Schweize , U s
A icle — Published Ve sion
G oup causa ion heo ies and de e ence o o ious ac s
Eu opean Jou nal o Law and Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Schweize , U s (2025) : G oup causa ion heo ies and de e ence o o ious ac s,
Eu opean Jou nal o Law and Economics, ISSN 1572-9990, Sp inge US, New Yo k, NY, Vol. 59, Iss. 3,
pp. 555-570,
h ps://doi.o g/10.1007/s10657-025-09838-y
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/330777
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Eu opean Jou nal o Law and Economics (2025) 59:555–570
h ps://doi.o g/10.1007/s10657-025-09838-y
G oup causa ion heo ies andde e ence o o ious ac s
U sSchweize 1
Accep ed: 31 Ma ch 2025 / Published online: 22 Ap il 2025
© The Au ho (s) 2025
Abs ac
This pape examines a model wi h mul iple ac o s, each o whom aces a bina y
ac ion choice. The ac ion choice imposes ha m on a ic im ha depends on he num-
be o ac o s who en e o , al e na i ely, who de ia e om a due ca e s anda d. The
pape examines incen i e and wel a e e ec s om pe -capi a liabili y unde simul-
aneous as well as sequen ial ac ion choice. The challenge is o cope wi h due ca e
s anda ds ha , o wha e e easons, need no be e icien . As i u ns ou , wel a e is
enhanced in any equilib ium as compa ed o he si ua ion whe e none o he ac o s
en e o , al e na i ely, whe e all o hem mee he due ca e s anda d. The ange o
pa ame e alues, howe e , whe e a wel a e maximizing ou come in equilib ium is
induced by pe -capi a liabili y u ns ou o be a he limi ed. As an al e na i e ule,
e icien pe -capi a liabili y is p oposed ha would lead o an e icien ou come qui e
gene ally o simul aneous as well as sequen ial ac ion choice.
Keywo ds G oup causa ion· Pe -capi a liabili y· E icien de e ence· Ine icien
due ca e s anda ds· Sequen ial ac ion choice
Ma hema ics Subjec Classi ica ion K13
1 In oduc ion
1.1 Scope and indings
To hold an ac o liable, he e should be a causal link be ween his o ious ac and
he ic im’s ha m. In some cases, he bu - o es es ablishes such a link. Wi h
mul iple o easo s, howe e , his es u ns ou being oo es ic i e. To ensu e
The au ho has bene i ed om ex ensi e discussions wi h J. Shaha Dillba y on issues ela ed o he
p esen pape and he is g a e ul o Mu a Mungan as well as o he edi o and wo e iewe s o e y
help ul commen s on an ea lie e sion o his pape .
* U s Schweize
schw[email p o ec ed]
1 Uni e si y o Bonn, Bonn, Ge many
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Eu opean Jou nal o Law and Economics (2025) 59:555–570
compensa ion o he ic im and o de e o ious ac s none heless, cou s ake
eso o g oup causa ion heo ies such as conce ed ac ion, concu en causes,
subs an ial ac o s, al e na i e liabili y, o e de e mined ha m and mul iple su i-
cien causes. In many such cases, he esul ing damages ule is pe -capi a liabil-
i y whe e negligen ac o s expec o sha e liabili y o he ic im’s ha m in equal
pa s.
Unde join and se e al liabili y, he ic im can eques he ull ha m om any
one o he o easo s. I hen becomes he esponsibili y o he o easo s o so
ou hei espec i e p opo ions o liabili y and paymen . Fo such cases, he Ge -
man Ci il Code, Sec ion426, explici ly s a es ha join and se e al deb o s a e
obliged in equal p opo ion in ela ion o one ano he . Thus, in Ge many, pe -
capi a liabili y ollows di ec ly om he law.
In a p i a e con e sa ion, Shaha Dillba y has old me ha , unde conce ed
ac ion and al e na i e liabili y, he e is no eal way o appo ion and, o ha ea-
son, pe -capi a is he way ou . Mo eo e , when liabili y is join and se e al, he
de endan could eco e om any one o mul iple o easo s and, in con as o
Ge man law, he paying pa y had no igh o go a e he o he s. None heless,
i he de endan picks a o easo andomly, in expec ed e ms, each de endan
would pe cei e pe -capi a liabili y in his case as well. F om he incen i e pe -
spec i e, expec ed damages only ma e .
In any case, Dillba y (2016) ques ions he jus i ica ion o g oup causa ion he-
o ies and pe -capi a liabili y in e ms o de e ence. In ac , dilu ion o liabili y
may encou age g oup w ongdoing. Ra he han de e ence, Dillba y o e s a jus-
i ica ion in e ms o wel a e. He iews pe -capi a liabili y as a so ing de ice.
Ac o s a e de e ed om engaging in wel a e dec easing acciden s bu hey a e
encou aged o engage in wel a e inc easing ones.
To suppo his claim, Dillba y o e s he ollowing wo nume ical examples.
The i s one e e s o he bene i s om d i ing o uously. A numbe N o ac o s
conside whe he o engage in a d ag ace. Each expec s a bene i
b=40
om
pa icipa ion. Independen o he numbe o o easo s, he ha m o he ic im
is assumed o be
H=90
. Since negligen ac o s sha e liabili y equally, each o n
o easo s is liable o 90/n and, hence, he ne bene i o each o easo amoun s
o
𝜋n=40 −90∕n
. The e is dilu ion o liabili y because indi idual liabili y
dec eases wi h he numbe n o o easo s in he ange
n>0
. Mo eo e , since
𝜋1=−50 <𝜋
2=−5<0<𝜋
3=10
, i ollows ha , o
N≤2
, no ac o would
pa icipa e.
Fo
N>2
, howe e , i is a Nash equilib ium when no a single ac o is de e ed
om pa icipa ion in he d ag ace. F om a wel a e pe spec i e, his is no wo -
isome as, in ac , wel a e is a i s maximum when all N ac o s pa icipa e. No e,
howe e , i would also be a Nash equilib ium when all ac o s a e de e ed. In
ac , o a gi en ac o , i no o he ac o pa icipa es, i is a bes esponse o him
no o pa icipa e ei he because, o he wise, his payo would be nega i e,
𝜋1<0
.
The e o e, in his example, while he e icien ou come is always a Nash equilib-
ium, a second equilib ium coexis s ha ea u es o e -de e ence.
The second example has h ee o easo s who lean on a ca causing he ca o all
o e he edge o a cli . The ca is des oyed, i s alue being
H=90
. I is assumed
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Eu opean Jou nal o Law and Economics (2025) 59:555–570
ha any o easo alone would no ha e exe ed enough o ce o push he ca o e
he edge bu any wo o easo s would ha e.
Based on he bu - o es , he e would be no causal link be ween any single ac o ’s
w ongdoing and he ic im’s ha m as he same le el o ha m would ha e occu ed
e en i , ce e is pa ibus, his ac o had no been leaning on he ca . In he ca exam-
ple, he bu - o causa ion equi emen would p e en compensa ion o he ic im
en i ely,1
As a subs i u e, he sui able g oup causa ion heo y would o e he equi ed
causal link. I seems o be consensus ha leaning on he ca is a o ious ac and
ha each o he h ee o easo s is liable o
H∕3=30
. Ne e heless, ac o s may be
emp ed o lean negligen ly on he ca as i comes wi h a p i a e bene i
b=40
o
each o easo .
When all h ee a e leaning on he ca , each o easo ends up wi h payo
𝜋3=b−H∕3=10
. This cons i u es a Nash equilib ium as no single ac o could
gain om no leaning on he ca . In ac , no leaning on he ca would dep i e he
o he bene i b and, while she would escape liabili y, he ne payo would be ze o.
Hence, all h ee leaning on he ca is a Nash equilib ium and so pe -capi a liabili y
would de e no a single ac o om he w ongdoing.
F om a wel a e pe spec i e, hough, he lack o de e ence would be less wo i-
some. Wel a e in his Nash equilib ium amoun s o
w3=3b−H=30
. When none
o hem had leaned on he ca , ha m would ha e been a oided bu he e also would
be no p i a e bene i s and, hence, wel a e would be
w0=0
. The e o e, while he
damages ule does no de e o ious ac s, i inc eases wel a e.
No e, howe e , he ou come in his Nash equilib ium alls sho o maximiz-
ing wel a e, i.e., i would no be e icien . Wel a e would a he be a i s maximum,
w1=b=40
, when a single o easo we e leaning on he ca . This ac o would
bene i wi hou causing any ha m o he ic im. Since
w1>w3
, he Nash equilib ium
wi h all h ee leaning on he ca is ine icien as i comes wi h unde -de e ence.
No e u he ha he Nash equilib ium in his example oo is no unique. The
e icien ou come i sel coexis s as a second Nash equilib ium. The single ac o lean-
ing on he ca ends up wi h ne payo
b=40
whe eas he o he wo ob ain ze o. Bu
none o hem could unila e ally imp o e he posi ion because, by a unila e al swi ch,
he payo would amoun o
b−H∕2=−5
and would be e en less. The e o e a sin-
gle ac o leaning on he ca is no only e icien . I e en cons i u es a second Nash
equilib ium.
To sum up, in bo h examples, he e icien ou come is a Nash equilib ium bu
ine icien equilib ia coexis , in one case, wi h o e -de e ence and, in he o he case,
wi h unde -de e ence. These examples s imula ed my cu iosi y and so I wan ed o
ind ou how obus he message d awn om hem can be.
To ind ou , he p esen pape o e s a se ing which goes beyond nume ical
examples bu emains su icien ly simple o allow dealing wi h he issues explic-
i ly. The e is a ini e se o ac o s and a ic im who is a ec ed by he ac o s’ ac ion
choice bu emains passi e o he wise. Ac ion choice is bina y, ze o/one. We ha e
1 To con i m he iew ha he bu - o s anda d o o e de e mined ha m is inadequa e, see Re hinking
Ac ual Causa ion in To Law Ha a d Law Re iew, June 2017.
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Eu opean Jou nal o Law and Economics (2025) 59:555–570
wo in e p e a ions in mind. Fi s , he decision is whe he o pa icipa e o no so ha
ze o/one means s aying ou o en e ing. Second, ze o means being in while mee ing
he due ca e s anda d o being in while de ia ing om i .
Fo he second in e p e a ion, he e is a as li e a u e on negligence ules ha
p o ide e icien p ecau ion incen i es. The co esponding con ibu ions ha e in
common ha due ca e s anda ds a e speci ied a e icien due ca e le els. Pe -capi a
liabili y as laid ou abo e, howe e , depa s om due ca e le els ha need no be
e icien . None heless, in he nume ical examples a leas , he e exis s an e icien
Nash equilib ium.
The compensa ion p inciple as es ablished and ex ended by he p esen au ho
shows ha Nash equilib ia unde pe -capi a liabili y, whe e de ia ions om due ca e
s anda ds occu , come wi h a wel a e le el no lowe as compa ed wi h he si ua ion
whe e all had me hese ine icien s anda ds. The p inciple emains silen , hough,
on he amoun , by which wel a e is enhanced in equilib ium. The abo e nume ical
examples sugges ha wel a e may be enhanced up o maximal wel a e so ha he
equilib ium ou come would be e icien . Un o una ely, he e iciency claim has lim-
i s e en in he p oposed se ing o bina y ac ion choice.
To allow o de e mining equilib ia and doing wel a e compa isons explici ly, i
is u he assumed ha bene i s om negligen beha io as well as ha m o he ic-
im depend on he numbe o en an s/ o easo s only bu no hei iden i y.
All esul s apply equally well o he i s in e p e a ion whe e he bina y decision
is in/ou . Fo no a ional simplici y, we e e in he ex mainly o he i s in e p e a-
ion. In any case, he damages ule is pe -capi a liabili y among en an s o among
negligen ac o s. In spi e o i s simplici y, he se ing exhibi s a su icien ly ich ana-
ly ic s uc u e o unco e he limi s o he e iciency claim.
Wi hin his se ing, he main indings a e as ollows. Fi s , qui e gene ally, he
numbe o en an s in Nash equilib ium wi h simul aneous ac ion choice is no
unique. The lowes and highes numbe s o en an s a e de i ed explici ly.
Second, unde sequen ial ac ion choice, he numbe o en an s in subgame pe -
ec equilib ium as ob ained by backwa d induc ion is (gene ically) unique. Mo e
su p isingly, he numbe o en an s in he subgame pe ec ou come is equal o he
highes numbe o en an s in Nash equilib ium.2 While backwa d induc ion leads o
a unique ou come, i need no be he e icien one hough (i an e icien Nash equi-
lib ium exis s a all).
Thi d, excep o bo de line cases, en y occu s in equilib ium and wel a e is
s ic ly enhanced bu need no be maximized.
Fou h, only i wel a e a ains i s maximum when all ac o s en e , hen he e i-
cien ou come is equal o he Nash equilib ium wi h he highes numbe o en an s
o su e.
While he abo e indings a e o posi i e na u e as pe -capi a liabili y, a ule used
in legal p ac ice, is examined, he i h con ibu ion o he pape is o no ma i e
na u e. A modi ied e sion o pe -capi a liabili y is p oposed ha would p o ide
2 Thus, in ui ion gained om Cou no e sus S ackelbe g is ho oughly misleading. The S ackelbe g
equilib ium unde sequen ial quan i y choice, while also being calcula ed by backwa d induc ion, would
ne e be a Cou no (Nash) equilib ium unde simul aneous quan i y choice.
559
Eu opean Jou nal o Law and Economics (2025) 59:555–570
e icien incen i es qui e gene ally. The ule, e e ed o as e icien pe -capi a liabil-
i y, denies liabili y o en an s (o o negligen ac o s) as long as ha m alls sho o
a sui ably chosen limi . This limi is equal o he ha m om he e icien numbe o
en an s. Beyond his limi , some sui ably chosen de ia o s will ha e o sha e ha m
beyond e icien ha m in equal pa s.
E icien pe -capi a liabili y may be a con lic wi h disc imina ion issues, a leas
when ac o s mo e simul aneously. When hey mo e sequen ially, howe e , i may be
jus i iable o disc imina e among hem acco ding o he s age when i is hei u n o
decide. In any case, e icien pe -capi a liabili y would p o ide e icien incen i es.
Ine icien ou comes would no occu , no ma e whe he ac ion choice is simul ane-
ous o sequen ial.
1.2 Rele an li e a u e
The e is a as li e a u e on damages ules o cases o mul iple o easo s. A p omi-
nen sou ce o a o mal analysis would be Ko nhause and Re esz (1989). Bu he e
also exis p ominen ex book ea men s o o law such as William and Posne
(1987) and Sha ell (1987, 2004). This li e a u e mainly deals wi h ules ha p o-
ide e icien incen i es. Fo negligence ules, his means ha due ca e s anda ds a e
speci ied a hei e icien le el.
Miceli and Sege son (1991) examine a se ing wi h wo ac o s who, a he i s
s age, simul aneously ake an en y decision. A he second s age, he ac o s who
ha e en e ed simul aneously decide on hei ac i i y le els. These au ho s do back-
wa d induc ion as he p esen pape does. By insis ing on a socially bes esponse
in e e y subgame, howe e , hey ind ou ha he subgame pe ec ou come ails
o maximize wel a e, a leas o he damages ule p oposed by hem. The p esen
pape , in con as , deals wi h he subgame pe ec ou come, no insis ing on e icien
eplies o he equilib ium pa h.
The main e e ences o he p esen pape , howe e , a e Dillba y (2013, 2016)
whe e due ca e s anda ds may be ine icien and whe e pe -capi a liabili y leads o
second bes p oblems ha , as such, a e no o iously mo e demanding o handle. The
compensa ion p inciple as de eloped o e ime by he p esen au ho , howe e , p o-
ides guidance.
A i s e sion o he compensa ion p inciple was in oduced in Schweize
(2005a) o a si ua ion wi h wo ac o s and in Schweize (2005b) o mul iple ac o s.
These ea ly e sions we e s ill con ined o e icien e e ence p o iles. The mos gen-
e al e sion so a can be ound in Schweize (2024). I allows o mul iple ac o s
as well as o ine icien e e ence p o iles and co e s he simul aneous mo e game
o he p esen se ing. The ex ension o sequen ial choice wi h mul iple ac o is p o-
ided by he p esen pape .
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Eu opean Jou nal o Law and Economics (2025) 59:555–570
1.3 O ganiza ional s uc u e
The p esen pape is o ganized as ollows. In Sec . 2, he basic model o bina y
ac ion choice unde pe -capi a liabili y is in oduced. In Sec .3, he Nash equilib-
ia o simul aneous ac ion choice a e iden i ied as unc ions o he pa ame e con-
igu a ion. In Sec .4, backwa d induc ion o sequen ial ac ion choice is ca ied ou .
Sec ion5 summa izes he wel a e s a emen s and p o ides nume ical examples. In
Sec .6, e icien pe -capi a liabili y is speci ied as a ule ha would p o ide e icien
incen i es qui e gene ally. Sec ion7 concludes. In he appendix, he ex ension o he
compensa ion p inciple is es ablished.
2 The model
Each ac o i ou o a ini e se
I={1, ..., N}
chooses an ac ion
xi
om he se
{0, 1}
o a ailable al e na i es. I ac o i decides
xi=0
we say ha i s ays ou . I he decides
xi=1
we say ha i en e s. An al e na i e in e p e a ion would be ha all ac o s a e
in. In his case,
i=0
means ha i mee s he due ca e s anda d while
xi=1
means
ha he de ia es. The esul s hold o bo h in e p e a ions. Fo simplici y, he ex
mainly e e s o he i s one.
An ac ion p o ile
x=(x1, ..., xN)
lis s exac ly one ac ion o each ac o . A ac ion
p o ile x, he numbe o en an s is
(x)=∑i∈Ixi
. By assump ion, ha m
H0=0
≤
Hn
o he ic im as well as he p i a e bene i
bn>0
be o e damages o each en an
depend on he numbe
0<n≤N
o en an s bu no hei iden i y i.
The d ag ace example om he in oduc ion has
H0=0<Hn=H=90
o
0<n≤N
and he ca example has
N=3
and
H0=H1=0<H2=H3=H=90
.
Bo h examples ha e
b0=0<bn=40
o all
0<n≤N
.
Unde pe -capi a liabili y, ac o s who s ay ou (o mee he s anda d) a e no lia-
ble whe eas en an s (o negligen ac o s) a e liable o o al ha m in equal sha es.
The ic im’s claim is ha m
Hn
when
n>0
ac o s ha e en e ed, in which case each
en an owes damages
hn=Hn∕n
o he ic im.
Wi h n en an s, wel a e amoun s o
w(0)=0
and, o all
0<n≤N
,
whe e
𝜋n=bn−hn
deno es he bene i ne o damages o each en an .
Unde pe -capi a liabili y, he payo o ac o i ne o damages amoun s o
as a unc ion o he chosen ac ion p o ile x. The payo unc ions (2) de ine a game.
Ra ional ac o s end up, when hey choose hei ac ions simul aneously, wi h a Nash
equilib ium o his game o , when ac ions a e chosen sequen ially, wi h a subgame
pe ec ou come. An ac ion p o ile is a Nash equilib ium when i consis s o mu u-
ally bes esponses. A subgame pe ec equilib ium is ob ained by backwa d induc-
ion. Fo mal de ails a e p o ided in he nex sec ions.
(1)
w
(n)=n⋅b
n
−H
n
=n⋅
(
b
n
−h
n)
=n⋅𝜋
n
(2)
𝜙i(x)=𝜋 (x)
⋅
xi
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Eu opean Jou nal o Law and Economics (2025) 59:555–570
The ic im is no an ac i e playe bu , as she is ully compensa ed, he payo
including damages amoun s o
𝜙 (x)=0
, no ma e , wha ac ions x a e chosen.
Rela i e o a gi en e e ence p o ile
xo
, a damages ule is called compensa o y
when no ac o who keeps o he e e ence p o ile su e s om de ia ions by o h-
e s, i.e.,
𝜙i(xo
i,x−i)≥𝜙i(xo)
holds o all i and all de ia ions
x
−i
by he o he ac o s
and i he ic im is compensa ed o any de ia ions, i.e.,
𝜙 (x)≥𝜙 (xo)
holds o
all ac ion p o iles. In his sense, pe -capi a liabili y is compensa o y ela i e o he
e e ence p o ile
xo=(0, …,0)
. By (1), his e e ence p o ile is e icien (i.e., wel-
a e maximizing) i and only i
max[𝜋1,…,𝜋N]≤0
. In his case, i ollows om he
compensa ion p inciple (see he appendix) ha any Nash equilib ium unde simul a-
neous ac ion choice as well as any subgame pe ec ou come unde sequen ial ac ion
choice mus be e icien .
In all o he cases, i.e., i
max[𝜋1,…,𝜋N]>0
, i ollows om he compensa ion
p inciple ha any Nash equilib ium and any subgame pe ec ou come comes wi h
a highe wel a e le el han unde he e e ence p o ile, in ou case
w(0)=0
. The
compensa ion p inciple does no ell us by wha amoun wel a e is inc eased hough.
Fo he se ing o bina y ac ion choice, his issue will be ully se led in he nex wo
sec ions.
3 Simul aneous ac ion choice
A Nash equilib ium in pu e s a egies consis s o an ac ion p o ile
x=(x1,…,xN)
so
ha no ac o can imp o e his payo by unila e ally de ia ing om i , i.e.,
holds o all ac o s i and o all de ia ions
xi≠xi
by ac o i. Le
x=1
n
=(1, …, 1, 0, …,0)
deno e he ac ion p o ile whe e ac o s
i≤n
en e bu he
emaining ones do no . Since ne bene i s o an en an depend only on he o al
numbe o en an s, he ollowing mus be ue: i
x=1
n
is a Nash equilib ium hen
any ac ion p o ile
x=(x1,…,xN)
wi h he same numbe
(x)=n
o en an s is also a
Nash equilib ium. This allows us o say he numbe
n
o en an s is Nash when
1
n
is
a Nash equilib ium.
Fini e games do no necessa ily ha e a Nash equilib ium in pu e s a egies. In ou
model, howe e , he e always exis such equilib ia. The nex wo p oposi ions show
how he lowes numbe
nL
and he highes numbe
nH
ha a e Nash can be ound in
a cons uc i e way.
P oposi ion 1 The lowes numbe
nL
ha is Nash is de e mined as ollows:
(i) I
𝜋1≤0
hen
nL=0
.
(ii) I
𝜋n+1≤0<min[𝜋1,…,𝜋n]
hen
nL=n
.
(iii) I
0<min[𝜋1,…,𝜋N]
hen
nL=N
.
𝜙i(xi,x−i)≥𝜙i(xi,x−i)
562
Eu opean Jou nal o Law and Economics (2025) 59:555–570
P oo To p o e claim (ii), suppose
𝜋n+1≤0<min[𝜋1,…,𝜋n]
and, in pa icu-
la ,
𝜋0=0<𝜋
1
. Then
n=0
canno be Nash. In ac , gi en p o ile
(0, …,0)
whe e
each ac o ecei es ze o payo , any one o hem, say,
i=1
could de ia e ending up
wi h a posi i e payo
𝜋1>0
. The e o e
(0, …,0)
canno be a Nash equilib ium. I
0<n<n
hen
𝜋n+1>0
. Such a numbe n can nei he be Nash. In ac , conside he
ac ion p o ile
1n=(1, …, 1, 0, …,0)
whe e ac o s
i≤n
de ia e bu ac o s
j>n
do
no . Since
n+1≤n
, i ollows ha
𝜋n+1>0
and, hence, ac o , say
j=n+1
, could
de ia e, ending up wi h a posi i e payo
𝜋n+1
compa ed wi h payo ze o unde
1n
.
The e o e no
0≤n<n
can be Nash.
The p o ile
1
n
, howe e , is Nash. In ac , any ac o
i≤n
makes posi i e payo
𝜋n>0
and would loose om de ia ing wi h
xi=0
. Ac o
j>n
makes ze o payo .
Ye , by de ia ing wi h
xj=1
, his payo
𝜋n+1≤0
would no be highe and, hence,
1
n
is a Nash equilib ium. Since
n<n
is no Nash, i ollows ha
n=nL
is he lowes
numbe o en an s (de ia o s) ha is Nash. Claim (ii) is es ablished. Claims (i) and
(iii) can be es ablished simila ly.
◻
This p oposi ion allows o he ollowing wel a e compa isons. Unde con-
igu a ion (i),
nL=0≤n∗
is ob iously ue and, hence, his Nash equilib ium
need no maximize wel a e. Unde con igu a ion (ii), wel a e is enhanced as
w(n
L
)=n
L⋅
𝜋
n
L
>0
holds o su e bu i need no maximize wel a e. Mo eo e ,
cases wi h o e -de e ence,
nL<n∗
as well as wi h unde -de e ence,
n∗<nL
,
can occu . Unde con igu a ion (iii), inally,
n∗≤nL=N
is ob iously ue. In his
case, wel a e is enhanced,
w(n
L
)=n
L⋅
𝜋
n
L
>0
, bu
nL
need no maximize wel a e.
Nume ical examples suppo ing hese claims a e p esen ed in sec ion5 below.
Fo he nex esul , o a oid case dis inc ions o li le in e es , we assume ha
pa ame e con igu a ions a e gene ic in he sense ha
𝜋n≠0
holds o all
0<n≤N
.3
P oposi ion 2 (gene ic pa ame e con igu a ions) The highes numbe
nH
ha is
Nash is de e mined as ollows:
(i) I
0<𝜋
N
hen
nH=N
.
(ii) I
max[𝜋n+1,…,𝜋N]<0<𝜋
n
hen
nH=n
.
(iii) I
max[𝜋1,…,𝜋N]<0
hen
nH=0
.
To economize on space, he p oo is omi ed. The a gumen is simila o he one
in he P oo o P oposi ion1.
As a i s co olla y o he abo e p oposi ion, i is easily seen ha , in Nash equi-
lib ia wi h he highes numbe o de ia o s, he e is unde -de e ence as, in ac ,
n∗≤nH
holds in all cases (i)–(iii). Unde -de e ence may be s ic .
Mo eo e , he p oposi ion allows o he ollowing wel a e compa ison. In case
(i), while wel a e is enhanced,
w(nH)=N
⋅
𝜋N>w(0)
, i need no maximize wel-
a e, i.e.,
n∗<nH
canno be uled ou . The same holds ue o case (ii). In case (iii),
3 The assump ion holds gene ically in he ollowing sense: when pa ame e alues a e d awn om a dis-
ibu ion ha has a densi y unc ion, hen he assump ion is me wi h p obabili y one.
569
Eu opean Jou nal o Law and Economics (2025) 59:555–570
holds as ollows om he compensa ion equi emen (3) wi h espec o playe
i=1
.
Gi en his subgame pe ec equilib ium, he subgame pe ec ou come
xsp
=(x
sp
1
,a1[x
sp
1])
will e ol e. I ollows ha , in his subgame pe ec ou come,
𝜙i(xsp)≥𝜙i(xo)
holds o all ac o s
i∈I
.
By adding up hese inequali ies, i ollows ha
holds o any subgame pe ec ou come
xsp
and so claim (iii) is es ablished. Claim
(i ) hen ollows om claim (iii) as claim (ii) ollows om claim (i). The gene al
e sion o he compensa ion p inciple is ully es ablished.
I emains o cons uc he unc ions
ai[x1,…,xi−1,xi]
explic-
i ly by backwa d induc ion. A he inal s age N, he e a e no subsequen
s ages le and, hence,
aN[x1,…,xN]
consis s o he emp y lis whe eas
aN−
1[x1,…,x
N−
1]=x
sp
N
(x1,…,x
N−
1
)
.
Suppose
ai[x1,…,xi]
has al eady been cons uc ed down o s age i. Take
xsp
i
{x1,…,x
i−
1
}
as de ined abo e. Then we speci y
a s age i. By epea ing his p ocess backwa ds, he unc ions
ai[x1,…,xi]
a e con-
s uc ed o
i=N,N−1, …,1
and we a e done.
◻
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