Time-Dependent Strategies in Repeated Asymmetric Public Goods Games

Hübne , Valen in; Hilbe, Ch is ian; S aab, Manuel; Kleshnina, Ma ia; Cha e jee,
K ishnendu
A icle — Published Ve sion
Time-Dependen S a egies in Repea ed Asymme ic Public
Goods Games
Dynamic Games and Applica ions
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Hübne , Valen in; Hilbe, Ch is ian; S aab, Manuel; Kleshnina, Ma ia; Cha e jee,
K ishnendu (2025) : Time-Dependen S a egies in Repea ed Asymme ic Public Goods Games,
Dynamic Games and Applica ions, ISSN 2153-0793, Sp inge US, New Yo k, NY, Vol. 15, Iss. 5, pp.
1617-1645,
h ps://doi.o g/10.1007/s13235-025-00627-5
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/330232
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Dynamic Games and Applica ions (2025) 15:1617–1645
h ps://doi.o g/10.1007/s13235-025-00627-5
Time-Dependen S a egies in Repea ed Asymme ic Public
Goods Games
Valen in Hübne 1·Ch is ian Hilbe2·Manuel S aab3·Ma ia Kleshnina4·
K ishnendu Cha e jee1
Accep ed: 26 Janua y 2025 / Published online: 7 Feb ua y 2025
© The Au ho (s) 2025
Abs ac
The public goods game is among he mos s udied me apho s o coope a ion in g oups. In
his game, indi iduals can use hei endowmen s o make con ibu ions owa ds a good ha
bene i s e e yone. Each indi idual, howe e , is emp ed o ee- ide on he con ibu ions o
o he s. He ein, we s udy epea ed public goods games among asymme ic playe s. P e ious
wo k has explo ed o which ex en asymme y allows o ull coope a ion, such ha playe s
con ibu e hei ull endowmen each ound. Howe e , by design ha wo k ocusses on
equilib ia whe e indi iduals make he same con ibu ion each ound. Ins ead, he e we conside
playe s whose con ibu ions along he equilib ium pa h can change om one ound o he
nex . We do so o h ee di e en models – one wi hou any budge cons ain s, one wi h
endowmen cons ain s, and one in which indi iduals can sa e hei cu en endowmen o be
used in subsequen ounds. In each case, we explo e wo key quan i ies: he wel a e and he
esou ce e iciency ha can be achie ed in equilib ium. Wel a e co esponds o he sum o
all playe s’ payo s. Resou ce e iciency ela es his wel a e o he o al con ibu ions made
by he playe s. Compa ed o cons an con ibu ion sequences, we ind ha ime-dependen
con ibu ions can imp o e esou ce e iciency ac oss all h ee models. Mo eo e , hey can
imp o e he playe s’ wel a e in he model wi h sa ings.
Keywo ds Social dilemmas ·Public goods games ·Inequali y ·Di ec ecip oci y
Ma hema ics Subjec Classi ica ion 91A05 ·91A06 ·91A10 ·91A20
Ma ia Kleshnina and K ishnendu Cha e jee ha e con ibu ed equally o his wo k.
BValen in Hübne
[email p o ec ed]
1Ins i u e o Science and Technology Aus ia, 3400 Klos e neubu g, Aus ia
2Max Planck Resea ch G oup Dynamics o Social Beha io , Max Planck Ins i u e o E olu iona y
Biology, 24306 Plön, Ge many
3School o Ma hema ical Sciences, Queensland Uni e si y o Technology, B isbane, QLD 4000,
Aus alia
4School o Economics, The Uni e si y o Queensland, B isbane, QLD 4067, Aus alia
1618 Dynamic Games and Applica ions (2025) 15:1617–1645
1 In oduc ion
Coope a ion is ypically concep ualised as a beha iou ha is cos ly o he indi idual bu
bene icial o he g oup [39]. Examples o coope a ion abound, anging om small a ou s
among iends o collec i e e o s o mi iga e clima e change. These coope a i e in e ac ions
can, and ha e been, desc ibed wi h game heo y [8,31]. This li e a u e has p oduced ich
p edic ions abou po en ial mechanisms ha can sus ain coope a ion [27,34]. One such
mechanism is di ec ecip oci y [5,7,40,44]. He e, indi iduals a e assumed o engage in
he same in e ac ion epea edly, o e many ounds. Repea ed in e ac ions allow playe s o
condi ion hei beha iou on he p e ious his o y o play. In his way, hey can en o ce mu ual
coope a ion despi e any sho - un emp a ions o ee- ide [13,14].
T adi ionally, many models o di ec ecip oci y, especially in he e olu iona y game
heo y li e a u e, assume ha in e ac ions a e symme ic [4,12,16,19,22,25,26,30,35,37,
38,41,42,46]. This means ha playe s a e comple ely in e changeable wi h espec o hei
ac ions and easible payo s. Mo e ecen ly, howe e , he e olu ion o coope a ion among
asymme ic playe s has ecei ed mo e a en ion [1–3,9,11,24,29,32,33,45]. This in e es
has also been spu ed by empi ical s udies ha explo e he ole o inequali y in con olled
expe imen s [6,10,17,21,28,36,43,47].
O en imes, hese s udies a e based on some a ia ion o he linea public goods game. In
his game, playe s ob ain hei ixed endowmen s in he beginning o each ound. Then hey
independen ly decide how much o hei endowmen hey wish o con ibu e o he public
good. Con ibu ions a e mul iplied by some p oduc i i y ac o , and he esul ing amoun is
e enly spli among all g oup membe s. The e a e a ious ways o allow o asymme y in his
game. Fo example, playe s may ha e unequal endowmen s, unequal p oduc i i ies, o bo h.
The main akeaway om he abo e-men ioned s udies is ha endowmen inequali y ends o
be de imen al o coope a ion. Howe e , as shown by [20]and[23], he e can be excep ions. I
indi iduals al eady di e in hei p oduc i i y, i can become easie o sus ain ull coope a ion
i hey also di e in hei endowmen s. As a ule o humb, a playe ’s endowmen ough o
be la ge he mo e p oduc i e ha playe is.
Howe e , he s udies o [20]and[23] conside a a he es ic ed ques ion. They ask:
Unde which condi ions a e he e subgame pe ec equilib ia in which all playe s con ibu e
hei ull endowmen in e e y ound? In pa icula , hey he eby only conside equilib ia
whose esul ing con ibu ion sequence along he equilib ium pa h is cons an . Ins ead, in he
ollowing we a e in e es ed in con ibu ion sequences ha can a y in ime. We ask: Once
indi iduals ha e he abili y o make ime-dependen con ibu ions along he equilib ium pa h,
o which ex en can hey achie e ou comes ha a e in easible wi h cons an con ibu ion
sequences?
To his end, we s udy h ee di e en bu ela ed models o public good p o ision. The
i s is mos con enien om a ma hema ical pe spec i e. He e, playe s can make a bi a y
(non-nega i e) con ibu ions each ound. In pa icula , con ibu ions a e no cons ained by
any endowmen s ha indi iduals migh ha e in ha ound. Ins ead, we only equi e ha
he playe s’ o e all discoun ed con ibu ions o e he en i e game a e bounded (wi h he
uppe bound being a bi a y). We e e o his model as he ‘base model’. The second model
ep oduces ypical public goods game models, such as he ones conside ed in [20]and[23].
He e, a playe ’s con ibu ion each ound is bounded by he playe ’s assigned endowmen .
Acco dingly, we speak o he ‘endowmen model’. Finally, he hi d model is a hyb id o
he i s wo. He e, playe s ob ain a ixed and cons an endowmen each ound. Bu now
hey can decide o deposi some o his endowmen in o a sa ings accoun , which is hen
Dynamic Games and Applica ions (2025) 15:1617–1645 1619
1
2
(a) Playe 1 con ibu es
an amoun o 1
12
(a) Playe 2 con ibu es
an amoun o 2
(b) Playe 1’s con ibu ion
is scaled up by 1
(b) Playe 2’s con ibu ion
is scaled up by 2
(c) Toge he , he scaled
con ibu ions o m he
public good
(d) Playe 1 ecei es one
hal o he public good
(d) Playe 2 ecei es one
hal o he public good
(e) Payo is equal o sha e om
public good minus con ibu ion
(e) Payo is equal o sha e om
public good minus con ibu ion
−
1=2=−
1
1
Fig. 1 A one- ound asymme ic public goods game. To illus a e he base model, we conside n=2 playe s.
They eely choose he size o hei con ibu ions, c1and c2. Con ibu ions a e enhanced by he indi idual
p oduc i i y ac o s, which a e 1=1.5and 2=1.1 in his example. The size o he public good is he sum
o all enhanced con ibu ions, c= 1c1+ 2c2. Each playe ecei es an equal sha e o his sum. The payo
o each playe hen equals hei sha e o he public good minus hei con ibu ion
a ailable in he nex ound. Unde his assump ion, con ibu ions each ound a e no bounded
by he playe s’ endowmen s anymo e. Ins ead, hey a e bounded by he playe s’ accumula ed
endowmen s up o ha poin . We call his he ‘sa ings model’.
Fo all h ee models, we conside he equilib ium ou comes ha can be achie ed wi h
ime-dependen con ibu ions. We compa e hem o he possible equilib ium ou comes when
playe s a e equi ed o make a ixed and cons an con ibu ion along he equilib ium pa h.
We make his compa ison based on wo key quan i ies. One quan i y is he g oup’s wel a e
in equilib ium ( he o al sum o he playe s’ payo s). As we discuss in mo e de ail below,
his quan i y is pa icula ly ele an in he endowmen model and in he sa ings model. The
o he quan i y is an equilib ium’s esou ce e iciency ( he a io o he g oup’s wel a e ela i e
o he playe s’ o al con ibu ions). This quan i y is ele an o all h ee models.
We cha ac e ise unde which condi ion ime-dependen con ibu ions allow o equilib ia
wi h la ge esou ce e iciency (compa ed o equilib ia based on cons an con ibu ions). We
do so o a bi a y discoun ac o s. In pa icula , we do no equi e ha playe s a e su icien ly
pa ien , as o en done in he classical olk heo em li e a u e [13,14]. Ou esul s depend on
he g oup size and on he numbe o playe s wi h he highes p oduc i i y. In pa icula , we ind
ha when he e is a unique playe wi h maximum p oduc i i y, ime-dependen con ibu ions
p o ide an ad an age. Wi h espec o wel a e maximisa ion, we ind a simila esul – bu
only o he sa ings model.
2 The Base Model
2.1 Model Se up
We s a wi h he base model (as illus a ed in Fig. 1), which we will use o de i e ou i s
esul s. These esul s will also ha e impo an implica ions o he o he wo models s udied
subsequen ly.
1620 Dynamic Games and Applica ions (2025) 15:1617–1645
In he base model, a g oup o n≥2 playe s in e ac s o an inde ini e sequence o ounds.
In e e y ound , each playe idecides which non-nega i e amoun ci( ) o con ibu e owa ds
he public good. The e is no limi on how much playe s may con ibu e. Hence, ci( )∈R≥0.
These indi idual con ibu ions can be collec ed in a ec o , c( )=(c1( ),...,cn( )).We
e e o a sequence c( ) o con ibu ion ec o s as a con ibu ion sequence, o as a ‘play’
o he game. I c( )=c(0) o all imes , he con ibu ion sequence is called cons an .
O he wise, i is ime-dependen .
Con ibu ions o each playe ia e mul iplied by hei p oduc i i y ac o iand added
o he public good. The o al public good is hen e enly sha ed among all playe s. This is
a sligh gene alisa ion o he s anda d o mula ion o he public goods game, acco ding o
which e e y playe has he same p oduc i i y. In he ollowing, we use =( 1,..., n) o
deno e he ec o o all p oduc i i ies. Based on his no a ion, we can w i e paye i’s payo
πi( )in ound as
πi( )=1
n c( )−ci( ). (1)
We assume p oduc i i ies sa is y 1< i<n o all playe s i. The i s inequali y i>1 ensu es
ha he g oup’s o al payo (ac oss all membe s) is inc easing in each playe ’s con ibu ions.
The second inequali y, i<n, on he o he hand, ensu es ha each indi idual is emp ed o
gi e as li le as possible. Toge he , hese wo inequali ies ende he game a social dilemma.
Fo each playe , he e is a con lic be ween hei p i a e in e es and he collec i e in e es o
he g oup.
To de ine he playe s’ payo s o e he en i e epea ed game, we assume playe s alue
each subsequen ound a a discoun o δ, wi h 0<δ<1. Acco dingly, when he con ibu ion
sequence c( ) is bounded, we de ine he o al payo o each playe as he weigh ed sum
o hei payo s each ound,
ˆπi=(1−δ)
∞

=0
δ πi( ).
He e, he e m 1 −δse es as a no malising ac o . I ensu es ha epea ed-game payo s a e
compa able o he game’s one-sho payo s. Because we assumed he con ibu ion sequence
o be bounded, he abo e sum is gua an eed o con e ge. We do no de ine a o al payo o
unbounded con ibu ion sequences.
In a common al e na i e in e p e a ion o epea ed games wi h discoun ing, which is
applicable o ou base model and model a ia ion I, bu no model a ia ion II, he numbe o
ounds is ini e and andom, wi h δbeing no a discoun ac o bu he ound-wise con inua ion
p obabili y. This means ha a e each ound, wi h p obabili y δ he game con inues o a
leas one mo e ound and wi h p obabili y 1 −δi ends. In ha in e p e a ion, all ounds
ha e equal alue gi en ha hey a e played, and he expec ed numbe o ounds is 1/(1−δ).
In ui i ely, his sugges s ha highe alues o δ, which mean longe games, a e conduci e o
coope a ion.
No a ion 1 In his base model, a game is ully speci ied by he playe s’ p oduc i i ies and
by he discoun ac o δ. We deno e he co esponding game as B( ,δ).
Fo ou subsequen analysis, i will be use ul o conside he weigh ed sum o a playe ’s
con ibu ions a e a gi en ime . Fo mally, hese con inua ion con ibu ions o playe ia e

Dynamic Games and Applica ions (2025) 15:1617–1645 1621
de ined as
¯ci( )=(1−δ)
∞

τ=0
δτci( +τ). (2)
One can also de ine a sequence ha collec s he espec i e con inua ion con ibu ions o
each ound, ¯
c( ) =¯
c(0), ¯
c(1), ¯
c(2),....Wecall¯
c( ) he con inua ion con ibu-
ion sequence associa ed wi h con ibu ion sequence (c( )) . E e y con ibu ion sequence
uniquely speci ies a con inua ion con ibu ion sequence and ice e sa. Analogously, we can
also de ine con inua ion payo s a ime ,
¯πi( )=(1−δ)
∞

τ=0
δτπi( +τ).
We w i e ¯
π( )=¯π1( ),..., ¯πn( ) o he espec i e ec o , and no e ha ˆ
π=¯
π(0). Simi-
la ly, we w i e ˆ
c=¯
c(0) o he con inua ion con ibu ions a ime ze o. We call ˆ
c he o al
con ibu ion ec o . By linea i y o he one- ound payo s (1), we ha e
ˆπi=1
n ˆ
c−ˆci.(3)
Tha is, each playe ’s o al payo is uniquely de e mined by he o al con ibu ion ec o . By
de ini ion, his payo ˆπiis he quan i y ha playe iaims o maximise.
Playe s make hei decisions based on hei s a egies. A s a egy σi o playe iis a
unc ion ha assigns o each ini ial con ibu ion sequence c(0), c(1),...,c( −1)anex
con ibu ion alue ci( )=σi(c(τ))τ< ∈R≥0. A s a egy is called bounded i i always
p oduces a bounded con ibu ion sequence (i espec i e o he co-playe s’ s a egies). An
assignmen o one s a egy o each playe (σi)iis called a s a egy p o ile. A s a egy p o ile
is bounded i all i s s a egies a e bounded.
2.2 Sus ainable Con ibu ion Sequences
In he ollowing, we a e pa icula ly in e es ed in hose s a egy p o iles ha o m a subgame
pe ec equilib ium [SPE, o ‘equilib ium’, see15]. We say a bounded s a egy p o ile is in
equilib ium when no playe has an incen i e o de ia e, a e no ini e sequence o mo es.
Fo mally, (σi)iis in equilib ium i he e is no ini ial con ibu ion sequence c(0),...,c( −1)
such ha some playe jcould ge a la ge payo by de ia ing owa ds ano he bounded
s a egy σ∗
ja e ha ime .Fo agi engameB( ,δ), we call a con ibu ion sequence
sus ainable i i is he con ibu ion sequence o some equilib ium s a egy p o ile. A o al
con ibu ion ec o ˆ
cis sus ainable i i is he o al con ibu ion ec o o a sus ainable
con ibu ion sequence.
To de i e ou main esul s, we make ex ensi e use o he p e iously published Theo-
em 1below. This heo em gi es us a com o able cha ac e isa ion o sus ainable con ibu ion
sequences.
Theo em 1 ([23]) Fo a gi en game B( ,δ), de ine an associa ed n×nma ixD=(Dij),
called he p oduc i i y ma ix in ze o-diagonal o m, by
Dij = j/(n− i)i i= j
0i i=j.(4)
1622 Dynamic Games and Applica ions (2025) 15:1617–1645
Then a con ibu ion sequence (c( )) is sus ainable i and only i he associa ed con inua ion
con ibu ions sa is y
¯
c( )≤δD¯
c( +1) o all .(5)
Wi h Theo em 1, we can de e mine whe he a gi en con ibu ion sequence is sus ainable
by checking i he associa ed con inua ion con ibu ion sequence (¯
c( )) ,asde inedby(2),
sa is ies (5) o all . The wo ollowing co olla ies a e immedia e consequences o his
heo em.
Co olla y 2 Fo any game B( ,δ), he se o sus ainable con ibu ion sequences (c( )) , he
se o sus ainable con inua ion con ibu ion sequences (¯
c( )) , and he se o sus ainable o al
con ibu ion ec o s ˆ
ca e closed unde addi ion and mul iplica ion by a non-nega i e scala
( ha is, hey a e con ex cones).
This con exi y esul implies ha whe he o no a con ibu ion sequence is sus ainable only
depends on he ela i e magni ude o he playe s’ con ibu ions. This esul holds because
payo s depend linea ly on con ibu ions. The e o e, scaling all con ibu ions up o down
by he same posi i e ac o does no a ec whe he o no he equilib ium condi ions a e
sa is ied.
Co olla y 3 In a gi en game B( ,δ), a cons an con ibu ion sequence o (ˆ
c) is sus ainable
i and only i
ˆ
c≤δDˆ
c.(6)
So we ha e a se o nlinea cons ain s ha de ines he se o easible cons an con ibu ion
sequences. We can use Co olla y 3 o de i e a e sion o he olk heo em, applied o ou
se up. The olk heo em amously ela es he possible equilib ium payo s in he epea ed
game o he p ope ies o he one-sho payo s [13,14]. To s a e ou e sion, we no e ha in ou
public goods game, playe s can always gua an ee a non-nega i e payo (by no con ibu ing
any hing). Hence, we say a con ibu ion ec o ˆ
c o he one-sho game is indi idually a ional
i i yields a non-nega i e payo o each playe .
Theo em 4 (Folk heo em o epea ed games) Cons an con ibu ions (ˆ
c) ∈R≥0a e
sus ainable in he game B( ,δ) o su icien ly la ge δi and only i ˆ
cis indi idually a ional.
The condi ion o a con ibu ion ec o ˆ
c o be indi idually a ional can be w i en as
max
iˆci≤1
n ˆ
c.(7)
An equi alen o mula ion o he olk heo em is he e o e: he cons an con ibu ion sequence
(ˆ
c) is sus ainable o su icien ly la ge δi and only i Eq. (7) holds.
The abo e esul s allow us o cha ac e ise he p ope ies o sus ainable con ibu ion
sequences. Pe haps one o he mos impo an p ope ies is whe he o no he con ibu ion
sequence en ails a leas some coope a ion. Mo e speci ically, we de ine a play o be non-
de ec i e i a leas one playe makes a posi i e con ibu ion in a leas one ound (i.e., ˆ
c=0).
O he wise we call he play de ec i e. I is easy o see ha o non-de ec ion o be sus ainable,
no jus one, bu a leas wo playe s ha e o make posi i e con ibu ions. This is because
a hypo he ical lone non-de ec o would bene i om de ia ing owa ds ull de ec ion. I
non-de ec ion is sus ainable in a gi en game B( ,δ), henwesay ha B( ,δ)allows o non-
de ec ion. Any p oduc i i y ec o (sa is ying he gene al equi emen i>1 o alli) allows
o non-de ec ion when δis su icien ly la ge [20, Supplemen a yIn o ma ion,P oposi ion 2].
Dynamic Games and Applica ions (2025) 15:1617–1645 1623
2.3 Wel a e and Resou ce E iciency
While he bina y dis inc ion be ween de ec ion and non-de ec ion is use ul, no all o ms
o non-de ec ion a e equally desi able. A e all, e en non-de ec i e con ibu ion sequences
migh esul in payo s a bi a ily close o he ull de ec ion payo o ze o. The e o e, in
he ollowing we in oduce wo o he key me ics o in e es . The i s me ic is he (o e all)
wel a e Wo a gi en play, which equals he sum o all payo s,
W=
n

i=1
ˆπi.(8)
This wel a e can be exp essed as a unc ion o he o al con ibu ion ec o ˆ
cas
W(ˆ
c)=( −1)ˆ
c.(9)
By his equa ion, non-de ec i e plays ha e W>0, whe eas de ec i e plays ha e W=0.
By Co olla y 2, any non-de ec i e con ibu ion sequence can be scaled a bi a ily wi hou
a ec ing hei sus ainabili y. I ollows ha ou base model allows o a bi a y wel a es. As
an al e na i e me ic ha is s ill ele an wi h unlimi ed esou ces, we measu e how e icien ly
he playe s a e able o use hem. The esou ce e iciency Eo a non-de ec i e play is de ined
as he sum o all payo s di ided by he sum o all con ibu ions,
E=n
i=1ˆπi
n
i=1ˆci
.(10)
Resou ce e iciency, oo, is a unc ion o he o al con ibu ion ec o :
E(ˆ
c)= ˆ
c
1ˆ
c−1.(11)
The i s e m on he igh hand side o Eq. (11) can be sligh ly ew i en, as
ˆ
c
1ˆ
c=
n

i=1
ci
c1+...+cn
· i.
This ep esen a ion allows us o in e p e his e m as a weigh ed mean o he playe s’ p oduc-
i i ies; he weigh s co espond o he playe s’ con ibu ions. In pa icula , his obse a ion
implies ha E(ˆ
c)is always in be ween mini i−1andmax
i i−1. Whe he o no he
uppe bound (o equi alen ly, he lowe bound) can be ealised depends on which playe s
con ibu e in equilib ium. Fo example, he uppe bound can be ealised i and only i he e
is an equilib ium in which only hose playe s iwi h i=maxj jmake a con ibu ion. Tha
is only possible i he e a e mul iple playe s wi h maximum p oduc i i y.
Example I is ins uc i e o illus a e hese concep s wi h a wo-playe game (which we
con inue o use h oughou his a icle). Conside he game B(1.5,1.1),0.9.Tha is,
he e a e n=2 playe s wi h p oduc i i ies 1=1.5and 2=1.1(asinFig.1), and he
discoun ac o is δ=0.9. In his example, he alue o he ma ix Dis
D=0 2/(2− 1)
1/(2− 2)0=011
/5
5
/30≈02.2
1.666 0 .
Suppose playe 1 makes a cons an con ibu ion ˆc1=7 in e e y ound, whe eas playe 2
makes he cons an con ibu ion ˆc2=5. I ollows ha he o al size o he public good is
ˆ
c=1.5·7+1.1·5=16. Thus, each playe ’s sha e o he public good is 8, and hei
1624 Dynamic Games and Applica ions (2025) 15:1617–1645
payo s acco ding o Eq. (3)a e ˆπ1=8−7=1and ˆπ2=8−5=3. Because bo h payo s
a e non-nega i e, he espec i e cons an con ibu ion sequences a e indi idually a ional.
Hence, by he olk heo em, hey a e sus ainable in he epea ed game o su icien ly la ge
δ. Fo his case o n=2, Eq. (6) in Co olla y 3 akes he o m o he ollowing sys em o
inequali ies:
ˆc1≤δ 2
2− 1
ˆc2(12)
ˆc2≤δ 1
2− 2
ˆc1(13)
Wi h hese, we can e i y ha he gi en discoun ac o δ=0.9 is indeed su icien ly la ge.
Acco ding o Eq. (8), he esul ing wel a e is W=3+1=4, and acco ding o Eq. (10),
esou ce e iciency is E=4/12 ≈0.333. I con ibu ions we e en imes la ge , wel a e
would inc ease o 40 bu he esou ce e iciency would emain he same.
Ins ead, suppose now ha he wo playe s con ibu e equal cons an amoun s, say ˆc1=
ˆc2=6. By he inequali ies (12–13), his con ibu ion ec o is also sus ainable. I yields a
payo o 1.8 o each playe . Thus, he wel a e is W=3.6, whe eas esou ce e iciency
is E=0.3. We conclude ha equal con ibu ions a e less esou ce e icien , compa ed o
he p e ious example wi h unequal con ibu ions. This is in ui i e because in he p e ious
example, he mo e p oduc i e playe 1 con ibu ed a la ge sha e.
In ligh o hese obse a ions, i is na u al o ask wha he op imal a io o he wo playe ’s
con ibu ions is, i we aim o maximise esou ce e iciency in equilib ium. Again om he
inequali ies (12–13), we see ha when 1> 2(as in ou example), his a io is gi en by
ˆc1
ˆc2
=δ 2
2− 1
.
I o example ˆc1=10, hen ˆc2=500/99 ≈5.051, which yields payo s o ˆπ1≈0.278 and
ˆπ2≈5.227. The esul ing wel a e is W≈5.505 and esou ce e iciency is Ec
sup ≈0.366. No e
ha his maximum esou ce e iciency depends on he discoun ac o δ.I δwe e la ge ,
e en highe e iciencies would be possible. In con as , i δwe e lowe , ei he Ec
sup would be
lowe , o he game migh no allow o non-de ec ion a all.
In he abo e example, we only conside ed he simple case o cons an con ibu ions.
We now add ess he ques ion o whe he we can achie e highe esou ce e iciency when
con ibu ion sequences a e allowed o be ime-dependen .
2.4 E iciency wi h Time-Dependen Con ibu ions
I we only conside he bina y dis inc ion be ween whe he o no a game allows o non-
de ec ion, hen cons an and ime-dependen con ibu ions a e equally e ec i e. Speci ically,
i a game has any non-de ec i e equilib ium a all, hen i also has a non-de ec i e equilib ium
wi h a cons an con ibu ion sequence [Co olla y 6 o 23]. Howe e , below we show ha
wi hin he space o non-de ec i e equilib ia, ime-dependen con ibu ions can indeed enable
ou comes ha a e no sus ainable o he wise. To his end, we i s in oduce some no a ion.
No a ion 2 Fo wo ec o s and w,wew i e ≤1wi i≤wi o all iand i=wi o
a mos one i.Ino he wo ds, ≤1wco esponds o ≤wwi h equali y in a mos one
componen .
Using his no a ion, we can cha ac e ise which o al con ibu ion ec o s a e sus ainable wi h
ime-dependen con ibu ion sequences.
Dynamic Games and Applica ions (2025) 15:1617–1645 1631
Round 0 Round 3 Round 4Round 2Round 1 Round 5
Con ibu ed ( )
Consumed ( )
Deposi ed ( )
×−1
×−1
×−1
×−1
×−1
Fig. 5 The sa ings model. He e, we depic he i s six ounds o a playe ’s gameplay. In each ound, he
playe no only has hei endowmen eia ailable, bu also wha e e esou ces hey deposi ed in he p e ious
ound, plus in e es a he a e δ−1−1. They decide which pa o ha hey wan o con ibu e, which pa o
consume, and which pa o deposi o he sa ings accoun
4 Model Va ia ion II: A Model wi h Sa ings
The sa ings model builds on he ea lie endowmen model. Again, each playe iob ains a
ixed endowmen eie e y ound. Howe e , now playe s ha e h ee op ions o how o spend
hei endowmen , a he han wo. They can con ibu e o he public good, consume pa s o
all o he endowmen p i a ely, o hey can make a deposi in o a sa ings accoun . Sa ings
pay in e es a he a e o (δ−1−1)pe ound – which exac ly co esponds o he ime alue
o money a he discoun ac o δ. These sa ings can hen be spen in u u e ounds, ei he o
con ibu e o he public good o o p i a e consump ion.
Mo e speci ically, each ound p oceeds as ollows. In he beginning o each ound ,
playe s ecei e an endowmen ei. In addi ion, hey ha e access o an amoun o si( )on hei
sa ings accoun (in he e y i s ound, sa ings a e se o ze o). Playe s hen decide which
amoun pi( ) o consume p i a ely, which amoun ci( ) o con ibu e o he public good, and
which amoun di( ) o deposi in o he sa ings accoun . These a iables need o sa is y he
budge cons ain ei+si( )=pi( )+ci( )+di( ). Con ibu ions o he public good and p i a e
consump ion di ec ly en e he playe ’s payo unc ion,
πi( )=1
n c( )+pi( ).
Deposi s, on he o he hand, de e mine a playe ’s sa ings in he beginning o he nex ound,
si( +1)=δ−1di( ). A ime +1, he p ocess is epea ed; indi iduals again ha e o decide
how much o consume, o con ibu e, and o sa e, see Fig. 5. To al payo s (ac oss all ounds),
wel a e, and esou ce e iciency a e hen de ined as in he p e ious wo models.
No a ion 4 The sa ings model uses he same pa ame e s as he endowmen model:
p oduc i i ies , endowmen s e, and he discoun ac o δ. We deno e he game as S( ,e,δ).
We make he ollowing obse a ions abou he sa ings model: Fi s , in ou implemen a ion
o his model, sa ings a e payo -neu al: The in e es ea ned o e one ound is exac ly o se
by a playe ’s discoun ing o u u e ewa ds. Second, wi hou he oppo uni y o sa ing,

1632 Dynamic Games and Applica ions (2025) 15:1617–1645
his model eco e s he endowmen model as discussed in he p e ious sec ion and s udied
in [20]. Thi d, he sa ings model is equi alen o saying ha endowmen s only apply as a
cons ain o he cumula i e con ibu ions. Tha is, each playe iis equi ed o play such ha

τ=0δτci( )≤
τ=0δτei o all , bu wi hou he s onge equi emen ha ci( )≤ei o all
. Finally, since esou ce e iciency is equal o su plus wel a e di ided by o al con ibu ions,
maximising esou ce e iciency in he base model is equi alen o maximising wel a e in he
sa ings model when he endowmen s a e also an op imisa ion a iable.
To s a e ou main esul s o his sec ion, we i s de ine he no ion o a wel a e sup emum
wi h and wi hou sa ings. Fo gi en pa ame e s ,e,δ, he wel a e sup emum wi h sa ings
Ws
sup(e)is he sup emum o wel a e o e all equilib ia o he game S( ,e,δ). I quan i ies
he maximum alue ha he g oup can de i e om coope a ion. Simila ly, we de ine he
wel a e sup emum wi hou sa ings, Wsup(e), as he sup emum o wel a e o e all equilib ia
ha sa is y di( )=0 o alliand . I quan i ies he maximum alue ha he g oup can de i e
wi hou e e sa ing any amoun . Equi alen ly, i co esponds o he wel a e sup emum o he
endowmen model, E( ,e,δ). We in e p e he di e ence Ws
sup(e)−Wsup(e)as he (posi i e
o ze o) ad an age ha sa ings can p o ide.
Theo em 12 Le ( ,δ) allow o non-de ec ion. Take any endowmen dis ibu ion e.Then
sa ings p o ide no ad an age (i.e. Ws
sup(e)=Wsup(e)), i and only i
e≤δDe.(18)
Wi hou sa ings, ˆ
c=e equi es cons an con ibu ions. So by Co olla y 3,Eq.(18)is
equi alen o ˆ
c=ebeing sus ainable wi hou sa ings. The e o e, Theo em 12 s a es ha
exac ly one o he ollowing is he case: Ei he ull con ibu ions a e sus ainable wi hou
sa ings, o sa ings p o ide an ad an age o wel a e. In pa icula , o any and e,when
δis su icien ly low, sa ings p o ide an ad an age. Al e na i ely, o any and δ,when
eis su icien ly unequal, sa ings p o ide an ad an age ( [20], Supplemen a y In o ma ion
P oposi ion 3).
Sa ings also p o ide an ad an age om he pe spec i e o a social planne who chooses
an endowmen dis ibu ion wi h he aim o maximising wel a e. To assess his, we conside
hesup emao Ws
sup(e)and Wsup(e)o e all possible endowmen dis ibu ions e.F om
Theo em 8, we can de i e he ollowing esul :
Theo em 13 Le ( ,δ)allow o non-de ec ion. Le max =maxi i, and le m be he numbe
o playe s wi h p oduc i i y max.ThensupeWs
sup(e)>supeWsup(e)i and only i
1+δ(m−1)· max <n.(19)
To unde s and he heo em in ui i ely, conside again he case ha he e is only a single
playe wi h maximum p oduc i i y max, i.e. ha m=1. Then Theo em 13 s a es ha s ic ly
be e wel a e is possible when playe s a e pe mi ed o sa e pa o all o hei endowmen
o la e ounds, compa ed o when hey a e no . This is because wi h sa ing, hey can play a
mo e esou ce-e icien ime-dependen con ibu ion sequence and s ill p oduc i ely use all
o hei endowmen , whe eas wi hou , hey a e es ic ed o make cons an con ibu ions in
o de o maximise wel a e. Con e sely, as ano he special case o Theo em 13, we conclude
ha sa ings ne e p o ide a wel a e ad an age i all playe s ha e he same p oduc i i y.
Example We e isi he same example as be o e, now in he sa ings model:
S((1.5,1.1),(0.2,0.8),0.9). In he endowmen model, we had o scale he op imally
esou ce-e icien con ibu ion sequence o c1(0)=0.2 so ha playe s do no exceed hei
Dynamic Games and Applica ions (2025) 15:1617–1645 1633
endowmen limi s. Wi h sa ings, i is enough ha a no poin in ime hei cumula i e con i-
bu ions exceed he endowmen limi . A simple example o a supe io con ibu ion sequence
is as ollows. In ound 0, playe 1 con ibu es and consumes no hing (c1(0)=p1(0)=0)
and deposi s e e y hing (d1(0)=e1=0.2). In ound 1, wi h he in e es ecei ed, playe 1
has sa ings o app oxima ely 0.222 and again ecei es an endowmen o 0.2, which makes
o a o al a ailable amoun o 0.422. O his, playe 1 con ibu es c1(1)≈0.222 and again
deposi s d1(1)=0.2. In ound 2, sa ings wi h in e es again make up 0.222, and playe 1
con inues wi h con ibu ing 0.222 and deposi ing 0.2 in e e y subsequen ound. Playe 2,
on he o he hand, om he beginning simply con ibu es c2( )≈0.333 in e e y ound and
p i a ely consumes he es , p2( )≈0.467, wi hou deposi ing any hing. The wel a e o his
con ibu ion sequence is W≈1.164, which is mo e han he op imal alue wi hou sa ing,
W=1.13. Theo em 12 p edic s ha sa ing p o ides an ad an age like his as long as ull
con ibu ions a e no sus ainable, which is he case he e.
The op imal wel a e o e all endowmen dis ibu ions, supeWs
sup(e), equi es he endow-
men dis ibu ion e=(11
/16,5
/16). This is exac ly he endowmen dis ibu ion a which
he maximally esou ce e icien con ibu ion sequence can be played in such a way ha all
endowmen s a e e en ually con ibu ed: Playe 1 con ibu es c1( )=11/16 in e e y ound.
Playe 2 deposi s e e y hing in ound 0 (d2(0)=e2=5/16). The ea e , playe 2 con ibu es
50/(16 ·9)and deposi s 5/16 in e e y ound. (The sequence o con ibu ions is iden ical o
ha o he ea lie example in Sec .2, up o escaling by a ac o o 160/11, and hus also
maximally esou ce e icien .) In his sequence, all esou ces a e used p oduc i ely and none
a e consumed p i a ely, which means ha he op imal esou ce e iciency also ansla es o
op imal wel a e. Indeed, he wel a e is W=supeWs
sup(e)=1+Esup =1.375. The ac ha
his is g ea e han he op imum wi hou sa ing o e all endowmen dis ibu ions, which by
P oposi ion 11 is supeWsup(e)=1+Ec
sup ≈1.366, is p edic ed by Theo em 13.
5 Discussion
The epea ed public good game is one o he majo models in (e olu iona y) game heo y o
unde s and coope a ion in g oups. This li e a u e desc ibes how indi iduals can use condi-
ionally coope a i e s a egies o sus ain ou comes ha a e in easible in one-sho encoun e s.
Ye when desc ibing he possible equilib ium ou comes, many p e ious s udies implici ly
es ic hei analysis o he case ha playe s make he same cons an con ibu ion each ound
[e.g.20,23]. Ins ead, he e we s udy he e ec o ime-dependen con ibu ions. Con a y o
many o he models o ecip oci y, we allow playe s o selec hei ac ions om a con in-
uum be ween ull de ec ion and ull coope a ion. We explo e o which ex en indi iduals can
ob ain be e ou comes (e.g., a be e esou ce e iciency o wel a e) when hey a e able o
a y hei con ibu ions along he equilib ium pa h.
F om he ou se , i is no clea whe he ime-dependen con ibu ions p o ide any sub-
s an ial ad an age a all. A e all, suppose playe s could achie e a supe io ou come wi h
con ibu ions c( ) ha a y in ime. Then playe s migh achie e jus he same ou come
by ins ead making a cons an con ibu ion ˆ
ceach ound, whe e ˆ
cis he app op ia e ( ime-
discoun ed) a e age con ibu ion pe ound, ˆ
c=(1−δ) δ c( ). Wi h espec o hei
payo implica ions, he wo sequences c( ) and (ˆ
c) a e iden ical. A e all, by Eq. (3),
payo s only depend on he playe s’ o al con ibu ions ac oss all ounds. As a esul , he wo
sequences gene a e he same esou ce e iciency and wel a e. Howe e , as we show in his
a icle, he wo sequences may di e in hei sus ainabili y. The e a e ins ances in which he
1634 Dynamic Games and Applica ions (2025) 15:1617–1645
ime-dependen sequence c( ) can be ealised by a subgame pe ec equilib ium, whe eas
he cons an sequence (ˆ
c) canno .
To make his poin , we s udy h ee di e en models: a base model, a model wi h endowmen
cons ain s, and a model wi h sa ings. In he base model, playe s a e allowed o make a bi a y
con ibu ions each ound ( he only equi emen is ha he sequence o con ibu ions does
no di e ge). This se up imposes minimal cons ain s on he playe s’ beha iou , and i is
con enien o wo k wi h ma hema ically. In con as , he o he wo models a e pe haps mo e
ealis ic (and hence hey ha e been s udied mo e equen ly). Fo example, he endowmen
model co esponds o he classical se up ha is also equen ly used in expe imen s [e.g.6,10,
21,28]. He e, con ibu ions a e cons ained by he endowmen s ha he playe s ecei e each
ound. The sa ings model is simila , bu in addi ion i allows playe s o (payo -neu ally)
ans e some o hei endowmen s o u u e ounds. In e es ingly, many o ou esul s o
hese las wo model a ian s a e di ec ly ela ed o ou indings in he base model. As an
example, wi h Theo em 13, we cha ac e ise unde which ci cums ances sa ings p o ide a
wel a e ad an age in he sa ings model. The espec i e esul is di ec ly ela ed o whe he o
no ime-dependen con ibu ions p o ide an ad an age in he base model, Theo em 8.These
simila i ies be ween hose heo ems highligh how se e al indings in he mo e abs ac base
model ca y o e o mo e applied se ings.
In e es ingly, he espec i e heo ems also sugges ha o ou esul s, some asymme-
y among playe s is c ucial. Speci ically, when playe s a e iden ical wi h espec o hei
p oduc i i ies, Theo em 8shows ha ime-dependen con ibu ions do no g an any ad an-
age. Any esou ce e iciency ha can be sus ained wi h ime-dependen con ibu ions can
al eady be sus ained wi h cons an con ibu ions. Bu once playe s di e in hei p oduc i i-
ies, i becomes ai ly easy o ime-dependen con ibu ions o be supe io . In ac , such an
ad an age is gua an eed when he g oup con ains a single playe whose p oduc i i y exceeds
e e yone else’s.
O e all, ou indings highligh he impac o a iable con ibu ions on esou ce e iciency
and, mo e gene ally, on he sus ainabili y o coope a ion.
They sugges ha by ocussing solely on cons an con ibu ions, we may o e look
impo an equilib ia ha can a ise in dynamic se ings.
Appendix A P oo s
A.1 P oo o Theo em 5
P oo o Theo em 5Clea ly ˆ
c=0is sus ainable, so i is su icien o show he s a emen o
ˆ
c= 0.
Fi s we will show ha any sus ainable ˆ
c= 0sa is ies ˆ
c≤1Dˆ
c. By Theo em 1,weha e
o any sus ainable ¯
c( ) ha
δ¯
c(1)≤¯
c(0)≤δD¯
c(1). (A1)
Le -mul iplying wi h he non-nega i e ma ix Don bo h sides o he i s inequali y, we
ob ain
δD¯
c(1)≤D¯
c(0)
wi h equali y in he i h componen exac ly i δcj(1)=cj(0) o all j= i.
Dynamic Games and Applica ions (2025) 15:1617–1645 1635
Toge he wi h he second inequali y o (A1), his gi es
¯
c(0)≤D¯
c(0),
o equi alen ly
ˆ
c≤Dˆ
c,(A2)
whe e equali y in he i h componen equi es δcj(1)=cj(0) o all j= i.
I (A2) has equali y in a leas wo componen s, hen δcj(1)=cj(0) o all j,soδc(1)=
c(0). Analogously, equali y in wo componen s equi es δc(2)=c(1), e c., so he sequence
(c( )) di e ges and is no a alid con ibu ion sequence. Tha is a con adic ion, so we can
ha e ˆci=(Dˆ
c)i o a mos one i.
Now we will show ha i some ˆ
c= 0sa is ies ˆ
c≤1Dˆ
c, henˆ
cis sus ainable wi h a
con inua ion con ibu ion sequence (¯
c( )) ha sa is ies ˆ
c≤¯
c( ) o all . Assume i s ha
he s onge condi ion 0<ˆ
c<Dˆ
cholds.
Take ε>0 such ha 1 +ε<δ
−1and (1+ε)x≤Dx.Le be he Pe on eigen ec o
o D, scaled so ha ≤x. Le inally
T=
δmin maxixi
i−minixi
i
ε(1−δmin)
o T=0, whiche e is la ge .
Le x=ˆ
c o a gi en ˆ
cwi h 0<ˆ
c<Dˆ
c. We de ine a con inua ion con ibu ion sequence
(¯
c( )) by
¯
c( )=((1+ε)δ)− (x+ε )
o all 0 ≤ ≤Tand
¯
c( )=(1+ε)−Tδ−(T+1)min
i
xi
i
+εT
o all >T. I we show ha i obeys (5) o all , we know i is sus ainable. Since ¯
c(0)=x,
ha is enough o p o e he i s s a emen o he p esen heo em. We also see ha (¯
c( )) is
non-dec easing. The e o e, ˆ
c=¯
c(0)≤¯
c( ) o all , which is he second s a emen o he
heo em.
Fo 0 ≤ <T, he i s inequali y, δ¯
c( +1)≤¯
c( ), ollows om
≤x
as ollows. Fi s , we mul iply wi h εand add ε2 , which is non-nega i e, on he igh -hand
side:
ε ≤εx+ε2
We add x+ε on bo h sides and ac o ou :
x+ε( +1) ≤(1+ε)(x+ε )
We mul iply wi h δ((1+ε)δ)−( +1)on bo h sides:
δ((1+ε)δ)−( +1)(x+ε( +1) )≤((1+ε)δ)− (x+ε )
This is equi alen o
δ¯
c( +1)≤¯
c( )
1636 Dynamic Games and Applica ions (2025) 15:1617–1645
by he de ini ion o ¯
c( ).
The second inequali y, ¯
c( )≤δD¯
c( +1), ollows om
=δmin D .
We eplace δmin wi h δ, which is la ge o equal, on he igh -hand side:
≤δD
By cons uc ion, 1 −ε<δ
−1, so we can w i e:
(1+ε) <D
Mul iply by ε on he le -hand side and by ε( +1)on he igh -hand side:
(1+ε)ε <ε( +1)D
Sum wi h he inequali y (1+ε)x≤Dx, which also holds by cons uc ion o ε:
(1+ε)(x+ε )<D(x+ε( +1) )
Mul iply by δ((1+ε)δ)−( +1)on bo h sides:
((1+ε)δ)− (x+ε )<δ((1+ε)δ)−( +1)D(x+ε( +1) )
This is equi alen o
¯
c( )≤δD¯
c( +1).
Nex , we conside =T: The i s inequali y ollows om
x+εT ≤(1+ε)(x+εT ).
Fi s , we eplace xby minixi
i , which is a mos as la ge in each componen :
min
i
xi
i
+εT ≤(1+ε)(x+εT )
Then, we mul iply by δ((1+ε)δ)−(T+1)on bo h sides:
δ((1+ε)δ)−(T+1)min
i
xi
i
+εT ≤((1+ε)δ)−T(x+εT )
This is by de ini ion equi alen o
δ¯
c(T+1)≤¯
c(T).
The second inequali y ollows om
δmin maxixi
i−minixi
i
ε(1−δmin)≤T,
which is by de ini ion o T. We emo e he ceiling unc ion and mul iply by ε(1−δmin)on
bo h sides:
δmin max
i
xi
i
−min
i
xi
i
≤(1−δmin)εT
We mo e minixi
i o he igh -hand side, δminεT o he le , and hen mul iply by δ−1
min :
max
i
xi
i
+εT ≤min
i
xi
i
+εTδ−1
min

Dynamic Games and Applica ions (2025) 15:1617–1645 1637
Now, we can eplace maxixi
i by x, which is a mos as la ge in each componen , and δ−1
min
by D , which is equal:
x+εT ≤min
i
xi
i
+εTD
We mul iply by ((1+ε)δ)−Ton bo h sides:
((1+ε)δ)−T(x+εT )≤δD(1+ε)−Tδ−(T+1)min
i
xi
i
+εT 
This is by de ini ion equi alen o
¯
c( )≤δD¯
c( +1).
Finally, o >T,weha e¯
c( )=¯
c( +1). The i s inequali y is i ially ue, and he
second inequali y is ue since ¯
c( )is a mul iple o .Soweha eshown ha ˆ
cis sus ainable
i 0<ˆ
c<Dˆ
c.
Now we will educe he mo e gene al case o ˆ
c= 0and ˆ
c≤1Dˆ
c o his in wo sequen ial
s eps.
Fi s , le some ˆ
c= 0ha e 0<ˆ
c≤1Dˆ
cwi h equali y o he weak inequali y in exac ly
one componen i.Le x(ε) =ˆ
c−εui,whe euiis he i h s anda d uni ec o . Since x(ε) →ˆ
c
as ε→∞and ˆcj<(Dˆ
c)j o all j= i, we can choose ε>0 su icien ly small such ha
ˆcj=xj(ε) < (Dx(ε))j o all j= ias well, and 0 <xi(ε).We henha exi(ε) < ˆci,bu
(Dx(ε))i=(Dˆ
c)i,soalsoxi(ε) < (Dx(ε))i. Hence x(ε) sa is ies 0<x(ε) < Dx(ε) and
ˆ
c≤Dx(ε). By he abo e esul , we can choose a con inua ion con ibu ion sequence (c∗( ))
such ha ˆ
c∗=x(ε). Now conside he sequence (ˆ
c,δ−1¯
c∗(0), δ−1¯
c∗(1), δ−1¯
c∗(2),...).By
de ini ion, ¯
c∗(0)=x(ε), and we ha e
δ(δ−1x(ε)) ≤ˆ
c≤δD(δ−1x(ε)),
so by Theo em 1, he sequence we cons uc ed is a sus ainable con inua ion con ibu ion
sequence. I begins wi h ˆ
c, so he s a emen o he Lemma holds o ˆ
c.
Now, he only case ha emains is ˆ
c≤1Dˆ
cand 0<ˆ
c= 0. In he de ini ion o he public
goods game, we imposed he condi ions 1 < iand i<n o all i.Weused i<nin he
p oo s abou sus ainabili y (i is necessa y o Dbeing well de ined and posi i e), bu 1 < i
was only used o p o e s a emen s abou maximal wel a e. So we can conside games ha
only sa is y >0ins ead o >1, and he same esul s abou sus ainabilil y will apply,
including he ones om his p oo .
Le ˆ
c= 0be any o al con ibu ion ec o sa is ying ˆ
c≤1Dˆ
c.Le nbe he numbe o non-
ze o componen s o ˆ
c. W.l.o.g. le hese be he i s ncomponen s o ˆ
c. Necessa ily n>1.
Le ˆ
cbe he i s ncomponen s o ˆ
c,andle be he n- ec o de ined by 
i=nn−1 i o
all 1 ≤i≤n. Conside he game B( ,δ) and i s ze o-diagonal p oduc i i y ma ix D.
We ha e D
ij =Dij o all 1 ≤i,j≤n.Sinceˆ
csa is ies ˆ
c≤1Dˆ
c, consequen ly ˆ
calso
sa is ies ˆ
c≤1Dˆ
c.Bu ˆ
caddi ionally sa is ies 0<ˆ
c. So by he case handled abo e, he e
is a sus ainable con inua ion con ibu ion sequence (c( )) s a ing wi h ˆ
c.Le (c( )) be he
n-playe sequence wi h c
i( )=ci( ) o all i≤nand all ,andc
i( )=0 o alli>nand
all .UsingEq.5, we can see ha sus ainabili y o (c( )) ollows i ially om sus ainabili y
o (c( )) . So we ha e ound a con inua ion con ibu ion sequence s a ing wi h ˆ
c.
We ha e hus shown in he mos gene al case ha i some ˆ
c= 0sa is ies ˆ
c≤1Dˆ
c, hen
ˆ
cis sus ainable. Toge he wi h he con e se, which we al eady showed, his comple es he
p oo .

1638 Dynamic Games and Applica ions (2025) 15:1617–1645
A.2 P oo o Theo em 8
P oo o Theo em 8Theo em 5says ha a o al con ibu ion ec o o ˆ
cis sus ainable exac ly
i
ˆ
c≤1Dˆ
c.
Co olla y 3says ha cons an con ibu ions o ˆ
ca e sus ainable exac ly i
ˆ
c≤δDˆ
c.
Le Fbe he egion de ined by ˆ
c≤1Dˆ
c.Le
Esup := sup
ˆ
c∈F {0}
E(ˆ
c)=sup
ˆ
c∈F {0}
E(ˆ
c),
whe e Fis he closu e o F, which is gi en by ˆ
c≤Dˆ
c. Fi s ly, no e ha Esup is well de ined,
since E(ˆ
c)is bounded abo e by maxi i. We obse e ha E(ˆ
c)a ains a maximum on F {0}:
We ha e
{E(ˆ
c)|ˆ
c∈F {0}} = {E(ˆ
c/ˆ
c)|ˆ
c∈F {0}} = {E(ˆ
c)|ˆ
c∈F∩Sn−1},
whe e Sn−1={x∈Rn|x=1}.SinceFis closed, F∩Sn−1is a compac se . So we can
w i e
Esup =max
ˆ
c∈F {0}
E(ˆ
c).
Now we a e eady o p o e he s a emen o he heo em. Bu ins ead o showing ha
a sus ainable ime-dependen con ibu ion sequence ha is mo e esou ce-e icien han all
cons an con ibu ion sequences exis s exac ly i
(δ(m−1)+1) max <n,
as s a ed in he heo em, we will show he ollowing equi alen s a emen : A cons an
con ibu ion sequence wi h esou ce e iciency Esup exis s exac ly i
n≤(δ(m−1)+1) max.(A3)
The ollowing s a emen s a e equi alen o ˆ
c∈Rn:
ˆ
c∈F
ˆ
c≤Dˆ
c
∀iˆci≤
j=i
j
n− i
ˆcj(A4)
∀inˆci≤
j
jˆcj.(A5)
Take any ˆ
c∈F {0}such ha E(ˆ
c)=Esup ha is sus ainable wi h a cons an con ibu ion
sequence, which we assume exis s. This ˆ
cmaximises E(ˆ
c)o e all ˆ
c= 0sa is ying he abo e
inequali y (A5). This implies he ollowing s a emen (∗): The e a e no i1,i2such ha i1> i2
and nˆci1<j jˆcjand ˆci2>0. O he wise, we could inc ease ˆci1and dec ease ˆci2by equal
amoun s o inc ease E(ˆ
c)while also s aying wi hin F {0}.
Dynamic Games and Applica ions (2025) 15:1617–1645 1639
Since ˆ
cis sus ainable wi h a cons an con ibu ion sequence, we ha e ˆ
c≤δDˆ
c,whichis
equi alen o
∀iˆci≤δ
j=i
j
n− i
ˆcj.(A6)
A leas wo ˆcimus be s ic ly posi i e, so j=i
j
n− iˆcj>0 o e e yi.Sinceδ<1, (A6)
implies
∀iˆci<
j=i
j
n− i
ˆcj.
So he s a emen (∗) simpli ies o: The e a e no i1,i2such ha i1> i2and ˆci2>0. This
means ha o any i,i ˆci>0, hen i= max.
F om (A6), we ge
∀i(n−(1−δ) i)ˆci≤δ
j
jˆcj,(A7)
whe e

j
jˆcj= max 
j
ˆcj.
Exac ly mcomponen s o ˆ
ca e non-ze o. So choose some isuch ha ˆci>0and
ˆci≥1
m
j
ˆcj.
Inse ing in o (A7), we ge
(n−(1−δ) max)1
m
j
ˆcj≤δ max 
j
ˆcj.
The ˆcjsum o 1. We mul iply wi h mon bo h sides and ge
n−(1−δ) max ≤δ maxm.
By simple ea angemen , his is equi alen o (A3). So (A3) being alse is a necessa y
condi ion o he exis ence o a cons an con ibu ion sequence wi h esou ce e iciency Esup,
which was ou only assump ion.
Now ins ead assume con e sely ha (A3) holds. Le ˆci=1 o allisuch ha i= max,
and le ˆci=0 o all o he i. We can check easily om he de ini ions ha ˆ
csa is ies
ˆ
c≤δDˆ
c, so i is sus ainable wi h a cons an con ibu ion sequence. The esou ce e iciency
o ˆ
cis E(ˆ
c)= max, so i is maximal. The e o e, (A3) is also a su icien condi ion, and
hence an exac condi ion, o he exis ence o a cons an con ibu ion sequence wi h esou ce
e iciency Esup.
A.3 P oo o Theo em 12
P oo o Theo em 12 We will show ha i e≤δDe, henWs
sup(e)=Wsup(e),andi e≤ δDe,
hen Ws
sup(e)>Wsup(e).
Fi s ly, i e≤δDe, hen by Co olla y 3, he cons an con ibu ion sequence (c( )) =(e)
is sus ainable. So he o al con ibu ion ec o ˆ
c=eis sus ainable wi hou sa ing. Since e
1640 Dynamic Games and Applica ions (2025) 15:1617–1645
is an uppe bound on he o al con ibu ion ec o , eis also an uppe bound on wel a e in
gene al, and we ha e Ws
sup(e)=Wsup(e)= e.
Now, assume ha e≤ δDe. By P oposi ion 11, he e is always a o al con ibu ion ec o
ˆ
c ha sa is ies W(ˆ
c)=Wsup(e)and is sus ainable wi h a cons an con ibu ion sequence in
E( ,e,δ), i.e. wi hou sa ing. Take such a ˆ
c. By Co olla y 3,weha eˆ
c≤δDˆ
c.Soˆ
c= e,
meaning he e is some isuch ha ˆci<ei. Fix such an i.
De ine ˆ
c(ε) =ˆ
c+ε(e−ˆ
c) o ε≥0. Clea ly ˆ
c(ε) →ˆ
cas ε→0. Since ˆ
c≤δDˆ
c,
since Dˆ
cis a posi i e ec o , and since δ<1, we ha e ˆ
c<Dˆ
c. So we can choose ε>0
su icien ly small such ha ˆ
c(ε) < Dˆ
c(ε) as well. By Theo em 5,ˆ
c(ε) is hus a sus ainable
o al con ibu ion ec o in S( ,e,δ).Bu ˆ
c(ε) ≥ˆ
cand ˆc(ε)i>ˆci.SoW(ˆ
c(ε)) > W(ˆ
c).
Consequen ly, Ws
sup(e)>Wsup(e).
A.4 P oo o Theo em 13
P oposi ion 14 Take any game B( ,δ) ha allows o non-de ec ion. Then o any ˆ
c∈Rn
≥0
sa is ying n
i=1ˆci≤1, he ollowing a e equi alen :
1. A o al con ibu ion ec o o ˆ
cis sus ainable in he game B( ,δ).
2. The e exis s an endowmen dis ibu ion esuch ha a o al con ibu ion ec o o ˆ
cis
sus ainable in he game S( ,e,δ).
In a game S( ,e,δ), a con ibu ion sequence (c( )) is called sus ainable i he e is a play
ha esul s in con ibu ion sequence (c( )) .
I is easy o check ha a con ibu ion sequence (c( )) is sus ainable in S( ,e,δ)i and
only i
ˆ
c≤δ e+δ +1¯
c( +1)(A8)
o all ≥0. Inequali y A8 simply s a es ha e e y playe ihas enough esou ces in e e y
ound in o de o make a con ibu ion o ci( )as long as hey ne e consume any o hei
a ailable esou ces p i a ely.
In he game B( ,δ), he G im s a egy p o ile G((c( )) ) o a con ibu ion sequence
(c( )) is he pu e s a egy p o ile G((c( )) )=(σi)ide ined as ollows: In each ound and
o each i he s a egy σicon ibu es ci( )i all playe s ha e so a also played acco ding o
(c( )) , bu o he wise con ibu es 0.
In he base model, a con ibu ion sequence (c( )) is sus ainable in a gi en game i and
only i i s associa ed G im s a egy p o ile G((c( )) )is a SPE o he game [23].
In he game B( ,e,δ), he G im s a egy p o ile Gs((c( )) ) o a con ibu ion sequence
(c( )) is de ined as ollows.
Fi s , we ecu si ely cons uc a deposi sequence (d( )) . Fo each and each i,le di( )
be he minimal dsuch ha he e is a play ha esul s in con ibu ion sequence (c( )) and an
ini ial deposi sequence o playe io di(0),…,di( −1),d. We can check ha a minimum is
indeed a ained. Then he e also exis s a play ha esul s in con ibu ion sequence (c( )) and
deposi sequence (d( )) . This also uniquely de e mines he p i a e consump ion sequence
(p( )) .
Now he G im s a egy Gs((c( )) )i o playe iplays as ollows. In each ound ,i all
playe s ha e so a played acco ding o Gs((c( )) ), hen playe icon ibu es ci( ), p i a ely
consumes pi( ), and deposi s di( ). O he wise, playe icon ibu es 0, deposi s 0, and p i a ely
consumes he en i e a ailable amoun .