This paper considers a realistic family of admission mechanisms, with multiple applications and application costs. Multiple applications impose serious coordination problems to colleges, but application costs restore stability. Without application costs and under incomplete information unstable allocations emerge.Application Costs; Matching Markets; Implementation.

Matteo Triossi

English

Research Papers in Economics

Application Costs in Sequential 
Admission Mechanisms
Matteo Triossi
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No.23, September 2006
www.carloalberto.orgApplication Costs in Sequential Admission Mechanisms1
Matteo Triossi
Collegio Carlo Alberto2
September 20063
1This paper is is the ￿fth chapter of my Ph.D. dissertation at Universidad Carlos III. Many thanks
to Antonio Romero-Medina, Luis Corh￿n, Jordi Mass￿, Maria Angeles de Frutos and Matthias Dahm
for very useful comments.
2Mailing address: Collegio Carlo Alberto, Via Real Collegio 30, 10024 Moncalieri (TO).
E-mail: matteo.triossi.@collegiocarloalberto.it, webpage: http://www.carloalberto.org/fellows/triossi
3 c   2006 by Matteo Triossi. Any opinions expressed here are those of the author and not those
of the Collegio Carlo Alberto.Abstract
This paper considers a family of admission mechanisms, with multiple applications and
application costs. Multiple applications impose serious coordination problems to colleges,
but application costs restore stability. Without application costs and under incomplete
information unstable allocations emerge.
JEL Classi￿cation: C78; D78
Keywords: Application Costs; Matching Markets; Implementation.1 Introduction
This paper analyzes a class of admission mechanisms: each student sends costly
applications to some colleges, then each college selects the applicants to accept,
￿nally each accepted student selects the college to join among the ones that ac-
cepted her. This procedure resembles many real world mechanisms, for instance
the admissions procedures to Graduate Schools and decentralized job markets.
The mechanism extends the Students-Propose-and-Colleges-Choose Mecha-
nism presented in Alcalde and Romero-Medina (2000), where application costs
were zero and each student was allowed to a unique application. Such a mech-
anism implements the stable correspondence in Subgame Perfect Equilibrium
(SPE). But in the real world applicants are rarely restricted to a unique appli-
cation and often application fees or information gathering costs are present.
With multiple applications each college selects a group of applicants and,
at the same time, proposes them like in the Colleges-Propose-and-Students-
Choose-Mechanism in Alcalde and Romero-Medina (2000). A problem of coor-
dination among colleges emerges. Regardless of the restriction on applications,
the set of equilibrium outcomes contains the stable set (Proposition 2) and with
positive application costs, they coincide (Theorem 1). If application costs are
zero or if information is incomplete (Examples 1 and 2, respectively) the coor-
dination problems among colleges induce unstable allocations. Future research
should clarify if the introduction of additional stages in which new applications
and o⁄ers are done might help to eliminate unstable allocations (see also So-
tomayor (2003) and Alcalde and Romero-Medina (2005)) and favor information
release.
The paper is organized as follows. Section 2 introduces the model and Section
3 contains the main results.
2 The Model
A bilateral matching market is represented by a triplet ():  = f1g
is the set of colleges,  = f1g is the set of students,  \  = ?,
 = (11) is the vector of agents￿preferences. Let  2 .
 represents college ￿ s preferences, a strict order on 2. Let 0 ￿ . The
choice set of  from 0 is (0) = argmax f00 : 00 ￿ 0g is the fa-
vorite group of students for college  among the ones belonging to 0. Any
student  such that ? is unacceptable to . Otherwise  is acceptable
to . The set of ￿ s acceptable students is denoted by (), ￿ s quota is
 = maxf0 : (0) 6= ?g. Let  2 .  denotes student ￿ s prefer-
ences, a strict order on  [ fg. Any  such that  is unacceptable to .
Otherwise  is acceptable to . Weak preferences are denoted by . For each
 2 ,  denotes a function,  :  [ fg ! R representing .
De￿nition 1 A matching on () is a function ￿ :  [  ! 2 [ , such
that, for every () 2  ￿ :
(i) ￿() 2 2.
1(ii) ￿() 2  [ fg.
(iii) ￿() =  ,  2 ￿().
De￿nition 2 ￿ is individually rational for  2  if ￿().
￿ is individually rational for  2  if ?, for all  2 ￿().
De￿nition 3 ￿ is blocked by a pair () 2  ￿  if:
(i) ￿().
(ii)  2 (￿() [ fgg.
A matching ￿ is unstable if there are a college  and a student  who are not
paired together but: (i)  would prefer to join  rather than her mate under ￿,
(ii)  would accept  among its students if it was given to choose its students
among the ones in ￿() [ fg.
De￿nition 4 ￿ is stable in market () if it is individually rational for
all  2  [  and if no pair blocks it. Otherwise ￿ is unstable.
The set of matchings that are stable in market () is the stable set,
denoted by ¬().
Two properties on all colleges￿preferences are imposed: substitutability and
separability. Substitutability assures the non-emptiness of the stable set. A
college￿ s preferences are substitutable if it wants to enroll a student even when
other students become unavailable.
De￿nition 5 Let  2 .  are said to be substitutable if, for each  2 2
and for all 0 2 ,  6= 0:
 2 () )  2 ( ¬ f0g).
Preferences are quota  separable when adding additional acceptable stu-
dents makes any given set of students of less than  elements a better one.
De￿nition 6 Let  2  and let   0 be a natural number.  are quota
-separable if for all 0 ￿ 
0     2 0,  2 () () (0 [ fg)0.
0   ) ?0.
This assumption, weaker than responsiveness, implies that the set of un-
matched students is the same in all stable matchings (Martinez and al (2000)),
a property used in the proof of Lemma 1.
The paper analyzes implementation in SPE. Let ￿ be a class of matching
markets and let  be a correspondence on the set of matchings on (). A
mechanism implements  in SPE if the set of SPE outcomes coincides with
the allocations prescribed by , for each () 2 ￿.
22.1 The Admission Mechanism
For all , let  representing the maximum number of colleges student  is
allowed to apply to. Let ￿ ￿ 0 be the cost that each student pays to apply to
each college. Application costs are assumed to be small: if ()  () then
() ¬ ￿  ()1.
The Sequential Admission Mechanism (SAM) with restriction  =
(1) is described by the following procedure
Stage 1: Application.  sends applications to at most  colleges.
Let 1(), 1() ￿  be the set of colleges  applies to. Let
1() =
S
21() fg the set of students who applied to  2 .
Stage 2: Acceptation.  accepts a subset of students, 2() ￿
1(). For each student  let 2() =
S
22() fg be the set of
colleges that accepted .
Stage 3: Matching. Student  decides which college to join among
the ones in 2().
Let ￿ be the matching resulting from such procedure. The payo⁄ of student
, is (￿()) ¬ ￿1().
Let 2 be the set of subgames starting at the second stage. Each 2 2 2
is characterized by the family of sets of students who applied to each college
f1(2)g2, or equivalently by the family of sets of colleges each student
applied to, f1(2)g2.
(2) denotes the following pro￿le of preferences:
(i) for each  2 : (2) = .
(ii) for each  2 : if   2 1(2) or if  then (2) If
0 2 1(2) then (2)0 i⁄ 0.
(2) coincide with  but for one aspect: each student ranks as unacceptable
all the colleges she did not apply to.
Let  = (1) be a restriction on the number of applications,  is assumed
to be public knowledge.
3 The Results
The ￿rst result characterizes the outcomes of the second stage subgames.
Lemma 1 For all 2 2 2, 2 implements ¬((2)) in SPE.
Proof. Let ￿ be a SPE outcome of 2. It is easily seen that ￿ is individually
rational for colleges and for students. Let () be a college-student pair. If ()
1If costs the same are higher the results of the paper apply to the market where, 8  2 ,
every college with () ¬ ￿  () is eliminated from ￿ s list of acceptable colleges.
3blocks ￿ let  accepting (￿()[fg). Such deviation would be pro￿table to
: at the last stage  accepts the best o⁄er she holds at each SPE.
Let ￿ 2 ¬((2)), then ￿ is a SPE outcome of 2. Consider the following
strategy for college : accept only the applicants in ￿(). Let students selecting
their best available college at the third stage. The stability of ￿ implies that no
college can pro￿tably deviate so ￿ is a  outcome of 2.
The result implies that the Colleges-Propose-And-Students-Choose-Mechanism
implements the stable set in SPE (Theorem 4.1 in Alcalde and Romero-Medina
(2000)).
The colleges become somehow ￿irrelevant￿ in the game. Indeed, to ana-
lyze the SAM it is su¢ cient to analyze the associated Reduced Admission
Mechanism (RAM) with restriction  denoted by . Here, only stu-
dents play and the outcomes are determined according their optimal stable
allocations. Let  be the set of the players. Each student ￿ s message space is

 = f0 ￿  : 0 ￿ g and the outcome function,  is de￿ned as follows.
Let  = (1) 2  =
 Q
=1

 . Let 2 = 2() be the second stage
subgame of the SAM induced by each student  applying to the colleges in .
Finally, set () = ￿
2, where ￿
2 is the students optimal stable matching of
((2)). The payo⁄ for player  is (()()) ¬ ￿.
Proposition 1 (i) If ￿￿ is the outcome matching of the SAM with restriction
 and ￿
2 2 2 is the second stage subgame on the equilibrium path, then ￿￿ is
the students￿optimal stable matching of ((￿
2)).
(ii) ￿￿ is a SPE outcome matching of the SAM with restriction  = (1)
if and only if it is a  outcome of 
Proof. (i) Let ￿ be the students￿optimal stable matching of ((￿
2)).
From Lemma 1 all students prefer ￿ to ￿￿. If ￿()￿￿() for some , con-
sider the following deviation:  applies only to ￿(). Let 2 be the second
stage subgame induced by such deviation. ￿ 2 ¬((2)), ￿() is the
unique ￿ s stable partner. From Lemma 1 the outcome of this deviation be-
longs to ¬((2)). Since colleges￿preferences are substitutable and quota
-separable,  is never unmatched in ¬((2)):  would reduce application
costs then the deviation would be pro￿table to her.
(ii) It is su¢ cient to prove that (￿
1(1)￿
1()) are ￿rst stage strategies
forming part of a  of the sequential game if and only if they constitute a
 of the RAM and that both equilibria yield the same outcome matching.
Let ￿￿ be a  outcome and let (￿
1(1)￿
1()) be a ￿rst stage 
strategy leading to ￿￿. From (i) it follows that it must be a  strategy for
the RAM. Now, let (￿
1(1)￿
1()) be a  of the RAM and let ￿￿ be
its outcome. Consider the following pro￿le of strategies for the players of the
sequential mechanism. At the ￿rst stage each student  applies to ￿
1(). At
the second stage, for each subgame 2, each college  accepts only the students
in ￿
2(), the students￿optimal stable matching of ((2)). At the third
stage students conform to  strategy. It is easily veri￿ed that such strategy
pro￿le constitute a SPE of the SAM yielding ￿￿ as outcome.
4First, a weak implementation result is proven.
Lemma 2 Let ￿ ￿ 0, and let ￿ be a stable matching. Then there exists a SPE
of the SAM yielding ￿ as outcome.
Proof. Consider (￿(1)￿()) as strategy pro￿le for the RAM. From the
stability of ￿ it follows that such strategies are a Nash Equilibrium of the RAM.
The claim follows from Proposition 1.
If  = (11), Lemma 2 implies that the SAM implements the stable set
in SPE, even when ￿ = 0, recovering the main result of Alcalde and Romero-
Medina (2000). When costs are positive the mechanism implements the stable
set in  independently on the restrictions on the number of applications.
Theorem 1 Let ￿  0. The SAM implements the stable set in .
Proof. At equilibrium each student applies to at most one college. Let
￿ = (￿
1(1)￿
1()) be a  strategy of the RAM and let ￿￿ be the
outcome matching of ￿. Let ￿
2 be the subgame induced by such strategy as
￿rst stage strategy of the SAM, then ￿￿ = ￿
2. If ￿￿() =  then ￿
1() = ?
otherwise  could save application costs by not applying to any college. If
￿￿() =  2  and ￿
1()  1, let 0
2 be the subgame obtained by the following
deviation 1() = fg. Then ￿￿ = ￿
2 = ￿
0
2. Such deviation is pro￿table to 
because she is enrolled by the same college and saves strictly positive costs, a
contradiction. The other part of the claim follows from Lemma 2.
Proposition 1 results helpful to prove that the SAM implements unstable
allocations if costs are zero2.
Example 1 Let ￿ = 0,  =  = 3  ￿ (212).
Let  = f123g and let  = f123g and set
1 = 123. 1 = 213.
2 = 312. 2 = 123.
3 = 123. 3 = 123.
Then 1 = 2 = 3 = 1, ¬() = f￿g, where :
￿ =
1 2 3
1 3 2
Let  ￿ (212). ￿ is a NE outcome of the RAM. Consider the following strategy
pro￿le of the RAM:
(1) = f12g, (2) = f3g, (3) = f12g.
It results in ￿
￿ =
1 2 3
3 1 2
￿ is blocked by (12). At ￿, 1 and 3 are matched with their ￿rst choices, so
they cannot pro￿tably deviate. Proposition 1, (i) implies that by including 3 in
2A longer direct proof is available upon request.
5her application 2 ends matched to 3 like in equilibrium, otherwise ends single,
so the proposed strategies are a NE of the RAM. Then the SAM implements
unstable allocation, too.
Incomplete information undermines the result, too.
Example 2 Let students￿preferences be public known and coinciding with the
ones de￿ned in the proof of Proposition 1. Let colleges preferences be the fol-
lowing with probability 1/2: 1 = 1, 2 = 3, 3 = 2. Assume that they are
like in the proof of Proposition 1 with probability 1/2. If ￿ is small enough there
exists a sequential equilibrium of the SAM with restriction in which the students
apply like in Proposition 1, and in which each college makes o⁄er to ￿
2. The
outcome is then unstable with probability 1/2.
4 References
Alcalde J. and A. Romero-Medina , 2000. Simple mechanisms to implement
the core of college admissions problems. Games and Economic Behavior 31,
294-302.
Alcalde J. and A. Romero-Medina , 2005. Sequential Decisions in the College
Admissions Problem. Economics Letters, 86,153-158.
Martinez R., Mass￿ J., Neme A. and J. Oviedo, 2000. Single Agents and the
Set of Many-to-one Stable Matchings. Journal of Economic Theory, 91, 91-105.
Sotomayor M., 2003. Reaching the core of the marriage market through a
non-revelation matching mechanism. International Journal of Game Theory 32,
241-251.
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