Smoo h mul ibidding mechanisms∗
Da id Pé ez-Cas illo†and Nicolas Qué ou‡
No embe 5, 2010
Abs ac
We p opose a smoo h mul ibidding mechanism o en i onmen s whe e a g oup
o agen s ha e o choose one ou o se e al p ojec s (possibly wi h he help o a social
planne ). Ou p oposal is ela ed o he mul ibidding mechanism (Pé ez-Cas illo
and We s ein, 2002) bu i is “smoo he ” in he sense ha small a ia ions in
an agen ’s bids do no lead o d ama ic changes in he p obabili y o selec ing a
p ojec . This mechanism is shown o possess se e al in e es ing p ope ies. Unlike
in he s udy by Pé ez Cas illo and We s ein (2002), he equilib ium ou come is
unique. Second, i ensu es an equal sha ing o he su plus ha i induces. Finally,
i enables eaching an ou come as close o efficiency as is desi ed.
JEL Classi ica ion numbe s: D78, D72
Keywo ds: mechanism design, NIMBY
∗We hank Inés Macho-S adle , Ped o Rey-Biel and Da id We s ein o help ul ema ks. Pa o his
esea ch was conduc ed while he second au ho isi ed he Ins i u e o Economic Analysis (IAE) and
he INRA-LAMETA. The hospi ali y o hese ins i u ions is g ea ly app ecia ed. Da id Pé ez-Cas illo
acknowledges he inancial suppo om ECO2009-7616, Consolide -Ingenio CSD2006-16, 2009SGR-169,
Ba celona Economics-Xa xa CREA and ICREA Academia. Da id Pé ez-Cas illo is MOVE ellow.
†Dep . o Economics & CODE, Uni e si a Au ònoma de Ba celona, 08193 Bella e a (Ba celona),
Spain. Email: da id.p[email p o ec ed]. Tel: +34 935811405. Fax: +34 935813767.
‡Queen’s Uni e si y Managemen School, Queen’s Uni e si y Bel as , Uni e si y Sq. 25, BT7 1NN
Bel as , UK. E-mail: n.[email p o ec ed]. Tel: +44 2890975024. Fax: +44 2890975156
1
1 In oduc ion
1.1 Con ibu ion
The design o mechanisms ha help agen s each decisions on con en ious issues is a
e y ele an and ac i e line o esea ch. A mechanism ha leads o an efficien p ojec
equi es in o ma ion abou agen s’ p e e ences o each possible decision. The mul ibidding
mechanism, p oposed by Pé ez-Cas illo and We s ein (2002) allows he agen s o exp ess
hei ela i e p e e ence o p ojec s. I p oceeds as ollows. Each agen submi s a ec o
o bids, one o each p ojec , wi h he sole es ic ion ha he sum o each agen ’s bids
is ze o. The e o e, bids measu e ela i e a he han absolu e alua ion. Each agen
also nomina es one o he p ojec s speci ically. The p ojec wi h he highes agg ega e
bid(sumo bidsmade o hisp ojec )ischosen. Incase he eismo e hanonesuch
p ojec , he e is a ule ha gi es p io i y o p ojec s ha ha e been nomina ed by some
agen . The winning p ojec is ca ied ou , agen s pay he p omised bid co esponding
o his p ojec , and any su plus is sha ed among he agen s, so ha he mechanism is
budge -balanced.
The main p ope y o he mul ibidding mechanism is ha all i s Nash (and s ong
Nash) equilib ium ou comes a e efficien . Howe e , in gene al en i onmen s, he mech-
anism has wo weak aspec s ha we add ess in he cu en pape . Fi s , i equi es he
ieb eaking ule ha , a equilib ium, is always used because all p ojec s’ equilib ium ag-
g ega e bids a e ze o. The e o e, he ieb eaking ule plays a c ucial ole. As Ehle s
(2009) highligh s, emo ing he agen s’ abili ies o nomina e one speci ic p ojec and us-
ing ieb eaking ules may p e en an equilib ium om exis ing. Because agen s do no
nomina e a speci ic p ojec in many eal-wo ld p ocesses such as auc ions, his lack o o-
bus ness cons i u es a weakness o he ini ial mechanism. Second, he se o equilib ium
ou comes is qui e la ge, as i consis s o all he ou comes whe e each agen ’s payoffis a
leas he expec ed payoffhe would ob ain in a si ua ion whe e all he p ojec s ha e he
same p obabili y o being de eloped. The e o e, almos any (“ easonable”) sha ing o he
su plus is an equilib ium ou come.
In he p esen pape , we ackle he issues highligh ed abo e by p oposing a smoo h
mul ibidding mechanism. I is close o he o iginal p oposal bu ou s is “smoo he ” in
2
he sense ha small a ia ions o an agen ’s bids do no lead o d ama ic changes in he
p obabili y o selec ing a p ojec . In he smoo h mechanism, each agen only submi s a
ec o o bids, wi hou nomina ing any p ojec . All p ojec s can be selec ed, wi h each
p ojec ’s p obabili y being a unc ion o i s agg ega e bid as well as he agg ega e bids
o he es o he p ojec s. P ojec s wi h a nega i e agg ega e bid ha e a e y low, bu
posi i e, ixed p obabili y o being selec ed (a unc ion o some pa ame e ). Each p ojec
wi h a posi i e agg ega e bid is selec ed wi h a p obabili y ha is a unc ion o he le el
o i s (and o he s’) posi i e agg ega e bid.
We i s show ha , o a gi en alue o , he equilib ium ou come is unique.This
p ope y is impo an because i highligh s ha he e is no coo dina ion issue wi h espec
o agen s’ expec a ion abou he inal ou come. We hen cha ac e ize he equilib ium
ou come. Al hough he e may be se e al equilib ium s a egies, he diffe ences among
hem only conce n bids o hose p ojec s ha , a equilib ium, end up wi h nega i e
agg ega e bids. We iden i y he se o p ojec s wi h posi i e equilib ium bids as well as
each agen ’s bids o any p ojec in his se . Only p ojec s ha a e efficien , o whose
o al alua ion is e y close o he efficien one, ul ima ely ecei e a posi i e agg ega e
bid. In case some non-efficien p ojec ecei es a posi i e agg ega e bid, i s le el e lec s
hedeg eeo inefficiency.
Second, he smoo h mul ibidding mechanism ensu es a ai sha e o he su plus ha i
induces. Indeed, an agen ’s equilib ium payoffin he mechanism is he sum o he alue o
he a e age p ojec plus his ai sha e o he emaining su plus. Tha is, agen s ob ain he
same le el o u ili y as in he o iginal mul ibidding mechanism, and he su plus is di ided
in equal pa s among he agen s. This ai ness p ope y ensu es ha he mechanism is
poli ically easible, which is an impo an cha ac e is ic o p ac ical implemen a ion.
Thi d, he mechanism does no ely on he use o ieb eaking ules and is immune
o he c i icism aised by Ehle s (2009). I can be hus hough o as a mo e na u al
mechanism han he ini ial one.
Finally, i is appa en om he p e ious desc ip ion ha he smoo h mul ibidding
mechanism does no achie e efficiency. I does, howe e , ge as close o efficiency as one
wishes. We show ha each agen ’s expec ed payoffinc eases as he alue o he pa ame e
dec eases; he e o e, o al efficiency inc eases as well. Mo eo e , he p obabili y o
3
choosing an inefficien p ojec con e ges o ze o as he alue o he pa ame e becomes
small. We can bound he le el o expec ed inefficiency as a unc ion o he pa ame e :
he maximum le el o inefficiency o a p ojec ha ecei es a posi i e agg ega e bid is a
linea unc ion o he squa e oo o .
To summa ize, he p esen mechanism exhibi s he in e es ing p ope ies o uniqueness
and ai ness o i s equilib ium ou come. The weakness compa ed o he ini ial mechanism
(which gua an ees ull efficiency) is only mino as a social planne elying on he new
p o ocol would be able o ge as close o ull efficiency as she wishes. The e o e, his
mechanism cons i u es an in e es ing op ion o p ac ical implemen a ions.
1.2 Applicabili y o he mechanism and ela ed li e a u e
The e a e many economic si ua ions whe e his mechanism can be success ully used. A
i s case conce ns he complex p oblem o he loca ion o noxious acili ies, such as p isons,
dump si es, nuclea was e eposi o ies, o ai po s. Many au ho s add ess his ype o
p oblem; we can e e among o he pape s o Kun eu he and Kleindo e , 1986; Rob,
1989; O’Sulli an, 1993; Ingbe man, 1995; Pé ez-Cas illo and We s ein, 2002; Mineha
and Neeman, 2002; and Lau en -Lucche i and Le oux, 2009. Whe eas he cons uc ion o
such acili ies may p o ide la ge global bene i s, hei cos is usually bo ne by he hos ing
agen . The si ing p oblems a e so se e e and so common ha an ac onym is used o
e e o hem:NIMBY(No InMyBackYa d).
Ano he sensi i e decision p oblem conce ns he loca ion o la ge in e na ional e-
sea ch in as uc u es. The decision abou he ci y ha should hos such a acili y is
always he subjec o ho deba e among he candida es and o he in e es ed coun ies and
ins i u ions. In 2002, he Eu opean Commission s a ed he Eu opean S a egy Fo um on
Resea ch In as uc u es (ESFRI) o suppo and acili a e mul ila e al ini ia i es leading
o a be e use and de elopmen o esea ch in as uc u es, including biological a chi es,
communica ion ne wo ks, esea ch essels, sa elli e and ai c a obse a ion acili ies, ele-
scopes, synch o ons, and pa icle accele a o s. Al hough i s 2006 Repo p esen ed a i s
oadmap iden i ying 35 p ojec s wi h he scien i icneeds o henex 10-20yea s,ESFRI
is silen abou how he in e es ed coun ies should de e mine he loca ion o he acili y.
4
Howe e , his is a e y difficul decision ha in ol es many scien i ic, economic, and social
issues. Fo each p ojec , suppo ing coun ies should wo k ou a p ocedu e o choose he
hos o he acili y. The e o e, hey mus i s decide on a mechanism and hen use he
p ocedu e o elec he hos ing ci y.
The p e ious examples belong o a gene al class o p oblems in which a g oup o
agen s has o choose one ou o se e al p ojec s. In some si ua ions, he se o p ojec s
coincides wi h he se o agen s, as is he case i a g oup o municipali ies mee o choose
one o hem o hos a dump si e o a hospi al. In ano he con ex , he se o agen s is
la ge han he se o p ojec s, as is ypically he case when coun ies o ins i u ions build
a la ge in e na ional esea ch in as uc u e: in such a si ua ion, all coun ies may no
ha e an own p oposal ega ding he speci ics o he p ojec o be ca ied ou . The main
objec i e o a mechanism in such si ua ions would be o maximize he agg ega e wel a e
o all he agen s (efficiency). Mo eo e , such decisions ypically equi e o compensa e
(some) agen s wi h mone a y ans e s. The p o ocol de ined in he p esen con ibu ion
can be conside ed a aluable op ion o be conside ed.
We ha e highligh ed ou con ibu ion o he li e a u e ha offe s mechanisms o decide
he loca ion o noxious acili ies o any o he join decision by a g oup o agen s. Ou
p oposal is also ela ed o pape s ha look o mechanisms ha agen s can use o choose
whe he o de elop a p ojec and which one o de elop (see, o ins ance, Moulin, 1984,
and Jackson and Moulin, 1992); o each good alloca ions in economic en i onmen s
wi h public goods and ex e nali ies (Va ian, 1994a and 1994b); o dissol e a pa ne ship
(McA ee, 1992); o sell (o no ) a p ojec o one agen when i affec s many (Jehiel e
al., 1996); o o awa d an indi isible good o one agen (in he spi i o King Solomon’s
dilemma; see, o ins ance, Glaze and Ma, 1989, and Pe y and Reny, 1999).
Ou con ibu ion can also expand he se o applica ions o he mul ibidding mech-
anism as pa o mo e complex mechanisms implemen ing solu ion concep s. Indeed,
a ian s o he mul ibidding mechanism (wi hou he need o eso o he ieb eaking
ule) ha e been used in se e al en i onmen s; see Pé ez-Cas illo and We s ein (2001),
Be gan iños and Vidal-Puga (2003, 2010), Macho-S adle e al. (2006), Po ei o (2007),
Slikke (2007), Ju e al. (2007), Kamijo (2008), Ehle s (2009), Ju and We s ein, (2009),
5

and Vesz eg (2010).1
Finally, ou pape can also be ela ed o he li e a u e on i ual (o −)implemen a-
ion (Ma sushima, 1988, and Ab eu and Sen, 1991) in he sense ha ou objec i e is no
o achie e an exac implemen a ion o an efficien and ai ou come bu o ge as “close”
as wished o ha alloca ion.
The pape is o ganized as ollows. In Sec ion 2, we p esen he en i onmen and he
smoo h mul ibidding mechanism. The equilib ium s a egies and ou come a e s a ed in
Sec ion 3. Sec ion 4 s udies he main p ope ies o he equilib ium ou come, including
he con e gence p ope ies when he pa ame e goes o ze o. We p o ide a simple
example in Sec ion 5. Finally, Sec ion 6 concludes he pape . All p oo s a e included in
he Appendix.
2 The en i onmen and he mechanism
We conside a se o agen s ={1}whichha e ochoosewhichp ojec willbe
ca ied ou o a se o possible p ojec s ={1}. The u ili y (payoff)o agen i
p ojec is selec ed is gi en by 
.
We deno e by ≡P∈
 he sum o agen s’ u ili ies i p ojec is implemen ed.
P ojec is efficien i ≥ o all ∈.Wedeno eby he se o efficien p ojec s,
ha is,
={∈≥ o all ∈}
In o ma ion abou all he alues 
is comple e among he agen s; ha is, each agen
knows no only he alue he assigns o he p ojec s bu also he alues assigned by
he o he agen s. Howe e , he planne does no ha e in o ma ion abou hese alues.
Al e na i ely, e en i she did ha e some in o ma ion, she would no wan o use i . The
planne is in e es ed in designing an impa ial mechanism ha will ea all he agen s in
a symme ic manne .
We p opose a smoo h mul ibidding mechanism h ough which agen s in luence he
p obabili y ha p ojec s a e selec ed. We now desc ibe he mechanism, which has a
unique s age.
1Fo u he discussions and applica ions, see Pé ez-Cas illo and Vesz eg (2007) and Vesz eg (2010).
6
Each agen ∈makes a ec o o bids ≡¡
¢∈in R,one o eachin wi h
P∈
=0. All agen s make hei decision simul aneously. Once he agen s ha e chosen
hei bids, he ou come o he smoo h mul ibidding mechanism is he ollowing.
Fo each ∈,≡P∈
deno es he agg ega e bid o p ojec and ≡()∈
deno es he ec o o agg ega e bids. The p obabili y ha p ojec be ca ied ou i he
ec o o agg ega e bids is is
()= ()
P∈()
whe e we conside he ollowing unc ion ():
()=  o all 0
+ o all ≥0
wi h 0. Finally, i p ojec is chosen, each agen ∈pays his bid o ha p ojec ,

, and he ecei es a ai sha e o he agg ega e bid, .The e o e,agen ’s u ili y i
p ojec is implemen ed is

−
+1

The smoo h mul ibidding mechanism bo ows om he mul ibbiding mechanism o
Pé ez-Cas illo and We s ein (2002) he idea o allowing he agen s o exp ess hei ela-
i e p e e ence o p ojec s h ough a ec o o bids. Howe e , unde he o iginal mecha-
nism, he p obabili y o selec ing any p ojec ab up ly jumps om 0 o 1 as he agg ega e
bid o his p ojec jus passes he maximum agg ega e bid o he o he p ojec s. Unde
ou p oposal, a highe (posi i e) agg ega e bid o a p ojec inc eases he p obabili y ha
i is selec ed, bu he inc ease is “smoo h”. This ea u e allows us o offe a mechanism
ha does no equi e ad hoc ieb eaking ules.
3 The equilib ia o he smoo h mul ibidding mecha-
nism
In his sec ion, we cha ac e ize he Nash equilib ia (NE) o he smoo h mul ibidding
mechanism. We p oceed as ollows. Fi s , we de i e se e al p ope ies ha a e necessa ily
7
sa is ied by NE o he game. Second, we use hese p ope ies o p o ide a cha ac e iza ion
o he se o NE.
Le us p oceed now wi h he analysis. Conside a ec o o agen s’ bids ()∈and le
deno e he se o p ojec s o which he agg ega e bid is posi i e unde his ec o o
s a egies, ha is, ≡{∈0}. Simila ly, deno e by ≡{∈0}and
≡{∈=0}so we ha e ∪= . Addi ionally, we deno e  he numbe
o p ojec s in .2The p obabili y ha p ojec ∈is chosen is gi en by
()=
+
+∈ o all ∈

+∈ o all ∈ 
Agen chooses his ec o o bids  o maximize his expec ed p o i s gi en he bids
chosen by he es o he agen s. Agen ’s p o i s a e
Π(
−)=X
∈
()∙
−
+1
¸
The e o e, agen chooses  o sol e he ollowing p og am, which we deno e by []:
X
∈
()∙
−
+1
¸
s. . X
∈

=0
To p oceed wi h ou analysis we no e i s , ha agen ’s p og am []is well beha ed
excep ha he de i a i e on he igh o unc ion ()wi h espec o (hence, wi h
espec o 
as well) is diffe en om i s de i a i e on he le , a he poin =0.
We in oduce he Fi s -O de Condi ions (FOCs) o he p og am. Deno ing by  he
Lag ange mul iplie o he cons ain , he FOC o [] o any ∈a e:



=Π


(
−)+=−(−1)
()+=0,(1)
whe eweha e akenin oaccoun ha ()
=0 o all ∈and ∈. I iswo hwhile
o no ice ha ()is he same o all ∈, which suppo s he ollowing p ope y:
2Al hough he se s ,,anddepend on he he ec o o agg ega e bids , we a oid using he
no a ions (),()(),and() o simplici y.
8
inc easing agen ’s bid o a p ojec in and dec easing ano he o his agen ’s bids o
adiffe en p ojec in does no ma e , as long as bo h p ojec s s ill ecei e a nega i e
agg ega e bid a e he changes.
The FOC o any ∈is



=()
∙
−
+1
¸−(−1)
()+
X
∈ {}
()
∙
−
+1
¸+X
∈ 
()
∙
−
+1
¸+=0 (2)
whe e
()

=(−1)+P∈ {}
¡ +P∈¢2(3)
()

=−+
¡ +P∈¢2 o all ∈ {}(4)
()

=−
¡ +P∈¢2 o all ∈ .(5)
Finally, o any ∈, i needs o be he case ha 

≥0on hele and 

≤0on
he igh . In ac , he de i a i e on he le is he same as he le -hand side o equa ion
(1), which is independen o . The e o e, he de i a i e 

≥0on he le always holds
(i holds wi h equali y). The e o e, we only ha e o add he ollowing condi ion:



=()
∙
−
+1
¸−(−1)
()+
X
∈
()
∙
−
+1
¸+X
∈ (∪{})
()
∙
−
+1
¸+≤0,(6)
o any ∈ whe e
()

=(−1)+P∈
¡ +P∈¢2(7)
and ()
is gi en by (4) o any ∈, and i is gi en by (5) o any ∈ (∪).
The p e ious FOCs a e necessa y (al hough no sufficien ) o cha ac e ize he NE o
he p oposed mechanism gi en ha any equilibi um mus be in e io .
Nex , we use he FOCs o each agen ’s p og am o cha ac e ize he se and he
NE agg ega e and indi idual bids o hese p ojec s. Lemma 1 con eys use ul in o ma ion
abou he equilib ium agg ega e bids o he p ojec s in .
9
When only efficien p ojec s a e selec ed by he mechanism, he o m o he agg ega e
bids is simple. Acco ding o his exp ession, he agg ega e bid will be highe as he
diffe ence be ween he alue o an efficien p ojec and hose o he o he p ojec s inc eases.
Mo eo e , all efficien p ojec s will be selec ed wi h equal p obabili y app oxima ely equal
o 1
as he pa ame e becomes a bi a ily small. In pa icula , he p obabili y ha an
efficien p ojec is selec ed con e ges o 1as  ends owa d 0.
P oposi ions 6 and 7 enable us o p o ide a inal esul on he ela i e efficiency o he
mechanism as he alue o he pa ame e becomes small. Speci ically, we show ha , o
any equilib ium, he p obabili y o implemen ing an inefficien p ojec con e ges o ze o.
The e o e, he ou come o he mechanism ge s as close o efficiency as one wishes as he
pa ame e  ends owa ds ze o.
P oposi ion 8 The ou come o he smoo h mul ibidding mechanism con e ges o ull
efficiency as he pa ame e con e ges o ze o. In o he wo ds, i p ojec ∈deno es
an inefficien p ojec , i s p obabili y o be implemen ed a he equilib ium con e ges o ze o
as becomes small.
The abo e esul con i ms ha he effec o a a ia iono hepa ame e is in ui i e.
Rega ding he ac ual implemen a ion o he mechanism, small alues o his pa ame e
will ensu e ha he chance o choosing an inefficien p ojec comes close o ze o.
5Example
Be o e concluding he pape , i migh be use ul o highligh he main p ope ies o he
mechanism wi h a simple example. Le us conside he ollowing si ua ion.
Two agen s (1and 2)ha e omakeacollec i edecisionon heimplemen a iono a
p ojec . The e a e ou po en ial choices co esponding o he se ={1234}whe e
he agen s’ bene i s a e: 1
1=6,2
1=3;1
2=4,2
2=6;1
3=2,2
3=1;and1
4=8,
2
4=2, espec i ely. The weigh ing pa ame e is posi i e; we will highligh how i s alue
in luences he ou come o he mechanism.
A equilib ium o he smoo h mul ibidding mechanism, P ojec 3 will ecei e a nega i e
agg ega e bid o any possible ( his ollows om Theo em 1 (a)). P ojec 1 will also
16

ecei e nega i e agg ega e bid as soon as 12. In his case, P oposi ion 7 p o ides
he exp ession o he agg ega e bids o p ojec s 2 and 4 ( he efficien p ojec s):
2=4=−+√2+40
and Theo em 1 (c) enables one o ind he equilib ium indi idual bids o p ojec s 1 and
2. Fo example, he bids ha agen s submi o p ojec 2 a e
1
2=−2+1
2h−+√2+4i
2
2=2+1
2h−+√2+4i.
The p obabili y ha p ojec s 2 and 4 a e selec ed a equilib ium is
2()=4()= √4+
2£√+√4+¤
he e o e, each 2()and 4()con e ges o 12as con e ges o ze o.
Finally, ega ding he agen s’ equilib ium payoffs, we know om P oposi ion 4 ha ,
o ins ance, agen 1’s payoffis gi en by he ollowing exp essions:
Π1=5+1
2"1
£√+√4+¤³10√4++6
√´−8#
which co esponds o his agen ’s alue o he a e age p ojec (5) plus his ai sha e o
he collec i e bene i s. The collec i e bene i s con e ge owa ds he o al alue 10 o an
efficien p ojec minus he o al alue o he a e age p ojec 8.The e o e,Π1con e ges
o 6as con e ges o 0.
6Conclusion
Relying on he main cha ac e is ics o he mul ibidding mechanism (Pé ez-Cas illo and
We s ein, 2002), we de eloped a new p ocedu e o choosing efficien p ojec s in si ua ions
whe e he social planne does no ha e in o ma ion on he agen s’ p e e ences.
E en hough he p esen p o ocol does no achie e efficiency, i has a numbe o in e -
es ing p ope ies compa ed o he mechanism de eloped in Pé ez-Cas illo and We s ein
(2002). Tha is, i implemen s a unique equilib ium ou come, sa is ies a ai ness p ope y
17
and is immune o he p oblems highligh ed by Ehle s (2009) as he use o ieb eaking ules
is a oided (by making he p obabili y o selec a gi en p ojec con inuous). Mo eo e , i
may come as close o ull efficiency as he social planne wishes.
As uniqueness and ai ness o he esul ing ou come a e impo an p ope ies o p ac-
ical implemen a ion (among o he hings, ai ness ensu es ha he mechanism will be
poli ically easible), his mechanism may be conside ed a aluable ool o such p oblems
o collec i e decision making.
7 Appendix
P oo o Lemma 1. Assume con ains a leas wo p ojec s, o he wise he lemma
holds i ially. The de i a i e o he payoff o any agen ,whenaddinganin ini esimal
 o 
and subs ac ing  om 
0is
1
¡ +P∈¢∙
−
+1
¸−(−1)

(+)
¡ +P∈¢−
1
¡ +P∈¢∙
0−
0+1
0¸+(−1)

(+0)
¡ +P∈¢
Thep e iousde i a i emus beze oa heop imum, ha is,

−
−(−2)
=
0−
0−(−2)
0.(11)
Summing o e we ge −(−1) =0−(−1) 0, which is equi alen o (8).
P oo o P oposi ion1. The FOC o any ∈implies =(−1)


(+∈).
The e o e, we w i e he FOC wi h espec o ∈(equa ion (2)) as (a e easy simpli i-
ca ions)
1
¡ +P∈¢2"Ã +X
∈
!∙
−
+1
¸−X
∈
∙
−
+1
¸#−

¡ +P∈¢2X
∈

−(−1)


¡ +P∈¢=0(12)
o Ã +X
∈
!∙
−
−(−2)
¸−X
∈
∙
−
+1
¸−X
∈

=0(13)
18
Summing (13) o e ∈we ob ain
Ã +X
∈
![−(−1) ]−X
∈
−X
∈
=0
i.e.,
−X
∈
− (−1) +X
∈
(−)−(−1) X
∈
=0(14)
No e ha we can w i e he las wo e ms in (14) as
X
∈
[(−)−(−1) ]=−1
−1X
∈
[(−)−(−1) ]2=
−1
−1X
∈
(−)2−(−1) 2
+2−2X
∈
,
whe eweha eusedequa ion(8).The e o e,(14)canbew i enas()=0.
P oo o Lemma 2. Fi s , suppose  con ains a leas wo p ojec s. Take p ojec s
∈ and ∈sa is ying ≥. We know ha o any agen ∈, changes in
¡
¢∈ do no in luence his p o i s as long as ≤0 o all ∈ is main ained.
The e o e, i 0 hen agen can inc ease 
 o 
=
−and dec ease 
 o

=
+ o some o he ∈ . The de i a i e o he payoff o any agen ,when
adding a posi i e in ini esimal  o 
and subs ac ing  om 
is
1
¡ +P∈¢£
−¡
−¢¤−(−1)


¡ +P∈¢−
1
¡ +P∈¢∙
−
+1
¸+(−1)

(+)
¡ +P∈¢
Thep e iousde i a i ecanno beposi i ea heop imum, ha is,
£
−
+¤−∙
−
−(−2)
¸≤0 o all ∈.
Summing he p e ious equa ion o e ,wege
−+(−1)≤0.(15)
Howe e , he las inequali y canno hold i ≥and 0
19
Second, suppose  ={}and pick such ha is he lowes among he elemen s
in . Taking in o accoun ha ≤, hen≤ o all ∈.Fo hisp ojec ,
0()|=0=−" (−1) + 2 X
∈
−2#≤− (−1) 0
and
()|=0=−X
∈
−1
(−1) X
∈
(−)2≤0.
The e o e, 0is no possible.
P oo o P oposi ion2. We i s p o e by con adic ion ha p ojec does no
belong o i (10) does no hold. We know ha , acco ding o Lemma 2, ⊂i ∈.
Deno e by  he p ojec in wi h he lowes o al alua ion: ≤ o all ∈.Then
()|=0=−X
∈
−1
(−1) X
∈
(−)2≤
−X
∈
−1
(−1)
2
X
∈
(−)≤0
Also, 0()|=0=−£ (−1) + 2 P∈−2¤0which, oge he wi h 00()
0implies ha ()0 o all posi i e . Howe e , his is no possible a equilib ium.
Second, we p o e ha (10) does no hold i ∈ .No e ha (10)canno happen
o i {}= . The e o e, we ake ∈ and and suppose ha he e a e a leas
wo p ojec s ou side .Conside some∈. By he same calcula ions as in he p oo
o Lemma2,weob ain(see(15))−+(−1)≤0, ha is,≤1
(−1) (−)
o , ()|=1
(−1) (−)≤0This is equi alen o
−1
(−1) (−)2−1
(−1) " (−1) + 2 X
∈
−2#(−)+
−X
∈
−1
(−1) X
∈
(−)2≤0,(16)
i.e.,
−X
∈
−1
(−1) X
∈
(−)2−2
(−1) X
∈
(−)+
2
(−1)(−)−1
(−1)(−)2≤0
20
Using ha −(−)2−2(−)=−2
−2
+2and 2(−)−(−)2=
2
−2
, he p e ious inequali y is equi alen o
−X
∈
−1
(−1) 2
−1
(−1) X
∈
2
+2
(−1)X
∈
+
1
(−1) 2
−1
(−1) 2
≤0
i.e.,
−X
∈
−1
(−1) X
∈
(−)2≤0(17)
Gi en ha ⊃ o any ∈, i is necessa ily he case ha equa ion (10) canno
hold, as we wan ed o p o e.
P oo o Theo em1.The necessi y o pa s ()and ()comes om p oposi ions 1
and 2. Fo pa (), no e ha om (11), we know ha

−
+1
=
−
+1
+(−1)
(−)
o any  ∈. The e o e, we can w i e (12) as
Ã +X
∈
!∙
−
+1
¸−X
∈
∙
−
+1
+(−1)
(−)¸−
X
∈

−−1
Ã +X
∈
!=0,
i.e.,
∙
−
+1
¸−X
∈

−1
"(−1) X
∈
2
+(−1) #=0(18)
We use (8) o show ha
X
∈
2
=X
∈∙+1
(−1) (−)¸2
=
2
+2
(−1)X
∈
−2
(−1)+1
(−1)2X
∈
(−)2
21

The e o e, (18) is equi alen o
∙
−
+1
¸−X
∈

−
1
"(−1) 2
+2X
∈
−2+1
(−1) X
∈
(−)2+(−1) #=0
and, using ha ()=0,weob ain
∙
−
+1
¸−X
∈

−1
"−X
∈
#=0,
and pa () ollows. Fo pa (), om he same calcula ions as in he p oo o Lemma
3, i ollows ha , o any agen , any p ojec ∈and any ∈we ha e



=1
¡ +P∈¢£
−
¤−1
¡ +P∈¢∙
−
−(−2)
¸≤0
as agen s do no ha e incen i es o de ia e. This implies he ollowing inequali y:

−
−∙
−
−(−2)
¸≤0(19)
Using ()and ew i ing, we check ha pa () ollows o any ∈.Pa ()is also
implied by () o any ∈when  is a single on,  ={},using
=−P∈
.
Finally, when  con ains a leas wo p ojec s, any agen can unila e ally amend his
bids ega ding he p ojec s in  o make any p ojec wi h an ini ially nega i e agg ega e
bid ge one equal o ze o. Mo eo e , he esul ing si ua ion is payoffequi alen o he
ini ial one. This implies ha condi ion (19) mus hold o all p ojec s ∈once we
inc ease 
 o 
so ha he =0, ha is,
=
−. The e o e, condi ion (d) mus
hold.
We now show ha () o ()a e also sufficien condi ions o NE. Conside any ec o
o bids ()∈sa is ying () o ().Wewillp o e ha ()∈is indeed a NE by showing
ha is a bes esponse o −.
Any bes esponse  o −mus sa is y he FOCs. We deno e by =
+P∈ 

o any ∈ and by , he se and numbe co esponding o he ec o o bids
(
−). Following he same calcula ions as in he p oo o Lemma 1, FOCs imply

−
−(−2)
=
0−
0−(−2)
0 o any 0∈.(20)
22
Also, when  has a leas wo elemen s, calcula ions simila o hose in Lemma 2 imply

−
+≤
0−
0−(−2)
0 o any ∈ 0∈.(21)
When  only con ains one elemen , ha is,  ={} o some ∈, hen(21)
also holds as i is implied by (20). Indeed, summing (20) o e ∈ and aking in o
accoun ha 
=−P∈
and =−P∈,weob ain
X
∈

+
+(−2)
=(−1)
0−(−1)
0−(−1)(−2)
0 o any 0∈,
ha is,

−
+=
0−
0−(−2)
0+
X
∈

+2(−1)
−
0+
0+(−2)
0 o any 0∈.
The e o e, (21) holds i

0−
0+1
0≥1
X
∈

+(−1)
∙2
+0¸ o some 0∈
Since =−P∈ , i is necessa ily he case ha 2
+0≤0 o some 0∈ .
Mo eo e , 
0−
0+1
0≥1
P∈
 o any 0∈ .3The e o e, (21) also holds o
when  ={}.
Now, we ake any 0∈and ew i e (20) and (21) as

+X
∈ 

−(2−2)
=
0+X
∈ 

0−(2−2)
0 o any 0∈,(22)

+X
∈ 

≤
0+X
∈ 

−(2−2)
0 o any ∈ 0∈.(23)
3Any bes esponse mus ensu e expec ed p o i s highe o equal han 1
P∈
,whichis
he le el ha agen can secu e wi h he “sa e” s a egy 
=−P∈ 
:p o i s unde a e
1
P∈£
−
¤=1
P∈
because =0 o all ∈. The e o e, all he p ojec s ∈
mus p o ide his le el o p o i s in case hey a e chosen; o he wise, agen woulddec easeall hebids
on hose p ojec s which p o ide less p o i s (he would also inc ease 
, s ill ensu ing ha is nega i e);
his would inc ease he p obabili y o success o hose p ojec s whose p o i s in case he e a e chosen is
highe o equal han 1
P∈
.
23
Equa ion (23) is a necessa y condi ion o  o be in  . Simila ly, because is
posi i e i ∈, a necessa y condi ion o  o be in is ( ollowing (22))

+X
∈ 



0+X
∈ 

0−(2−2)
0(24)
The e o e, i 0∈, hen∈i and only i (24) holds. Equa ion (24) implies ha i
0∈, hen∈i 
+P∈ 
is la ge han 
0+P∈ 
0. An implica ion is ha
∈i and only i 
+P∈ 
is la ge han some h eshold. Also no ice ha his is
also necessa ily ue o he se (possibly wi h a diffe en h eshold). The e o e, ei he
⊂o ⊂.
We go back o (20), which we ew i e as (25)
(2−2)
¡
0−
¢=
0−
−(−2)
X
∈ ¡
0−
¢ o any 0∈.(25)
Taking in o accoun ha (25) also holds o (ins ead o )i 0∈, hen

0−
=
0−
 o any 0∈∩,(26)
ha is, 
=
+(and also =+), o some ∈R, o all∈∩.
Take some ∈. The FOC wi h espec o 
is (see (13))
⎛
⎝ +X
∈
⎞
⎠∙
−
−(−2)
¸−X
∈
∙
−
+1
¸−X
∈

=0(27)
Fi s , suppose ha ≤0Then, (24) is mo e limi ing o  han o ; he e o e, ⊂.
Equa ion (27) becomes
⎛
⎝ +X
∈
+⎞
⎠∙
−
−(−2)
−2(−1)
¸−
X
∈
(+)∙
−
+1
−(−1)
¸−X
∈

=0
24
which we w i e as
Ã +X
∈
!∙
−
−(−2)
¸−X
∈
∙
−
+1
¸−X
∈

−
⎛
⎝X
∈ 
⎞
⎠∙
−
−(−2)
¸+X
∈ 
∙
−
+1
¸+
⎛
⎝∙
−
−(−2)
¸−2(−1)
⎛
⎝ +X
∈
⎞
⎠−X
∈∙
−
+1
¸+(−1)
X
∈
⎞
⎠+
2µ−2(−1)
+(−1)
¶=0(28)
The sum o he i s h ee e ms in (28) is equal o ze o, as i co esponds o he FOC o
. Then a e some calcula ions, (28) becomes
−⎛
⎝(−1)
[(+)+2 +X
∈
]+X
∈µ∙
−
+1
¸−∙
−
+1
¸¶⎞
⎠+
X
∈ 
µ∙
−
+1
¸−∙
−
+1
¸¶+⎛
⎝X
∈ 
⎞
⎠(−1)
=0(29)
We no ice ha because o condi ion (),£
−
+1
¤−£
−+1
¤=1
[−]
o any  ∈(in pa icula , his is also ue i  ∈). Mo eo e , Lemma 1 implies
ha −+(−1)=(−1)o any  ∈. The e o e, (29) can be w i en as
(−1)
X
∈ 
2
−(−1)
⎡
⎣+2 +2X
∈
⎤
⎦=0(30)
The i s e m in (30) is non-nega i e; in ac , i is ze o i and only i  is emp y.
Mo eo e , +2 +2P∈is posi i e. Taking in o accoun ha ≤0,(29)only
holds i  is emp y, ha is, =and =0.
Second, suppose ha ≥0, which implies ( ollowing (24)) ha ⊃.We ake
∈and we ew i e (27):
⎛
⎝ +X
∈
+ +X
∈ 
⎞
⎠∙
−
−(−2)
−2(−1)
¸−
X
∈
(+)∙
−
+1
−(−1)
¸−X
∈ 
∙
−
+1
¸−X
∈

=0
25
Bound can be ew i en as ollows:
=(−1)(−1)
2⎡
⎢
⎣
u
u
u
4
(−1)2(−1) ⎡
⎣(−1)∗−X
∈{}
⎤
⎦+1−1⎤
⎥
⎦≤
(−1)(−1)
2
u
u
u
4
(−1)2(−1) ⎡
⎣(−1)∗−X
∈{}
⎤
⎦≤
≡(−1)(−1)
2s4
(−1)2(−1)(−1)∗
The e o e, ∗−≤, which gi es he exp ession s a ed in he P oposi ion.
P oo o P oposi ion7.The exp ession o  ollows immedia ely om ()=0
once we ake in o accoun ha = o any  ∈when =. I is also immedia e
ha con e ges o 0as  ends owa ds 0.Finally,
()= +
 +P∈
=1

(2−)+ 22+4
(−1) ³−P∈´
 + 22+4
(−1) ³−P∈´
which con e ges o 1 as  ends owa ds 0.
P oo o P oposi ion8. Le ∈deno e a second-bes p ojec and ≡∗−0
deno e he diffe ence be ween he alue o an efficien p ojec and ha o .Weha e
∗−
∗=
∗0Le us conside ha he pa ame e  akes alues such ha
³
∗´21
(−1)(−1)
Then, by P oposi ion 6 we deduce ha p ojec does no belong o  o he abo e
alues o he pa ame e , which implies ha any inefficien p ojec is in  as well.
The e o e, o small enough alues o ,=and, acco ding o P oposi ion 7, he
p obabili y o selec ing an efficien p ojec con e ges o 1as he pa ame e  ends o
ze o, which ensu es con e gence o an efficien ou come as  ends o ze o.
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