On bargaining norms as solutions to cost-minimization problems

Ta u , Tymon
A icle
On ba gaining no ms as solu ions o cos -minimiza ion
p oblems
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Ta u , Tymon (2024) : On ba gaining no ms as solu ions o cos -minimiza ion
p oblems, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol.
19, Iss. 4, pp. 1443-1472,
h ps://doi.o g/10.3982/TE4451
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/320271
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Theo e ical Economics 19 (2024), 1443–1472 1555-7561/20241443
On ba gaining no ms as solu ions o cos –minimiza ion
p oblems
Tymon Ta u
Depa men o Economics, Uni e si y o Bonn
This pape s udies ba gaining ou comes in economies in which agen s may be
able o impose ou comes ha de ia e om he ele an social no ms bu incu
cos s when hey do so. I cha ac e izes ba gaining ou comes ha a e easies o
a socie y o sus ain as pa o a social no m ha e e ybody will wan o ollow.
Depending on echnological assump ions, he app oach yields he Nash ba gain-
ing solu ion, he Kalai–Smo odinsky solu ion, he equal mone a y spli , and o he
ba gaining solu ions. Se - alued solu ion concep s a e de i ed ha a e ele an i
one is unable o unwilling o make speci ic echnological assump ions.
Keywo ds. Ba gaining, social no ms, sanc ions, in e nalized no ms, ba gaining
solu ions.
JEL classi ica ion. C71, C78.
1. In oduc ion
Following Nash (1950,1953) much o he mode n heo y on ba gaining can be spli in o
wo b anches. The axioma ic app oach, bo n wi h Nash (1950), p edic s ba gaining ou -
comes based on assump ions abou how ou comes in di e en ba gaining si ua ions di -
e . The s a egic app oach, also concei ed by Nash (1953), conside s a single ba gaining
si ua ion in isola ion and uses a non-coope a i e game o model he s a egic incen i es
playe s may ace when nego ia ing an ag eemen . Once an ag eemen is eached, he
game ypically ends. In con as , his pape is conce ned wi h ba gaining ou comes in
socie ies whe e coope a ion is go e ned by social no ms, including e hical and mo al
no ms. The idea ha social no ms may be ele an in ba gaining is no new. Fo in-
s ance, A ow (1971, p. 22) a gued ha a p ima y eason o he exis ence o social no ms
may be o acili a e coope a ion by c ea ing en i onmen s whe e indi iduals can us
each o he , he eby making mo e e icien coope a ion possible:
I is a mis ake o limi collec i e ac ion o s a e ac ion... I wan o [call] a en ion o a less
isible o m o social ac ion: no ms o social beha io , including e hical, and mo al ones.
I sugges as one possible in e p e a ion ha hey a e eac ions o socie y o compensa e
o ma ke ailu e. I is use ul o indi iduals o ha e some us in each o he ’s wo d. In
he absence o us , i would become e y cos ly o a ange o al e na i e sanc ions and
Tymon Ta u : [email p o ec ed]
I hank Philip S ack, Fa uk Gul, S ephen Mo is, An onio Pen a, Wol gang Pesendo e , Juuso Valimaki,
F ancoise Fo ges, Ma ek Pycia, S ephan Laue mann, Benny Moldo anu, La y Samuelson, Paul Heidhues,
and h ee anonymous e e ees o hei help ul ques ions and commen s. I am also g a e ul o Jonas
Neuschä e o poin ing ou a numbe o ypos and mino mis akes in an ea lie d a o he pape .
©2024 The Au ho . Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE4451
1444 Tymon Ta u Theo e ical Economics 19 (2024)
gua an ees, and many oppo uni ies o mu ual bene icial coope a ion would ha e o be
o gone.
Wha alloca ion o su plus should we expec o be imposed as pa o a social no m i
such no ms a e used o a oid ine iciencies ha will occu in he absence o such no ms?
To unde s and he basic idea o he app oach p oposed in his pape , conside he
ollowing s ylized example. Two agen s can collabo a e o cos lessly p oduce a good
ha is wo h $1. The e is ba gaining o e how o di ide he c ea ed $1 su plus. In o he
wo ds, agen s can ag ee on any alloca ion o su plus (x1,x2)∈{[0, 1]2:x1+x2=1},
whe e x1is he su plus ecei ed by playe 1 and x2is he su plus ecei ed by playe 2.
Howe e , in he spi i o he ine iciencies men ioned by A ow, assume such ag eemen s
a e no easily en o ceable. Be o e he coope a ion is comple e and he inal p oduc can
be sold, each side will epea edly ha e an oppo uni y o “s eal” he un inished good,
which can be sold o $0.60. No e ha no ma e wha alloca ion o su plus he wo
agen s ag ee on, in he absence o e hical o o he no ms, a leas one pa y will ha e
an incen i e o b eak he ag eemen i doing so gua an ees hem $0.60. Mo eo e , i an
agen expec s he o he agen o s eal he un inished p oduc a he i s oppo uni y—
lea ing hem wi h no hing— ha agen would ha e an incen i e o s eal he un inished
good hemsel be o e he o he agen does. Thus, we can expec ha each agen will y
o s eal he un inished good i gi en he chance. Hence, he good will ne e be com-
ple ed and he su plus di ided be ween he wo agen s will be a mos $0.60 ins ead o
$1. This is ine icien in he sense ha i bo h agen s could us each o he ’s assu ances
ha hey will comple e he p ojec , hey could c ea e a su plus o $1 ins ead and di ide
i in such a way ha each o hem would be s ic ly be e o . Pe haps i was, among
o he s, his so o si ua ion ha A ow had in mind when he w o e ha “i is use ul o
indi iduals o ha e some us ” and “in he absence o us ” i may be he case ha
“many oppo uni ies o mu ual bene icial coope a ion would ha e o be o gone.”1
Imagine ha a social no m is in place manda ing ha he p ojec be comple ed when
ag eed upon and whe e s ealing he un inished p oduc would iola e he no m. A e
he p oduc is inished, playe 1 ecei es x1and playe 2 ecei es x2dolla s, whe e x1
and x2a e nonnega i e numbe s ha add up o 1. Fu he mo e, suppose ha i one
playe iola es he no m, ha playe will incu a “de ia ion cos ” o mdolla s. In he
case o an in e nalized no m, mcould cap u e cos s ha he agen needs o incu o
o e come he anxie y a e b eaking he no m. I obse ed de ia ions om he no m
esul in sanc ions, mcould ep esen he cos s he agen needs o incu o a oid hose
sanc ions o he disu ili y expe ienced when acing hose sanc ions. I is clea ha i m
1The example illus a es why ine iciencies may a ise in p olonged collabo a ions when us is absen .
I is pe haps less ob ious ha no ms may also p e en ine iciencies in si ua ions whe e comple e con ac s
speci ying any di ision o su plus a e a ailable. Conside , o ins ance, a buye and a selle who can d aw
up a legally binding con ac speci ying he e ms o deli e y and he p ice o he goods sold, knowing ha
cou s will en o ce bo h he deli e y and paymen o he goods i necessa y. As was poin ed ou by C aw o d
(1982), in said si ua ions, ine iciencies will o en occu i agen s can impe ec ly commi o ba gaining
posi ions be o e ba gaining s a s. Social no ms can also help elimina e such ine iciencies: i de ia ions
om he no m esul in sanc ions o o he cos s, incen i es o impe ec ly commi o a ba gaining posi ion
in an a emp o ge a di ision mo e a o able han he one speci ied by he no m will dec ease.
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Theo e ical Economics 19 (2024) On ba gaining no ms 1445
is su icien ly la ge (in he conside ed example, a leas $0.60), any alloca ion xcan be
sus ained as a no m. While any alloca ion o su plus can be sus ained as a no m i he
“punishmen ” mis su icien ly la ge, he minimal mneeded o sus ain us will depend
on he alloca ion. Indeed, i playe ige s xiunde he alloca ion x, hen o hemno o
be willing o b eak he no m and lea e wi h $0.60, i mus be ha 0.60 −m≤xi.Thus,
o any alloca ion x, he se o punishmen s m o which xcan be sus ained as a no m is
gi en by
S(x)=m∈[0, ∞):m≥0.60 −min(x1,x2).
In o he wo ds, i we iden i y no ms wi h pai s (x,m),whe exis he alloca ion o su plus
pos ula ed by he no m and mspeci ies how de ia o s a e sanc ioned, hen he no ms
ha can be sus ained a e exac ly hose no ms (x,m) o which m∈S(x).I suchano m
is used, which will i be?
Sus aining a social no m ha p e en s indi iduals om s ealing he un inished good
will, o cou se, in ol e ce ain social cos s. Fo ins ance, i he no m is sus ained by pun-
ishing de ia o s, po en ial de ia o s need o be moni o ed and he indi iduals punish-
ing de ia o s need o be incen i ized. Imagine ha κ(m)is he minimal cos ha socie y
needs o incu o sus ain a no m wi h an en o cemen echnology in which de ia o s
incu a cos o m. I appea s na u al o assume ha κ(m)inc eases in m.Conside now
he p oblem o inding he cheapes sus ainable no m (x,m)among all he no ms ha
can be sus ained, namely he p oblem
min
xmin
m∈S(x)κ(m).(1)
In he example conside ed abo e, his p oblem is easy o sol e. Since he minimal pun-
ishmen s needed o sus ain an alloca ion xa e gi en by 0.60−min(x1,x2), i is clea ha
among all alloca ions x ha a e sus ainable as pa o a no m, he e is a unique one ha
is he cheapes o main ain, and ha is he alloca ion xe en in which each playe ecei es
$0.50, ha is, whe e he dolla is spli e enly. I is no coincidence ha his alloca ion
xe en also has he p ope y ha o any x= xe en,i is hecase ha S(x)S(xe en),
namely, ha he se o punishmen s wi h which xe en can be sus ained is s ic ly la ge
han he se o punishmen s wi h which any o he alloca ion xcan be sus ained.
In his pape , we will analyze he alloca ion o su plus in ba gaining p oblems by
asking which alloca ion o su plus is leas cos ly o socie y o sus ain as a social no m.
We will see ha in he case o wo-playe ba gaining unde comple e in o ma ion,2 his
app oach yields di e en p edic ions depending on he na u e o he assumed punish-
men s. In pa icula , s anda d solu ion concep s like he Kalai–Smo odinsky solu ion,
he Nash ba gaining solu ion, and he equal mone a y spli , can all be unde s ood as
unique solu ions o ou cos -minimiza ion p oblem o na u al punishmen echnolo-
gies. We also show how ou app oach can be used o yield se - alued solu ion con-
cep s ha gene alize he ba gaining solu ions men ioned abo e, and a e use ul i li le is
known abou he way in which no ms a e en o ced and abou he unc ion κ.
2A companion pape s udies ba gaining among mo e playe s.
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1446 Tymon Ta u Theo e ical Economics 19 (2024)
The pape is o ganized as ollows. Sec ion 2in oduces he model and basic con-
cep s. Sec ion 3conside s se e al examples and, in pa icula , shows how o app op i-
a e no m en o cemen echnologies, ou app oach yields he Kalai–Smo odinsky solu-
ion and he Nash ba gaining solu ion. Sec ion 4de i es some mo e gene al esul s. The
main esul s he e a e Theo ems 1and Theo em 2, which cha ac e ize all alloca ions ha
a e he unique solu ion o some cos -minimiza ion p oblem and, a he same ime, cha -
ac e ize all alloca ions ha a e he easies o sus ain o some en o cemen echnology.
Sec ion 5discusses ex ensions. Sec ion 6concludes and he Appendix con ains p oo s.
Since his pape p o ides al e na i e ounda ions o concep s like he Nash ba gain-
ing solu ion, he Kalai–Smo odinsky solu ion, and mo e, ou wo k can be seen as pa
o a la ge body o li e a u e discussing ounda ions o hose and ela ed concep s. Ou
app oach, howe e , di e s om ypical pape s using he axioma ic app oach (e.g., Nash
(1950), Kalai and Smo odinsky (1975), o Rubins ein (1982)), as a single ype o ba gain-
ing p oblem is conside ed in isola ion and no assump ions a e made abou how ba -
gaining ou comes will change i some aspec s o he ba gaining si ua ion—like he se o
al e na i es o he p e e ences o he playe s—a e modi ied. Ou app oach also di e s
om pape s using he s a egic app oach (e.g., Nash (1953), Rubins ein (1982), Ab eu
and Gul (2000), Comp e and Jehiel (2010), and Pe y and Reny (1994)) and, mo e gene -
ally, pape s using non-coope a i e game heo y, gi en ha we do no selec ou comes
based on s anda d solu ion concep s used in non-coope a i e game heo y.
I one hinks abou social no ms ha a e in e nalized (i.e., pa o he agen ’s p e -
e ences) he p oposed app oach seems ela ed o a li e a u e s udying he e olu ion o
p e e ences in educed models in which na u e designs p e e ences o a oid ce ain in-
e iciencies, such as in Samuelson (2004)o Samuelson and Swinkels (2006). Pape s ha
use e olu iona y game heo y o selec Nash equilib ia in non-coope a i e ba gaining
games (e.g., Young (1993)) appea less ela ed because he me hodology is again e y
di e en .
2. Model
Conside he p oblem o wo agen s who can engage in some ac i i y ha c ea es a mon-
e a y su plus. Fo he sake o conc e eness, we will assume ha he su plus is equal o
$1.
Le
X=(x1,x2)∈[0, 1]2:x1+x2=1
be he se o possible e icien alloca ions o he mone a y su plus, whe e (x1,x2)∈X
is in e p e ed as an alloca ion whe e playe 1 ecei es x1and playe 2 ecei es x2.I an
alloca ion o su plus x∈Xis implemen ed, playe s ecei e on Neumann–Mo gens e n
u ili ies u1(x1)and u2(x2), espec i ely, whe e ui o i=1, 2 a e di e en iable u ili y
unc ions sa is ying u
i>0andu
i≤0. In he ollowing discussion, we will assume ha
playe s a e symme ic in all aspec s excep hei u ili y unc ions ui.3
3See Sec ion 5.1 o some ema ks abou he case whe e playe s a e asymme ic in o he aspec s.
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Theo e ical Economics 19 (2024) On ba gaining no ms 1447
In he example conside ed in he In oduc ion, we assumed ha du ing he coope -
a ion, agen s ha e he op ion o s eal an un inished p oduc ha can be sold o $0.60.
We hen conside ed social no ms ha we iden i ied wi h pai s (x,m),whe exwas he
alloca ion o su plus pos ula ed by he no m and mwas a eal numbe desc ibing he
consequences an agen had o ace when de ia ing om he no m.
In his sec ion, we will conside a mo e gene al amewo k and, in pa icula , allow
he agen s o a emp o “g ab” any ac ion o he p oduced su plus. We will also al-
low o mo e sub le en o cemen mechanisms whe e he consequences ha an agen
aces a e a de ia ion can, among o he hings, depend on how a hey de ia ed om
he social no m in place. While we will s ill be able o iden i y social no ms wi h pai s
(x,c),whe exis he alloca ion o su plus pos ula ed by he no m and cdesc ibes he
consequences agen s ha e o ace a e a de ia ion, he pa ame e c ha desc ibes wha
happens a e a de ia ion will no longe be a eal numbe , bu a mo e complica ed ob-
jec . In pa icula , he pa ame e cwill be equal o a pai o unc ions c=(p,m),whe e
p:[0, 1]2→[0, 1]and m:[0, 1]2→[0, 1]a e bo h nondec easing in each o hei wo
a gumen s.
2.1 No ms
Le Ebe a se whose elemen s a e pai s o unc ions (p,m),whe ep:[0, 1]2→[0, 1]
and m:[0, 1]2→[0, 1]a e bo h nondec easing in each o hei wo a gumen s. We will
in e p e Eas he se o possible ways in which social no ms can be en o ced in a gi en
socie y and call E he se o possible no m en o cemen s o he en o cemen echnology
se .
Asocial no m will be iden i ied wi h a pai (x,c),whe ex∈Xis he alloca ion o
su plus speci ied by he no m and c∈Edesc ibes wha happens i agen s de ia e om
he no m.
We wan o conside he case whe e each agen ican a emp o g ab any ac ion
x
i∈[0, 1]o he p oduced su plus. I he no m (x,c)∈X×Eis in place and he agen
a emp s o g ab x
i∈[0, 1]o he join ly p oduced su plus, hen hei a emp is suc-
cess ul wi h a p obabili y o 1 −p(x
i−xi,xi)and is de ec ed by he o he playe wi h a
p obabili y o p(x
i−xi,xi). I he a emp is success ul, hen he agen ecei es x
i,bu
incu s a mone a y cos o m(x
i−xi,xi), which, o ins ance, could ep esen he cos s
o o e coming anxie y a e b eaking an in e nalized no m, he disu ili y om sanc ions
ha de ia o s ace, o cos s incu ed o a oid such sanc ions. Wha happens i he agen
is unsuccess ul and hei a emp is de ec ed? Fo now, we will assume ha i he o he
agen de ec s ha hei pa ne wan s o beha e in a way ha is inconsis en wi h he
alid no ms, hey will cease any coope a ion and playe iwho a emp ed o de ia e om
he no m will ecei e no hing, meaning hei u ili y is ui(0).4No e ha we said no hing
abou he payo s ha he o he playe ecei es when playe ia emp s o de ia e om
he beha io speci ied by he no m. Thus, ou modeling app oach, o ins ance, can
handle si ua ions whe e i playe ig abs x
i, his lea es he o he playe wi h no hing
4See Sec ion 5.2 o a b ie discussion o al e na i es.
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1448 Tymon Ta u Theo e ical Economics 19 (2024)
(e.g., in he example discussed in he In oduc ion) and si ua ions whe e i playe ig abs
x
i<1, his s ill lea e some posi i e sha e o su plus o he o he playe .
I he no m speci ies a di ision x∈Xand de ia ion cos s a e cap u ed by c=(p,m),
agen iwill ha e no incen i e o use he ac ion ha gi es hem x
ii and only i
ui(xi)≥1−px
i−xi,xi·uix
i−mx
i−xi,xi+px
i−xi,xi·ui(0).
This mo i a es he ollowing de ini ion.
De ini ion 1. An alloca ion x∈Xcan be sus ained as pa o a no m wi h c=(p,m)∈E
i and only i
ui(xi)≥1−px
i−xi,xi·uix
i−mx
i−xi,xi+px
i−xi,xi·ui(0)(2)
holds o all playe s i∈{1, 2}and all x
i∈[0, 1]. Fo any alloca ion x∈X,deno e hese
o c∈E o which xcan be sus ained by SE(x).
We es ic ed a en ion o unc ions pand m, which a e non-dec easing in bo h o
hei a gumen s. The ac ha pis nondec easing in he i s a gumen cap u es he idea
ha o any gi en no m, i is (weakly) ha de o g ab la ge sha es o he su plus wi hou
being de ec ed. The ac ha pis non-dec easing in he second a gumen cap u es he
idea ha i is (weakly) ha de o g ab a ixed amoun om he o he playe i he o he
playe is ge ing e y li le. I mcap u es he eeling o anxie y a e b eaking a no m,
hen he assump ion ha mis nondec easing in he i s a gumen cap u es he idea ha
la ge de ia ions cause (weakly) mo e anxie y. Finally, he ac ha mis nondec easing
in he second a gumen cap u es he idea ha a de ia ion om he no m by a ixed
amoun may be seen as mo e jus i ied i one ge s a li le ins ead o a lo .
Rema k 1. No e ha he unc ion SEin oduced in De ini ion 1only depends on he
p e e ences o bo h playe s and, he e o e, no on which u ili y unc ion is used o ep-
esen hose p e e ences. Thus, SEwill no be a ec ed i posi i e a ine ans o ma ions
a e applied o he u ili y unc ions u1and u2. As a esul , he same is ue o he de-
i ed concep s de ined in e ms o SE(e.g., De ini ion 3and De ini ion 4in he nex
subsec ion). Thus, we can, wi hou loss o gene ali y, assume ha u1(0)=u2(0)=0and
u1(1)=u2(1)=1 whene e his is con enien .
De ini ion 2. An en o cemen echnology se Eis egula i and only i , o any (p,m)∈
E, i is he case ha he unc ions pand ma e con inuous.
Regula echnology se s ha e he p ope y ha o any c∈E, he se o alloca ions
x∈X ha can be sus ained as pa o a no m using ha cis closed in X.
2.2 An induced pa ial o de on X
The unc ion SEcan be used o compa e di e en alloca ions o su plus in e ms o how
la ge he se o en o cemen echnologies is o which a gi en alloca ion can be sus ained
as pa o a no m.
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Theo e ical Economics 19 (2024) On ba gaining no ms 1449
De ini ion 3. Fix an en o cemen echnology se E. An alloca ion x∈Xis easie o
sus ain as pa o a no m han y∈X(we will also use he no a ion xEy) i and only i
SE(y)SE(x).
The abo e de ini ion immedia ely implies ha he bina y ela ion Eon Xis i e lex-
i e (i.e., he e is no xwi h xEx) and ansi i e (i.e., o x,y,z∈X,yExand zEy
implies zEx). Thus, Eis a s ic pa ial o de on X.5
I will be use ul o in oduce some language o desc ibe alloca ions ha a e he g ea -
es elemen s wi h espec o he pa ial o de E.
De ini ion 4. Fix an en o cemen echnology se E. An alloca ion x∈Xis he easies o
sus ain as pa o a no m i and only i SE(y)SE(x) o all alloca ions y∈Xsa is ying
y= x.
I x∈Xis he easies alloca ion o sus ain o some en o cemen echnology se E,
hen he se o c∈E o which a playe would wan o de ia e is smalle han o any
o he alloca ion y= x. Thus, an alloca ion ha is he easies o sus ain can be seen as
one ha is s ic ly mo e obus han any o he alloca ion.
2.3 The cos -minimiza ion p oblem
In he case o he example conside ed in he In oduc ion, he alloca ion in which each
playe ecei ed hal a dolla was no only he alloca ion ha was he easies o sus ain as
pa o a no m in he sense de ined abo e: I was also he unique alloca ion ha sol ed
a ce ain cos -minimiza ion p oblem.
We will say ha a unc ion κ:E→[0, ∞)is nondec easing i and only i κ(p,m)≥
κ(p,m)whene e p≥pand m≥m.Le κ:E→[0, ∞)be some nondec easing unc-
ion and conside he p oblem
min
x∈X:SE(x)=∅ in κ(c):c∈SE(x).(3)
We will call his minimiza ion p oblem he cos -minimiza ion p oblem o he en o ce-
men echnology se Eand cos unc ion κ, and will be in e es ed in he ques ion whe he
he e exis s a unique x∈Xwi h SE(x)= ∅ ha minimizes in {κ(c):c∈SE(x)}.Ino he
wo ds, we will be in e es ed in he ques ion whe he he e is an alloca ion ha is cheap-
es o sus ain in ha he cos is lowes i we conside he in imum o e all en o cemen
echnologies c ha can be used o sus ain x. The eason why we ake in {κ(c):c∈SE(x)}
a he han minc∈SE(x)κ(c)is ha addi ional assump ions on he se Ea e equi ed o
gua an ee ha he la e exis s.
De ini ion 5. Fix an en o cemen echnology se Eand a nondec easing cos unc ion
κ:E→[0, ∞). We will say ha xis he unique solu ion o he cos -minimiza ion p oblem
( o ha en o cemen echnology se and cos unc ion) i and only i xis he unique
solu ion o he p oblem (3), in he sense ha SE(x)= ∅ and o any y∈{z∈X:SE(z)= ∅}
wi h y= x,i is hecase ha in {κ(c):c∈SE(x)}<in {κ(c):c∈SE(y)}.
5A bina y ela ion ha is i e lexi e and ansi i e is called a s ic pa ial o de .
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1450 Tymon Ta u Theo e ical Economics 19 (2024)
We will la e see ha he e is a ela ionship be ween he se o alloca ions sa is ying
De ini ion 4 o some en o cemen echnology Eand he se o alloca ions sa is ying he
abo e De ini ion 5 o some en o cemen echnology se Eand some nondec easing cos
unc ion κ:E→[0, ∞).
3. Examples
In he In oduc ion, we saw an example whe e he alloca ion in which each playe
ecei ed hal a dolla was he easies o sus ain and a unique solu ion o he cos -
minimiza ion p oblem o some nondec easing cos unc ion κ.6The ollowing exam-
ples will yield wo o he p ominen ba gaining solu ions.
To simpli y no a ion, we will assume in his sec ion ha ui(0)=0 o each playe i.
Unde Rema k 1, his is wi hou loss o gene ali y.
3.1 Kalai–Smo odinsky solu ion
Le Ebe he se o pai s (p,m)such ha he unc ion m:[0, 1]2→[0, ∞)sa is ies m≡0
and he unc ion p:[0, 1]2→[0, 1]sa is ies p≡ o some ∈[0, 1]. This means ha
de ia ions do no esul in any mone a y cos s. Ins ead, i an agen ies o g ab a la ge
sha e o he su plus han speci ied by he no m, he e is an exogenously ixed p oba-
bili y ha he ba gaining p ocess will pe manen ly end and each playe will ge hei
disag eemen payo .
P oposi ion 1. Fo he en o cemen echnology se Econside ed in his subsec ion, he e
exis s an alloca ion o su plus ha is easie o sus ain as a no m han any o he alloca-
ion o su plus: he Kalai–Smo odinsky solu ion, meaning he unique alloca ion xK.S.,in
which u1(xK.S.
1)
u1(1)=u2(xK.S.
2)
u2(1).
P oo . Conside an a bi a y alloca ion x. Since does no depend on x,whene e
inequali y (2) is no sa is ied o some playe iand x
i∈[0, 1], i will also no hold o ha
playe iand x
i=1. Thus, (p,m)∈SE(x)i and only i
ui(xi)≥1−p(1, xi)·ui(1)
o i∈{1, 2}.Thismeans ha
SE(x)=(p,m)∈E:p(1, 0)∈1−minu1(x1)
u1(1),u2(x2)
u2(1),1
.
6I is s aigh o wa d o ep oduce his example in ou amewo k whe e an agen can a emp o g ab
any amoun o su plus. Le ¯
Ebe he se o pai s (p,m)such ha p:[0, 1]2→[0, 1]and m:[0, 1]2→[0, ∞),
and bo h unc ions a e nondec easing in each. De ine Eas he se o (p,m)∈¯
Esuch ha mis a cons an
unc ion and pis gi en by he equi emen ha p(,xi)=0 o xi+≤0.60 and p(,xi)=1 o xi+>
0.60. This means ha he agen can a emp o g ab any amoun o su plus, bu a emp s o g ab mo e han
$0.60 will be unsuccess ul wi h p obabili y 1—and, hus, ne e some hing he agen will wan o conside —
and a emp s o g ab $0.60 o less will always be success ul. The easoning om he In oduc ion can hen
be applied unchanged o conclude ha he alloca ion in which each playe ecei es hal a dolla is bo h
easies o sus ain and he unique solu ion o he cos -minimiza ion p oblem o a na u al cos unc ion κ.
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Theo e ical Economics 19 (2024) On ba gaining no ms 1457
De ini ion 9. An alloca ion y∈Xdomina es an alloca ion x∈Xi and only i , o any
egula en o cemen echnology se E,i is hecase ha SE(x)⊂SE(y)and he e ex-
is s a egula en o cemen echnology se Esuch ha SE(x)SE(y). An alloca ion is
undomina ed i he e is no alloca ion y∈X ha domina es i .
I an alloca ion ydomina es an alloca ion x, hen o any egula en o cemen ech-
nology se Eand cos unc ion κ,whene e xis a solu ion o he minimiza ion p oblem
(8), so is y.Mo eo e , o anyc∈SE(x), implemen ing (y,c)ins ead o (x,c)would
ha e he ad an age ha i achie es he same cos , bu is mo e obus . Fo ins ance, i
agen s a e occasionally con used abou he en o cemen echnology used, he e would
be an ad an age in using y, because he e a e en o cemen echnology se s E o which
SE(x)SE(y), bu he e is no downside, as SE(x)⊂SE(y)always holds.
Theo em 3. An alloca ion x∈Xis undomina ed i and only i xlies in {x∈X:x1≥
¯
x1and x2≥¯
x2},whe e ¯
xi=supx∈Dixi>0 o i=1, 2 and Dia e de ined as in De ini ion 8.
Le Xundom be he se o alloca ions ha a e undomina ed. Fo Theo em 1 o imply
Theo em 3, one jus needs o show ha Xundom =Xeasies .
Since an alloca ion xlies in Xeasies i and only i i is easies o implemen o some
egula echnology se E, he de ini ion o Xundom immedia ely implies ha Xeasies ⊂
Xundom. Howe e , Xundom ⊂Xeasies does no ollow immedia ely om he de ini ions
o he wo se s and, o ins ance, will no hold in gene al i he de ini ion o a egula
echnology se (De ini ion 2in Sec ion 2.1) is made su icien ly mo e es ic i e. Why?
The he de ini ion o Xundom gua an ees ha i x∈Xundom holds, hen he e is no ysuch
ha (i) o any egula E,i is hecase ha SE(x)⊂SE(y)and (ii) he e exis s a egula
Esuch ha SE(x)SE(y). Howe e , i could po en ially be ha o some x∈Xundom,
he e is a y= xsuch ha o any egula E,i is hecase ha SE(x)=SE(y).I hiswe e
he case, hen such a x∈Xundom would no be an elemen o x∈Xeasies ,asxcan ne e
be easies o implemen i SE(x)=SE(y)always holds. The p oo in he Appendix shows
ha Xundom −Xeasies is emp y by showing ha i an alloca ion xis no in Xeasies , hen
he e exis s an alloca ion y ha domina es x.
5. Discussion and ex ensions
5.1 Asymme ic en o cemen echnologies
In ou analysis we assumed ha playe s a e symme ic in all ways excep in hei u ili y
unc ions ui. This is na u al i we wan o compa e ou esul s wi h symme ic ba gaining
solu ions like he symme ic Nash ba gaining o he Kalai–Smo odinsky solu ion.
O cou se, he e a e also si ua ions whe e i is na u al o conside asymme ic en-
o cemen echnologies. To show how he app oach p oposed in his pape can be ap-
plied o s udy hese si ua ions analogously, conside again he example discussed in he
In oduc ion, whe e wo playe s can p oduce a good wo h $1 wi hou cos s, bu ha e
oppo uni ies o s eal he un inished good be o e p oduc ion is comple e and can sell
i o $0.60 i hey do so. Imagine now ha because he playe s ha e di e en oles in
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1458 Tymon Ta u Theo e ical Economics 19 (2024)
he p oduc ion p ocess o due o di e en skill se s, i is easie o playe 1 o s eal he
un inished good han o playe 2. Speci ically, assume ha while playe 1 can s eal he
un inished good wi hou incu ing any cos s, playe 2 needs o incu a cos o $0.05 o
s eal i . Thus, i he no m speci ies a di ision o su plus x=(x1,x2), and s ealing he
good will esul in a punishmen ha co esponds o mdolla s, playe 1 would p e e o
s eal he good i and only i x1<0.60 −m, and playe 2 would p e e o s eal he good i
and only i x2<0.60 −0.05 −m. Thus, no agen will ha e an incen i e o s eal he good
i and only i
m≥max(0.60 −x1,0.60−0.05 −x2)=0.60 −min(x1,x2+0.5).
Analogously as in he In oduc ion, o any alloca ion x,le S(x)be he se o nonnega-
i e numbe s msuch ha he alloca ion xis sus ainable as a no m in he sense ha no
playe would ha e an incen i e o s eal he good, i.e., he abo e inequali y holds o i=1
and i=2. Then
S(x)=[0.60 −min(x1,x2+0.5),∞).
Clea ly, he alloca ion in which playe 1 ecei es $0.525 and playe 2 ecei es $0.475 is
easies o implemen in he sense ha o any alloca ion ywi h y= x,S(y)S(x)holds.
As in he In oduc ion, i he cos s a socie y incu s a e inc easing in m, his would also
be he alloca ion ha would be he cheapes o sus ain as pa o a no m.
The e o e, in he abo e example, ou model p edic s ha mo e skilled indi iduals
(o o o he easons he playe o whom i is easie o s eal a ac ion o he su plus)
will ecei e a highe sha e o he su plus unde he cos -minimizing no m. O cou se,
we expec his o hold much mo e gene ally.
I is also no di icul o use he p oposed app oach o gene a e some well known
asymme ic ba gaining solu ions like he asymme ic Nash ba gaining solu ion. To see
his, de ine Eas in Sec ion 3.2, bu imagine ha i (p,m)∈Eis used, he no m speci ies
an alloca ion x, and i playe 1 ies o g ab x
1, his a emp will be unsuccess ul wi h
p obabili y
γ1·px
1−x1,x1,
while i playe 2 ies o g ab x
2, his a emp will be unsuccess ul wi h p obabili y
γ2·px
2−x2,x2,
whe e γ1,γ2∈(0, 1)a e cons an s. Gene alizing he ideas om Sec ion 2.1, we can hen
say ha an alloca ion x∈Xcan be sus ained as pa o a social no m i and only i
u1(x1)≥1−γ1·px
1−x1,x1·u1x
1−mx
1−x1,x1+γ1·px
1−x1,x1·u1(0)(9)
holds o all x
1∈[0, 1]and
u2(x2)≥1−γ2·px
2−x2,x2·u2x
2−mx
2−x2,x2+γ2·px
2−x2,x2·u2(0)(10)
holds o all x
2∈[0, 1]. Using exac ly he same a gumen s as in Sec ion 3.2, i is s aigh -
o wa d o e i y ha i we de ine Sγ1,γ2
E(x)as being he se o all alloca ions x∈Xsuch
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Theo e ical Economics 19 (2024) On ba gaining no ms 1459
ha (9)and(10) a e sa is ied, he e will be a unique alloca ion x∗ ha is he easies o
sus ain—in he sense ha Sγ1,γ2
E(x)Sγ1,γ2
E(x∗) o any alloca ion x= x—and ha allo-
ca ion is he asymme ic Nash ba gaining solu ion o weigh s α=γ2
γ1+γ2and β=γ1
γ1+γ2,
meaning he unique alloca ion sol ing
max
x∈Xu1(x1)
γ2
γ1+γ2·u2(x2)
γ1
γ1+γ2.
The abo e wo examples sugges ha , concep ually, i is easy o gene alize he p oposed
app oach o si ua ions in which en o cemen echnologies a e asymme ic.
5.2 O he symme ic en o cemen echnologies
The undamen al goal o his pape is o desc ibe a no el way in which one can hink
abou ou comes in ce ain ba gaining si ua ions. The e is no ques ion ha o speci ic
applica ions, i migh be wo hwhile o conside en o cemen echnologies ha a e di -
e en om hose conside ed he e.
Fo ins ance, in Sec ion 2.1, we assumed ha i an a emp o g ab a sha e x
io
he su plus was unsuccess ul, hen he esul would be disag eemen . Al e na i ely,
one could imagine ha in such a si ua ion no only would coope a ion be pe manen ly
ended, bu he de ia o would also ace some sanc ion such as, o ins ance, incu ing a
cos o K>0. Analogously o Sec ion 2.1, we could hen say ha an alloca ion x∈Xcan
be sus ained as pa o a no m i and only i
uix
i≥1−px
i−xi,xi·uix
i−mx
i−xi,xi+px
i−xi,xi·ui(−K)
holds o all playe s iand all x
i∈[0, 1]. The analysis could hen be ca ied ou along
he same lines as was done in Sec ion 4,andTheo em1could be de i ed o simila ly
de ined se s Diwhe e, howe e , in De ini ion 8, hepayo u(0)would be eplaced wi h
he payo u(−K). O cou se, he esul s would hen be a bi ha de o compa e wi h
s anda d solu ion concep s, as he de i ed solu ion concep would depend on ui(−K),
he disu ili y om a sanc ion ha is no di ec ly ela ed o he “disag eemen payo ”
ha playe s’ ecei e i hey decide no o coope a e.
The echnical eason why his ex ension would be ela i ely s aigh o wa d is ha a
undamen al mono onici y p ope y used in ou analysis would s ill hold. The ele an
p ope y is ha o any en o cemen echnology c, a playe ihas weake incen i es o de-
ia e i he no m p esc ibes o hem a highe a he han lowe sha e o he su plus. (Fo
he amewo k conside ed in his pape , his p ope y is o mally p o en in Lemma 1in
he Appendix.) Fo some o he modi ica ions ha one could conside , o ins ance, i he
u ili y o a playe iwho a emp s o g ab x
iand ails is gi en by ui(−m(x
i−xi,xi)),ad-
di ional assump ions on he u ili y unc ions uia e needed o gua an ee ha he a o e-
men ioned mono onici y p ope y holds.
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1460 Tymon Ta u Theo e ical Economics 19 (2024)
6. Conclusion
In he con ex o wo-playe ba gaining unde symme ic in o ma ion,16 his pape s ud-
ied ba gaining ou comes in socie ies whe e coope a ion is go e ned by social no ms. I
p oposed a new app oach in which ba gaining ou comes a e analyzed based on how
cos ly hey a e o sus ain as pa o a social no m. I showed ha well known ba gain-
ing solu ions like he Nash ba gaining solu ion o he Kalai–Smo odinsky solu ion can
be unde s ood as unique solu ions o he p oblem o choosing an alloca ion ha is he
cheapes o sus ain as a no m o some en o cemen echnologies and cos unc ions.
Mo eo e , all alloca ions wi h he p ope y o being a unique solu ion o some such cos -
minimiza ion p oblem we e cha ac e ized.
The ques ions o whe he and when no ms would o m was no add essed in his
pape . This is an impo an ques ion. Howe e , i an impo an ole o social no ms is
indeed o compensa e o ma ke ailu es, hen add essing his ques ion will mos likely
equi e assump ions ega ding he social cos s o hose ma ke ailu es. In con as , ou
analysis equi ed no such assump ions. Since we only analyzed he incen i es o agen s
o de ia e om a no m, i was i ele an o know how he payo s o o he agen s a e
a ec ed by de ia ions.
Appendix
A.1 P oo o Theo em 1
De ine ¯
x1and ¯
x2as in he s a emen o he heo em. As no ed in he main pa o he
pape , o p o e he equi alence o (i), (ii), and (iii), i is enough o show
(1) Xeasies ⊂{x∈X:x1≥¯
x1and x2≥¯
x2}
(2) {x∈X:x1≥¯
x1and x2≥¯
x2}⊂Xcos -min
(3) Xcos -min ⊂Xeasies .
Unde Rema k 1i is enough o p o e (1), (2), and (3) o he case whe e ui(0)=0 o
i∈{1, 2}. Assume ui(0)=0 o i∈{1, 2}.
Fo any playe i∈{1, 2}and alloca ion x∈X,le Si
E(x)be he se o (p,m)∈Ewi h
he p ope y ha
ui(xi)≥1−px
i−xi,xi·uix
i−mx
i−xi,xi (11)
holds o all alloca ions x∈Xwi h x
i>x
i. On an in ui i e le el, Si
E(x)is exac ly equal
o he se o c∈E o which playe iwould no wan o impose some al e na i e ou come
x∈X. The de ini ion o SE(x)immedia ely implies ha SE(x)=S1
E(x)∩S2
E(x).
The ollowing lemma o malizes he in ui ion ha a playe who ecei es mo e will
ha e smalle incen i es o impose an al e na i e alloca ion.
Lemma 1. Le i∈{1, 2}.I x,y∈Xsa is y xi<y
i, henSi
E(x)⊂Si
E(y).
16In a companion pape , ba gaining be ween mo e playe s is conside ed.
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Theo e ical Economics 19 (2024) On ba gaining no ms 1461
P oo .Le x,y∈Xbe such ha xi<y
i. We need o show ha Si
E(x)⊂Si
E(y). Assume
his is no he case, meaning he e exis s a c=(m,p)∈Ewi h he p ope y ha c∈Si
E(x)
and c/∈Si
E(y).Asc/∈Si(y), he eexis say∈Xwi h y
i>y
isuch ha
ui(yi)<1−py
i−yi,yi·uiy
i−my
i−yi,yi (12)
o , equi alen ly,17
ui(yi)·1
1−py
i−yi,yi<u
iy
i−my
i−yi,yi.
Sub ac ing ui(yi) om bo h sides, we see ha he abo e inequali y (and, he e o e, also
inequali y (12)) is equi alen o
ui(yi)·1
1−py
i−yi,yi−1<u
iy
i−my
i−yi,yi−ui(yi). (13)
Le x∈Xbe gi en by x
i−xi=y
i−yi.18 No e he ollowing s a emen s:
(a) We ha e ui(xi)<u
i(yi),asuiis inc easing and xi<y
i.
(b) We ha e (1
1−p(x
i−xi,xi)−1)≤(1
1−p(y
i−yi,yi)−1),asp(x
i−xi,xi)=p(y
i−yi,xi)≤
p(y
i−yi,yi).19
(c) We ha e ui(x
i−m(x
i−xi,xi)) −ui(xi)≥ui(y
i−m(y
i−yi,yi)) −ui(yi). Indeed,
he ac ha uiis conca e oge he wi h x
i−xi=y
i−yiimplies ui(x
i−m(x
i−
xi,xi))−ui(xi)≥ui(y
i−m(y
i−yi,xi))−ui(yi).Bu ui(y
i−m(y
i−yi,xi))−ui(yi)≥
ui(y
i−m(y
i−yi,yi)) −ui(yi)holds as yi>x
i,mis nondec easing in i s second
a gumen , and uiis inc easing.
Inequali y (13) oge he wi h (a)–(c) imply ha
ui(xi)·1
1−px
i−xi,xi−1<u
ix
i−mx
i−xi,xi−ui(xi)
o , equi alen ly
ui(xi)<1−px
i−xi,xi·uix
i−mx
i−xi,xi
which con adic s c∈Si(x).
The nex lemma cha ac e izes he se s Di o i=1, 2.
Lemma 2. Le i∈{1, 2}. The e exis s a numbe ¯
xi∈(0, 1
2]wi h he p ope y ha ei he
Di={x∈X:xi<¯
xi}
17No e ha since ui(yi)≥0, he las inequali y implies 1 −p(y
i−yi,yi)>0.
18Such xexis s since xi<y
iholds and y
i≤1.
19The ela ionship p(y
i−yi,xi)≤p(y
i−yi,yi) ollows om he ac ha pis nondec easing in i s a gu-
men s and xi<y
i.
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1462 Tymon Ta u Theo e ical Economics 19 (2024)
o
Di={x∈X:xi≤¯
xi}.
In pa icula , he se Diis nonemp y.
P oo .Le i∈{1, 2}. We will o ganize he a gumen in se e al s eps.
S ep 1. No e ha he de ini ion o he se Diimmedia ely implies ha Dicon ains he
alloca ion xwi h xi=0. Since Di⊂Xis nonemp y, we can de ine ¯
xiby ¯
xi=supx∈Dixi.
S ep 2. No e ha i x∈Di, hen also x∈Di o any x∈Xwi h x
i<x
i. To see ha his
is he case, conside he de ini ion o he se Diand no e ha , o q∈(0, 1)and ∈[0, 1],
ui(xi)<q·ui(xi+)+(1−q)·ui(0)
is equi alen o
ui(xi)−ui(0)
ui(xi+)−ui(xi)<q
1−q
and, simila ly,
uj(xj)≤q·uj(xj+)+(1−q)·uj(0)
is equi alen o
uj(xj)−uj(0)
uj(xj+)−uj(xj)≤q
1−q.
Ou claim now ollows immedia ely om he obse a ion ha since he u ili y unc ions
o bo h playe s a e inc easing and conca e, o k∈{1, 2},uk(xk+)−uk(xk)is nonin-
c easing in xkand uk(xk)−uk(0)is inc easing in xk.
S eps 1 and 2 oge he imply ha o ¯
xi=supx∈Dixi,ei he Di={x∈X:xi<¯
xi}o
Di={x∈X:xi≤¯
xi}and ¯
xi≥0. All ha emains o be shown is ha 0 <¯
xi≤1
2.
S ep 3. To show ha ¯
xi≤1
2, assume ha i is no he case and le x∈Dibe an alloca-
ion wi h xi>1
2.Le j∈{1, 2}wi h j= i. No e ha o =xiand q=uj(xj)/uj(xj+),
we ha e xj+=1≤1anduj(xj)=q·uj(xj+). Since x∈Di, his implies ha
xi+=2·xi≤1. Howe e , 2 ·xi≤1 con adic s xi>1
2.
S ep 4. To show ha ¯
xi>0, no e ha o any alloca ion xsuch ha 0 <x
i<1
2and
ui(2·xi)−ui(0)
u
i(1)<
uj1
2−uj(0)
u
i(0), (14)
i will be he case ha x∈Di.
To see his, assume xis an alloca ion sa is ying he abo e condi ions and ha
xj+≤1anduj(xj)≤q·uj(xj+)+(1−q)·uj(0)(15)
holds o some q∈(0, 1)and ∈(0, 1).Thenxj=1−xi oge he wi h he le inequali y
in (15)andxi<1
2implies
≤xiand xi+≤1. (16)
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Theo e ical Economics 19 (2024) On ba gaining no ms 1463
Now no e ha he igh inequali y in (15) is equi alen o
1−q≤uj(xj+)−uj(xj)
uj(xj+)−uj(0). (17)
Using he conca i y o u1and u2,xi<1
2,(14), and (16)weob ain
uj(xj+)−uj(xj)
uj(xj+)−uj(0)≤·u
j(0)
uj1
2−uj(0)
<·u
i(1)
ui(2·xi)−ui(0)≤ui(xi+)−ui(xi)
ui(xi+)−ui(0)(18)
Combining (16), (17), and (18), we ob ain
xi+≤1andui(xi)<q·ui(xi+)+(1−q)·ui(0). (19)
This p o es ha o any alloca ion xsuch ha 0 <x
i<1
2and (14)holds,i willbe he
case ha x∈Di.
The nex lemma ela es he se s Di o S1
Eand S2
E. This lemma is he key obse a ion
in he p oo o s a emen (1).
Lemma 3. Fix an en o cemen echnology se E.Le i,j∈{1, 2}wi h i= j. Fo any x∈Di,
i is he case ha
Si
E(x)⊂Sj
E(x).
P oo . We will p o e he s a emen in he lemma o he case whe e i=1andj=2. The
a gumen o he case whe e j=1andi=2 is analogous. To show ha o any x∈D1i
is hecase ha
S1
E(x)⊂S2
E(x),
i is enough o show ha (p,m)/∈S2
E(x)implies (p,m)/∈S1
E(x). Assume, he e o e,
(p,m)/∈S2
E(x).
Since (p,m)/∈S2
E(x), he emus exis ax
2∈(x2,1
]such ha
u2(x2)<1−px
2−x2,x2·u2x
2−mx
2−x2,x2. (20)
No e ha (20) implies ha x
2−m(x
2−x2,x2)>x
2.Se =x
2−m(x
2−x2,x2)−x2and
q=1−p(x
2−x2,x2).No e ha x
2−m(x
2−x2,x2)≤1impliesx2+≤1. We can now
ew i e (20)as
u2(x2)<q·u2(x2+).
Since x∈D1, he las inequali y oge he wi h x2+≤1 implies ha x1+≤1and
u1(x1)<q·u1(x1+). (21)
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1464 Tymon Ta u Theo e ical Economics 19 (2024)
Le x be he alloca ion cha ac e ized by x
1−x1=x
2−x2. No e ha such an al-
loca ion does indeed exis , as x1≤x2 ollows om Lemma 2gi en ha x∈D1.No e
ha x
1−x1=x
2−x2 oge he wi h x1≤x2implies ha =x
2−m(x
2−x2,x2)−x2=
x
1−m(x
1−x1,x2)−x1≤x
1−m(x
1−x1,x1)−x1and q=1−p(x
2−x2,x2)=1−p(x
1−
x1,x2)≤1−p(x
1−x1,x1). Thus, inequali y (21)implies
u1(x1)<1−px
1−x1,x1·u1x
1−mx
1−x1,x1.
Since x
1−x1=x
2−x2>0andx
1=x1+x
2−x2≤x
2≤1, his p o es ha (p,m)/∈S1
E(x),
whichiswha wewan ed oshow.
Lemma 4. Le x,y∈Diwi h xi<y
i,whe ei∈{1, 2}.ThenSE(x)⊂SE(y)holds o any
en o cemen echnology se E.
P oo . We will p o e he lemma o he case whe e i=1. The a gumen o he case
i=2 is analogous.
Acco ding o Lemma 3,
S1
E(y)⊂S2
E(y)
and
S1
E(x)⊂S2
E(x).
No e ha acco ding o Lemma 1,S1
E(x)⊂S1
E(y)and S2
E(y)⊂S2
E(x).Thus,
SE(x)=S1
E(x)∩S2
E(x)=S1
E(x)⊂S1
E(y)=S1
E(y)∩S2
E(y)=SE(y).
This p o es ha SE(x)⊂SE(y).
The nex lemma cha ac e izes he elemen s o he se {x∈X:x1≥¯
x1and x2≥¯
x2}.
I will be used o p o e ha s a emen (2) holds.
Lemma 5. Assume he u ili y unc ions a e no malized such ha u1(0)=u2(0)=0.20 Fo
any x∗∈{x∈X:x1≥¯
x1and x2≥¯
x2}, a leas one o he ollowing s a emen s is ue:
(a) The a iable x∗sa is ies x∗
1=x∗
2=1
2.
(b) The e exis i,j∈{1, 2}such ha x∗
i<x
∗
jand
uix∗
i
uix∗
i+≥ujx∗
j
ujx∗
j+(22)
o some ∈(0, 1 −x∗
j].
(c) The a iable x∗is he symme ic Nash ba gaining solu ion.
20I his we e no he case, he o mula in s a emen (b) would need o be adjus ed.
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Theo e ical Economics 19 (2024) On ba gaining no ms 1465
P oo .Le x∗∈{x∈X:x1≥¯
x1and x2≥¯
x2}. Clea ly, he s a emen o he lemma is
ue i x∗=(1
2,1
2). We will p o e he lemma o he case whe e x∗
1<x
∗
2. The a gumen
o he case whe e x∗
2<x
∗
1is analogous.
As x∗∈{x∈X:x1≥¯
x1and x2≥¯
x2}implies ha x∗
1≥¯
x1,i mus be ha ei he x∗
1>
¯
x1o x∗
1=¯
x1.
Conside i s he case whe e x∗
1>¯
x1. No e ha in his case, by Lemma 2,x∗/∈D1.
Howe e , x∗/∈D1implies he e mus exis q∈(0, 1)and ∈(0, 1)so ha he s a emen
x∗
2+≤1andu2x∗
2≤q·u2x∗
2+(23)
holds, bu he s a emen
x∗
1+≤1andu1x∗
1<q·u1x∗
1+(24)
does no . No e now ha since we assumed x∗
1<x
∗
2, he inequali y x∗
2+≤1 om(23)
implies x∗
1+≤1 om(24). Thus, i (24) does no hold, i mus be ha
u1x∗
1≥q·u1x∗
1+.
Combining he las inequali y wi h he second inequali y om (23), we ob ain
u1x∗
1
u1x∗
1+≥u2x∗
2
u2x∗
2+.
Thus, in his case, x∗sa is ies condi ion (b) in he s a emen o he lemma.
All ha emains is o p o e he lemma o he case whe e x∗
1=¯
x1.To hisend,le xn
be a sequence o alloca ions such ha 1
2>x
n
1>¯
x1and xn→¯
x1. No e ha since xn
1>¯
x1,
he same easoning ha yielded (22) o x∗wi h 1
2>x
∗
1>¯
x1will yield ha o each n,
he e exis s n∈(0, xn
1]such ha
u1xn
1
u1xn
1+n≥u2xn
2
u2xn
2+n. (25)
Gi en ha n∈[0, 1] o all nand [0, 1]is compac , he e exis s a con e gen subse-
quence nk.Le =limk→∞ nk∈[0, ¯
x1].
I >0, inequali ies (25) oge he wi h he ac ha u1and u2a e con inuous implies
ha , in he limi ,
u1x∗
1
u1x∗
1+≥u2x∗
2
u2x∗
2+.
Thus, in his case, x∗sa is ies condi ion (b) in he s a emen o he lemma.
I =0, hen (25)implies
u1x∗
1
u
1x∗
1≥u2x∗
2
u
2x∗
2.
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1466 Tymon Ta u Theo e ical Economics 19 (2024)
No e ha i he las inequali y is binding, hen x∗sa is ies condi ion (c) in he s a e-
men o he lemma.21 On he o he hand, i he las inequali y is s ic , hen o all su i-
cien ly small posi i e h,i willbe hecase ha
u1x∗
1
u1x∗
1+h≥u2x∗
2
u2x∗
2+h.
Thus, in his case, x∗sa is ies condi ion (b) in he s a emen o he lemma.
We a e now eady o p o e Theo em 1by showing ha (1), (2), and (3) s a ed a he
beginning o his sec ion hold.
P oo o S a emen (1). To show ha Xeasies ⊂{x∈X:x1≥¯
x1and x2≥¯
x2}, assume
he e is an alloca ion x∗∈Xeasies such ha x∗/∈{x∈X:x1≥¯
x1and x2≥¯
x2}.
I x∗/∈{x∈X:x1≥¯
x1and x2≥¯
x2}, henx∗
i<¯
xiholds o some playe i.Le y∈Xbe
an alloca ion such ha x∗
i<y
i<¯
xi. Unde Lemma 4, o any en o cemen echnology
se E,i mus be ha
SEx∗⊂SE(y). (26)
Howe e , x∗∈Xeasies implies ha he e is an en o cemen echnology se Esuch ha x∗
is he easies o sus ain, meaning SE(y)SE(x∗)holds o any y= x∗. This con adic s
(26).
P oo o S a emen (2). We will now show ha o any x∗∈{x∈X:x1≥¯
x1and x2≥
¯
x2}, he e exis s an en o cemen echnology se Eand a nondec easing cos unc ion κ
such ha x∗is he unique solu ion o he cos -minimiza ion p oblem o ha en o ce-
men echnology se and cos unc ion.
Recall ha , in keeping wi h Rema k 1, we can, wi hou loss o gene ali y, es ic
a en ion o he case whe e ui(0)=0 o i∈{1, 2}.
Le x∗∈{x∈X:x1≥¯
x1and x2≥¯
x2}.No e ha x∗mus hen sa is y (a), (b), o (c) in
Lemma 5. We will conside he h ee cases sepa a ely.
S ep 1. Conside i s he case whe e x∗=(1
2,1
2), i.e., condi ion (a) in Lemma 5is
sa is ied. Fo ε∈(0, 1
2), de ine pε:[0, 1]2→[0, 1]by
pε(h,xi)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1 o h∈[0, 1],xi∈1
2,1

ε+xi−1
2
ε o h∈[0, 1],xi∈[1
2−ε,1
2)
0 o h∈[0, 1],xi∈[0, 1
2−ε).
21The unique alloca ion sa is ying
u1x∗
1
u
1x∗
1=u21−x∗
1
u
21−x∗
1
is he Nash ba gaining solu ion, as he abo e equa ion is he i s -o de condi ion o he p oblem
maxx1∈[0,1]u1(x1)·u2(1−x1).
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