Dong, Xiaoge
Wo king Pape
Will ul igno ance in legal con ex s: A mechanism design
app oach
Cen e o Ma hema ical Economics Wo king Pape s, No. 752
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Dong, Xiaoge (2025) : Will ul igno ance in legal con ex s: A mechanism design
app oach, Cen e o Ma hema ical Economics Wo king Pape s, No. 752, Biele eld Uni e si y, Cen e
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752
Sep embe 2025
Will ul Igno ance in Legal Con ex s: A Mechanism
Design App oach
Xiaoge Dong
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
e-mail: [email p o ec ed]
uni-biele eld.de/zwe/imw/ esea ch/wo king-pape s
ISSN: 0931-6558
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A ibu ion 4.0 In e na ional (CC BY) license. Fu he in o ma ion:
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Will ul Igno ance in Legal Con ex s: A Mechanism
Design App oach
Xiaoge Dong∗
Sep embe 24, 2025
Abs ac
We s udy “will ul igno ance” - choosing no o lea n whe he a ask is illegal
- in a lawmake -p incipal-agen game and cha ac e ize he penal y policies ha
implemen wel a e-maximizing beha io . The model deli e s an implemen abili y
on ie : which equilib ium beha io s can exis and be selec ed by penal ies. Wi h
pe ec inqui y, his on ie is aligned wi h he wel a e o de ing, so he lawmake
can make he wel a e-maximizing beha io bo h exis and be p e e ed by all pa -
ies. Wi h impe ec inqui y, noise b eaks ha alignmen and p oduces wo ailu es:
inqui y ha is socially desi able may be in easible a any penal y, and inqui y ha
is socially undesi able may pe sis because i canno be swi ched o . We compa e
ha m-based, compliance-based, and dual-penal y ules: ha m-based ules p ese e
con ol bu igh ens easibili y; compliance-based ules elax easibili y bu sac i ices
con ol; dual penal y ules eco e bo h le e s subjec o simple bounds. The ame-
wo k yields p ac ical guidance o calib a ing penal ies o ha m, inqui y accu acy,
and inqui y cos s. I also implies ha igno ance canno se e as a shield: he ab-
sence o knowing c ime in equilib ium is d i en by incen i es a he han mo ali y,
making non-inqui y he ue s a egic ma gin o liabili y design.
JEL Codes: K14, K42, D82, D86, H23.
Keywo ds: will ul igno ance, os ich ins uc ion, law and economics, asymme ic
in o ma ion.
∗Zeppelin Uni e si ä F ied ichsha en, Fallenb unnen 3, 88045 Ge many.
I hank Niels Boissonne , Y es B ei mose , He be Da id, Manuel Fö s e , Ru eyda Gozen, Ma ina
Mio o, F ank Riedel, and Ge ald Willmann o aluable commen s and sugges ions. Ea lie e sions o
his pape we e p esen ed a he BiGSEM colloquium a Biele eld Uni e si y, and I hank he pa icipan s
o hei help ul eedback. Financial suppo om he DFG h ough he p ojec RUTHLESS is g a e ully
acknowledged. All emaining e o s a e my own.
1
1 In oduc ion
Will ul igno ance - also called “will ul blindness” o “delibe a e igno ance” - a ises when a
de endan in en ionally a oids lea ning ac s ha would igge legal liabili y (Ki el and
Hannikainen 2023). Since Uni ed S a es . Jewell (1976), cou s ha e widely adop ed
he will ul-igno ance doc ine, ea ing ailu e o inqui e as an amoun o knowledge
(Cha low 1991; A. Sa ch 2018; Hellman 2009; Luban 1998). Ye deba e pe sis s abou
i s ounda ions and he p ope design o penal ies; de ini ions a y ac oss ju isdic ions
and en o cemen p ac ice is une en. Simons (2021) u ges cau ion un il clea e , b oadly
accep ed s anda ds eme ge. Agains his backg ound, he policy ques ion we ake up is:
how should penal ies be s uc u ed so ha equilib ium beha io aligns wi h social wel a e
when ac o s may s a egically a oid in o ma ion?
We s udy will ul igno ance in a lawmake -p incipal-agen game. The lawmake se s
a penal y ule ex an e. Na u e hen d aws he p incipal’s ype (good o bad). The
p incipal’s ype de e mines he kind o ask he o e s: a good p incipal o e s a legal ask,
while a bad p incipal o e s an illegal one. The p incipal s a egically p oposes a ans e
o he ask o he agen o induce pe o mance, an icipa ing he agen ’s esponses and he
legal ule. The agen decides whe he o inqui e in o he legali y o he ask (a a cos )
and whe he o pe o m i ; i an illegal ask is pe o med, a penal y is imposed. This
amewo k applies di ec ly o compliance domains such as an i-money-launde ing due
diligence, expo con ols, and p oduc -sa e y egula ion, whe e inqui y du ies a e cen al
and penal ies a y in s uc u e. Ou objec i e is o cha ac e ize he “implemen abili y
on ie ”-which equilib ium beha io s can bo h exis and be selec ed by penal ies-and o
map ha on ie in o wel a e and policy guidance.
Ou main esul s can be summa ized as ollows. 1) Wi h pe ec inqui y, he lawmake
can always implemen he wel a e-maximizing beha io . Penal ies ule ou unin o med
ac ion; whe he he ma ke emains ac i e hen depends only on whe he he good ype’s
su plus co e s he cos o inqui y. I so, he agen pe o ms only legal asks; i no , he
ma ke shu s down. In his se ing, he agen ne e knowingly pe o ms an illegal ask, so
ans e s canno signal ypes and no sepa a ing equilib ium a ises. Penal ies de e mine
ma ke composi ion, while inqui y cos de e mines whe he aluable ac i i y su i es-an
alignmen ha gua an ees wel a e op imiza ion. The absence o knowing c ime is hus
no e idence o highe mo ali y bu an endogenous ou come o incen i es: all ha m ul
conduc lows h ough delibe a e igno ance, unde sco ing ha igno ance canno se e as
a shield in liabili y design.
2) Wi h impe ec inqui y, wel a e and implemen abili y can di e ge. Noise gene a es
alse posi i es and nega i es, lowe ing he alue o sc eening and p oducing wo ailu es:
(i) inqui y may be wel a e-maximizing bu in easible, i good ypes canno bea inqui y
cos s plus esidual isk; (ii) sc eening may pe sis e en when shu -down would be be e ,
since penal ies canno ully swi ch i o . How alse posi i es a e ea ed becomes pi o al.
Unde ha m-based ules, agen s emain liable a e inqui y, p ese ing le e age o de e
sc eening when i is wel a e domina ed, bu his can also make desi able sc eening in-
easible. Unde compliance-based ules, mee ing he inqui y s anda d elimina es esidual
isk, sus aining sc eening, bu emo es he lawmake ’s o -swi ch. A dual-penal y scheme
se s one penal y o non-inqui y and ano he o pos -inqui y exposu e, combining he
ad an ages o bo h app oaches. Equalizing he wo eco e s ha m-based ules; se ing he
pos -inqui y penal y o ze o eco e s compliance-based ules. Dual penal ies hus es o e
con ol, allowing he lawmake bo h o shu o sc eening when exclusion is op imal and
2
o sus ain i when inqui y is wel a e-maximizing.
Rela ion o he li e a u e. This pape con ibu es o he legal deba e on will ul
igno ance and he “os ich ins uc ion” by p o iding a ac able economic model ha
complemen s no ma i e and ju isp uden ial analyses. Addi ionally, i ela es o he li e -
a u e on s a egic igno ance in economics, bu di e s by ocusing on penal y design and
implemen abili y a he han on social p e e ences o sel -image. In he end, i speaks o
he op imal-de e ence adi ion in law and economics, ex ending s anda d p esc ip ions
o en i onmen s whe e igno ance i sel is he s a egic ma gin. A ull discussion o ela ed
wo k appea s in Sec ion 2.
Con ibu ion. This pape makes h ee con ibu ions. Fi s , i de elops he i s
ac able lawmake -p incipal-agen amewo k o will ul igno ance, embedding he in-
qui y decision in a mechanism-design se ing whe e penal ies a e chosen ex an e. In
doing so, i o malizes he doc ine ha delibe a e igno ance may be ea ed like knowl-
edge ( he “os ich ins uc ion”) and shows how his a ec s implemen abili y and wel a e.
Second, i cha ac e izes he lawmake ’s implemen abili y on ie , iden i ying when he
wel a e-maximizing beha io can be bo h sus ained and selec ed, and when easibili y and
desi abili y di e ge. This ex ends he li e a u e on s a egic igno ance by showing how
liabili y ules-ha m-based, compliance-based, and dual-map in o equilib ium ou comes.
Thi d, i ansla es he analysis in o ope a ional guidance: calib a e penal ies di ec ly o
he cu o condi ions ha sus ain he socially desi able beha io ; use ha m-based ules
when inqui y canno be e i ied; ese e compliance-based exemp ions o en i onmen s
wi h audi able inqui y and ole ance o pe sis en sc eening; and deploy dual penal ies
whe e lexibili y is needed o de e non-inqui y while keeping sc eening easible. When
en o cemen abso bs eal esou ces, implemen he a ge beha io wi h he minimal ex-
pec ed penal y and a oid on-pa h sanc ions. Toge he , hese con ibu ions ex end he
op imal-de e ence adi ion o en i onmen s whe e igno ance i sel is he s a egic ma -
gin, and o e a p ac ical menu o legal design.
Roadmap. Sec ion 2 e iews he li e a u e. Sec ion 3 p esen s he model. Sec ion 4
de elops he esul s o pe ec and impe ec inqui y and compa es liabili y designs.
Sec ion 5 discusses applica ions in legal p ac ice. Sec ion 6 collec s ex ensions. Sec ion 7
concludes. All p oo s a e p esen ed in he Appendix.
2 Rela ed Li e a u e
The will ul-igno ance doc ine add esses a p ac ical ac ic: de endan s who s a egically
a oid lea ning inc imina ing ac s. Since Uni ed S a es . Jewell, 532 F.2d 697 (9 h Ci .
1976), cou s ha e pe mi ed ju ies o ea delibe a e igno ance as knowledge- he so-
called “os ich ins uc ion.” Legal schola ship has deba ed he legi imacy o his doc ine
(Sa ch 2014; Simons 2021; Hellman 2009; Luban 1998), bu exis ing analyses emain
la gely no ma i e. Ou model p o ides he i s ac able mechanism-design amewo k
o analyze will ul igno ance as a policy ins umen , showing how ea ing igno ance as
culpable a ec s implemen abili y and wel a e, and he eby ex ending he ju isp uden ial
deba e in o a o mal economic analysis.
3
Closes o ou analysis is Ya e (2018), who o e s a no ma i e de ense o will ul igno-
ance and shows ha ailu e o inqui e indica es dis ega d o o he s’ in e es s. Ya e’s
model builds in social-p e e ence conce ns, ea ing igno ance as culpable because i e-
lec s de icien ega d o o he s. Ou app oach di e s in p imi i es and ques ion. We
keep playe s ully a ional and sel -in e es ed, and show ha will ul igno ance a ises as an
equilib ium esponse o incen i es— ans e s, penal ies, and inqui y cos s— a he han
om diminished conce n o o he s. This shi allows us o cha ac e ize implemen abili y
and o analyze how liabili y ules map in o wel a e. Social conce n en e s in an ex ension
in Sec ion 6, as a pa ame e ha subs i u es o legal penal ies unde pooling.
Empi ical wo k speaks o pe cep ions a he han design. Ki el and Hannikainen
(2023) show ha will ully igno an ac o s a e judged mo e an isocial han unsuspec ing
ones bu less han knowing iola o s (see also Al e e al. 2007). These indings in o m
he legi imacy and likely accep ance o os ich ins uc ions, bu hey a e o hogonal o
he penal y-design p oblem we s udy. Ou analysis akes incen i es as p ima y and, in
ex ensions, allows o non-ma e ial inqui y bu dens and social conce n as dis inc wel a e
p imi i es.
Ou pape is also ela ed o economic models o s a egic igno ance. Ka ik e
al. (2007) de elop a heo e ical model o mo i a ed in o ma ion a oidance, whe e agen s
p e e o emain unin o med in o de o p ese e plausible deniabili y in communica ion.
G ossman and Van De Weele (2017) s udy will ul igno ance in social decisions, showing
ha indi iduals a oid in o ma ion abou he consequences o hei ac ions o p o ec hei
sel -image. Rela ed expe imen al wo k by Dana e al. (2007) demons a es how p incipals
delibe a ely emain igno an abou payo s o excuse exploi a i e choices. These pape s
highligh he beha io al logic o igno ance bu do no s udy op imal penal y design. We
di e in ocusing squa ely on legal en o cemen : he lawmake se s penal ies ex an e,
and igno ance is ea ed as a s a egic ma gin in equilib ium. This mechanism-design
pe spec i e allows us o de i e implemen abili y condi ions and o compa e al e na i e
liabili y ules, which is absen in he exis ing in o ma ion-acquisi ion li e a u e.
Finally, ou en o cemen esul s connec o he op imal-de e ence adi ion (see Polin-
sky and Sha ell 2000). When se e i y is esou ce cos ly, wel a e a o s he minimal
penal y ha implemen s he desi ed beha io ; when en o cemen in ensi y is cos ly, wel-
a e a o s he minimal in ensi y. Unde impe ec inqui y, hese p esc ip ions in e ac
wi h esidual legal isk a e inqui y, explaining why pa i y can ende inqui y in easible
when i is desi able, why exemp ion can make inqui y ha d o swi ch o when exclusion is
be e , and how dual penal ies can es o e con ol by sepa a ing he le e s o unin o med
ac ion and pos -inqui y exposu e.
3 Model
The game. We s udy a ma ke en i onmen wi h h ee pa ies: he lawmake (L), he
p incipal (P), and he agen (A). The lawmake d a s and announces a penal y ule: i
an illegal ask is pe o med, he agen will be con ic ed and penalized a le el T∈R+.
Na u e (N) hen d aws he p incipal’s ype ω∈ {G, B}, wi h P (G) = θ∈(0,1) common
knowledge.1The p incipal p i a ely obse es ωand o e s he agen a co esponding ask
1. We assume 0< θ < 1, so ha bo h good and bad ypes lie in he suppo o belie s. This
cap u es he doc inal idea o “subs an ial suspicion”: he agen a aches posi i e p obabili y o illegali y,
so ha non-inqui y is a s a egic choice a he han innocen igno ance. Legal schola ship di e ges on
4
wi h ans e X∈R+. By a sligh abuse o no a ion, we use G( esp. B) o deno e bo h
he good ( esp. bad) p incipal and he legal ( esp. illegal) ask he o e s.
The agen does no di ec ly obse e legali y bu can conduc an inqui y a cos k > 0.
We deno e he belie ha he ask is legal by µ∈[0,1]. A e obse ing Xand T, she
chooses wo s a egies in sequence: i) an inqui y s a egy σi
A(X)∈ {i, ni}, whe e ideno es
inqui y and ni deno es no inqui y; and ii) an ac ion s a egy σa
A(X, ·)∈ {a, na}, whe e
adeno es accep ance o he ask and na non-accep ance. In he baseline game, belie
is ully de e mined by he in o ma ion s uc u e: wi hou inqui y, µ=θ, e lec ing he
p io p obabili y ha he ask is legal; wi h pe ec inqui y, µ∈ {0,1}, as he inqui y
e eals ype wi h ce ain y.
Timing. The sequence o mo es is summa ized in Figu e 1.
= 0 = 1 = 2 = 3 = 4
Lchooses T;N
d aws ω∈ {G, B}
Plea ns T, ω;
o e s X o A
Achooses inqui y
σi
A∈ {i, ni}
I i, she lea ns
ω; hen chooses
ac ion σa
A∈ {a, na}
I illegal and
accep ed, con ic ion
a penal y T
Figu e 1: Timing o he game
No es: The igu e summa izes he sequence o mo es in he lawmake -p incipal-agen game. A = 0,
he lawmake se s he penal y Tand na u e chooses he p incipal’s ype. A = 1, he p incipal o e s
a ask; a = 2, he agen decides whe he o inqui e; a = 3, he agen decides whe he o pe o m
he ask; and a = 4, con ic ion occu s i he ask is illegal.
Payo s. I he legal ask is pe o med, he good p incipal ge s yG−Xand he agen ge s
X−k·1{i}. I he illegal ask is pe o med and illegal, he bad p incipal ge s yB−Xand
he agen ge s X−T−k·1{i}; socie y bea s ha m H. I he ask is no pe o med, bo h
p incipal and agen ge 0, and he agen pays he inqui y cos i she inqui ed. T ans e s
and penal ies a e pu e edis ibu ions be ween p incipal and agen .
As e e ence, we o malize he in e ac ion in he game illus a ed in Figu e A.1, which
we e e o as he LPA game. This game models o enses in ol ing inculpa o y p opo-
si ions o he o m: “...whe e he unde lying ac ion would no be independen ly w ong-
ul absen he de endan ’s knowledge o he inculpa o y p oposi ion.” The delega ed
ask-such as anspo ing a subs ance o de eloping so wa e-is no inhe en ly illici bu
becomes so when pe o med wi h knowledge ha he subs ance is con aband o he so -
wa e acili a es money launde ing. We assume pe ec de ec ion o illegal ac s, bu only
he agen bea s he legal consequences. We elax hese assump ions in sec ion 6.
how demanding his h eshold should be. Some a gue ha any posi i e p obabili y su ices o g ound
culpabili y (e.g. Luban 1998; Hellman 2009), while o he s emphasize awa eness o a high p obabili y
o w ongdoing as he co ec doc inal es (e.g. Cha low 1991; A. F. Sa ch 2014; Simons 2021). Ou
baseline collapses he h eshold o i s minimal alue, e ec i ely s= 0. One could al e na i ely in oduce a
suspicion h eshold s∈(0,1) and ea igno ance as culpable only i 1−˜
θ≥sgi en he agen ’s subjec i e
belie ˜
θ. A highe senla ges he ange in which igno ance is excused, na owing implemen abili y, bu
he co e logic o penal y design emains unchanged.
5
Equilib ium concep . Gi en a ixed penal y T, we analyze he con inua ion game
induced by he lawmake ’s mo e. We e e o any pe ec Bayesian equilib ium o his
subgame as a con inua ion equilib ium. Such an equilib ium sa is ies sequen ial a io-
nali y and Bayesian consis ency. In he analysis ha ollows, we assume ha i he
agen ecei es an o -equilib ium o e X′, she in e s i comes om a bad ype p incipal,
e ains om inqui y, and accep s he ask only i he legal penal y is ully o se . Fo
o -equilib ium o e s X′, we deno e he belie by ˆµ. Fo mally, his implies:
ˆµ(G|X′) = 0, σi
A(X′) = ni, and σa
A(X′, ni) = a⇐⇒ X′≥T.
This speci ica ion o o -pa h belie s maximizes he equilib ium se and is adop ed wi hou
loss o gene ali y.2Mo e op imis ic con en ions (ˆµ > 0) sh ink he exis ence windows bu
do no al e he quali a i e wel a e ankings, so ou main esul s a e obus o nea by
belie speci ica ions. We adop s anda d ie-b eaking in a o o he p eceding mo e :
indi e en agen s accep ; indi e en p incipals o e he smalles ans e ha induces
accep ance. Al e na i e ie-b eaking ules shi only kni e-edge bounda ies and do no
a ec he wel a e ankings epo ed below. Gi en a con inua ion equilib ium, we classi y
he s uc u e o play in o ou ypes:
i. Pooling: Bo h ypes o p incipals o e he same ans e ; he agen does no inqui e
and pe o ms he ask. Belie s a e no upda ed. Fo mally pu : XB=XG,σi
A(·) =
ni,σa
A(·) = a.
ii. Semi-pooling: Bo h ypes o e he same ans e ; he agen inqui es be o e deciding.
Belie s a e upda ed. Fo mally pu : XB=XG,σi
A(·) = i,σa
A(·)∈ {a, na}. Because
he key ea u e o his equilib ium is ha he agen pays he inqui y cos and
condi ions he ac ion on he inqui y esul , we will e e o his equilib ium ype
as sc eening in he analysis ha ollows. “Semi-pooling” and “sc eening” a e hus
in e changeable e ms in wha ollows.
iii. Sepa a ing: Types sepa a e ia ans e ; he agen does no inqui e and decides
based on he o e . Belie s a e no upda ed. Fo mally pu : XB> XG,σi
A(·) = ni,
σa
A(·)∈ {a, na}.
i . Inac i e: Bo h ypes o e below-penal y ans e ; he agen does no inqui e and
e uses he ask. Belie s a e no upda ed. Fo mally pu : XB, XG∈[0, T),σi
A(·) =
ni,σa
A(·) = na.
Non i ial mixed-s a egy equilib ia do no exis in ou en i onmen ; equilib ium
beha io is exhaus ed by he pu e ypes we analyze.3
Social wel a e and benchma ks. We use a u ili a ian wel a e measu e: he sum
o expec ed payo s ac oss playe s minus expec ed social ha m. We include he bad
p incipal’s payo wi h a no ma i e weigh τ∈[0,1] o cap u e di e en policy iews: τ
nea 0 i s p eda o y o c oss-bo de c imes whe e he o ende ’s gain is no alued; τnea
1 i s local ex e nali ies (e.g., pollu ion aba emen ailu es) whe e he p incipal’s ou pu
s ill coun s. Once a beha io is ixed, ans e s and penal ies a e pu e edis ibu ions and
do no a ec SW (penal ies a e esou ce-cos less; see Sec ion 6 o cos ly en o cemen ).
2. Fo mal bounds o gene al ˆµ∈[0,1] a e p o ided in Appendix C.2.
3. See Appendix C.3 o a o mal a gumen .
6
We will compa e ou esul s o wo benchma ks: (i) he Pe ec In o ma ion Equilib-
ium (PIE), whe e he p incipal’s ype is common knowledge; and (ii) he No Inqui y
Equilib ium (NIE), whe e ype is unknown and playe s ne e inqui e.
Lemma 3.1 (Benchma k wel a e unde PIE and NIE).Unde PIE,
SWPIE =(θ yG+ (1 −θ)(τyB−H),i τyB−H≥0,
θ yG,i τyB−H < 0,
and unde NIE,
SWNIE =(θ yG+ (1 −θ)(τyB−H),i θyG+ (1 −θ)(τyB−H)≥0,
0,o he wise.
Mo eo e , SWNIE ≤SWPIE, wi h s ic inequali y whene e τyB−H < 0.
P oo . All p oo s a e elega ed o he Appendix.
In ui ion. I he bad ype’s ne con ibu ion is nonnega i e (τyB−H≥0), bo h PIE and
NIE ea u e bo h ypes, so wel a e coincides. I i is nega i e, PIE can exclude he bad
ype while NIE shu s he ma ke down (wel a e 0< θyG= SWPIE). A ull p oo appea s
in Appendix B.1.
Implemen a ion objec i e. Le E={pooling,semi-pooling,inac i e}deno e he se
o equilib ium beha io s, and w i e E(T) o he beha io s sus ained unde penal y T. Fo
each e∈ E, le SW(e)deno e he associa ed le el o social wel a e. We call an equilib ium
beha io socially desi able i i maximizes wel a e,
e⋆∈a g max
e∈E SW(e).
The lawmake chooses T o implemen he socially desi able beha io , i.e. o ensu e ha
e⋆∈ E(T). When E(T)con ains mul iple beha io s, we adop a minimal implemen a ion
con en ion: whene e possible, we pick Tin he in e io o he penal y egion ha sus-
ains e⋆ o secu e uniqueness; i uniqueness canno be o ced, we c edi e⋆as implemen ed
only when i is uni o mly p e e ed (Pa e o-dominan ) among coexis ing beha io s (e e y
playe weakly p e e s e⋆ o any al e na i e and a leas one playe s ic ly p e e s i ). This
con en ion is no an addi ional ins umen o he lawmake , bu me ely a ie-b eaking
de ice o exposi ion. When Pa e o dominance does no hold, we adop he conse a i e
s ance and ea he desi able beha io as non-implemen able.
4 Analysis
Equilib ium p elimina ies. Two obse a ions will be used h oughou . Fi s , wi h
pe ec inqui y (α= 1), no equilib ium sepa a es in ans e s: i XB=XG, he agen
would skip inqui y, in e ype om he o e , and he bad ype would p o i ably mimic
he good ype. Second, whene e an illegal ask is pe o med wi h posi i e p obabili y on
he pa h, i occu s only wi hou inqui y (will ul igno ance): i he agen inqui ed and s ill
pe o med he illegal ask, she would also pe o m he legal one a e inqui y, so inqui y
7
easibili y by limi ing esidual exposu e a e inqui y.
P oposi ion 4.4 (Impe ec inqui y wi h dual penal ies).Conside impe ec inqui y and
a dual-penal y ule di e en ia ing will ul igno ance and knowing c ime. The lawmake can
implemen he socially desi able con inua ion beha io in all cases, excep when inqui y
is desi able bu he good ype canno inance he expec ed inqui y cos .
Rela i e o single-penal y egimes, dual penal ies weakly expand implemen abili y and
weakly aise he maximal a ainable wel a e; wel a e emains (weakly) below he pe ec -
inqui y benchma k.
In ui ion. Wi h wo le e s he lawmake sepa a es he wo ma gins. The no-inqui y
penal y Tnp ices unin o med ac ion: aising Tnmakes accep ance wi hou inqui y ex-
pensi e and can always shu down pooling; lowe ing Tnp ese es i when desi ed. The
a e -inqui y penal y Tip ices he agen ’s esidual legal exposu e om alse posi i es;
lowe ing Ti educes he expec ed ans e she needs o be willing o inqui e, and aising
Timakes inqui y una ac i e. Unde a semi-pooling o e (bo h ypes pos he same
ans e and he agen inqui es), he minimal pe -pe o mance paymen equals
semi =k+(1−θ)(1−α)Ti
φ,whe e φ:= θα + (1 −θ)(1 −α)
is he p obabili y ha inqui y leads o pe o mance. The good p incipal pays semi only
when his ask is ac ually pe o med (p obabili y α), so semi-pooling is p i a ely easible
i yG≥ semi, i.e. φyG≥k+ (1 −θ)(1 −α)Ti. By se ing Tias low as needed (down
o 0), he lawmake makes easibili y easies ; hus inqui y is implemen able i φyG> k.
Con e sely, i inqui y is no desi ed, choosing Tila ge pushes semi > yGso no one unds
sc eening; Tn hen selec s be ween pooling and inac i i y. Hence dual penal ies can always
“swi ch sc eening on o o ” excep in he single obs uc ion φyG≤k. Because (Tn, Ti)
nes ha m-based (Ti=Tn)and compliance-based (Ti= 0) ules, he maximal wel a e
a ainable wi h dual penal ies weakly domina es wha ei he single-penal y egime can
achie e, and emains (weakly) below he pe ec -inqui y benchma k when α < 1.
Nume ical Example (dual-penal y). Unde dual penal ies, he ealized wel a e a a gi en
equilib ium is as unde he o he egimes ( ans e s and penal ies cancel in wel a e);
he gain comes om which equilib ium is implemen able and selec ed. The lawmake
calib a es penal ies by se ing Tnhigh enough o de e unin o med ac ion while keeping
Tilow enough o main ain inqui y easibili y.
4.3 Calib a ion and compa a i e analysis unde impe ec in-
qui y
Rela i e o he pe ec -inqui y benchma k, he placemen o penal ies changes in wo
ways. Fi s , he lowe bound ha makes inqui y a ac i e ises as accu acy alls: wi h
noisie signals, he agen equi es highe compensa ion o co e he expec ed exposu e o
penal ies when she in es iga es. Fo mally, he indi e ence cu o k/[θ(1 −θ)] is eplaced
by
L(α) = k
θ(1−θ)(2α−1), α ∈(1/2,1),
so L′(α)<0: highe accu acy educes he penal y needed o igge inqui y. Second,
sc eening is now subjec o a easibili y cons ain ,
U(α) = φ(α)yG−k
(1−θ)(1−α), φ(α) = θα + (1 −θ)(1 −α),
14
0.0
0.5
1.0
1.5
2.0
−4 −3 −2 −1 0 1
Ne social ha m (τ * yB − H)
Social wel a e
Pe ec In o ma ion Equilib ium
No Inqui y Equilib ium
Pe ec Inqui y
Impe ec Inqui y (dual penal y)
Figu e 5: Social wel a e unde dual-penal y impe ec inqui y
No es: Pa ame e s a e θ= 0.4,yG= 4,k= 0.5,α= 0.8, and τ= 1. The x-axis plo s ne social ha m,
τyB−H; he y-axis plo s social wel a e. Cu es compa e Pe ec In o ma ion Equilib ium (PIE), No
Inqui y Equilib ium (NIE), Pe ec Inqui y (PIq), and Impe ec Inqui y (dual-penal y), whe e he
lawmake can se sepa a e penal ies o ac ion wi hou inqui y and o pos -inqui y iola ions. Fo
τyB−H≥0, pooling is e icien and all cu es coincide. As ne ha m u ns nega i e, PIq swi ches
om pooling o sc eening. The dual-penal y cu e also selec s sc eening whe e easible bu lies weakly
below PIq because noisy inqui y bo h le s some illegal asks slip h ough and blocks some legal ones.
When τyB−His su icien ly nega i e, sc eening ceases o be p o i able unde noise and he
dual-penal y cu e alls o ze o (inac i e), while PIq can s ill sus ain posi i e wel a e by sc eening.
so ha i accu acy is oo low o inqui y cos s oo high, his cons ain binds and he
inqui y window collapses e en when wel a e would a o sc eening. By con as , he
cu o o accep ance wi hou inqui y, T≤yG/(1 −θ), and he equi emen ha Tbe a
leas as la ge as he bad ype’s ou pu yB emain he same as unde pe ec inqui y.
These h esholds deli e anspa en compa a i e s a ics. A highe αenla ges he
inqui y window (lowe L, highe U), making sc eening easie o implemen . A highe k
aises Land lowe s U, squeezing he inqui y window and pushing he economy owa d
pooling o inac i i y. G ea e θo highe yGexpand easibili y by elaxing he accep ance
cap and aising U, while a la ge yBmakes de e ence ha de unde ha m-based ules and
aises he cu o o inac i i y. Ne ha m (τyB−H)a ec s only wel a e ankings, no
easibili y.
Ac oss egimes, he calib a ion logic is uni ied. Wi h a single penal y, he lawmake
calib a es Tdi ec ly in o he ange ha sus ains he socially desi able ou come: aising
i o he indi e ence cu o when inqui y mus be induced, o lowe ing i o he easibili y
bound when inqui y mus be de e ed. Unde ha m-based ules, inqui y can always be
de e ed by aising Tbu no always sus ained when desi able because he easibili y cap
may bind. Unde compliance-based ules, bo h p oblems appea : i φ(α)yG≤k, inqui y
is desi able bu in easible; con e sely, once unin o med ac ion is de e ed and φ(α)yG> k,
inqui y pe sis s o all highe T, so exclusion canno be es o ed by penal ies alone. Dual
penal ies sepa a e he ma gins: Tn egula es inqui y e sus no-inqui y, while Tip ices
esidual alse-posi i e isk. This lexibili y allows he lawmake o sus ain inqui y when-
15
e e φ(α)yG> k and o swi ch i o when exclusion is op imal, hough a he p ac ical
cos o speci ying and en o cing wo dis inc penal ies. All explici cu o exp essions and
compa a i e s a ics a e collec ed in Appendix B.4.
5 Applica ions
The p eceding analysis de i ed wel a e compa isons and illus a ed hem wi h diag ams
based on s ylized pa ame e alues. These igu es al eady showed how di e en penal ies
map in o pooling, sc eening, o exclusion, and how wel a e ankings depend on θ,α,k,
yG, and yB. Wha emains is o connec hese p edic ions o eal-wo ld se ings. This
sec ion illus a es he model wi h legal cases whe e a ia ion in pa ame e s such as ha m
H, economic con ibu ion τ, o inqui y accu acy αplayed a decisi e ole.
En i onmen al en o cemen : Rese e Mining (1975) s. HF Sinclai Na ajo
(2025)
Rese e Mining (D. Minn. 1974; 8 h Ci . 1975). Rese e Mining discha ged
aconi e ailings in o Lake Supe io , eleasing asbes os-like ibe s in o he d inking wa e
o Dulu h and nea by communi ies. The dis ic cou , a e e iewing epidemiological
e idence, concluded ha he cance and espi a o y isks we e in ole able e en unde
unce ain y, and o de ed an immedia e shu down, s essing ha “human heal h mus
come i s .”5On appeal, he Eigh h Ci cui so ened his s ance, holding ha “no ha m
o he public heal h has been shown o ha e occu ed o his da e and he dange o
heal h is no imminen .”6Ins ead o closu e, he company was o de ed o p epa e on-
land disposal acili ies a an es ima ed cos o $243-300 million (a ound $1.3-1.6 billion in
2024 dolla s).7A he ime, oxicological s udies placed inges ion isks only sligh ly abo e
andom in e ence (α≈0.55-0.6), and moni o ing cos s kwe e high because long- e m
sampling campaigns equi ed millions in expendi u e.8Regional economic con ibu ion
was subs an ial: 3,050 di ec jobs and 12,000 dependen jobs.9
This sequence illus a es will ul igno ance: he company had in e nal wa nings bu
a oided commissioning comp ehensi e s udies, elying on scien i ic unce ain y o con-
inue ope a ions. The dis ic cou p io i ized ha m Has ca as ophic and imminen ,
while he appella e cou emphasized economic con ibu ion τand downg aded H o “no
imminen .” Wi h αlow and khigh, he easibili y o semi-pooling (sc eening) was enu-
ous. In he model, his would place inqui y nea he bounda y o easibili y; in p ac ice,
he cou s ole a ed con inued ope a ion in he sho un bu coupled i wi h phased
compliance obliga ions. This esembles p esen -pe iod pooling combined wi h an o de
ha compels ans o ma ion in o compliance in subsequen pe iods.
5. Uni ed S a es . Rese e Mining Co., 380 F. Supp. 11, 26-28 (D. Minn. 1974).
6. Rese e Mining Co. . EPA, 514 F.2d 492, 538 (8 h Ci . 1975). EPA = En i onmen al P o ec ion
Agency.
7. Rese e Mining Co. . He bs , 262 N.W.2d 596, 605 (Minn. 1977) (“o e $300 million” o on-land
disposal); 514 F.2d a 505 ($243m es ima e o Milepos 7 p ojec ).
8. See Ge ald Ma kowi z & Da id Rosne , Decei and Denial: The Deadly Poli ics o Indus ial Pol-
lu ion (2002), ch. 6.
9. TIME, Oc . 22, 1973, “En i onmen : C isis in Sil e Bay.”
16
HF Sinclai Na ajo (D.N.M. 2025). In 2025, HF Sinclai Na ajo Re ining en e ed
a consen dec ee equi ing a $35 million ci il penal y and $137 million in injunc i e com-
pliance in es men s, including la e gas eco e y, was ewa e upg ades, and ad anced
moni o ing h ough Con inuous Emissions Moni o ing Sys ems (CEMS) and Leak De-
ec ion and Repai (LDAR).10 Managemen had ecei ed epea ed iola ion no ices bu
delayed ins alling a ailable moni o ing echnology un il o ced by en o cemen .
He e, inqui y was easible bu delibe a ely a oided. Mode n moni o ing p ecision
was high (α≈0.9), wi h EPA quali y assu ance p o ocols o CEMS a ge ing ela i e
accu acy wi hin 10-20%.11 Inqui y cos s kwe e mode a e, wi h EPA manuals es ima ing
CEMS ope a ion and main enance a $13,000-27,000 pe uni -yea and LDAR p og ams
cos ing ens o housands annually.12 Sec o al compliance was high (θ≈0.8-0.9), as mos
U.S. e ine ies had adop ed CEMS/LDAR by 2020.13 The ha m Hwas well documen ed:
benzene and o maldehyde exposu e om e ine y emissions ca y signi ican cance and
espi a o y isks acco ding o EPA’s Na ional Ai Toxics Assessmen .14
A hese alues, ha m-based sc eening was easible, and compliance-based ules p edic
ha once penal ies elimina e he no-inqui y b anch, inqui y pe sis s. The scale o he
dec ee easily clea ed his h eshold. In s a ic e ms, he ou come esembles semi-pooling;
in p ac ice, he consen dec ee ansla ed his in o a dynamic emedy, combining sho - un
ole ance wi h manda ed ansi ion in o ull compliance.
A inal nuance conce ns he di e ence be ween ou one-sho model and eal-wo ld
emedies. In he model, he bad ype is ei he ole a ed (pooling) o excluded (inac-
i i y). By con as , en i onmen al en o cemen o en allows con inued ope a ion con-
di ional on ans o ma ion— i ms a e o de ed o in es in moni o ing and echnology
ha make u u e iola ions impossible. This can be unde s ood as he dynamic coun-
e pa o ou amewo k: in he i s pe iod, he bad ype is punished bu ole a ed; o
emain in he ma ke in subsequen pe iods, i mus ans o m in o a complian good
ype. Consen dec ees and mi iga ion o de s he e o e unc ion as exclusion ollowed by
condi ional e-en y, an ex ension ha unde sco es how ou s a ic analysis maps on o
epea ed egula o y p ac ice.15
5.1 Empi ical p edic ions
The case s udies illus a e how judicial ou comes align wi h he model’s logic, bu he
amewo k also yields es able compa a i e s a ics ha go beyond single dispu es. These
10. U.S. Depa men o Jus ice (DOJ), P ess Release, Jan. 17, 2025; EPA Se lemen Summa y, Ap .
25, 2025, Uni ed S a es . HF Sinclai Na ajo Re ining LLC (D.N.M. 2025). EPA = En i onmen al
P o ec ion Agency.
11. EPA, Clean Ai Ma ke s: Con inuous Emission Moni o ing Sys ems (CEMS), echnical guidance.
12. EPA, Technical Suppo Documen o CEMS Cos s, 2016 (O&M $13-27k pe moni o -yea ); EPA,
Leak De ec ion and Repai : A Bes P ac ices Guide, 2007.
13. EPA, Ai Facili y Sys em Compliance Repo s (2022).
14. U.S. EPA, Na ional Ai Toxics Assessmen (2018 cycle).
15. See, e.g., U.S. EPA, Consen Dec ee: Cummins Inc. (1998, amended 2006) ( equi ing phased engine
moni o ing and compliance upg ades o e mul iple yea s), a ailable a h ps://19janua y2021snapsho .
epa.go /si es/s a ic/ iles/2013-09/documen s/cumminscd.pd (Las isi ed: 21s Sep 2025); Joseph A.
Hes e , “Consen Dec ees as Eme gen En i onmen al Law,” 85 Mo. L. Re . 2020 (discussing how
consen dec ees o en subs i u e o s a u o y p ecision by imposing phased compliance obliga ions); see
also U.S. EPA, Consen Dec ee: The Williams Companies Inc. (2023) ( equi ing compliance ce i ica ion
and moni o ing wi hin 180 days), a ailable a h ps://www.epa.go /sys em/ iles/documen s/2023-04/
hewilliamscompaniesinc-cd.pd . (Las isi ed: 21s Sep 2025)
17
p edic ions can be aken o da a in a ious legal a eas.
Fi s , he model p edic s ha inqui y is mo e likely when complian ac o s a e common
(θhigh), legal su plus is la ge (yG), moni o ing is accu a e (αhigh), and cos s a e low
(k). Sec o s wi h highe baseline compliance and cheape moni o ing echnologies should
he e o e display highe a es o inqui y.
Second, liabili y design lea es dis inc empi ical oo p in s. Unde ha m-based ules,
sc eening should collapse when moni o ing cos s ise o accu acy alls: o example, a-
cili ies may educe inqui y in ensi y du ing pe iods o equipmen ailu e o igh ened QA
p o ocols. By con as , unde compliance-based ules, once he sa e-ha bo h eshold is
c ossed, inqui y pe sis s e en i exclusion would be mo e e icien . This p edic s a di e -
gence: sc eening a es should be mo e sensi i e o cos shocks unde ha m-based egimes
han unde compliance-based ones.
Thi d, he model p edic s ha obse ed penal ies should align wi h implemen abili y
h esholds. In compliance-based egimes, sanc ions ha induce inqui y should all jus
abo e he h eshold T≥k/[φ(1 −θ)], since highe penal ies no longe a ec pos -inqui y
incen i es. In ha m-based egimes, penal ies ha allow pooling o pe sis despi e iola-
ions should emain a o below T≤yG/(1 −θ), while penal ies chosen o induce inqui y
should exceed his cu o . In dual-penal y egimes, on -end penal ies Tnshould be se
jus abo e yG/(1 −θ) o elimina e pooling, while back-end penal ies Tia e uned o go -
e n inqui y quali y. Empi ically, one would he e o e expec clus e ing o penal ies nea
hese cu o s: egula o s and cou s end o impose he minimum su icien sanc ion o
induce he desi ed equilib ium, o o hold penal ies below he le el ha would des abilize
an equilib ium hey wish o ole a e.
Toge he , hese p edic ions imply c oss-sec ional co ela ions be ween compliance
a es and inqui y, di e ences in sensi i i y o cos shocks ac oss liabili y egimes, and
penal y magni udes ha line up wi h heo e ical h esholds. This p o ides a b idge om
he heo e ical model o en o cemen da a, enabling sys ema ic es s beyond quali a i e
case s udies.
6 Fu he Discussion
6.1 Endogenous inqui y p ecision.
Now assume he agen chooses he p ecision o inqui y, α∈(1/2,1) as an addi ional
s a egy, acing an inc easing cos k(α)(k(1/2) = 0,k(1) = ∞,k′>0, k′′ ≥0). In
semi-pooling, he agen ’s esidual legal exposu e a e inqui y is (1 −θ)(1 −α)mul i-
plied by he penal y ha applies a e inqui y: unde ha m-based ules his is T, unde
compliance-based ules i is 0, and unde dual penal ies i is Ti. The agen ’s p i a e
choice o p ecision he e o e sol es a simple ade-o be ween he ma ginal sa ing in ex-
pec ed esidual exposu e and he ma ginal cos k′(α). Two implica ions ollow. Fi s ,
p ecision is inc easing in he penal y ha p ices esidual exposu e (none unde compli-
ance, Tunde ha m-based, Tiunde dual): compliance yields he lowes p i a ely chosen
p ecision; dual penal ies allow he lawmake o aise p ecision by a ge ing Tiwi hou
simul aneously in la ing he no-inqui y ma gin. Second, he socially op imal p ecision
balances he social bene i o be e sc eening ( ewe alse posi i es and alse nega i es)
agains k′(α); wi h dual penal ies he lawmake can implemen his a ge (subjec o he
good ype’s p o i abili y) by se ing Ti o ma ch ha i s -o de condi ion, while using
Tnonly o egula e he no-inqui y b anch. This a angemen p ese es he selec ion le e
18
( ia Tn) and aligns inqui y quali y ( ia Ti). Howe e , e i iabili y is s ic ly equi ed: α
(o a su icien p oxy) and due-diligence e o mus be audi able so ha ans e s and
liabili y can condi ion on he chosen p ecision. Fo mal s a emen s and p oo s appea in
Appendix C.1. A close analogue a ises in compliance law, whe e egula o s adjus s an-
da ds upwa d as inqui y can be made mo e accu a e a simila cos . Fo example, U.S.
en i onmen al ules now manda e con inuous emissions moni o ing once he echnology
became easible, eplacing ea lie sel - epo ing schemes.
6.2 Psychological inqui y cos s and social p e e ences
Beyond angible e o , inqui y o en ca ies non-ma e ial bu dens-emba assmen , ea
o social disapp o al, and epu a ional loss (Hellman 2009; Alexande and Fe zan 2009;
G ossman and Van De Weele 2017). Some agen s also in e nalize o he s’ wel a e o
a ying deg ees (Ya e 2018).
Psychological (non-ma e ial) cos s o inqui y. Le he agen ’s p i a e inqui y cos
emain k, bu le he lawmake place weigh σ∈[0,1] on i s non-ma e ial componen in
wel a e. P i a e incen i es o inqui e a e unchanged ( hey depend on k, no on σ); he
social alua ion o inqui y is a enua ed: he semi-pooling wel a e e m is θyG−σk a he
han θyG−k. Hence, as σ alls, he egion whe e inqui y is socially p e e ed expands,
ye implemen abili y is una ec ed-inducing inqui y s ill equi es uling ou ac ion wi hou
inqui y and ensu ing he legal su plus co e s k. Unde impe ec inqui y, he same logic
holds; wi h dual penal ies one simply eads he no-inqui y side wi h Tnand he inqui y
side wi h he agen ’s esidual exposu e go e ned by Ti.16
Social p e e ence (mo al) conce n unde pooling. Suppose an agen who pe -
o ms an illegal ask su e s an in e nal mo al cos m≥0. Unde pooling (no inqui y),
accep ance depends on expec ed disu ili y, so he minimal ans e ha induces accep ance
is
X∗= (1 −θ)T+m
(wi h a dual scheme, eplace Tby he no-inqui y penal y Tn). Thus mac s like an addi i e
expec ed penal y: when he lawmake aims o de e ac ion wi hou inqui y, a la ge m
pe mi s a lowe legal penal y o achie e he same de e ence; when he lawmake ins ead
p e e s o p ese e pooling (e.g., when he bad ype’s ne con ibu ion is non-nega i e
o only mildly nega i e), a la ge mcan make pooling in easible by pushing X∗abo e a
ype’s payo . Unde impe ec inqui y, hese easibili y e ec s a e unchanged because m
bi es only on he no-inqui y ma gin.
In e ac ion and obus ness. Psychological inqui y cos s and social p e e ences pull
on di e en le e s. Discoun ing non-ma e ial inqui y bu dens in wel a e (σ < 1) shi s
social desi abili y owa d inqui y bu does no change he agen ’s p i a e cu o o inqui e;
mo al conce n (m > 0) igh ens he easibili y o pooling by aising he ans e needed
o no-inqui y accep ance. Wi h impe ec inqui y (α < 1), hese di ec ions a e in ac :
noise lowe s he wel a e o inqui y-based sc eening and na ows i s exis ence window,
bu σand ma ec he same ma gins as unde pe ec inqui y. The compa a i e s a ics
16. Fo mal h esholds wi h (σ, T)unde a single penal y and (σ, Tn, Ti)unde dual penal ies a e col-
lec ed in Appendix C.4.
19
abo e do no depend on adop ing any pa icula liabili y scheme; hey desc ibe how σ
and men e he lawmake ’s ade-o s unde each egime. Fo mal windows and wel a e
compa isons a e in he appendix C.4.
6.3 En o cemen in ensi y s. penal y se e i y.
Assume ha con ic ion is no necessa ily pe ec , bu wi h p obabili y p∈(0,1]. I p
is exogenous and en o cemen is cos less, we can simply ead Tas he expec ed penal y
pT h oughou ; unde dual penal ies ead (Tn, Ti)as (pTn, pTi). Now le us elax he as-
sump ion o “cos less” con ic ion and/o en o cemen , and in oduce cos in o he penal y
sys em.
Case A: pcos less (p= 1), Tcos ly. When en o cemen is cos less ( hus p= 1) bu
penal y se e i y Tabso bs eal esou ces (e.g., inca ce a ion), only equilib ia ha impose
penal ies on he pa h educe wel a e by expec ed esou ce cos . Unde pe ec inqui y,
pooling bea s (1 −θ)C(T)while semi-pooling (sc eening) and inac i e bea none. Unde
impe ec inqui y wi h ha m-based liabili y, semi-pooling addi ionally bea s (1 −θ)(1 −
α)C(T)( alse posi i es); unde compliance-based liabili y, inqui ing agen s a e exemp
and semi-pooling again bea s no esou ce cos ; unde impe ec inqui y wi h dual-penal y
liabili y, semi-pooling addi ionally bea s (1 −θ)(1 −α)C(Ti)( alse posi i es). Hence he
lawmake should always choose he minimal T ha implemen s he desi ed con inua ion
beha io and a oid on-pa h penal ies when C(·)is la ge.
Case B: Tcos less, pcos ly. I ins ead in ensi y pis esou ce-cos ly (wi h con ex
K(p)) and se e i y Tis ee, he conclusions a e symme ic: equilib ia ha impose penal-
ies on he pa h now bea expec ed Cp(p); he wel a e-maximizing choice is he minimal p
ha implemen s he desi ed equilib ium; and designs ha a oid on-pa h penal ies (semi
unde compliance-based ules; inac i e) a e s ic ly mo e a ac i e when Cp(p)is la ge.
Case C: bo h pand Tcos ly. When bo h ins umen s a e cos ly, choose he cheap-
es mix o (p, T)(o (p, Tn, Ti)unde dual penal ies) ha sa is ies he ele an incen-
i e/exis ence cons ain s. Wi h con ex cos s CT(·)and Cp(p), an in e io solu ion equal-
izes ma ginal esou ce cos pe uni o expec ed penal y; co ne solu ions a ise when one
ins umen is much cheape . Unde dual penal ies, Tnp ices he no-inqui y ma gin and
Tip ices esidual isk a e inqui y, so i is e icien o se he unused componen o ze o
whene e i does no bind.
6.4 Ma ke s uc u e and liabili y sha ing
Ou baseline assumes a single p incipal makes a ake-i -o -lea e-i o e o a single agen .
The main compa a i e s a ics and implemen abili y logic ex end o al e na i e ma ke
s uc u es and liabili y sha ings wi h only cosme ic changes o he easibili y h esholds.
We summa ize ou use ul a ian s and e e o Appendix C.5 o o mal s a emen s and
p oo s.
Single p incipal wi h unce ain y (one side, one payo ). I nei he pa y knows
legali y ex an e, se yG=yB≡yand in e p e ype unce ain y as unce ain y abou
he legal s a e. The axonomy and wel a e anking mi o he baseline. Unde pe ec
inqui y, pooling (no inqui y) is easible i y≥(1 −θ)Tn;sc eening (inqui y) is easible i
θy ≥k. Unde impe ec inqui y wi h dual penal ies, eplace he sc eening h eshold by
20
φ y ≥k+ (1 −θ)(1 −α)Ti, whe e φ:= θα + (1 −θ)(1 −α). Hence he lawmake ’s design
p oblem and he penal y le e s ca y o e e ba im.
No dis inc p incipal (sel -solici a ion). In se ings like solici ing con aband se -
ices, he “p incipal” and “agen ” collapse in o he same decision-make ; ans e s anish
and only he decision o inqui e (a cos k) e sus ac wi hou inqui y emains. The
sc eening condi ion becomes he agen ’s p i a e su plus es (as abo e wi h y), and he
pooling condi ion becomes he accep ance o expec ed legal isk. Quali a i ely, his is he
single-p incipal case wi h ze o en s: inqui y occu s i he legal-su plus e m clea s k, and
will ul igno ance a ises i expec ed penal ies a e (p i a ely) co e able. The lawmake ’s
penal y le e s and wel a e compa ison a e unchanged.
Agen ba gaining powe (o compe i ion among p incipals). I he agen can
ex ac mo e han he minimal accep ance ans e (e.g., Nash ba gaining, o many p in-
cipals bidding), pooling easibili y sh inks (p incipals mus co e highe ans e s) while
sc eening easibili y expands ( he agen can be unded o co e kand any esidual expo-
su e). Implemen abili y he e o e il s owa d inqui y. When inqui y is socially desi able,
ba gaining powe helps he lawmake (lowe penal y su ices o ule ou no-inqui y ac-
ion); when inclusion o bo h ask ypes ia pooling is desi able, s ong agen powe can
make pooling in easible. The lawmake hen ades o desi abili y agains easibili y as
in he baseline, bu wi h he windows shi ed owa d sc eening.
Liabili y alloca ion as “jus ans e s.” The same Coase-s yle logic behind ba gain-
ing powe ca ies o e o who bea s legal penal ies. I pa ies a e isk-neu al, penal ies
a e ( ines) cos less o socie y, and indemni y p omises a e en o ceable, hen shi ing li-
abili y sha es be ween he p incipal (he) and he agen (she)-o allowing he p incipal
o wa an y he agen ’s penal ies-simply eassigns ans e s wi hou changing wel a e.
Wha can change is implemen abili y: p incipal-side liabili y can es o e a le e o de e
no-inqui y beha io when compliance-based ules would o he wise make sc eening ha d
o swi ch o , whe eas ull indemni y o he agen e ec i ely eplica es an exemp ion
and emo es ha le e . In sho , he logic ha “nego ia ion powe ansla es in o pu e
ans e s” ex ends o sha ed-liabili y and wa an y a angemen s as well; when penal ies
o en o cemen a e cos ly, o indemni y is non-con ac ible, his equi alence b eaks and
design ade-o s eappea (ou side ou baseline).
6.5 Robus ness o unknown θ.
I he agen does no know he ue θ(sha e o good p incipals), le he ac on a pe cei ed
alue ˜
θ(pos e io mean unde Bayes; wo s case θunde ambigui y a e sion). Then
all p i a e h esholds in ou analysis hold wi h θ eplaced by ˜
θ(e.g., he no-inqui y
accep ance ans e becomes (1−˜
θ)Tn, and sc eening is easible unde impe ec inqui y i
˜φ yG≥kwi h ˜φ:= ˜
θα+(1−˜
θ)(1−α)). Social desi abili y s ill e alua es wel a e a he ue
θ. Ambigui y (lowe ˜
θ) simul aneously makes no-inqui y less a ac i e (highe expec ed
penal y) and educes he expec ed upside om inqui y (lowe ˜φ), so implemen abili y
can shi ei he way; ou compa a i e s a ics o he wise ca y h ough e ba im. Dual
penal ies emain he mos obus ins umen : se Tnhigh o de e no-inqui y ac oss he
ele an ˜
θ ange, and keep Tilow o a oid o e bu dening inqui y when easible.
21
This o mula ion also aligns wi h he doc ine’s “su icien suspicion” equi emen : a
cou can impose a minimum suspicion h eshold on 1−˜
θwi hou al e ing ou equilib-
ium axonomy. I also cla i ies how di e en mens ea ca ego ies map in o he model.
In ou baseline, non-inqui y is a ully a ional choice and hus co esponds o delibe a e
igno ance. By con as , negligence could be ep esen ed by mis aken belie s (sys ema -
ically mispe cei ing θ), and ecklessness by agen s who ecognize isk bu unde weigh
i ela i e o incen i es. These cases lie ou side he a ional baseline bu illus a e how
he model can accommoda e he doc inal dis inc ion be ween negligen , eckless, and
delibe a e igno ance.
6.6 Ju isp uden ial conce ns.
Se e al doc inal deba es a ound will ul igno ance can be mapped in o ou amewo k.
Fi s , he pa i y ques ion asks whe he delibe a e igno ance should be punished like
knowledge. In equilib ium, knowing c ime is s ic ly domina ed by non-inqui y once ig-
no ance is punished a pa i y o abo e, so he equal-culpabili y deba e has no bi e excep
in he implausible case whe e igno ance is punished mo e se e ely han knowledge. Re-
la edly, some a gue culpabili y should depend on whe he knowledge would ha e changed
beha io . Ou model shows his c i e ion is moo : inqui y ollowed by knowing iola ion
is ne e op imal, so all ha m ul conduc lows h ough igno ance.
Second, he legali y p inciple (nulla poena sine lege) cau ions agains punishing unde
ague o shi ing s anda ds. In ou amewo k, agueness is cap u ed by educed accu-
acy α: inqui y may no yield a clea answe . Compliance-based o dual-penal y ules
accommoda e his by lowe ing esidual isk once inqui y is documen ed in good ai h.
Thi d, conce ns abou o e b ead h and chilling e ec s co espond di ec ly o ou im-
plemen abili y on ie : excessi ely high penal ies on non-inqui y can make sc eening
in easible and collapse aluable ma ke s. Finally, doc ine dis inguishes negligence, eck-
lessness, and delibe a e igno ance. Ou baseline assumes a ional agen s, so non-inqui y
is delibe a e; negligence and ecklessness could be modeled as sys ema ic mispe cep ion o
θo unde weigh ing o ecognized isk. Some cou s and commen a o s also place will ul
igno ance be ween ecklessness and knowledge, on he iew ha he ac o “in ac ” does
no know. We ea delibe a e igno ance a pa i y wi h knowledge, e lec ing i s s a egic
na u e. This choice has no e ec on equilib ium ou comes, since knowing c ime is s ic ly
domina ed and ne e a ises.
7 Conclusion
This pape o e s a ac able amewo k o analyzing will ul igno ance unde asymme ic
in o ma ion and legal penal ies. The cen al message is an implemen abili y one. Wi h
pe ec inqui y, penal ies can be used o sc een ou ha m ul beha io while p ese ing
aluable ac i i y, so he wel a e-maximizing beha io can always be made o exis and
be selec ed. Wi h impe ec inqui y, noise simul aneously lowe s he alue o sc eening
and decouples easibili y om desi abili y; o non-emp y egions o p imi i es, inqui y is
ei he una ainable when desi able o di icul o swi ch o when exclusion is p e e ed.
We show how liabili y design media es ha ade-o . Ha m-based ules keep a s ong
selec ion le e bu can choke o desi able sc eening; compliance-based ules elax easi-
bili y bu isk coexis ence and misselec ion; dual-penal y ules sepa a e hese oles and
22
weakly domina e single-penal y ules in implemen abili y, ailing only when he legal su -
plus canno co e expec ed inqui y cos . These insigh s ansla e in o simple guidance:
calib a e penal ies di ec ly o he cu o condi ions implied by accu acy and inqui y cos s.
When accu acy is impe ec , use sepa a e penal ies o unin o med ac ion and o esidual
w ongdoing a e inqui y. In his sense, igno ance is no a shield: he absence o knowing
c ime in equilib ium is an endogenous consequence o incen i es a he han e idence o
a highe mo al s anda d, and liabili y ules mus he e o e a ge he s a egic ma gin o
non-inqui y.
The amewo k’s s eng hs a e i s cla i y abou wha penal ies can and canno ac-
complish and i s po abili y: i p o ides a empla e o s udying inqui y echnologies, e -
iden ia y s anda ds, and en o cemen ic ions. Fu u e wo k could ex end he amewo k
o dynamic and he e ogeneous en i onmen s-e.g., epea ed in e ac ions wi h epu a ion,
he e ogenei y in inqui y cos s, subsidies o inqui y and in o ma ion disclosu e egula-
ion. Toge he , hese can deepen he wel a e ounda ions o egula ing will ul igno ance
in p ac ice.
Re e ences
Alexande , La y, and Kimbe ly Kessle Fe zan. 2009. C ime and culpabili y: A heo y o
c iminal law. Camb idge Uni e si y P ess.
Al e , Adam L, Julia Ke nochan, and John M Da ley. 2007. “Mo ali y in luences how
people apply he igno ance o he law de ense.” Law & Socie y Re iew 41 (4): 819–
864.
Cha low, Robin. 1991. “Wil ul igno ance and c iminal culpabili y.” Tex. L. Re . 70:1351.
Dana, Jason, Robe o A Webe , and Jason Xi Kuang. 2007. “Exploi ing mo al wiggle
oom: expe imen s demons a ing an illuso y p e e ence o ai ness.” Economic The-
o y, 67–80.
G ossman, Zacha y, and Joel J Van De Weele. 2017. “Sel -image and will ul igno ance
in social decisions.” Jou nal o he Eu opean Economic Associa ion 15 (1): 173–217.
Hellman, Debo ah. 2009. “Will ully blind o good eason.” C iminal Law and Philosophy
3:301–316.
Ka ik, Na in, Ma co O a iani, and F ancesco Squin ani. 2007. “C eduli y, lies, and
cos ly alk.” Jou nal o Economic heo y 134 (1): 93–116.
Ki el, La a, and I a R Hannikainen. 2023. “Why blame he os ich? Unde s anding
culpabili y o will ul igno ance.” K., P ochownik, S. Magen,(Eds.), Ad ances in ex-
pe imen al philosophy o law, 75–98.
Luban, Da id. 1998. “Con i ed igno ance.” Geo. LJ 87:957.
Polinsky, A Mi chell, and S e en Sha ell. 2000. “The economic heo y o public en o ce-
men o law.” Jou nal o economic li e a u e 38 (1): 45–76.
Sa ch, Alexande . 2018. “Will ul igno ance in law and mo ali y.” Philosophy Compass 13
(5): e12490.
23
B.4 P oo o P oposi ion 4.2
Lemma B.3 (Th eshold geome y unde ha m-based liabili y wi h impe ec inqui y).
Le
φ:= θα + (1 −θ)(1 −α), p3:= (1−θ)α
(1−θ)α+θ(1−α), p4:= (1−θ)(1−α)
φ,
and de ine he bad- ype incen i e-compa ibili y h eshold
TIC
B:= φαyB+(1−α)k
α(1−α+αθ).
Then unde ha m-based liabili y wi h α∈(1
2,1), he con inua ion equilib ium exis ence
condi ions a e:
Pool 1 (insu ance-s yle pooling) exis s o k
θ(1−θ)(2α−1) ≤T≤yG
p3+k
(1−θ)α,
Pool 2 (minimal-accep ance pooling) exis s o T≤minnyG
1−θ,k
θ(1−θ)(2α−1)o,
Semi (inqui y; ac only on g)exis s o T > maxnk
θ(1−θ)(2α−1), TIC
B,yG
1−θoand T≤φ yG−k
(1−θ)(1−α),
Inac i e exis s o T > maxnyB,yG
1−θoand (1 −θ)(1 −α)T+k > φ yG.
Consequen ly, mul iplici y a ises only in:
maxnyB,yG
1−θo< T ≤yG
p3+k
(1−θ)α,(1−θ)(1−α)T+k > φyG⇒Pool 1 and Inac i e coexis .
In pa icula , whene e Semi is easible (i.e. k < θ(2α−1)yG), i s lowe bound exceeds
Pool 1’s uppe bound, so Semi and Pool 1 canno coexis .
P oo o P oposi ion 4.2 (implemen a ion unde ha m-based ules, impe ec
inqui y).
P oo . De ini ions. Minimal ans e s:
Xpool
min = maxn(1 −θ)T, p3T−k
1−φo, Xsemi =(1−θ)(1−α)T+k
φ=p4T+k
φ.
As in he main ex , ans e s/penal ies a e edis ibu i e; wel a e is T-in a ian wi hin
a b anch:
SWpool =θyG+(1−θ)(τyB−H), SWsemi =θα yG+(1−θ)(1−α)(τyB−H)−k, SWinac = 0.
Le eSW ∈ {Pool,Semi,Inac i e}deno e a wel a e-maximizing ype (igno ing imple-
men abili y).
Case 1: e⋆=Pool (all ades desi able).
(a) Insu ance-s yle pooling (Pool 1). Choose Ts ic ly inside he egion:
T∈hk
θ(1−θ)(2α−1),yG
p3+k
(1−θ)αi.
We do no need o ule ou Inac i e: unde ou implemen a ion con en ion, Inac-
i e is weakly Pa e o-domina ed by Pool 1 and he e o e no c edi ed when bo h
30
coexis .17
(b) Minimal-accep ance pooling (Pool 2). Pick any
T≤minyG
1−θ,k
θ(1−θ)(2α−1),
Semi ails by T≤k
θ(1−θ)(2α−1) and we do no need o ule ou Inac i e.
Case 2: e⋆=Inac i e (no ade desi able). Choose Ts ic ly inside he inac i i y
egion and ou side bo h pooling egions, e.g.
T > M := max(yB,yG
1−θ,φ yG−k
(1−θ)(1−α),yG
p3+k
(1−θ)α).
Then Pool 2 ails by T > yG
1−θ, Pool 1 ails by T > yG
p3+k
(1−θ)α, and Semi ails by
(1 −θ)(1 −α)T+k > φ yG. Hence Inac i e is unique.
Case 3: e⋆=Semi (inqui y desi able). Semi mus be easible, i.e.
maxnk
θ(1−θ)(2α−1), TIC
B,yG
1−θo<φ yG−k
(1−θ)(1−α).
Since Semi’s egion is s ic ly abo e yG
1−θwhile Pool 1’s uppe bound is below yG
1−θwhene e
Semi is easible, Semi and Pool 1 canno coexis . Also Semi and Inac i e canno coex-
is because hei easibili y inequali ies o (1 −θ)(1 −α)T+ka e mu ually exclusi e.
The e o e, selec ing any
T∈maxnk
θ(1−θ)(2α−1), TIC
B,yG
1−θo,φ yG−k
(1−θ)(1−α)
pu s Tin he in e io o Semi’s egion and ou side all compe i o s. Hence Semi is unique
i easible.
Wel a e compa ison and conclusion. Wi hin each equilib ium ype, SW is inde-
penden o T; hence he lawmake ’s choice o Tonly selec s be ween ypes. Pooling and
Inac i e equilib ia deli e exac ly he same wel a e as in he No-Inqui y benchma k, while
Semi yields
SWsemi =θα yG+ (1 −θ)(1 −α)(τyB−H)−k,
which is s ic ly below i s pe ec -inqui y coun e pa θyG−kbu s ic ly abo e he No-
Inqui y ou come whene e Semi is easible. Thus he lawmake can always implemen
pooling o inac i i y o de e inqui y when i is no socially desi able. Howe e , he e exis
non-emp y pa ame e anges (lowe αo highe k) in which Semi is wel a e-maximizing
ye in easible, so inqui y canno be induced by any penal y. In all cases, maximal wel a e
unde impe ec inqui y is weakly below ha unde pe ec inqui y and weakly abo e
ha unde No Inqui y.
B.5 P oo o P oposi ion 4.3
P oo . De ini ions. Le
φ:= θα + (1 −θ)(1 −α),
17. Non-emp iness: he in e al k
θ(1−θ)(2α−1) ,yG
p3+k
(1−θ)αis non-emp y i k≥θ(2α−1)yG.
31
Minimal ans e s:
Xsemi =k
φ, Xpool = (1 −θ)T.
Agen u ili ies:
Ui=φX −k, Uni =X−(1 −θ)T.
Good- ype easibili y o semi: φyG≥k.
Wel a e is T-in a ian wi hin b anches:
SWsemi =θαyG+(1−θ)(1−α)(τyB−H)−k, SWpool =θyG+(1−θ)(τyB−H), SWinac = 0.
Case 1: e⋆=Semi (inqui y desi able). Semi equi es he agen o s ic ly p e e
inqui y. A X=k/φ,
Ui= 0, Uni =k
φ−(1 −θ)T.
Thus inqui y is op imal whene e
T≥k
φ(1−θ).(C1)
Good- ype easibili y equi es
φyG> k. (C2)
Toge he , (C1)–(C2) de ine he egion in which inqui y is implemen able. In his egion,
Semi is unique: Pool ails by (C1), and Inac i e ails by good- ype de ia ions. Hence
Semi is implemen ed whene e i is easible.
Case 2: e⋆=Inac i e (no ade desi able). Suppose wel a e would p e e no ac ion.
I φyG≤k, Semi is in easible by (C2), and he lawmake can se T > yB o ule ou
Pool, implemen ing Inac i e uniquely. I φyG> k, howe e , Semi emains easible a
X=k/φ o any T≥k/[φ(1 −θ)]. Because compliance-based liabili y exemp s inqui y
om penal ies on alse posi i es, no choice o Tcan emo e he inqui y equilib ium. Thus
when exclusion is desi able bu (C2) holds, inqui y pe sis s in he equilib ium se .
Case 3: e⋆=Pool (no inqui y desi able). Wi hou inqui y, he agen accep s i
Uni ≥0, i.e. X≥(1 −θ)T. Minimal ans e is X= (1 −θ)T. The lawmake can always
se Tsmall enough ha X≤yG, yB; hence Pool can be implemen ed whene e pooling
is wel a e-maximizing.
Wel a e compa ison and conclusion. Pooling and Inac i e equilib ia yield exac ly
he same wel a e as he No-Inqui y benchma k. Semi yields
SWsemi =θαyG+ (1 −θ)(1 −α)(τyB−H)−k,
which con e ges o he pe ec -inqui y benchma k θyG−kas α→1, and is s ic ly below
i o α < 1. Thus, unde compliance-based liabili y, inqui y can always be sus ained
when i is easible bu canno always be de e ed when socially undesi able. Maximal
wel a e is he e o e weakly below pe ec inqui y and may in case (ii) all s ic ly below
he No-Inqui y benchma k.
32
B.6 P oo o P oposi ion 4.4
P oo . P imi i es and no a ion. Le
φ:= θα + (1 −θ)(1 −α),1−φ=θ(1 −α) + (1 −θ)α,
he p obabili y (unde pooling ypes) ha inqui y yields a “good” signal and he ask is
pe o med. Unde dual penal ies we deno e by Tn he penal y i he agen did no inqui e
and he ask is illegal, and by Ti he penal y i he agen did inqui e ye an illegal ask
is pe o med ( alse posi i e).
S ep 1 (Minimal ans e s). I bo h ypes o e he same pe -pe o mance ans e
and he agen inqui es and pe o ms only on a good signal, he expec ed u ili y is
Ui=φ −k−(1 −θ)(1 −α)Ti.
By ie-b eaking, he minimal ans e ha induces inqui y is
semi =k+(1−θ)(1−α)Ti
φ.(B.1)
I ins ead she accep s wi hou inqui y, he minimal lump-sum ans e ha induces ac-
cep ance equals he expec ed penal y:
Xpool = (1 −θ)Tn.(B.2)
S ep 2 (Tu ning inqui y o when i is no desi able). Make semi in easible by
choosing Tila ge so ha he good ype canno und semi:
semi > yG⇐⇒ Ti>φ yG−k
(1−θ)(1−α).
Since α > 1/2, he denomina o is posi i e, so such Tiexis s whene e desi ed. Wi h
inqui y in easible, use Tn o selec he no-inqui y b anch: implemen pooling by keeping
Xpool ≤min{yG, yB}( he agen hen weakly p e e s no inqui y a he minimal ans e ),
o implemen inac i i y by aking Tn>max{yG, yB}/(1 −θ).
S ep 3 (Tu ning inqui y on when i is desi able). Minimize he semi ans e by
lowe ing Ti(down o 0i needed), so semi becomes as small as possible. Sus aining semi
equi es h ee condi ions:
(i) Good- ype easibili y.
semi ≤yG⇐⇒ (1 −θ)(1 −α)Ti+k≤φ yG.(F)
(ii) Inqui y op imal o he agen a semi.A semi we ha e Ui= 0 by cons uc ion,
while Uni = semi −(1 −θ)Tn. Hence Ui≥Uni i
(1 −θ)Tn≥ semi ⇐⇒ Tn≥k+(1−θ)(1−α)Ti
φ(1−θ).(AO)
(iii) Bad- ype IC agains he de ia ion X′=Tn.O pa h he agen accep s any
X′≥Tnwi hou inqui y, yielding yB−Tn o ype B. On-pa h unde semi, Bea ns
(1 −α)(yB− semi). De e ing he de ia ion equi es
yB−Tn≤(1 −α) (yB− semi)⇐⇒ Tn≥α yB+ (1 −α) semi.(BIC)
33
I φyG> k, ake Ti= 0 o minimize semi =k/φ, and hen choose
Tn≥max nk
φ(1−θ), αyB+ (1 −α)k
φo,
which sa is ies (AO) and (BIC) wi h slack; (F) holds since φyG> k. Pooling is hen
i ele an because he agen s ic ly p e e s inqui y. I φyG≤k, e en Ti= 0 gi es
semi ≥yG, so semi is p i a ely in easible.
S ep 4 (Wel a e and dominance o e single-penal y egimes). The dual-penal y
pai (Tn, Ti)nes s ha m-based (Tn=Ti)and compliance-based (Ti= 0) ules. The e-
o e he maximal wel a e a ainable unde dual penal ies weakly domina es he maximal
wel a e a ainable unde ei he single-penal y design. Since α < 1implies alse posi-
i es/nega i es unde inqui y, he maximal wel a e unde dual penal ies is (weakly) below
he pe ec -inqui y benchma k. The implemen abili y on ie s a ed in he p oposi ion
ollows om S ep 2 (de e ing inqui y) and S ep 3 (sus aining inqui y) ia he linea
bounds (F), (AO), and (BIC).
Impe ec inqui y: calib a ion and compa a i e s a ics
This appendix collec s he ull cu o exp essions and compa a i e s a ics ha unde lie
he b ie discussion in Sec ion 4. They make p ecise how penal y placemen depends on
signal accu acy α, inqui y cos k, he p io θ, and he ype-dependen ou pu s yG, yB.
Assume impe ec inqui y wi h p ecision α∈(1/2,1) and de ine
φ(α) := θα + (1 −θ)(1 −α).
Lemma B.4 (Calib a ion unde impe ec inqui y by liabili y ule).Le e⋆∈ {pooling,semi,inac i e}
be he socially desi able beha io .
(a) Ha m-based single penal y T.
(i) Semi-pooling. Implemen able i
maxnyB,k
θ(1−θ)(2α−1)o< T < φ(α)yG−k
(1−θ)(1−α)and φ(α)yG> k.
(ii) Pooling. Implemen able i
T≤minnyB,yG
1−θowhen θyG≤k,T=k
θ(1−θ)(2α−1), T ≤yBwhen θyG> k.
(iii) Inac i e. Implemen able i
T > maxnyB,yG
1−θ,φ(α)yG−k
(1−θ)(1−α)o.
(b) Compliance-based single penal y T(sa e ha bo a e inqui y).
(i) Semi-pooling. Implemen able i
T≥k
φ(α)(1−θ)and φ(α)yG> k.
34
(ii) Pooling. Implemen able i
T≤minnyB,yG
1−θowhen θyG≤k,T < yG
1−θwhen θyG> k.
(iii) Inac i e. Implemen able i
φ(α)yG≤kand T > yB.
(c) Dual penal ies (Tn, Ti).
(i) Semi-pooling. Implemen able i
Tn>yG
1−θand (1 −θ)(1 −α)Ti+k≤φ(α)yG.
(ii) Pooling. Implemen able i
Tn≤minnyB,yG
1−θo,
o , i θyG> k,
Tn=k
θ(1−θ)(2α−1), Tn≤yB,and (1 −θ)(1 −α)Ti+k > φ(α)yG.
(iii) Inac i e. Implemen able i
Tn> yBand (1 −θ)(1 −α)Ti+k > φ(α)yG.
P oposi ion B.1 (Compa a i e s a ics o impe ec -inqui y cu o s).Fo he ha m-based
ule (and he Ti-a m o he dual-penal y ule), de ine
L(α) := k
θ(1−θ)(2α−1), U(α) := φ(α)yG−k
(1−θ)(1−α).
1. L′(α)<0: highe accu acy lowe s he penal y equi ed o induce inqui y.
2. U′(α)>0i θyG> k;U′(α)=0i θyG=k;U′(α)<0i θyG< k.
3. ∂L/∂k > 0,∂U/∂k < 0;∂U/∂yG>0while Ldoes no depend on yG.
4. L(θ)is minimized a θ= 1/2;U(α) ises in θwhene e αyG> k.
5. Limi s: as α→1,L(α)→k/[θ(1 −θ)] and, i θyG> k,U(α)→+∞; as α↓1/2,
he in e al (L(α), U(α)) sh inks and may anish unless yGis la ge ela i e o k.
6. Unde compliance-based ules, once unin o med ac ion is de e ed, sc eening is ea-
sible i φ(α)yG> k. Pooling and inac i i y cu o s in yG/(1 −θ)and yB emain
unchanged.
Appendix C Ex ensions-P oo s and Technical No es
C.1 Endogenous Inqui y P ecision
Se up and assump ions. In semi-pooling, a e ag eeing o inqui e he agen chooses
p ecision α∈(1/2,1) a cos k(α). Assume:
35
A1.α∈(1/2,1) and k: (1/2,1) →R+is C1, s ic ly inc easing and s ic ly con ex,
wi h k(1/2) = 0 and limα↑1k(α) = ∞.
A2. T ans e s and liabili y can condi ion on (audi able) inqui y and i s eco ded p eci-
sion (o a e i iable p oxy).
A3. Tie-b eaking: indi e en agen s accep ; indi e en p incipals o e he smalles
ans e ha induces accep ance.
Le λdeno e he penal y ha p ices he agen ’s esidual exposu e a e inqui y:
λ≡
T(ha m-based single penal y),
0(compliance-based exemp ion),
Ti(dual-penal y: a e -inqui y a m).
In semi-pooling, esidual exposu e occu s wi h p obabili y (1 −θ)(1 −α). Wi h e i ia-
bili y, he good p incipal (he) pays he minimal ans e ha co e s he agen ’s p i a e
inqui y bill k(α)plus expec ed esidual exposu e (1 −θ)(1 −α)λ.
Lemma C.1 (Agen ’s p i a ely op imal p ecision).Fix a semi-pooling ou come and a
esidual-exposu e p ice λ≥0. The agen ’s p i a ely op imal p ecision α∗(λ)is he unique
solu ion o
k′(α) = (1 −θ)λ,
and sa is ies α∗(λ)∈(1/2,1),α∗′(λ)>0,limλ↓0α∗(λ) = 1/2, and limλ↑∞ α∗(λ)=1.
P oo . In semi-pooling he agen minimizes k(α) + (1 −θ)(1 −α)λon (1/2,1). S ic
con exi y o kgi es a unique minimize ; he FOC is k′(α) = (1 −θ)λ. Mono onici y and
limi s ollow om A1.
Wel a e wi h semi-pooling. I inqui y is unde aken wi h p ecision α, he wel a e
con ibu ion ( ans e s cancel) is
SWsemi(α) = θ α yG+ (1 −θ)(1 −α) (τyB−H)−k(α).(C.3)
The de i a i e is
d
dα SWsemi(α) = θ yG−(1 −θ)(τyB−H)−k′(α),
hence he socially op imal p ecision bα(when semi-pooling is he a ge beha io ) sol es
k′(bα) = θ yG−(1 −θ)(τyB−H).(C.4)
By A1,bα∈(1/2,1) is unique whene e he igh -hand side is posi i e.18
P oposi ion C.1 (Ins umen powe ac oss liabili y egimes).Le α∗(λ)be as in Lemma C.1,
and bαsa is y (C.4). Then:
18. I θ yG≤(1 −θ)(τyB−H)(ne bene i o accu acy non-posi i e), he lawmake p e e s he lowes
p ecision in he admissible se ; unde A1 his is 1/2. The in e es ing case o sc eening has τyB−H < 0,
making he RHS s ic ly la ge han θyG.
36
(a) compliance-based: λ= 0 implies α∗= 1/2. The lawmake canno aise p ecision
ia penal ies; inqui y accu acy is p i a ely minimized.
(b) ha m-based: λ=Tso α∗is inc easing in T. Raising Talso igh ens he no-
inqui y ma gin, po en ially elimina ing pooling e en when inclusion is desi able.
(c) Dual: λ=Tiso α∗is inc easing in Tiwhile he no-inqui y ma gin is go e ned
by Tn. Thus Ti a ge s inqui y quali y wi hou colla e al e ec s on pooling, and Tn
con ols selec ion on he no-inqui y b anch.
P oo . Immedia e om Lemma C.1 and he mapping o λ o he penal y by egime.
P oposi ion C.2 (Implemen ing he socially op imal p ecision unde dual penal ies).
Suppose semi-pooling is (socially) he a ge beha io and is easible a bα, i.e.,
θbα yG≥k(bα) + (1 −θ)1−bαTi.(C.5)
Unde dual penal ies, se ing
T⋆
i=k′(bα)
1−θand choosing any Tn ha p ese es he desi ed selec ion on he no-inqui y b anch
induces α∗=T⋆
i7→ bαand implemen s he socially op imal p ecision.
P oo . By Lemma C.1, α∗sol es k′(α) = (1 −θ)Ti. Wi h T⋆
ias abo e he agen ’s p i a e
FOC coincides wi h (C.4), hence α∗=bα. Feasibili y is gua an eed by (C.5). Tncan be
uned (independen ly) o p ese e o elimina e pooling as desi ed.
Co olla y C.1 (Limi s unde single-penal y egimes).Unde compliance-based ules,
α∗= 1/2and he lawmake canno implemen bα > 1/2 ia penal ies. Unde ha m-based
ules, he lawmake can aise α∗by inc easing T, bu doing so simul aneously a ec s he
no-inqui y b anch, c ea ing selec ion ade-o s ha dual penal ies a oid.
Exis ence and wel a e. When semi-pooling is socially desi able, easibili y wi h en-
dogenous α equi es ha he good ype’s expec ed legal su plus a α∗(λ)co e s he
p i a ely chosen inqui y bill:
θ α∗(λ)yG≥kα∗(λ)+ (1 −θ)1−α∗(λ)λ,
wi h λmapped o T,0, o Tiby egime. Wel a e a he induced p ecision ollows om
(C.3). Dual penal ies allow he lawmake o (i) selec beha io on he no-inqui y b anch
ia Tnand (ii) align inqui y quali y wi h bα ia Ti, subjec only o he easibili y condi ion
abo e.
C.2 O -pa h belie s and implemen abili y
Le ˆµ∈[0,1] deno e he agen ’s belie ha he ask is legal i she accep s an unexpec ed
(o -pa h) o e wi hou inqui y. Unde a dual-penal y ule, le Tnbe he penal y i she
did no inqui e, and Ti he penal y i she did inqui e bu s ill pe o ms an illegal ask
(e.g., a alse posi i e). Unde single-penal y ule, bo h Tnand Time ge in o T. Le
α∈(1/2,1) deno e inqui y p ecision, θ∈(0,1) he p io p obabili y o a good p incipal
(he), and k > 0 he inqui y cos .
37
Lemma C.2 (O -pa h h esholds).(i) I he agen accep s an o -pa h o e X′wi hou
inqui y, he expec ed u ili y is X′−(1−ˆµ)Tn, so (by ie-b eaking) she accep s he smalles
such o e X′
min = (1 −ˆµ)Tn. (ii) A de ia ing bad p incipal ea ns yB−(1 −ˆµ)Tna X′
min;
he is de e ed i
Tn>yB
1−ˆµ.
(iii) Suppose he on-pa h ou come is sc eening (semi-pooling): he good p incipal unds
inqui y and he agen ac s only on a “legal” signal. The minimal on-pa h ans e ha
induces inqui y equals
Xsemi =k+ (1 −θ)(1 −α)Ti,
i.e., he inqui y cos plus he agen ’s esidual expec ed exposu e unde impe ec inqui y.
To de e a good- ype o -pa h de ia ion o “no inqui y” accep ance, i su ices ha
(1 −ˆµ)Tn≥Xsemi ⇐⇒ Tn≥k+(1−θ)(1−α)Ti
1−ˆµ.
Unde pe ec inqui y (α= 1), his educes o Tn≥k/(1 −ˆµ).
P oo . Pa (i) ollows om he agen ’s o -pa h accep ance condi ion wi h no inqui y:
X′−(1 −ˆµ)Tn≥0, minimized a equali y. Pa (ii) plugs X′min in o he de ia ing bad
ype’s payo yB−X′min and equi es i <0. Pa (iii) equa es he o -pa h accep ance
h eshold o he on-pa h paymen ha exac ly co e s p i a e inqui y cos kplus esidual
expec ed penal y (1 −θ)(1 −α)Ti. The pe ec -inqui y simpli ica ion is immedia e when
(1 −α) = 0.
The de e ence bound o he bad ype, yB/(1−ˆµ), is (weakly) inc easing in ˆµ; he no-
de ia ion bound o he good ype, k+(1−θ)(1−α)Ti/(1−ˆµ), is also (weakly) inc easing
in ˆµ. Hence mo ing om he pessimis ic benchma k ˆµ= 0 o mo e op imis ic belie s
(ˆµ > 0)weakly sh inks he penal y windows ha suppo ei he pooling o sc eening.
This is he sense in which he pessimis ic con en ion maximizes implemen abili y.
C.3 Mixed s a egies by he p incipal
Fix any liabili ies (single-penal y o dual-penal y), α∈(1/2,1], and k > 0. Conside a
candida e p o ile whe e he bad ype andomizes o e wo ans e s, Xℓ< Xh, while he
good ype plays a single ans e . Le ˆµ(X)deno e he agen ’s pos e io ha he ask
is legal upon obse ing X(Bayes-consis en on-pa h; a bi a y o -pa h subjec o he
con en ion abo e).
P oposi ion C.3 (Ex eme-poin bes eply o he bad ype).Fo any ixed o -pa h
belie s and he agen ’s bes esponse, he bad ype’s expec ed payo as a unc ion o his own
ans e Xis piecewise linea wi h a mos one kink ( he poin a which he agen swi ches
om “inqui e/ ejec when illegal” o “accep wi hou inqui y”). Hence his bes eply is
a ained a an ex eme poin : ei he he lowes X ha s ill induces accep ance wi hou
inqui y, o he highes easible X ha yields accep ance. Gene ically, he maximize is
unique and pu e.
P oo . Fo each obse ed X, he agen ’s bes esponse is h eshold in Xand ˆµ(X):
below a cu o , she inqui es (and ejec s when illegal); abo e i , she accep s wi hou
inqui y. The e o e he accep ance p obabili y as a unc ion o Xis a s ep unc ion wi h
38
a single jump. The bad ype’s expec ed payo is yB−Xwhen accep ance occu s and 0
o he wise, so i is piecewise linea in Xwi h a mos one kink a he jump. A piecewise-
linea unc ion wi h one kink a ains i s maximum a an endpoin unless pa ame e s a e
kni e-edge; hus he bad ype’s bes eply is pu e excep on a measu e-ze o se .
Implica ion. Apa om he i ial inac i e case ( ejec ion ega dless o o e s), equi-
lib ium beha io is exhaus ed by he pu e ypes analyzed in he main ex (pooling o
semi-pooling). Allowing he bad ype o “mix” be ween legal and illegal asks wi h some
p obabili y qsimply escales pos e io s ˆµ(X)and lea es he h eshold logic unchanged;
i does no gene a e new equilib ium ypes.
C.4 Psychological Inqui y Cos s and Social P e e ences
Le he agen ’s p i a e inqui y cos be k, bu he lawmake place weigh σ∈[0,1] on i s
non-ma e ial componen in wel a e. Le m≥0be an in e nal mo al cos he agen su e s
when she pe o ms an illegal ask wi hou inqui y. Unde dual penal ies, Tnapplies o
no-inqui y ac s and Ti o pos -inqui y illegal ac s ( alse posi i es).
Lemma C.3 (Wel a e and easibili y wi h (σ, m)).(i) The p i a e inqui y condi ion is
unchanged by σand m; h esholds shi only wi h k,Tn,Ti, and α. (ii) The semi-pooling
wel a e e m becomes θyG−σk (pe ec inqui y) and θαyG+(1−θ)(1−α)(τyB−H)−σk
(impe ec inqui y). (iii) Unde pooling (no inqui y), he minimal accep able ans e is
X∗= (1 −θ)(Tn+m)( eplace Tnby Twi h a single penal y). Feasibili y o pooling
equi es yG≥X∗and yB≥X∗.
P oo . (i) σis a lawmake ’s weigh and does no en e he agen ’s p i a e p oblem. mis
bo ne only when he agen pe o ms an illegal ask unde no inqui y; i does no a ec he
inqui y b anch. (ii) T ans e s cancel in wel a e; he only change om σis he scaled k.
Unde impe ec inqui y, he s anda d wel a e e m o semi-pooling is educed by noise; σ
scales kiden ically. (iii) Unde pooling, he agen ’s expec ed disu ili y is (1−θ)(Tn+m);
ie-b eaking gi es he h eshold X∗and he easibili y condi ions s a ed.
Implica ions. Lowe σexpands he egion whe e inqui y is socially p e e ed bu lea es
implemen abili y unchanged. Highe msubs i u es o Tnwhen de e ing no-inqui y
ac ion is he objec i e, bu i also makes sus aining pooling ha de when inclusion o
bo h ypes is desi able. Dual penal ies keep he le e s dis inc : une Tn(wi h m) o he
no-inqui y ma gin and Ti o esidual exposu e a e inqui y.
C.5 Addi ional no es on ma ke s uc u e
Th oughou , “sc eening” e e s o he semi-pooling equilib ium in which he agen inqui es
and pe o ms only when he signal (o belie ) indica es legali y. Penal y no a ion unde
impe ec inqui y: Tnapplies when he agen ac s wi hou inqui y; Tiapplies o pos -
inqui y illegal pe o mance ( alse posi i es). P ecision α∈(1/2,1) and φ:= θα + (1 −
θ)(1 −α).
A. Single p incipal wi h unce ain y abou legali y
39
