Pseudoma ke s wi h P io i ies
in La ge Random Assignmen Economies
An onio Mi alles
Uni e si a Au onoma de Ba celona
Ba celona G adua e School o Economics
and Eu opean Uni e si y Ins i u e
Fi s e sion, May 2010. This e sion, Ma ch 2011.
Abs ac
I s udy la ge andom assignmen economies wi h a con inuum o agen s and a …ni e
numbe o objec ypes. I conside he exis ence o weak p io i ies disc imina ing among
agen s wi h espec o hei igh s conce ning he …nal assignmen . The espec o
p io i ies ex an e (ex-an e s abili y) usually p ecludes ex-an e en y- eeness. The e o e
I de…ne a new concep o ai ness, called no unjus i…ed lowe chances: p io i ies wi h
espec o one objec ype canno jus i y di¤e en achie able chances ega ding ano he
objec ype. This concep , which applies o he assignmen mechanism a he han o
he assignmen i sel , implies ex-an e en y- eeness among agen s o he same p io i y
ype. I p opose a a ia ion o Hylland and Zeckhause ’s (1979) pseudoma ke ha mee s
ex-an e s abili y, no unjus i…ed lowe chances and ex-an e e¢ ciency among agen s o he
same p io i y ype. Assuming enough ichness in p e e ences and p io i ies, he con e se
is also ue: any andom assignmen wi h hese p ope ies could be achie ed h ough
an equilib ium in a pseudoma ke wi h p io i ies. I p io i ies a e acyclical ( he o de ing
an[email p o ec ed]. I am g a e ul o Yeon-Koo Che, Pie o Go a di, Jo di Massó, Massimo
Mo elli and Fe nando Vega-Redondo o p e ious commen s and sugges ions. Fo he same eason,
I would also like o hank he a endan s a semina s a he Eu opean Uni e si y Ins i u e, he Uni-
e si y o Ba celona and he Uni e si y o Siena. I also acknowledge he …nancial suppo p o ided
by he Spanish Minis y o Science and Inno a ion (ECO2009-06946 and ECO2008-04756).
1
o agen s is he same o each objec ype), his pseudoma ke achie es ex-an e e¢ cien
andom assignmen s.
Keywo ds: Random Assignmen ; Fai ness; S abili y; School Choice.
1 In oduc ion
In 1979, Hylland and Zeckhause p oposed a pseudoma ke mechanism ha sol ed he ex-an e
e¢ cien andom assignmen p oblem. This esul could be compa ible wi h ex-an e en y- eeness
should he agen s ace iden ical budge s. F om Thomson and Zhou (1993) i was unde s ood ha
any ex-an e e¢ cien and en y- ee andom assignmen could be ob ained h ough a pseudoma ke
wi h iden ical budge s in la ge economies (con innum o agen s). This pape cons i u es an ex ension
o pseudoma ke s o en i onmen s whe e p io i ies, o agen s’ igh s conce ning he …nal assignmen ,
exis ha ha e o be espec ed.
Conce ns abou jus ice and e¢ ciency a ise in many assignmen p oblems. Examples include he
assignmen o child en o public schools, s uden s o college esidences, pa ien s o public heal h
ca e se ices, e c. In his pape , I conside he ac ha he exis ence o p io i ies in many assign-
men p oblems p ecludes s anda d ai ness deside a a such as he absence o ex-an e en y (i.e. no
agen should p e e any o he agen ’s assignmen p obabili ies o he own) om being achie able.
The e o e I cons uc a weake no ion o ai ness ha is compa ible wi h he espec o p io i ies.
I is based on he idea ha p io i ies wi h espec o one objec ype canno jus i y di¤e ences in
chances wi h espec o ano he objec ype. I call i no unjus i…ed lowe chances. I p opose a a i-
a ion o he pseudoma ke mechanism à la Hylland and Zeckhause (1979) ha espec s p io i ies
while mee ing he new ai ness condi ion. Unde acyclical p io i y s uc u es, ha is, when agen s
a e equally p io i y-o de ed wi h espec o e e y objec ype, his pseudoma ke achie es ex-an e
e¢ cien andom assignmen s.
The p esence o p io i ies is a ecu en ea u e in assignmen p oblems. An agen has p io i y
o e ano he agen wi h espec o an objec ype when a uni o his objec ype mus be gi en
o he o me a he han he la e i bo h agen s claim o he same uni . Examples o p io i ies
a e hose gi en by he p esence o a sibling a and li ing a walking dis ance om he school in
child en- o-school assignmen p oblems, senio i y and exis ing enan s in esidence assignmen and
medical eme gency in heal h ca e se ices. Respec o p io i ies is unde s ood as he s abili y o
any ex-pos (o …nal) assignmen . The e m s abili y, om he ma iage ma ke li e a u e (Gale and
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Shapley, 1962), indica es ha an agen wi h p io i y o e ano he agen o some objec ype mus
ob ain a uni o ha objec ype o a p e e ed one in he case ha he la e agen ob ains a uni
o ha objec . Since p io i ies could be ega ded as p i ileges wi h ega d o he …nal assignmen ,
ex-an e en y- eeness canno be gua an eed.
The new concep o ai ness I p opose he e, no unjus i…ed lowe chances, a¤ec s he assignmen
mechanism i sel a he han he andom assignmen . I s a es ha any agen should be able o
ob ain, i she wishes, a leas he assignmen p obabili ies o any o he agen , igno ing hose objec
ypes o which he la e has p io i y o e he o me . Mo e o mally, o each easible andom
assignmen , he assignmen mechanism p o ides each agen wi h a menu o assignmen p obabili y
ec o s om which he agen eely chooses. Acco ding o he p oposed concep , his menu should
include any o he agen ’s assigned p obabili ies (o highe ), igno ing he objec ypes o which he
la e has p io i y o e he o me . This concep implies ex-an e en y- eeness among agen s o he
same p io i y ype (i.e. hose who ha e equal p io i y le els o each objec ype).
In his pape , I p opose a pseudoma ke wi h p io i ies, a a ia ion o Hylland and Zeckhause ’s
sugges ed mechanism whe e he p ices each agen pays depending on he p io i y ype. Fo each
objec ype, he p io i y ype o an agen can be summa ized in o one o he ollowing h ee s a uses:
gua an eed, pi o al and banned. Gua an eed agen s pay ze o p ice, pi o al agen s pay he ma ke
p ice and banned agen s pay an in…ni e p ice. A s able …nal assignmen always a ises om any
equilib ium in a pseudoma ke wi h p io i ies. Mo eo e , a pseudoma ke wi h p io i ies gua an ees
no unjus i…ed lowe chances. I also ob ains ex-an e e¢ cien andom assignmen s among agen s o
he same p io i y ype.
In some scena ios, p io i y o de ings coincide ac oss objec ypes. Tha is, p io i ies a e acyclical.
Senio i y igh s in esidence assignmen , o low-income p io i ies in school choice, a e examples o
such a p io i y s uc u e. In such cases, a pseudoma ke wi h p io i ies can be unde s ood as a
sequen ial pseudoma ke : hose agen s wi h he highes p io i y le el a end he pseudoma ke , buy
hei assignmen p obabili ies and lea e; hen hose wi h he second-highes p io i y le el a end he
pseudoma ke o he emaining objec uni s, and so on. A sequen ial pseudoma ke ob ains ex-an e
e¢ cien andom assignmen s. The eason is ha he p ice disc imina ion is such ha no agen o a
highe p io i y ype would ha e an incen i e o ade wi h agen s o lowe ypes.
Rela ed li e a u e and commen s.
The deba e: Since he seminal pape by Abdulkadi o¼
glu and Sönmez (2003), he e has been
a li ely deba e on he ela i e alue o he assignmen mechanisms ha a e used in p ac ice. Mos
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o he in e es has been cen e ed on school choice mechanisms (i.e. he assignmen o child en
o public schools), and mo e speci…cally on he compa ison be ween s a egy-p oo mechanisms
(wi h u h- elling as weakly dominan s a egy) such as De e ed Accep ance and non-s a egy-
p oo mechanisms such as he Bos on Mechanism.1The s a egic simplici y o De e ed Accep ance
inspi es bo h a jus ice a gumen (p o ec ion o naï e agen s, see Abdulkadi o¼
glu, Pa hak, Ro h and
Sömez, 2006, and Pa hak and Sönmez, 2008) and also an e¢ ciency a gumen (a oiding coo dina ion
ailu es, see E gin and Sönmez, 2006) agains non-s a egy-p oo mechanisms.2Howe e , s a egy-
p oo ness may come a oo high a cos in e ms o ex-an e e¢ ciency, as e idenced in Abdulkadi o¼
glu,
Che and Yasuda (2008), Mi alles (2008) and Abdulkadi o¼
glu, Che and Yasuda (2009). Some non-
s a egy-p oo mechanisms such as he Bos on Mechanism and ela ed mechanisms achie e be e
quali a i e ex-an e e¢ ciency esul s because hey impose some ma ke (o ade-o¤) incen i es.
Inspi ed by his idea, I conside he e how we can combine ma ke incen i es wi h espec o p io i ies.
Jus ice and e¢ ciency: The main ecen e e ence is he su ey by Thomson (2007). Fo
la ge economies wi h a con inuum o agen s, Va ian (1976) and Zhou (1992) p o ide gene al esul s
ela ing ai ness and e¢ ciency o Wal asian ma ke s wi h equal endowmen s. Va ian assumes s ic ly
conca e p e e ences and Zhou analyzes s ic ly posi i e consump ion se s. Bo h au ho s conclude
ha he se o ai alloca ions coincides wi h he se o alloca ions a ising om Wal asian ma ke
equilib ia wi h equal endowmen s.
The no-en y equi emen is one o se e al concep s o jus ice ha one could use. Egali a ian
equi alence (Pazne and Schmeidle , 1978), ha is, e e yone’s indi¤e ence o some (no necessa ily
easible) equal spli alloca ion, could al e na i ely ha e been chosen among o he concep s. Thomson
and Zhou (1993) gi e a wide esul on ha subjec : all egali a ian (wi h espec o equal spli ),
consis en (i.e. holding o any subse o agen s) and e¢ cien alloca ions a e ob ained by Wal asian
ma ke s wi h equal endowmen s, e en i p e e ences a e sa ia ed. I easily ex end ha esul o he
kind o assignmen p oblems I analyze. Assignmen p obabili ies a e he goods in his economy.
The ac ha p obabili ies mus add up o one could be modeled as a case o sa ia ion. In his
1Bo h mechanisms a e anking mechanisms in ha pa en s ( he agen s) a e … s eques ed o submi a anking o e
he schools ( he objec ypes). The assignmen algo i hm uses he p o ided in o ma ion in se e al ounds. In he … s
ound, s uden s a e conside ed o he schools pa en s anked … s . In schools wi h excess demand some s uden s a e
ejec ed ( ollowing p io i y c i e ia and ie-b eaking lo e ies) and go o he nex ound, whe e hey a e conside ed o
he schools ha we e anked in second posi ion. Accep ed s uden s a e de…ni ely accep ed in he Bos on Mechanism,
whe eas in De e ed Accep ance hey a e only econside ed o ha school in he nex ound. The algo i hms likewise
ollow a …ni e numbe o ounds un il all s uden s a e …nally accep ed a some school.
2Fo hese easons, he Bos on Mechanism was eplaced by De e ed Accep ance in Bos on (!).
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economy, ex-an e en y- eeness is equi alen o consis ency and egali a ian equi alence wi h espec
o equal spli . Thus, among agen s o he same p io i y ype, ex-an e en y- eeness and e¢ ciency
a e ob ained h ough a Wal asian ma ke wi h equal endowmen and p io i y-dependen p ices. The
pseudoma ke wi h p io i ies I p opose mee s hese p ope ies.
Ex an e and ex pos : The impo ance o ex-an e e¢ ciency (in which andom assignmen s a e
compa ed o each o he ) compa ed o ex-pos e¢ ciency (whe e …nal assignmen s a e compa ed) is
s essed when p io i ies a e a he coa se, ha is, when massi e se s o agen s belong o he same
p io i y ype. An example is he case o elemen a y school choice in Bos on, whe e he e a e only
ou p io i y ca ego ies (combining sibling and walking zone p io i ies) o mo e han one housand
new en an s a yea . P io i y ies a e ypically sol ed by some so o lo e y and hus we alk
abou andom assignmen s when his unce ain y has no ye been esol ed. In his con ex , ex-
an e e¢ ciency is a e…nemen o ex-pos e¢ ciency. Since any easible andom assignmen could be
unde s ood as a lo e y o e easible su e assignmen s, i is easy o see ha no ex-pos e¢ ciency
implies no ex-an e e¢ ciency. The con e se is no ue. An example is he mechanism known as
andom se ial dic a o ship, in which agen s a e s ic ly anked acco ding o an e en lo e y, and
hen he op- anked agen picks a uni om he mos -p e e ed objec ype, he second- anked
agen picks among he a ailable uni s, and so on. This mechanism always ob ains ex-pos e¢ cien
assignmen s. Howe e , i can be shown ha i is ex-an e ine¢ cien (see o ins ance Bogomolnaia
and Moulin, 2001).
Ex-an e en y- eeness is howe e a weake concep han ex-pos en y- eeness. In e¤ec , a andom
assignmen could be ex-an e, ye no ex-pos , en y- ee, whe eas he absence o ex-pos en y implies
i s absence ex an e. Howe e , he concep o ex-pos en y- eeness is so igh ha i could ei he
be una ainable o yield non-sensible assignmen s. Fo ins ance, i an objec ype is popula (i.e.
he numbe o agen s who p e e i o e e e y o he objec ype exceeds he numbe o uni s o his
ype), hen ex-pos en y- eeness implies ha no uni o ha objec ype could be assigned o any
agen who p e e s i . In a simila way, he concep o no unjus i…ed lowe chances makes sense only
when applied o assignmen p obabili ies. In conclusion, he ex-an e app oach o e¢ ciency and no
en y seems ecommendable, and his mo i a es my ocus on andom assignmen s.
P io i ies and ai ness: In se e al cases, p io i y s uc u es a e designed o p o ide agen s
wi h a chance o p o e he in ensi y o hei p e e ences. In school choice, ha ing a sibling a he
school and li ing nea by is posi i ely co ela ed wi h he pa en s’p e e ence o he school, and his
jus i…es gi ing p io i y on he basis o hese a iables o he sake o a mo e e¢ cien alloca ion.
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Simila ly, p e ious enan s may ha e a p e e ence o s aying whe e hey a e, conce ning esidence
alloca ion. Ne e heless, in many o he cases p io i ies a ise o o he easons. As an example,
he San F ancisco school au ho i y gi es p io i y o hose applican s who "depa mo e" om he
a e age pool wi h espec o some socioeconomic indica o s, in o de o lessen he concen a ion
o mino i y s uden s in a ew schools. My esul s in his pape indica e ha when p io i ies a e
single-o de ed (e.g. senio i y igh s), i is possible o ob ain a andom assignmen sa is ying ex-pos
s abili y, no unjus i…ed lowe chances and ex-an e e¢ ciency. In he li e a u e on ex-pos assignmen s
(e.g. E gin, 2002), single-o de ed p io i y ules o simila equi emen s (e.g. "one s ep away om
single o de ing") a e also needed o sa is y jus ice and e¢ ciency p ope ies.
Mo e cen ally ela ed o he p esen pape is he ecen wo k by Kes en and Ün e (2010).
Acknowledging he need o espec p io i ies, hey concei e a weak no ion o ai ness which hey
call no ex-an e disc imina ion. The e is ex-an e disc imina ion o an agen wi h espec o ano he
agen and a ce ain objec ype i : 1) bo h agen s a e a he same p io i y le el conce ning ha
objec ype, 2) he o me agen ob ains lowe chances han he la e o be assigned an uni o ha
objec ype, and 3) he o me agen has posi i e p obabili y o be assigned an uni o an objec
ype ha is less p e e ed o he p e ious objec ype. I is a sound concep o ai ness in ha
agen s wi h same p io i y le el o some objec ype should ha e he same chances wi h espec o
ha objec ype unless he e is "enough" compensa ion. Mo eo e , i implies ex-an e en y- eeness
among agen s o he same p io i y ype. Based on ha concep , he au ho s p opose a modi…ca ion
o he De e ed Accep ance algo i hm ha mee s ex-an e s abili y and no ex-an e disc imina ion
while Pa e o-domina ing all o he assignmen s mee ing he same condi ions.
Ne e heless, he concep o no ex-an e disc imina ion is no exemp o discussion. A … s poin
is, wha hey unde s and as "enough compensa ion" (no being possibly assigned a wo s objec
ype) is no he only way o concei e i . Fo ins ance, "enough compensa ion" could jus mean ha
he p obabili y o being assigned o he conside ed objec ype o a p e e ed one should no be less
ha he p obabili y ha he o he agen has o be assigned o ha objec ype.
A second poin is ha his concep o jus ice could come a a high p ice in e ms o (ex-an e)
e¢ ciency. Conside he ex eme case whe e he e is only one p io i y le el o all schools, ha is,
he no-p io i y case. In such an en i onmen , and wi h a con inuum o agen s, any ex-an e en y-
ee and e¢ cien andom assignmen is ob ained h ough a pseudoma ke wi h equal budge s (once
again ci ing Thomson and Zhou, 1993). Howe e , no ex-an e disc imina ion is igh e han no ex-an e
en y, hus in many cases he pseudoma ke assignmen s a e p ecluded and ex-an e e¢ ciency is no
6
achie able. A simple example con ains h ee objec ypes a; b; c wi h equilib ium p ices 3=2;1=2;0
espec i ely. Agen s ha e uni budge s. Depending on p e e ences, some agen s would buy 2=3
p obabili y a aand 1=3a c, and some o he s would buy 1=2p obabili y a aand 1=2a b. The
la e agen s p e e a o b, o he wise buying su e assignmen a bwould ha e been a be e op ion.
Consequen ly, he la e agen s would be ex-an e disc imina ed wi h espec o he o me agen s
and objec ype a. The al e na i e no ion o ai ness p oposed he e, no unjus i…ed lowe chances, is
in ead compa ible wi h hese pseudoma ke s.
P io i ies and p ope y igh s: A las obse a ion conce ning he p esence o p io i ies is
he ac ha hey could be conside ed as ex e nali ies. I could hen be concei able, à la Coase, o
con e he p io i ies in o p ope y igh s and o le hen he agen s ade hem (Abdulkadi o¼
glu and
Sönmez, 2003). Howe e , apa om legal issues ( his p ocedu e does no gua an ee ex-pos s abili y,
and uns able assignmen s ha e been legaly dispu ed), o he heo e ical conce ns a ise he e. Fi s
o all is he ques ion on how p io i ies a e con e ed in o p ope y igh s, ha is, in o p obabili y
endowmen s ha agen s ade. E en hoguh a sa is ac o y answe is possible, a second conce n
a ises when assignmen p obabili ies a e aded ins ead o bough using " ake" mone a y income.
Hylland and Zeckhause (1979) ha e a gued ha a pseudoma ke equilib ium may no exis i agen s
ade p obabili y endowmen s. Key in hei a gumen is he ac ha each agen ends up wi h a
p obabili y bundle ha mus add up o one. As an example, conside an en i onmen wi h wo
objec ypes aand bwhe e a se o agen s who p e e a o ba e gi en an endowmen o a su e
assignmen (p obabili y 1) o b. As long as he p ice o blies below he p ice o a, hese agen s
canno ade p obabili ies o b o p obabili ies o a(demanded p obabili ies would no add up o
one). As soon as he p ices equal each o he , howe e , hese agen s ade all hei endowmen o a
su e assignmen a a. This gene a es an hemi-discon inui y in agg eg a e demand, hus Kaku ani’s
…xed-poin heo em may no apply. The example could be easily ex ended o scena ios wi h mo e
han wo objec ypes.
The pape is s uc u ed as ollows. The second sec ion p esen s he assignmen p oblem wi h and
wi hou p io i ies, and he concep s o jus ice and e¢ ciency a e b ough in o play. The hi d sec ion
in oduces and analyzes an ex ended e sion o Hylland and Zeckhause ’s (1979) pseudoma ke s.
The ou h sec ion p esen s he esul s and discussion. The las sec ion concludes.
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2 The ( andom) assignmen p oblem
The e is a …ni e se So J > 2objec ypes S= 1; :::; Jg. Fo simplici y I assume ha he e is no
ou side op ion.3Each objec ype jhas capaci y mass j>0. The o al sum o capaci ies ac oss
objec ypes is a leas 1. Le ~ = (1; :::; J). The e is a mass 1 o agen s x2X[0;1] whe e
Xis he se o agen s endowed wi h he Lebesgue (uni o m) measu e . The e is a measu able on
Neumann-Mo gens ein ( NM) alua ion unc ion :X!VRJ
+, whe e (x)=( 1(x); :::; J(x))
deno es agen x’s alua ions o objec ypes 1 o J. Each agen is indi¤e en be ween any wo
uni s o he same objec ype. The unc ion is assumed o ha e a ange ha in e sec s wi h any
posi i e ay om he o igin (i.e. all ela i e p e e ences belong o he image). I is also assumed
ha ( x2X: (x)2V0g) = 0 whene e dim(V0)<dim(V).4
A andom assignmen is a measu able unc ion q:X!(S), whe e q(x)=(q1(x); :::; qJ(x))
deno es agen x’s assignmen chances o objec ypes 1 o J. A andom assignmen qis easible a
( ;~)i RXq(x)d ~. An (ex-pos o …nal)assignmen is a measu able unc ion a:X!S. Le A
be he amily o all easible …nal assignmen s (i.e. he mass o agen s who a e assigned o each objec
ype jdoes no exceed j). By he Bi kho¤- on Neumann heo em, any easible andom assignmen
can be implemen ed as a (no necessa ily unique) lo e y l2(A)o e easible assignmen s. In
some esul s, we selec lsuch ha i pu s posi i e weigh only on hose …nal assignmen s awhose
assignmen equencies coincide wi h he andom assignmen p obabili ies: 8X0X; (X0)>0;
RX0a(x)d =RX0q(x)d. This is ega ded as he equency-p obabili y condi ion. Le Fdeno e he
se o all easible andom assignmen s (which ob iously depends on ~).
Each agen x’s expec ed payo¤ om he andom assignmen qis equal o q(x) (x). A easible
andom assignmen is ex-pos e¢ cien a ( ;~)i i can be implemen ed as a lo e y o e Pa e o-
op imal assignmen s. Tha is, o any possible lo e y ou come, he esul ing assignmen is Pa e o-
op imal. A easible andom assignmen is ex-an e e¢ cien a ( ;~)i he e is no o he easible
andom assignmen a ( ;~) ha p o ides each agen wi h a weakly highe expec ed payo¤ and a
posi i e-measu e se o agen s ob ains a s ic ly highe payo¤. Ex-an e e¢ ciency implies ex-pos
e¢ ciency, bu he con e se may no be ue. The concep could be ex ended o g oups: o ins ance,
qis ex-an e e¢ cien wi hin X0Xi he e is no easible eassignmen a¤ec ing only agen s in X0
such ha he e is a Pa e o-imp o emen .
A andom assignmen qis ex-an e en y- ee i o any x; y 2X; q(x) (x)q(y) (x). An
3The esul s p esen ed he e could easily be ex ended o include ha op ion.
4I am abusing no a ion: he dimension o a se is in eali y aken wi h espec o he closu e o i s in e io .
8
assignmen is en y- ee i o any x; y 2X; a(x)(x) a(y)(x). A andom assignmen is ex-pos en y
ee i o any lo e y ou come, any …nal assignmen is en y- ee. Ex-pos en y- eeness implies ex-
an e en y- eeness. Howe e , i is easy o see ha ex-pos absence o en y is oo a ha sh condi ion.
Fo any se o agen s p e e ing he same objec , ei he none o all o hem ha e o be assigned o
ha objec ype. A wi hin-g oups e sion o he en y- eeness de…ni ion could also apply he e.
Ap io i y s uc u e is a unc ion P:X!Yj2S 0; :::; Gjg, whe e P(x) = (P1(x); :::; PJ(x))
deno es agen x’s p io i y ype wi h espec o objec ypes 1 o J, and Gj2Nis a bi a ily la ge.
We say ha agen xhas p io i y o e agen ywi h espec o objec ype ji Pj(x)> Pj(y). The
iple ( ;~; P)de…nes he economy. Le 2deno e a gene ic p io i y ype, = (1; :::; J), and
le Xbe he se o agen s o p io i y ype . In some esul we will use he ollowing assump ion:
De…ni ion 1 The p io i y s uc u e Psa is…es he p io i y-b idge condi ion i o any pai ; 02
; (X); (X0)>0;and o any iple i; j; k 2Ssuch ha i> 0
i; j=0
jand k< 0
k; he e
exis s 00 2; (X00 )>0;such ha i=00
i; j=0
j=00
jand 0
k=00
k.
So i wo p io i y ypes a e ied wi h espec o some objec ype, and one p io i y ype has
highe p io i y wi h espec o a second objec ype while he o he p io i y ype has highe p io i y
wi h espec o a hi d objec ype, hen he e is a "b idge" p io i y ype which has he highes
p io i y o he wo in bo h h ee objec ypes. The p io i y s uc u e is equi ed o be ich in ha
sense. A pa icula case o a p io i y s uc u e sa is ying he p io i y-b idge condi ion would be he
join-semila ice s uc u e: o any pai ; 02; (X); (X0)>0;we ha e (X_0)>0.
Fu he mo e, we will impose an assump ion on he p e e ence p o…le in some o he esul s. In
wo ds, he assump ion s a es ha all p e e ences a e possible ega dless he p io i y ype.
De…ni ion 2 The economy ( ;~; P)sa is…es he p e e ence- ichness condi ion i o any 2
such ha (X)>0;we ha e (X) = V.
We say ha an assignmen ais s able (o i espec s p io i ies) gi en Pi 8x; y 2X; Pj(x)> Pj(y)
and a(y) = j=) a(x)(x) j(x). A andom assignmen is ex-pos s able i i can be de…ned as a
lo e y l2(A)whose suppo is cons i u ed by s able assignmen s. Following Kes en and Ün e
(2010), a andom assignmen qis de…ned as ex-an e s able gi en Pi and only i 8x; y 2X; Pj(x)>
Pj(y)and qj(y)>0g=) qi(x)=08i2S: i(x)< j(x)g. Tha is, i agen xhas p io i y o e
agen ywi h espec o objec ype jand yob ains chances o being assigned o j, hen xcanno be
possibly assigned o an objec ype ha is less p e e ed han j. Ex-an e s abili y implies ex-pos
s abili y. Fo he con e se, he equency-p obabili y condi ion is needed.
9
In he nex wo pa ag aphs, I show ha o any school j2Sand any wo p io i y ypes
; 02Xsuch ha j=0
j, we mus ha e (WLOG) pj() = pj(0). Le us suppose pj()> pj(0).
I pj()>1, some agen x2X ha p e e s j o any o he objec ype will be able o buy less
assignmen p obabili y a j han some agen y2X0who op imally chooses o spend he budge
on jand comple es he bundle wi h w( he "wo s " objec ype, wi h ze o p ice). This iola es no
unjus i…ed lowe chances. I pj()1, I conside se e al cases. Fi s , i ei he xo y(o bo h)
is acing p ices ha a e ei he no highe han 1 o 1, hen hese no -abo e-one p ices could be
no malized making pj() = pj(0)wi h no al e a ion o he budge se (WLOG). Second, i he e is
ano he objec ype h:h=0
h,ph() = ph(0)>1, (we mus ha e ph() = ph(0)as seen be o e
in he case pj()>1) hen pj()> pj(0)implies, unde he p e e ence- ichness assump ion, ha
some agen y2X0op imally buys a bundle wi h 1pj(0)
ph()pj(0)>1pj()
ph()pj()p obabili y uni s o
being assigned a hand he emaining p obabili y a j. This bundle does no belong o he budge
se o agen s in X, hence iola ing no unjus i…ed lowe chances.
Thus I conside a las se o scena ios whe e j=0
jand 9i; k 2SnE0:1> pi()>1;
1> pk(0)>1, whe e E0 h2S:0
h=hg. He e I use he p io i y-b idge condi ion imbedded
in he ichness condi ion. The e is a p io i y ype 00 such ha i; j 2E00  h2S:00
h=hg
and j; k 2E000  h2S:0
h=00
hgwi h a posi i e mass o agen s o ha ype. By he p e ious
pa ag aph we mus ha e bo h pj(00) = pj(0)and pj(00) = pj(0), con adic ing pj()> pj(0).
The necessi y a gumen is also seen he e: i he e we e no such a 00, he e would be a p e e ence
p o…le such ha a ec o o equilib ium p ices exis s sa is…ng pj()> pj(0)(while all he s a ed
p ope ies a e me ).
A second ques ion he e is whe he he equency-p obabili y condi ion is ele an . This condi ion
p o ides some s uc u e o ex-pos assignmen s. I has o be said, hough, ha his condi ion educes
he se o ex-an e assignmen s ha a e ex-pos s able unde a p ope y designed lo e y o e …nal
assignmen s. Conside he ollowing example wi h h ee objec ypes a; b; c whe e a(wi h capaci y
1/2) is p e e ed o b(wi h capaci y 1/4) and b o c(wi h capaci y 1/4) by e e y agen . The e
is a measu e 1/2 o agen s o p io i y ype and ano he measu e 1/2 o agen s o p io i y ype
0. The only di¤e ence be ween and 0is ha hose agen s o ype ha e p io i y o e hose o
ype 0wi h espec o objec ype b. Conside a andom assignmen in which all agen s ob ain he
same assignmen p obabili ies 1/2,1/4,1/4 o a; b and c espec i ely. Acco ding o he equency-
p obabili y condi ion, his andom assignmen is no ex-pos s able. Howe e , conside an ex-pos
implemen a ion whe e wi h p obabili y 1/2, all agen s o ype a e assigned o aand he o he agen s
16

a e andomly spli be ween band c, and wi h p obabili y 1/2 all agen s o ype 0a e assigned o
awhe eas hose o ype a e andomly spli be ween band c. All he ex-pos assignmen s he e
a e s able. The example could be sligh ly modi…ed o acommoda e he ichness assump ion. This
sugges s ha a elaxa ion o he equency-p obabili y condi ion could en ich he analysis o ex-pos
s able andom assignmen s.
4.2 Acyclical p io i ies
Conside a speci…c p io i y s uc u e Psuch ha o any x; y 2X; i; j 2S; Pi(x)> Pi(y),Pj(x)>
Pj(y). This would be ega ded as acyclical p io i ies, since he e is a unique o de ing among agen s
ha applies o all objec ypes. One example in eal scena ios could be he assignmen o college
s uden s o esidences, whe e senio i y may gi e p io i y. In hese cases, I could hen gene ally
alk abou op- anked agen s, second- anked agen s and so on, wi h no men ion o objec ypes.
Le us de…ne a sequen ial pseudoma ke in he ollowing way: … s , op- anked agen s a end a
pseudoma ke wi h equal budge s, buy hei assignmen p obabili ies a equilib ium p ices and
lea e; hen, second- anked agen s a end a pseudoma ke wi h equal budge s o he emaining
objec uni s; and so on. I is easy o see ha he sequen ial pseudoma ke is equi alen o he mo e
gene al pseudoma ke wi h p io i ies when hese a e acyclical.
Theo em 3 Fix an acyclical p io i y s uc u e P. Unde he equency-p obabili y condi ion, a
sequen ial pseudoma ke gua an ees s abili y and ob ains ex-an e e¢ cien andom assignmen s.
P oo . Le 0, and conside any x2Xand any y2X0. Conside any objec ype j om
which ybuys assignmen p obabili ies in he ma ke equilib ium. Then i mus be he case ha
pj(x) = 0, since ha objec ype was necessa ily o e supplied o  anked agen s. This gua an ees
ex-pos s abili y, since xwill op imally buy assignmen p obabili ies om objec ypes ha a e a
leas as p e e ed as j. Since ex-an e e¢ ciency is gua an eed among agen s o he same p io i y ype,
po en ially mu ually bene…cial ade could only a ise be ween p io i y g oups. Once again, howe e ,
no se o agen s x2Xcould ha e an in e es in ading wi h any se o agen s y2X0. All objec
ypes om which any such ybuys a e (weakly) less p e e ed o x han he objec ypes om which
xbuys, since xis acing a ze o p ice o he o me goods. No e ha any ade has o keep e e y
agen wi h assignmen p obabili ies adding up o one.
No ice he e ha we ha e no men ioned he ai ness condi ion p oposed in his pape , no un-
jus i…ed lowe chances. I u ns ou ha his p ope y is no in o ma i e he e, since i holds only
17
i ially. In ac , o  > 0, his p ope y jus implies ha agen s in 0ob ain a non-nega i e
assignmen p obabili y ec o in he menu o¤e ed, which is he budge se .
5 Conclusion
In his pape , I ha e analyzed la ge andom assignmen economies wi h a con inuum o agen s and a
…ni e numbe o objec ypes. I ha e in oduced he ealis ic assump ion ha some p io i y c i e ia
migh al e he symme y o agen s in hei igh s ega ding he …nal assignmen . Wi hou hese
p io i ies, ex-an e e¢ ciency and en y- eeness is cha ac e ized by he equilib ium ou comes om a
pseudoma ke wi h equal budge s, à la Hylland and Zeckhause (1979), as poin ed ou by Thomson
and Zhou (1993). In a pseudoma ke , each agen is endowed wi h a …c i ious budge ha allows
he o buy assignmen p obabili ies, whe e each objec ype has i s own p ice. In his pape , I ha e
p oposed an ex ension o his pseudoma ke mechanism ha akes he exis ence o p io i ies in o
accoun . Fo each objec ype, he e is a p io i y le el ha is pi o al, ha is, agen s a ha le el
pay he ma ke p ice. Agen s a a highe p io i y le el (gua an eed) pay ze o p ice, and hose a
lowe le els (banned) pay in…ni e p ice.
Once p io i ies a e conside ed, ex-an e en y- eeness is ypically no achie able. I ha e al e na-
i ely sea ched o a no ion o "maximal ai ness" subjec o espec o p io i ies. I ha e based his
no ion on he ac ha p io i ies ega ding one objec ype canno jus i y any p i ileges conce ning
ano he objec ype, and I ha e named he obse a ion o his deside a um no unjus i…ed lowe
chances. This p ope y implies ex-an e en y eeness among agen s o he same p io i y ype. I
s a es ha , o any easible andom assignmen , each agen mus be o¤e ed a menu o assignmen
p obabili y ec o s such ha each o he agen ’s assignmen p obabili ies (o highe ) a e included in
ha menu, igno ing he objec s o which he la e agen has p io i y o e he o me . I will be
no iced ha his ai ness condi ion no only a¤ec s andom assignmen bu also he p ocess ( he
mechanism) gene a ing i . The menu is a educed o m o he assignmen p obabili ies ha he
agen can ob ain gi en he s a egy space and he o he agen s’s a egy p o…le.
In gene al scena ios whe e I do no impose any s uc u e on p io i ies, I show ha a pseudoma -
ke wi h p io i ies espec s p io i ies (i is ex-pos s able), gua an ees no unjus i…ed lowe chances
and ob ains ex-an e e¢ cien alloca ions among agen s o he same p io i y ype. Assuming enough
ichness in p e e ences and p io i ies, he con e se is also ue: any andom assignmen wi h hese
p ope ies could be achie ed h ough an equilib ium in a pseudoma ke wi h p io i ies. I p io i ies
18
a e cons ained o be acyclical, ha is, he anking o agen s’p io i y le els does no a y ac oss
objec ypes, hen he pseudoma ke wi h p io i ies can be implemen ed ia a sequen ial pseudo-
ma ke . In such a mechanism, agen s wi h he highes p io i y le el a end he pseudoma ke , buy
hei assignmen p obabili ies and lea e. Agen s a he second-highes p io i y le el a end he
pseudoma ke o he emaining slo s and lea e. And so on. This sequen ial pseudoma ke ob ains
in equilib ium o e all ex-an e e¢ cien andom assignmen s, since agen s om a highe p io i y ype
would ne e ha e an incen i e o ade hei assigned p obabili ies wi h agen s in lowe p io i y
le els.
The e a e se e al o he in e es ing ea u es in assignmen p oblems ha ha e been skipped he e
in o de o ob ain mo e concise esul s. Among hem, he …ni eness o he numbe o agen s,
and he p esence o pee -g oup e¤ec s on agen s’p e e ences. The … s elemen is o en ound in
he heo e ical li e a u e. Following Hylland and Zeckhause ’s (1979) seminal wo k, he esul s
ound in he p esen pape do no subs an ially di¤e om wha we could expec in en i onmen s
wi h su¢ cien ly many agen s pe objec ype. Howe e , in e es ing depa u es om compe i i e
ma ke beha io dese e mo e ca e ul analysis, when he numbe o agen s pe objec ype is low
enough. Fo ins ance, in he child en- o-school assignmen p oblem, is he easonable 25-30 new
s uden s pe school and yea a io su¢ cien ly high? The answe will come om empi ical e idence.
The second elemen I men ion is specially in e es ing in ha i has no ecei ed p ope a en ion
om he heo e ical li e a u e despi e he empi ical e idence. I could be a gued ha a model
ha inco po a es pee -g oup e¤ec s adds much mo e complexi y o al eady cumbe some p oblems.
Howe e , u he inclusion o his ea u e is ele an since i may a¤ec wha we know om he
mechanism I sugges in his pape and o he mechanisms ha ha e been sugges ed in he li e a u e.
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