n?u=RePEc:ris:qmetal:2016_002&r=pke

MQITE WORKING PAPER SERIES

WP # 16-02

(In)visible Hands in Matching Markets.

José Alcalde.

(In)visible Hands in Matching Markets ∗

José Alcalde †

∗ This work is partially supported by the Spanish Ministerio de Economía y Competitividad, under project

ECO2013-43119-P.

† IUDESP, University of Alicante. jose.alcalde@ua.es

Abstract

This paper explores sequential mechanisms for (many-to-one) two-sided matching problems.

In these mechanisms, agents belonging to a side of the market determine an eligibility

restriction; and the agents on the other side select their preferred mates, constrained by the

above eligibility.

We find out some asymmetries, as well as some coincidences, related to the mechanisms in

which the first decision is made by the individuals or the institutions. In particular, for the

two classes of mechanisms, it is likely that the outcome is stable. This can be interpreted

as if the sequentiality in which agents decide takes the place of the coordination among

the whole society. As a main difference, it is found that an extra coordination between

individuals is exercised when they impose the eligibility restriction. A consequence of such

an ‘over-coordination’ is that the likely outcome coincides with the individuals optimal stable

matching. The dual result yielding the institutions’ optimal stable allocation does not hold

when these agents are the ones to impose eligibility.

Keywords: Sequentiality, (Pairwise) Stability, Matching Markets

Journal of Economic Literature Classification Numbers: C78, D61, D61, D78.

1 Introduction

Individual decisions, guided by the selfish objective of maximizing her own utility, might induce

optimal allocations for the whole society. This paradigm, coined as the invisible hand

since Smith (1776), helps to support the applicability of some rooted allocation procedures.

More than that, some systems survive because the presence of this “hand” allows the agents

to be pleased with its use. As a counterpart, certain allocation systems have been abandoned

because individual decisions are difficult to be coordinated. This is, for instance,

what happened in Boston when the local educative authority decided in 2005 to move from

the former “Boston procedure” to the newly adopted “deferred-acceptance mechanism” to

distribute school places among the newcomer students (see, v.g., Abdulkadiroğlu et al.,

2005).

This paper tackles the analysis of allocation procedures in (many-to-one) two-sided

matching problems. In particular, we investigate how real-life procedures survive because,

even when not explicit coordination is considered, agents in both sides behave as if they

were co-operating. In other words, agents, when faced to these mechanisms, are conducted

by an “invisible hand”.

Two-sided matching problems are characterized by the presence of two disjoint groups

of agents, understood as institutions (firms, hospitals, schools, military, . . . ) and individuals

(workers, interns, students, soldiers, . . . ) that have to complete a joint activity. A relevant,

formal difference between the two sides is that each institution can be attached to several

individuals, whereas no individual can serve to more than one institution.

Matching rules are systematic procedures solving the problem of which individuals

should be connected to each institution. Its analysis has been addressed both by a theoretical

and a practical perspectives (see, v.g., Sönmez and Switzer, 2013, related to the US

Army). In this line, Alcalde and Romero-Medina (2000, 2005) design sequential mechanisms

yielding stable matching as expected outcomes. In a close framework, Alcalde et al.

(1998) point out that sequentiality also helps to attain stable configurations in job markets,

when salary is a relevant variable to be considered.

Sotomayor (2003) explored the design of (non-revelation) sequential mechanisms for

the marriage problem implementing stable allocations. The idea she underlined has been

recently extended by Romero-Medina and Triossi (2014) to the more general case of manyto-one

matchings framework. These authors identify two mechanisms guaranteeing that

agents are indirectly coordinated thanks to the sequentiality in which their decisions are

adopted. The mechanisms they proposed mimic, among others, the allocation procedures

that are employed in Spain to allocate civil servants to specific places, as illustrated in the

following instance. Each year some individuals enter the public educative system as teachers.

Each school has some needs to be covered by the newly assigned teachers. Therefore, it

1

is reasonable to allow the schools to restrict which candidates might be incorporated in it. 1

Once this distinction between acceptable and unacceptable new-members is settled, the applicants

are sequentially required, according an exogenous priority, to select their workplace

among the ones declaring them as acceptable.

The results obtained by Romero-Medina and Triossi (2014) are conditioned by an assumption

made on the institutions’ preferences. It is supposed that they fulfill substitutability

(Roth, 1984). Nevertheless, their analysis excludes any predictability when such an assumption

is not fulfilled. It particular, stability of the outcomes reached at each equilibrium

is not guaranteed when individuals declare which institutions are admissible, and institutions’

preferences fail to satisfy substitutability (see, vg. Example 3). Therefore, the rules

proposed by Romero-Medina and Triossi (2014) deserves a further, more detailed analysis.

This exhaustive study is developed in the present paper.

To be consequent with the above observations, and for the sake of concreteness, in

our exposition we devote one section to each mechanism, depending on which category

of agents, institutions or individuals, are the first to select which mates are admissible and

which are not. This selection limits the options that are electable by the agents belonging

to the alternative category.

The first mechanism that we analyze corresponds to the case where institutions announce

which individuals are acceptable and which are not. We found that this mechanism

admits (at least) one Subgame Perfect Nash equilibrium, SPNE hereafter, whenever the

problem admits stable outcomes. More than that, we demonstrate that the set of outcomes

supported by a SPNE coincides with that of stable allocations. As a by-product, it is shown

that conditions delimiting which preferences might exhibit the institutions so that a stable

allocation exists, independently of the individuals descriptions, can be characterized as

properties on institutions’ characte**ris**tics ensuring the existence of SPNE for this mechanism.

The results proposed here strength Proposition 2 in Romero-Medina and Triossi (2014) in

the sense that we also analyze how the mechanism behaves when no condition is imposed

to guarantee the existence of stable allocations. Even though our contribution related to this

mechanism is minor, this analysis complements the study of the ‘dual’ mechanism where the

roles of the two groups of agents -institutions and individuals- are reversed.

In the second mechanism, the individuals are consulted to declare which institutions are,

in their opinion, acceptable and which are not. The analysis of agents’ behavior yields to

see that the mechanism partially implements, in SPNE, the set of pairwise stable allocations.

A further examination of this mechanism, when the institutions’ preferences satisfy responsiveness,

and individuals exhibit sophisticated behavior, yields to a surp**ris**ing result. Even

though any pairwise stable matching can be supported by a SPNE, there is only one matching

that should be expected: the stable matching which all the individuals unanimously

1 Think of a school needing some instructor(s) to teach Latin courses. It is fair to allow it to block the

allocation of new teachers whose specialty is Physics to that school.

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prefer. This is the worst stable allocation from the institutions point of view. In this mechanism,

the institutions select according a given ordering, the group of individuals that they

want to hire. The intuitive result, based on this sequentiality, might yield to anticipate that

the allocation should be optimal from the institutions’ point of view. Nevertheless, what we

obtain is just the opposite, namely the optimality from the individuals’ perspective. Notre

that such a reversibility is not new in this framework. For instance, Alcalde (1996) pointed

out a similar “counter-intuitive” result when stable mechanisms are employed.

The rest of the paper is organized as follows. Section 2 describes the framework and

supplies the main standard concepts. In particular, Subsection 2.1 explicitly states what

we understand by sequential matching mechanisms. Section 3 explores the mechanism

where each individual selects her preferred institutions, among the ones declaring her as

acceptable. We point out that any stable allocation can be supported as an equilibrium of

the mechanism. The ‘dual’ mechanism in which the institutions select the individuals, restricted

to acceptability, is analyzed in Section 4. We demonstrate that any stable allocation

can be supported by an equilibrium. We find out (Subsection 4.1) that this result fails to

be true when individuals select undominated strategies. This observation leads to explore

agents’ behavior in some restricted domains. Subsection 4.2 is devoted to show that, when

individuals exhibit a sophisticated behavior, the only expectable allocation is the optimal

from their point of view, among the ones being stable. Finally, Section 5 concludes.

2 The Model and Main Definitions

There are two finite, disjoint groups of agents, namely W = {w 1 , . . . , w i , . . . , w n } denoting

the set of individuals, to be also referred as ‘workers’, and the set of institutions,

F =

f 1 , . . . , f j , . . . , f m , that for interpretative purposes will be named ‘firms’.

Each worker w i has preferences ≻ i describing a complete, linear ordering on F ∪ {w i }.

Similarly, each firm f j has preferences ≻ j over the set of groups of workers, describing a

complete, linear ordering on 2 W . Given a set of workers W ′ ⊆ W , and a firm f j , C

W ′ , ≻ j

denotes f j choice on W ′ according ≻ j , i.e. W ′′ = C

W ′ , ≻ j whenever (i) W ′′ ⊆ W ′ and,

(ii) for each W ⊆ W ′ , W ≠ W ′′ , W ′′ ≻ j

W . For agent x ∈ W ∪ F, x stands for her weak

preferences; i.e. y x z means that either y ≻ x z or y = z.

A specific problem can be described as P =

(≻ i ) wi ∈W ;

≻ j . A solution for P, also

called a matching, is a correspondence µ : W ∪ F ↠ W ∪ F such that

(a) For each worker w i , µ (w i ) ∈ F ∪ {w i };

(b) For each firm f j , µ (f i ) ⊆ W ; and

(c) For each worker-firm pair w i , f j

, µ (wi ) = f j if, and only if, w i ∈ µ f j

.

Given a problem P, we say that matching µ is

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f j ∈F

(1) Individually Rational if

(i) there is no worker w i such that w i ≻ i µ (w i ); and

(ii) for each firm f j , µ f j

= C

µ

f j

, ≻j

.

(2) Pairwise Stable if is individually rational and there is no worker-firm pair w i , f j

such that f j ≻ i µ (w i ) and w i ∈ C µ f j

∪ {wi } , ≻ j

.

(3) Stable if is individually rational and there is no firm f j and (non-empty) team of

workers W ′ such that

(i) for each worker w i ∈ W ′ , f j ≻ i µ (w i ); and

(ii) W ′ ⊆ C µ f j

∪ W ′ , ≻ j

.

For a given problem P,

• IR (P) is the set of its individually rational matchings;

• PC (P) denotes the set of its pairwise stable matchings; and

• C (P) represents the set of its stable matchings.

It is well-known that for each given problem P, C (P) ⊆ PC (P), whereas the opposite might

not be true. Moreover, there are some instances having no pairwise stable allocation, and

thus not stable matching.

Roth (1984) explored the possibility of finding environments where all the problems

have stable allocations. He defined substitutability, a property guaranteeing the existence

of such matchings. It requires that the selection of some worker by a firm is not reversed

when some workers become unelectable.

Definition 1 [Substitutability]

We say that ≻ j , the preferences of firm f j , satisfy substitutability whenever for each set of

workers W ′ ⊆ W , if w i ∈ C W ′ , ≻ j

, then wi ∈ C W ′ \ {w k } , ≻ j

for any wk ≠ w i .

A problem is said substitutable when all the preferences of all its firms exhibit substitutability.

Under substitutability, the following results hold (see Roth, 1984).

Lemma 1 Let P a substitutable problem, then PC (P) = C (P) ≠ .

Lemma 2 Let P a substitutable problem. Then, there is µ W O ∈ C (P) such that for each

w i ∈ W and µ ∈ C (P), µ W O (w i ) i µ (w i ). Such an allocation is known as the Workers-

Optimal Stable matching.

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The next condition, known as responsiveness (Roth, 1985), connects each firm’s opinion

on isolated workers with its consideration about the teams of workers. To properly describing

responsiveness, it is assumed that each firm has a maximal number of workers to be

hired, called its ‘quota’. The idea underlying responsiveness is very simple. Given a set of

workers, a firm will select the best ones up to fulfill its quota. Moreover, a worker that is

considered as unacceptable (i.e., to contract her is worse than not to contract any worker)

will never be selected, no matter whether the quota is exhausted or not.

Definition 2 [Responsiveness]

Let f j be a firm with quota q j ≥ 1. We say that its preferences over groups of workers ≻ j

satisfy responsiveness if the following holds.

(i) For each W ′ ⊂ W , C

W ′ , ≻ j

≤ q j ; 2

(ii) for each W ′ ⊂ W , with |W ′ | < q j and any two workers w i and w h not in W ′ , W ′ ∪

{w i } ≻ j W ′ ∪ {w h } if, and only if {w i } ≻ j {w h }; and

(iii) for each W ′ ⊂ W , with |W ′ | ≤ q j and any worker w i ∈ W ′ , w i ∈ C W ′ , ≻ j

if, and only

if, {w i } ≻ j .

A problem is said responsive if all its firms preferences satisfy responsiveness.

It is well known that Lemmata 1 and 2 above are still valid if we replace substitutability

by responsiveness.

2.1 Sequential Matching Mechanisms

We now deal with the description of agents’ behavior when facing to sequential matching

mechanisms. It is adapted to the class of mechanisms to be introduced in the next sections.

We want to stress that we concentrate on a complete information scenario. That is, each

agent knows all the agents’ preferences, and this is public information. Moreover, given the

ordinal description of the model, we restrict attention to pure strategy equilibria.

The way in which agents interact can be described as follows. There are k steps, 1 < k ≤

n + m. At step k a group of agents must leave a message. No agent is called to do it in two

different steps. Given an agent x ∈ W ∪ F, we denote by B (x) the set of its predecessors,

namely the group of agents leaving a message before x is required to do it.

A strategy for agent x is a function specifying, for each possible combination of messages

leaved by her predecessors, a message for agent x. Therefore, given the sequentiality of the

mechanism, each profile of strategies S = (S x ) x∈W ∪F induces a unique messages profile

M (S) = (M x (S)) x∈W ∪F .

A sequential matching mechanism ϕ is a function associating each message profile M

with a matching µ = ϕ (M).

2 Given a set T, |T| denotes the number of elements it has.

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Definition 3 Let ϕ be a sequential matching mechanism, and P a problem. Strategy profile

S = (S x ) x∈A∪F is an equilibrium if there is no agent x and strategy for such an agent, S ′ x

such that

µ ′ (x) ≻ x µ (x)

where µ = ϕ (M (S)) is the matching obtained when the strategy profile is S; and µ ′ =

ϕ M S −x , S x ′ refers to the case where agent x selects strategy S

′

, and the remaining

x

agents are still selecting the strategies described through S.

Definition 4 Consider a sequential matching mechanism ϕ and a problem P. Strategy S x

is dominated by S ′ x if

(a) there are strategies for the remaining agents S −x =

S y , such that

y≠x

µ ′ (x) ≻ x µ (x)

where µ = ϕ (M (S)) and µ ′ = ϕ M S −x , S x ′ are the matchings obtained when all

the agents but x select strategies S y and x’s strategy is either S x or S ′ , respectively;

x

and

(b) there are no strategies S

S ′ = ′ for the remaining agents such that

−x y

y≠x

µ (x) ≻ x µ ′ (x)

where µ is associated to the case where x selects S x and µ ′ to her selection of S ′ , x

being the remaining agents strategies S ′ in the two instances.

y

A strategy for x which is not dominated by any alternative one is called an undominated

strategy. A strategy dominating each other alternative is named a dominant strategy.

Definition 5 A Subgame Perfect Nash Equilibrium for sequential matching mechanism ϕ

at problem P is an equilibrium S where no agent having predecessors selects a dominated

strategy. 3

Definition 6 Consider a sequential matching mechanism ϕ, and a problem P. Let S x denote

the set of strategies for agent x, and S = Π x∈W ∪F S x the set of strategy profiles. Now, reduce

the set of strategies that are electable by each agent through an iterative elimination of the

dominated strategies. That is, for each agent x, S 1 ⊆ S x x denotes the set of her undominated

strategies; and denote S 1 = Π x∈W ∪F S 1. Then, define, for each agent x S2 ⊆ x

S1 her set of

x

undominated strategies, given that each agent y is selecting an strategy in S 1 . We can

y

then define S 2 = Π x∈W ∪F S 2 . Let repeat this procedure until some step t where no strategy

x

is dominated; that is, t is such that S t−1 S t = S t+1 . A sophisticated equilibrium for

mechanism ϕ at P is an equilibrium S where each agent’s strategy survives to the iterated

deletion of dominated strategies; i.e., for each x, S x ∈ S t . x

3 Note that, since each agent is called to send a message at once, the description of an equilibrium at each

subgame is equivalent to the selection of undominated strategies for each agent x such that B (x) ≠ .

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3 Implementing the Set of Stable Allocations

The aim of this section is to explore how firms can achieve that workers’ decisions result

in stable allocations. To reach this goal no co-ordination among firms and/or workers is

needed. The mechanism that we explore exhibits some similarities with that introduced

in Romero-Medina and Triossi (2014) as the “Colleges Admit Students Sequentially Choose

mechanism”. In particular, in both mechanisms firms declare which workers are ‘acceptable’

from their point of view. Then each worker selects the firm -if any- to which she wants to be

engaged. The main formal difference between the two mechanisms is that Romero-Medina

and Triossi (2014) requires that workers’ decisions should be made sequentially, whereas

in the present paper such a sequentiality disappears. Given that each worker’s preferences

depend on which is the firm she is engaged to, and it is independent from the colleagues

attached to the same institution, there is no need to require workers to play sequentially to

reach our results.

Definition 7 [The Firms Demand and Workers Select Mechanism]

This mechanism, to be denoted as ϕ F , operates as follows. At step 0, each firm is required

to select a (possibly empty) set of workers. Then, at step 1, all the workers, knowing

firms’ messages -or strategies-, select a firm (or her being unmatched option). Given

agents’ strategies S, inducing the messages profile M (S), the mechanism produces output

ϕ F (M (S)) = µ such that, for each worker w i ,

µ (w i ) =

f j i f M i (S) = f j and w i ∈ M j (S)

w i

otherwise

In other words, each worker is matched to the firm she selected at step 1 - if any - unless

that firm declared at the very beginning that it does not admit this worker. In the latter case,

the worker remains unmatched, and thus is not engaged to any firm.

Note that the strategies selected by the workers, at any SPNE, are easily identifiable.

When, at Step 1, each worker has to select her message, she knows the messages sent by

all the firms. Let M F ≡

M j be the vector describing the messages sent by all the firms

f j ∈F

at

Step 0. Recall

that, for each firm f j , M j ⊆ W . For each given worker, say w i , F i (M F ) =

f j ∈ F : w i ∈ M j describes the group of institutions declaring that wi is an ‘admissible’

worker. Then, the optimal message for worker w i is to select the best option, according her

preferences ≻ i , in F i (M F ) ∪ {w i }. If we denote it as B i (M F ), the strategy played by worker

w i in a SPNE is the function S i associating to each possible M F her best achievable mate

B i (M F ).

Before introducing our main result related to this mechanism, we propose two illustrative

examples.

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Example 1 A problem with non-substitutable preferences

Consider the following 2-workers-2-firms problem. W = {a, b}, and F = {1, 2}. Agents’

preferences are 4 ≻ a := 1, 2 ≻ 1 =: {a, b} , {b}

≻ b := 2, 1 ≻ 2 =: {a, b} , {a}

It is easy to see that the preferences by the firms do not satisfy substitutability. 5 It can be

also seen that this problem has two stable outcomes, namely µ and µ ′ characterized by

µ (1) = {a, b}, and µ ′ (2) = {a, b}.

To describe a SPNE for ϕ F we just need to specify which are the messages selected by the

firms. For the workers, the description of their optimal behavior just argued characterizes

their strategies, which is enough to describe their messages. Let us observe that this mechanism

has two SPNE, whose messages are M (S) = (M x ) x∈W ∪F , and M (S ′ ) =

M ′ x x∈W ∪F

respectively, with

M 1 = {a, b} M 2 = M a = 1 M b = 1

M ′ 1 = M ′ 2 = {a, b} M ′ a = 2 M ′ b = 2

As a conclusion, this example suggests that each stable matching can be supported by a

SPNE.

Example 2 A problem with no stable allocation

This instance is borrowed from Roth and Sotomayor (1990, Example 2.7). It is useful to

illustrate how the absence of stable outcomes affects the predictability of the agents’ actions.

Consider the following 3-workers-2-firms problem. W = {a, b, c}, and F = {1, 2}.

Agents’ preferences are

≻ a := 2, 1 ≻ 1 =: {a, c} , {a, b} , {b, c} , {a} , {b}

≻ b := 2, 1 ≻ 2 =: {a, c} , {b, c} , {a, b} , {c} , {a} , {b}

≻ c := 1, 2

It can be observed that this problem has no stable allocation. Now we check that ϕ F has no

SPNE for this problem. For, consider the following cases:

(a) If c ∈ S 1 , then µ (c) = 1, therefore any optimal strategy for firm 2, say S 2 , should verify

that {a, b} ⊆ S 2 , guaranteeing that µ (2) = {a, b}.

4 As usual, preferences are described as an ordered list of admissible mates. In this example, the fact that

firm 2 does not list worker a means that this institution prefers not to hire any worker rather that being

attached to {a}.

5 In particular, note that a ∈ C ({a, b} , ≻ 1 ), but a /∈ C ({a} , ≻ 1 ) = C ({a, b} \ {b} , ≻ 1 ).

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(b) If {a, b} ⊆ S 2 . Then µ (a) = µ (b) = 2. Therefore, c /∈ S 1 because, otherwise, µ (1) =

{c}. Note that 1, by declaring S ′ 1 = gets = µ′ (1) ≻ 1 {c} = µ (1).

(c) If c /∈ S 1 , then 2’s optimal strategy is S 2 = {a, c} because in such a case it obtains its

best set of workers; i.e. µ (2) = {a, c}.

(d) If S 2 = {a, c}, then the optimal strategy for 1 is to select strategy S 1 such that {b, c} ⊆ S 1

because it guarantees µ (1) = {b, c}. Since c ∈ S 1 , the postulate of case (a) above

applies.

Note that the four cases above exhaust all the possibilities and incur in a contradiction with

any assumption about the existence of an equilibrium.

Therefore, what this example suggest is that the existence of SPNE for the ‘Firms Demand

and Workers Select’ mechanism, FDWSm in short, is related to the fact that whether C (P)

is empty or not.

The next result synthesizes the universality of conclusions in Examples 1 and 2 above:

ϕ F implements in SPNE the set of stable outcomes for each problem P.

Theorem 1 Let P be a problem. Then, mechanism ϕ F has a SPNE if, and only if, C (P) ≠ .

Moreover, for each SPNE, say S, ϕ F (M (S)) ∈ C (P). Similarly, associated to each µ ∈ C (P)

there is a strategies profile S describing a SPNE with µ = ϕ F (M (S)).

Proof

Given the agents’ preferences, and thus problem P, let S be a SPNE for ϕ F . Assume that

µ = ϕ F (M (S)) /∈ C (P).

First, note that µ must be individually rational. Otherwise, S trivially fails to be a SPNE.

Therefore there should be a firm f j and a non-empty set of workers W ′ such that W ′ ≻ j µ

f j

and for each w i ∈ W ′ \ µ

f j , f j ≻ i µ (w i ). Moreover, since µ is individually rational, there

should be a worker in W ′ , say w i , not matched to f j under µ. Now, consider strategy

(and message) S ′ for firm f

j j declaring acceptable all the workers in W ′ , i.e. S ′ = W ′ .

j

Since workers in W ′ play optimally, each worker w i in W ′ set as message M i

S − j , S ′ = f

j j ,

yielding matching µ ′ such that µ ′ f j = W ′ . This contradicts that S was a SPNE. Note that,

in particular, it is also proven that when C (P) = , no strategy profile S constitutes a SPNE

for ϕ F .

Now, let µ be a stable matching for problem P, and consider the following strategies.

Each f j ∈ F declares acceptable only the workers in µ

f j . Then, each worker’s strategy

determines to select the best institution, according her preferences, amongst those declaring

her as acceptable (or her being unmatched option, if none of them is acceptable under her

true preferences). It is straightforwardly verifiable that these strategies induce a messages

profile M (S) yielding ϕ F (M (S)) = µ. In can be also checked that these strategies constitute

a SPNE.

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Theorem 1 above points out that the sequentiality in which agents take their decisions

is enough to guarantee that, without imposing any coordination between them, stable allocations

are reached.

Some authors have been worried about the conditions that firms’ preferences must satisfy

to guarantee the existence of stable allocations (see, v.g. Martínez et al., 2000). This problem

has been partially solved by resorting to conditions very related to substitutability and/or

responsiveness, which are sufficient, but not necessary, to ensure that some stable matching

exists. The generic question to be addressed is: Given a family of problems, say P, when

can be ensured that it is stable -i.e. for each P ∈ P, C (P) ≠ ? Theorem 1 allows to describe

the set of problems admitting stable allocations as the family of instances under which the

FDWSm has some SPNE.

Corollary 1 Let P be a collection of problems. P is a stable family if, and only if, for each

P ∈ P the Firms Demand and Workers Select mechanism admits, at least, one SPNE.

4 Workers’ Decisions and Pairwise Stability

This section introduces a procedure, to be named “Workers Apply and Firms Select mechanism”,

WAFSm in short, in which the agents’ roles are exchanged with respect to that

exhibited in the FDWSm, explored in Section 3.

A first difference between the “Firms Demand and Workers Select” and the “Workers Apply

and Firms Select” mechanisms is that the former allows workers to play simultaneously,

whereas the latter requires a sequentially in the description of firms’ decisions. The main

reason justifying it is that this sequentiality helps to unambiguously obtain a matching as

the result of agents’ interaction.

We now introduce the mechanism. Its essence is the following. Workers are required to

declare which firms are acceptable and which are not. Then, firms sequentially select the

group of workers among those that have marked it as admissible and has not been previously

engaged to another firm. Note that each order in which firms are called to manifest their

selection defines a different mechanism. In our definition, for the sake of simplicity, we refer

to the case where institutions are called to declare their messages according the ordering

given by their labels.

Definition 8 [The Workers Apply and Firms Select Mechanism]

This mechanism, to be denoted as ϕ W , operates as follows. At step 0, each worker is

required a message M i ⊆ F consisting of a (possibly empty) set of firms. At step j, j =

1, . . . , m, firm f j selects a subset of workers, 6 so that M j ⊆ W . Given agents’ strategies S,

6 It is assumed that the firm selecting a group of workers at this step knows the messages sent by all the

workers, as well as the selections made by the previous firms (if any).

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inducing messages profile M (S), the mechanism produces output ϕ W (M (S)) = µ such that,

for each worker w i and firm f j , µ (w i ) = f j if an only if

(a) f j ∈ M i ;

(b) w i ∈ M j ; and

(c) for each j ′ < j either f j ′ /∈ M i , or w i /∈ M j ′.

Previous to analyze which outcomes should be expected when ϕ W is employed, we

explore the optimal behavior by the firms. At step 0 each worker declares which firms

she considers to be acceptable. This is the information contained in workers’ strategies

S W = (S i ) i∈W . At step 1 when firm f 1 has to decide which workers to select, we can identify

its set of available workers, namely

A 1 (S) = {w i ∈ W : f 1 ∈ S i } .

Therefore, the message sent by firm f 1 should be a set of workers W ′ such that W ′ ∩A 1 (S) =

C

A 1 (S) , ≻ j . Note that this yields that firm f1 is engaged to µ (f 1 ) = C (A 1 (S) , ≻ 1 ). Applying

an iterative reasoning, we have that for each j > 1, firm f j should select among a

group of ‘available’ workers being the ones that declared that f j is acceptable for her, not

being selected by a precedent firm, namely

A j (S) = ⋃

w i ∈ W : f j ∈ S i \ µ (f h ) ,

so that any group of workers W ′ such that W ′ ∩A j (S) = C A j (S) , ≻ j

constitutes an optimal

message for f j given the messages sent by all the workers as well as the firms preceding it.

Note that, in such a case µ f j

= C

Aj (S) , ≻ j

.

Observe that the (many) ‘optimal messages’ that a firm, say f j , might exhibit are obtained

by enlarging A j (S). It is done throughout the inclusion of some unavailable workers, i.e.

workers that have been previously assigned to another firm and/or individuals that declared

that f j is an unacceptable firm. Nevertheless, as we have already argued, all these messages

yield the same outcome, µ f j

= C

Aj (S) , ≻ j

. This is equivalent to say that we do loss

generality by assuming that, when a firm plays optimally, its message coincides with its

choice on A j (S). So in what follows, we maintain this assumption.

We succinctly describe now the main results in this Section. First, Example 3 shows that

pairwise unstable allocations might be supported by some SPNE. Then Proposition 1 establishes

each pairwise stable matching can be supported as a SPNE of this mechanism. Finally,

we address the analysis of the equilibria that are more expectable to hold. To develop this

study, we concentrate on equilibria where agents’ strategies are undominated. A first result

h

in this framework is that the order in which institutions are called to choose is determinant

to characterize which outcomes are more likely to hold. We then concentrate on a restricted

framework, namely the case where firms’ preferences satisfy responsiveness. We find that,

under such a consideration, the set of attainable equilibria drastically reduces: the only

outcome that is supported by agents’ interaction is the Workers-Optimal Stable matching.

Example 3 Consider the following 5-workers-2-firms problem, with W = {a, b, c, d, e} and

F = {1, 2}. Agents’ preferences are

≻ a := 2, 1 ≻ d := 1, 2 ≻ 1 =: {a, b, c} , {a, b} , {a, c} , {a} , {d, e}

≻ b := 1, 2 ≻ e := 1, 2 ≻ 2 =: {a, d, e} , {a, d} , {a, e} , {d, e} , {b, c}

≻ c := 1, 2

Assume that the strategy for each worker is S i = {1, 2}. If firm 1 is the first to select,

its election will be µ (1) = {a, b, c}. Then firm 2 will be matched to µ (2) = {d, e}. The

strategies above described constitute a SPNE. Note that firms act optimally, therefore we

just need to explore workers’ behavior. First, since b and c are engaged to their preferred

firm, they have no incentive to modify their strategy. Given the strategies for a, b and c

there is no room for agents d and e to improve their allocation. Therefore, the unique agent

that might have incentives to deviate is worker a. Let S ′ (a) an alternative strategy for this

worker. Note that if 1 ∈ S ′ (a), then a should be still matched to firm 1; if S ′ (a) = , then

a will be unmatched; i.e. µ ′ (a) = a. Finally, if S ′ (a) = {2} then matching µ ′ = ϕ

W S a , S ′ a

will be described as µ ′ (a) = a; µ ′ (b) = µ ′ (c) = 2; and µ ′ (d) = µ ′ (e) = 1. This shows that

strategy S constitutes a SPNE.

Nevertheless, matching µ = ϕ W (S) is pairwise unstable. Note that 2 ≻ a 1 = µ (a) and

a ∈ C ({a, d, e} , ≻ 2 ).

Therefore, this example points out that the employ of ϕ W might induce allocations failing

to be pairwise stable.

Proposition 1 Let P be a problem. Let µ be a pairwise stable matching for P. Then, there

is a SPNE, namely S, inducing messages profile M (S) with µ = ϕ W (M (S)).

Proof

Let P be a problem, and µ be a pairwise stable matching for P. Consider the following

strategies. Each w i ∈ W selects strategy S i = µ (w i ) ∩ F. 7 Then, each firm strategy determines

to select the best workers, according its preferences, among the available ones. Note

that, for each firm, say f j the set of its available workers is A j (S) = µ

f j .

Assume that the strategies profile S above is not an SPNE. Since firms are selecting

their optimal strategies, there should be a worker, say w i , and strategy S ′ for her such that

i

7 That is, S i = whenever µ (w i ) = w i .

12

µ ′ (w i ) ≻ i µ (w i ), with µ ′ = ϕ W S i , S ′ i

. Since µ is pairwise stable, and thus individually

rational, then there is f j ∈ F such that µ ′ (w i ) = f j . This implies that w i is available for f j

and w i ∈ C µ f j

∪ {wi } , ≻ j

, which contradicts that µ ∈ PC (P).

4.1 Undominated Strategies for ϕ W

Proposition 1 established that any pairwise stable outcome can be supported by a SPNE.

Nevertheless, such a conclusion might be not expected whenever the agents exhibit some

minimal rationality when selecting their strategies. The main argument is related to what

Lemma 3 below reports. Consider a worker, say w i . If her strategy S i does not include her

best firm as an acceptable one, then she is employing a dominated strategy. The next example

points out that, when the agents do not select dominated strategies, there are pairwise

stable allocations that should not be expected as the consequence of agents’ interactions.

Example 4 Consider the four-worker-two-firm problem with W = {a, b, c, d}, and F =

{1, 2}, with agents’ preferences

≻ a := 1, 2 ≻ 1 =: {c, d} , {a, b}

≻ b := 1, 2 ≻ 2 =: {a, b} , {a} , {b} , {c, d}

≻ c := 2, 1

≻ d := 2, 1

This problem has two pairwise stable matchings, namely µ and µ ′ , with

• µ (1) = {a, b} and µ (2) = {c, d}; and

• µ ′ (1) = {c, d} and µ ′ (2) = {a, b}.

Now, consider mechanism ϕ W (with firm 1 being the first to select its workers). According

the arguments above, that constitute the essence of Lemma 3 below, any strategy for the

workers in which these agents employ undominated strategies, say S W = (S a , S b , S c , S d ),

must verify that 1 ∈ S a ∩ S b and 2 ∈ S c ∩ S d .

Note that strategies profile S, where S W is such that S a = S b = {1} and S c = S d = {2}, and

firms select their optimal message for the decisions taken by their predecessors, constitutes

a SPNE yielding ϕ W (M (S)) = µ. Now, suppose that there is a SPNE, S ′ , yielding matching

µ ′ . Since µ ′ (1) = {c, d}, if workers c and d select undominated strategies, it must be the

case that S ′ = c S′ = {1, 2}. Moreover, since a and b are selecting undominated strategies,

d

1 ∈ S ′ (a) ∩ S ′ (b). Now, assume that d selects strategy S ′′ = {2}. In such a case, the best

d

selection for firm 1 yields message M 1 = {a, b}, and thus firm 2 selects workers c and d.

Since 2 ≻ d 1, S ′ does not constitute a SPNE. Therefore, when workers select undominated

strategies, the conclusions of Proposition 1 are not still valid.

13

Consider the variant of ϕ W where firm 2 is the first one to select which ‘available’ workers

it hires. It is easy to see that, in such a case, strategy profile S where S i = {1, 2} for each

worker w i and firms select optimally, constitutes a SPNE in which all the agents select undominated

strategies. These strategies yield matching µ ′ as the mechanism outcome. Note

that if worker a selects another strategy, say S ′ , the only possibility to modify the firm she is

a

engaged to is by declaring that 2 is a non-acceptable firm (i.e., 2 /∈ S ′ ). In such a case, given

a

others’ strategies, worker a will remain unmatched, and 2 = µ ′ (a) ≻ a a. Similar arguments

yield to ensure that worker b cannot profitably modify her strategy. Given that a and b are

selecting 2 as an acceptable firm, and 2 is the first firm to select, workers c and d must be

either engaged to firm 1 or remain unmatched. Therefore, neither c nor d can profitably

modify their strategy.

Notice that this example illustrates that

• when agents select undominated strategies, the set of matchings which are supported

by a SPNE might be smaller than that obtained without such a requirement; and

• the allocations that can be supported by some equilibrium might depend on the order

in which firms select the workers they hire.

To introduce the next result we need some additional notation. We say that x ∈ F ∪{w i }

is achievable by worker w i at mechanism ϕ W if there is a profile of strategies S such that

µ (w i ) = x, with µ = ϕ W (M (S)).

Lemma 3 Let P be a given problem, and let B (≻ i ) denote the best achievable firm -if anyfor

worker w i according her preferences ≻ i . Then, when applying mechanism ϕ W , strategy

S i is dominated for worker w i if either

(i) S i ≠ and B (≻ i ) = w i ; or

(ii) B (≻ i ) /∈ S i when w i ≠ B (≻ i ).

Proof

First, note that if w i = B (≻ i ), and S i ≠ then, there should be a firm f i ∈ F ∩ S i . Assume

that such a firm selects strategy S j = {w i }. This guarantees that at any strategy profile 8 S

where w i selects strategy S i and f j employs S j worker w i will be matched to some firm (and

thus will not be unmatched). Notice that, when w i selects strategy S ′ = , for any strategy

i

selected by the remaining agents, w i will be unmatched, that is preferred by her to stay with

any firm in F because w i = B (≻ i ).

Now, suppose that there is some f j ∈ F being the best achievable firm for w i . Related

to this worker, consider strategy S i such that f j /∈ S i , and denote S ′ = S

i i ∪

f j . For any

8 Note that, at this stage, we do not require that S constitutes a SPNE. Therefore f j might not play optimally.

14

strategy selected by the remaining agents, S −i , we denote µ = ϕ W (M (S −i , S i )) and µ ′ =

ϕ W M S −i , S i ′ . Note that, by construction, if µ (wi ) ≠ µ ′ (w i ) then µ ′ (w i ) = f i and, by

hypothesis, f j ≻ i µ (w i ). Consider strategies for the firms such that, independently from

the information their precedents have transmitted to them, w i ∈ S j and, for each f h ≠ f j ,

w i /∈ S h . Note that, for any such strategies we have that µ ′ (w i ) = f i ≻ i w i = µ (w i ).

Since we are interested in SPNE and, as previously argued, each firm exhibits a dominant

strategy, hereafter we restrict attention to strategy profiles in which each firm is selecting

her dominant strategy. Moreover, we are interested in stable, and thus individually rational

allocations. For this reason we assume, from now on, that any problem satisfies a ‘mutual

acceptability’ condition. To be precise, we concentrate on problems P such that for each

worker-firm pair

w i , f j , f j ≻ i w i if, and only if, there is a set of workers W ′ such that

w i ∈ C

W ′ ∪ {w i } , ≻ j . Note that we do not lose generality by invoking this property. In

particular, under such an hypothesis we have not to be worried about whether a worker

includes unacceptable firms 9 in her strategy or not. Because of the mutual acceptability

condition, and the fact that firms play optimally, we know that no worker will never be

selected by an unacceptable firm.

4.2 The WAFSm under Responsiveness

Example 4 above has pointed out that, when workers agree on which is the best stable

allocation, the selection of undominated strategies can be enough to guarantee that such a

matching is the unique to be supported as a SPNE. Unfortunately, this conclusion cannot be

considered as a general result. In particular, if no structure is assumed on firms’ preferences,

we cannot be sure about such a coincidence of interests from the workers’ perspective. This

is, for instance, the case illustrated in Example 1. Note that this example has two stable

matchings and workers’ interest conflict when asking which is the allocation they prefer.

Throughout this section we assume that firms’ preferences satisfy responsiveness. This

assumption allows to guarantee the existence of a stable allocation which is unambiguously

considered as the best allocation that all the workers can achieve, provided that it has to

satisfy stability.

What this section concludes is that under iterated elimination of dominated strategies,

the unique matching that is expected to hold is the Workers-Optimal Stable matching.

To introduce our results we need some additional notation. Imagine that each agent is

constrained to select one among a set of strategies, say S x , and let S ≡ (S x ) x∈W ∪F . B (S; ≻ i )

denotes the best allocation that worker w i can achieve -when ϕ W is applied- provided that

each agent x chooses strategies in S x .

9 As usual, firm f j is said to be unacceptable to worker w i whenever w i ≻ i f j . Otherwise, this firm is said

acceptable by w i .

15

Lemma 3 reported that any worker selecting an undominated strategy should declare, as

acceptable, her preferred firm. 10 Similarly, when exploring the optimal behavior by firms at

any SPNE we argued that her message should include the best set of workers that declared

it as an acceptable firm and have not been assigned to a precedent firm.

Let S 1 be the set of undominated strategies for agent x ∈ W ∪ F, as previously described.

x

Note that for each firm, say f j , all the strategies in S 1 induce the same outcome. Therefore,

j

even though S 1 might not be a singleton, all the strategies it contains can be considered as

j

equivalent. This implies that to analyze which strategies in S 1 =

S 1 are dominated

x x∈W ∪F

we just need to concentrate on strategies selected by the workers.

In what follows, we will illustrate how to obtain the set of undominated strategies for a

worker. We want to stress that Lemmata 4 and 5 do not require any assumption on firms

preferences.

Lemma 4 Let P be a given problem. For each worker w i such that B S 1 ; ≻ i

≠ wi strategy

S i ∈ S 1 i

is dominated whenever B S 1 ; ≻ i

/∈ Si .

Proof

First, note that B S 1 ; ≻ i

= wi is the consequence of some of the following facts.

(i) Remaining unmatched is the preferred alternative for w i . Accordingly the arguments

in footnote 10, in such a case w i has a dominant strategy, namely S i = . Therefore

S 1 i

= .

(ii) w i will not be hired by any firm. This is because for each firm f j such that f j ≻ i w i ,

there is a set of workers W ′ not containing w i satisfying that w i /∈ C W ′ ∪ {w i } ; ≻ j

and B (≻ h ) = f j for each w h ∈ W ′ .

Now, assume that B

S 1 ; ≻ i = f j ∈ F; and f j /∈ S i . Consider strategy S ′ = S i i∪

f j . Since f j is

the best achievable firm for w i , provided that the remaining agents are selecting strategies

in S 1 ≡

S 1 , we have that for each profile of strategies S ∈ S 1 , f

−i x x≠w j ≻ i µ (w i ), with

i

µ = ϕ W (M (S)). Moreover, when w i selects strategy S ′ we have that µ ′ (w

i i ) ∈ {µ (w i ) , f i },

with µ ′ = ϕ W M S −i , S i ′ . Therefore, the firm associated to wi when declaring S ′ is as least

i

as good as that obtained when selecting S i . Since f j = B

S 1 ; ≻ i we conclude the proof.

The idea underlying Lemma 4 can be replicated in an iterative way. I.e., we can describe

S 2 =

S 2 ⊆ x x∈W ∪F S1 as the set of strategies profiles in which no agent x selects dominated

strategies, provided that the remaining agents are selecting strategies in S 1 . And so forth.

−x

As a general statement, for t > 1, let S t =

S t denote the set of strategies profiles

x

x∈W ∪F

10 Except if her preferred alternative is to remaining unmatched. Recall that in such a case, as Lemma 3

establishes, it is a dominant strategy for this worker not to select any firm. I.e., for such a worker, say w i ,

S i ∈ S 1 i

if, and only if, S i = .

16

in which no agent x selects dominated strategies, provided that her rivals are selecting

strategies in S t−1 . In such a case, an extension of Lemma 4 can be established, whose proof

−x

is omitted because it is similar to that of Lemma 4.

Lemma 5 Let P be a given a problem. For each worker w i such that B (S t ; ≻ i ) ≠ w i strategy

S i ∈ S t i

is dominated whenever B (S t ; ≻ i ) /∈ S i .

Recall that, under responsiveness, associated to each problem there is one stable matching

µ W O being (weakly) preferred by all the workers to any other stable matching. Note

that this process of iterative elimination of dominated strategies yields to describe the set

of strategies profiles in which each worker, say w i declares acceptable all the firms being as

least as desirable as µ W O (w i ). 11 This allows to establish that there is some T, in which the

process specified in Lemma 5 does not allow to eliminate more strategies, i.e. S T = S T+1 .

In such a case, S T =

S T is characterized by

x

x∈W ∪F

(i) each firm f j has a dominant strategy S j , which consists in selecting its best group of

available workers;

(ii) each worker w h such that µ W O (w h ) = w h selects strategy

S h = f j ∈ F : f j ≻ h w i

; and

(iii) for each worker w i such that µ W O (w i ) ≠ w i , S i ∈ S T i

if, and only if

(a) µ W O (w i ) ∈ S i and, for each f j ≻ i µ W O (w i ), f j ∈ S i ; 12 and

(b) for each f j ∈ F such that w i ≻ i f j , f j /∈ S i . 13

We now proceed to see that, under responsiveness, the unique outcome of WAFSm being

supported by a sophisticated equilibrium is the Workers-Optimal Stable allocation, µ W O .

To simplify the exposition, for a given problem P satisfying responsiveness, and worker

w i , let S i be the set of firms being at least as preferred as µ W O (w i ), i.e.

S i = F \ f j ∈ F : µ W O (w i ) ≻ i f i

.

Notice that any strategies profile S = (S x ) x∈W ∪F in S T verifies that for each worker, say w i ,

S i ⊇ S i .

The next result is useful to prove Theorem 2 below. It establish the stability of each

outcome supported by a SPNE for the WAFSm.

11 Recall the mutual acceptability assumption made after Lemma 3.

12 See footnote 11.

13 This condition comes from Lemma 3.

17

Proposition 2 Let P be a responsive problem, and S a SPNE for WAFSm.

ϕ W (M (S)) ∈ C (P).

Then

Proof

Let S be a SPNE for WAFSm, and µ = ϕ F (M (S)). Assume that µ is not stable. Since P is

responsive µ neither is pairwise-stable. Since S is a SPNE, µ must be individually rational.

This implies that there is a worker-firm pair, say w i , f j

such that (a) f j ≻ i µ (w i ) and (b)

w i ∈ C µ f j

∪ {wi } , ≻ j

. Assume that wi selects strategy S ′ i = {f h ∈ F : f h ≻ i f i }∪{f i }, and

denote µ ′ = ϕ F M S i , S ′ i

. Consider the following cases, that exhaust all the possibilities:

(i) µ (w i ) = w i . This implies that no firm selects worker w i under the initial strategies S.

Now, when w i selects S ′ , there are two options,

i

(1) w i is available for f j . Therefore, no firm preceding f j has selected w i . This implies

that, for each j ′ < j, µ f j ′

= µ

′ f j ′

. Since wi ∈ C µ f j

∪ {wi } , ≻ j

, it must

be the case that µ ′ (w i ) = f j . This contradicts that S is a SPNE.

(2) w i is not available for f j . This implies that w i has been selected by some f j ′, with

j ′ < j; i.e., µ ′ (w i ) ∈ S ′ i \ f j

. Thus µ ′ (w i ) ≻ i f j . Since f j ≻ i µ (w i ), transitivity

implies that µ ′ (w i ) ≻ i µ (w i ), which contradicts that S were a SPNE.

(ii) µ (w i ) = f h , with h > j. Note that since w i ∈ C µ

f j ∪ {wi } , ≻ j , it must be the case

that f j /∈ S i . Therefore, the arguments provided in case (i) above can be replicated

here to see that S cannot be a SPNE for WAFSm.

(iii) µ (w i ) = f h , with h < j. Since f j ≻ i f h , when applying the WAFSm for messages

M

S −i , S i ′ , wi /∈ A h S−i , S i ′ . This implies that

µ ′ (w i ) ≠ f h . Note that, by the description

of WAFSm, for each k, h < k < j, A k S−i , S i ′ \ {wi } ⊆ A k (S) \ {w i }. Moreover,

since (a) f j ∈ S ′; (b) w i i ∈ C µ

f j ∪ {wi } , ≻ j ; and (c) ≻ j satisfies responsibility,

either w i ∈ µ

′ f j = C Aj S−i , S i ′ , ≻ j - contradicting that S is a SPNE -, or

w i /∈ A j S−i , S i ′ . Note that, in the latter case, if wi is not available for f j is because she

has been attached to a preceding firm. But all the firms that w i declares as acceptable

are preferred by this worker to f h . A contradiction.

Theorem 2 reports that, under responsiveness, WAFSm has, at least, one sophisticated

equilibrium. Moreover, each such an equilibrium yields, as outcome, the Workers-Optimal

Stable matching.

Theorem 2 Let P be a responsive problem. Then, WAFSm implements in sophisticated

equilibria its Workers-Optimal Stable matching.

18

Proof

First, note that the profile of strategies S ∗ where each firm select its dominant strategy and

for each worker w i , S ∗ i

= S i constitutes a sophisticated equilibrium yielding messages M (S ∗ )

such that ϕ F (M (S ∗ )) = µ W O .

Now, consider a sophisticated equilibrium S yielding messages M (S) such that

ϕ F (M (S)) = µ ≠ µ W O . Note that, since S is a sophisticated equilibrium, each worker’s

strategy S i must satisfy that S i ⊇ S i . Moreover, by Proposition 2, µ has to be a stable

matching for P. This implies that for each worker w i either (a) µ (w i ) = µ W O (w i ) or (b)

µ W O (w i ) ≻ i µ (w i ), where condition (b) is fulfilled by, at least, one worker. It is also wellknown

that for each w i ∈ W , µ W O (w i ) ∈ F if, and only if, µ (w i ) ∈ F (see, v.g., Theorems

2.22 and 5.12 in Roth and Sotomayor, 1990), and for each w i satisfying condition (b) above,

µ f j

≻ j µ W O f j

for f j = µ W O (w i ) ∈ F.

Let f j1

be the firm such that µ W O f j1

≠ µ

f

j1

and, for each j ′ < j 1 , µ W O f j ′

= µ

f j ′

;

and let w i1

be a worker in µ f j1

\ µ

W O f j1

. Since µ

W O f j1

≻i µ f j1

, S i1

S i1

. Now,

consider the profile of strategies S ′ = S −i1 , S i1

, and let µ ′ = ϕ F (M (S ′ )).

Note that, since S is a sophisticated equilibrium, µ ′ w i1

= wi1 . Otherwise, µ ′ w i1

i1

µ W O w i1

≻i1 µ w i1

, which contradicts that S is a sophisticated equilibrium.

Observe that for each j ′ < j 1 , µ ′ f j ′

= µ

f j ′

= µ

W O f j ′

. Moreover, for firm f

j1

, since

µ W O f j1

⊆ Aj1 (S ′ ), and ≻ j1

satisfies substitutability, for each w i ′ ∈ µ ′ f j1

\ µ

W O f j1

and

w i ′′ ∈ µ W O f j1

\ µ

′ f j1

, wi ′ ≻ j1

w i ′′. For each firm succeeding f j1

, say f j ′′, j ′′ > j 1 , the

following holds.

(a) If for each k < j ′′ , µ ′ (f k ) ∩ µ f j ′′

= , then µ

′ f j ′′

= µ

f j ′′

.

(b) There is, at most, one worker, say w i ′ being in µ f j ′′

but not in µ

′ f j ′′

. Moreover, if

such a worker exists, then there is k < j ′′ such that µ ′ (w i ′) = f k .

Let consider again matching µ ′ . Since µ ′ w i1 = wi1 , there should be a firm, f j ′, with

j ′ > j 1 , such that either µ

′

f j ′ < q j ′ or there is some worker, say w i ′ ∈ µ ′ f j ′

such that

µ W O (w i ′) = w i ′. Note that, it implies that there is a worker w i2

∈ µ W O f j ′ \ µ

′

f j ′

such

that w i2

∈ C µ ′ f j ′

∪ wi2 , ≻ j ′

.

Given f j ′ and w i2

as described above, consider now strategy profile S ′′ =

S −i2 , S i2 , and

let µ ′′ = ϕ F (M (S ′′ )). Note that there should be j 2 < j ′ such that µ

w i2 = f j2

, and thus

f j ′ ≻ i2

f j2

. This implies that µ ′′ w i2 ≠ f j2

. Consider the following two cases, that exhausts

all the possibilities,

(i) j 2 = j 1 . Note that in this case, µ ′′ f j1

\ µ

f

j1

= µ

′ f j1

\ µ

f

j1

. This implies that for

each j, j 1 < j < j ′ , A j (S ′ ) = A j (S ′′ ), and thus f j ′ = µ ′′ w i2

= µ

W O w i2

≻i2 µ w i2

,

contradicting that S was a sophisticated equilibrium.

(ii) j 1 < j 2 < j ′ . Since S is a sophisticated equilibrium, it must be the case that µ ′′ w i2

=

w i2

.

19

Note that we can argue again, related to f j2

in a similar way as we did in relation with f j1

.

That means that an iterative argument over the (finite) set of firms will necessarily yield to

the existence of a worker, say w ih

such that µ W O w ih

≻ih µ w ih

and, µ

h w ih

= µ

W O w ih

,

with µ h = ϕ F M S −ih , S ih

, contradicting that S is a sophisticated equilibrium.

5 Conclusions

This paper inquiries on the survival of some matching mechanisms selecting, when agents’

reports are sincere, unstable allocations. These mechanisms exhibit a natural sequentiality:

agents on one side declare which mates are admissible, and that belonging to the other side

select their mates, observing such a restriction.

We found that the crucial reason justifying the persistence of these mechanisms is that

the sequentiality of agents’ declarations is enough to guarantee that the outcome can be

seen as the result of a collective co-ordination. This is because any expected allocation is

stable. However, this coordination needs not to be formalized.

We further explore whether the order in which this sequentiality is described determines

which allocations are more likely to occur. Our analysis points out an asymmetry between

the two sides of the market. When institutions determine acceptability, any stable allocation

is equal likely. On the contrary, when agents decide the eligibility, an extra veiled coordination

occurs: The unique allocation that should be expected is the optimal stable from the

agents’ point of view.

References

Abdulkadiroğlu, A., Pathak, P.A., Roth, A.E., Sönmez, T., 2005. The Boston Public School

Match. American Economic Review, Papers and Proceedings 95, 368 – 371.

Alcalde, J., 1996. Implementation of Stable Solutions to Marriage Problems. Journal of

Economic Theory 69, 240 – 254.

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Alcalde, J., Romero-Medina, A., 2000. Simple Mechanisms to Implement the Core of College

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20

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Roth, A.E., 1985. The College Admissions Problem is not Equivalent to the Marriage Problem.

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21

PUBLISHED ISSUES

WP1101

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WP1103

’Fair School Placement’

J. Alcalde, A. Romero-Medina.

’Does Stock Return Predictability Affect ESO Fair Value’

J. Carmona, A. Leon, A. Vaello-Sebastia.

’Competition for Procurement Shares’

J. Alcalde, M. Dahm.

WP1201

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WP1207

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’The Minimal Overlap Rule: Restrictions on Mergers for Creditors’ Consensus’

J. Alcalde, M. Marco-Gil, J.A. Silva.

’Mediation in Bankruptcy Problems’

J. Pe**ris**, J.M. Jimenez-Gomez.

’Affirmative Action and School Choice’

J. Alcalde, B. Subiza

’M-stability: A Reformulation of Von Neumann-Morgenstern Stability’

J. Pe**ris**, B. Subiza

’Stability Versus Rationality in Choice Functions’

B. Subiza, J. Pe**ris**.

’Stable Sets: A Descriptive and Comparative Analysis’

J. Pe**ris**, B. Subiza.

’A Proportional Approach to Bankruptcy Problems with a Guaranteed Minimum.’

J. Pe**ris**, J.M. Jimenez-Gomez.

’Solidarity and Uniform Rules in Bankruptcy Problems.’

J. Pe**ris**, J.M. Jimenez-Gomez.

’The Cooperative Endorsement of a Strategic Game.’

P. Hernández, J.A. Silva-Reus.

’Strategic Sharing of a Costly Network.’

P. Hernández, J. E. Pe**ris**, J.A. Silva-Reus.

’Tax Burden Degree as a Tool to Design Tax Systems.’

J. Alcalde, M.C. Marco-Gil, J.A. Silva-Reus.

’Fair Bounds Based Solidarity.’

J. Pe**ris**, J.M. Jimenez-Gomez.

’A Concessions-Based Mechanism for Meta-Bargaining Problems.’

M.C. Marco-Gil, J. Pe**ris**, Begoña Subiza.

WP1214

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’Neoclassical Growth and the Natural Resource Curse Puzzle.’

María Dolores Guilló, Fidel Pérez-Sebastián.

’Perron’s Eigenvector for Matrices in Distribution Problems’

B. Subiza, J.A. Silva, J. Pe**ris**.

’Executive Stock Options and Time Diversification’

J. Carmona, A. Leon, A. Vaello-Sebastia.

’Technology Diffusion and its Effects on Social Inequalities’

M. Magalhaes, C. Hellström.

WP1301

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WP1305

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’A Pareto Eficient Solution for General Exchange Markets with

Indivisible Goods when Indifferences Are Allowed’

B. Subiza, J. Pe**ris**.

’From Bargaining Solutions to Claims Rules: A Proportional Approach’

José-Manuel Giménez-Gómez, Antonio Osorio, Josep E. Pe**ris**.

’Allocating via Priorities’

José Alcalde, J.A. Silva-Reus.

’Relative Injustice Aversion’

Luis José Blas Moreno-Garrido.

’Random Housing with Existing Tenants’

José Alcalde.

’Cost sharing solutions defined by non-negative eigenvectors’

B. Subiza, J.A. Silva, J. Pe**ris**.

WP1401

WP1402

WP1403

WP1404

WP1405

’Strategy-Proof Fair School Placement’

José Alcalde, Antonio Medina-Romero.

’Compromise Solutions for Bankruptcy Situations: A Note’

José-Manuel Giménez-Gómez, Antonio Osorio and Josep E. Pe**ris**.

’Conflicting Claims Problem Associated with Cost Sharing of a Network’

José-Manuel Giménez-Gómez, Begoña Subiza and Josep E. Pe**ris**.

’Heterogeneity, Endogeneity, Measurement Error and Identification of the

Union Wage Impact’

Georgios Marios Chrysanthou.

’A Consensual Committee Using Approval Balloting’

Begoña Subiza and Josep E. Pe**ris**.

WP1501

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’Algunas Propuestas para la Reforma del Sistema de Financiación de las

Comunidades Autónomas de Régimen Común.’

Ángel de la Fuente.

’La Financiación Autonómica: Una Historia Interminable.’

Marta Espasa.

’Lower Partial Moments under Gram Charlier Distribution: Performance

Measures and Efficient Frontiers.’

Ángel León and Manuel Moreno.

’Convergence in a Dynamic Heckscher-Ohlin Model with Land.’

María Dolores Guilló and Fidel Pérez-Sebastián.

’Rationalizable Choice and Standards of Behavior.’

Josep E. Pe**ris** and Begoña Subiza.

’Cognitive (Ir)reflection: New Experimental Evidence.’

Carlos Cueva, Iñigo Iturbe-Ormaetxe, Esther Mata-Pérez, Giovanni Ponti,

Marcello Sartarelli, Haihan Yu and Vita Zhukova.

’Folk Solution for Simple Minimum Cost Spanning Tree Problems.’

Begoña Subiza, José-Manuel Giménez-Gómez and Josep E. Pe**ris**.

WP1601

WP1602

’Dual Sourcing with Price Discovery.’

José Alcalde and Matthias Dahm.

’(In)visible Hands in Matching Markets.’

José Alcalde.