Efficient incentives with social preferences

Daske, Thomas; Ma ch, Ch is oph
A icle
E icien incen i es wi h social p e e ences
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Daske, Thomas; Ma ch, Ch is oph (2024) : E icien incen i es wi h social
p e e ences, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol.
19, Iss. 3, pp. 975-999,
h ps://doi.o g/10.3982/TE5335
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Theo e ical Economics 19 (2024), 975–999 1555-7561/20240975
E icien incen i es wi h social p e e ences
Thomas Daske
Depa men o Economics and Policy, Technical Uni e si y o Munich
Ch is oph Ma ch
Depa men o Economics, Uni e si y o Bambe g
We explo e mechanism design wi h ou come-based social p e e ences. Agen s’
social p e e ences and p i a e payo s a e all subjec o asymme ic in o ma ion.
We assume quasi-linea u ili y and independen ypes. We show how he asym-
me y o in o ma ion abou agen s’ social p e e ences can be ope a ionalized o
sa is y agen s’ pa icipa ion cons ain s. Ou main esul is a possibili y esul o
g oups o a leas h ee agen s: Any such g oup can esol e any gi en alloca ion
p oblem wi h an ex pos budge -balanced mechanism ha is Bayesian incen i e-
compa ible, in e im indi idually a ional, and ex pos Pa e o-e icien .
Keywo ds. Mechanism design, social p e e ences, Bayesian implemen a ion,
pa icipa ion cons ain s, pa icipa ion s imula ion, con es s, money pump.
JEL classi ica ion. C72, C78, D62, D82.
1. In oduc ion
How can alloca ion p oblems be esol ed in an e icien and mu ually accep able way?
The li e a u e on mechanism design has pos ula ed ou desi able p ope ies o incen-
i e mechanisms: incen i e compa ibili y, ex pos Pa e o e iciency, ex pos budge bal-
ance, and in e im indi idual a ionali y. Bayesian implemen a ion is sui able o achie e
he i s h ee o hese p ope ies (see, e.g., A ow (1979), d’Asp emon and Gé a d-
Va e (1979)). O en, howe e , Bayesian mechanisms iola e agen s’ pa icipa ion con-
s ain s.1
Thomas Daske: [email p o ec ed]
Ch is oph Ma ch: [email p o ec ed]
An ea lie e sion ci cula ed unde he i le “E icien incen i es in social ne wo ks: Gami ica ion and he
Coase heo em” and is a ailable a h p://hdl.handle.ne /10419/222527.
Fo hei help ul commen s and c i ical ema ks, we hank Claude d’Asp emon , Jacques C éme , Benny
Moldo anu, Ma co Sahm, Klaus Schmid , Johannes Schneide , Roland S ausz, and Robe on Weizsäcke
as well as pa icipan s a he Eu opean Win e Mee ing o he Econome ic Socie y in Milan, he Wo ld
Cong ess o he Game Theo y Socie y in Maas ich , he Annual Mee ing o he Associa ion o Public Eco-
nomic Theo y in S asbou g, he Annual Cong ess o he In e na ional Ins i u e o Public Finance in Glas-
gow, he Eu opean Mee ing o he Econome ic Socie y in Manches e , he Annual Cong ess o he Ge man
Economic Associa ion in Leipzig, he i ual Econome ic Socie y Wo ld Cong ess, and he Annual Cong ess
o he Socie y o he Ad ancemen o Economic Theo y in Canbe a. We a e pa icula ly g a e ul o se e al
anonymous e e ees o hei pa ience and e y help ul sugges ions.
1Fo se ings wi h independen p i a e signals see, e.g., Mye son and Sa e hwai e (1983), Maila h and
Pos lewai e (1990), Williams (1999), and Segal and Whins on (2016).
©2024 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE5335
976 Daske and Ma ch Theo e ical Economics 19 (2024)
Bayesian mechanisms ha econcile all ou p ope ies exis i agen s’ p i a e sig-
nals (o ypes) a e su icien ly co ela ed. C éme and McLean (1985,1988) show ha
he designe can exploi his co ela ion o alida e he agen s’ epo s, ex ac all in o -
ma ion en s, and ensu e pa icipa ion en passan .2
Mezze i (2004) shows ha he logic
o C éme and McLean (1985,1988) can be ex ended o he case o independen p i a e
signals i he designe is pe mi ed o implemen a wo-s age mechanism. The alloca ion
p oblem can be esol ed wi h unanimous pa icipa ion by sequen ially adminis e ing a
social al e na i e and ans e s, wi h agen s i s epo ing hei p e e ence ypes and
hen hei sa is ac ion wi h he chosen al e na i e be o e inally ecei ing (o paying)
ans e s.
The p esen s udy en iches he se o possibili y esul s. We assume ha ypes a e
independen (in con as o C éme and McLean (1985,1988)) and ha he e is only one
ound o epo ing (in con as o Mezze i (2004)). Speci ically, we conside agen s wi h
ou come-based social p e e ences ha a e p i a ely known (nex o p i a ely known
p e e ences o consump ion). Tha is, agen s ca e abou he o e all dis ibu i e e ec s
o a mechanism, and hei dis ibu i e p e e ences a e p i a e in o ma ion. We show
how his kind o in o ma ion asymme y can be ope a ionalized o sa is y agen s’ pa -
icipa ion cons ain s.
Ou main esul , Theo em 1, s a es ha any g oup o a leas h ee agen s can e-
sol e any gi en alloca ion p oblem wi h an ex pos budge -balanced mechanism ha
is Bayesian incen i e-compa ible, in e im indi idually a ional, and ex pos Pa e o-
e icien . I builds on he ollowing insigh s. In quasi-linea en i onmen s, a mechanism
can be designed such ha he incen i es o e eal payo ypes and social ypes a e sep-
a a ed. While he alloca ion p oblem can be esol ed h ough payo - ype condi ional
budge -balanced ans e s, pa icipa ion can be s imula ed h ough addi ional budge -
balanced ans e s ha condi ion on agen s’ social ypes. The la e is possible o mo e
han wo agen s when le e aging he di e ences in agen s’ o he - ega ding conce ns.
Technically, we exploi ha each agen ’s u ili y is a linea combina ion o all agen s’
p i a e payo s, which a e weigh ed acco ding o ha agen ’s o he - ega ding conce ns.
This linea i y enables us o ende he agen s’ social ypes s a egically inope a i e in he
payo - ype condi ional mechanism, so we can use hem in a sepa a e, social- ype con-
di ional mechanism o c oss-subsidize he o me . In his manne , ou solu ion bundles
wo s a egically independen mechanisms. (Ou bundling o wo mechanisms esem-
bles Mezze i (2004). We de ail he di e ences be ween his and ou s udy in Sec ion 6.4.)
Un il ecen ly, he li e a u e on e icien design has ei he neglec ed social p e e -
ences al oge he o assumed hem o be common knowledge.3An excep ion is Bie -
b aue and Ne ze (2016), who s udy mechanism design when agen s ha e p i a ely
known in en ion-based social p e e ences. They show ha his so o social p e e ences
allows o e icien , indi idually a ional design i and only i all agen s a e (commonly
known o be) condi ionally p o-social. Ou s udy di e s om hei s in he kind o social
p e e ences unde conside a ion as well as in he condi ions o and he d i ing o ces
2Likewise, McA ee and Reny (1992), McLean and Pos lewai e (2004), Kosenok and Se e ino (2008).
3See, e.g., Desi aju and Sapping on (2007), Kucuksenel (2012), and Tang and Sandholm (2012).
Theo e ical Economics 19 (2024) E icien incen i es 977
behind he possibili y esul . Fi s , we conside uncondi ional ou come-based a he
han in en ion-based social p e e ences, and nex o al uism and sel ishness, we allow
o an i-social p e e ences such as spi e.4Second, he e ela ion p inciple holds in ou
se up, bu no in Bie b aue and Ne ze ’s (2016), as hei agen s’ p e e ences depend
on he se o ac ions (i.e., messages) a ailable in he mechanism. Indeed, he indepen-
dence o agen s’ p e e ences om he mechanism dis inguishes ou pape om a i-
ous o he s on mechanism design wi h in en ion-based social p e e ences (e.g., An le
(2015), Kozlo skaya and Nicoló (2019)). Finally, he possibili y esul o Bie b aue and
Ne ze (2016) exploi s he mechanism dependence o p e e ences by in oducing ad-
di ional messages ha a e no chosen in equilib ium, bu manipula e he kindness o
u h- elling; his cons uc ion only wo ks in he absence o sel ish ypes. In con as ,
ou esul exploi s he asymme y o in o ma ion abou agen s’ social p e e ences.
No ably, ou s udy ela es o he li e a u e on money pumps (o du ch books). This
li e a u e has a long adi ion in indi idual-choice heo y. I shows how non a ional in-
di idual decision-making can be exploi ed o pull agen s in o ansac ions hey s and
o lose om (see, e.g., Bo de and Segal (1994)andRubins ein and Spiegle (2008); o
asu ey,seeYaa i (1998)). In he mul i-agen e sion, a g oup o agen s is subjec o
a money pump i an ou side pa y is “able o ex ac money om he agen s wi hou
pu ing any money a isk” (Nau (1992, p. 380)). Ou s udy shows ha a g oup o a leas
h ee agen s wi h p i a ely known social p e e ences can be o e ed an ex pos budge -
balanced (nonze o) ans e scheme ha all o hem accep ex in e im. This implies ha
a ans e scheme can be cons uc ed ha ex ac s money om he g oup ia pa icipa-
ion ees and is s ill unanimously accep ed, and, hus, becomes a money pump. While
he li e a u e has ocused on non a ional expec a ions (see, e.g., Eliaz and Spiegle (2007,
2009), Chen, Micali, and Pass (2015), We ne (2022)), we show ha mul i-agen money
pumps can be g ounded in nons anda d a ional p e e ences. As in An le (2023) o
non a ional expec a ions, we equi e su icien ly many agen s, a leas h ee in ou case.
The ollowing example p o ides a basic in ui ion o how asymme ic in o ma ion
abou agen s’ social p e e ences can be exploi ed o gene a e a money pump. Conside
wo agen s, each o whom is ei he sel ish (ca ing only abou he p i a e payo ) o al-
uis ic (weigh ing he o he ’s payo hal as much as he own). Types a e independen
and equally likely. I bo h epo sel ish (al uis ic), each is axed ( ewa ded) 1 dolla ; i
hey epo opposi e ypes, he al uis mus pay he sel ish 2 dolla s. Clea ly, epo -
ing sel ish always yields a highe p i a e payo , incen i izing u h- elling o sel ish
agen s. Repo ing al uis always yields a conside ably la ge payo o he opponen
han epo ing sel ish, incen i izing u h- elling o al uis s. As unanimous pa icipa-
ion yields each ype an in e im-expec ed u ili y gain (as compa ed o a s a us quo o
ze o ans e s), agen s a e willing o pay o playing his game. Thus, a hi d agen can
o e o inance he game by balancing he budge (i.e., o ax o ewa d acco ding o
he ules) in e u n o a uni o m pa icipa ion ee. As ans e s a e ze o ex an e, a su -
4The beha io al ele ance o uncondi ional ou come-based social p e e ences has been well es ablished.
Fo e idence on al uism and sel ishness, see And eoni and Mille (2002), Cha ness and Rabin (2002), and
B uhin, Feh , and Schunk (2019). Fo e idence on spi e, see Saijo and Nakamu a (1995), Feh , Ho , and
Kshe amade (2008), and P edige , Vollan, and He mann (2014).
978 Daske and Ma ch Theo e ical Economics 19 (2024)
icien ly small ee gua an ees ha all h ee agen s a e wan ing ex in e im o pa icipa e
in he ex ended game. Con o ming his scena io o he quo ed money-pump de ini ion
o “ex ac ing money om he agen s wi hou pu ing any money a isk” esembles a
go e nmen selling a casino license: An ou side pa y may en e he scene and o e ou
“ hi d agen ” he pla o m on which she can le o he s play ou game in e u n o hal
o he pa icipa ion ees.
In his example, when looking a he ac ual playe s (sel ish o al uis ic), money is e-
dis ibu ed ex in e im o hose agen s who ca e leas abou o he s. On he o he hand, a
p o-social agen in e im-expec s o impose a posi i e mone a y ex e nali y on he oppo-
nen , and his ex e nali y o e compensa es he emo ionally o in e im-expec ed mon-
e a y losses. These dis ibu i e e ec s a e a gene al ea u e o he a ious money pumps
we de elop in his pape , al hough he no ions o ca ing leas and p o-sociali y will bea
mo e in ica e meanings. (We p esen his example in mo e de ail in Sec ion 5.1.)
The pape p oceeds as ollows. Sec ion 2ou lines he model amewo k. Sec ion 3
s a es and in e p e s ou main esul . Sec ion 4de ails he p oo . Sec ion 5illus a es
he in ui ion behind ou pa icipa ion-s imula ing ans e s. Finally, Sec ion 6 e lec s
upon he assump ions ha a e c i ical o ou esul , dis inguishes ou mechanism om
Mezze i’s (2004), and illus a es how pa icipa ion s imula ion can be implemen ed in
p ac ice. The Appendix p o ides addi ional p oo s.
2. The model
2.1 The alloca ion p oblem
The e is a g oup I={1, ,n}o n≥2 agen s and he e is a ini e se Ko social al-
e na i es. F om al e na i e k∈Kand a ans e i∈R,agen igains a p i a e payo
i(k, i|θi)=πi(k|θi)+ i,wi hπi:K×i→R.Agen i’s payo ype θibelongs o a ini e
se i,wi h|i|≥2. The collec ion o agen s’ payo ypes is deno ed by θ=(θi,θ−i)∈
=ii,whe eθ−i=(θj)j=i. Agen s exhibi social p e e ences in he o m o al uism
o spi e: F om he collec ion o p i a e payo s (j)j∈I,agen ide i es ex pos u ili y
uik,( j)j∈I,θ−i|θi,δi=
j∈I
δij j(k, j|θj),
whe e he alue δij ha iassigns o j’s payo , j=i, belongs o a closed (p ope ) in e al
ij =[δmin
ij ,δmax
ij ]⊂(−1/(n−1);1
), while δii =1 o alli. We e e o δij as i’s deg ee o
al uism owa d j, o he collec ion δi=(δij )j=i∈i=j=iij as i’s social ype, and o
he pai (θi,δi)as i’s ype.
The in o ma ion s uc u e is as ollows. Each agen is p i a ely in o med abou
he payo ype and social ype. Hence, he e is a ype dis ibu ion on ×(whe e
=ii) wi h s ic ly posi i e a iance o payo ypes and social ypes. Type eal-
iza ions a e independen ac oss agen s. An agen ’s payo ype and social ype ealize
independen ly acco ding o s ic ly posi i e densi ies, bu he a ious deg ees o al u-
ism de e mining his agen ’s social ype may co ela e. We assume ha agen s will ob-
se e each o he ’s payo s ex pos . (We make he implici assump ion o con inuous
social- ype dis ibu ions o keep he exposi ion simple, bu all esul s a e equally alid i
a social- ype se con ains mass poin s; see Sec ion 5.1.)

Theo e ical Economics 19 (2024) E icien incen i es 979
A ew ema ks a e app op ia e. Fi s , he in e al (−1/(n−1);1
)is he maximum
ange o al uism, o spi e, o which agen s ca e abou o e all ma e ial e iciency while
s ill being sel ish o he ex en ha e e y one o hem p e e s a dolla o be he own
a he han ha ing ha same dolla dis ibu ed among he o he s. Second, despi e he
asymme y o in o ma ion, i can s ill be common knowledge who is a iend and who is
a oe. Fo ins ance, i δmax
k ,δmax
k <0<δ
min
ij ,δmin
ji , hen, in compa ison, iand ja e iends,
whe eas kand a e oes. Likewise, i can be common knowledge ha ilikes jmo e
han k,whichis hecasei δmax
ik <δ
min
ij . Finally, while we assume ha he a iance o
e e y δij is s ic ly posi i e, i is allowed o be a bi a ily small. Recip ocal social p e e -
ences can hus be cap u ed by le ing ij =ji and δmin
ij ≈δmax
ij .
The agen s’ p oblem is o choose a social al e na i e kand ans e s ( i)i∈Isuch ha
he esul ing alloca ion, i.e., he collec ion o p i a e payo s, is ex pos Pa e o e icien .
We equi e ha agen s mus do so wi hou ha ing access o an ou side sou ce o money,
such ha ans e s mus be weakly budge -balanced: i∈I i≤0.
2.2 Re ela ion mechanisms
A di ec e ela ion mechanism in ol es he agen s in a s a egic game o incomple e in-
o ma ion in which hey a e asked o epo hei ypes u h ully. Types a e epo ed
simul aneously. Based on hei epo s, a social al e na i e is chosen and ans e s a e
made. As he e ela ion p inciple applies o he p esen se up (Mye son (1979)), he e is
no loss o gene ali y in conside ing only di ec mechanisms. Fo mally, a di ec mecha-
nism is gi en by a pai k,Twi h alloca ion unc ion k:×→Kand ans e scheme
T=( i)i∈I:×→Rn. No ice ha ans e s may ake a bi a y nega i e alues.
Deno e by Ui(ˆ
θi,ˆ
δi|θi,δi)agen i’s in e im-expec ed u ili y om epo ing (ˆ
θi,ˆ
δi)
i he ue ypeis(θi,δi)while all he o he agen s epo hei ypes u h ully:
Ui(ˆ
θi,ˆ
δi|θi,δi)=j∈Iδij [¯πij (ˆ
θi,ˆ
δi)+¯
ij (ˆ
θi,ˆ
δi)],whe e ¯πij (θi,δi)=Eθ−i,δ−i[πj(k(θ,δ)|
θj)] and ¯
ij (θi,δi)=Eθ−i,δ−i[ j(θ,δ)]. Fo con enience, Ui(θi,δi)=Ui(θi,δi|θi,δi).Then
he mechanism k,Tis Bayesian incen i e-compa ible i , o all i∈Iand all (θi,δi)∈
i×i,weha eUi(θi,δi)=max(ˆ
θi,ˆ
δi)∈i×iUi(ˆ
θi,ˆ
δi|θi,δi).5
2.3 E iciency and pa icipa ion
The ollowing lemma links ma e ial e iciency ( he maximum su plus o p i a e payo s)
o Pa e o e iciency. I allows us o ocus on alloca ion unc ions ha a e ex pos ma e-
ially e icien , k(θ,δ)=k(θ)∈a gmaxk∈Ki∈Iπi(k|θi), and ans e s ( i)i∈I ha a e
(s ic ly o ex pos ) budge -balanced, i∈I i=0.
Lemma 1. A mechanism is ex pos Pa e o-e icien only i ans e s a e ex pos budge -
balanced. I |δij |<1/(2n−3) o all iand all j= i, hen an ex pos ma e ially e icien
5Bayesian implemen a ion has been c i icized o assuming ha he dis ibu ion o agen s’ ypes is com-
mon knowledge. Be gemann and Mo is (2005)ha ep oposedex pos implemen a ion o en i onmen s
wi h in e dependen u ili ies, equi ing ha u h ul e ela ion o ypes cons i u es a Nash equilib ium.
Howe e , Jehiel, Meye - e Vehn, Moldo anu, and Zame (2006) show ha ex pos implemen a ion is “gene -
ically” no easible in he p esence o in o ma ional ex e nali ies, a inding ex ended by Zik (2021) oou
p esen con ex .
980 Daske and Ma ch Theo e ical Economics 19 (2024)
alloca ion unc ion is also ex pos Pa e o-e icien ; mo eo e , no ex pos budge -balanced
ans e scheme ex pos Pa e o-domina es ano he .
The in ui ion behind Lemma 1is his: I agen s swi ch om a social al e na i e ha
is ma e ially e icien o one ha is no o swi ch om one budge -balanced ans e
scheme o ano he , hen a leas one agen mus incu a ma e ial loss. Now conside he
agen whose ma e ial loss is la ges ; i his agen iis su icien ly sel ish, |δij |<1/(2n−3)
o all j=i, hen she would also incu a loss u ili y-wise. In con as , he Pa e o on ie
can be inde ini e o combina ions o social ypes sa is ying |δij |≥1/(2n−3),inwhich
case a subg oup o agen s migh be willing o ans e a bi a y amoun s o money o
hei join a o i e agen .6
Finally, k,Tis in e im indi idually a ional i i gains all agen s’ app o al a he in-
e im s age (i.e., unanimous app o al cons i u es a Bayes–Nash equilib ium a he s age
whe e agen s’ ypes a e p i a e in o ma ion). Following Segal and Whins on (2016), we
ep esen ese a ion u ili ies by he in e im-expec ed u ili ies ha agen s’ de i e om
a Bayesian mechanism k◦,T◦,wi hk◦:×→Kspeci ying “p ope y igh s” and
T◦=( ◦
i)i∈I:×→Rnspeci ying “liabili y ules.”
3. A possibili y esul
We es ablish ou main esul wi h he help o wo concep s: p e e ence-sepa a ing
mechanisms and pa icipa ion-s imula ing ans e s.
De ini ion 1 (P e e ence Sepa a ion and Pa icipa ion S imula ion). A p e e ence-
sepa a ing mechanism k,Tconsis s o he ex pos ma e ially e icien alloca ion
unc ion k:→K,wi hk(θ)∈a g maxk∈Ki∈Iπi(k|θi), and an ex pos budge -
balanced ans e scheme T=( 
i)i∈I:×→Rnde ined by

i(ˆ
θ,ˆ
δ)=
j=iEθ−iπjk(ˆ
θi,θ−i)|θj−Eθ−jπik(ˆ
θj,θ−j)|θi
 
he e ms o ade
+s
i(ˆ
δ)

pa icipa ion-
s imula ing
ans e s
,
whe e pa icipa ion-s imula ing (PS) ans e s s=(s
i)i∈I:→Rna e de ined by
join ly sa is ying he ollowing condi ions:
(i) PS ans e s sa e s a egy-p oo . Fo all i∈I,allδ∈,andall ˆ
δi∈i,

j∈I
δij s
j(δ)≥
j∈I
δij s
j(ˆ
δi,δ−i).
(ii) PS ans e s sa e ex pos budge -balanced. Fo all δ∈,

j∈I
s
j(δ)=0.
6An example is he g oup o h ee agen s wi h δ13 =δ23 >1/3, δ12 =δ21 =−1/3, and δ31 =δ32 =0, in
which agen s 1 and 2 a e willing o join ly ans e a bi a y indi idual amoun s >0 oagen 3.
Theo e ical Economics 19 (2024) E icien incen i es 981
(iii) F om unanimous pa icipa ion in s, each agen de i es a s ic ly posi i e
in e im-expec ed u ili y gain: Fo all i∈Iand all δi∈i,

j∈I
δij Eδ−is
j(δ)>0.
Theo em 1(Efficien Implemen a ion Wi h a Leas Th ee Agen s). I n≥3, hen
he e exis s a p e e ence-sepa a ing mechanism k,T ha is Bayesian incen i e-
compa ible, in e im indi idually a ional, ex pos budge -balanced, and ex pos ma e-
ially e icien . I |δij |<1/(2n−3) o all iand all j=i, henk,Tis necessa ily ex pos
Pa e o-e icien .
Be o e we p o e Theo em 1, we shall discuss he inne logic o ou mechanism. No-
ice i s ha , despi e he decoupling o incen i es o e eal payo ypes and social ypes,
ou mechanism asks agen s o epo hese ypes simul aneously.
Conside he e ms o ade, which ope a e on agen s’ payo ypes. As we will see, he
e ms o ade a e social-p e e ence obus in ha hey lea e agen s’ social p e e ences
s a egically i ele an . This is achie ed by applying he mu ual-concessions p inciple
o he dyadical expec ed-ex e nali y mechanism (A ow (1979); d’Asp emon and Gé a d-
Va e (1979)) o each and e e y single dyad. Fo he ma e ially e icien social al e na i e
k(ˆ
θ), he ans e o agen i o e e y o he jequals j’s expec a ion o i’s ma e ial payo
when j epo s payo ype ˆ
θj; ha is,i ans e s Eθ−j[πi(k(ˆ
θj,θ−j)|θi)] o jand ecei es
Eθ−i[πj(k(ˆ
θi,θ−i)|θj)] om j.
Fo wo o he - ega ding agen s, Bie b aue and Ne ze (2016) show ha he expec -
ed-ex e nali y mechanism is social-p e e ence obus . Agen s a e incen i ized o beha e
as i hey a e sel ish. I −i epo s he payo ype u h ully, hen Eθ−i[π−i(k(ˆ
θi,θ−i)|
θ−i)+ 
−i(ˆ
θi,θ−i)] =Eθ[πi(k(θ)|θi)]; he eby,i’s deg ee o al uism is ende ed s a egi-
cally i ele an . Bie b aue and Ne ze (2016, p. 570) also show ha , in hei amewo k,
he con en ional n-agen s expec ed-ex e nali y mechanism (see Mas-Colell, Whins on,
and G een (1995, pp. 886)) is social-p e e ence obus only unde an addi ional sym-
me y condi ion. In ou amewo k, social-p e e ence obus ness can be es ablished o
g oups o a bi a y size wi hou any symme y equi emen s.
Consequen ly, he e ms o ade p ese e agen s’ p i a ely known social p e e ences
as a s a egic deg ee o eedom, which is u ilized by pa icipa ion-s imula ing ans e s.
Those a e independen o he ac ual alloca ion p oblem and se e he pu pose o s imu-
la ing agen s’ pa icipa ion in he e ms o ade. While being ex pos budge -balanced,
PS ans e s yield agen s an in e im-expec ed Pa e o imp o emen upon he e ms o
ade. I his in e im-expec ed Pa e o imp o emen is ampli ied su icien ly h ough
uni o mly scaling up he PS ans e s, hen agen s’ in e im-expec ed u ili ies om unan-
imous pa icipa ion will ou weigh hei ese a ion u ili ies. No ice ha he scaling-up
is only possible i we allow ans e s o ake a bi a y nega i e alues.
Finally, we no e ha ou pa icipa ion-s imula ion app oach canno succeed in
dyads.
P oposi ion 1. Pa icipa ion-s imula ing ans e s do no exis i n=2.
982 Daske and Ma ch Theo e ical Economics 19 (2024)
P oo . Suppose he opposi e is ue. Then De ini ion 1(iii) equi es ha 0 <
Eδ−i[s
i(δ)] +δiEδ−i[s
−i(δ)] o bo h i∈{1, 2}and all δi∈i⊂(−1, 1), while s
−i(δ)=
−s
i(δ)due o ex pos budge balance. Hence, 0 <(1−δi)Eδ−i[s
i(δ)], implying ha
0<Eδ−i[s
i(δ)] o all i,δi.Bu hen0<Eδ[s
i(δ)] o bo h i, con adic ing ex pos budge
balance.
The in ui ion behind P oposi ion 1is s aigh o wa d: Unanimous pa icipa ion e-
qui es each social ype o in e im-expec a u ili y gain, bu as budge s mus be balanced
ex pos while each agen alues he own ma e ial wellbeing mo e han he o he ’s, his
equi es each social ype o in e im-expec a ma e ial bene i . These in e im expec a-
ions canno be mu ually consis en o all social ypes, ega dless o he speci ica ion
o ans e s s; o he wise, bo h agen s would bene i ma e ially ex an e, con adic ing
budge balance.
4. P oo o Theo em 1
The p oo o Theo em 1p oceeds in a se ies o lemmas. Th oughou , n≥3.
Lemma 2. P e e ence-sepa a ing mechanisms a e Bayesian incen i e-compa ible and ex
pos ma e ially e icien . I |δij |<1/(2n−3) o all iand all j= i, hey a e also ex pos
Pa e o-e icien .
P oo .Incen i e Compa ibili y. Suppose he agen s o he han i e eal hei ypes
u h ully. Then he ans e s ha iin e im-expec s o he sel and e e y o he j ead
¯
ii(ˆ
θi,ˆ
δi)=
=i
Eθ−iπk(ˆ
θi,θ−i)|θ−(n−1)Eθπik(θ)|θi+Eδ−is
i(ˆ
δi,δ−i),
¯
ij (ˆ
θi,ˆ
δi)j=i
=
=j
Eθ−i,θ−jπk(θ)|θ−
=i,j
Eθ−i,θ−πjk(θ)|θj
−Eθ−iπjk(ˆ
θi,θ−i)|θj+Eδ−is
j(ˆ
δi,δ−i)
=
∈I
Eθπk(θ)|θ−(n−1)Eθπjk(θ)|θj
−Eθ−iπjk(ˆ
θi,θ−i)|θj+Eδ−is
j(ˆ
δi,δ−i).
Agen i’s in e im-expec ed u ili y om epo ing (ˆ
θi,ˆ
δi) hus sa is ies
Ui(ˆ
θi,ˆ
δi|θi,δi)=
j∈I
δij Eθ−iπjk(ˆ
θi,θ−i)|θj+¯
ij (ˆ
θi,ˆ
δi)
=Eθ−i
∈I
πk(ˆ
θi,θ−i)|θ+
j=i
δij Eθ
∈I
πk(θ)|θ
−(n−1)Eθ
j∈I
δij πjk(θ)|θj+
j∈I
δij Eδ−is
j(ˆ
δi,δ−i).(1)
Theo e ical Economics 19 (2024) E icien incen i es 989
Figu e 2. The u ili y gain gi(δi)=Eδ−i[s
i(δ)] +δiEδ−i[s
−i(δ)] >0 ha asocial ypeδi
in e im-expec s unde he ans e scheme so (11), o wo di e en ype dis i-
bu ions, δi∈[δmin
i,δmax
i]=[−4/5, 4/5],Eδi[δi]=∓2/5, and Va δi[δi]=1/5, such ha
Eδ−i[s
i(δ)] =9/25 −δ2
i,Eδ−i[s
−i(δ)] =2δi±4/5, and gi(δi)=(δi±2/5)2+1/5.
abou M han abou he o he s in e im-expec s o in oke a edis ibu ion om he o h-
e s o M( om M o he o he s), which o e compensa es he emo ionally o in e im-
expec ed mone a y losses.
6. Discussion
6.1 Wha i social ypes a e common knowledge?
Asymme ic in o ma ion abou agen s’ social p e e ences is a key assump ion in he
abo e analysis. We can easily ule ou ha pa icipa ion s imula ion in he manne o
De ini ion 1would wo k o commonly known social ypes: Unde common knowledge,
De ini ion 1(iii) would ans o m in o he equi emen ha pa icipa ion-s imula ing
ans e s ex pos Pa e o-domina e he ans e scheme (si=0)i∈Io ex pos budge -
balanced ze o ans e s (i.e., j∈Iδij s
j(δ)>0 o alliand all δ), which is impossible
due o Lemma 1.
We shall also discuss wha is easible i social ypes a e common knowledge. Plau-
sibly, i agen s a e su icien ly al uis ic (i.e., δij →1 o alli,j= i), hen indi idual a-
ionali y is sa is ied o ma e ially e icien alloca ion unc ions and budge -balanced
ans e s; see also Kucuksenel (2012). Seeking solu ions ha wo k o a bi a y so-
cial ypes, le us conside he ollowing example, which we owe o an anonymous e -
e ee.

990 Daske and Ma ch Theo e ical Economics 19 (2024)
Example 1. Suppose he e a e h ee agen s and i is commonly known ha δ12 =δ23 =
δ31 =1/10 <δ
13 =δ21 =δ32 =1/5. Now conside he ollowing liabili y ule:I agen 1
e uses o pa icipa e while he o he agen s ag ee, hen agen 3 mus pay x>0 o
agen 2; i 2 e uses while he o he s ag ee, hen 1 mus pay x o 3; and i 3 e uses while
he o he s ag ee, hen 2 mus pay x o 1. Unde his liabili y ule, assuming he espec i e
o he agen s pa icipa e, an agen who e uses incu s a u ili y loss o x/10. Le ing xbe
su icien ly la ge, e e y mechanism becomes indi idually a ional in Bayes–Nash equi-
lib ium. ♦
In Example 1, an agen who e uses o pa icipa e is (emo ionally) penalized by o c-
ing he agen she likes mo e o subsidize he agen she likes less. Ob iously, his s a -
egy wo ks o e e y g oup in which each agen ip e e s some agen jio e some o he
agen i. Commonly known social p e e ences can hus be exploi ed o push, a he han
pull, agen s in o pa icipa ion. Example 1 ela es o he b anch o li e a u e ha consid-
e s mo e gene al p ope y igh s and liabili y ules, allowing o edis ibu ion e en i
some agen s e use o pa icipa e (Segal and Whins on (2016)) o allowing he designe
o impose o he h ea s agains nonpa icipa ion (Jehiel and Moldo anu (2006, p. 108)).
A ca ea o such pa icipa ion-en o cemen s a egies is ha hey p esume subs an-
ial ba gaining powe o he designe . Mo eo e , he concep has a la o o edundancy:
Would he co esponding p ope y igh s and liabili y ules no equi e agen s’ app o al
in ad ance, po en ially uling ou pa icipa ion in he o e all mechanism by backwa d
induc ion? In con as , ou pa icipa ion-s imula ion app oach wo ks o any speci i-
ca ion o p ope y igh s and liabili y ules. I he eby accoun s o bo h he designe ’s
limi ed ba gaining powe and he agen s’ ee will.
6.2 P ac ical implemen a ion
An impo an ques ion ega ding possibili y esul s conce ns hei p ac ical ele ance;
i.e., whe he hey show how e icien design is a ainable in p ac ice o whe he hey
se e o poin ou p ac ical di icul ies in he manne o a “ educ io ad absu dum c i-
ique.” We shall he e o e discuss he possibili ies o and limi a ions o p ac ically im-
plemen ing ou pa icipa ion-s imula ing ans e s.
We a gue ha , in an abs ac way, pa icipa ion s imula ion can be seen in p ac ice.
Obse e ha ou PS ans e s only equi e agen s o epo a one-dimensional su icien
s a is ic o hei social ype. Thus, epo ing social ypes ansla es in o agen s selec ing
one-dimensional s a egies in a s a egic game. I is his s a egic game ha ende s pa -
icipa ion a ac i e. In wha ollows, we illus a e how pa icipa ion may be s imula ed
h ough a ious game o ms.
The idea is o exploi he agen s’ social p e e ences by ha ing hem choose among
di e en le els o a one-dimensional s a egic a iable and he eby impose posi i e o
nega i e ex e nali ies on each o he . Fo his pu pose, de ine o each agen i=Mase
o dedica ed suppo e s Si⊆I {i,M}and deno e by S−i=I ({i,M}∪Si) he se o
i’s dedica ed opponen s. Fo ins ance, i we le Si=I {i,M} o all i= M, henou
o malism shall cap u e a public-good game among I {M}, whe eas Si=∅ o each i=
Theo e ical Economics 19 (2024) E icien incen i es 991
Mshall cap u e a compe i ion be ween he agen s o he han M.Fo gi ense s(Si)i=M,
pa icipa ion s imula ion can be implemen ed as ollows.
P oposi ion 3. Pa icipa ion s imula ion can be implemen ed wi h an indi ec mech-
anism unde which agen s i=Min es xi≥0 o ecei e ne e u ns ˆ
si((xj)j=M)=−xi+
ci+2μ√xi+2j∈Si√xj−2j∈S−i√xj o app op ia e cons an s μand (ci)i=M,while
ˆ
sM=−i=Mˆ
si.
In hegameo P oposi ion3, agen s’ in es men s may ake he o m o mone a y
in es men s, labo e o , o physical exe ion. An agen ’s in es men imposes a pos-
i i e (nega i e) payo ex e nali y on hose o he agen s o whom she is a dedica ed
suppo e (opponen ). I Si=I {i,M} o each i= M, agen s a e in ol ed in a si u-
a ion o eam-pe o mance pay, e ec i ely a game o p i a e con ibu ions o a public
good o I {M}. Con e sely, le ing Si=∅ o each i= Myields a con es -like si ua-
ion wi h ela i e-pe o mance pay. Pa icipa ion s imula ion hus becomes a p incipal–
agen scena io wi h M aking he ole o he p incipal. Mix u es o ela i e- and eam-
pe o mance pay a e easible oo. Fo ins ance, he pa i ion I=I1∪I2∪{M}wi h
Si=I {i} o all i∈Iand ∈{1, 2}leads o a eam compe i ion be ween eams I1
and I2. In all hose cases, each i=Mhas he dominan s a egy o in es xi=(μ+δS
i)2,
whe e
δS
i=
=i:i∈S
(δi −δiM )−
=i:i∈S−
(δi −δiM )
δii −δiM
, (12)
while le ing μ=maxj=M,δ∈|δS
j|ensu es ha he mechanism is well de ined.
Agen i’s in es men is s ic ly inc easing in δS
i, and i inc eases (dec eases) in i’s
ela i e p o-sociali y owa d hose agen s o whom iis a dedica ed suppo e (oppo-
nen ). Hence, whe he a dedica ed suppo e (opponen ) u ns ou o be an ac ual sup-
po e (opponen ) depends on ha agen ’s social p e e ences. Mo eo e , i’s in es men
inc eases (dec eases) in i’s p e e ence o Mi he e a e mo e (less) agen s o whom i
is a dedica ed opponen a he han suppo e . The ans e ha iin e im-expec s o
he sel is maximal i δS
i=0(see(14) in Appendix A.2). Money is hus edis ibu ed ex
in e im o hose agen s who a e (nea ly) indi e en abou any o m o edis ibu ion be-
ween h ee pa ies: hose hey a e mean o suppo , hose hey a e mean o oppose,
and, inally, M. On he o he hand, an agen who has s ong conce ns abou he dis-
ibu i e e ec s o and be ween hese h ee pa ies will obey (i δS
i0) o disobey (i
δS
i0) he dedica ed oles; his esul s in an in e im-expec ed mone a y loss, which is
o e compensa ed o emo ionally.
We accompany P oposi ion 3wi h a eal-wo ld example. Think o a communi y o -
ganizing a und aise in suppo o hei elemen a y school (e.g., o und a new baske -
ball cou ). The ha d-co e alloca ion p oblem unde lying his e en is ob iously one o
public-good p o ision, and he mechanism o esol e i , i only second bes , is ac u-
ally qui e simple, ealis ically speaking: “Once you’ e in, you ha e o gi e” as a ma e
992 Daske and Ma ch Theo e ical Economics 19 (2024)
o social no m. E en s o his so a e o en complemen ed wi h some so -co e incen-
i e de ice, like awa ding he bes -d essed gues . The majo pu pose o such an add-
on con es is no o make gues s d ess well, bu a he o supp ess “ ee- iding a he
doo s ep” by compensa ing pa icipan s o hei mone a y losses ( he los e u ns om
ee- iding) wi h he social u ili y hey de i e om engaging in he con es . Awa ding he
bes -d essed gues p o ides pa icipan s wi h a pla o m o li e ou hei p opensi ies o
compe e, and i is his a ac ion ha helps pull hem o e he doo s ep.9
6.3 Model limi a ions
F om he o he angle, hough, we mus sc u inize he assump ions ha ende pa ici-
pa ion s imula ion possible. As is s anda d in mechanism-design heo y, we assume ha
ans e s may ake a bi a y nega i e alues. This p esumes ha agen s a e endowed o
pay hese ans e s. As in e pe sonal ans e s play an impo an ole in ou s udy be-
yond s anda d heo y, i is wo hwhile o discuss he impac o budge cons ain s. As is
e iden om (5), an agen ’s paymen (i.e., nega i e ans e ) inc eases wi h ha agen ’s
al uism owa d ha special agen Mwho is designa ed o balance he budge , and as
δiM →1(allelse ixed),agen i’s paymen would exceed all limi s. Hence, in oducing
budge cons ain s would con lic wi h allowing o a bi a y social- ype se s. This aises
he ques ion o whe he pa icipa ion s imula ion would s ill wo k o bounded ans-
e s i we con ined social- ype se s app op ia ely. In ac , his is no he case, o any
ype-se con inemen : As is ob ious om ou de i a ion o PS ans e s in Sec ion 5.2,
and om Figu e 2in pa icula , indi idual paymen s a e likely la ges o he ex eme
social ypes. On he o he hand, na owing he suppo ends o dec ease he a iance
(which is he minimum alue ha he in e im-expec ed u ili y gain om pa icipa ion
s imula ion can ake), so PS ans e s mus be ampli ied e en u he h ough he ac-
o αin he p oo o Theo em 1. Simila a gumen s hold o ou a ious e sions o
PS ans e s, so we mus conclude ha budge cons ain s limi he scope o pa icipa-
ion s imula ion (as we cons uc ed i ). A way o esol e his p oblem would be o mee
budge cons ain s wi h cons ain s on agen s’ ese a ion u ili ies.
Ou assump ion ha agen s’ social p e e ences ex end o each o he s’ ans e s is
c i ical o ou main esul , and i dis inguishes ou s om o he pape s on mechanism
design wi h social p e e ences. I implies ha agen s ca e abou he o e all dis ibu i e
e ec s o he mechanism, bu i equi es ha agen s lea n all o he agen s’ ull p i a e
payo s ex pos . As ou lined by Sobel (2005, p. 400), he domain o social p e e ences
is c i ical in models wi h in e dependen p e e ences. Ye , he li e a u e p o ides li le
guidance in his ega d. Ve y ecen expe imen al s udies sugges ha some subjec s
some imes apply hei social p e e ences na owly, bu hey conclude ha mo e wo k is
needed o explo e he ex en and he d i e s o na ow dis ibu i e conce ns (see Ellis
and F eedman (2024); Exley and Kessle (2024)).
Ou assump ion ha p i a e payo s a e quasi-linea while u ili y is linea in p i a e
payo s is c ucial o bo h p e e ence sepa a ion and pa icipa ion s imula ion. I im-
plies ha agen s a e isk-neu al wi h espec o ans e s. We know om Sec ion 6.1
9A simila poin is equen ly made in concep ual esea ch on how o o ganize und aise s; see, e.g.,
Webbe (2004) and Peloza and Hassay (2007).
Theo e ical Economics 19 (2024) E icien incen i es 993
ha pa icipa ion s imula ion elies on agen s accep ing a gamble o e he composi ion
o social ypes a play. Plausibly, hen, isk-a e se agen s a e less suscep ible o pa ic-
ipa ion s imula ion. We con end ha when elaxing hese assump ions, pa icipa ion
s imula ion, now gene ally unde s ood as complemen ing a mechanism wi h an un e-
la ed s a egic game, may s ill p o e help ul in a aining indi idually a ional second-
bes implemen a ion. We lea e his o u u e wo k.
6.4 Rela ion o Mezze i (2004)
Ou bundling o wo mechanisms esembles he app oach o Mezze i (2004; hence o h
Mezze i). The key di e ences be ween his s udy and ou s a e he ollowing.
In ou model, agen s’ social p e e ences, and hus he alloca ional and in o ma ional
ex e nali ies associa ed wi h hem, ex end o all agen s’ ans e s. Mezze i’s agen s can
be o he - ega ding wi h espec o social al e na i es, bu mus dis ega d o he agen s’
ans e s; ha is, hey do no accoun o he o e all dis ibu i e e ec s o a mechanism.
As we will see, Mezze i’s mechanism is hus no incen i e-compa ible in ou model.
While we conside a speci ic amewo k o one-dimensional alloca ional and in o -
ma ional ex e nali ies, Mezze i conside s a mo e gene al amewo k in which hese ex-
e nali ies can be mul i-dimensional. Jehiel and Moldo anu (2001)hadshown ha ,
wi h mul i-dimensional ex e nali ies, he e exis s no mechanism ha is bo h incen i e-
compa ible and e icien , bu hey es ic ed a en ion o one ound o epo ing mech-
anisms. Mezze i shows ha he conclusion changes when conside ing a wo-s age
mechanism: In he i s ound o epo ing, each agen signals he p e e ence ype
ega ding a se o social al e na i es; based on hese epo s, he designe ul ima ely
chooses an al e na i e ha maximizes agg ega e u ili y. In he second ound o epo -
ing, each agen signals he payo she ealizes unde his al e na i e, and in e pe sonal
ans e s a e de e mined based on hese epo s. Speci ically, he second-s age ans e
scheme u ilizes he p inciple o he VCG (Vick ey (1961); Cla ke (1971); G o es (1973))
mechanism: Each agen is ans e ed he sum o all o he agen s’ epo ed ou come-
decision payo s; since his ans e is independen o one’s own epo , each agen has
he weakly dominan s a egy o epo he ou come-decision payo u h ully. By back-
wa d induc ion, his mechanism makes each agen a esidual claiman o he ull su plus
and he eby incen i izes u h- elling in he i s epo ing s age.
Ha ing ske ched Mezze i’s mechanism, we can ule ou ha i would be incen i e-
compa ible in ou model. I is app op ia e o conside wo e sions o his mechanism.
The i s is a one- o-one adap ion o ou amewo k. In he i s s age, agen s epo
bo h hei payo ypes and social ypes; based on hese epo s, he designe chooses
he al e na i e k ha maximizes agg ega e u ili y (which is a weigh ed sum o all agen s’
p i a e payo s unde k). In he second s age, each agen epo s he u ili y she de-
i es unde k; based on hese epo s, she ecei es a ans e ha equals he sum o
all he o he agen s’ epo ed u ili y le els. This mechanism is clea ly no incen i e-
compa ible in he second epo ing s age: An agen ’s epo ed u ili y le el a ec s e e y
o he ’s ans e , which she alues acco ding o he social ype; she is indi e en only
i he sum o he deg ees o al uism owa d he o he s equals ze o and would o he -
wise unde - o o e s a e he ou come-decision u ili y le el. The second e sion shall
994 Daske and Ma ch Theo e ical Economics 19 (2024)
accoun o ou ocusing on social al e na i es ha condi ion on payo ypes. Whe eas
we obse ed ha ou e ms o ade implemen he ma e ially e icien al e na i e, i is
na u al o ask whe he a e sion o Mezze i’s mechanism ha me ely ope a es on pay-
o ypes would achie e he same. Howe e , he e, oo, he second-s age ans e scheme
is no incen i e-compa ible: T ans e ing o each agen he sum o he o he s’ epo ed
ou come-decision payo s gi es almos all social ypes he incen i e o unde - o o e -
s a e hese payo s.
Finally, Mezze i’s and ou mechanism di e in he way hey a ac pa icipa ion
and allow he designe o ex ac he esul ing su plus. (These issues a e no discussed
in Mezze i (2004), bu a e conside ed in Mezze i (2003,2007).) When applied o se -
ings in which he su plus om he mechanism is s ic ly posi i e o any ealiza ion o
ypes, Mezze i’s mechanism can be ende ed indi idually a ional h ough app op ia e
lump-sum ans e s (see Mezze i (2003, P oposi ion 3)). Deploying side be s ha le e -
age he co ela ion in agen s’ second-s age payo epo s (simila o hose in C éme
and McLean (1985,1988)), he designe may ex ac nea ly he ull su plus (see Mezze i
(2007, Theo em 4)). In ou model, in con as , pa icipa ion can be a ac ed whene e
social- ype dis ibu ions ha e s ic ly posi i e a iance while ans e s may ake a bi-
a y nega i e alues. By le e aging he di e ences in agen s’ o he - ega ding conce ns,
he designe can gene a e a money pump and ex ac a mo e han he gains om ade.
Appendix
A.1 P oo o Lemma 1
Ha ing equi ed weak budge balance, Pa e o e iciency implies s ic budge balance.
Suppose i∈I i=− o some >0. Then a Pa e o imp o emen can be achie ed
h ough ans e s ( i+/n)i∈I, since j∈Iδij >0 by assump ion.
In he ollowing discussion, le |δij |<1/(2n−3) o all iand all j=i. Suppose ha , o
any ixed ans e s ( i)i∈I, he e exis s a social al e na i e k◦(θ) ha Pa e o-domina es
he al e na i e k(θ)∈a g maxk∈Ki∈Iπi(k|θi)while i∈Iπi(k◦|θi)<i∈Iπi(k|θi).
Then he e mus exis agen s iwho make s ic ma e ial losses when swi ching om k
o k◦; ha is,πi(k◦|θi)−πi(k|θi)=−i<0. Le ibe one o he agen s o whom his
ma e ial loss is la ges . Agen iis no wo se o u ili y-wise unde k◦ han unde ki and
only i she is emo ionally compensa ed h ough he dis ibu i e e ec s on he o he s:
j=iδij[πj(k◦|θj)−πj(k|θj)] ≥i. We show ha his is impossible.
Fi s suppose δij≤0 o allj=i.Theniob ains he maximum emo ional com-
pensa ion easible i each j= ialso ealizes he maximum ma e ial loss o −iwhen
swi ching om k o k◦; ha is,i πj(k◦|θj)−πj(k|θj)=−i<0. Bu e en hen,
j=iδij[πj(k◦|θj)−πj(k|θj)] =j=iδij(−i)<
i, since 0 ≥δij>−1/(2n−3)≥
−1/(n−1).
Now suppose maxj=iδij>0andle j∈a g maxj=iδijbe he a o i e agen
o i.Theniob ains he maximum emo ional compensa ion easible i j ealizes
a maximum ma e ial gain when swi ching om k o k◦unde he cons ain ha
j∈Iπj(k◦|θj)<j∈Iπj(k|θj).Thisis hecasei eachj=i,jalso ealizes he max-
imum ma e ial loss o −iwhile agg ega e losses, amoun ing o (n−1)i, se e as

Theo e ical Economics 19 (2024) E icien incen i es 995
a subsidy o agen j; ha is,i πj(k◦|θj)−πj(k|θj)=−i<0 o allj= i,jwhile
πj(k◦|θj)−πj(k|θj)=(n−1)i. Bu e en hen, j=iδij[πj(k◦|θj)−πj(k|θj)] =
j=i,jδij(−i)+δij(n−1)i<
i(n−2)/(2n−3)+i(n−1)/(2n−3)=i, since
|δij|<1/(2n−3) o all j= i.Hence,agen iis wo se o unde k◦ han unde k,
implying kis Pa e o-e icien .
I emains o show ha , o any ixed social al e na i e k, no ex pos budge -balanced
ans e scheme ex pos Pa e o-domina es ano he i |δij |<1/(2n−3) o all iand all
j= i. Suppose he opposi e is ue and ha ans e s ( ◦
i)i∈Iex pos Pa e o-domina e
ans e s ( 
i)i∈I, while bo h a e ex pos budge -balanced. Then he e is an agen iwho
su e s he maximum mone a y loss when swi ching om ( 
i)i∈I o ( ◦
i)i∈I.F omhe e,
he p oo p oceeds exac ly as abo e.
A.2 P oo o P oposi ion 3
Fo any gi en se s (Si)i=M, we ob ain pa icipa ion-s imula ing ans e s by modi ying
he ans e scheme (2)–(5)as
s
M(δ)=−
j=M
s
j(δ)(13)
s
j(δ)=−C+gjδS
j−δS
jg
jδS
j+
=j,M
(−1)1S−j()·g
δS
 o j=M(14)
gjδS
j=Va δjδS
j+δS
j−EδjδS
j2(15)
δS
j=
=j,M
(−1)1S−(j)·(δj −δjM )
δjj −δjM
(16)
o some cons an C>0, whe e 1A(x)is he indica o unc ion (i.e., 1A(x)=1i x∈A
and 1A(x)=0i x/∈A). To see his, we ollow he p oo o Lemma 4.
S a egy P oo ness.Unde s, each agen j= M epo s a social ype ˆ
δj,whichis
s a egically equi alen o epo ing some signal ˆ
δS
j∈R. He ex pos u ili y is gi en by

=M
(δj −δjM )s
(ˆ
δ)
=(δjj −δjM )gjˆ
δS
j−ˆ
δS
jg
jˆ
δS
j+
=j,M
(−1)1S−j()·g
ˆ
δS

+
=j,M
(δj −δjM )gˆ
δS
−ˆ
δS
g
ˆ
δS
+
=,j,M
(−1)1S−()·g
ˆ
δS

+
=j,M
(−1)1S−(j)·(δj −δjM )g
jˆ
δS
j−C
=M
(δj −δjM ).
Hence, when subs i u ing o δS
j==j,M(−1)1S−(j)·(δj −δjM )/(δjj −δjM ),agen j
maximizes gj(ˆ
δS
j)+(δS
j−ˆ
δS
j)g
j(ˆ
δS
j)o e hechoiceo ˆ
δS
j.Asg
j>0, each j=Mhas he
996 Daske and Ma ch Theo e ical Economics 19 (2024)
s ic ly dominan s a egy o epo ˆ
δS
j=δS
j.Asagen Mis no in ol ed s a egically, she
has he weakly dominan s a egy o epo he ue social ype δM.
Ex Pos Budge Balance.Thisisimmedia e om(13).
In e im-Expec ed Pa e o Imp o emen . When subs i u ing o δS
jand Eδ[g
(δS
)] =
0=Eδ[g(δS
)−δS
g
(δS
)], due o Lemma 3,j’s in e im-expec ed u ili y om sis

=M
(δj −δjM )Eδ−js
(δ)=(δjj −δjM )gjδS
j−C
=M
(δj −δjM )
=(δjj −δjM )gjδS
j−C(δjj −δjM )−C
=j,M
(δj −δjM )
=(δjj −δjM )gjδS
j−C1+δ
j
o δ
i==i,M(δi −δiM )/(δii −δiM ). Recall ha δjj =1>δ
jM and gj(δS
j)≥Va δj[δS
j]>
0, and ha δ
j<n−2, since δjj −δjM >δ
j −δjM o all = j,M. We hus ob ain ha
each j= Mde i es posi i e in e im-expec ed u ili y om unanimous pa icipa ion i
we le C≤minj=MVa δj[δS
j]/(n−1). Finally, due o Lemma 3again, also M’s in e im-
expec ed u ili y is posi i e i all agen s pa icipa e: i∈IδMiEδ−M[s
i(δ)] =j=M(δMj −
1)Eδ[s
j(δ)] =Cj=M(1−δMj )>0.
To implemen s:→Rwi h an indi ec mechanism ˆ
s:[0, ∞)n→R,weobse e
ha
s
j(δ)=2
∈SjδS
−EδδS
−2
∈S−jδS
−EδδS
+EδjδS
j2−δS
j2−C
=2ˆ
cj−μ+δS
j2+2μμ+δS
j+2
∈Sjμ+δS
−2
∈S−jμ+δS

=cj−xj+2μxj+2
∈Sj
√x−2
∈S−j
√x
=ˆ
sj(x)=M
when le ing √x=μ+δS
 o μ=maxj=M,δ∈|δS
j|while le ing cj=2ˆ
cj o
ˆ
cj=μ·|S−j|−μ·|Sj|−1
2μ2+1
2EδjδS
j2−
∈Sj
EδδS
+
∈S−j
EδδS
−1
2C.
Since agen j= Mhas he s ic ly dominan s a egy o epo δS
junde sand since
dxj/dδS
j>0, she also has he dominan s a egy o in es xj=(μ+δS
j)2unde ˆ
s.
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