QcZdGwFmM=">AAACGnicbVDLSgMxFM3UV62vUZdugkVwNczUgi6LblxWsA/olJJJ77ShmQfJHaEM/Q43/oobF4q4Ezf+jeljoa0HQg7n3EtyTpBKodF1v63C2vrG5lZxu7Szu7d/YB8eNXWSKQ4NnshEtQOmQYoYGihQQjtVwKJAQisY3Uz91gMoLZL4HscpdCM2iEUoOEMj9WzPjxgOOZN5fdKrUF9CiNTPqa8yCbnLo0nuOtULcxtJDIbo9Oyy67gz0FXiLUiZLFDv2Z9+P+FZBDFyybTueG6K3ZwpFFzCpORnGlLGR2wAHUNjFoHu5rNoE3pmlD4NE2VOjHSm/t7IWaT1OArM5DSIXvam4n9eJ8PwqpuLOM0QYj5/KMwkxYROe6J9oYCjHBvCuBLmr5QPmWIcTZslU4K3HHmVNCuO5zreXbVcu17UUSQn5JScE49ckhq5JXXSIJw8kmfySt6sJ+vFerc+5qMFa7FzTP7A+voBHsmgQw==</latexit><latexit sha1_base64="O4U8xMJ682nJERKvp8QcZdGwFmM=">AAACGnicbVDLSgMxFM3UV62vUZdugkVwNczUgi6LblxWsA/olJJJ77ShmQfJHaEM/Q43/oobF4q4Ezf+jeljoa0HQg7n3EtyTpBKodF1v63C2vrG5lZxu7Szu7d/YB8eNXWSKQ4NnshEtQOmQYoYGihQQjtVwKJAQisY3Uz91gMoLZL4HscpdCM2iEUoOEMj9WzPjxgOOZN5fdKrUF9CiNTPqa8yCbnLo0nuOtULcxtJDIbo9Oyy67gz0FXiLUiZLFDv2Z9+P+FZBDFyybTueG6K3ZwpFFzCpORnGlLGR2wAHUNjFoHu5rNoE3pmlD4NE2VOjHSm/t7IWaT1OArM5DSIXvam4n9eJ8PwqpuLOM0QYj5/KMwkxYROe6J9oYCjHBvCuBLmr5QPmWIcTZslU4K3HHmVNCuO5zreXbVcu17UUSQn5JScE49ckhq5JXXSIJw8kmfySt6sJ+vFerc+5qMFa7FzTP7A+voBHsmgQw==</latexit>

P1

n<latexit sha1_base64="D2CfLn6rzP9sMtq7dBV+EVzwwU0=">AAACGnicbVDLSgMxFM34rPVVdekmWARXw4wWdFl047KCfUBnKJn0ThuaeZDcEcrQ73Djr7hxoYg7cePfmLaz0NYDIYdz7iU5J0il0Og439bK6tr6xmZpq7y9s7u3Xzk4bOkkUxyaPJGJ6gRMgxQxNFGghE6qgEWBhHYwupn67QdQWiTxPY5T8CM2iEUoOEMj9SquFzEccibzxqTnUk9CiNTLqacyCbnDo0nu2LULcxtJDIZo9ypVx3ZmoMvELUiVFGj0Kp9eP+FZBDFyybTuuk6Kfs4UCi5hUvYyDSnjIzaArqExi0D7+SzahJ4apU/DRJkTI52pvzdyFmk9jgIzOQ2iF72p+J/XzTC88nMRpxlCzOcPhZmkmNBpT7QvFHCUY0MYV8L8lfIhU4yjabNsSnAXIy+T1rntOrZ7V6vWr4s6SuSYnJAz4pJLUie3pEGahJNH8kxeyZv1ZL1Y79bHfHTFKnaOyB9YXz8dIaBC</latexit><latexit sha1_base64="D2CfLn6rzP9sMtq7dBV+EVzwwU0=">AAACGnicbVDLSgMxFM34rPVVdekmWARXw4wWdFl047KCfUBnKJn0ThuaeZDcEcrQ73Djr7hxoYg7cePfmLaz0NYDIYdz7iU5J0il0Og439bK6tr6xmZpq7y9s7u3Xzk4bOkkUxyaPJGJ6gRMgxQxNFGghE6qgEWBhHYwupn67QdQWiTxPY5T8CM2iEUoOEMj9SquFzEccibzxqTnUk9CiNTLqacyCbnDo0nu2LULcxtJDIZo9ypVx3ZmoMvELUiVFGj0Kp9eP+FZBDFyybTuuk6Kfs4UCi5hUvYyDSnjIzaArqExi0D7+SzahJ4apU/DRJkTI52pvzdyFmk9jgIzOQ2iF72p+J/XzTC88nMRpxlCzOcPhZmkmNBpT7QvFHCUY0MYV8L8lfIhU4yjabNsSnAXIy+T1rntOrZ7V6vWr4s6SuSYnJAz4pJLUie3pEGahJNH8kxeyZv1ZL1Y79bHfHTFKnaOyB9YXz8dIaBC</latexit><latexit sha1_base64="D2CfLn6rzP9sMtq7dBV+EVzwwU0=">AAACGnicbVDLSgMxFM34rPVVdekmWARXw4wWdFl047KCfUBnKJn0ThuaeZDcEcrQ73Djr7hxoYg7cePfmLaz0NYDIYdz7iU5J0il0Og439bK6tr6xmZpq7y9s7u3Xzk4bOkkUxyaPJGJ6gRMgxQxNFGghE6qgEWBhHYwupn67QdQWiTxPY5T8CM2iEUoOEMj9SquFzEccibzxqTnUk9CiNTLqacyCbnDo0nu2LULcxtJDIZo9ypVx3ZmoMvELUiVFGj0Kp9eP+FZBDFyybTuuk6Kfs4UCi5hUvYyDSnjIzaArqExi0D7+SzahJ4apU/DRJkTI52pvzdyFmk9jgIzOQ2iF72p+J/XzTC88nMRpxlCzOcPhZmkmNBpT7QvFHCUY0MYV8L8lfIhU4yjabNsSnAXIy+T1rntOrZ7V6vWr4s6SuSYnJAz4pJLUie3pEGahJNH8kxeyZv1ZL1Y79bHfHTFKnaOyB9YXz8dIaBC</latexit><latexit sha1_base64="D2CfLn6rzP9sMtq7dBV+EVzwwU0=">AAACGnicbVDLSgMxFM34rPVVdekmWARXw4wWdFl047KCfUBnKJn0ThuaeZDcEcrQ73Djr7hxoYg7cePfmLaz0NYDIYdz7iU5J0il0Og439bK6tr6xmZpq7y9s7u3Xzk4bOkkUxyaPJGJ6gRMgxQxNFGghE6qgEWBhHYwupn67QdQWiTxPY5T8CM2iEUoOEMj9SquFzEccibzxqTnUk9CiNTLqacyCbnDo0nu2LULcxtJDIZo9ypVx3ZmoMvELUiVFGj0Kp9eP+FZBDFyybTuuk6Kfs4UCi5hUvYyDSnjIzaArqExi0D7+SzahJ4apU/DRJkTI52pvzdyFmk9jgIzOQ2iF72p+J/XzTC88nMRpxlCzOcPhZmkmNBpT7QvFHCUY0MYV8L8lfIhU4yjabNsSnAXIy+T1rntOrZ7V6vWr4s6SuSYnJAz4pJLUie3pEGahJNH8kxeyZv1ZL1Y79bHfHTFKnaOyB9YXz8dIaBC</latexit>

m1 : u1(m1) = 1



















11 = 1<latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit><latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit><latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit><latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit>

11 = 1<latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit><latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit><latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit><latexit sha1_base64="EipQaaiF7oqDFYQ3L7UKOVgqp9k=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hR0RzEUIePEYwTwgWcPspDcZMvtgZlaJS/7DiwdFvPov3vwbJ8keNLGgoajqprvLT6TQxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR9dRvPaDSIo7uzDhBL2SDSASCM2Ol++4TGtbLKJ2QK0J7pbJbcWcgy4TmpAw56r3SV7cf8zTEyHDJtO5QNzFexpQRXOKk2E01JoyP2AA7lkYsRO1ls6sn5NQqfRLEylZkyEz9PZGxUOtx6NvOkJmhXvSm4n9eJzVB1ctElKQGIz5fFKSSmJhMIyB9oZAbObaEcSXsrYQPmWLc2KCKNgS6+PIyaZ5XqFuhtxflWjWPowDHcAJnQOESanADdWgABwXP8ApvzqPz4rw7H/PWFSefOYI/cD5/APRlkXM=</latexit>







25 = 0<latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit><latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit><latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit><latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit>

25 = 0<latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit><latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit><latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit><latexit sha1_base64="YqQyc9nbgcEumjR7Zbc0pEfjC1Q=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd2gmIsQ8OIxgnlAsobZySQZMju7zPQqccl/ePGgiFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3321lZXVvf2Mxt5bd3dvf2CweHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqNx+4NiJSdziOuR/SgRJ9wSha6b7zxJF20/LFhFwRt1souiV3BrJMvIwUIUOtW/jq9CKWhFwhk9SYtufG6KdUo2CST/KdxPCYshEd8Laliobc+Ons6gk5tUqP9CNtSyGZqb8nUhoaMw4D2xlSHJpFbyr+57UT7Ff8VKg4Qa7YfFE/kQQjMo2A9ITmDOXYEsq0sLcSNqSaMrRB5W0I3uLLy6RRLnluybs9L1YrWRw5OIYTOAMPLqEKN1CDOjDQ8Ayv8OY8Oi/Ou/Mxb11xspkj+APn8wf6j5F3</latexit>

R

8>><>>:


P2


P1


m2 : u2(m2) = 3



















15 = 0<latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit><latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit><latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit><latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit>

15 = 0<latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit><latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit><latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit><latexit sha1_base64="okgIFMm6ERYBYTVmlJ11kWouLAs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYi5CwIvHCOYByRpmJ51kyOzsMjOrxCX/4cWDIl79F2/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwup76zQdUmkfyzoxj9EM6kLzPGTVWuu88oaHd1LuYkCvidoslt+zOQJaJl5ESZKh1i1+dXsSSEKVhgmrd9tzY+ClVhjOBk0In0RhTNqIDbFsqaYjaT2dXT8iJVXqkHylb0pCZ+nsipaHW4zCwnSE1Q73oTcX/vHZi+hU/5TJODEo2X9RPBDERmUZAelwhM2JsCWWK21sJG1JFmbFBFWwI3uLLy6RxVvbcsnd7XqpWsjjycATHcAoeXEIVbqAGdWCg4Ble4c15dF6cd+dj3ppzsplD+APn8wf5BZF2</latexit>

24 = 0<latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit><latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit><latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit><latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit>

24 = 0<latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit><latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit><latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit><latexit sha1_base64="xw18vWCINyb0tEj7wZrT1w0IJRg=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hd0QMBch4MVjBPOAJIbZSW8yZHZ2mZlV4pL/8OJBEa/+izf/xkmyB00saCiquunu8mPBtXHdb2dtfWNzazu3k9/d2z84LBwdN3WUKIYNFolItX2qUXCJDcONwHaskIa+wJY/vp75rQdUmkfyzkxi7IV0KHnAGTVWuu8+oaH9tFyZkivi9gtFt+TOQVaJl5EiZKj3C1/dQcSSEKVhgmrd8dzY9FKqDGcCp/luojGmbEyH2LFU0hB1L51fPSXnVhmQIFK2pCFz9fdESkOtJ6FvO0NqRnrZm4n/eZ3EBNVeymWcGJRssShIBDERmUVABlwhM2JiCWWK21sJG1FFmbFB5W0I3vLLq6RZLnluybutFGvVLI4cnMIZXIAHl1CDG6hDAxgoeIZXeHMenRfn3flYtK452cwJ/IHz+QP5BpF2</latexit>





R

8>><>>:


P2


P1


m12: Reward for I1 & I2

FIGURE 6. Credible Punishments and Rewards

2. Congested Segment—Self-Fulfilling Dynamic Motivation:

Both I1 and I2, by contrast, face a congested market—even without the participation from

I3 or I4, I1 and I2 can effectively punish each other in a bidding war. Consider punishing

I1 through m1 and I2 through m2, while rewarding both through m12 (Fig. 6): u1(m1) <

u1(m12) and u2(m2) < u2(m12), and none of these matchings rely on I3 or I4 offering

29

excessive wage compensation. Both I1 and I2 can there be motivated through varying future

punishments and rewards. This, in turn, ensures the credibility of the punishments m1 and

m2, since m1 relies on the participation from I2, and vice versa—the interdependence in

payoffs ensures the internal consistency of promised punishments. After identifying I4 and I3

as uncongested, the congestion hierarchy now classifies I1, I2 as the congested segment.

3. Constructing Credible Stage-Game Outcomes:

Apart from parsing out institutions that can be motivated dynamically, the congestion hi-

erarchy is also important for identifying credible stage-game behaviors. In the current exam-

ple, I1 and I2 can be motivated dynamically. By the classical folk-theorem logic, any stage-

game behaviors that generate payoffs above u1(m1) and u2(m2) for I1 and I2, respectively,

can be dynamically enforced when firms are patient. Since I3 can always obtain maximum

payoff regardless of the behaviors of I1 and I2, once the matches for I1 and I2 are determined,

I3 must be matched to one of the remaining unmatched workers among a1, a3, a4, a5 (the

ones that can generate maximum total surplus for I3), and offer its employee 0 wage. An

inductive argument then identifies the possible matches for I4.

Example 2 illustrated the construction of the congestion hierarchy in a specialized, one-to-one matching environment. I now extend the idea to general one-to-many matchings.

Let χ(Ik, A) denote the total surplus Ik can extract from matching with agents A. That is,

χ(Ik, A) ≡ uk(A) +∑

aj∈A

vj(Ik, A).

In particular, this leaves every agent inAwith 0 payoff. Such an arrangement is feasible whenIk is protected under a no-poaching agreement, and need not worry about losing members ofA to other institutions’ offers.

For each institution Ik, define

πk ≡ maxB⊆A:|B|≤qk

χ(Ik, B).

πk is the highest possible payoff Ik can obtain from an individually rational stage-gamematching.

For a set of institutions I ′ ⊆ I, I use Q(I ′) ≡ ∑ll∈I′ ql to denote their combined hiringcapacity. As illustrated in Example 2, not every institution can be motivated dynamically topartake in a bidding war. For a set of institutions I ′ ⊆ I containing Ik, if everyone in I ′\Ik

30

can be motivated, then

πk(I ′) ≡ minA⊆A:|A|≤Q(I′)

maxB⊆A\A

χ(Ik, B).

is the most severe punishment that can be imposed on Ik, in a bidding war, by I ′\Ik andIk’s employees.

To understand the definition of πk(I ′), recall that in a bidding war, as in Example 2,I ′\Ik can conspire to bid away a total of (Q(I ′) − qk) agents from Ik; meanwhile, Ikitself is matched to qk agents, all of whom are employed at premium wages. If Ik were torejects such an arrangement, its best alternative is to abandon the current employees, and hirefrom outside of the combined Q(I ′) agents. The minmax operation in πk(I ′) captures Ik’sbest response to the most effective punishment. Finally, observe that πk(I ′) ≤ πk for everyinstitution Ik and I ′ ⊆ I such that Ik ∈ I ′.

A procedure analogous to the one in Example 2 produces the congestion hierarchy in thegeneral, one-to many matchings environment.

Definition 4.1. The congestion hierarchy is the partition O = P1, . . . ,PG,R over institu-tion produced when the following algorithm terminates:

1. Set P0 = ∅.

2. For g ≥ 1:

• if Ik ∈ I\ ∪g−1i=0 Pi : πk = πk(I\ ∪g−1

i=0 Pi) 6= ∅, set Pg = Ik ∈ I\ ∪g−1i=0 Pi :

πk = πk(I\ ∪g−1i=0 Pi), and set g = g + 1.

• if Ik ∈ I\ ∪g−1i=0 Pi : πk = πk(I\ ∪g−1

i=0 Pi) = ∅, setR = I\ ∪g−1i=1 Pi and stop;

In Example 2, I4 is the segmentP1: these are institutions that can obtain their maximumpayoff through blocking and surplus extraction, even while punished in a bidding war. Theseinstitutions cannot be disciplined dynamically, and therefore cannot be expected to partakein a bidding war. Given P1, . . . ,Pg−1, Pg then captures all the institutions that cannotbe punished in a bidding war without the cooperation from some institution in ∪g−1

i=1Pi—inExample 2, I3 is the segment P2. The iterative procedure stops when an empty Pg is found,and classifies all remaining institutions intoR.

Notice that by construction, if R is nonempty, then every institution Ik ∈ R satisfiesπk > πk(R): as illustrated in Example 2, this ensures that dynamic motivation amongmembers of R can be self-fulfilling. Meanwhile, again by construction, no matter how

31

R,PG, . . . ,Pg+1 are matched, every Ik ∈ Pg can find ways to obtain πk: even thoughinstitutions in ∪Gi=1Pi cannot be motivated dynamically, they can always adapt so there is noteven myopic incentives for deviation.

With the congestion hierarchy O = P1, . . . ,PG,R in place, we are now ready to char-acterize the set of self-enforcing matching processes in the flexible-wage environment. Dueto the lack of dynamic motivation in A and ∪Gi=1Pi, define

M ≡ m ∈M : vj(m) ≥ vj(∅, ∅) ∀aj ∈ A; uk(m) = πk ∀Ik ∈ I\R.

Matchings in M ensure that no players in A or ∪Gi=1Pi have static incentive to deviate.Define

Λ ≡ λ ∈ ∆(M) : Eλ[uk(m)] > πk(R) ∀Ik ∈ R, supp(λ) is bounded.

Λ is the set of lotteries over M with bounded support, which gives each institution inR anexpected payoff higher than its punishment payoff πk(R).

As Theorem 3 and Theorem 4 demonstrate, the sets M and Λ provide a tight character-ization for the set of self-enforcing matching processes: every element in Λ can be sustainedas the stationary outcome in a self-enforcing matching process with patient institutions; onthe other hand, matchings in M are the only ones that can be played in any self-enforcingmatching process for any patience level.

Theorem 3. For every λ ∈ Λ, there is a δ such that for every δ ∈ (δ, 1), there exists a

self-enforcing matching process that randomizes over stage-game matchings according to λ

in every period.

The proof for Theorem 3 is divided into three cases, based on the cardinality of R. IfR = ∅, all matchings in M are static stable, so Lemma 3 delivers the result. If R 6= ∅,institutions in R are disciplined using the “Folk Theorem” construction as in Fudenberg andMaskin (1986) and Abreu et al. (1994): note that the condition Eλ[uk(m)] > πk(R) ∀Ik ∈ Rin the definition of Λ ensures that the equilibrium payoffs for all players inR are above theirrespective minmax levels; institutions in I\R, on the other hand, are always granted theirmaximum surplus so no dynamic enforcement is necessary. Finally, depending on if R is asingleton set, a slightly different equilibrium construction is needed.

Theorem 4. For every 0 ≤ δ < 1 and in every self-enforcing matching process µ:

32

1. regardless of history, institutions facing uncongested market must obtain their maxi-

mum stage-game payoffs:

µ(h) ∈M ∀h ∈ H;

2. institutions facing congested market are no worse off than when targeted in a bidding

war:

uk(µ) ≥ π(R) ∀Ik ∈ R.

The proof of the first statement exploits the construction of hierarchyO = P1, . . . ,PG,R:since institutions in P1 always extract maximum surplus from their matched agents, their em-ployees must be getting zero payoff. As a result, when institutions in P2 contemplates adeviation, P1’s matched agents are vunerable to poaching just like unmatched agents. Byconstruction, institutions in P2 can always obtain maximum surplus even after all the agentsemployed by I\P1 are removed, so P2 must also be extracting maximum surplus, whichleaves their matched agents available for poaching to P3. An inductive procedure like thisensures that every Ik ∈ ∪Gg=1Pg obtains πk. The second statement follows since only institu-tions inR can be motivated to participate in bidding wars.

4.2 Existence: Static vs Dynamic

In static matching markets where agents do not care about peer effects, Kelso and Crawford(1982) showed that stable matchings exist if firm preferences satisfy the gross substitutescondition. In particular, for a vector of wages for the agents ζk = (ζk1, . . . , ζkJ), define

Chk(ζk) ≡ arg maxB⊆A

uk(B)−

∑

aj∈B

ζkj

.

For every subset of agents B ⊆ A and pair of wage vectors ζ1k and ζ2

k such that ζ2kj ≥ ζ1

kj forall aj ∈ A, define Tk(B, ζ1

k , ζ2k) ≡ aj ∈ B : ζ2

kj = ζ1kj. Firm preferences are said to satisfy

the gross substitutes condition if B ∈ Chk(ζ1k) implies the existence of B ∈ Chk(ζ2

k) suchthat Tk(B, ζ1

k , ζ2k) ⊆ B.

However, when institutions’ preferences do not conform to the gross substitutes condition,or when agents can have non-trivial preferences over which colleagues they work with, it iswell known that static stable matchings may fail to exist.

In fact, in a very general model of matching with contracts, Hatfield and Milgrom (2005)and Hatfield and Kojima (2008) showed that if an institution violates a variant of the gross

33

substitutes condition, it is always possible to construct a market where all other institutionssatisfy the gross substitutes condition, but there is no stable matching. Pycia (2012) showedthat when coalitions can divide matching surplus through transfers, stable matchings are onlyguaranteed to exist when players are restricted to dividing surplus through a generalized classof Nash bargaining rules.

The following examples illustrate the non-existence problem introduced by complemen-tarity and peer effects.

Example 3. 3 (Complementarity in Institution Preferences) There are two identical firms and

three identical workers. Absent wages, workers are indifferent between being employed or

not. Each firm has a hiring quota for 2 workers, and generates a gross profit of 3 only when

both positions are filled; otherwise, firms produce 0 profit. Firms’ gross profits before wage

payments are as follows:

a1, a2 a2, a3 a1, a3 a1 a2 a3

I1 3 3 3 0 0 0

I2 3 3 3 0 0 0

TABLE 4. Gross Profit for Firms

Note that firms’ preferences do not satisfy the gross substitutes condition. To see this

market has no static stable matchings, notice that no matter how players are matched, there

must be at least one firm and one worker, say I1 and a1, who are obtaining zero payoff, and

at least one another worker, say a2, whose payoff is less than 1.5. I1, a1 and a2 can then form

a blocking coalition.

Example 4. (Peer Effects in Agent Preferences) There are two firms and two workers. Firm

I1 has 2 hiring quotas, while I2 can only hire one worker. Each firm can make a gross

profit of 3 if it hires at least one worker. Before wage compensation, workers’ disutility from

employment depends on the their co-worker in the following way:

I1, aj I1 I2

ai 0 −3 −1

TABLE 5. Worker Disutility from Employment

3I thank Henrique de Oliveira for a conversation that inspired this example.

34

To see this market has no static stable matching, there are two cases to consider:

1. If I1 employs both a1 and a2, then one of them, say a1, must be getting a wage less that

1.5. I2 and a1 can then form a blocking coalition, where a1 is awarded a wage of 2 and

I2 gets to keep the remaining surplus of 1;

2. If either a1 or a2 is not matched to I1, then both agents must be getting 0 payoffs: I1

does not find it individually rational to pay a wage above 3; I2 must be paying zero

wage, for otherwise it will block with the other agent and extract all surplus. However,

in this case I1 will form a blocking coalition with a1, a2.

Whenever the market has no stable outcomes, the “problem” always stems from the con-gested segment R: it is always possible to find matchings that are individually rational forall agents; moreover, based on the analysis in the previous section, institutions in the uncon-gested segments can always be matched in ways that create no blocking incentives—R istherefore the source of (static) instability.

Indeed, in the congested segment, institutions’ opportunities for surplus extraction are in-terdependent. As illustrated in Example 3 and Example 4, whenever one institution exhaustsits blocking opportunity, new blocking opportunities arise for some other institution. Whenthis happens, competition among institutions lead to instability.

In repeated matching markets, however, payoff dependence in the congested segmentalso gives rise to intertemporal tradeoffs that can dampen the institutions’ competitive incen-tives. The next result, Theorem 5, ensures that when institutions are patient, any intratempo-ral incentives for blocking can be disciplined through future punishments. Essentially, thisamounts to proving the existence of a collusive agreement, in the form of a (randomized)matching λ, which is preferred by everyone in R when compared to the consequences ofbidding wars.

Theorem 5. The sets M and Λ are non-empty.

Together with Theorem 3, Theorem 5 delivers the existence result.Theorem 5 is proved by direct construction. First, let institutions in R randomize over

priorities, then take turns to extract full surplus from their favorite agents according to therealized priority. In expectation, this gives every Ik ∈ R a payoff higher than πk. Next, bythe construction of O = P1, . . . ,PG,R, no matter how R are matched, it will not preventPG from obtaining maximum surplus. Once everyone inR∪PG is matched, every Ik ∈ PG−1

can be matched to obtain πk as well. The matchings in M and Λ can then be constructedby carrying on this procedure through to P1.

35

To sum up, in the flexible-wage environment, institutions in the congested segment can bemotivated dynamically, while those in the uncongested segments cannot—this is in parallelwith the fixed-wage environment, where players may or may not be motivated dynamically,depending on if they belong to the top coalition sequence. Using the congestion hierarchy, Ishow that the destabilizing incentives in static environments can be disciplined by intertem-poral tradeoffs—self-enforcing matching processes always exist when institutions are patient.

5 Conclusion

This paper considers stability in two-sided matching markets, where one side is long-livedand the other is short-lived. The analysis builds on elements from both the matching litera-ture and the repeated games literature. In both fixed-wage and flexible-wage environments,some players can be motivated through intertemporal tradeoffs, while others are immune todynamic enforcement. These results have normative implications for the design of matchingprograms; they also help describe market outcomes when static solution concepts fail to makeany meaningful predictions.

The current paper provides a general framework for understanding history dependencein repeated matching markets, but much remains unanswered about these environments. Forfuture work, more study is merited in order to better understand matching processes underfixed patience levels, or when players may have a restricted class of preferences that reflect thespecific matching environment being analyzed. Another important avenue for future researchis the communication of private information in matching processes. In static environments,the deferred acceptance algorithm is strategy-proof for one side of the market. In repeatedmatching markets, if a matching algorithm goes beyond proposing static stable outcomes, itremains an open question whether such algorithms can still be made strategy-proof, or haveto rely on weaker incentive-compatibility conditions. I hope to address these issues in futureresearch.

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39

A Appendix

A.1 Proofs in Section 2

Lemma 5. Let µ be a self-enforcing matching process. For any institution Ik, the set of recommended

stage-game payoffs from µ across all possible ex post histories, uk(µ(h)) : h ∈ H, is bounded.

Proof. Fix an institution Ik and let µ = (ψ, ξlKl=1) be a self-enforcing matching process.

I first show the set uk(µ(h)) : h ∈ H is bounded from above. Note that by the definition of self-

enforcing matching process, at every ex post history h ∈ H, the recommended matching µ(h) must

satisfy vj(µ(h)) ≥ 0 for all aj ∈ A. In particular, vj(Ik, ψ(Ik|h)) + ξkj(h) ≥ 0 for all aj ∈ ψ(Ik|h).

This implies that, at every ex post history h ∈ H,

uk(µ(h)) = uk(ψ(Ik|h))−∑

aj∈ψ(Ik|h)

ξkj(h)

≤ uk(ψ(Ik|h)) +∑

aj∈ψ(Ik|h)

vj(Ik, ψ(Ik|h))

≤ maxB⊆A

uk(B) +∑

aj∈Bvj(Ik, B) ≡ bk

So uk(µ(h)) : h ∈ H is bounded above by bk.

In order to show uk(µ(h)) : h ∈ H is bounded from below. I will first establish that the

set of continuation values for Ik across all ex ante histories is bounded from above, while the set of

continuation values following ex post histories is bounded from below, which then delivers the desired

claim.

Fix any ex ante history h ∈ H, the continuation matching process at µ|h

satisfies

uk(µ|h(h)) = uk(µ(h, h)) ≤ bk

for all ex post histories h ∈ H. Since the above inequality holds for all h, it must hold for ev-

ery stage-game matching along every possible outcome path generated by µ|h. So the continuation

value uk(µ|h), as an expectation over the discounted sum of these stage-game payoffs, must satisfy

uk(µ|h) ≤ bk. Since this holds uniformly for every h ∈ H, the set uk(µ|h) : h ∈ H is bounded

above.

Next I show the set uk(µ|h) : h ∈ H is bounded from below. In particular, I will show that for

all h ∈ H,

uk(µ|h) ≥ minA⊆A:|A|≤

∑Kl=1 ql

maxB⊆A\A

uk(B) +∑


Suppose by contradiction that uk(µ|h) < bk for some h ∈ H. We will show that Ik has a feasible

40

profitable deviation plan from µ|h, which is a contradiction to µ being a self-enforcing matching

process.

Consider the following deviation plan (d′k, ξ′k) from µ|

h= (ψ, ξlKl=1) : for any h ∈ H,

d′k(h) = arg maxB⊆A\(∪Kl=1ψ(Il|h))

uk(B) +∑

aj∈Bvj(Ik, B).

If d′k(h) 6= ∅, then

ξ′kj(h) =

−vj(Ik, d′k(h)) + 1

2|d′k(h)| [bk − uk(µ|h)] if aj ∈ d′k(h);

0 otherwise.

If d′k(h) = ∅, define

ξ′kj(h) = 0 ∀aj ∈ A.

Note that by construction,

uk(d′k(h)) +

∑

aj∈d′k(h)

vj(Ik, d′k(h))

= maxB⊆A\(∪Kl=1ψ(Il|h))

uk(B) +∑

aj∈Bvj(Ik, B)

≥ minA⊆A:|A|≤

∑Kl=1 ql

maxB⊆A\A

uk(B) +∑

aj∈Bvj(Ik, B) = bk.

(1)

The inequality above follows from∣∣∣(∪rl=1ψ(Il|h))

∣∣∣ ≤∑K

l=1 ql.

We first verify that the deviation plan (d′k, ξ′k) is feasible. By construction, at every ex post history

h ∈ H, d′k(h) ⊆ A\(∪Kl=1ψ(Il|h)). Since every agent in aj ∈ A\ ∪Kl=1 ψ(Il|h)) is unmatched, we

have vj(µ|h(h)) = 0 for all aj ∈ A\ ∪Kl=1 ψ(Il|h)). So vj(µh(h)) = 0 for all aj ∈ d′k(h).

Meanwhile, at every ex post history h and for all aj ∈ d′k(h),

vj(Ik, d′k(h)) + ξ′kj(h)

= vj(Ik, d′k(h))− vj(Ik, d′k(h)) +

1

2∣∣d′k(h)

∣∣ [bk − uk(µ|h)]

=1

2∣∣d′k(h)

∣∣ [bk − uk(µ|h)] > 0 = vj(µ|h(h))

So at every possible ex post history h, every agent in d′k(h) finds himself strictly better off by joining

the deviation, which ensures the feasibility of the deviation (d′k, ξ′k).

To see that (d′k, ξ′k) is profitable, observe that at every ex post history h, institution Ik’s stage-game

41

payoff from the manipulated static matching [µ|h(h), (d′k(h), ξ′k(h))] is

uk(d′k(h))−

∑

aj∈d′k(h)

ξ′kj(h)

= uk(d′k(h)) +

∑

aj∈d′k(h)

vj(Ik, d′k(h))− 1

2[bk − uk(µ|h)]

≥ bk −1

2[bk − uk(µ|h)]

=1

2bk +

1

2uk(µ|h) > uk(µ|h).

The second inequality above follows from inequality (1). Since this is true for every ex post history h,

it must hold along every possible outcome paths generated by the manipulated matching [µ|h, (d′k, ξ

′k)].

Therefore, Ik’s total expected discounted payoff from the deviation plan satisfies

uk([µ|h, (d′k, ξ′k)]) =

1

2bk +

1

2uk(µ|h) > uk(µ|h).

The deviation plan (d′k, ξ′k) is both feasible and profitable for institution Ik, which is a contradiction

to the self-enforcement of µ. So uk(µ|h) ≥ bk for all h ∈ H. The set uk(µ|h) : h ∈ H is bounded

from below.

Now, at every ex post history h ∈ H, we have

uk(µ|h) = (1− δ)uk(µ(h)) + δuk(µ|h,µ(h)),

or

uk(µ(h)) =uk(µ|h)− δuk(µ|h,µ(h)

)

1− δ .

uk(µ|h) is an element in uk(µ|h) : h ∈ H, which is bounded from below by bk; uk(µ|h,µ(h)) is an

element in uk(µ|h) : h ∈ H, which is bounded from above by bk. uk(µ(h)) : h ∈ H is therefore

bounded uniformly from below.

Lemma 6. Let µ be a matching process that satisfies vj(µ(h)) ≥ 0 for all aj ∈ A and all h ∈ H. If

Ik has a profitable, feasible deviation plan from µ, then Ik must have one such plan (d′k, ξ′k) that has

bounded per-period payoffs, i.e. the setutk([µ, (d′k, ξ

′k)])

: t ≥ 0

is bounded.

Proof. Let (d′k, ξ′k) be a feasible, profitable deviation plan from µ for institution Ik. First we show

thatutk([µ, (d′k, ξ

′k)])

: t ≥ 0

is bounded from above.

By the feasibility of the deviation plan (d′k, ξ′k), we have, for all h ∈ H,

vj(Ik, d′k(h)) + ξ′kj(h) ≥ vj(µ(h)) ≥ 0 for all aj ∈ d′k(h) ∩ ψ(Ik|h)

42

and

vj(Ik, d′k(h)) + ξ′kj(h) > vj(µ(h)) ≥ 0 for all aj ∈ d′k(h)\ψ(Ik|h).

So

vj(Ik, d′k(h)) + ξ′kj(h) ≥ 0 for all aj ∈ d′k(h).

for all h ∈ H. This implies

uk([µ, (d′k, ξ

′k)](h)

)= uk(d

′k(h))−

∑

aj∈d′k(h)

ξ′kj(h)

≤ uk(d′k(h)) +∑

aj∈d′k(h)

vj(Ik, d′k(h))

≤ maxB⊆A

uk(B) +∑


for all h ∈ H, so

utk([µ, (d′k, ξ′k)]) = EµHt

[uk([µ, (d′k, ξ

′k)](h)

)]≤ bk for all t.

The setutk([µ, (d′k, ξ

′k)])

: t ≥ 0

is bounded from above by bk.

Ifutk([µ, (d′k, ξ

′k)])

: t ≥ 0

is also bounded from below, there is nothing left to prove. If,

however,utk([µ, (d′k, ξ

′k)])

: t ≥ 0

is not bounded from below, I will show that it is possible to

construct another feasible, profitable deviation plan (d′′k, ξ′′k) such that

utk([µ, (d′′k, ξ

′′k)])

: t ≥ 0

is

bounded from below.

Supposeutk([µ, (d′k, ξ

′k)])

: t ≥ 0

is not bounded from below, then the set

t ≥ 0 : utk

([µ, (d′k, ξ

′k)])< − δbk

1− δ

must be nonempty. Set

t = min

t ≥ 0 : utk

([µ, (d′k, ξ

′k)])< − δbk

1− δ

.

By construction,

utk([µ, (d′k, ξ

′k)])≥ − δbk

1− δfor all 0 ≤ t < t, and

(1− δ)utk([µ, (d′k, ξ

′k)])

+ δbk < 0, (2)

Inequality (2) implies that if institution Ik follows the stage-game matchings prescribed by [µ, (d′k, ξ′k)]

43

in period t, even if Ik obtains its highest possible continuation payoff bk from period (t+ 1) onwards,

it would still be better off to instead remain autarkic from period t. Therefore, Ik must have a feasible

and even more profitable deviation plan from µ by enforcing autarky from period t. Formally, define

the deviation plan (d′′k, ξ′′k) by

d′′k(h) =

d′k(h) if h ∈ ∪t−1

t=0Ht∅ if h ∈ ∪∞

t=tHt

and

ξ′′k(h) =

ξ′k(h) if h ∈ ∪t−1

t=0Ht0 if h ∈ ∪∞

t=tHt

By construction, (d′′k, ξ′′k) is feasible. Moreover,

uk([µ, (d′′k, ξ′′k)]) = (1− δ)

[ t−1∑

t=0

δtutk([µ, (d′′k, ξ

′′k)])

+ δt · 0]

> (1− δ)[ t−1∑

t=0

δtutk([µ, (d′′k, ξ

′′k)])]

+ δt[(1− δ)utk

([µ, (d′k, ξ

′k)])

+ δbk

]

= (1− δ)[ t−1∑

t=0

δtutk([µ, (d′k, ξ

′k)])]

+ δt[(1− δ)utk

([µ, (d′k, ξ

′k)])

+ δbk

]

= (1− δ)[ t−1∑

t=0


′k)])

+ δtutk([µ, (d′k, ξ

′k)])

+ δt+1 bk1− δ

]

= (1− δ)[ t∑

t=0


′k)])

+

∞∑

t=t+1

δtbk

]

≥ (1− δ)[ ∞∑

t=0


′k)])]

= uk([µ, (d′k, ξ′k)])

The second line above follows from inequality (2), the third from the construction of (d′′k, ξ′′k), the last

from the fact thatutk([µ, (d′k, ξ

′k)])

: t ≥ 0

is bounded above by bk. Since (d′k, ξ′k) is profitable, it

follows that uk([µ, (d′′k, ξ′′k)]) > uk([µ, (d

′k, ξ′k)]) > uk(µ), so (d′′k, ξ

′′k) is both feasible and profitable.

Lastly, by construction,utk([µ, (d′′k, ξ

′′k)])

: t ≥ 0

is bounded from below by min− δbk1−δ , 0.

This completes the proof.

Proof of Lemma 1. The “only if” direction is immediate from the definition and Lemma 5.

To prove the other direction, let µ = (ψ, ξlKl=1) be a matching process that satisfies:

• uk(µ(h)) : h ∈ H is bounded,

44

• vj(µ(h)) ≥ 0 for all aj ∈ A and all h ∈ H.

We will prove that the existence of a feasible, profitable deviation plan for an institution Ik from the

matching process µ implies the existence of a feasible, profitable one-shot deviation plan for the same

institution.

Suppose (d′k, ξ′k) is a feasible, profitable deviation plan by institution Ik from the matching process

µ. By the definition of feasibility, we have

1. vj(Ik, d′k(h)) + ξ′kj(h) > vj(µ(h)) for all aj ∈ dk(h)\ψ(Ik|h), and vj(Ik, d′k(h)) + ξ′kj(h) ≥vj(µ(h)) for all aj ∈ dk(h) ∩ ψ(Ik|h) at every history h ∈ H;

2. uk([µ, (d′k, ξ′k)]) > uk(µ).

Let ε = uk([µ, (d′k, ξ′k)])−uk(µ). By Lemma 6, we can without loss assume the set

utk([µ, (d′′k, ξ

′′k)])

:

t ≥ 0

is bounded. By assumption, uk(µ(h)) : h ∈ H is bounded, so the setutk(µ) : t ≥ 0

is

also bounded. We can therefore find a number x such that

x > sup∣∣utk

([µ, (d′′k, ξ

′′k)])− utk(µ)

∣∣ : t ≥ 0.

Let T be large enough such that δTx < ε/2. By the definition of ε,

(1− δ)T−1∑

t=0

δtutk([µ, (d′k, ξ′k)]) + (1− δ)

∞∑

t=T

δtutk([µ, (d′k, ξ′k)])− ε

= (1− δ)T−1∑

t=0

δtutk(µ) + (1− δ)∞∑

t=T

δtutk(µ)

(3)

By the definition of T , we have

∣∣∣∣∣(1− δ)∞∑

t=T

δtutk([µ, (d′k, ξ′k)])− (1− δ)

∞∑

t=T

δtutk(µ)

∣∣∣∣∣ <ε

2

which, when combined with Eq. (3), implies

(1− δ)T−1∑

t=0

δtuk([µ, (d′k, ξ′k)])− (1− δ)

T−1∑

t=0

δtuk(µ) >ε

2(4)

In addition, since δTx < ε/2, the impact of different continuation paths after period T , after

discounted by δT , is strictly less than ε/2. This along with Eq. (4) implies that the T -period deviation

(dTk , ξTk ) defined by

dTk (h) =

d′k(h) if h ∈ ∪T−1

t=0 Htψ(Ik|h) if h ∈ ∪∞t=THt

45

and

ξTk (h) =

ξ′k(h) if h ∈ ∪T−1

t=0 Htξk(h) if h ∈ ∪∞t=THt

is a profitable T -period deviation, i.e. uk([µ, (dTk , ξTk )]) > uk(µ). There are two cases to consider:

1. uk([µ, (dTk , ξTk )]|

h) = uk([µ|h, (dTk |h, ξTk |h)]) > uk(µ|h) for some h ∈ HT−1. Since (dTk , ω

Tk )

is a T -period deviation from µ, (dTk |h, ωTk |h) is a one-shot deviation from µ|h. (dTk |h, ωTk |h) a

profitable one-shot deviation from µ|h.

2. uk([µ, (dTk , ξTk )]|h) = uk([µ|h, (dTk |h, ξTk |h)]) ≤ uk(µ|h) for all h ∈ HT−1: then

EµHT−1

[uk([µ, (d

Tk , ξ

Tk )]|h)

]≤ EµHT−1

[uk(µ|h)

].

Consider the (T − 1)-period deviation (dT−1k , ξT−1

k ) defined by

dT−1k (h) =

d′k(h) if h ∈ ∪T−2

t=0 Htψ(Ik|h) if h ∈ H\ ∪T−2

t=0 Ht

and

ξT−1k (h) =

ξ′k(h) if h ∈ ∪T−2

t=0 Htξk(h) if h ∈ H\ ∪T−2

t=0 Ht

The payoff Ik receives from the manipulated matching process [µ, (dT−1k , ξT−1

k )] is

uk([µ, (dT−1k , ξT−1

k )]) = (1− δ)[T−2∑

t=0

δtutk

([µ, (dT−1

k , ξT−1k )]

)]+ δT−1EµHT−1

[uk(µ|h)

]

≥ (1− δ)[T−2∑

t=0

δtutk

([µ, (dT−1

k , ξT−1k )]

)]+ δT−1EµHT−1

[uk([µ, (d

Tk , ξ

Tk )]|h)

]

= (1− δ)[T−2∑

t=0

δtutk

([µ, (dTk , ξ

Tk )])]

+ δT−1EµHT−1

[uk([µ, (d

Tk , ξ

Tk )]|h)

]

= uk([µ, (dTk , ξ

Tk )])

> uk(µ)

So (dT−1k , ξT−1

k ) is a feasible, profitable (T − 1)-period deviation plan.

Proceeding in this way, we must end up with a profitable one-shot deviation.

Proof of Lemma 2. Letm = (φ, ζlKl=1), [m, (Ik1 , A′k1, ζ ′k1)] = (φ, ζ lKl=1), and [m, (Ik2 , A

′k2, ζ ′k2)] =

46

(φ, ζlKl=1). Towards a contradiction, suppose Ik1 6= Ik2 , but [m, (Ik1 , A′k1, ζ ′k1)] = [m, (Ik2 , A

′k2, ζ ′k2)] 6=

m, then it must be that φ = φ, and ζ l = ζl for every l = 1, . . . ,K.

There are two cases to consider. I will show there is a contradiction in each of these cases.

1. If A′k1 = φ(Ik1) and A′k2 = φ(Ik2): since [m, (Ik1 , A′k1, ζ ′k1)] 6= m, it must be that ζ ′k1 6= ζk1 .

Then ζk1 = ζ ′k1 6= ζk1 = ζk1 , a contradiction.

2. If A′k1 ⊆ φ(Ik1) and A′k2 ⊆ φ(Ik2) but, without loss of generality, A′k1 6= φ(Ik1): then

φ(Ik1) = A′k1 6= φ(Ik1) = φ(Ik1), so φ 6= φ, a contradiction.

3. If, without loss of generality, A′k1 * φ(Ik1): let aj ∈ A′k1\φ(Ik1). φ(aj) = Ik1 , whereas

φ(aj) ∈ φ(aj), Ik2. Since aj /∈ φ(Ik1) and Ik1 6= Ik2 , we have Ik1 /∈ φ(aj), Ik2. So

φ 6= φ, a contradiction.

Therefore, [m, (Ik1 , A′k1, ζ ′k1)] = [m, (Ik2 , A

′k2, ζ ′k2)] 6= m implies Ik1 = Ik2 .

Lemma 7. Automaton representation


Lemma 8. Let U∗ =Eλ[u(φ)] : λ ∈ Λ∗

. For every vector u ∈ U∗, there exist vectors uk : k ∈

κ(I\T ) ⊆ U∗ such that

ukk < uk

for all k ∈ κ(I\T ), and

ukk < uk′k

for all k 6= k′ ∈ κ(I\T ).

Proof. The proof relies on similar techniques as in Abreu et al. (1994). Whenever possible, I will cite

intermediate results in Abreu et al. (1994) without reproducing their proofs. Compared to Abreu et al.

(1994), the main difference is that instead of imposing NEU, it is shown that a matching environment

satisfying the non-triviality assumption (Assumption 2) always satisfies NEU.

I will first establish that the set ΦT satisfies what Abreu et al. (1994) called the non-equivalent

utilities (NEU) condition for institutions in I\T . By Assumption 2, for every k ∈ κ(T ), there exists

aj(k) ∈ A\T such that vj(k)(Ik) > 0. Define the assignment φ0 by

φ0(Il) =

Al if Il ∈ T∅ otherwise

47

For each k ∈ κ(I\T ), define the assignment φk by

φk(Il) =

Al if Il ∈ Taj(k) if Il = Ik

∅ otherwise

Clearly φ0 ∈ ΦT . Since vj(k)(Ik) > 0 for all k ∈ κ(I\T ), it follows that φk ∈ ΦT for all k ∈ κ(I\T )

as well.

By Assumption 2, there are at least two elements in κ(I\T ). For each pair of k 6= k′ ∈ κ(I\T ),

uk(φ0) = uk(φ

k′) = 0, while by strictness of institution preferences, uk′(φ0) = 0 6= uk′(aj(k′)) =

uk′(φk′). Therefore, for each pair k 6= k′ ∈ κ(I\T ), there do not exist scalars α > 0 and β such

that uk(φ) = αuk′(φ) + β for all φ ∈ ΦT . This verifies the NEU condition in Abreu et al. (1994) for

indices in κ(I\T ).

Since co(UT ) = co(u(φ) : φ ∈ ΦT ), Lemma 1 and Lemma 2 in Abreu et al. (1994) then ensure

the existence of vectors uk : k ∈ κ(I\T ) ⊆ co(UT ) that satisfy ukk < uk′k for all k 6= k′ ∈ κ(I\T ).

Since u(φ) : φ ∈ ΦT is a finite set, for each k ∈ κ(I\T ), there exists an element uk in u(φ) : φ ∈ΦT that minimizes Ik’s payoff.

For an arbitrary vector u ∈ U∗, define

uk = ε(1− η)uk + ηεuk + (1− ε)u

for each k ∈ κ(I\T ). Observe that if εη > 0, then ukk < uk′k ; for all 0 < ε < 1 and 0 < η < 1,

uk ∈ co(UT ); for small enough ε > 0, ukk > uTk ; and finally, for small enough η > 0, it must be true

that ukk < uk. Therefore, there must exist uk : k ∈ κ(I\T ) ⊆ U∗ such that

ukk < uk

for all k ∈ κ(I\T ), and

ukk < uk′k

for all k 6= k′ ∈ κ(I\T ).

Lemma 9. For each institution Ik ∈ I\T , there exists φk∈ ΦT such that uk(φk) = uTk , and

maxB∈DAk (φ

k)uk(B) = uTk

Lemma 10. For each institution Ik ∈ T and every φ ∈ ΦT ,

maxB∈DAk (φ)

uk(B) = uk(φ)

48

Lemma 11. Let Φ′ ⊆ Φ be a subset of static assignments, and ψ : HF → Φ′ be a self-enforcing

assignment process. Fix an institution Ik ∈ I. If for every φ ∈ Φ′, there exists a subset of agents

Aφ ∈ DAk (φ) that satisfies uk(Aφ) ≥ uk, then

uk(ψ) ≥ uk

Proof. Suppose by contradiction that ψ is a self-enforcing assignment process, but uk(ψ) < uk. I will

show that institution Ik has a feasible profitable deviation plan from ψ, so ψ must not be self-enforcing.

Consider the following deviation plan d′k from ψ: for every ex post history h ∈ HF , define

d′k(h) = Aψ(h).

By assumption, ψ(h) ∈ Φ′ for all h ∈ HF , so d′k is well-defined and feasible. To see that d′k is prof-

itable, observe that at every ex post history h, institution Ik’s stage-game payoff from the manipulated

static assignment [ψ(h), (Ik, d′k(h))] is uk(Aψ(h)) ≥ uk > uk(ψ). Since this is true for every ex post

history h, Ik’s total discounted payoff from the deviation plan satisfies

uk([ψ, (Ik, d′k)]) > uk(ψ).

The deviation plan d′k is both feasible and profitable for institution Ik, which is a contradiction to

the self-enforcement of ψ. So uk(ψ) ≥ uk.

Proof of Theorem 2. Let φk

: k ∈ κ(I\T ) be the static assignments as constructed in Lemma 9.

Fix λ0 ∈ Λ∗ and define u0 = Eλ0 [u(φ)]. By Lemma 8, there exist vectors uk : k ∈ κ(I\T ) ⊆ U∗,such that

ukk < u0k

for every k ∈ κ(I\T ), and

ukk < uk′k

for all k, k′ ∈ κ(I\T ), k 6= k′. For each k ∈ κ(I\T ), let λk ∈ Λ∗ be the distribution over ΦT that

give rise to the payoff vectors uk.

By Lemma 7, it suffices to consider the matching process represented by the automaton (Θ, p0, f, γ),

where

• Θ =θ(e, φ) : e ∈ κ(I\T ) ∪ 0, φ ∈ ΦT

∪θ(k, t) : k ∈ κ(I\T ), 0 ≤ t < L

is the set

of all possible states;

• p0 is the initial distribution over states, which satisfies p0(θ(0, φ)) = λ0(φ) for all φ ∈ ΦT ;

49

• f : Θ→ Φ is the output function, where f(θ(e, φ)) = φ and f(θ(k, t)) = φk;

• γ : Θ× Φ→ ∆(Θ) is the transition function. For states θ(k, t)|0 ≤ t < L− 1, γ is defined

as

γ(θ(k, t), φ′

)=

θ(k′, 0) if φ′ 6= φ

k; φ′ = [φ

k′, (Ik′ , B)] for some k′ ∈ κ(I\T ) and B ∈ DAk′(φk′)

θ(k, t+ 1) otherwise

For states θ(k, L− 1), the transition is defined as

γ(θ(k, L−1), φ′

)=

θ(k′, 0) if φ′ 6= φ

k; φ′ = [φ

k′, (Ik′ , B)] for some k′ ∈ κ(I\T ) and B ∈ DAk′(φk′)

pk otherwise

where pk is the distribution over states that satisfies pk(θ(k, φ)) = λk(φ) for all k ∈ κ(I\T )

and φ ∈ ΦT .

For states θ(e, φ), the transition is

γ(θ(e, φ), φ′

)=

θ(k′, 0) if φ′ 6= φ; φ′ = [φ, (Ik′ , B)] for some k′ ∈ κ(I\T ) and B ∈ DAk′(φ)

pe otherwise

Note that owing to the identifiability of deviating institution (Lemma 2), for any θ ∈ Θ and assignment

φ′ 6= f(θ) which can result from an institution’s deviation, we can uniquely identify the institution, so

the transition above is well-defined. Any φ′ 6= f(θ) that cannot possibly result from a deviation by an

institution is ignored by the transition.

The assignment process represented by the above automaton randomizes over ΦT according to λ0

in every period. It remains to check that it is self-enforcing, or equivalently,

1. every agent’s payoff is greater than or equal to 0 in all automaton states,

2. no institution has any profitable one-shot deviation in any automaton state,

3. for every institution, the stage-game payoffs across all automaton states are bounded.

Point 3 above follows because the set of states Θ is finite. In every state θ ∈ Θ, the recommended

assignment f(θ) ∈ ΦT is individually rational for all agents, so point 1 follows as well. It remains to

verify point 2.

For every state θ, I use U(θ) = (U1(θ), . . . , UK(θ)) to denote the discounted expected payoff

profile for the institutions in state θ. By construction,

U(θ(e, φ)) = (1− δ)u(φ) + δue, for all e ∈ κ(I\T ) ∪ 0,

50

and

U(θ(k, t)) = (1− δL−t)u(φk) + δL−tuk, for all k ∈ κ(I\T ), 0 ≤ t ≤ L− 1.

In addition, since ul(φ) = ul(Al) for every Il ∈ T and all φ ∈ ΦT , the above equalities simplify to

Ul(θ(e, φ)) = Ul(θ(k, t)) = ul(Al) (5)

for every Il ∈ T .

I now verify no institution has profitable one shot deviations in any automaton states.

For statesθ(e, φ) : e ∈ κ(I\T ) ∪ 0, φ ∈ ΦT

: there are three cases to consider.

Case 1: Ik′ ∈ T . By Eq. (5), the continuation value of Ik′ is identical across states. By Lemma 10,

no feasible deviation for Ik′ can improve its stage-game payoff. Ik′ does not have any profitable one

shot deviation.

Case 2: Ik′ ∈ I\T , k′ 6= e. Choose a number Z > supφ∈Φ,k∈κ(I\T ) uk(φ). For any assignment

φ ∈ ΦT , the manipulated assignment resulting from a feasible deviation must still be an assignment

in Φ. So Z is larger than the payoff any institution can obtain in any feasible deviation from an

assignment in ΦT .

Consider a one-shot deviation (Ik′ , B) by institution Ik′ . Without deviation, Ik′ has value (1 −δ)uk′(φ) + δuek′ . After deviation, Ik′ yields less than

(1− δ)Z + δUk′(θ(e, 0)) = (1− δ)Z + δ(1− δL)uTk′ + δL+1uk′k′

There is no profitable one-shot deviation for Ik′ if

(1− δ)uk′(φ) + δuek′ ≥ (1− δ)Z + δ(1− δL)uTk′ + δL+1uk′k′

As δ → 1, the LHS converges to uek′ while the RHS converges to uk′k′ . By construction, uek′ > uk

′k′ . It

follows that such deviations are not profitable for δ high enough.

Case 3: Ik′ ∈ I\T , k′ = e. Without deviation, Ik′ has value (1 − δ)uk′(φ) + δuk′k′ . After deviation,

Ik′ yields less than

(1− δ)Z + δUk′(θ(k′, 0)) = (1− δ)Z + δ(1− δL)uTk′ + δL+1uk

′k′ .


(1− δ)uk′(φ) + δuk′k′ ≥ (1− δ)Z + δ(1− δL)uTk′ + δL+1uk

′k′ .

51

The inequality is equivalent to

Z − uk′(φ) ≤ δ(1 + . . .+ δL−1)[uk′k′ − uTk′ ]

By construction, uk′k′ − uTk′ > 0. Choose L large enough so that L(uk

′k′ − uTk′) > Z − uk′(φ). As

δ → 1, the LHS remains unchanged while the RHS converges to L(uk′k′ − uTk′), so such deviations are

not profitable for δ high enough.

For statesθ(k, t) : k ∈ κ(I\T ), 0 ≤ t ≤ L− 1


Case 1: Ik′ ∈ T . By Eq. (5), the continuation value of Ik′ is identical across states. By Lemma 10,


shot deviation.

Case 2: Ik′ ∈ I\T , k′ 6= k. Without deviation, institution Ik′ has payoff

(1− δL−t)uk′(φk) + δL−tukk′

With any deviation, Ik′ has payoff less than

(1− δ)Z + δ(1− δL)uTk′ + δL+1uk′k′


(1− δL−t)uk′(φk) + δL−tukk′ ≥ (1− δ)Z + δ(1− δL)uTk′ + δL+1uk′k′

Observe that as δ → 1, the LHS converges to ukk′ for all t such that 0 ≤ t ≤ L, while the RHS

converges to uk′k′ . By construction ukk′ > uk

′k′ . So the above inequality holds for sufficiently high δ.

Case 3: Ik′ ∈ I\T , k′ = k. Without deviation, institution Ik′ has payoff

(1− δL−t)uTk′ + δL−tuk′k′ .

When deviating from φk′

, by Lemma 9, Ik’s stage-game payoff is at most uTk . So Ik′’s discounted

expected payoff from deviation is at most

(1− δ)uTk′ + δ(1− δL)uTk′ + δL+1uk′k′ = (1− δL+1)uTk′ + δL+1uk

′k′

Institution Ik′ has no profitable deviation if

(1− δL−t)uTk′ + δL−tuk′k′ ≥ (1− δL+1)uTk′ + δL+1uk

′k′ , (6)

52

or

uk′k′ ≥ uTk′ ,

which is true by construction. So Ik′ has no profitable one-shot deviation.

We have verified that there is no profitable one-shot deviation in any states of the automaton. This

completes the proof.

Proof of Theorem 1. To prove the result, it is sufficient to show that if ψ is a self-enforcing assign-

ment process, then

1. ψ(h) ∈ ΦA at every ex post history h ∈ HF ;

2. ψ(Ik|h) = Ak for every Ik ∈ T at every ex post history h ∈ HF ; and

3. uk(ψ) ≥ uTk for every Ik ∈ I\T .

I now establish these claims in order.

1. This follows from the definition of self-enforcing matching process.

2. If T = ∅, there is nothing to prove. Suppose T = (Ik1 , Ak1), . . . , (IkG , AkG). The proof

proceeds by induction.

First I establish that in every self-enforcing assignment process ψ, ψ(Ik1 |h) = Ak1 for all

h ∈ HF . Suppose by contradiction that ψ(Ik1 |h) 6= Ak1 for some self-enforcing assignment

process ψ at some h ∈ HF .

By the construction of T , uk1(B) ≤ uk1(Ak1) for all B ⊆ A. By the strictness of institutions’

preferences, uk1(ψ(Ik1 |h)) < uk1(Ak1). In addition, since uk1(Ak1) is the highest possible

stage-game payoff for Ik1 , uk1(ψ) ≤ uk1(Ak1) for every assignment process ψ. Together these

imply

uk1(ψ|h) = (1− δ)uk1(ψ(Ik1 |h)) + δuk1(ψ|

h,ψ(h))

< (1− δ)uk1(Ak1) + δuk1(Ak1) = uk1(Ak1)(7)

Since ψ is a self-enforcing assignment process, ψ|h

must also be self-enforcing, so ψ|h(h) ∈

ΦA at every h ∈ HF . In addition, for every φ ∈ ΦA, by the construction of T and the strictness

of agents’ preferences, vj(Ik1) > vj(φ) for every aj ∈ Ak1\φ(Ik1). So Ak1 ∈ DAk1(φ) for all

φ ∈ ΦA. By Lemma 11,

uk1(ψ|h) ≥ uk1(Ak1) (8)

Inequalities (7) and (8) cannot be true at the same time, a contradiction. So ψ(Ik1 |h) = Ak1 for

all h ∈ HF .

53

Suppose it has been shown that in every self-enforcing assignment process ψ, ψ(Iki |h) = Aki

for i = 1, . . . , g− 1 at every ex post history h ∈ HF . I show that this implies that in every self-

enforcing assignment process ψ, ψ(Ikg |h) = Akg at every ex post history h ∈ HF . Suppose

by contradiction that ψ(Ikg |h) 6= Akg for some self-enforcing assignment process ψ and some

h ∈ HF .

Since by the inductive hypothesis ψ(Iki |h) = Aki for all 1 ≤ i ≤ g−1 and all h ∈ HF , it must

be that ψ(Ikg |h) ⊆ A\ ∪g−1i=1 Aki at all h ∈ HF . From the construction of T , ukg(ψ(Ikg |h)) ≤

ukg(Akg) for all h ∈ HF , it follows that at every ex post history h ∈ HF ,

ukg(ψ|h) ≤ Akg

In addition, by the strictness of institutions preferences, ψ(Ikg |h) 6= Akg implies that ukg(ψ(Ikg |h)) <

ukg(Akg). Together these imply

ukg(ψ|h) = (1− δ)ukg(ψ(Ikg |h)) + δukg(ψ|h,ψ(h))

< (1− δ)ukg(Akg) + δukg(Akg) = ukg(Akg)(9)

Since ψ is a self-enforcing assignment process, ψ|h

must also be self-enforcing. Let Φg = φ ∈ΦA : φ(Iki) = Aki for i = 1, . . . , g − 1. By the inductive hypothesis, ψ|

h(h) ∈ Φg at every

h ∈ HF . In addition, for every φ ∈ Φg, by the construction of T and the strictness of agents’

preferences, vj(Ikg) > vj(φ) for every aj ∈ Akg\φ(Ikg). So Akg ∈ DAkg(φ) for all φ ∈ Φg. By

Lemma 11,

uk1(ψ|h) ≥ uk1(Ak1) (10)

Inequalities (9) and (10) cannot be true at the same time, a contradiction. So ψ(Ikg |h) = Akg

for all h ∈ HF , completing the induction argument.

3. Let ψ be a self-enforcing assignment process. ψ(h) ∈ ΦT for all ex post histories h ∈ HF .

For every institution Ik ∈ I\T , from the definition of uTk , there exists Aφ ∈ DAk (φ) for every

φ ∈ ΦT such that

uk(Aφ) ≥ uTk .

By Lemma 11, uk(ψ) ≥ uTk for every Ik ∈ I\T .

Proof of Corollary 2. Suppose all agents share a common ordinal ranking Ik1 Ik2 . . . IkK

54

over the institutions. For each institution Ikg , 1 ≤ g ≤ K, define

Akg = arg maxB⊆A\∪g−1

i=1 Aki

ukg(B)

Then T = (Ik1 , Ak1), . . . , (IkK , AkK ) is the top coalition sequence of the matching market. Let φTbe the static assignment that satisfies φT (Ik) = Ak for k = 1, . . . ,K.

Claim 1. φT (Ik) = Ak is the unique static stable matching.

The proof is by induction. It is clear that in every static stable assignment, Ik1 must be matched

to Ak1 , otherwise Ak1 is a feasible profitable deviation for Ik1 . Suppose Ik1 , . . . , Ikg−1 is matched to

Ak1 , . . . , Akg−1 , respectively, in every static assignment. Let φ be an arbitrary static stable assignment.

I prove that φ(Ikg) = Akg .

Suppose by contradiction that φ(Ikg) 6= Akg , then by the inductive hypothesis, ψ(aj) ∈ I\Ik1 , . . . , Ikg−1for every aj ∈ Akg\φ(Ikg). By the definition of top coalition sequence and the strictness of agents’

preferences, vj(Ikg) > vj(φ) for every aj ∈ Akg\φ(Ikg), so Akg ∈ DAkg(φ). Similarly, by the induc-

tive hypothesis, φ(Ikg) ⊆ A\ ∪g−1i=1 Aki , so ukg(Akg) > ukg(φ) by the definition of a top coalition

sequence and the strictness of institutions’ preferences. This implies that Akg is a profitable and fea-

sible deviation, a contradiction to φ being a stable assignment. So φ(Ikg) = Akg . This completes the

inductive step.

Claim 2. There is a unique self-enforcing assignment process ψT , where ψT (h) = φT for all h ∈ HF .

Since I ⊆ T , ΦT = φ : φ(Ik) = Ak ∀k = 1, . . . ,K. So φT is the unique element of the set

ΦT . The claim then follows from Theorem 1.


Definition A.1. Fix the hierarchy O = P1, . . . ,PG,R. For each Ik ∈ Pg, g = 1, . . . , G, define

P(Ik) ≡ ∪Gi=g+1Pi ∪ Il ∈ Pg : l < k

Definition A.2. Fix the hierarchyO = P1, . . . ,PG,R and a set of agentsC. For each Ik ∈ Pg, g =

1, . . . , G, the sets of agent AOk (C) is defined by

AOk (C) = arg maxB⊆A\

[C∪(∪Il∈P(Ik)

AOl (C))]uk(B) +

∑

aj∈Bvj(Ik, B)

Definition A.3. Fix the hierarchyO = P1, . . . ,PG,R and a set of agentsC. For each Ik ∈ Pg, g =

1, . . . , G, define

πOk (C) = uk(AOk (C)) +

∑

aj∈AOk (C)

vj(Ik, AOk (C))

55

Lemma 12. For each Ik ∈ ∪Gg=1Pg, and all C ⊆ A such that |C| ≤ Q(R),

πOk (C) = πk (11)

In addition, for every Ik ∈ R,

πk > πk(R)

Proof. Fix any Ik ∈ Pg for some g = 1, . . . , G. By the construction ofO, we have πk = πk(I\∪g−1i=1

Pi). Recall that

πOk (C) = uk(AOk (C)) +

∑

aj∈AOk (C)

vj(Ik, AOk (C))

= maxB⊆A\

[C∪(∪Il∈P(Ik)

AOl (C))]uk(B) +

∑

aj∈Bvj(Ik, B)

≤ maxB⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

= πk

(12)

Note that since C ≤ Q(R) and AOl (C) ≤ ql for all Il ∈ ∪Gg=1Pg, we have

∣∣C ∪(∪Il∈P(Ik) A

Ol (C)

)∣∣ ≤ Q(R) +

G∑

i=g

Q(Pi) = Q(I\ ∪g−1i=1 Pi)

So

πOk (C) = maxB⊆A\

[C∪(∪Il∈P(Ik)

AOl (C))]uk(B) +

∑

aj∈Bvj(Ik, B)

≥ minA⊆A:|A|≤Q(I\∪g−1

i=1 Pi)maxB⊆A\A

uk(B) +

∑

aj∈Bvj(Ik, B)

= πk(I\ ∪g−1i=1 Pi)

(13)

Combining inequalities (12) and (13), we have

πk ≥ πOk (C) ≥ πk(I\ ∪g−1i=1 Pi) = πk,

so πOk (C) = πk.

For every Ik ∈ R, πk ≥ πk(R). If πk = πk(R), then by the construction of O, Ik /∈ R, a

contradiction. So πk > πk(R).

I use κ(R) ≡ k : Ik ∈ R to denote the indices of institutions that are in R, and Γ(R) ≡σ :

1, 2, . . . , |κ(R)| → κ(R)

to denote the set of ordering of the numbers in κ(R). For all k ∈ κ(R),

56

define

Γk(R) = σ ∈ Γ(R) : σ(1) = k.

Γk(R) is the set of permutations in Γ(R) that ranks k in the first place.

Definition A.4. Given a permutation σ ∈ Γ(R), the sets of agent Aσk : k ∈ κ(R) is defined by

Aσσ(l) = arg maxB⊆A\∪l−1

i=1Aσσ(i)

uσ(l)(B) +

∑

aj∈Bvj(Iσ(l), B)

for each 1 ≤ l ≤ |κ(R)|.

Definition A.5. For each institution Ik ∈ R and σ ∈ Γ(R), define

πk(σ) = uk(Aσk) +

∑

aj∈Aσk

vj(Ik, Aσk)

Lemma 13. For any institution Ik ∈ R, and all σ ∈ Γk(R),

πk(σ) = πk

Proof. For all σ ∈ Γk(R), we have σ(1) = k. By definition,

Aσk = Aσσ(1) = arg maxB⊆A\∪1−1

i=1Aσσ(i)

uσ(1)(B) +

∑

aj∈Bvj(Iσ(1), B)

= arg maxB⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

,

so


∑

aj∈Aσk

vj(Ik, Aσk)

= maxB⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

= πk

Lemma 14. For any σ ∈ Γ(R) and any institution Ik ∈ R,

πk(σ) ≥ πk(R)

Proof. For every institution Ik and permutation σ ∈ Γ(R), define Ck(σ) = ∪1≤σ−1(l)<σ−1(k)Aσl .

57

Then

Aσk = arg maxB⊆A\Ck(σ)

uk(B) +

∑

aj∈Bvj(Ik, B)

By definition,


∑

aj∈Aσk

vj(Ik, Aσk)

= maxB⊆A\Ck(σ)

uk(B) +

∑

aj∈Bvj(Ik, B)

Note that for all σ ∈ Γ(R) and k ∈ κ(R), |Ck(σ)| ≤ Q(R). So

πk(σ) = maxB⊆A\Ck(σ)

uk(B) +

∑

aj∈Bvj(Ik, B)

≥ minA⊆A:|A|≤Q(R)

maxB⊆A\A

uk(B) +

∑

aj∈Bvj(Ik, B)

= πk(R)

Proof of Theorem 5. There are two cases to consider.

Case 1: R = ∅. It suffices to check that M is nonempty, as any degenerate lottery on M will be in

Λ. Define the matching m∗ = (φ∗, w∗kKk=1) by

φ∗(Ik) = AOk (∅) ∀k = 1, . . . ,K

and

w∗kj =

−vj(Ik, AOk (∅)) if aj ∈ AOk (∅)0 otherwise

By construction,

uk(m∗) = πOk (∅)

for all 1 ≤ k ≤ K.

Since R = ∅, we have r = 0, and |∅| = 0 = Q(R). By Lemma 12, uk(m∗) = πOk (∅) = πk for

all 1 ≤ k ≤ K. In addition, all employed agents have zero payoff, so m∗ is individually rational for

agents. Therefore, m∗ ∈M.

58

Case 2: R 6= ∅. First I show that M is nonempty. For each k ∈ κ(R), fix

σk ∈ Γk(R)

and define the matchings mk = (φk, wkl Kl=1) by

φk(Il) =

Aσkl if Il ∈ RAOl (∪i∈κ(R)A

σki ) if Il ∈ ∪Gg=1Pg

and

wklj =

−vj(Il, φk(Il)) if aj ∈ φk(Il)0 otherwise

By construction, for each mk and every Ik′ ∈ ∪Gg=1Pg,

uk′(mk) = πOk′(∪l∈κ(R)Aσkl ) = πk′

where the second equality follows from Lemma 12 and the fact that∣∣∣∪l∈κ(R)A

σkl

∣∣∣ ≤ Q(R). In

addition, in each mk, all agents are getting zero payoff. So mkk∈κ(R) ⊆M.To see Λ is nonempty, first observe that for each k ∈ κ(R),

uk(mk) = uk(Aσkk )−

∑

aj∈Aσkk

wkkj

= uk(Aσkk ) +

∑

aj∈Aσkk

vj(Ik, Aσkk )

= πk(σk) = πk > πk(R)

The first two equalities above follow from the definition of the matching mk, the third equality follows

from the definition of πk(.), the fourth follows from Lemma 13. Lastly, πk > πk(R) follows from

Lemma 12.

Second, for all k, k′ ∈ κ(R), k 6= k′,

uk′(mk) = uk′(Aσkk′ )−

∑

aj∈Aσkk′

wkk′j

= uk′(Aσkk′ ) +

∑

aj∈Aσkk′

vj(Ik′ , Aσkk′ )

= πk′(σk) ≥ πk′(R)

59

where the last inequality follows from Lemma 14.

Fix a set of weights λl : l ∈ κ(R) that satisfy

λl > 0 ∀l ∈ κ(R), and∑

l∈κ(R)

λl = 1.

Let λ0 ∈ ∆(M) be the lottery over M that is defined by

λ0 =∑

l∈κ(R)

λlml

I will show Eλ0 [uk(m)] > πk(R) ∀k ∈ κ(R), which proves λ0 ∈ Λ and therefore the nonemptyness

of Λ.

Fix any k ∈ κ(R), then uk(ml) ≥ πk(R) for all l ∈ κ(R). In addition, uk(mk) = πk > πk(R).

So Eλ0 [uk(m)] =∑

l∈κ(R) λluk(ml) > πk(R), and λ0 ∈ Λ.

Lemma 15. For every m ∈M and every Ik ∈ ∪Gg=1Pg,

sup(A′k′ ,ζ′k′ )∈Dk′ (m)

uk′(m, [A′k′ , ζ

′k′ ]) = uk′(m),

so no institution in ∪Gg=1Pg has profitable feasible deviations.

Proof. Consider any feasible deviation (A′k, ζ′k) ∈ Dk(m). For the deviation to be feasible, each

aj ∈ A′k must be individually rational, so vj(Ik, A′k) + ζ ′kj ≥ 0. This implies

uk(A′k)−

∑

aj∈A′k

ζ ′kj ≤ ul(A′k) +∑

aj∈A′k

vj(Ik, A′k)

≤ maxA⊆A

uk(A) +∑

aj∈Avj(Ik, A)

= πk = uk(m)

Lemma 16. SupposeR = ∅, then every matching in m ∈M is a static stable matching

Proof. Since R = ∅, ∪Gg=1Pg = I. By Lemma 15, no institutions in I has any profitable feasible

deviations. By construction, every agent aj ∈ A has utility vj(m) ≥ 0, so no agent has any profitable

deviation. The matching m is a static stable matching.

Lemma 17. SupposeR 6= ∅, then there exist static matchings mkk∈κ(R) ⊆M such that

60

1. for all k ∈ κ(R), uk(mk) ≤ πk(R);

2. for all k ∈ κ(R),

sup(A′k,ζ

′k)∈Dk(mk)

uk(mk, [A′k, ζ′k]) ≤ πk(R);

Proof. For each k ∈ κ(R), I use Ak to denote the set of agents

Ak = arg minA⊆A:|A|≤Q(R)

maxB⊆A\A

uk(B) +

∑

aj∈Bvj(Ik, B)

,

and bk to denote the value

bk = maxB⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

− πk(R).

Note that bk ≥ 0 for all k ∈ κ(R). For each institution k ∈ κ(R), let Alkl∈κ(R) be a partition of Aksuch that

∣∣Alk∣∣ ≤ ql for all l ∈ κ(R). This is possible because |Ak| ≤ Q(R) by construction. Define

the static matching mk = (φk, ζklKl=1) by

φk(Il) =

Alk if l ∈ RAOl (Ak) if l ∈ ∪Gg=1Pg

and

ζklj

=

bk − vj(Il, φk(Il)) if l ∈ R, and aj ∈ φk(Il)−vj(Il, φk(Il)) if l ∈ ∪Gg=1Pg, and aj ∈ φk(Il)0 otherwise

By construction, ul(mk) = πOl (Ak) for all l ∈ ∪Gg=1Pg. Since |Ak| ≤ Q(R), by Lemma 12, we have

ul(mk) = πOl (Ak) = πl for all l ∈ ∪Gg=1Pg, so mk ∈M.

1. For each l ∈ R, if Akk = ∅, then

uk(mk) = uk(∅)

≤ maxB:B⊆A\A

uk(B) +

∑

aj∈Bvj(Ik, B)

61

for all A such that |A| ≤ Q(R). So

uk(mk) ≤ minA⊆A:|A|≤Q(R)

maxB⊆A\A

uk(B) +

∑

aj∈Bvj(Ik, B)

= πk(R)

If Akk 6= ∅, then

uk(mk) = uk(Akk)−

∑

aj∈Akk

ζkkj

= uk(Akk) +

∑

aj∈Akk

vj(Ik, Akk)−

∣∣∣Akk∣∣∣ · bk

≤ uk(Akk) +∑

aj∈Akk

vj(Ik, Akk)− bk

= uk(Akk) +

∑

aj∈Akk

vj(Ik, Akk)−max

B⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

+ πk(R)

≤ πk(R)

The first two equalities above follow from the definition of the matchingmk; the first inequality

follows from∣∣Akk∣∣ ≥ 1 and the fact that bk ≥ 0.

2. Consider any feasible deviation (A′k, ζ′k) ∈ Dk(mk). Suppose A′k ⊆ A\Ak. By feasibility,

each aj ∈ A′k must at least find the deviation individually rational, so vj(Ik, A′k) + ζ ′kj ≥ 0.

This implies

uk(A′k)−

∑

aj∈A′k

ζ ′kj ≤ uk(A′k) +∑

aj∈A′k

vj(Ik, A′k)

≤ maxB⊆A\Ak

uk(B) +∑

aj∈Bvj(Ik, B)

= πk(R)

where the second inequality above follows from the fact that A′k ⊆ A\Ak, and the equality

follows the definition of Ak.

Suppose instead A′k * A\Ak. Fix ai ∈ A′k ∩ Ak. By the construction of mk, vi(mk) = bk.

For the deviation to be feasible, ai must obtain a payoff weakly higher than in mk (and strictly

higher than in mk if ai /∈ Akk) , so vi(Ik, A′k) + ζ ′ki ≥ bk, or −ζ ′kj ≤ vi(Ik, A′k) − bk; at the

same time, every other agent aj ∈ A′k needs to at least find the deviation individually rational,

62

so vj(Ik, A′k) + ζ ′kj ≥ 0 for all aj ∈ A′k, aj 6= ai. We have

uk(A′k)−

∑

aj∈A′k

ζ ′kj = uk(A′k)− ζ ′ki −

∑

aj∈A′k,aj 6=ai

ζ ′kj

≤ uk(A′k) +∑

aj∈A′k

vj(Ik, A′k)− bk

= uk(A′k) +

∑

aj∈A′k

vj(Ik, A′k)−max

B⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

+ πk(R)

≤ πk(R).

We have shown that uk(mk, [A′k, ζ′k]) ≤ πk(R) regardless of whether A′k ⊆ A\Ak, so

sup(A′k,ζ

′k)∈Dk(mk)

uk(mk, [A′k, ζ′k]) ≤ πk(R).

Lemma 18. Let U =Eλ[u(m)] : λ ∈ Λ

. If |R| ≥ 2, for every vector u ∈ U, there exist vectors

uk : k ∈ κ(R) ⊆ U such that

ukk < uk

for all k ∈ κ(R), and

ukk < uk′k

for all k 6= k′ ∈ κ(R).

Proof. The proof of this lemma is based on Abreu et al. (1994). Compared to Abreu et al. (1994), the

changes are:

1. instead of imposing NEU condition like Abreu et al. (1994), it is shown that the matching

environment always satisfies NEU;

2. since the set M can be potentially infinite, a different method of construction for uk is needed;

3. the current proof operate in a cooperative game environment without action spaces; and

4. only a subset of indices, κ(R), are concerned.

Whenever possible, I will cite intermediate results in Abreu et al. (1994) without reproducing their

proofs. I will first establish that the set M satisfies what Abreu et al. (1994) called the non-equivalent

utilities (NEU) condition for institutions inR. Fix an arbitrary m0 = (φ0, ζ0l Kl=1) ∈M and define

bk = maxB⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

− πk(R).

63

Note that bk > 0 for all k ∈ κ(R). For each k ∈ κ(R), define the matching mk = (φk, ζkl Kl=1) by

φk = φ0, and

ζklj =

ζ0kj + bk if l = k and aj ∈ φk(Ik)ζ0lj otherwise

Now I establish a few properties about the matchings mk.

Clearly, vj(mk) = vj(m0) for all j ∈ A\φk(Ik), and vj(mk) > vj(m

0) for all j ∈ φk(Ik), so

mk is individually rational for all agents. In addition, uk(mk) = uk(m0) = πk for all Ik ∈ ∪Gg=1Pg.

It follows that mk ∈M for all k ∈ κ(R).

By the construction of mk : k ∈ κ(R), for each k ∈ κ(R),

uk(mk) = uk(φ

k(Ik))−∑

aj∈φk(Ik)

ζkkj

= uk(φ0(Ik))−

∑

aj∈φ0(Ik)

ζ0kj −

∣∣φ0(Ik)∣∣ · bk

≤ uk(φ0(Ik))−∑

aj∈φ0(Ik)

ζ0kj − bk

≤ uk(φ0(Ik)) +∑

aj∈φ0(Ik)

vj(Ik, φ0(Ik))− bk

= uk(φ0(Ik)) +

∑

aj∈φ0(Ik)

vj(Ik, φ0(Ik))−max

B⊆A

uk(B) +

∑

aj∈Bvj(Ik, B)

+ πk(R)

≤ πk(R).

(14)

The first two equalities follow from the definition of mk; the first inequality follows because bk > 0;

the second inequality followsm0 is individually rational for agents and therefore ζ0kj+vj(Ik, φ

0(Ik)) ≥0; the third equality follows from the definition of bk; the last inequality follows by construction.

For each pair of k 6= k′ ∈ κ(R), uk(m0) = uk(mk′), while uk′(m0) > uk′(m

k′). Therefore, for

each pair k 6= k′ ∈ κ(R), there do not exist scalars α > 0 and β such that uk(m) = αuk′(m) + β for

all m ∈M. This verifies the NEU condition in Abreu et al. (1994) for indices in κ(R).

Lemma 1 and Lemma 2 in Abreu et al. (1994) then ensure the existence of vectors uk : k ∈κ(R) ⊆ Eλ[u(m)] : λ ∈ ∆(M) that satisfy ukk < uk

′k for all k 6= k′ ∈ κ(R).

For an arbitrary vector u0 ∈ U, define

uk = ε(1− η)u(mk) + ηεuk + (1− ε)u

for each k ∈ κ(R).

First observed that by the construction of mk : k ∈ 0∪κ(R), uk(mk) < uk(m0) = uk(m

k′),

64

so ukk < uk′k if εη > 0. Second, since mk : k ∈ κ(R) ⊆ M, uk ∈ co(UT ) for all 0 < ε < 1 and

0 < η < 1. Third, for small enough ε > 0, ukk > πk(R). Finally, by inequality (14), for small enough

η > 0, it must be true that ukk < uk. Therefore, there must exist uk : k ∈ κ(R) ⊆ U such that

ukk < uk

for all k ∈ κ(R), and

ukk < uk′k

for all k 6= k′ ∈ κ(R).

Lemma 19. If a matchingm = (φ, ζlKl=1) ∈M is individually rational for all agents, then uk(m) ≤πk for all Ik ∈ I

Proof. Since m is individually rational for agents, it must be that vj(Ik, φ(Ik)) + ζkj ≥ 0, or equiva-

lently, −ζkj ≤ +vj(Ik, φ(Ik)) for all aj ∈ φ(Ik). So

uk(m) = uk(φ(Ik))−∑

aj∈φ(Ik)

ζkj

≤ uk(φ(Ik)) +∑

aj∈φ(Ik)

vj(Ik, φ(Ik))

≤ maxB⊆A

uk(B) +∑

aj∈Bvj(Ik, B)

= πk.

Proof of Theorem 3. There are three cases to consider.

Case 1: R = ∅. By Lemma 16, all λ ∈ Λ are lotteries over static stable matchings. By Lemma 3,

the matching process µ∗ defined by

µ∗(h) = λ0 ∀h ∈ H

is a self-enforcing matching process.

Case 2: |R| = 1. Suppose R = Ik. Fix any λ0 ∈ Λ and define u0 ≡ Eλ0 [u(m)]. Let mk ∈ Mbe the static matching constructed in Lemma 17. Note that mk ∈M, and

uk(mk) ≤ πk(R) (15)

65

Let ρ ∈ (0, 1) be such that

(1− ρ)uk(mk) + ρu0k > πk(R), (16)

for all ρ ∈ [ρ, 1]. When the post-minmaxing phase payoffs are discounted at ρ, inequality (16) implies

that compared to obtaining its minmax value forever, institution Ik would rather live with the lower

payoff uk(mk) during the minmax phase, with the promise of transitioning back into the lottery λ0;

The above inequalities holds at ρ = 1, so there exists a value of ρ ∈ (0, 1) such that the inequality

holds for all ρ ∈ [ρ, 1]. Let L(δ) ≡⌈

log ρlog δ

⌉where d·e is the ceiling function. Below, we use the

property that limδ→1 δL(δ) = ρ.4


where

• Θ =θ(m) : m ∈M

∪θ(t) : 0 ≤ t < L(δ)

is the set of all possible states;

• p0 is the initial distribution over states, which satisfies p0(θ(m)) = λ0(m) for all m ∈M;

• f : Θ→M is the output function, where f(θ(m)) = m and f(θ(t)) = mk;

• γ : Θ×M → ∆(Θ) is the transition function. For states θ(t)|0 ≤ t < L(δ)−1, γ is defined

as

γ(θ(t),m′

)=

θ(0) if m′ 6= mk; m′ = [mk, (Ik, A

′k, ζ′k)]

θ(t+ 1) otherwise

For states θ(L(δ)− 1), the transition is defined as

γ(θ(L(δ)− 1),m′

)=

θ(0) if m′ 6= mk; m′ = [mk, (Ik, A

′k, ζ′k)]

p0 otherwise

For states θ(m), the transition is

γ(θ(m),m′

)=

θ(0) if m′ 6= m; m′ = [m, (Ik, A

′k, ζ′k)]

p0 otherwise

Note that owing to the identifiability of deviating institution (Lemma 2), for any θ ∈ Θ and matching

m 6= f(θ) which can (but not necessarily does) result from Ik’s deviation, we can uniquely identify

Ik as the deviating institution, so the transition above is well-defined. Any m 6= f(θ) that cannot

possibly result from a deviation by Ik is ignored by the transition.

4To see why, observe that δL(δ) ∈ [δlog ρlog δ , δ

log ρlog δ+1] = [δρ, ρ]→ ρ as δ → 1

66

The matching process represented by the above automaton randomizes over M according to λ0





Point 3 above follows by the construction of λ0. In every state θ ∈ Θ, the recommended assign-

ment f(θ) ∈M is individually rational for all agents, so point 1 follows as well. It remains to verify

point 2.



Uk(θ(m)) = (1− δ)u(m) + δu0,

and

Uk(θ(t)) = (1− δL(δ)−t)u(mk) + δL(δ)−tu0

In addition, since ul(m) = πl for every Il 6= Ik and all m ∈M, the above equalities simplify to

Ul(θ(m)) = Ul(θ(t)) = πl (17)

for every Il 6= Ik.

I now verify no institution Ik′ has profitable one shot deviations in any automaton states.

For statesθ(m) : m ∈M

: there are two cases to consider.

Case a: Ik′ 6= Ik. By Eq. (17), the continuation value of Ik′ is identical across states. By Lemma 15,


shot deviation.

Case b: Ik′ = Ik. Without deviation, Ik has value (1− δ)uk(m) + δu0k. After deviation, Ik yields less

than

(1− δ)Z + δ(1− δL(δ))uk(mk) + δL(δ)+1u0k.

There is no profitable one-shot deviation for Ik if

(1− δ)uk(m) + δu0k ≥ (1− δ)Z + δ(1− δL(δ))uk(mk) + δL(δ)+1u0

k.


(1− δ)(Z − uk(m)) ≤ δ(1− δL(δ))[u0k − uk(mk)]

67

As δ → 1, the LHS converges to 0 while the RHS converges to (1−ρ)(u0k−uk(mk)). By construction,

0 < (1− ρ)(u0k − uk(mk)), so such deviations are not profitable for δ high enough.

For statesθ(t) : 0 ≤ t ≤ L(δ)− 1

: there are two cases to consider.

Case a: Ik′ 6= Ik. By Eq. (17), the continuation value of Ik′ is identical across states. By Lemma 15,


shot deviation.

Case b: Ik′ ∈ R, k′ = k. Without deviation, institution Ik has payoff

(1− δL(δ)−t)uk(mk) + δL(δ)−tu0k

With any deviation, Ik has payoff less than

(1− δ)πk(R) + δ(1− δL(δ))uk(mk) + δL(δ)+1u0k

There is no profitable one-shot deviation for Ik if, for all 0 ≤ t ≤ L(δ),

(1− δL(δ)−t)uk(mk) + δL(δ)−tu0k ≥ (1− δ)πk(R) + δ(1− δL(δ))uk(mk) + δL(δ)+1u0

k

Since uk(mk) ≤ π(R) < ukk, the LHS of the above inequality is monotonically increasing in t. As a

result, to show that the above inequality holds for all 0 ≤ t ≤ L(δ), it is sufficient to show that it holds

for t = 0.

When t = 0, the above inequality is equivalent to

(1− δL(δ))uk(mk) + δL(δ)u0k ≥ uk(R)

By inequality (16), this is satisfied for δ suffifiently close to 1.

Case 3: |R| ≥ 2. Fix any λ0 ∈ Λ. Let mkk∈κ(R) ⊆ M be the static matchings as constructed in

Lemma 17. Note that mkk∈κ(R) ⊆M, and

uk(mk) ≤ πk(R) (18)

for k ∈ κ(R). Define u0 ≡ Eλ0 [u(m)]. By Lemma 18, there exist payoff vectors ukk∈κ(R) ⊆ U

such that ukk < u0k for all k ∈ κ(R), and ukk < uk

′k for all k 6= k′, k, k′ ∈ κ(R). For each k ∈ κ(R),

let λk ∈ Λ be the distribution over M that give rise to the payoff vector uk.

Given ukk∈κ(R) and mkk∈κ(R), let ρ ∈ (0, 1) be such that

1. for every k ∈ κ(R) and every ρ ∈ [ρ, 1],

(1− ρ)uk(mk) + ρukk > πk(R), (19)

68

and

2. for all k, k′ ∈ κ(R), k′ 6= k, and every ρ ∈ [ρ, 1],

(1− ρ)uk(mk′) + ρuk′k > (1− ρ)uk(mk) + ρukk. (20)

When the post-minmaxing phase payoffs are discounted at ρ, inequality (19) implies that com-

pared to obtaining its minmax value forever, institution Ik would rather live with the lower payoff

uk(mk) during the minmax phase, with the promise of transitioning into its institution specific pun-

ishment; inequality (20) implies that institution Ik′ is willing to bear the cost of minmaxing institution

Ik with the promise of transitioning into institution Ik’s specific punishment rather than its own.

The above inequalities holds at ρ = 1 for each k and k′ 6= k. Since the set of players is finite,

there exists a value of ρ ∈ (0, 1) such that the inequality holds for all ρ ∈ [ρ, 1], k, k′ ∈ κ(R)

and k 6= k′. Let L(δ) ≡⌈

log ρlog δ

⌉where d·e is the ceiling function. Below, we use the property that

limδ→1 δL(δ) = ρ.5


where

• Θ =θ(e,m) : e ∈ κ(R) ∪ 0,m ∈ M

∪θ(k, t) : k ∈ κ(R), 0 ≤ t < L(δ)

is the set

of all possible states;

• p0 is the initial distribution over states, which satisfies p0(θ(0,m)) = λ0(m) for all m ∈M;

• f : Θ→M is the output function, where f(θ(e,m)) = m and f(θ(k, t)) = mk;

• γ : Θ ×M → ∆(Θ) is the transition function. For states θ(k, t)|0 ≤ t < L(δ) − 1, γ is

defined as

γ(θ(k, t),m′

)=

θ(k′, 0) if m′ 6= mk; m′ = [mk′ , (Ik′ , A′k′ , ζ

′k′)] for some

k′ ∈ κ(R) and (A′k′ , ζ′k′) ∈ Dk′(mk′)

θ(k, t+ 1) otherwise

For states θ(k, L(δ)− 1), the transition is defined as

γ(θ(k, L(δ)− 1),m′

)=

θ(k′, 0) if m′ 6= mk; m′ = [mk′ , (Ik′ , A′k′ , ζ

′k′)] for some

k′ ∈ κ(R) and (A′k′ , ζ′k′) ∈ Dk′(mk′)

pk otherwise

5To see why, observe that δL(δ) ∈ [δlog ρlog δ , δ

log ρlog δ+1] = [δρ, ρ]→ ρ as δ → 1

69

where pk is the distribution over states that satisfies pk(θ(k,m)) = λk(m) for all k ∈ κ(R)

and m ∈M.

For states θ(e,m), the transition is

γ(θ(e,m),m′

)=

θ(k′, 0) if m′ 6= m; m′ = [m, (Ik′ , A′k′ , ζ

′k′)] for some

k′ ∈ κ(R) and (A′k′ , ζ′k′) ∈ Dk′(m)

pe otherwise

Note that owing to the identifiability of deviating institution (Lemma 2), for any θ ∈ Θ and matching

m 6= f(θ) which can (but not necessarily does) result from an institution’s deviation, we can uniquely

identify the institution, so the transition above is well-defined. Any m 6= f(θ) that cannot possibly

result from a deviation by an institution is ignored by the transition.

The matching process represented by the above automaton randomizes over M according to λ0





Point 3 above follows by the construction of λ0. In every state θ ∈ Θ, the recommended assign-

ment f(θ) ∈M is individually rational for all agents, so point 1 follows as well. It remains to verify

point 2.



U(θ(e,m)) = (1− δ)u(m) + δue, for all e ∈ κ(R) ∪ 0,

and

U(θ(k, t)) = (1− δL(δ)−t)u(mk) + δL(δ)−tuk, for all k ∈ κ(R), 0 ≤ t ≤ L(δ)− 1.

In addition, since ul(m) = πl for every Il ∈ ∪Gg=1Pg and all m ∈ M, the above equalities simplify

to

Ul(θ(e,m)) = Ul(θ(k, t)) = πl (21)

for every Il ∈ ∪Gg=1Pg.

I now verify no institution Ik′ has profitable one shot deviations in any automaton states.

For statesθ(e,m) : e ∈ κ(R) ∪ 0,m ∈M


70

Case a: Ik′ ∈ ∪Gg=1Pg. By Eq. (21), the continuation value of Ik′ is identical across states. By

Lemma 15, no feasible deviation for Ik′ can improve its stage-game payoff. Ik′ does not have any

profitable one shot deviation.

Case b: Ik′ ∈ R, k′ 6= e. Choose a number Z > supm∈M supm′∈Dk(m) uk(m′). For any matching

m ∈ M, the manipulated matching resulting from a feasible deviation must still be individually

rational. So Z is finite by Lemma 19, and is larger than the payoff any institution can obtain in any

feasible deviation from a matching in M.

Consider a one-shot deviation (Ik′ , A′k′ , ζ

′k′) by institution Ik′ . Without deviation, Ik′ has value

(1− δ)uk′(m) + δuek′ . After deviation, Ik′ yields less than

(1− δ)Z + δUk′(θ(e, 0)) = (1− δ)Z + δ(1− δL(δ))uk′(mk′) + δL(δ)+1uk′k′


(1− δ)uk′(m) + δuek′ ≥ (1− δ)Z + δ(1− δL(δ))uk′(mk′) + δL(δ)+1uk′k′

As δ → 1, the LHS converges to uek′ while the RHS converges to (1 − ρ)uk′(mk′) + ρuk′k′ . By

construction, uek′ > uk′k′ and uk

′k′ > πk′(R) for all k′ such that k′ ∈ κ(R) and k′ 6= e. From Eq. (18),

it follows that uek′ > uk′k′ > (1− ρ)πk′(R) + ρuk

′k′ ≥ (1− ρ)uk′(mk′) + ρuk

′k′ , so such deviations are

not profitable for δ high enough.

Case c: Ik′ ∈ R, k′ = e. Without deviation, Ik′ has value (1− δ)uk′(m) + δuk′k′ . After deviation, Ik′

yields less than

(1− δ)Z + δUk′(θ(k′, 0)) = (1− δ)Z + δ(1− δL(δ))uk′(mk′) + δL(δ)+1uk

′k′ .


(1− δ)uk′(m) + δuk′k′ ≥ (1− δ)Z + δ(1− δL(δ))uk′(mk′) + δL(δ)+1uk

′k′ .


(1− δ)(Z − uk′(m)) ≤ δ(1− δL(δ))[uk′k′ − uk′(mk′)]

As δ → 1, the LHS converges to 0 while the RHS converges to (1 − ρ)(uk′k′ − uk′(mk′)). By con-

struction, 0 < (1− ρ)(uk′k′ − uk′(mk′)) for k ∈ κ(R), so such deviations are not profitable for δ high

enough.

For statesθ(k, t) : k ∈ κ(R), 0 ≤ t ≤ L(δ)− 1


Case a: Ik′ ∈ ∪Gg=1Pg. By Eq. (21), the continuation value of Ik′ is identical across states. By

71

Lemma 15, no feasible deviation for Ik′ can improve its stage-game payoff. Ik′ does not have any

profitable one shot deviation.

Case b: Ik′ ∈ R, k′ 6= k. Without deviation, institution Ik′ has payoff

(1− δL(δ)−t)uk′(mk) + δL(δ)−tukk′

With any deviation, Ik′ has payoff less than

(1− δ)Z + δ(1− δL(δ))uk′(mk′) + δL(δ)+1uk′k′


(1− δL(δ)−t)uk′(mk) + δL(δ)−tukk′ ≥ (1− δ)Z + δ(1− δL(δ))uk′(mk′) + δL(δ)+1uk′k′ (22)

We prove that this inequality is satisfied if δ is sufficiently high. Examining the LHS of inequality

(22), observe that for all t such that 0 ≤ t ≤ L(δ),

limδ→1

[(1− δL(δ)−t)uk′(mk) + δL(δ)−tukk′

]= lim

δ→1

[(1− ρ

δt

)uk′(mk) +

ρ

δtukk′]

= (1− ρ)uk′(mk) + ρukk′

for some ρ ∈ [ρ, 1].6

Examining the RHS of inequality (22), observe that

limδ→1

[(1− δ)Z + δ(1− δL(δ))uk′(mk′) + δL(δ)+1uk

′k′

]= lim

δ→1

[(1− δL(δ))uk′(mk′) + δL(δ)uk

′k′

]

= (1− ρ)uk′(mk′) + ρuk′k′

≤ (1− ρ)uk′(mk′) + ρuk′k′ ,

where the first equality follows from taking limits, the second from limδ→1 δL(δ) = ρ, and the weak

inequality follows from ρ ≥ ρ and uk′(mk′) ≤ πk′(R) < uk′k′ . Since ρ ∈ [ρ, 1], inequality (20)

delivers that (1 − ρ)uk′(mk) + ρukk′ is strictly higher than (1 − ρ)uk′(mk′) + ρuk′k′ . This guarantees

that inequality (22) holds for sufficiently high δ.

Case c: Ik′ ∈ R, k′ = k. Without deviation, institution Ik has payoff

(1− δL(δ)−t)uk(mk) + δL(δ)−tukk

6In the second equality, we use ρ rather than ρ because τ is any integer between 0 and L(δ).

72

With any deviation, Ik has payoff less than

(1− δ)πk(R) + δ(1− δL(δ))uk(mk) + δL(δ)+1ukk

There is no profitable one-shot deviation for Ik if, for all 0 ≤ t ≤ L(δ),

(1− δL(δ)−t)uk(mk) + δL(δ)−tukk ≥ (1− δ)πk(R) + δ(1− δL(δ))uk(mk) + δL(δ)+1ukk

Since uk(mk) ≤ π(R) < ukk, the LHS of the above inequality is monotonically increasing in t. As a

result, to show that the above inequality holds for all 0 ≤ t ≤ L(δ), it is sufficient to show that it holds

for t = 0.

When t = 0, the above inequality is equivalent to

(1− δL(δ))uk(mk) + δL(δ)ukk ≥ uk(R)

By inequality (19), this is satisfied for δ suffifiently close to 1.

Lemma 20. Let µ = (ψ, ξkKk=1) be a self-enforcing matching process, then uk(µ) ≤ πk for all

Ik ∈ I.

Proof. By Lemma 1, vj(µ(h)) ≥ 0 for all aj ∈ A, at every ex post history h. So by Lemma 19,

uk(µ(h)) ≤ πk at every h ∈ H. Therefore, as the discounted sum of per-period utilities, uk(µ) ≤ πk

as well.

Lemma 21. Let m = (φ, ζlKl=1) be a static matching that satisfies vj(m) ≥ 0 for all aj ∈ A. If

uk(m) = uk(φ(Ik)) +∑

aj∈φ(Ik)

vj(Ik, φ(Ik)),

for some Ik ∈ I, then vj(m) = 0 for all aj ∈ φ(Ik).

Proof. For every aj ∈ φ(Ik), since vj(m) = vj(Ik, φ(Ik)) + ζkj ≥ 0, we have −ζkj ≤ vj(Ik, φ(Ik)).

Suppose by contradiction that vi(m) > 0 for some ai ∈ φ(Ik), then −ζki < vi(Ik, φ(Ik)). It follows

that

uk(m) = uk(φ(Ik)) +∑

aj∈φ(Ik)

ζkj

= uk(φ(Ik)) +∑

aj∈φ(Ik),j 6=i

ζkj + ζki

< uk(φ(Ik)) +∑

aj∈φ(Ik)

vj(Ik, φ(Ik)),

73

a contradiction.

Lemma 22. Let M ′ ⊆M be a subset of stage-game matchings, and µ : H →M ′ be a self-enforcing

matching process. Fix an institution Ik ∈ I. If for every m ∈ M ′, Ik has a feasible deviation

(Amk , ζmk ) ∈ Dk(m) that satisfies uk([m, (Ik, Amk , ζ

mk )]) ≥ uk, then

uk(µ) ≥ uk

Proof. Suppose by contradiction that µ is a self-enforcing matching process, but uk(µ) < uk. I will

show that institution Ik has a feasible profitable deviation plan from µ, so µmust not be self-enforcing.

Consider the following deviation plan (d′k, ξ′k) from µ: for every ex post history h ∈ H, define

d′k(h) = Aµ(h)k and ξ′k(h) = ζµk

By assumption, µ(h) ∈ M ′ for all h ∈ H, so (d′k, ξ′k) is well-defined and feasible. To see that

(d′k, ξ′k) is profitable, observe that at every ex post history h, institution Ik’s stage-game payoff from

the manipulated static assignment [µ(h), (Ik, d′k(h), ξ′k(h))] is uk([µ(h), (Ik, A

µ(h)k , ζ

µ(h)k )]) ≥ uk >

uk(µ). Since this is true for every ex post history h, Ik’s total discounted payoff from the deviation

plan satisfies

uk([µ, (Ik, d′k, ξ′k)]) > uk(µ).

The deviation plan (d′k, ξ′k) is both feasible and profitable for institution Ik, which is a contradic-

tion to the self-enforcement of µ. So uk(µ) ≥ uk.

Proof of Theorem 4. It suffices to show that any self-enforcing matching process µ must satisfy:

1. vj(µ(h)) ≥ 0 for all aj ∈ A, at any history of the market h;

2. uk(µ(h)) = πk for all Ik ∈ ∪Gg=1Pg, at any history of the market h;

3. IfR 6= ∅, then uk(µ) ≥ πk(R) for all Ik ∈ R;

I now establish these claims in order.

1. This follows from Lemma 1.

2. The proof proceeds by induction.

First I establish that uk(µ(h)) = πk for all Ik ∈ P1 and all ex post history h ∈ H. Suppose

by contradiction that uk(µ(h)) < πk for some Ik ∈ P1 and ex post history h ∈ H. Since µ

74

is self-enforcing and so is µ|h,µ(h), by Lemma 20, uk(µ|h,µ(h)) ≤ πk. So Ik’s payoff from the

continuation matching process µ|h is

uk(µ|h) = (1− δ)uk(µ(h)) + δuk(µ|h,µ(h)) < πk (23)

For every m = (φ, ζkKk=1) ∈M , consider the feasible deviation (Amk , ζmk ) defined by

Amk = arg maxB⊆A\(∪Kl=1φ(Il))

χ(Ik, B),

ζmkj =

−vj(Ik, Amk ) + 1

2|Amk | [πk − uk(µ|h)] if aj ∈ Amk ;

0 otherwise.

if Amk 6= ∅, and ζmkj = 0 for all aj ∈ A if Amk = ∅. Note that by construction,

χ(Ik, Amk ) = max

B⊆A\(∪Kl=1φ(Il))

χ(Ik, B). (24)

Recall that since Ik ∈ P1, πk(I) = πk. Moreover,∣∣(∪Kl=1φ(Il))

∣∣ ≤ Q(I), so

maxB⊆A\(∪Kl=1φ(Il))

χ(Ik, B) ≥ minA⊆A:|A|≤Q(I)

maxB⊆A\A

χ(Ik, B)

= πk(I) = πk = maxB⊆A

χ(Ik, B) ≥ maxB⊆A\(∪Kl=1φ(Il))

χ(Ik, B).(25)

Combining Eq. (24) and Eq. (25), it follows that

χ(Ik, Amk ) = πk. (26)

I now verify that (Amk , ζmk ) is a feasible deviation that satisfies uk([m, (Ik, Amk , ζ

mk )]) ≥ 1

2 πk +12uk(µ|h). Since Amk ⊆ A\(∪Kl=1φ(Il)), vj(m) = 0 for all aj ∈ Amk . Meanwhile, for all

aj ∈ Amk ,

vj(Ik, Amk ) + ζmkj

= vj(Ik, Amk )− vj(Ik, Amk ) +

1

2∣∣Amk

∣∣ [πk − uk(µ|h)]

=1

2∣∣Amk

∣∣ [πk − uk(µ|h)] > 0 = vj(m)

So every agent in Amk finds himself strictly better off by joining the deviation, which ensures

75

the feasibility of (Amk , ζmk ). To see uk([m, (Ik, Amk , ζ

mk )]) ≥ 1

2 πk + 12uk(µ|h), observe

uk(Amk )−

∑

aj∈Amk

ζmkj = uk(Amk ) +

∑

aj∈Amk

vj(Ik, Amk )− 1

2[πk − uk(µ|h)]

= πk −1

2[πk − uk(µ|h)]

=1

2πk +

1

2uk(µ|h),

where the second equality above follows from Eq. (26). Since for every m ∈ M , (Amk , ζmk )

is a feasible deviation that satisfies uk([m, (Ik, Amk , ζmk )]) ≥ 1

2 πk + 12uk(µ|h), it follows from

Lemma 22 that µ|h, as a self-enforcing matching process, must satisfy uk(µ|h) ≥ 12 πk +

12uk(µ|h), which is a contradiction to Eq. (23). So uk(µ(h)) = πk for all Ik ∈ P1 and all ex

post history h ∈ H.

Now, suppose uk(µ(h)) = πk for every Ik ∈ ∪n−1g=1Pg and all h ∈ H, I will show uk(µ(h)) =

πk for every Ik ∈ Pn and all h ∈ H. Let Mn−1 ≡ m ∈ M : uk(m) = πk for all Ik ∈∪n−1g=1Pg, vj(m) ≥ 0 for all aj ∈ A. By the inductive hypothesis, and because µ is self-

enforcing, µ must be a mapping fromH to Mn−1.

Suppose by contradiction that uk(µ(h)) < πk for some Ik ∈ Pn and h ∈ H. This implies, in

the same way inequality (23) is established, that

uk(µ|h) = (1− δ)uk(µ(h)) + δuk(µ|h,µ(h)) < πk (27)

For every m = (φ, ζkKk=1) ∈Mn−1, consider the feasible deviation (Amk , ζmk ) defined by

Amk = arg maxB⊆A\φ(I\∪n−1

g=1Pg)

χ(Ik, B).

ζmkj =

−vj(Ik, Amk ) + 1

2|Amk | [πk − uk(µ|h)] if aj ∈ Amk ;

0 otherwise.

if Amk 6= ∅, and ζmkj = 0 for all aj ∈ A if Amk = ∅. Note that by construction,

χ(Ik, Amk ) = max

B⊆A\φ(I\∪n−1g=1Pg)

χ(Ik, B). (28)

Recall that since Ik ∈ Pn, πk(I\ ∪n−1g=1 Pg) = πk. Moreover,

∣∣∣φ(I\ ∪n−1g=1 Pg)

∣∣∣ ≤ Q(I\ ∪n−1g=1

76

Pg), so

maxB⊆A\φ(I\∪n−1

g=1Pg)

χ(Ik, B) ≥ minA⊆A:|A|≤Q(I\∪n−1

g=1Pg)maxB⊆A\A

χ(Ik, B)

= πk(I\ ∪n−1g=1 Pg) = πk = max

B⊆Aχ(Ik, B) ≥ max

B⊆A\φ(I\∪n−1g=1Pg)

χ(Ik, B).(29)

Combining Eq. (28) and Eq. (29), it follows that

χ(Ik, Amk ) = πk. (30)

I now verify that (Amk , ζmk ) is a feasible deviation that satisfies uk([m, (Ik, Amk , ζ

mk )]) ≥ 1

2 πk +12uk(µ|h). Since Amk ⊆ A\φ(I\ ∪n−1

g=1 Pg), or equivalently,

Amk ⊆ φ(∪n−1g=1Pg) ∪

[A\φ(I)

].

Since m = (φ, ζkKk=1) ∈ Mn−1, vj(m) ≥ 0 for all aj ∈ φ(∪n−1g=1Pg) and ul(m) = πl for all

Il ∈ ∪n−1g=1Pg. Lemma 21 then delivers that vj(m) = 0 for every aj ∈ φ(∪n−1

g=1Pg). Since every

agent in aj ∈ A\φ(I) is unmatched, vj(m) = 0 for all aj ∈ A\φ(I) as well. So vj(m) = 0

for all aj ∈ Amk .

Meanwhile, for all aj ∈ Amk ,

vj(Ik, Amk ) + ζmkj

= vj(Ik, Amk )− vj(Ik, Amk ) +

1

2∣∣Amk

∣∣ [πk − uk(µ|h)]

=1

2∣∣Amk

∣∣ [πk − uk(µ|h)] > 0 = vj(m)

So every agent in Amk finds himself strictly better off by joining the deviation, which ensures

the feasibility of (Amk , ζmk ). To see uk([m, (Ik, Amk , ζ

mk )]) ≥ 1

2 πk + 12uk(µ|h), observe

uk(Amk )−

∑

aj∈Amk

ζmkj = uk(Amk ) +

∑

aj∈Amk

vj(Ik, Amk )− 1

2[πk − uk(µ|h)]

= πk −1

2[πk − uk(µ|h)]

=1

2πk +

1

2uk(µ|h),

where the second equality follows from Eq. (30).

Since for everym ∈Mn−1, (Amk , ζmk ) is a feasible deviation that satisfies uk([m, (Ik, Amk , ζ

mk )]) ≥

12 πk + 1

2uk(µ|h), it follows from Lemma 22 that µ|h : H → Mn−1, as a self-enforcing match-

77

ing process, must satisfy uk(µ|h) ≥ 12 πk + 1

2uk(µ|h), which is a contradiction to Eq. (27). So

uk(µ(h)) = πk for all Ik ∈ Pn and all ex post history h ∈ H. This completes the inductive

argument

3. Suppose by contradiction that uk(µ) < πk(R) for some Ik ∈ R. I will show that institution Ikhas a feasible profitable deviation plan from µ = (ψ, ξlKl=1), so µ must not be self-enforcing.

Consider the following deviation plan (d′k, ξ′k) from µ:

For any h ∈ H,

d′k(h) = arg maxB⊆A\ψ(R|h)

uk(B) +∑

aj∈Bvj(Ik, B).

If d′k(h) 6= ∅, then

ξ′kj(h) =

−vj(Ik, d′k(h)) + 1

2|d′k(h)| [πk(R)− uk(µ)] if aj ∈ d′k(h);

0 otherwise.

If d′k(h) = ∅, define

ξ′kj(h) = 0 ∀aj ∈ A.

Note that by construction,

uk(d′k(h)) +

∑

aj∈d′k(h)

vj(Ik, d′k(h))

= maxB⊆A\ψ(R|h)

uk(B) +∑

aj∈Bvj(Ik, B)

≥ minA⊆A:|A|≤Q(R)

maxB⊆A\A

uk(B) +∑

aj∈Bvj(Ik, B) = πk(R).

(31)

The inequality above follows from |ψ(R|h)| ≤ Q(R).

We first verify that the deviation plan (d′k, ξ′k) is feasible. At any history h ∈ H, d′k(h) ⊆

A\ψ(R|h), or equivalently,

d′k(h) ⊆ ψ(∪Gg=1Pg|h) ∪[A\ψ(I|h)

].

At every history of the market h, since µ is self-enforcing, by Lemma 1, vj(µ(h)) ≥ 0 for all

aj ∈ ψ(∪Gg=1Pg|h). In addition, from the second claim of this proof, ul(µ(h)) = πl for all

Il ∈ ∪Gg=1Pg. Lemma 21 then delivers that vj(µ(h)) = 0 for every aj ∈ ψ(∪Gg=1Pg|h). Since

every agent in aj ∈ A\ψ(I|h) is unmatched, we have vj(µ(h)) = 0 for all aj ∈ A\ψ(I|h) as

well. So vj(µ(h)) = 0 for all aj ∈ d′k(h).

78

Meanwhile, at every history h and for all aj ∈ d′k(h),

vj(Ik, d′k(h)) + ξ′kj(h)

= vj(Ik, d′k(h))− vj(Ik, d′k(h)) +

1

2∣∣d′k(h)

∣∣ [πk(R)− uk(µ)]

=1

2∣∣d′k(h)

∣∣ [πk(R)− uk(µ)] > 0 = vj(µ(h))

So at every possible history h, every agent in d′k(h) finds himself strictly better off by joining

the deviation, which ensures the feasibility of the deviation (d′k, ξ′k).

To see that (d′k, ξ′k) is profitable, observe that at every history h, institution Ik’s stage-game

payoff from the manipulated static matching [µ(h), (Ik, d′k(h), ξ′k(h))] is

uk(d′k(h))−

∑

aj∈d′k(h)

ξ′kj(h)

= uk(d′k(h)) +

∑

aj∈d′k(h)

vj(Ik, d′k(h))− 1

2[πk(R)− uk(µ)]

≥ πk(R)− 1

2[πk(R)− uk(µ)]

=1

2πk(R) +

1

2uk(µ) > uk(µ).

The second inequality above follows from inequality (31). Since this is true for every history

h, Ik’s total discounted payoff from the deviation plan satisfies

uk([µ, (Ik, d′k, ξ′k)]) =

1

2πk(R) +

1

2uk(µ) > uk(µ).

The deviation plan (d′k, ξ′k) is both feasible and profitable for institution Ik, which is a contra-

diction to the self-enforcement of µ. So uk(µ) ≥ πk(R) for all Ik ∈ R.

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stability in repeated matching...

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