No Prices No Games!: Four Economic Models

Rich e , Michael; Rubins ein, A iel
Book
No P ices No Games!: Fou Economic Models
P o ided in Coope a ion wi h:
Open Book Publishe s
Sugges ed Ci a ion: Rich e , Michael; Rubins ein, A iel (2024) : No P ices No Games!: Fou Economic
Models, ISBN 978-1-80511-310-2, Open Book Publishe s, Camb idge,
h ps://doi.o g/10.11647/OBP.0404
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/305343
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
NO PRICES NO GAMES!
FOUR ECONOMIC MODELS
NO PRICES NO GAMES!
FOUR ECONOMIC MODELS
Michael Rich e
Ba uch College
Royal Holloway, Uni e si y o London
A iel Rubins ein
Tel A i Uni e si y
New Yo k Uni e si y

c2024 Michael Rich e and A iel Rubins ein
This wo k is licensed unde a C ea i e Commons A ibu ion-NonComme cial-NoDe i s
license (CC BY-NC-ND 4.0). This license allows you o sha e, copy, dis ibu e, and
ansmi he wo k p o iding you do no modi y he wo k, you do no use he wo k o
comme cial pu poses, you a ibu e he wo k o he au ho s, and you p o ide a link o
he license. A ibu ion should no in any way sugges ha he au ho s endo se you o
you use o he wo k and should include he ollowing in o ma ion.
Michael Rich e and A iel Rubins ein, No P ices No Games!. Camb idge, UK: Open
Book Publishe s, 2024, h ps://doi.o g/10.11647/OBP.0404
Fu he de ails abou CC BY-NC-ND licenses a e a ailable a h ps://c ea i ecommons.
o g/licenses/by-nc-nd/4.0/. All ex e nal links we e ac i e a he ime o publica-
ion unless o he wise s a ed and ha e been a chi ed ia he In e ne A chi e Wayback
Machine a h ps://a chi e.o g/web.
Digi al ma e ial and esou ces associa ed wi h his olume a e a ailable a h ps://
doi.o g/10.11647/OBP.0404# esou ces.
ISBN Pape back: 978-1-80511-308-9
ISBN Ha dback: 978-1-80511-309-6
ISBN Digi al (PDF): 978-1-80511-310-2
DOI: 10.11647/OBP.0404
Co e image: A iel Rubins ein
Co e concep : Michael Rich e and A iel Rubins ein
Co e design: Jee anjo Kau Nagpal
Con en s
Pe sonal No e ii
No a ion and Te minology ix
0 In oduc ion 1
0.1 The Book 1
0.2 The No ion o an Economy 3
0.3 Examples o Economies 4
0.4 Equilib ium Concep s 8
1 Equilib ium in he Jungle 13
1.1 The Housing Jungle: Model and Equilib ium 15
1.2 The Jungle Equilib ium: Wel a e 18
1.3 Compa ison o he Compe i i e Equilib ium 21
1.4 Commen s on he Jungle Equilib ium 25
1.5 The Di ision Jungle 28
1.6 The Di ision Jungle: Commen s on Wel a e 33
1.7 A Didac ic Pe spec i e 34
2 The Pe missible and he Fo bidden 37
2.1 The Y-Equilib ium Concep 39
2.2 Y-Equilib ium, Pa e o Op imali y, and En y-F eeness 43
2.3 Euclidean Economies 45
2.4 The “Koshe ” Economy 46
2.5 Con ex Y-Equilib ium 49
2.6 Pa e o Op imali y and Exis ence o Con ex Y-Equilib ium 51
2.7 A S uc u e Theo em o Con ex Y-equilib ium 53
2.8 The Di ision Economy 55
2.9 The Gi e-and-Take Economy 59
2.10 The S ay Close Economy 61
i Con en s
3 S a us and Indoc ina ion 65
3.1 S a us Equilib ium 67
3.2 S a us Equilib ium – Examples 68
3.3 A De ou : Con ex P e e ences 71
3.4 P imi i e Equilib ium 76
3.5 A Fi s Wel a e Theo em 79
3.6 A Second Wel a e Theo em 81
3.7 P imi i e Equilib ium – Examples 83
3.8 Ini ial S a us Equilib ium 85
4 Biased P e e ences Equilib ium 91
4.1 The Economy and he Equilib ium Concep 92
4.2 The Gi e-and-Take Economy 98
4.3 The Fixed-P ices Exchange Economy 100
4.4 Housing-Type Economies 104
5 A Compa ison o Game Theo y 111
5.1 The Ma ching Economy 112
5.2 The Jungle Equilib ium 115
5.3 Res ic ing Pa ne ships: Pai wise Y-equilib ium 120
5.4 P es ige by Pa ne : S a us Equilib ium 122
5.5 P es ige by Sel : Ini ial S a us Equilib ium 124
5.6 A Compa ison o App oaches 127
5.7 The Majo i y Vo ing Economy 127
5.8 Con ex Y-equilib ium 128
5.9 Biased P e e ences Equilib ium 130
5.10 The Majo i y Vo ing Game and Nash Equilib ium 131
5.11 Compa ing ou App oaches wi h Nash Equilib ium 133
Re e ences 134
Pe sonal No e
We eel some dissa is ac ion wi h cu en ends in Economic Theo y. No el y
has allen by he wayside. Models ha e become o e ly complica ed and
excessi ely sophis ica ed ma hema ically. Pape s a e oo long and con ain ew
new undamen al ideas. Au ho s go o g ea leng hs o masque ade heo e ical
wo k as being applied.
This book con ains a collec ion o models in Economic Theo y ha a e
simple in hei app oach and s aigh o wa d ma hema ically. They include
new concep s and a e p esen ed concisely, wi hou any p e end claims
ega ding hei di ec applied use ulness. A bes , he models ha e helped us o
unde s and a ious social ins i u ions, such as powe , s a us, social no ms, and
p e e ence biases as a means o achie e ha mony in economic en i onmen s.
Needless o say, we do no ad oca e o he adop ion o any o hese ins i u ions
bu , a he , in es iga e hei a ionales. Ou main objec i e is o dis up he
con en ion ha e e y economic model should be ei he a ma ke wi h p ices
o a s a egic game.
This p ojec began jus o e a decade ago and de eloped om a se ies o
pape s, mos o which we w o e join ly. The book b ings he pape s oge he in
a uni ied language and an accessible s yle. I can be used o each a uni in an
ad anced Economic Theo y cou se o as a sou ce o independen s udy.
We wish o acknowledge gene ous assis ance om wo ou s anding indi id-
uals: Ma in Osbo ne, who was kind, as always, and sha ed wi h us he o ma
o he book which he o iginally designed (so, i you a e happy wi h he o ma ,
you should hank him) and Á on Tóbiás, who con ibu ed so much o his ime
o ca e ully e iewing a d a o he book and sa ed us (and you) om a la ge
numbe o e o s. We a e also g a e ul o Tu al Danenbe g o his commen s.
MR: I am g a e ul o he suppo o my wi e, Emel Yildi im-Rich e , wi hou
whom his p ojec would ne e ha e been comple ed.
URLs: h ps://m ich e .co and h ps://a iel ubins ein. au.ac.il
4 Chap e 0. In oduc ion
•F⊂XNis a non-emp y se o easible p o iles.
A choice p o ile (xi)i∈Nspeci ies an elemen xi∈X o each agen
i∈N. The se XNis comp ised o all choice p o iles. No all p o iles a e
easible, and he easibili y cons ain is gi en by a se F⊂XN. Unless
s a ed o he wise, we assume ha Fis closed unde all pe mu a ions
(i.e. he easibili y cons ain is anonymous and does no disc imina e
be ween agen s). We usually abb e ia e (xi)i∈Nas (xi).
An economy wi hou p e e ences, ‹N,X,F›, is called an en i onmen .
Some imes, we conside an ex ended e sion o an economy which
speci ies o each agen ian elemen eiin X, wi h he in e p e a ion ha i
always has he igh o choose ei. The ec o (ei)is equi ed o be in F, namely
he alloca ion o hese ini ial igh s is easible. The ole o he ec o (ei)is
analogous o ha o he p o ile o ini ial endowmen s in he s anda d exchange
economy.
De ini ion: Ex ended Economy
An ex ended economy is a uple ‹N,X,(%i)i∈N,F,(ei)i∈N›whe e:
• ‹N,X,(%i)i∈N,F›is an economy.
•(ei)i∈Nis a easible ini ial p o ile.
0.3 Examples o Economies
We now in oduce some economies which appea h oughou he book. As
men ioned, some o he examples a e adi ional economic se ings while
o he s demons a e he amewo k’s abili y o model a a ie y o al e na i e
social si ua ions.

0.3 Examples o Economies 5
Example: The Housing Economy
The se Xcon ains ndis inc elemen s called houses ( ecall ha nis he
numbe o agen s) and each agen ihas p e e ences %io e he houses.
Each agen chooses a house, bu no wo agen s can occupy he same
one. Tha is, Fis he se o p o iles ha assigns a dis inc house o e e y
agen . This economy is he iconic model o Shapley and Sca (1974). The
model is a ac i e due o i s simplici y and i s use ulness as a pla o m
o in oducing a ich a ie y o concep s.
I each agen ’s ideal is dis inc , hen he si ua ion is “bliss”, he e a e
no con lic ing desi es, and so he e is no need o a social ins i u ion o
achie e ha mony in he socie y. Howe e , bliss does no usually exis ,
and, he e o e, we need social ins i u ions o esol e he con lic be ween
agen s’ desi es and socie al easibili y.
Example: The Di ision Economy
The e a e Kcommodi ies, and he se o al e na i es X=RK
+consis s
o he non-nega i e bundles o hose commodi ies. P e e ence ela ions
a e mono onic, con inuous, and con ex. As in s anda d ma ke
se ings, he e a e limi ed esou ces, and he se o easible p o iles
F={(xi)|Σixi=e}is he se o all pa i ions o a o al endowmen
e∈RK
+among he agen s. I we would add ini ial endowmen s o
he model, hen we would ob ain he classical amewo k used by
economis s since Edgewo h (1881) o discuss olun a y exchange and
compe i i e equilib ium. Bliss is always impossible, unlimi ed wan s
mus be cons ained in he ace o limi ed esou ces, and achie ing social
ha mony equi es some social ins i u ion.
6 Chap e 0. In oduc ion
Example: The Gi e-and-Take Economy
The e a e si ua ions in li e in which edis ibu ion is imposed by an
au ho i y ha o ces indi iduals o comply, and he e a e o he s in which
edis ibu ion is accomplished by means o olun a y exchange be ween
indi iduals. The e a e u he si ua ions (e.g. a soup ki chen) in which
exchange is ca ied ou by unila e al ac ions: some indi iduals gi e while
o he s ake wi hou any exe cise o powe , commi men s o “ e u n he
a ou ”, o coe cion by an au ho i y. These ac ions a e sel -mo i a ed:
some people like o gi e, while o he s like o ake. Bu ypically, such
mo i es will no balance each o he ou , and social no ms a e needed o
achie e ha mony.
Fo mally, we conside he ollowing gi e-and- ake economy, which
was i s s udied by Sp umon (1991). Le X= [−1,1], whe e a posi i e
x ep esen s a wi hd awal o x om a social und (i.e. aking) and a
nega i e x ep esen s a con ibu ion o |x| o he social und (i.e. gi ing).
P e e ences a e assumed o be con inuous and s ic ly con ex ( ha is,
single-peaked) bu need no be mono onic. Feasibili y equi es ha he
social und is balanced, ha is, F={(xi)|Σixi=0}.
Example: The Clubs Economy
The se Xconsis s o a ini e se o clubs (see Buchanan (1965)). Each
agen chooses a single club o become a membe o . Agen s ha e
p e e ences o e he clubs and no o e he clubs’ membe s. The
easibili y cons ain is de ined by he limi s on how many people can
belong o each club. Speci ically, he e is a ec o o posi i e in ege s
(qx)x∈Xwhe e qxis he quo a o club x( o non- i iali y, we equi e ha
he sum o he quo as is a leas n). The se o easible p o iles a e hose
o which no club is chosen by mo e people han allowed by i s capaci y.
0.3 Examples o Economies 7
Example: The S ay Close Economy
This example illus a es he po en ial o ou abs ac concep o expand
he scope o classical economic analysis. I does no in ol e goods
bu none heless i s squa ely in o ou concep o an economy. In his
example, Xis a se o loca ions in some geog aphical a ea. Each agen
chooses a loca ion in Xand has p e e ences o e he loca ions. No e e y
p o ile o loca ions is easible because he socie y is unde h ea and i s
su i al depends upon he abili y o i s membe s o quickly each one
ano he in he case o dange . The e o e, all membe s need o li e close
enough o each o he so ha whene e one o hem is a acked he o he s
can quickly come o his de ence. Fo mally, he easibili y cons ain
F equi es ha he dis ance be ween any wo agen s does no exceed
some cons an d. When dis e y la ge, e e y agen can choose his ideal
loca ion, bu when dis small, his is no longe easible.
We e e o he special case when d=0 as he consensus economy.
This i s, o example, he si ua ion o a poli ical pa y whose membe s
need o p esen a uni ed on . Tha is, in o de o main ain cohesion, all
membe s o he pa y need o exp ess he same posi ion.
Example: The Ma ching Economy
Ma ching p oblems a e classics o Coope a i e Game Theo y. Agen s
ha e o ind a ma ch, and each agen has a p e e ence ela ion o e his
po en ial pa ne s. This si ua ion i s ou amewo k by le ing he se
o al e na i es Xbe he se o agen s N. Tha is, each agen chooses
a pa ne , which can be himsel . Each has a p e e ence ela ion on X
ha places himsel a he bo om. The easibili y cons ain Fs ipula es
ha o any iand j, i ichooses j, hen jmus choose i. No e ha his
easibili y cons ain di e s om hose in he p e ious examples in ha
Fis no closed unde all pe mu a ions.
8 Chap e 0. In oduc ion
Example: The Sequen ial P oduc ion Economy
A g oup o nagen s wo ks in nshi s o ans o m an ini ial p oduc x0
in o a di e en p oduc . Each wo ks one shi , and he agen s may wo k
in any o de . An agen ’s abili y o p oduce a p oduc , which migh be jus
an in e media e p oduc , depends on he ou pu o he p e ious shi .
The g oup possesses a echnology ha enables ce ain ans o ma ions
o one p oduc in o ano he .
Mo e p ecisely, Xis a se o p oduc s ha includes x0. Each agen has
p e e ences o he p oduc ha he p oduces ( a he han o he inal
p oduc ). The common p oduc ion echnology is a co espondence T
om X o Xwhe e T(x)is he se o ou pu s which xcan be ans o med
in o. Any agen can choose o be “idle” and no ans o m he p oduc
p oduced in he p e ious shi , ha is x∈T(x). Thus, Fis he se
o all pe mu a ions o p o iles (x1,...,xn)such ha xm∈T(xm−1) o
m=1,...,n.
0.4 Equilib ium Concep s
This book in oduces and analyzes se e al solu ion concep s and applies
hem o a a ie y o economic en i onmen s. In gene al, a solu ion concep
ela es o some domain o economic en i onmen s and de e mines o each
en i onmen a se o ha monious ou comes. These ou comes a e ha monious
in he sense ha he assumed o ces ha may dis u b ha mony a e neu alized.
In ou se ing, he domain o a solu ion concep is a class o economies and a
candida e o equilib ium ypically includes wo componen s:
(i) A p o ile o choices — one choice o each agen .
(ii) A speci ica ion o ce ain pa ame e s ha sys ema ically in luence
ei he agen s’ choice p oblems o hei p e e ence ela ions.
Ha mony is achie ed in equilib ium as ollows: agen s make indi idually
op imal choices, and he pa ame e s es ic hei choice se s (o , in one case,
0.4 Equilib ium Concep s 9
biases hei p e e ences) o be compa ible in he sense ha he esul ing p o ile
o choices is easible. The concep s will di e in he pa ame e s and in how
hey es ic agen s’ choice se s.
The solu ion concep s discussed in he book can be di ided in o wo
g oups. In he choice g oup, each agen ’s choice se depends on a p ice-like
equilib ium pa ame e bu no on he equilib ium p o ile o choices. Such
choices mus be indi idually op imal and compa ible. These concep s a e
simila in s uc u e o he no ion o compe i i e equilib ium whose pa ame e s
a e p ices and each agen ’s choice se (budge se ) is de e mined solely by his
ini ial endowmen and he p ices.
Th ee o ou solu ion concep s belong o his g oup:
Y-equilib ium (Chap e 2). The p ice-like pa ame e in a Y-equilib ium is a
se o al e na i es which is in e p e ed as he se o “pe missible” al e na i es
ha uni o mly binds all agen s. When making a choice, an agen only needs
o know he se o pe missible al e na i es and no hing else. In equilib ium,
he pe missible se is a maximal se o al e na i es om among hose which
sa is y he ollowing p ope y: i e e y agen chooses a p e e ence-maximizing
al e na i e om his se , hen he esul ing choice p o ile is easible.
Ini ial S a us Equilib ium (Chap e 3). This concep ela es o an ex ended
economy whe ein he no ion o an economy is en iched wi h an addi ional
elemen : a easible p o ile o al e na i es, one o each agen , in which he
al e na i e designa ed o an agen is in e p e ed as one ha he always has he
igh o choose. The p ice-like pa ame e in an ini ial s a us equilib ium is an
o de ing o he al e na i es ha can be in e p e ed as “s a us” o “ alue”. An
agen ’s choice se is comp ised o all al e na i es which ha e a weakly lowe
s a us han his endowmen . In equilib ium, a s a us o de ing p e ails such
ha each agen ’s designa ed al e na i e is his mos p e e ed om among his
choice se , namely he se o all al e na i es ha a e o weakly lowe s a us han
his ini ial al e na i e. As always, an equilib ium p o ile o choices has o be
easible.

10 Chap e 0. In oduc ion
Biased P e e ences Equilib ium (Chap e 4). The p ice-like pa ame e in a
biased p e e ences equilib ium is a ec o ha sys ema ically biases agen s’
p e e ences. In his model, agen s’ choice se s a e ixed and una ec ed by he
pa ame e s. Ra he , in an equilib ium, a sys ema ic bias p e ails such ha each
agen chooses a mos -p e e ed al e na i e om his choice se , acco ding o his
biased p e e ences, and he p o ile o choices is easible.
In he de ia ion g oup o solu ion concep s, an equilib ium is a p o ile o
choices ha is immune o any single agen ’s de ia ion om his p esc ibed al-
e na i e o any al e na i e in a se de e mined by he equilib ium pa ame e s.
This is he app oach aken in Game Theo y. Fo example, a Nash equilib ium is
a p o ile o ac ions such ha , o each agen , he ou come o ha p o ile is no
wo se o him han any o he ou come he can achie e gi en he o he playe s’
choices in he p o ile.
Two o ou solu ion concep s all in o his g oup:
Jungle Equilib ium (Chap e 1). In his case, he economy is ex ended wi h
an exogenous powe anking o he agen s; bu , in an equilib ium, he e a e
no addi ional pa ame e s. In he jungle, an agen can s eal om hose ha a e
weake han himsel ; he e o e, his choice se is de e mined by his equilib ium
choice as well as he choices o hose who a e weake han him. A jungle
equilib ium is a p o ile o choices such ha each agen ’s assigned choice is
p e e ence-maximal om among he se o he al e na i es he can ob ain by
s ealing esou ces om weake agen s.
S a us Equilib ium (Chap e 3). Again, he p ice-like equilib ium pa ame e is
an o de ing o e he al e na i es ha conno es s a us (o alue). Howe e , in
his case, an agen ’s choice se depends no only on his pa ame e bu also
on his own equilib ium al e na i e. In de ail, his choice se is he se o all
al e na i es which a e weakly lowe - anked han his equilib ium al e na i e
( a he han his ini ial al e na i e). An equilib ium is a s a us o de ing and a
p o ile o op imal choices such ha he p o ile o choices is easible.
0.4 Equilib ium Concep s 11
The book analyzes each o hese solu ion concep s bo h in he abs ac , by
means o gene al p oposi ions, and mo e conc e ely, by applying he solu ion
concep s o a a ie y o economic en i onmen s (some amilia and some
no el).
1Equilib ium in he Jungle
The s anda d economic app oach ea s economic ac i i y as olun a y: all
in ol ed pa ies a e doing wha e e hey do o hei own ee will. When
analyzed using he compe i i e equilib ium app oach, economic agen s
ope a e wi hin bounds se by a p ice sys em ha hey ake as gi en, bu hei
decisions a e ee — no one o ces hem o ac . When analyzed using he game-
heo e ical app oach, agen s beha e s a egically, and in equilib ium hey bes
espond o co ec p edic ions abou he o he agen s’ beha io , and, again, no
one can o ce anyone o ake a pa icula ac ion.
Howe e , li e is no jus a se ies o olun a y ac ions. An agen (o a g oup
o agen s) migh use powe o seize asse s om o he s o o o ce o he s o
do hings agains hei will. Resou ces a e o en ans e ed om one agen
o ano he based on he exe cise o powe , a he han due o he sa is ac ion o
mu ual wan s. While an agen can use powe o o ce ano he o beha e agains
his bes in e es s, he e is o en no need o ac ually use powe since he me e
h ea o doing so can be su icien o pe suade a weake agen o gi e in.
Economic Theo y ypically igno es he use o powe as a d i e o social
ac i i y. In he wo ds o Hi shlei e (1994) (see also Bowles and Gin is (1992)
and G ossman (1995) who exp ess simila sen imen s):
... he mainline Ma shallian adi ion has ... almos en i ely
o e looked wha I will call he da k side o he o ce — o wi ,
c ime, wa , and poli ics. ... App op ia ing, g abbing, con isca -
ing wha you wan — and, on he lip side, de ending, p o ec ing,
seques e ing wha you al eady ha e — ha ’s economic ac i i y
oo.
As he i le o he book p omises, we conside economic in e ac ions ha
a e ha monized wi hou he eme gence o a p ice sys em o he use o s a egic
©2024 Michael Rich e and A iel Rubins ein, CC BY-NC-ND 4.0
h ps://doi.o g/10.11647/OBP.0404.01
20 Chap e 1. Equilib ium in he Jungle
and de ine ik+1 o be he agen who holds ik’s a ou i e house (ik+16=ik
because no agen ’s a ou i e house is his cu en house). Since Nis ini e,
he e will e en ually be some lsuch ha k>l≥0 and ik+1=il. Then,
assigning yij=xij+1 o each l≤j≤k, and keeping yj=xj o all o he
agen s, we ob ain a easible alloca ion (yi)which Pa e o-domina es (xi).
We cons uc a powe ela ion Bas ollows: Le i1be an agen o
whom xi1is his i s -bes house and make him he mos powe ul agen .
Now emo e i1 om he se o indi iduals and xi1 om he se o houses.
The induc i e p ocess con inues as ollows: a he beginning o he
k+1s s age, kagen s ha e been assigned powe . The alloca ion o
he emaining houses among he emaining agen s is Pa e o op imal;
he e o e, iden i y an agen ik+1 o whom xik+1is his a ou i e house
om among X−{xi1,...,xik}and make him he (k+1)s -mos powe ul
indi idual.
By cons uc ion, o each agen i, he house xiis p e e ed by io e
e e y house ha is alloca ed o an indi idual weake han him acco ding
o B. Thus, (xi)is a jungle equilib ium o ‹N,X,(%i)i∈N,F,B›.
Ex e nali ies: To inco po a e ex e nali ies, we modi y he model by de ining
he agen s’ p e e ences o e he se o easible p o iles ( a he han he se o
houses) and by allowing indi e ences. The de ini ion o a jungle equilib ium
also needs o be modi ied. When deciding whe he o con isca e a house, an
agen compa es he cu en p o ile o he one ha would esul i he does so.
One way o p oceed is by in e p e ing iBj o mean ha agen ican o ce
j o exchange houses: i akes o e he house occupied by j and o ces j o
accep he house ip e iously occupied. Thus, an equilib ium o he jungle
wi h ex e nali ies ‹N,X,(%i)i∈N,F,B›is a easible p o ile (ai)such ha o no
wo agen s j,j0∈Nis i he case ha jBj0and (bi)j(ai), whe e (bi)is he
alloca ion ha di e s om (ai)only in he ac ha bj=aj0and bj0=aj.

1.3 Compa ison o he Compe i i e Equilib ium 21
In he model wi h ex e nali ies, a jungle equilib ium does no necessa ily
exis . Fo example, conside a case wi h 3 agen s whe e 1 B2B3 and
X={a,b,c}. Think o he houses as being loca ed clockwise on a ci cle:
a→b→c→a. Suppose ha agen 1 op- anks he h ee p o iles whe e he
is he clockwise neighbou o 2. Likewise, agen 2 op- anks he h ee p o iles
whe e he is he clockwise neighbou o 1. The e is no equilib ium because in
any p o ile, agen 3 is he clockwise neighbou o ei he agen 1 o 2, in which
case he o he agen desi es agen 3’s posi ion and is s onge han him. I is also
easy o ind an example wi h h ee indi iduals in which a jungle equilib ium
exis s bu is no Pa e o op imal.
1.3 Compa ison o he Compe i i e Equilib ium
Shapley and Sca (1974) used he ex ended housing economy o s udying
he no ion o compe i i e equilib ium in a simple se ing wi h disc e e goods.
Recall ha he ex ended housing economy is a uple ‹N,X,(%i)i∈N,F,(ei)i∈N›
whe e ‹N,X,(%i)i∈N,F›is a housing economy and (ei)is a easible p o ile
which is in e p e ed as an ini ial alloca ion o he houses. Thus, ins ead o a
powe ela ion, he housing economy model is en iched wi h he speci ica ion
o an ini ial endowmen o each agen . Shapley and Sca (1974) de ine a
compe i i e equilib ium o his ex ended economy o be a p o ile o p ices (one
eal numbe o each house) and a p o ile o houses such ha : (i) each agen
p e e s his assigned house o any ha is no mo e expensi e han his ini ial
endowmen and (ii) he housing assignmen is easible. Fo mally:
De ini ion: Compe i i e Equilib ium
Acompe i i e equilib ium o an ex ended housing economy is a uple
‹(px)x∈X,(xi)i∈N›whe e (px)x∈Xis a p o ile o p ices and (xi)is a p o ile o
houses such ha :
(i) Fo e e y indi idual i, he house xiis %i-maximal in {x|pei≥px}.
(ii) The p o ile (xi)is in F.
22 Chap e 1. Equilib ium in he Jungle
The ollowing p oposi ion, due o Shapley and Sca (1974), shows ha a
compe i i e equilib ium exis s. The p oo , due o Da id Gale, uses an algo i hm
which is based on he no ion o a op- ading cycle. Gi en any g oup o agen s
wi h ini ial endowmen s, a op- ading cycle is a cycle o agen s all o whom
mos p e e he house o he nex agen in he cycle om among hose ha
he g oup membe s a e endowed wi h. I an agen p e e s his own house o all
o he s hen he makes a cycle o leng h one. We will see ha a op- ading cycle
always exis s.
The op- ading cycle algo i hm p oceeds as ollows: a each s age, a op-
ading cycle is iden i ied. Each agen in he cycle is exclusi ely assigned he
house o he nex agen in he cycle (which he p e e s om among he houses
ha we e no assigned p e iously). All houses in he cycle a e assigned he
same p ice, which is lowe han he p ices o all p e iously assigned houses,
and bo h he assigned agen s and he assigned houses a e emo ed.
P oposi ion 1.5: Exis ence o Compe i i e Equilib ium
Fo any ex ended housing economy, a compe i i e equilib ium exis s.
P oo :
Le ‹N,X,(%i)i∈N,F,(ei)i∈N›be an ex ended housing economy. We i s
show ha a op- ading cycle exis s o e e y g oup o agen s G. S a
a bi a ily wi h an agen i0∈G, and de ine ik+1∈Gas he ini ial holde
o ik’s a ou i e house om he se o houses belonging o G. Since he
g oup is ini e, he e will e en ually be some lsuch ha k≥l≥0 and
ik+1=il. Then, he sequence (il,...,ik)cons i u es a op- ading cycle.
See Figu e 1.1 o an illus a ion o he a gumen whe e l=2 and k=5.
i0i1i2=i6i3i4i5
wan s wan s wan s wan s wan s
wan s
Figu e 1.1 The Top–T ading Cycle algo i hm.
1.3 Compa ison o he Compe i i e Equilib ium 23
The algo i hm cons uc s a pa i ion {I1,...,Il,...,IL}o Nas ollows:
Fi s , ind a op- ading cycle om he g oup o all agen s. Se I1 o be he
se o membe s o his cycle and assign o each o hem he house he mos
p e e s. Con inue induc i ely: a s age l+1, ind a op- ading cycle om
among he g oup N−I1−...−Iland o each membe o he cycle assign
he house which he mos p e e s om among hose ini ially held by he
g oup. Se Il+1 o be he se o membe s in he cycle. Con inue in his
ashion un il a pa i ion is comple ed. Choose a sequence o numbe s
p1>p2>... >pL>0 and, o each x∈X, de ine px=plwhe e he
agen who ini ially occupies xis in Il. The assigned p o ile (xi) oge he
wi h he p ice ec o (px)cons i u es a compe i i e equilib ium because
(xi)∈F, and e e y agen iin Ilchooses his a ou i e house om wi hin
his “budge se ”, namely he se o houses ini ially held by he membe s
o Il∪...∪IL.
Compa ing he abo e cons uc ion o ha o he jungle equilib ium cla i ies
he sou ce o powe in he ma ke s. he sou ce o powe in he jungle. In Gale’s
cons uc ion, in each ound some agen s ob ain hei a ou i e house om
among hose no alloca ed in p e ious ounds. So oo in he jungle equilib ium.
Howe e , in he case o compe i i e equilib ium, he o de is de e mined by he
exis ence o a “ op- ading cycle” which indica es he pa ies’ join in e es in
making an exchange, whe eas in he jungle he o de is de e mined by powe ,
independen ly o he agen s’ p e e ences.
Gi en ha he p e e ence ela ions a e assumed o be s ic , he e is
a unique compe i i e equilib ium alloca ion ( o a p oo , see Osbo ne and
Rubins ein (2023)). Howe e , his alloca ion can be suppo ed by many p ice
sys ems, and i can e en be ha one house is mo e expensi e han ano he in
one equilib ium p ice sys em bu less expensi e in ano he .
The wo undamen al wel a e heo ems hold o he compe i i e equilib-
ium in his model:
24 Chap e 1. Equilib ium in he Jungle
(a) Any compe i i e equilib ium ‹(px),(xi)›is Pa e o-op imal since i (yi)∈F
Pa e o domina es (xi) hen pyi≥pxi o all iwi h s ic inequali y o any agen
i o whom yiixiand hus Σi∈Npyi>Σi∈Npxiwhich con adic s he ac ha
he wo sums mus be equal.
(b) Fo any Pa e o-op imal alloca ion (xi) he e is a p ice ec o (px)such
ha ‹(px),(xi)›is a compe i i e equilib ium. By P oposi ion 1.5 a compe i i e
equilib ium exis s o he ex ended economy wi h he ini ial alloca ion (xi). I s
alloca ion (yi)is weakly Pa e o supe io o (xi)and since (xi)is Pa e o-op imal
i mus coincide wi h (xi). The e o e, i we s a wi h (ei)=(xi) he p oo
cons uc s a compe i i e equilib ium in which each agen ikeeps xi.
Powe and Weal h: Since he jungle equilib ium is Pa e o op imal, i can
be suppo ed by p ices as a compe i i e equilib ium. This in i es a na u al
ques ion: wha is he ela ionship be ween powe and weal h?
Fi s , he e is always a p ice sys em in which “s onge ” in he jungle
economy means “ iche ” in he compe i i e equilib ium o he ex ended hous-
ing economy wi h he ini ial endowmen p o ile being he jungle equilib ium
o he jungle economy. Fo mally, le (xi)be he jungle equilib ium in he
housing economy jungle ‹N,X,(%i)i∈N,F,B›. The ex ended housing economy
‹N,X,(%i)i∈N,F,(ei=xi)i∈N›has a compe i i e equilib ium ‹(px),(xi)›whe e
pxi>pxjwhene e ij. Howe e , o he equilib ium p ice ec o s may exis .
Fo example, i he s onges agen op- anks his own house while all o he
agen s bo om- ank i , hen he e also exis s a compe i i e p ice ec o in which
he s onges agen is he poo es .
In ac , i we modi y he economy somewha , hen he e may be no jungle
equilib ium in which he s a emen “s onge = iche ” holds. Fo example,
ecall he clubs economy whe e each agen chooses one club om he se X,
and no mo e han qxagen s can choose club x. Conside he economy wi h 4
agen s, whe e X={a,b}and qa=qb=2. I he p e e ences a e such ha agen
1 p e e s aand all o he agen s p e e b, hen he unique jungle equilib ium is
(a,b,b,a). Howe e , in his equilib ium, e e y agen ob ains his i s -bes club
excep o agen 4 and o p e en agen 4 om ge ing wha he wan s i mus
1.4 Commen s on he Jungle Equilib ium 25
be ha pb>pa. Thus, any p ice ec o which suppo s he jungle equilib ium
alloca ion mus ha e he p ope y ha he s onges agen is he poo es .
1.4 Commen s on he Jungle Equilib ium
Compa a i e s a ics: The jungle equilib ium sa is ies he expec ed compa a-
i e s a ics p ope y ha ad ancing an agen in he powe anking canno hu
he agen . To see his, ecall ha he e is a unique jungle equilib ium and i can
be calcula ed ia a se ial dic a o ship p ocedu e. When an indi idual agen
becomes s onge , all agen s who a e s ill s onge han him will con inue o
make he same choices, while he indi idual now ge s o choose ea lie and,
he e o e, has a s ic ly la ge se o houses o choose om.
On he o he hand, in he case o compe i i e equilib ium, imp o ing an
agen ’s ini ial house endowmen , acco ding o his own p e e ences, migh
make him wo se o in equilib ium. Al hough he new house is be e o him,
i migh be una ac i e o o he agen s. Thus, when applying he op- ading
cycle algo i hm, i could be ha he ini ially appea ed in he i s cycle and, a e
he “imp o emen ”, he now appea s in he las cycle and, he e o e, ends up
wo se o in he new equilib ium han in he old one.
Manipulabili y: The jungle equilib ium is immune o p e e ence mis ep esen-
a ions by an agen . Again, he unique jungle equilib ium can be calcula ed by
he se ial dic a o ship algo i hm. When i is an agen ’s u n o choose, he se
o al e na i es ha he chooses om is una ec ed by his decla ed p e e ences,
and, hus, he can do no be e by mis ep esen ing his p e e ences. This non-
manipulabili y p ope y also holds o compe i i e equilib ia.
Indi e ences: E en i some o he agen s’ p e e ences a e no s ic , he se ial
dic a o ship p ocedu e s ill p oduces a jungle equilib ium. Howe e , i is no
necessa ily unique since, when an agen has o make a choice, he migh ha e
mo e han one maximal op ion and each p oduces a di e en equilib ium.
No e ha indi e ences can also c ea e a mul iplici y o compe i i e equilib-
ium p o iles in he housing economy ma ke .

26 Chap e 1. Equilib ium in he Jungle
Equilib ium and Dynamics: The jungle equilib ium concep is s a ic, like
mos solu ion concep s in Economic Theo y. The ollowing is an example o
dynamics ha lead o a jungle equilib ium: A he beginning, all agen s a e
assigned o be “homeless”. A s age +1, gi en he assignmen o he agen s
a s age o X∪{homeless}, e e y homeless agen chooses his a ou i e house
om among hose ha , a he end o s age , a e ei he : i) acan o ii) assigned
o an agen weake han him. E e y agen who cu en ly occupies a house
chooses o s ay he e. A he end o s age +1, i a house is chosen by only one
agen , hen he se les he e. I mo e han one agen chooses he same house,
hen he s onges among hem se les he e and all he es emain homeless.
P oposi ion 1.6: Equilib ium Dynamics
The abo e dynamics con e ges in a mos ns ages o he jungle
equilib ium.
P oo :
Le H be he se o homeless agen s a he beginning o s age and i
be he mos powe ul among hem. I he e a e any homeless agen s a
s age +1, hen i i +1: To see why, no e ha a s age ,i will ob ain
a home because all homeless agen s a e weake han him and so he will
win a any home which he app oaches. Fu he mo e, all agen s s onge
han i emain in hei homes as no one challenges hem. Thus, in he
beginning o s age +1, all homeless agen s mus be weake han i .
The e o e, a e a mos ns ages, all agen s ha e a home and he p ocess
e mina es a a p o ile (xi).
Suppose ha (xi)is di e en han he jungle equilib ium p o ile (yi).
Take i o be he s onges agen o whom xi6=yi. Thus, iBjwhe e jis
he agen who holds yi, i.e. xj=yi.
By P oposi ion 1.2, yiis i-maximum in X− {y1,...,yi−1}=X−
{x1,...,xi−1}and he e o e yiixi. A he s age in he algo i hm whe e i
selec ed xii mus be ha yiwas being held by someone s onge han
1.4 Commen s on he Jungle Equilib ium 27
i. Bu , in he algo i hm, when a house changes hands, i can only go o
someone s onge so as i e en ually eaches ji mus be ha jBi, a
con adic ion.
A di e en powe ela ion o each house: A key assump ion in he jungle
model is he uni o mi y o he powe ela ion: i an agen iis able o e ic agen
j om one house, hen he is able o e ic him om any house. An ex ension o
he model allows o dependence o he powe ela ion on he house in dispu e.
Suppose ha , o each house x∈X, he e is a s ic powe o de ing Bxwhe e
iBxjmeans ha agen iis s onge han agen jin a igh o e house x. Tha
is, i agen joccupies xand iBxj, hen agen ican con isca e xi he wishes o
do so. An equilib ium in he economy wi h house-dependen powe ela ions
‹N,X,(%i)i∈N,F,(Bx)x∈X›is a p o ile (xi)such ha he e a e no wo agen s iand
jsuch ha ip e e s he house occupied by j o he house he occupies (xjixi)
and iis s onge han j ega ding xj(iBxjj).
As commen ed on in Rubins ein and Yıldız (2022), he no ion o a jungle
equilib ium in ‹N,X,(%i)i∈N,F,(Bx)x∈X›is equi alen o pai wise s abili y in he
wo-sided ma ching p oblem be ween Nand Xwhe e each agen i∈Nhas he
p e e ence %io e Xand each house x∈Xhas he p e e ence ela ion Bxo e
N. An assignmen (xi)is pai wise s able i he e is no pai iand xjsuch ha
ip e e s xjo e xi(xjixi) and xj“p e e s” io e j(iBxjj). The e o e, an
assignmen is pai wise s able in he auxilia y ma ching p oblem i and only i i
is a jungle equilib ium wi h house-dependen powe ela ions.
Gale and Shapley (1962) showed, using he de e ed accep ance algo i hm,
ha a pai wise s able ma ching exis s in any wo-sided ma ching p oblem.
Thus, in he jungle wi h house-dependen powe ela ions, a jungle equilib-
ium also exis s. Since he pai wise s able ma ching need no be unique,
nei he is he jungle equilib ium when he powe ela ion is house-dependen .
Finally, Gale and So omayo (1985)’s analysis implies ha he e is always a
jungle equilib ium (xi)which is weakly Pa e o op imal, in he sense ha he e
is no assignmen (zi)such ha ziixi o e e y i∈N.
28 Chap e 1. Equilib ium in he Jungle
1.5 The Di ision Jungle
We now apply he jungle concep o a e sion o he di ision economy. To he
de ini ion o a di ision economy om Chap e 0, we add a p o ile (Xi)i∈No
pe sonal consump ion se s, which ep esen bounds on each agen ’s abili y
o consume. These se s can be hough o as ei he physical limi s on wha
a pe son can consume o wha possessions he can p o ec . No e ha , in he
housing economy, he e is an implici assump ion o a simila na u e, namely
ha an agen can hold only one house. The ollowing is he o mal de ini ion o
a jungle di ision economy ( h oughou , when compa ing bundles, he no a ion
x≤ymeans ha xk≤yk o e e y commodi y k):
De ini ion: Jungle Di ision Economy
Ajungle di ision economy is a uple ‹N,(Xi)i∈N,(%i)i∈N,F,B›whe e:
•N={1,...,n}is he se o agen s.
•Xi⊆RK
+is agen i’s pe sonal consump ion se in a K-commodi y
wo ld. The se s Xia e assumed o be compac , con ex, and sa is y
ee disposal ( ha is, i xi∈Xi,y∈RK
+and y≤xi, hen y∈Xi).
•%ia e p e e ences o e Xiand assumed o sa is y con inui y, s ic
mono onici y, and s ic con exi y.
•Fis he se o all p o iles o bundles (xi)such ha :
(i) xi∈Xi o all i, and
(ii) Σi∈Nxi≤ewhe e e∈RK
+is an agg ega e bundle a ailable
o dis ibu ion among he agen s.
•is a s ic powe o de ing o e N.
Gi en a p o ile (xi), deno e he “le o e ” bundle e−Σi∈Nxias x0.
We now u n o modi ying he de ini ion o a jungle equilib ium o i he
di ision jungle. The e a e (a leas ) wo possible de ini ions ha coincide wi h
ha o he housing economy. The i s is a s ong jungle equilib ium which
1.5 The Di ision Jungle 29
is a easible p o ile such ha no agen can assemble a p e e able bundle by
combining his own bundle wi h all bundles held by weake agen s and he
le o e bundle. By his de ini ion, he s abili y o a p o ile is dis u bed by he
possibili y ha an agen can a ack mo e han one weake agen . The second
de ini ion is a weak jungle equilib ium, which is a easible p o ile such ha no
agen can assemble a p e e able bundle by combining his own bundle wi h one
o he ha is ei he held by a weake agen o is he le o e bundle. Fo mally:
De ini ion: S ong Jungle Equilib ium
As ong jungle equilib ium is a easible p o ile (xi)wi h he p ope y
ha he e is no agen iand bundle yi∈Xisuch ha :
(i) yiixi.
(ii) yi≤xi+ ΣiBjxj+x0( he agen akes om weake agen s and om
he le o e bundle and po en ially disposes o some o his possessions).
De ini ion: Weak Jungle Equilib ium
Aweak jungle equilib ium is a easible p o ile (xi)wi h he p ope y ha
he e is no agen iand bundle yi∈Xisuch ha :
(i) yiixi.
(ii) Ei he (a) o (b) holds.
(a) yi≤xi+xj o some j o whom iBj( he agen s eals om a
single weake agen and hen may dispose o some o his possessions);
o
(b) yi≤xi+x0( he agen akes om he le o e bundle and hen
may dispose o some o his possessions).
No e ha he abo e de ini ions use inequali ies a he han equali ies. This
is because, when a s onge agen seizes o he esou ces, he migh be pu
ou side o his consump ion se and, hus, needs ei he o ake less o o dispose
o some goods in o de o emain in his consump ion se . Ob iously, any s ong
jungle equilib ium is also a weak jungle equilib ium.
36 Chap e 1. Equilib ium in he Jungle
Howe e , i he ini ial weal h is alloca ed un ai ly, dishones ly
o a bi a ily, hen we migh no a ou he ma ke sys em.
Simila ly, i powe is desi able hen we migh ad oca e o he
jungle sys em, bu i he dis ibu ion o powe e lec s b u e o ce
ha h ea ens li es hen we would clea ly no be in a ou .

2The Pe missible and he Fo bidden
Pic u e in you mind a amily consis ing o nmembe s. The g andpa en s ha e
p epa ed a holiday eas and all a e si ing happily a ound a long able. When
he main dish is se ed, he g andpa en s ac as dic a o s, pu ing a po ion o
i on each amily membe ’s pla e and making su e hey ea i o he las bi e.
And hen, desse a i es and wi h i a d ama ic u n o e en s. G andma and
G andpa en e he oom wi h hei amous homemade pie. E e yone lo es hei
pie and gazes eage ly a i s en ance. Gi en he chance, each amily membe
would gladly ea mo e han 1/no he pie. A his poin , he g andpa en s
decla e ha hey will no in e e e in he di ision o he pie and will le he
younge gene a ion use hei academic knowledge o decide how he pie is
di ided.
One membe o he amily, an economis , sugges s ha each amily
membe should be endowed wi h 1/no he pie and — since some pe haps
app ecia e he pie mo e, while o he s pe haps less — a ma ke should ope a e
unde he able whe e membe s can exchange slices o he pie o money.
Ano he membe o he amily, a game heo is , sugges s ha he g andpa en s
conduc an auc ion. He claims ha his migh be un and, mo e impo an ly,
he pie will be di ided op imally. Hope ully, in you amily, nei he ma ke s no
auc ions a e used o esol e such a con lic and, ins ead, ha mony is achie ed
by means o a social no m: each amily membe does no da e o e en conside
aking mo e han he socially accep able amoun , say q, o he pie.
Ob iously, no e e y qwill b ing ha mony o he amily. I q>1/n, hen
a amily c isis would e up since he e would no be enough pie o sa is y he
amily membe s. All amily membe s would ace o ge hei slice, and some
will be disappoin ed because hey a e unable o ealize hei an icipa ion o
ea ing qo he pie. I q<1/n, hen no con lic a ises, bu he membe s o he
amily would eel uneasy looking a he le o e s on he able and, nex yea ,
©2024 Michael Rich e and A iel Rubins ein, CC BY-NC-ND 4.0
h ps://doi.o g/10.11647/OBP.0404.02
38 Chap e 2. The Pe missible and he Fo bidden
would eel jus i ied in aking a bi mo e. I q=1/n, hen ha mony p e ails. I is
op imal o each amily membe o ake q, and any loosening o he no m will
lead o demands which canno be sa is ied.
We hink o a bound on he po ion ha one can ake as an example o a
na u al social no m ha speci ies wha is conside ed pe missible (“done”) and
o bidden (“no done”). Such a no m esol es he amily’s alloca ion p oblem
bu no wi h p ices o games.
Following Rich e and Rubins ein (2020), we analyze he Y-equilib ium
concep . I is de ined as a se o pe missible al e na i es (which is he same o
all agen s) combined wi h a p o ile o choices (one o each agen ) such ha :
(i) each agen ’s choice is op imal om among he pe missible al e na i es;
(ii) he p o ile o choices is easible; and
(iii) he se o pe missible al e na i es is maximal in he sense ha he e is no
supe se o pe missible al e na i es om which a p o ile sa is ying (i) and
(ii) can be ound.
By his de ini ion, wo o ces make a pe missible se uns able: he i s
modi ies he pe missible se in he case ha he p o ile o (in ended) choices
is no easible, while he second loosens es ic ions on he pe missible se as
long as a new p o ile o op imal choices is easible.
The Y-equilib ium concep e lec s a decen alized ins i u ion o achie ing
ha mony in a socie y. We en ision ha , wi hou a cen al au ho i y, he same
in isible hand ha calcula es equilib ium p ices so “e ec i ely” is also able
o de e mine a maximal se o pe missible al e na i es ha a e compa ible
wi h sel -maximizing beha io . The abo e o ces adjus he social no m un il
ha mony is achie ed. While we do no p o ide a gene al dynamic p ocess
ha con e ges o Y-equilib ium, in Rich e and Rubins ein (2020), o se e al
examples, we demons a ed na u al â onnemen -like p ocesses ha lead o a
Y-equilib ium.
We now p oceed o he o mal de ini ion o he equilib ium no ion.
2.1 The Y-Equilib ium Concep 39
2.1 The Y-Equilib ium Concep
Recall ha an economy is a uple ‹N,X,(%i)i∈N,F›whe e Nis he se o agen s,
Xis he se o al e na i es ha each agen chooses om, %iis agen i’s
p e e ences on X, and F⊆XNis he se o easible choice p o iles.
A candida e o an equilib ium is a con igu a ion which consis s o a subse
o X, called a pe missible se , oge he wi h a p o ile o choices:
De ini ion: Con igu a ion
Acon igu a ion is a pai ‹Y,(yi)i∈N›whe e Y⊆Xand (yi)i∈Nis a p o ile
o elemen s in Y. We e e o Yas a pe missible se and o (yi)i∈Nas an
ou come.
As explained in Chap e 0, a candida e o a solu ion in his book has a s uc u e
analogous o ha o a compe i i e equilib ium. I is comp ised o a p o ile o
choices (one o each agen ) and an addi ional pa ame e . In a con igu a ion,
he addi ional pa ame e is a pe missible se , ha is aken by all agen s as gi en
and uni o mly binds he choices o all agen s. Analogously, in a compe i i e
equilib ium, he addi ional pa ame e is a p ice sys em, ha is aken by all
agen s as gi en and uni o mly binds he exchanges o all agen s.
Be o e de ining he equilib ium concep , we need an addi ional concep :
a pa a-equilib ium is a con igu a ion whe e each indi idual maximizes his
in e es s gi en he pe missible se and he esul ing choice p o ile is easible.
De ini ion: Pa a-equilib ium
Apa a-equilib ium is a con igu a ion ‹Y,(yi)›sa is ying:
(i) Fo all i,yiis a %i-maximal al e na i e in Y.
(ii) The p o ile (yi)is in F.
A Y-equilib ium is a pa a-equilib ium such ha any expansion o he pe missi-
ble se will lead o a iola ion o easibili y i agen s sel -maximize wi h espec
o he expanded pe missible se .
40 Chap e 2. The Pe missible and he Fo bidden
De ini ion: Y-equilib ium
AY-equilib ium is a pa a-equilib ium ‹Y,(yi)›such ha he e is no pa a-
equilib ium ‹Z,(zi)› o which Zis a s ic supe se o Y.
As men ioned ea lie , we iew he pe missible se no as being de e mined
by an au ho i y bu , a he , as e ol ing h ough an in isible-hand-like p ocess
wi h wo o ces: Fi s , i he p o ile o in ended choices om he pe missible
se is no easible, hen al e na i es a e emo ed o added o he pe missible
se . Second, when he p o ile o chosen al e na i es is easible, addi ional
al e na i es a e added o he pe missible se as long as ha mony is no
dis u bed. No e ha (yi)can di e om (zi), ha is, when assessing he
exis ence o a la ge pe missible se , choices can adap o he loosening.
We ake he pe missible se o be uni o m o all agen s, al hough we
a e awa e ha he e a e si ua ions in li e whe e no ms a e nonuni o m,
such as allowing handicapped d i e s o pa k in places whe e o he s a e no
pe mi ed. The uni o mi y o he pe missible se in ou model is analogous
o he uni o mi y o he p ice sys em in models o compe i i e equilib ium
(al hough p ices a e o en no uni o m in eal li e). In some ci cums ances,
uni o mi y can be iewed as an exp ession o equali y o oppo uni y. I also is
a simplici y p ope y: in o de o be ollowed, no ms mus be simple and clea ,
and no ms a e simple when hey do no dis inguish be ween agen s based on
hei names o p e e ences.
Example: A Housing Economy
Conside he housing economy wi h N={1,2},X={a,b,c,d,e}, and
p e e ences a1b1c1d1eand a2c2b2e2d. One
pa a-equilib ium is Y={d,e},y1=d,y2=e. This is no a Y-equilib ium
since Y={b,c,d,e}wi h y1=b,y2=cis also a pa a-equilib ium wi h
a la ge pe missible se . The la e is he unique Y-equilib ium since he
al e na i e acanno be a membe o any pa a-equilib ium pe missible
2.1 The Y-Equilib ium Concep 41
se as i is he op- anked o bo h agen s. Inciden ally, he Y-equilib ium
ou come is no Pa e o-op imal because ais le unassigned.
Exis ence: No e e y economy has a Y-equilib ium. In any housing economy,
i a leas wo agen s ha e he same s ic p e e ences o e he houses, hen no
Y-equilib ium exis s. This is because, wha e e he pe missible se is, hose wo
agen s will pick he same house, which iola es easibili y. This demons a es
ha social no ms ega ding “ he pe missible and he o bidden” do no esol e
con lic s when agen s ha e simila p e e ences ye easibili y equi es hem o
make di e en choices.
Example: A Single Pie
Conside he g andpa en s’ pie economy discussed in he beginning o
he chap e . The e a e n amily membe s, and a pie o size 1 is o be
di ided among hem. The se o al e na i es is X= [0,1]whe e x∈Xis a
sha e o he pie. Each agen p e e s o ge as la ge a sha e as possible. The
easibili y cons ain s a es ha he sum o hei choices canno exceed 1
( hough some pie can be le o e ).
To see ha his economy has a unique Y-equilib ium, no ice i s
ha he pai ‹Y= [0,1/n],(yi≡1/n)›is a pa a-equilib ium. The e
is no pa a-equilib ium wi h a poin abo e 1/nin he pe missible se
since, hen, e e y agen would choose a poin abo e 1/n, which is no
easible. The e o e, he abo e pai is a Y-equilib ium. The e is no o he Y-
equilib ium since he pe missible se in any pa a-equilib ium is a subse
o [0,1/n].
Example: The Quo um Economy
Conside an economy wi h a ini e se o clubs, X. Agen s ha e
p e e ences o e he clubs (wi hou ega d o he clubs’ membe ships).
In o de o ope a e, each club xneeds a minimal quo um o mx≤n

42 Chap e 2. The Pe missible and he Fo bidden
( a he han ha ing a maximal capaci y as in he clubs economy). Tha is,
easibili y equi es ha each club xis ei he emp y o chosen by a leas
mxmembe s. A special case is he consensus economy whe e mx=n
o all x, ha is, easibili y equi es ha all agen s make he same choice.
In gene al, i e e y agen we e o choose his a ou i e club, hen he e
would be non-emp y clubs wi h less han a quo um. The ole o he
pe missible se is o help he agen s o coo dina e hei choices while
imposing minimal es ic ions on he pe missible clubs.
A Y-equilib ium always exis s: Fi s , a pa a-equilib ium exis s because
any con igu a ion Y={x}combined wi h all agen s choosing xis a
pa a-equilib ium. Second, since he se o subse s o Xis ini e, he e
is a pa a-equilib ium wi h a pe missible se ha canno be expanded.
Howe e , Pa e o op imali y is no gua an eed, as illus a ed by he
ollowing example. Le n=6, X={a,b,c}, and mx=3 o all x.
Two agen s ha e he p e e ences abc, wo ha e he p e e ences
bca, and wo ha e he p e e ences cab. Ob iously, he e
is no pa a-equilib ium wi h Y=X. Fu he mo e, he e is no pa a-
equilib ium wi h exac ly wo pe missible clubs since ou o he agen s
would choose one club and only wo would choose he o he , iola ing
easibili y. As abo e, ha ing a single club open is a pa a-equilib ium and
since he e a e no mul i-club pa a-equilib ia, i is a Y-equilib ium. Thus,
he e a e h ee Y-equilib ia, each wi h a single di e en club open. Each
Y-equilib ium ou come is no Pa e o-op imal since he e is an unopened
club ha is s ic ly p e e ed by ou agen s and, he e o e, he e is a
Pa e o imp o emen whe e exac ly h ee o hose ou agen s swi ch o
ha mo e-p e e ed club.
The Y-equilib ium concep is no mean o be no ma i e in any sense.
Howe e , i has wo ai ness p ope ies:
(i) All agen s ace he same choice se . Analogously, in he s anda d compe i i e
equilib ium, all agen s ace he same ading oppo uni ies.
2.2 Y-Equilib ium, Pa e o Op imali y, and En y-F eeness 43
(ii) I is en y- ee (see Foley (1966) and Va ian (1974)). En y- eeness ensu es
ha no agen can complain ha someone else is assigned an al e na i e ha
he p e e s.
De ini ion: En y- eeness
A p o ile (yi)i∈Nis en y- ee i , o all i6=j,yi%iyj.
The concep s o pa a-equilib ium and en y- eeness a e closely ela ed. A
p o ile is en y- ee i and only i i is he ou come o some pa a-equilib ium:
Fi s , any pa a-equilib ium ou come is en y- ee (no agen can en y ano he ’s
choice since all agen s choose om he same se ). Second, i a p o ile (yi)is
en y- ee, hen ‹{y1,...,yn},(yi)›is a pa a-equilib ium.
2.2 Y-Equilib ium, Pa e o Op imali y, and En y-F eeness
We ha e seen ha Y-equilib ium p o iles need no be o e all Pa e o-op imal.
None heless, hey s ill sa is y some e iciency c i e ion. We now show ha he
Y-equilib ium p o iles a e p ecisely hose which a e Pa e o op imal om among
he se o easible en y- ee p o iles.
P oposi ion 2.1: Y-equilib ium Ou come Cha ac e iza ion
A p o ile is a Y-equilib ium ou come i and only i i is Pa e o-op imal
among all easible en y- ee p o iles.
P oo :
Le ‹Y,(yi)›be a Y-equilib ium. The p o ile (yi)is easible and en y-
ee. I i is no Pa e o-op imal among he easible en y- ee p o iles, hen
he e is a easible en y- ee p o ile (zi) ha Pa e o-domina es (yi). The
con igu a ion ‹Y∪{z1,...,zn},(zi)›is a pa a-equilib ium (since zi%izj
o all i,j, and zi%iyi%iy o all iand y∈Y). By Pa e o dominance,
ziiyi o a leas one agen iand he e o e zi/∈Y. Thus, Y∪{z1,...,zn}
is a s ic supe se o Y, con adic ing he de ini ion o Y-equilib ium.
44 Chap e 2. The Pe missible and he Fo bidden
In he o he di ec ion, le (yi)be Pa e o-op imal among he easible
en y- ee p o iles. Le Ybe he se o all elemen s in his p o ile plus
any elemen which is weakly in e io o yi o e e y agen i, namely,
Y=Si{yi}∪{x| o all i,yi%ix}. The con igu a ion ‹Y,(yi)›is a
pa a-equilib ium. In o de o show ha i is also a Y-equilib ium, we
need o in alida e he exis ence o a pa a-equilib ium ‹Z,(zi)› o which
Z)Y. I i exis s, hen, zi%iyi o all iand (zi)is en y- ee. Le
x∈Z−Y. By he de ini ion o Y, he e is an agen j o whom xjyj
and, consequen ly, zj%jxjyj. The e o e, (zi)is a easible en y-
ee p o ile ha Pa e o-domina es (yi), con adic ing (yi) being Pa e o-
op imal among he easible en y- ee p o iles. Thus, no such pa a-
equilib ium ‹Z,(zi)›exis s and, he e o e, ‹Y,(yi)›is a Y-equilib ium.
We do no ake o e all Pa e o op imali y as a necessa y condi ion o he
plausibili y o desi abili y o a solu ion concep . S ill, a na u al ques ion is:
Wha condi ion gua an ees ha any Y-equilib ium ou come is o e all Pa e o-
op imal (and no jus among he en y- ee p o iles)? One such condi ion is he
imi a ion p ope y:Fsa is ies he imi a ion p ope y i , whene e a p o ile is
in F, so is any p o ile o which one agen adop s he al e na i e chosen by
ano he agen ins ead o his own. Tha is, o any (ai)∈Fand any i,j∈N, he
p o ile whe e aiis eplaced wi h ajis also in F. An example whe e he imi a ion
p ope y holds is he s ay close economy (desc ibed in Chap e 0) since, i one
agen adop s ano he ’s posi ion, he maximal dis ance be ween any wo agen s
does no inc ease.
P oposi ion 2.2: The Imi a ion P ope y and Pa e o Op imali y
Assume ha Fsa is ies he imi a ion p ope y. Then, a p o ile is a Y-
equilib ium ou come i and only i i is o e all Pa e o op imal.
2.3 Euclidean Economies 45
P oo :
Le ‹Y,(yi)›be a Y-equilib ium. Assume by con adic ion ha he e is a
easible p o ile (zi)which Pa e o-domina es (yi). We cons uc a p o ile
(xi)as ollows: Assign x1, a %1-maximal al e na i e om {z1,...,zN}, o
agen 1. Assign x2, a %2-maximal al e na i e om {x1,z2,...,zN}, o
agen 2, and so on. In his cons uc ion, he p o ile selec ed a each
s age is easible (due o he imi a ion p ope y) and xi%izi o all i.
Fu he mo e, o e e y agen i, he al e na i e xiis %i-maximal om
{x1,...,xi−1,zi,...,zN}⊇{x1,...,xN}. Thus, (xi)is easible and en y-
ee. I weakly Pa e o-domina es (zi)and hus Pa e o-domina es (yi),
con adic ing P oposi ion 2.1.
The o he di ec ion ollows immedia ely om P oposi ion 2.1
because, unde he imi a ion condi ion on F, e e y Pa e o-op imal
p o ile is en y- ee and, he e o e, is also Pa e o-op imal among he
en y- ee alloca ions.
2.3 Euclidean Economies
In many common economic models, such as Wal asian economies, he se
o al e na i es is aken o be a subse o a Euclidean space wi h s anda d
closedness, con exi y, and di e en iabili y es ic ions on he al e na i es, he
p e e ence ela ions, and he easibili y se . We now conside ou amewo k in
a Euclidean se ing.
De ini ion: Euclidean Economy
AEuclidean economy is an economy ‹N,X,(%i)i∈N,F›such ha :
(i) The se Xis a closed subse o some Euclidean space.
(ii) Fo each i, he p e e ences %ia e con inuous.
(iii) The easibili y se Fis anonymous (closed unde pe mu a ions),
compac , and con ains a leas one cons an p o ile.
52 Chap e 2. The Pe missible and he Fo bidden
Gi en a chain Co elemen s in P, le Ube he union o he se s in C.
Clea ly, Uis an uppe bound on C, and we now show ha Uis in P. The
se Uis con ex since o any wo poin s x,y∈U, he e is some Y∈ C
such ha x,y∈Yand, since any con ex combina ion o xand yis in Y,
i is also in U. To show ha he uple ‹U,(yi)›is a pa a-equilib ium, i
su ices o show ha , o each i, he elemen yiis %i-maximal in U. I
he e is an x∈Usuch ha xiyi o some i, hen he e is Y∈C such
ha x∈Y, con adic ing ha ‹Y,(yi)›is a pa a-equilib ium.
Le Y∗be a maximal elemen o P. I is le o show ha ‹Y∗,(yi)›is a
Y-equilib ium. Suppose ha he e is a con ex pa a-equilib ium ‹Z,(zi)›
such ha Z)Y∗. I mus be ha zi%iyi o all i. Since (yi)is Pa e o-
op imal om among he con ex pa a-equilib ium ou comes, i mus be
ha zi∼iyi o all i. Then, ‹Z,(yi)›is also a con ex pa a-equilib ium,
con adic ing he maximali y o Y∗.
Fo Euclidean economies, we ha e al eady shown ha a Y-equilib ium always
exis s (P oposi ion 2.3). The ollowing p oposi ion demons a es ha a con ex
Y-equilib ium also exis s.
P oposi ion 2.5: Exis ence o a Con ex Y-equilib ium
E e y con ex Euclidean economy has a con ex Y-equilib ium.
P oo :
Le Obe he se o con ex pa a-equilib ium ou comes. The se Ois
no emp y since Fcon ains a cons an p o ile (yi≡y∗)and he pai
‹{y∗},(yi≡y∗)›is i ially a con ex pa a-equilib ium.

2.7 A S uc u e Theo em o Con ex Y-equilib ium 53
The se Ois compac . To see his, since O⊆Fand Fis compac ,
i su ices o show ha Ois closed. Take a sequence ‹Y ,(yi
)›o pa a-
equilib ia such ha (yi
)con e ges o (zi)as → ∞. Le Z⊆Xbe
he con ex hull o he limi alloca ions {z1,...,zn}. The con igu a ion
‹Z,(zi)›is a con ex pa a-equilib ium since i he e is an agen jand a
con ex combina ion o he {z1,...,zn}such ha Σi∈Nλizijzj, hen by
con inui y, o some la ge enough ,Σi∈Nλiyi
jyj
. Since Y is con ex,
i holds ha Σi∈Nλiyi
∈Y , bu his iola es ‹Y ,(yi
)›being a con ex
pa a-equilib ium.
Since Ois compac , he same a gumen as in P oposi ion 2.3 implies
he exis ence o a p o ile ha is Pa e o-op imal in Oand, by P oposi ion
2.4, i is a con ex Y-equilib ium ou come.
2.7 A S uc u e Theo em o Con ex Y-equilib ium
Much o Economic Theo y deals wi h es ablishing condi ions ha gua an ee
he exis ence o a solu ion concep . Theo ems abou he s uc u e o equilib-
ium a e less common, al hough, in ou opinion, a e mo e in e es ing. We now
show ha ou assump ions on he economy, oge he wi h a di e en iabili y
condi ion, gua an ee ha he pe missible se o con ex equilib ia is an
in e sec ion o a mos nhal -spaces ( ecall ha nis he numbe o agen s).
Thus, he equi emen ha he pe missible se is con ex implies ha he
con ex Y-equilib ium pe missible se akes a ela i ely simple o m.
P oposi ion 2.6: The S uc u e o Con ex Y-equilib ia
Le ‹Y,(yi)›be a con ex Y-equilib ium in a di e en iable Euclidean
economy. Le J={i|yiis no he %i-global maximum in X}. Then,
he e is a p o ile o closed hal -spaces (Hj)j∈J, such ha Y=∩j∈JHj.
54 Chap e 2. The Pe missible and he Fo bidden
y1
y4
y5
y2
y3
%4
%5
%1
Y
Figu e 2.1 An illus a ion o P oposi ion 2.6 (no e ha J={1,4,5})
P oo :
Fi s , no e ha i J=;, ha is, e e y agen is assigned his i s -bes ,
hen Y=X(which is he degene a e case whe e Yis he in e sec ion
o an emp y se o hal -spaces). O he wise, o e e y j∈J, le Hjbe he
unique hal -space o al e na i es con aining yjsuch ha yjis s ic ly
p e e ed o all o he elemen s in Hj. I s exis ence is gua an eed by
he assump ions o di e en iabili y and s ic con exi y o he agen s’
p e e ence ela ions.
We i s show ha Yis a subse o ∩j∈JHj: Suppose ha o some
j∈J he e is an al e na i e wj∈Y−Hj. By he di e en iabili y and
s ic con exi y o j’s p e e ences, and o small " > 0, i holds ha
"wj+(1−")yjjyj. By con exi y o Yi holds ha "wj+(1−")yj∈Y.
The e o e, yjis no %j-maximal in Y, a con adic ion.
To show ha he pe missible se Yis equal o ∩j∈JHj, i emains o
be shown ha ‹∩j∈JHj,(yi)›is a con ex pa a-equilib ium. This ollows
om:
2.8 The Di ision Economy 55
(i) The se ∩j∈JHjis con ex.
(ii) Fo each agen i,yi∈Y⊆∩j∈JHj.
(iii)Fo each j∈J,yjis he %j-maximum in Hjand, hus, also in ∩j∈JHj.
(i ) Fo each i/∈J,yiis he %i-global maximum and, hus, also in ∩j∈JHj.
2.8 The Di ision Economy
A leading economic p oblem is he di ision o a bundle among he membe s
o a socie y. The g andpa en s single pie economy is i s simples e sion. The
only con ex Y-equilib ium is he in ui i ely appealing no m ha o bids aking
mo e han 1/n h o he pie. Fo he mul i-good di ision economy, he analogous
no m which allows an agen o ake up o 1/n h o he o al bundle is ypically
no a Y-equilib ium pe missible se because i does no allow any ades. We
p oceed by explo ing he p ope ies o con ex Y-equilib ia in a di e en iable
di ision economy, o mally de ined as:
De ini ion: Di e en iable Di ision Economy
Adi e en iable di ision economy ‹N,X,(%i)i∈N,F›is a di e en iable
Euclidean economy such ha :
(i) The se o al e na i es is all bundles wi h mcommodi ies, i.e. X=Rm
+.
(ii) E e y p e e ence ela ion %iis s ic ly mono onic (besides being
con inuous, s ic ly con ex, and di e en iable).
(iii) The e is a bundle e∈Rm
++ such ha (xi)∈Fi and only i Σixi≤e.
The ollowing claim d aws a connec ion be ween con ex Y-equilib ium and
egali a ian compe i i e equilib ium (see Foley (1966) and Va ian (1974)) which
is a compe i i e equilib ium o he exchange economy in which each agen is
ini ially endowed wi h 1/no he o al bundle. We will see ha e e y egali a ian
compe i i e equilib ium ou come is a con ex Y-equilib ium ou come and, i a
leas one agen selec s an in e io bundle, hen i s pe missible se is iden ical
o he egali a ian compe i i e equilib ium’s common budge se .
56 Chap e 2. The Pe missible and he Fo bidden
Claim: Egali a ian Compe i i e Equilib ia and Con ex Y-equilib ia
Le ‹p,(yi)›be an egali a ian compe i i e equilib ium in a di e en iable
di ision economy. Then, he e is a con ex Y-equilib ium wi h he same
alloca ion ‹Y,(yi)›. Fu he mo e, i a leas one o he bundles yjis
s ic ly posi i e, hen Ymus be B={y|p∙y≤p∙e/n}.
P oo :
The pai ‹B,(yi)›is a con ex pa a-equilib ium and (yi)is o e all Pa e o-
op imal by he s anda d i s wel a e heo em. Thus, by P oposi ion 2.4,
(yi)is a con ex Y-equilib ium ou come.
I ‹Y,(yi)›is a con ex Y-equilib ium, hen by P oposi ion 2.6,
Y=∩i∈NHi, whe e Hiis he lowe hal -space o %ia yi(since no agen
has his i s -bes , i holds ha J=N). Fo all i,B⊆Hi, since o he wise
he e exis s zi∈B Hiand, by di e en iabili y and s ic con exi y, yi
would no be %i-op imal in B. I o some j he bundle yjhas a ze o
coo dina e, hen i can be ha B(Hj, bu i o any j he bundle yjis
s ic ly posi i e, hen Hj=Band, he e o e, Y=∩iHi=B.
Commen s:
E e y o e all Pa e o-op imal in e io con ex Y-equilib ium p o ile is an
egali a ian compe i i e equilib ium alloca ion:
Le ‹Y,(yi)›be a con ex Y-equilib ium such ha each bundle yiis in e io .
By mono onici y, he al e na i e yiis ne e %i-globally maximal and hus,
by P oposi ion 2.6, Y=∩i∈NHiwhe e Hiis he lowe hal -space o %ia yi
and, by mono onici y, he e is a posi i e ec o piand a posi i e numbe wi
such ha Hi={x|pi∙x≤wi}. Since e e y yiis in e io and he alloca ion is
Pa e o op imal, he hal -spaces mus be pa allel (o he wise, any wo agen s
on non-pa allel hal -spaces could make a Pa e o-imp o ing local exchange)
ha is, he e is a posi i e ec o psuch ha pi=p o all i. I ollows ha
Y={x|p∙x≤w} o some posi i e ec o pand a posi i e numbe w. By
2.8 The Di ision Economy 57
mono onici y, p∙yi=w o all i. Since p∙e=p∙Σi∈Nyi=nw , we ha e
p∙yi=w=p∙(e/n). Thus, (yi)is a compe i i e egali a ian equilib ium
alloca ion wi h p ice ec o p.
The e can exis a non-in e io Pa e o-op imal con ex equilib ium ou come
ha is no an egali a ian compe i i e equilib ium alloca ion:
He e is a simple example: Le n=3, m=2, e= (5,5)and he agen s’ p e e ences
be ep esen ed by he u ili y unc ions speci ied in Figu e 2.2, panel (a) (a
sligh modi ica ion o he p e e ences will make he p e e ence ela ions s ic ly
con ex):
u1(x1,x2) = x1
u2(x1,x2) = x1+x2
u3(x1,x2) = x2
(a) U ili y unc ions
1
2
3y3
y2
y1
Y
123
u3
u2
u1
(b) Illus a ion
Figu e 2.2 A con ex Y-equilib ium wi h a non-egali a ian Pa e o-op imal ou come.
Le y1= (3,0),y2= (2,2)and y3= (0,3)(Figu e 2.2., panel (b)). The
alloca ion (yi)is Pa e o-op imal: I (zi)Pa e o-domina es (yi), hen zi
1+zi
2≥
yi
1+yi
2 o all iwi h a leas one inequali y. Thus, Σi(zi
1+zi
2)>Σ(yi
1+yi
2) = 10,
which is no easible. The se Yis he in e sec ion o (Hi)whe e each Hiis a
hal -space o bundles below i’s indi e ence cu e, which includes yi.
The pai ‹Y,(yi)›is a con ex pa a-equilib ium and, by P oposi ion 2.4, (yi)
is a con ex Y-equilib ium ou come. To see his di ec ly, no e ha i he e we e a
la ge con ex pa a-equilib ium, ‹Z,(zi)›, henZwould con ain an elemen ha
is no in Y. Any such elemen is s ic ly p e e ed o yi o a leas one agen i.
Thus, (zi) would Pa e o-domina e (yi).

58 Chap e 2. The Pe missible and he Fo bidden
The e can exis a non Pa e o-op imal in e io con ex equilib ium ou come:
Conside he economy (depic ed in Fig-
u e 2.3) wi h wo agen s, wo goods, o-
al bundle e= (3,3), and kinked u ili y
unc ions as depic ed (a small de ia-
ion could make hem s ic ly con ex).
Agen 1’s indi e ence cu e has slope
−1.25, and agen 2’s indi e ence cu e
has slope −0.8. The depic ed alloca ion
y1= (2,1)and y2= (1,2)is no Pa e o-
op imal since i is mu ually bene icial
o ha e agen 1 ge one addi ional uni
o good 1 and one uni ewe o good 2.
1
2
y1= (2,1)
y2= (1,2)
Y
1 2
u2
u1
Expansion Pa h2
Expansion Pa h1
Figu e 2.3 A non Pa e o-op imal con-
ex equilib ium.
The con igu a ion ‹Y,(yi)›is a con ex pa a-equilib ium. In any la ge
con ex pa a-equilib ium, ‹Z,(zi)›, he con ex se Zincludes a bundle ha is
s ic ly be e o a leas one o he agen s and, he e o e, z16=y1and z26=y2.
Gi en he agen s’ indi e ence cu es, in z1agen 1 ecei es mo e o good 1
and less o good 2 han in y1. Fu he mo e, agen 1 mus no p e e ano he
bundle on he line segmen be ween z1and he co ne (2,2). This means ha
he slope be ween hese wo poin s has o be a leas −1.25 o , in o he wo ds,
z1
2−2≥−1.25z1
1+2.5, which implies z1
2+z1
1≥4.5 −0.25z1
1>3 whe e he las
inequali y is due o 3 ≥e1≥z1
1. Likewise, agen 2’s o al demands a e g ea e
han 3, and such demands a e in easible.
Ini ial Endowmen s: Recall ha a di ision economy di e s om he s anda d
exchange economy as i does no speci y an ini ial dis ibu ion o he goods.
One way o inco po a e ini ial endowmen s in o ou amewo k is by he
ollowing no ion o a ade economy. Le (ei)be an ini ial endowmen p o ile.
Le X=Rmwhe e a membe o Xis in e p e ed as a ade (and hus includes
nega i e componen s as well). Se F o include all p o iles o ades ( i)such
ha Σi i=0 and o e e y agen i, he pos - ade bundle i+ei≥0. As o he
p e e ences, assume ha each agen ihas a basic p e e ence ela ion %i
co e
2.9 The Gi e-and-Take Economy 59
he se o bundles (sa is ying he s anda d di ision economy assump ions).
Among ades ha gi e an agen a non-nega i e amoun o e e y good, agen
i’s p e e ences %ion Xa e induced om hei basic p e e ences by i%isii
i+ei%i
csi+ei. E e y agen p e e s he no-exchange op ion 0 o any ade
which lea es hem wi h a nega i e amoun o any good.
Analogous esul s o he p e ious claims o he di ision economy also hold
o he ade economy: (i) he p o ile o ades in any compe i i e equilib ium
in he s anda d exchange economy is a con ex Y-equilib ium ou come in his
ade economy, and (ii) any Pa e o-op imal con ex Y-equilib ium ou come in
he ade economy, whe e a leas one agen has a s ic ly posi i e pos - ade
alloca ion, is a p o ile o ades in a compe i i e equilib ium o he s anda d
exchange economy.
2.9 The Gi e-and-Take Economy
Recall ha in he gi e-and- ake economy, he se o al e na i es is X= [−1,1],
whe e a posi i e x ep esen s a wi hd awal o x om a social und while a
nega i e x ep esen s a con ibu ion o −x. Feasibili y equi es ha he social
und be balanced, ha is, (xi)∈Fi Σixi=0. All agen s ha e s ic ly con ex
p e e ences o e Xwi h agen i’s ideal deno ed by peaki. As men ioned ea lie ,
he gi e-and- ake economy is an economic si ua ion in which he ma ke plays
no ole. We will see ha no ms ega ding wha is pe missible and wha is
o bidden can se e as an e ec i e non-ma ke ool o achie ing ha mony.
The case Σipeaki=0 is “bliss”: e e y hing is pe mi ed and ‹X,(peaki)›
is a con ex Y-equilib ium. Howe e , in gene al, he e is ension be ween
easibili y and he agen s’ desi es. The ollowing claim cha ac e izes he con ex
Y-equilib ium o he case whe e he sum o wha people ideally wan o ake
is g ea e han wha people ideally wan o gi e. We will see now ha , in his
case, he e is a unique con ex Y-equilib ium. In i , people a e allowed o gi e
as much as hey wan bu he e is a bound on he maximum ha can be aken,
and i s ou come is Pa e o op imal.
60 Chap e 2. The Pe missible and he Fo bidden
Claim: A Cha ac e iza ion o he Con ex Y-equilib ium
Conside a gi e-and- ake economy wi h Σpeaki>0. The e is a unique
con ex Y-equilib ium ‹Y,(yi)›. The se Y akes he o m [−1,m] o some
m>0, and (yi)is Pa e o op imal.
P oo :
Conside a pe missible se o he o m [−1,m]. I m<0, hen all agen s
mus gi e. I m≥0, hen e e y agen who wan s o gi e will selec his
peak, and e e y agen who wan s o ake is ei he a his peak o has a
peak o he igh o mand makes do wi h aking m. Le D(m)be he sum
o all agen s’ choices gi en he pe missible se [−1,m]. The unc ion Dis
con inuous, s ic ly inc easing o any msmalle han max{peaki}, and is
cons an wi h alue Σipeaki>0 o any la ge m. In pa icula , D(0)≤0
and D(1)>0. Thus, he e is a unique m∗≥0 o which D(m∗) = 0.
The pe missible se [−1,m∗], oge he wi h he agen s’ op imal
choices om ha se , cons i u es a con ex pa a-equilib ium. I is also a
con ex Y-equilib ium because he e is no con ex pa a-equilib ium wi h
a la ge pe missible se . I he e we e, i would ha e he o m [−1,m]
whe e m>m∗, bu hen agen s would ake oo much (since D(m)>0).
The p o ile (yi)is Pa e o op imal: Fo each i,yiis a o o he le o his
peak. Thus, i (zi)∈FPa e o-domina es (yi), hen yi≤zi o all iwi h a
leas one s ic inequali y, hus 0 = Σyi<Σzi, iola ing easibili y.
To p o e uniqueness o he con ex equilib ium, i emains o be
shown ha any closed con ex pa a-equilib ium pe missible se [x,y]is
included in [−1,m∗]. In o de o he social und o be balanced, i mus
be ha x≤0≤y. In equilib ium, agen s who wish o gi e will do so a
ei he hei peak o a xi peaki<x. The e o e, he o al gi ing in [x,y]
is no mo e han ha in [−1,m∗]. Since he social und is balanced, he
o al aking in [x,y]mus also be less han o equal o ha in [−1,m∗],
and he e o e y≤m∗. Thus, [x,y]⊆[−1,m∗].
2.10 The S ay Close Economy 61
Commen : Fo his economy, while con ex Y-equilib ia a e Pa e o-op imal, a
Y-equilib ium ou come need no be. A de ailed example appea s in Rich e
and Rubins ein (2020). The essence o he example is as ollows: Le
X={−2,−1,0,1,2}and n=2. The agen s’ “con ex” p e e ence ela ions a e
1101−11−212 and 1 22202−12−2. The “con ex” pe missible
se Y={−2,−1,0}, oge he wi h he p o ile y1=y2=0, is a “con ex”
Y-equilib ium wi h a Pa e o-op imal ou come. Howe e , i is easy o e i y ha
he non-con ex pe missible se Y={−2,2}wi h he p o ile y1=−2,y2=2 is a
Y-equilib ium whose ou come is Pa e o domina ed by z1=−1, z2=1.
2.10 The S ay Close Economy
The s ay close economy is a con ex Euclidean economy in which Xis a closed
con ex se o loca ions and Fis he se o p o iles o which he dis ance
be ween any wo agen s is a mos d∗. Tha is, each membe o he g oup
chooses a posi ion ( o example, a poli ical s ance o a geog aphical loca ion),
and he g oup’s su i al equi es ha he membe s “s ay close” o each o he . As
always, each agen has s ic ly con ex p e e ences o his own loca ion wi hou
ega d o he loca ion o o he s. The po en ial sou ce o con lic is ha he
g oup membe s ha e a di e se se o ideal loca ions which ails he closeness
equi emen . No e ha he se Fsa is ies he imi a ion condi ion de ined in
Sec ion 2.3. When d∗=0, his economy is called a consensus economy.
In a cen alized socie y, he au ho i ies can coe ce agen s in o occupying
loca ions ha gua an ee su i al. In a ma ke , membe s would ha e o pay
each o he o s ay close by. The Y-equilib ium idea is ha he e a e no ms
ha de e mine he bo de s o he pe missible loca ions and s ike a balance
be ween socie al ha mony and indi idual libe y. Each agen chooses his mos
p e e ed loca ion wi hin he bo de s, and he ou come is ha hey all li e close
enough o one ano he . The bo de s a e maximally libe al in he sense ha i
he bo de s a e enla ged in any way, hen he esul ing indi idual choices would
no be “close enough”.
68 Chap e 3. S a us and Indoc ina ion
P oo :
Le (ai)be a Pa e o-op imal p o ile. De ine he bina y ela ion Don
A={a1,...,an}by xDy i xis desi ed by a holde o y, ha is, he e a e
iand jsuch ha x=aijaj=y. I Dhas a cycle, hen he e is a se o
agen s who can pe mu e hei al e na i es among hemsel es ( ecall ha
Fis closed unde pe mu a ions) so ha all o hem a e s ic ly be e o ,
con adic ing (ai)being Pa e o-op imal. Since Dhas no cycles, i can be
ex ended o a comple e o de ing o e A. Then, Dcan be ex ended o a
s ic o de ing Pon he en i e se Xby pu ing all elemen s in X−Aabo e
all elemen s in A(making all unassigned elemen s “una o dable”) and
a bi a ily anking he elemen s in X−Aamong hemsel es. Pe sonal
op imali y holds since, o e e y agen i, he al e na i e aiis op imal in
{x|aiPx}(i aiPx, hen x=aj o some j, and i iwe e o p e e i , hen
xDai, which con adic s aiPx since Pex ends D).
By he same p oo , any easible p o ile (Pa e o-op imal o no ) o which
he ela ion Ddoes no ha e cycles is a s a us equilib ium p o ile. In pa icula ,
in he consensus economy, whe e all agen s ha e o make he same choice,
any p o ile ha assigns he same elemen x∗ o all agen s is suppo ed by any
public o de ing ha anks x∗as he unique lowes elemen in Xand hus all
o he al e na i es a e “blocked”. Such a p o ile migh be no Pa e o-op imal.
Thus, any Pa e o-op imal p o ile is a s a us equilib ium p o ile, bu a s a us
equilib ium p o ile does no ha e o be Pa e o-op imal.
3.2 S a us Equilib ium – Examples
Example: The Jobs Economy
Le Xbe a non-single on se o ypes o jobs. Each agen holds s ic
p e e ences on X. Feasibili y is gi en by a ec o (nx)x∈Xwhe e nxis
he numbe o a ailable jobs o ype x(non-emp iness o F equi es ha
Σx∈Xnx≥n).

3.2 S a us Equilib ium – Examples 69
A public o de ing in his example has a na u al in e p e a ion o social
s a us, which is o en associa ed wi h a job. Once an agen is assigned
o a job, he canno swi ch o a highe -s a us job bu he can swi ch o
any job o equal o lowe s a us (e.g. a p o esso can mo e o a lowe -
anked uni e si y bu no o a highe - anked one). The housing model o
Shapley and Sca (1974) is he special case whe e nx≡1 and |X|=n.
Claim: The ollowing holds o he jobs economy:
(i) I Σxnx=n, hen he Fi s Wel a e Theo em holds: e e y s a us
equilib ium p o ile is Pa e o-op imal.
(ii) I Σxnx>n, hen he Fi s Wel a e Theo em ails: he e is always a
s a us equilib ium p o ile ha is no Pa e o-op imal.
P oo : (i) Le ‹P,(xi)›be a s a us equilib ium. Assume by con adic ion
ha he easible p o ile (yi)Pa e o-domina es (xi). Le jbe an agen o
whom xjis P-maximal om among {xi|yi6=xi}. Since p e e ences a e
s ic , i mus be ha yjjxjand, he e o e, yjPx j. Since Σxnx=n, i
mus be ha in any easible p o ile, all jobs a e illed. The e o e, he e
is ano he agen whose o iginal job is yjand whose new job is no ,
con adic ing he P-maximali y o xj om among {xi|yi6=xi}.
(ii) Le (xi)be a Pa e o-op imal p o ile. By P oposi ion 3.1, he e is a
public o de ing such ha ‹P,(xi)›is a s a us equilib ium. Le zdeno e
a job wi h spa e capaci y, and le jbe an agen who does no ha e job
z(which exis s since Σxnx>nand |X|>1). Le (yi)be he easible
p o ile ob ained om (xi)by mo ing j om xj o z. Since (xi)is Pa e o-
op imal, e e y agen who does no ha e job zs ic ly p e e s his assigned
job o zand hus, (yi)is no Pa e o-op imal. Le P0be he public o de ing
ob ained om Pby mo ing z o he bo om ank. The pai ‹P0,(yi)›is
clea ly a s a us equilib ium.
70 Chap e 3. S a us and Indoc ina ion
Example: R-Mono onic P e e ences
Le Rbe a s ic pa ial o de ing (i e lexi e, ansi i e, and an i-
symme ic bu no necessa ily comple e) on X. A p e e ence ela ion
%is R-mono onic i abwhene e aRb. Fo example, le Xbe a se
o bundles and Rbe de ined by xRy i he bundle xcon ains weakly
mo e han yo e e y good and s ic ly mo e o a leas one. In his case,
R-mono onici y is he s anda d no ion o s ong mono onici y.
I will now be shown ha , o any economy wi h R-mono onic
p e e ences, any s a us equilib ium p o ile can also be suppo ed as
a s a us equilib ium wi h an R-mono onic public o de ing. Thus, a
s onge assump ion on agen s’ p e e ences (R-mono onici y) leads o
s onge conclusions abou he equilib ium public o de ing (being R-
mono onic).
Claim: Le Rbe a s ic pa ial o de ing and le ‹N,X,(%i)i∈N,F›be an
economy whe e e e y p e e ence %iis R-mono onic. I ‹P,(xi)i∈N›is a
s a us equilib ium, hen he e is an R-mono onic o de ing Qsuch ha
‹Q,(xi)i∈N›is also a s a us equilib ium.
P oo : De ine he desi e bina y ela ion Das y Dz i he e is an agen who
is assigned zand s ic ly p e e s y(and hus, i mus be ha y Pz s ic ly).
Le S=R∪D. The ela ion Sis acyclic: i no , le z1S1z2S2z3S3...zmSmz1
be a minimal cycle whe e each Siis ei he Ro D.
•I canno be ha all Sia e Rbecause Ris acyclic.
•I canno be ha all Sia e Dsince zi−1Dziimplies zi−1Pzis ic ly and
hus, a D-cycle implies a s ic P-cycle, which is impossible.
•I canno be ha he cycle S1,...,Smcon ains bo h Dand R. This is
because, i i did, hen i would con ain an R ollowed by a D. Howe e ,
i aRbDc, hen he e is a jsuch ha c=xjand bjc. Since j
ex ends R, i ollows ha ajband he e o e, ajcand so aDc.
The e o e, he cycle can be sho ened.
3.3 A De ou : Con ex P e e ences 71
Thus, Sis a s ic pa ial o de ing. Ex end S o an o de ing Q. Since S
ex ends R, so does Qand hus, Qis R-mono onic. Since Sex ends D, so
doesQand hus, ‹Q,(xi)›is a s a us equilib ium.
3.3 A De ou : Con ex P e e ences
In Sec ion 3.4, we will e ine he no ion o a s a us equilib ium by imposing
some s uc u e o he public o de ing. In p epa a ion, we make a de ou o he
concep o con ex p e e ences.
One con en ional de ini ion o con ex p e e ences o Euclidean spaces
equi es ha i ais weakly p e e ed o b, hen any con ex combina ion o a
and bis also weakly p e e ed o b. This de ini ion is equi alen o equi ing
ha all uppe con ou s (se s o he ype {x|xa}) a e con ex se s. Bo h
o hese de ini ions e e o he e m “con ex combina ion”, which i sel uses
an algeb aic s uc u e on he space o al e na i es and so does no apply o
economies whe e he se Xlacks such a s uc u e.
Following Rich e and Rubins ein (2019), we sugges an al e na i e de ini-
ion o con ex p e e ences which gene alizes he s anda d Euclidean no ion
and is also applicable o spaces wi hou algeb aic s uc u e. A co ne s one o
his app oach is he iew ha p e e ences a e buil om p imi i e building
blocks. He e, we ake he building blocks o be he membe s o a se o
o de ings Λ, which we call p imi i e o de ings. Each p imi i e o de ing is a
comple e, e lexi e, and ansi i e bina y ela ion o e he se X(indi e ences
a e allowed). We in e p e he p imi i e o de ings as exp essions o objec i e
a ibu es o he al e na i es ha a e in he ocabula y o all agen s.
The assump ion behind his de ini ion is ha , when hinking abou
eplacing an al e na i e b∈X, an agen has in mind a necessa y c i e ion
(p imi i e o de ing) ha is c i ical, in he sense ha , o an al e na i e o be
be e han b, i mus be be e by his c i e ion. No e ha he c i ical c i e ion
can depend on b.
72 Chap e 3. S a us and Indoc ina ion
Fo example, imagine a depa men chai who is con empla ing eplacing
b, who is a weak eache . In his case, he c i ical conside a ion may be
pedagogical abili y, and any eache who is pedagogically wo se han bwill be
ejec ed. Howe e , his does no mean ha any candida e who is pedagogically
be e han bwill be p e e ed. Again, he c i ical c i e ion can a y om one
al e na i e o ano he : when he depa men chai conside s eplacing c, who
is a g ea eache and a poo esea che , he may eel ha esea ch abili y is now
c i ical, and hus, any candida e who is a wo se esea che han cwill be judged
o be a wo se candida e han c.
De ini ion: Λ-con ex P e e ences
Le Xbe a se o objec s and Λbe a se o o de ings on X e e ed o as
p imi i e o de ings. The symbol D ep esen s a gene ic membe o Λ.
A p e e ence ela ion %on Xis Λ-con ex i :
∀b∈X,∃D∈Λsuch ha o x6=bi is necessa y o xb ha xBb.
A p e e ence ela ion %on Xis Λ-s ic ly con ex i :
∀b∈X,∃D∈Λsuch ha o x6=bi is necessa y o x%b ha xBb.
In bo h de ini ions, he o de ing Dis called a c i ical di ec ion a b( he e
can be mul iple c i ical di ec ions).
Th ee commen s:
(i) E e y (s ic ) p imi i e o de ing in Λis Λ-(s ic ly) con ex: o each
al e na i e, he p imi i e o de ing i sel is a c i ical di ec ion.
(ii) A “Pa e o” p ope y holds: I band ca e dis inc , bDc o e e y D∈Λ,
and %is Λ-con ex, hen b%c. This is because he e is a c i ical o de ing D
a ached o band bDcand he e o e, ccanno be s ic ly p e e ed o b. Fo
Λ-s ic ly con ex p e e ences, he conclusion is s onge , namely, bc.
(iii) In Rich e and Rubins ein (2019), we also sugges ed o he simila
de ini ions o con ex p e e ences and discussed hei connec ion o Edelman
and Jamison (1985)’s no ion o “abs ac con exi y”.
3.3 A De ou : Con ex P e e ences 73
Unde pinning ou con exi y no ion is he ab-
s ac ion o a concep ha plays a undamen al
ole in economic analysis when we alk abou
con ex p e e ences on a Euclidean space: o
each al e na i e, he e is a hype plane which
con ains i , such ha all weakly p e e ed al e -
na i es lie on one side o he hype plane. In he
same spi i , ou no ion o con ex p e e ences
equi es ha o e e y al e na i e he e is an
o de ing ha pu s all p e e ed al e na i es on
one side o he o de ing.
%
D
Figu e 3.1 A suppo ing hy-
pe plane and i s co espond-
ing c i ical di ec ion
The de ini ion o con ex p e e ences is a ac i e o se e al easons:
(a) I is compelling as a p ocedu al assump ion o p e e ence o ma ion.
(b) I emphasizes and allows o he dependence o he con exi y p ope y on
he speci ica ion o he conside a ions used o cons uc p e e ences.
(c) I gene alizes s anda d con exi y o Euclidean spaces, as will be shown
la e .
(d) I does no equi e any algeb aic s uc u e.
Example: Le and Righ
Le X= [0,1], and suppose ha Λcon ains wo o de ings: he igh is DR
(which anks elemen s o he igh highe ) and he le is DL(which anks
elemen s o he le highe ). A p e e ence ela ion is single-peaked i :
(i) i has a unique maximum poin (peak) in X; and
(ii) i is s ic ly inc easing below he peak and s ic ly dec easing abo e i .
Claim: Le Λ = {DL,DR}and X= [0,1]. A con inuous p e e ence ela ion
is Λ-s ic ly con ex i and only i i is single-peaked.
P oo : Suppose %is singled-peaked. A any b>peak, he o de ing DL
is c i ical, while a any b<peak he o de ing DRis c i ical. A he peak,
bo h o de ings a e c i ical.

74 Chap e 3. S a us and Indoc ina ion
Suppose %is Λ-s ic ly con ex. Since he p e e ences a e con inuous
and Xis compac , he e is a %-maximal elemen . I is unique since i
he e a e wo %-maximal elemen s y<z, and xis be ween yand z, hen
yDLxand zDRxand bo h yand za e weakly p e e ed o x. The e o e,
he e is no c i ical di ec ion a x.
Le Mbe he %-maximal elemen . Fo e e y y<x<M, he c i ical
o de ing a xmus be DRand hus y≺x. The e o e, %is s ic ly
inc easing o he le o M. Likewise, o e e y x>M, he c i ical o de ing
mus be DL, and %is s ic ly dec easing o he igh o M. Thus, he
p e e ences a e single-peaked wi h he peak a M.
The nex example shows ha , o con inuous p e e ences, he Λ-con exi y
no ion used he e gene alizes he s anda d no ion o con ex p e e ences on
Euclidean spaces.
Example: Euclidean Space wi h Algeb aic Linea O de ings
Le Xbe an open con ex subse o a Euclidean space. Fo any ec o
6=0, de ine he algeb aic linea o de ing ≥ by x≥ yi ∙x≥ ∙y. Le
Ψbe he se o all algeb aic linea o de ings.
Claim: Le %be a con inuous p e e ence ela ion on X. Then:
%is con ex by he s anda d de ini ion i and only i %is Ψ-con ex.
P oo : Assume %is con ex by he s anda d de ini ion. Tha is, o e e y
b∈X, he se U(b) = {z|zb}is con ex. Since %is con inuous, by he
sepa a ing hype plane heo em, he e exis s ≥ ∈Ψsuch ha o e e y
x∈U(b)i holds ha x> b. Tha is, ≥ is a c i ical di ec ion.
Assume %is Ψ-con ex. Le a,cbe elemen s in Xsuch ha a,cb,
and le zbe an elemen on he line be ween aand c. By Ψ-con exi y,
he e is a c i ical di ec ion ≥ a z. Then, z≥ ao z≥ co bo h, and
since &is Ψ-con ex, i ollows ha z%ao z%c, and hus zb.
3.3 A De ou : Con ex P e e ences 75
Cons uc ion o Con ex P e e ences: Fo a ini e se X, i all o de ings in Λ
a e s ic , hen he ollowing p ocedu e builds a Λ-con ex p e e ence ela ion:
Take an al e na i e x1which is a he bo om o one o he p imi i e o de ings,
and place i a he bo om o %. Then, le x2be an al e na i e a he bo om o
X−{x1}wi h espec o one o he p imi i e o de ings, and place i (s ongly o
weakly) abo e x1. Con inue his p ocedu e un il all al e na i es a e exhaus ed.
The cons uc ed p e e ence is Λ-con ex since he posi ion o each b∈Xin %
was de e mined when bwas a he bo om o some p imi i e o de ing, which
is hen a c i ical di ec ion a bsince any s ic ly p e e ed al e na i e is anked
s ic ly highe han bby ha o de ing.
I Xis ini e and all o de ings in Λa e s ic , hen e e y Λ-con ex p e e ence
ela ion %can be cons uc ed by he p ocedu e desc ibed abo e (see Rich e
and Rubins ein (2019)): To apply he cons uc ion o ob ain %, a e e y s age we
mus iden i y an al e na i e and a p imi i e o de ing Dso ha he al e na i e
is bo h %-minimal and D-minimal om among he emaining al e na i es. To
s a , pick x∈Xwhich is %-minimal, and le D∈Λbe a c i ical di ec ion a x. I
xis D-minimal, hen se x1=x. I no , hen pick ywhich is minimal acco ding
o he same D. The al e na i e yis also %-minimal since Dis a c i ical di ec ion
a xand xDy, and hen se x1=y. Con inue induc i ely wi h he emaining
al e na i es.
U ili y Rep esen a ion: We say ha a p e e ence ela ion %o e Xhas a Λ-
maxmin ep esen a ion i he e is a p o ile o unc ions (UD)D∈Λsuch ha o
e e y D∈Λ he unc ion UDis a u ili y ep esen a ion o Dand he unc ion
U(x) = minDUD(x)is well-de ined and ep esen s %.
I Λis ini e and %has a Λ-maxmin ep esen a ion (UD), hen %is Λ-con ex:
Fo any b∈X, ake he o de ing D∈Λ o whichUD(b)is minimal. The o de ing
Dis a c i ical di ec ion a bbecause bDximplies UD(b)≥UD(x)and hus,
U(b)≥U(x)and b%x. In Rich e and Rubins ein (2019), i is shown ha o
ini e X, he con e se is also ue: any Λ-s ic ly con ex p e e ence ela ion has
aΛ-maxmin ep esen a ion.
76 Chap e 3. S a us and Indoc ina ion
The exis ence o such a ep esen a ion means ha we can iden i y e e y
al e na i e in he se Xby a ec o o numbe s in RΛsuch ha :
(i) o e e y p imi i e o de ing, he alues ha a e a ached o he elemen s in
Xa he co esponding coo dina e a e consis en wi h ha p imi i e o de ing’s
anking; and
(ii) he p e e ences a e ep esen ed by he minimum alue a ached o an
al e na i e ac oss he di e en dimensions.
3.4 P imi i e Equilib ium
In he canonical consume model, he se o al e na i es (bundles) is a subse
o a Euclidean space and he ollowing holds:
(i) Agen s ha e s anda d con ex p e e ences.
(ii) The “mo e expensi e han” o de ing is induced by a linea p ice sys em.
In he language o his chap e :
(i) Agen s ha e Ψ-con ex p e e ences whe e Ψis he se o all algeb aic linea
o de ings (as shown in Sec ion 3.3).
(ii) The “mo e expensi e han” o de ing ≥pon he se o al e na i es (de ined
by x≥pyi p∙x≥p∙y) is a membe o Ψ.
An impo an poin is ha he same se o p imi i e o de ings appea s in
(i) and (ii) abo e. This sugges s wo new de ini ions. Fi s , we en ich he
no ion o an economy wi h a se o p imi i es o de ings Λand equi e ha
all agen s’ p e e ence ela ions a e Λ-con ex. We e e o such an economy by
he e m con ex economy. Second, we e ine he s a us equilib ium no ion and
equi e ha he public o de ing is one o he p imi i e o de ings in Λ. We e e
o such an equilib ium as a p imi i e equilib ium. Fo mally:
3.4 P imi i e Equilib ium 77
De ini ion: Con ex Economy
Acon ex economy is a uple ‹N,X,(%i)i∈N,F,Λ›whe e ‹N,X,(%i)i∈N,F›
is an economy, Λis a se o p imi i e o de ings o e X, and all p e e ences
a e Λ-con ex.
De ini ion: P imi i e Equilib ium
Le ‹N,X,(%i)i∈N,F,Λ›be a con ex economy. A p imi i e equilib ium is
a s a us equilib ium ‹D,(xi)i∈N›whe e D∈Λ.
Ob iously, any p imi i e equilib ium is a s a us equilib ium, and when Λis
he se o all o de ings, any s a us equilib ium is a p imi i e equilib ium.
Example: The Gi e-and-Take Con ex Economy
We e u n o he gi e-and- ake economy. Recall ha X= [−1,1]and F
is he se o all p o iles ha sum up o 0. Le Λconsis o he wo na u al
o de ings: he igh is DR(which a ou s aking) and he le is DL(which
a ou s gi ing). Assume ha e e y agen iholds con inuous Λ-s ic ly
con ex p e e ences (wi h a single peak deno ed by peaki).
When Σipeaki=0, he e is no con lic o in e es in he economy and
ei he p imi i e o de ing, oge he wi h all agen s choosing hei peaks,
is a p imi i e equilib ium. In ac , hese a e he only p imi i e equilib ia
(i ‹DL,(xi)i∈N›is an equilib ium, hen peaki≤xi o all iand he e o e,
xi=peaki o all i).
A mo e in e es ing case is Σipeaki>0 whe e agen s wish o ake mo e
han hey wish o gi e. Le F≤be he se o all easible p o iles wi h all
agen s a o o he le o hei peaks. We now e i y ha F≤is equal
o he se o all Pa e o-op imal p o iles. Any (xi)∈F≤is Pa e o-op imal
because any p o ile (yi) ha Pa e o-domina es i mus ank xi≤yi o all
i, wi h a leas one s ic inequali y; howe e , such a p o ile is in easible
because 0 = Σxi<Σyi. On he o he hand, i (xi)is easible and no
84 Chap e 3. S a us and Indoc ina ion
u ili y unc ion (Θ) = Σz∈Θ (z). Tha is, (Θ) is he sum o he -
alues a ached o he indi idual i ems in he se Θ. Le Λbe he se o
such s ic o de ings. I u ns ou (see Rich e and Rubins ein (2019))
ha he Λ-con ex p e e ences a e exac ly all p e e ences ha a e weakly
mono onic wi h espec o he inclusion ela ion. We assume ha all
agen s’ p e e ence ela ions a e s ic and Λ-con ex.
In his economy, a p imi i e equilib ium has he in e p e a ion ha
a p ice is a ached o each good and he p ice o a collec ion o goods is
he sum o he p ices o he goods in he collec ion. In con as , a s a us
equilib ium has he in e p e a ion ha he e is a p ice o each collec ion.
Claim: Fo he se alloca ion economy: he se o p imi i e equilib ium
p o iles ⊆ he se o Pa e o-op imal p o iles ⊆ he se o s a us equilib-
ium p o iles, and hese inclusions can be s ic .
P oo : To es ablish he i s inclusion, by P oposi ion 3.2 i su ices o
e i y ha condi ion Dholds. Take a p imi i e o de ing D . Fo any wo
dis inc easible p o iles, (Θi)and (Φi), i holds ha Σi (Θi) = Σi (Φi) =
(Z). Thus, i canno be ha (Θi)≥ (Φi) o all iwi h a leas one s ic
inequali y.
Howe e , he e can be Pa e o-op imal p o iles ha a e no p imi i e
equilib ium p o iles. Fo example, le Z={a,b,c,d}and n=2. Bo h
agen s ha e p e e ences ha ank any ca dinally la ge se highe and a e
he e o e, Λ-con ex. To simpli y no a ion, deno e he se o goods {x,y}
as xy . Table 3.1 depic s he agen s’ p e e ences o e wo-elemen se s:
%1%2
ac,bd ad ,bc
ab cd
ad ,bc,cd ab,ac,bd
Table 3.1 P e e ences wi h a Pa e o-op imal p o ile ha is no a p imi i e
equilib ium p o ile (highligh ed).

3.8 Ini ial S a us Equilib ium 85
The p o ile (x1,x2)=(ab,cd )is Pa e o-op imal. Howe e , he e is no
public o de ing ≥ ha suppo s his p o ile as a p imi i e equilib ium.
I he e we e, hen ac > ab ( o ensu e ha ab is op imal o agen 1),
which implies ha (c)> (b). Simila ly, we can conclude ha (b)>
(d)> (a)> (c), a con adic ion.
Fo he second inclusion, ecall ha by P oposi ion 3.1, any Pa e o-
op imal p o ile is a s a us equilib ium p o ile. Howe e , he e a e
se alloca ion economies wi h s a us equilib ium p o iles ha a e no
Pa e o-op imal. Fo example, suppose ha Z={a,b,c,d},n=2, and
bo h agen s ha e he same Λ-con ex p e e ences %∗sa is ying ha he
se s ac and bd a e %∗-supe io o ab and cd . Then, ‹P=%∗,(ab,cd )›is
a s a us equilib ium ha is no Pa e o-op imal. 
This example demons a es a s a k con as be ween equilib ia wi h
i em-p icing (whe e he p ice o a bundle is he sum o he indi idual
i ems’ p ices) and hose wi h bundle-p icing (whe e a p ice is a ached
o each bundle). The ollowing able summa izes he abo e claim:
I em-p icing Bundle-p icing
equilib ia equilib ia
Fi s Wel a e Theo em ØX
Second Wel a e Theo em XØ
Table 3.2 Depic ion o he Claim
3.8 Ini ial S a us Equilib ium
In his sec ion, we ex end he de ini ion o a s a us equilib ium o co e
ex ended economies. To emind he eade , an ex ended economy is
an economy wi h he speci ica ion o an addi ional easible p o ile (ei)i∈N
in e p e ed as an “ini ial p o ile”. I speci ies an al e na i e o each agen which
86 Chap e 3. S a us and Indoc ina ion
he has he absolu e igh o choose, independen ly o o he agen s’ choices
and o he equilib ium pa ame e s. When he al e na i es a e asse s, he ini ial
p o ile can be hough o as speci ying ini ial owne ship.
De ini ion: Ini ial S a us Equilib ium
Gi en an ex ended economy ‹N,X,(%i)i∈N,F,(ei)i∈N›:
An ini ial s a us equilib ium is a pai ‹P,(xi)i∈N›whe e Pis an o de ing
on Xand (xi)is a easible p o ile such ha e e y agen i’s assigned
al e na i e xiis %i-op imal in his “budge se ” B(P,ei) = {x∈X|eiPx}.
In an ini ial s a us equilib ium, an agen ’s choice se consis s o all al e na i es
ha a e weakly P-in e io o his ini ial al e na i e. In con as , in a s a us
equilib ium, an agen ’s choice se consis s o all al e na i es ha a e weakly
P-in e io o his equilib ium al e na i e.
Two commen s:
(i) I xiis %i-maximal in B(P,ei), hen xiis also %i-maximal in B(P,xi).
Thus, any ini ial s a us equilib ium o an ex ended economy is also a s a us
equilib ium o he unde lying economy.
(ii) I ‹P,(xi)›is a s a us equilib ium, hen o e e y s ic o de ing P0which
is a ieb eaking o P, he pai ‹P0,(xi)›is also a s a us equilib ium. This is
no he case o an ini ial s a us equilib ium: Conside he ex ended housing
economy wi h wo houses aand b, wo agen s, ini ial p o ile (e1,e2) = (a,b),
and p e e ence ela ions b1aand a2b. The public o de ing ha equally
anks aand band he p o ile (b,a)cons i u e an ini ial s a us equilib ium o
he ex ended economy. Howe e , b eaking his indi e ence will in alida e he
equilib ium since one o he wo agen s will no be able o “a o d” he o he
house.
E en hough a s a us equilib ium exis s when a Pa e o-op imal p o ile does
(P oposi ion 3.1), he ollowing example demons a es ha he exis ence o an
ini ial s a us equilib ium is no gua an eed e en o ini e ex ended economies.
3.8 Ini ial S a us Equilib ium 87
Example: An Ex ended Jobs Economy
Conside he jobs economy o Sec ion 3.2 wi h 3 agen s, wo jobs aand
b, and capaci ies na=2 and nb=1. Assume ha agen s 1 and 2 p e e b
and agen 3 p e e s a.
The e a e wo Pa e o-op imal p o iles (b,a,a)and (a,b,a), bo h o
which a e s a us equilib ium p o iles (wi h he public o de ing bPa). I
ei he o hose p o iles is he ini ial p o ile, hen i is also an ini ial s a us
equilib ium p o ile.
Howe e , i he ini ial p o ile is (a,a,b), whe e each agen s a s wi h
he al e na i e he dislikes, hen an ini ial s a us equilib ium does no
exis . To see why, no e ha an equilib ium public o de ing canno ank
aweakly abo e b, because hen agen s 1 and 2 would bo h choose b,
iola ing easibili y. No can i be ha bis anked s ic ly abo e a,
because hen all h ee agen s would choose a, again iola ing easibili y.
The “p oblem” is ha he ini ial s a us equilib ium concep does no
allow o he exchange o aand bbe ween 1 and 3 (o be ween 2 and
3) due o he equilib ium concep ’s inabili y in allowing di e en budge
se s o wo agen s wi h he same ini ial al e na i e.
The eade may wonde why no equilib ium exis s in his ex ended
economy whe eas an equilib ium does exis in he s anda d compe i i e
ma ke model. The eason is ha , in he s anda d compe i i e ma ke
model he e is also money in he economy and a mone a y amoun
can be a ached o he ansac ion o exchanging a o bso ha a
leas one o he wo agen s who p e e b o awould be indi e en
be ween conduc ing he ansac ion o e aining om i . Then, he
public o de ing in he s anda d ma ke is no me ely o dinal bu ca dinal,
indica ing he mone a y amoun equi ed o exchange a lowe - anked
good o a highe - anked one.
Thus, he exis ence o an ini ial s a us equilib ium is no gua an eed when he
ini ial p o ile assigns iden ical elemen s o di e en agen s. Howe e , whene e
88 Chap e 3. S a us and Indoc ina ion
e e y agen has a dis inc ini ial al e na i e, he ollowing p oposi ion es ab-
lishes he exis ence o an ini ial s a us equilib ium. Fu he mo e, i shows
ha i in addi ion he e is a s ic pa ial o de ing Rsuch ha all indi idual
p e e ences a e R-mono onic ( ha is, i aRb hen aib o all i), hen he e
is an R-mono onic equilib ium public o de ing. Taking R o be he emp y
bina y ela ion gi es he baseline esul o Shapley and Sca (1974) (p esen ed
in Sec ion 1.3).
P oposi ion 3.4: Exis ence o an Ini ial S a us Equilib ium
Any ex ended economy ‹N,X,(%i)i∈N,F,(ei)i∈N›whe e all ini ial al e -
na i es a e dis inc has an ini ial s a us equilib ium. I , in addi ion,
all p e e ence ela ions a e R-mono onic wi h espec o a s ic pa ial
o de ing R, hen he public o de ing can be aken o be R-mono onic as
well.
P oo :
Le Ybe he se o al e na i es in he ini ial p o ile (ei). Fo any Z⊆Y,
de ine M(Z) = {i|ei=z o some z∈Z} o be he se o agen s ini ially
assigned o al e na i es in Z. Following he cons uc ion
in P oposi ion 1.5, selec a sequence o op ading
cycles, B1,...,BT. De ine a pa ial o de ing Pon Yby
aPb i a∈B ,b∈Bs, and ≤s(all elemen s in he same
B a e P-indi e en ). We need o ex end P o all o X.
Pa i ion X Yin o se s A1,...,AT+1as ollows: Fo any
x∈X Y, le x∈A whe e is he smalles index such
ha he e is an agen i∈M(B )who s ic ly p e e s x
o e all elemen s in B . I he e is no such , hen le
x∈AT+1. Place he elemen s in any A below B −1and
abo e B . De ine Pon A as any a bi a y expansion o
R. To see ha Pexpands Re e ywhe e, conside aand
bsuch ha aRb (and hus, all agen s p e e a o b).
AT+1
BT
AT
B2
A2
B1
A1
Figu e 3.4
The Con-
s uc ion
3.8 Ini ial S a us Equilib ium 89
•I b∈Y, hen b∈B o some . I a∈Y, hen i mus belong o
an ea lie ading cycle because no agen would op- ank bwhen ais
p esen , and hus, aPb. I a/∈Y, hen he agen who op- anks b∈B
p e e s a o all elemen s o B . The e o e, a∈Aswi h s≤ , hus, aPb.
•I b∈A o some ≤T, hen o some i∈M(B )i holds ha biy
o all y∈B . Thus, ialso p e e s ao e all y∈B . I a∈Y, hen a
belongs o a p e ious ading cycle and i a/∈Y, hen i belongs o As
wi h s≤ . In ei he case aPb ( o he case ha a,b∈A , ecall ha P
expands Ron A ).
•I b∈AT+1and a/∈AT+1, hen aPb and i a∈AT+1, hen aPb because
Pexpands Ron AT+1.
The p o ile (yi), which assigns o each agen i∈M(B ) he elemen yi
ha eipoin s o in he op ading cycle B , oge he wi h P, cons i u es
an ini ial s a us equilib ium: Fi s , (yi)is easible since i is a pe mu a ion
o he ini ial p o ile. Second, o e e y i,eiPy ibecause yiand eia e in
he same cycle. Thi d, suppose ha ziyi o some agen i. Then,
i z∈Y, i belongs o an ea lie cycle. I z/∈Y, hen ip e e s z o all
elemen s o B , and so z∈As o s≤ . In ei he e en , zPy i.
Example: The Ex ended Gi e-and-Take Economy
Ex end he gi e-and- ake economy by adding an ini ial p o ile (ei)i∈N
ha is easible, Σei=0. Each agen i o whom ei>0 has he igh o
ake ei om he public und, while each agen i o whom ei<0 has he
igh o con ibu e −ei. Remembe ha e e y agen ihas con inuous
and s ic ly con ex (and hus single-peaked) p e e ences wi h a peak a
peaki. As be o e, we ocus on he case whe e Σpeaki>0. He e, we adop
he in e p e a ion ha aPb means ha bis mo e socially bene icial han
a. Each agen chooses how much o gi e o ake om he al e na i es
ha a e mo e socially bene icial han his ini ial assignmen .

90 Chap e 3. S a us and Indoc ina ion
The exis ence o an ini ial s a us equilib ium o his ex ended
economy is gua an eed by P oposi ion 3.4 only o he case ha all ei
a e dis inc . He e, we cons uc a simple ini ial s a us equilib ium wi h
an a ac i e s uc u e which also demons a es exis ence e en when he
eia e no dis inc .
Le Pzbe he o de ing ha places all al e na i es be ween −1 and
zequally a he bo om and is s ic ly inc easing om z o 1. E e y
agen i aces he in e al budge se [−1,max{z,ei}]and so has a unique
op imal choice which is con inuous in z, weakly inc easing, and s ic ly
inc easing o z∈[ei,peaki](in he case ha ei<peaki). Gi en he o al
indi e ence o de ing P1, e e y agen would choose peaki, and he sum o
hei chosen ac ions would be Σipeaki>0. Gi en he s ic ly inc easing
o de ing P−1, e e y agen ichooses an al e na i e xi≤eiand he sum o
he chosen al e na i es is non-posi i e since Σixi≤Σiei=0. Thus, by
he con inui y o he agen s’ choices in z, he e is a z∗∈[−1,1] o which
he sum o he chosen elemen s is 0. The o de ing Pz∗ oge he wi h he
p o ile o op imal choices om he co esponding budge se s is an ini ial
s a us equilib ium.
4Biased P e e ences Equilib ium
In any economy, he co e ension is be ween agen s’ wan s and socie al
easibili y, and an equilib ium no ion inds a balance be ween hem. In
Chap e s 2 and 3, we in es iga ed equilib ium no ions ha in oke a ious
social mechanisms o achie e ha balance: no ms eme ge ha a ec agen s’
oppo uni y se s such ha i e e y agen op imizes his p e e ence ela ion, hen
he p o ile o op imal choices is easible.
This chap e akes a di e en app oach. We ollow Rubins ein and Wolinksy
(2022) who p opose a solu ion concep which cap u es a di e en social
mechanism ha can esol e he undamen al con lic be ween wan s and
easibili y: agen s’ p e e ence ela ions a e sys ema ically biased. The bias does
no a ec he agen s’ oppo uni y se s bu , a he , hei p e e ences, which a e
sys ema ically biased in such a way ha he p o ile o agen s’ biased op imal
choices is easible.
Recall Aesop’s classic able ( ansla ion om Gibbs (2002)):
D i en by hunge , a ox ied o each some g apes hanging high
on he ine bu was unable o, al hough he leaped wi h all his
s eng h. As he wen away, he ox ema ked “Oh, you a en’ e en
ipe ye ! I don’ need any sou g apes.”
In his able, he e is one agen , he Fox, and wo al e na i es, “picking he
g apes” and “no picking he g apes”. The economic p oblem is ha he Fox
ini ially p e e s he o me al e na i e bu only he la e al e na i e is easible.
The con lic in he able is esol ed no by es ic ing he Fox’s oppo uni ies
bu , a he , by biasing his p e e ences so ha he now p e e s no o pick he
g apes (which in his mind a e u ned o “sou g apes”).
©2024 Michael Rich e and A iel Rubins ein, CC BY-NC-ND 4.0
h ps://doi.o g/10.11647/OBP.0404.04
92 Chap e 4. Biased P e e ences Equilib ium
P e e ence biases a e no jus a ma e o ables. In ospec ion ells us ha
easibili y o en in luences ou p e e ences in e e yday li e. We o en assign
g ea e alue o wha we can ob ain (such as being an economis ) and less o
wha we canno (such as being a ma hema ician). Howe e , we do no deny
ha he e a e also ci cums ances whe e he opposi e is ue, and he mo e
unob ainable some hing is he mo e desi able i becomes.
The Fox biased his p e e ences, and ha mony was achie ed. Simila ly,
we en ision biases as a mechanism o b inging ha mony o a mul i-agen
economy. These biases, like p ices, will be sys ema ic and apply uni o mly
o all agen s. E e y agen ’s inal p e e ences a e de e mined by bo h he
commonly sha ed bias and his ini ial p e e ences. Thus, in con as o a
compe i i e equilib ium whe e p ices a ec choice se s and p e e ences a e
ixed, in a biased p e e ences equilib ium biases a ec p e e ences and choice
se s a e ixed. This illus a es he dual oles played by p ices and p e e ences in
s anda d economic se ings.
No e he di e ence be ween his chap e ’s app oach and he one aken by
o he economic models. In some o hose models, he change in p e e ences is
a side e ec o an agen ’s ac ion ( o example, smoking may in luence he desi e
o smoke in he u u e, as modelled by Becke and Mu phy (1988)). In o he s,
he change in p e e ences is he ou come o a delibe a e ac ion by an in e es ed
pa y ( o example, ad e ise s seek o in luence cus ome s’ p e e ences o
hei own ad an age, as modelled by Bagwell (2007)). By con as , his chap e
models social si ua ions in which p e e ences in isibly espond o easibili y
p essu es, jus as p ice adjus men s achie e ha mony in a compe i i e ma ke .
4.1 The Economy and he Equilib ium Concep
In his chap e , he no ion o an economy is modi ied o accommoda e
modelling sys ema ic p e e ence biases.
4.1 The Economy and he Equilib ium Concep 93
De ini ion: An Economy
An economy is a uple ‹N,(Xi)i∈N,K,((ui
k)k∈K)i∈N,F›whe e:
•Fo each agen i,Xiis his ixed pe sonal choice se .
•The se Kis a se o conside a ions common o all agen s.
•Fo each agen i,(ui
k)k∈Kis a uple o conside a ion unc ions o e Xi
such ha i’s u ili y unc ion o e Xiis Σkui
k(x).
•The se o easible p o iles, F, is a subse o Πi∈NXi.
This de ini ion modi ies ou no ion o an economy in wo ways. Fi s , and
less impo an ly, di e en agen s can ha e di e en choice se s. This allows o
modelling a a ie y o se ings. Fo example, an exchange economy wi h a se
o goods K, a ixed p ice ec o p, and ini ial endowmen p o ile (ei) can be
modelled by se ing Xi={x∈RK
+|p∙x=p∙ei}and F={(xi)|Σxi= Σei}.
Ano he example is a wo-sided ma ching ma ke wi h wo equally-sized
popula ions Aand B. This can be modelled by se ing Xi=B o any i∈A
and Xj=A o any j∈B, while Fis he se o all p o iles (xi) o which o e e y
i,j,xi=jimplies xj=i.
The second and mo e impo an modi ica ion o he o iginal de ini ion
o an economy is he use o a di e en no ion o p e e ences. Ra he han
speci ying an o dinal p e e ence ela ion o e he se o al e na i es, we use he
ollowing ype o u ili y unc ion ha enables us o model sys ema ic biases.
All agen s sha e he same se o conside a ions K. Each agen iis cha ac e ized
no by an o dinal p e e ence ela ion, bu by a ec o o conside a ion unc ions
ui= (ui
k)k∈Kwhe e ui
k(x) ep esen s he impac o conside a ion kon his
o e all e alua ion o he al e na i e x. The conside a ion unc ions a e no
cons an and, whe e applicable, a e di e en iable. Agen i’s o e all u ili y om
an al e na i e xis he sum o he u ili ies ob ained om hose conside a ions,
i.e. Σk∈Kui
k(x).
A p e e ence bias is modelled as a sys ema ic and uni o m change in he
weigh s placed on he conside a ions. Le Λ = RK
++ be he se o biases. A bias
100 Chap e 4. Biased P e e ences Equilib ium
agen s a e assigned o al e na i es weakly below hei peaks). In con as , in
a Y-equilib ium, he e is a uni o m cap on wi hd awals and only he g eedies
agen s a e impac ed, and in a jungle equilib ium, only he weakes agen s a e
es ic ed.
4.3 The Fixed-P ices Exchange Economy
The nex example is ela ed o he li e a u e on economies wi h ixed p ices (see
Benassy (1986) and he e e ences he ein). Le X=RK
++ be he se o bundles
in a wo ld wi h a se o goods K. E e y agen ihas an ini ial endowmen
ei∈X, and exchange akes place acco ding o a ixed p ice ec o p= (pk).
Acco dingly, Xi={x∈X|p∙x=p∙ei}, and he se o easible p o iles is
F={(xi)∈ΠiXi|Σixi= Σiei}. All agen s sha e he same conside a ions, one
o each good. Each conside a ion unc ion ui
k(x)is a unc ion o only xk, which
is assumed o be inc easing, wice-di e en iable, and s ic ly conca e.
In economies wi h ixed p ices, a ioning is ypically he mechanism used
o achie e ha mony. Tha is, uppe bounds a e es ablished on he consumable
quan i y o each good. In con as , in a biased p e e ences equilib ium,
economic ha mony is achie ed by means o a sys ema ic adjus men o
p e e ences.
P oposi ion 4.2: Biased P e e ences Equilib ia in Exchange Economies
wi h Fixed P ices
In any exchange economy wi h ixed p ices:
(i) A biased p e e ences equilib ium exis s.
(ii) All biased p e e ences equilib ium ou comes a e p e-Pa e o op imal.
P oo :
(i) To illus a e, conside he wo-good wo-agen case, which can be
depic ed using an Edgewo h Box (see Figu e 4.1). Assume ha agen s
do no like consuming on he bounda y, i.e. he de i a i e o e e y

4.3 The Fixed-P ices Exchange Economy 101
conside a ion unc ion ui
ka 0 is in ini y. Le (x1,x2)be a Pa e o-op imal
easible alloca ion o e1+e2, which always exis s. Then, a (x1,x2), bo h
agen s ha e he same ma ginal a e o subs i u ion μ. I μ=p1/p2,
hen no bias is needed, ha is, ‹λ= (1,1),(xi)›is a biased p e e ences
equilib ium. I μ6=p1/p2, hen he bias λ= (p1,p2μ)modi ies bo h
p e e ences so ha he MRS1,2 o he biased p e e ences o each ia xi
is μλ1/λ2=p1/p2. Thus, ‹λ,(xi)›is a biased p e e ences equilib ium.
(x1,x2)
p1
p2
u1u1
u2
u2
(e1,e2)
λ(x1,x2)
p1
p2
T(u1,λ)
T(u2,λ)(e1,e2)
Figu e 4.1 Equilib ium in an Edgewo h Box
Fo he case o mo e han wo goods and any numbe o agen s, Keiding
(1981) ( ollowing Balasko (1979)) showed ha he e is a ec o q= (qk)
and an alloca ion (xi)such ha p∙xi=p∙ei o all i, and i yiixi, hen
q∙yi>q∙xi. The e o e, o e e y agen i, any good l ha he consumes
and any o he good k, i holds ha MRSk,la xiis bounded om abo e
by qk/ql. Consequen ly, by se ing λ= (pk/qk)k∈K, i holds ha o agen
i’s biased p e e ences and any good l ha he consumes, he MRSk,la xi
is bounded om abo e by pk/pl=λkqk/λlql( he bound is an equali y
i xi
k>0). The e o e, o e e y agen i, gi en he p ice ec o pand
he ini ial bundle ei, he bundle xiis op imal o i’s biased p e e ences.
Thus, ‹λ,(xi)›is a biased p e e ences equilib ium.
(ii) Le ‹λ,(xi)›be a biased p e e ences equilib ium. Then, o each agen
iand any good l ha he consumes, he MRSk,lo he biased p e e ences
T(λ,ui)a xiis bounded om abo e by pk/pl. The e o e, he MRSk,lo
102 Chap e 4. Biased P e e ences Equilib ium
his ini ial p e e ences a xiis bounded om abo e by pk/λk
pl/λl. Thus, (xi)
is a Wal asian equilib ium ou come in he unbiased economy wi h p ice
ec o pk/λkand ini ial endowmen (xi). The e o e, by he s anda d
Fi s Wel a e Theo em, (xi)is p e-Pa e o op imal.
We now conside an example wi h wo goods and linea p e e ences whe e
he biased p e e ences equilib ium can easily be calcula ed.
Example: Linea P e e ences
Suppose ha he e a e wo agen s, wo goods, and ha o e e y agen
i, he wo conside a ion unc ions a e linea , ha is, ui
1(x1) = x1and
ui
2(x2) = αix2whe e e e y αiis a posi i e numbe . Conside he
con igu a ion depic ed in Figu e 4.2:
T(λ,u1)T(λ,u2)u1u2u1u2T(λ,u1)T(λ,u2)
p1
p2
u1
u1
u2
u2
T(u1,λ)
T(λ,u1)
T(u2,λ)
T(λ,u2)
(e1,e2)
con ac cu e
biased p e e ences equilib ium
Figu e 4.2 Biased linea p e e ences in an Edgewo h Box (dashed line = he
budge lines; black solid lines =ini ial p e e ences; ed solid lines =biased
p e e ences; blue line = he con ac cu e o he ini ial p e e ences)
In his example (o he con igu a ions can be analyzed simila ly):
(i) Agen 2 likes good 2 mo e han agen 1 does, ha is, α2> α1.
(ii) The a io p1/p2is g ea e han bo h agen s’ (cons an ) pe sonal
ma ginal a es o subs i u ion, ha is, p1/p2>1/α1>1/α2.
4.3 The Fixed-P ices Exchange Economy 103
(iii) In any easible alloca ion, bo h agen s mus consume posi i e
amoun s o good 1.
The economy is no in ha mony because, gi en (ii), bo h agen s wish o
pu chase only good 2.
In any biased p e e ences equilib ium, he e is an agen iwho consumes
good 2, and by (iii), he also consumes good 1. By he linea i y o he
p e e ences, agen imus be indi e en be ween all al e na i es in Xi,
ha is, p1/p2=λ1/(αiλ2). I i=1, hen by (i), agen 2 does no consume
good 1, iola ing (iii). Thus, i mus be ha i=2, and he bias sa is ies
p1/p2=λ1/(α2λ2).
Such a bias is pa o he equilib ium depic ed in Figu e 4.2, whe e
agen 1 consumes only good 1, and agen 2 (who is indi e en be ween
all bundles in his budge se ) consumes all o good 2 and he emainde o
good 1. I ollows ha his is he unique biased p e e ences equilib ium.
Failu e o Indi idual Ra ionali y: An in e es ing ea u e o a biased p e e -
ences equilib ium is ha , e en hough i is p e-Pa e o op imal, “Indi idual
Ra ionali y” can ail: in an equilib ium, an agen migh choose a bundle ha
is in e io o his endowmen bundle when judged by his ini ial p e e ences, as
in he p e ious example. By his o iginal p e e ences, agen 1 is wo se o in he
equilib ium han he was wi h his ini ial endowmen , since he ades some o his
good 2 endowmen o good 1, bu ex-an e he would p e e o do he opposi e.
Example: Non-Con ex P e e ences
In he s anda d exchange economy wi h non-con ex p e e ences, a
compe i i e equilib ium may no exis : he e may be no p ice ec o o
which he sum o he demands equals he o al bundle. Ne e heless,
he e may be a p ice ec o o which a biased p e e ences equilib ium
exis s. Thus, p ices and biased p e e ences oge he may achie e ha -
mony when he s anda d compe i i e equilib ium ools ail o do so.
104 Chap e 4. Biased P e e ences Equilib ium
To illus a e, conside he di ision economy whe e bo h agen s ha e
he non-con ex p e e ences ep esen ed by (x1)2+2(x2)2and he ini ial
endowmen s a e e1= (1,1)and e2= (2,2). The e is no s anda d
compe i i e equilib ium. Gi en any p ice ec o , each agen will
consume only one o he wo goods, and since he agen s ha e he
same p e e ences, any equilib ium p ice ec o mus make each agen
indi e en be ween he wo goods, i.e. p= (1,p2). Bu , hen agen 2 will
demand mo e han 3 uni s o one o he goods.
In con as , a biased p e e ences equilib ium exis s. Le p= (2,1)and
λ= (8,1). Each agen ’s biased u ili y unc ion is 4(x1)2+ (x2)2. Agen 1’s
op imal bundles a e (1.5,0)and (0,3), and agen 2’s op imal bundles a e
(3,0)and (0,6). Thus, he bias λ, oge he wi h he alloca ion x1= (0,3)
and x2= (3,0), is a biased p e e ences equilib ium in he exchange
economy wi h ixed p ices p.
No e ha agen 1 is ini ially poo e han agen 2, bu in he
equilib ium, agen 1 is ac ually be e o acco ding o he ini ial
p e e ences!
4.4 Housing-Type Economies
We e u n o he classic housing economy o Shapley and Sca (1974), in which
he e is a se No agen s and an equally-sized se Ho houses. Each agen i
chooses a single house, ha is, Xi=H. Le i(h)>0 be agen i’s alua ion o
house h. The model can be en iched o i ou amewo k by aking he se o
conside a ions o be Hand se ing ui
h(xi) = i(h)i xi=hand 0 o he wise.
Gi en a bias ec o (λh), an agen ide i es u ili y λh i(h) om house h.
Example:
The ollowing able p esen s he conside a ion unc ion alues in a
housing economy wi h wo agen s.
4.4 Housing-Type Economies 105
h1h2
1(h)4 3
2(h)3 1
Table 4.2 House u ili ies
Bo h agen s ini ially p e e house h1. To achie e ha mony, he bias mus
boos h2so ha one agen will choose i , bu no o he ex en ha
bo h will. Fo example, a biased p e e ences equilib ium is ob ained
by he bias (1,2), which esul s in agen 1 choosing h2and agen 2
choosing h1. O cou se, o he biases a e possible bu , in all biased
p e e ences equilib ia, agen 1 ge s h2and agen 2 ge s h1. No e ha ,
in he biased p e e ences equilib ium p o ile, he p oduc o he ex-an e
alues (3∙3=9) is la ge han ha in he o he assignmen (4 ∙1=4). We
will see below ha his is no a coincidence.
We say ha a easible p o ile (xi)is Nash maximal i i maximizes Πi∈N i(xi)
o e all easible p o iles. We now show ha he se o biased p e e ences
equilib ium p o iles is p ecisely he se o Nash-maximal p o iles and hus, any
biased p e e ences equilib ium p o ile is p e-Pa e o op imal. The p oo is a
di ec applica ion o Shapley and Shubik (1971) (see also Gale (1984) o a p oo
using he KKM Lemma).
P oposi ion 4.3: Biased P e e ences Equilib ium =Nash Maximali y
In he housing economy, he se o biased p e e ences equilib ium
p o iles is he se o Nash-maximal p o iles.
P oo :
Le (hi)i∈Nbe a Nash-maximal p o ile, ha is, i maximizes Σi∈Nln( i(xi))
o e all easible assignmen s. By Shapley and Shubik (1971), he e exis s
a p ice ec o (ph)so ha o each agen i, he house hiis a maximize o

106 Chap e 4. Biased P e e ences Equilib ium
ln( i(xi)) −pxi, and he e o e, i is also a maximize o i(xi)/epxi. Thus,
‹(λh=1/eph)h∈H,(hi)i∈N›cons i u es a biased p e e ences equilib ium.
In he o he di ec ion, le ‹λ,(hi)›be a biased p e e ences equilib ium
and (xi)be any o he assignmen . Fo each i,λhi i(hi)≥λxi i(xi)and
he e o e, Πiλhi i(hi)≥Πiλxi i(xi). Since Πiλhi= Πiλxi, i ollows ha
Πi i(hi)≥Πi i(xi), ha is, (hi)is Nash maximal.
We p oceed wi h wo modi ica ions o he housing economy:
Example: The Pa ne ship Economy
As in Shapley and Shubik (1971), he agen s a e composed o wo equally-
sized popula ions, Aand B. Each agen chooses a unique pa ne om
he o he popula ion, ha is, Xi=B o any i∈Aand Xj=A o any
j∈B. A p o ile is easible i o e e y i,j, i ichooses j, hen jchooses
i. An agen i’s alua ion o a pa ne ship wi h jis i(j)>0. Impo an ly,
he ex-an e alua ions a e assumed o be symme ic, ha is, i(j) = j(i),
bu he biased alua ions migh no be.
Example: Le A={1,2}and B={3,4}. Table 4.3 p esen s he o iginal
alua ions (le bi-ma ix) and he equilib ium biased alua ions ( igh
bi-ma ix). Each cell gi es he alues o iand jo being ma ched. The
Nash-maximal ma ching is 1↔4 and 2↔3 (depic ed), which is a biased
p e e ences equilib ium wi h he bias λ= (2,1,2,1).
3 4
1 1,1 3,3
2 3,3 4,4
T(∙,λ)
→
3 4
1 2,2 3,6
2 6,3 4,4
Table 4.3 A Biased P e e ences Equilib ium
Claim: In he pa ne ship economy, he se o biased p e e ences
equilib ium ou comes is he se o all Nash-maximal p o iles.
4.4 Housing-Type Economies 107
P oo : Le (ai)be a Nash-maximal p o ile, ha is, one ha maximizes
Σi∈Nln( i(xi)) o e F. Since he u ili y unc ions a e symme ic, i.e.
( i(j) = j(i)), he p o ile (ai)also maximizes bo h Σi∈Aln( i(xi)) and
Σi∈Bln( i(xi)) o e F. Taking he agen s o be Aand he houses o be
B,Shapley and Shubik (1971) showed ha a p ice ec o (pj)j∈Bexis s,
such ha o e e y agen i∈A, he choice o aimaximizes ln( i(j)) −pj
o e all j∈Xi=Band he e o e, maximizes i(j)/epjas well. Re e sing
oles, he e is a p ice ec o (pj)j∈Awi h analogous op imali y p ope ies.
The e o e, ‹(λj=1/epj)j∈N,(ai)i∈N›cons i u es a biased p e e ences
equilib ium.
In he o he di ec ion, le ‹(λj)j∈N,(ai)i∈N›be a biased p e e ences
equilib ium, and le (xi)∈F. Fo e e y i, i holds ha λai i(ai)≥
λxi i(xi). The e o e, Πi[λai i(ai)] ≥Πi[λxi i(xi)]. Since Πiλai= Πiλxi,
i ollows ha Πi i(ai)≥Πi i(xi). Tha is, (ai)is Nash maximal. 
The condi ion ha he alue o a ma ch be ween any wo agen s is he
same o bo h o hem is su icien o he A-Nash-maximal ma ching o
be B-Nash-maximal as well. Wi hou his condi ion, a biased p e e ences
equilib ium may no exis :
Example: Conside an assignmen economy whe e A={1,2},B={3,4},
and 3 14, 1 42, 4 23, 2 31. No u ili y p esen a ion o hese
p e e ences is consis en wi h he assump ion ha he alue o a ma ch
is iden ical o bo h pa ne s (since i equi es ha 1(3)> 1(4) = 4(1)>
4(2) = 2(4)> 2(3) = 3(2)> 3(1) = 1(3)).
Suppose ha 1↔3 is a ma ch in a biased p e e ences equilib ium.
Then, λ1> λ2(so ha agen 3 chooses agen 1 o e agen 2). Bu hen,
agen 4 will also choose agen 1, iola ing easibili y. Likewise, 1↔4
canno be a ma ch in a biased p e e ences equilib ium.
108 Chap e 4. Biased P e e ences Equilib ium
Example: A P oduc ion Economy
In he p oduc ion economy ( ela ed o A akan e al. (2023)), he e is a se
o indi isible goods Kand wo equally-sized g oups o agen s: consume s
(C) and p oduce s (P). E e y i∈Cconsumes exac ly one uni o a single
good, ha is Xi=K, and ui
k>0 is consume i’s u ili y om consuming
good k(which he wishes o maximize). E e y p oduce i∈P mus
p oduce exac ly one uni o a single good, ha is, Xi=K, and ci
k>0
is p oduce i’s u ili y-cos om p oducing good k(which he wishes o
minimize). The se Fconsis s o all p o iles sa is ying ha , o e e y good
k, he numbe o i s consume s is equal o he numbe o i s p oduce s.
No e ha his economy di e s om he pa ne ship economy in ha
consume s and p oduce s choose a good a he han a pa ne and he
biases a e applied o he goods a he han o he agen s.
A bias ec o λ= (λk)k∈Kal e s consume i’s u ili y ec o om
(ui
k)k∈K o (λkui
k)k∈Kand p oduce i’s u ili y-cos ec o om (ci
k)k∈K o
(λkci
k)k∈K. Thus, a bias λsimul aneously escales bo h he consume s’
u ili y and he p oduce s’ u ili y-cos s o good kby he same ac o λk.
Thus, an inc ease in λkis analogous o ha o a dec ease in he p ice o
good kin a egula exchange economy: i makes he good mo e desi able
o buye s and less desi able o selle s. Unde lying a bias could be some
ai such as quali y: a high bias, like a high quali y le el, makes he good
mo e desi able o consume s and inc eases he u ili y-cos o p oduce i .
The ollowing claim again uses a Shapley and Shubik (1971)-s yle
a gumen o cha ac e ize he biased p e e ences equilib ium p o iles. I
implies ha hey exis and a e p e-Pa e o op imal.
Claim: In he p oduc ion economy, he biased p e e ences equilib ium
p o iles a e p ecisely he solu ions o :
max
(xi)∈F
Π
i∈Cui
xi
Π
i∈Pci
xi
(*)
4.4 Housing-Type Economies 109
P oo : Le ‹λ,(xi)›be a biased p e e ences equilib ium, and le (yi)∈F.
I ollows ha λxiui
xi≥λyiui
yi o e e y i∈Cand λxici
xi≤λyici
yi o
e e y i∈P. Combined wi h he equali y Πi∈Cλzi= Πi∈Pλzi, which holds
o all (zi)∈F, we conclude ha :
Π
i∈Cui
xi
Π
i∈Pci
xi
=
Π
i∈Cλxiui
xi
Π
i∈Pλxici
xi≥
Π
i∈Cλyiui
yi
Π
i∈Pλyici
yi
=
Π
i∈Cui
yi
Π
i∈Pci
yi
and he e o e (xi)is a solu ion o (*).
In he o he di ec ion, le (xi)be a solu ion o (*). Le (xi
k)be he
alloca ion ma ix wi h a ow o each agen and a column o each good,
whe e xi
k=1 i ichooses kand xi
k=0 o he wise. The ma ix sol es he
ollowing linea maximiza ion p oblem:
max
(mi
k)
ΣkΣi∈C[ln(ui
k)mi
k]+ΣkΣi∈P[−ln(ci
k)mi
k]
such ha Σi∈Cmi
k−Σi∈Pmi
k=0∀k(μk)
mi
k≥0∀i,k(γi
k)
Σkmi
k=1∀i(ψi)
The abo e p oblem always has a solu ion which is a bina y ma ix ( ha
is, xi
k=0 o 1, o e e y i,k). To unde s and why, Bi kho (1946) (and his
ex ensions in Budish e al. (2013)) shows ha any ma ix o eal numbe s
ha sa is ies he abo e cons ain s is a con ex combina ion o bina y
ma ices ha also sa is y hem. Since he a ge unc ion is linea , any
such bina y ma ix is also a solu ion o he linea p og amming p oblem.
The cons ain s in he abo e op imiza ion a e labelled by hei
shadow alues, which appea in he pa en hesis o he igh . Le
λ= (eμk)k∈K. We will now e i y ha ‹λ,(xi)›is a biased p e e ences
equilib ium.
116 Chap e 5. A Compa ison o Game Theo y
In a J1-equilib ium pai ing, i can be ha an agen p e e s ano he o his
cu en pa ne . Bu , he agen is p ohibi ed om making such an app oach
because he agen o be app oached is s onge han him. This con as s wi h
pai wise s abili y whe e wha p e en s him om ac ing is ha he agen o be
app oached will ejec him.
The ollowing p oposi ion compa es he J1-equilib ium concep wi h ha
o pai wise s abili y. I is ound ha he J1-equilib ium concep is s ic e : any
J1-equilib ium ou come is pai wise s able (and he e o e also Pa e o-op imal).
Since pai wise-s able pai ings do no always exis , nei he will J1-equilib ia.
P oposi ion 5.1: J1-equilib ium P ope ies
(i) E e y J1-equilib ium pai ing is pai wise s able.
(ii) A pai wise-s able pai ing migh no be a J1-equilib ium pai ing.
P oo :
(i) Le ‹B,(xi)›be a J1-equilib ium. Suppose ha he e a e wo agen s
iand jwho s ic ly p e e each o he o hei cu en pa ne s. One o
hem mus be B-s onge han he o he , and he p e e s he weake agen
o e his cu en pa ne , hus iola ing he J1-equilib ium condi ion.
(ii) In a J1-equilib ium, he s onges agen is ma ched wi h his i s -bes
choice. In he ollowing ma ching economy, he pai ing 1↔2 and 3↔4 is
pai wise s able, bu he e is no agen who is ma ched wi h his i s bes :
Agen 1 2 3 4
1s P e e ence 4 3 1 2
2nd P e e ence 2 1 4 3
3 d P e e ence 3 4 2 1
Table 5.2 P e e ences wi h a pai wise-s able pai ing (highligh ed) ha is no a
J1-equilib ium ou come.

5.2 The Jungle Equilib ium 117
Example: The Common- anking Two-sided Ma ching Economy
The pai ing {i1↔j1,i2↔j2,∙∙∙,in/2↔jn/2}combined wi h any powe o -
de ing  ha sa is ies i1,j1i2,j2∙∙∙in/2,jn/2is a J1-equilib ium.
The e is no o he J1-equilib ium pai ing: Since e e y agen in N2 op-
anks i1, agen i1mus be s onge han e e yone in N2excep pe haps
his pa ne . This means ha , in any equilib ium, i1has o be ma ched
wi h his i s -bes , namely j1. Simila ly, j1mus be s onge han all
membe s o N1, excep possibly i1. This pa e n con inues down he
anking. Among he emaining agen s, i2and j2a e ma ched, and i2
mus be mo e powe ul han {j3,...,jn/2}while j2mus be s onge han
{i3,...,in/2}and so on.
In he jungle model (Chap e 1), he abili y o one agen o ake he house o
ano he and, likewise, in a J1-equilib ium he abili y o one agen o app oach
ano he , depends solely on he powe ela ionship be ween he wo agen s.
Howe e , in he con ex o he ma ching economy, any app oach in ol es no
only he agen who ini ia es he app oach and he app oached agen bu also
hei pa ne s. The ollowing wo solu ion concep s ake his in o accoun . In a
J2-equilib ium, an agen can app oach ano he in a di e en pai only i he is
s onge han bo h he desi ed agen and ha agen ’s pa ne .
De ini ion: J2-Equilib ium
AJ2-equilib ium is a uple ‹B,(xi)›in which he e a e no wo agen s i
and jsuch ha ip e e s jo e his cu en pa ne ( ha is, jixi) and
is mo e powe ul han bo h j and j’s pa ne ( ha is, iBjand iBxj).
Like he J1-equilib ium, he J2-equilib ium does no allow o pai ings in
which no agen ge s his i s bes : in any J2-equilib ium, he s onges agen
is ma ched wi h his mos -p e e ed pa ne . Ob iously, e e y J1-equilib ium
is also a J2-equilib ium. We will now see ha a J2-equilib ium always exis s,
unlike a J1-equilib ium.
118 Chap e 5. A Compa ison o Game Theo y
P oposi ion 5.2: J2-equilib ium P ope ies
(i) A J2-equilib ium always exis s.
(ii) E e y J2-equilib ium pai ing is Pa e o-op imal.
(iii) A Pa e o-op imal pai ing (e en i i is pai wise s able) need no be a
J2-equilib ium pai ing.
P oo :
(i) Choose an a bi a y agen i1, and make him he s onges agen .
Call his i s -bes pa ne j1, pai hem oge he , and make j1 he
weakes agen . Con inue in his manne o ob ain a pai ing in which
e e y agen ikis pai ed wi h jk, who is ik’s a ou i e pa ne om
N−{i1,j1,...,ik−1,jk−1}, and se he powe ela ion o be i1Bi2B∙∙∙B
j2Bj1. Thus, e e y ikis s onge han e e y jl.
This p ocedu e gene a es a J2-equilib ium. Any agen who migh be
p e e ed by iko e jkmus ha e been pai ed ea lie , and hus, is ei he
s onge han iko has a pa ne who is. No jlcan app oach any o he
agen because e e y o he couple, ik↔jk, has a leas one membe who
is s onge han him, namely ik.
(ii) Le ‹B,(xi)›be a J2-equilib ium. Assume ha (yi)Pa e o-domina es
(xi). Le jbe he s onges agen in D≡ {i|xi6=yi}. By Pa e o-
dominance, yjjxj( ecall ha p e e ences a e s ic ). Agen yjand yj’s
o iginal pa ne xyja e bo h in D. The e o e, jyj,xyj, which iola es
‹B,(xi)›being a J2-equilib ium.
(iii) Fo he economy depic ed in Table 5.2, he highligh ed pai ing
{1↔2,3↔4}is pai wise s able bu is no a J2-equilib ium pai ing since
no agen ge s his i s -bes .
5.2 The Jungle Equilib ium 119
Example: The Common- anking Two-sided Ma ching Economy
E e y mixed pai ing (xi)is a J2-equilib ium pai ing suppo ed by
assigning he powe ela ions o agen s in each side by he ank o hei
ma ches ( ha is, o e e y wo membe s iand j om he same side
assign iBji xiis highe - anked han xj).
While e e y mixed pai ing is pa o a J2-equilib ium, no e e y powe
ela ion is. Fo example, o he case o ou agen s, he e is no J2-
equilib ium wi h he powe ela ion j2i1i2j1. This is because
j2is he mos powe ul, and mus be ma ched wi h i1. Thus, he
only candida e pai ing is {i1↔j2,i2↔j1}. Bu his is no a J2-equilib ium
because i1p e e s j1o e j2, and is s onge han bo h i2and j1.
Finally, in a J3-equilib ium, an agen can o ce a pa ne ship wi h jonly i he is
s onge han j,j’s pa ne , and his own abandoned pa ne .
De ini ion: J3-Equilib ium
AJ3-equilib ium is a uple ‹B,(xi)› o which he e a e no iand jsuch
ha jixiand iBj,xi,xj.
Ob iously, e e y J2-equilib ium is also a J3-equilib ium. The J3-equilib ium
equi es s onge condi ions o an agen o be able o dis u b socie y’s
ha mony, and we will see ha e e y powe ela ion is pa o some J3-
equilib ium (unlike he J2-equilib ium case). None heless, in e ms o
equilib ium pai ings, he J2- and J3-equilib ium no ions a e equi alen .
P oposi ion 5.3: J3-equilib ium P ope ies
(i) Fo e e y powe ela ion B, he e is a J3-equilib ium ‹B,(xi)›.
(ii) The se o J3-equilib ium pai ings is equal o he se o J2-equilib ium
pai ings ( hus, e e y J3-equilib ium pai ing is Pa e o-op imal, hough
no e e y Pa e o-op imal pai ing is a J3-equilib ium pai ing).
120 Chap e 5. A Compa ison o Game Theo y
P oo :
(i) Le be a s ic o de ing. We induc i ely cons uc a pai ing
(xi) o which ‹,(xi)›is a J3-equilib ium using a “gene alized se ial
dic a o ship” p ocedu e: Fi s , he B-s onges agen picks his mos -
p e e ed pa ne and hey a e ma ched. In each subsequen s ep, he
B-s onges emaining agen is ma ched wi h his mos -p e e ed pa ne
om among hose emaining. By his p ocedu e, hal o he agen s “make
a choice” while he o he hal “a e chosen”. Any agen who “makes a
choice” can only p e e agen s who ma ch be o e him, i.e. hose who a e
s onge han him o a e pai ed wi h a s onge pa ne . Any agen who
“is chosen” is neu alized because he is ma ched wi h a s onge pa ne .
(ii) As men ioned, any J2-equilib ium is also a J3-equilib ium. We
now show ha any J3-equilib ium pai ing is a J2-equilib ium pai ing
(pe haps wi h a di e en powe ela ion). Conside a J3-equilib ium
‹B,(xi)›. In e e y couple, he e is a s onge agen and a weake one.
Le Sbe he se o n/2 s onge agen s and Wbe he se o n/2 weake
ones. De ine a new powe ela ion B0by p ese ing BonSand on Wand
pushing all membe s o Wbelow all membe s o S. The uple ‹B0,(xi)›
is a J2-equilib ium: I no , hen he e would be iand jsuch ha jixi
and iB0xj,j. I mus be ha i∈Ssince iis B0-s onge han a pai o
agen s, j↔xj. Thus, iBxi. Since he powe ela ion is p ese ed on S,
iis -s onge han he -s onge agen in {j,xj}who is in S. Thus,
ixi,j,xj, con adic ing ‹B,(xi)›being a J3-equilib ium.
5.3 Res ic ing Pa ne ships: Pai wise Y-equilib ium
We now adjus he Y-equilib ium concep (Chap e 2) o i he ma ching
economy. Since e e y agen needs a pa ne , uni o mly es ic ing he se o
pe mi ed pa ne s will lea e some agen s wi hou a pa ne . Ins ead, he social
no m de e mines which pai s a e pe mi ed and which a e o bidden.
5.3 Res ic ing Pa ne ships: Pai wise Y-equilib ium 121
De ini ion: Y-Equilib ium
Le Mbe he se o all double ons (se s o size 2). A pa a-Y-equilib ium
is a uple ‹Y,(xi)›whe e Y⊆Mand (xi)is a pai ing such ha , o e e y
agen i,xiis i-maximal in {j|{i,j}∈Y}. A Y-equilib ium is a pa a-Y-
equilib ium such ha he e is no o he pa a-Y-equilib ium ‹Z,(yi)›wi h
Y⊂Z.
Any se o pe missible pai s Yinduces, o each agen i, a choice se o
pe missible pa ne s {j| {i,j} ∈ Y}. Thus, unlike he Y-equilib ium no ion o
Chap e 2, he e he Y-equilib ium no ion ea s agen s asymme ically in he
sense ha di e en agen s ace di e en choice se s wi h he es ic ion ha i
jis pe missible o i, hen iis also pe missible o j. The adap ed Y-equilib ium
no ion equi es ha , o any la ge pe missible se , he e is an iand jsuch ha
iwould choose jbu jwould no choose i.
The ollowing p oposi ion shows ha he se o Y-equilib ium pai ings is
he se o all Pa e o-op imal pai ings. This implies ha , in e m o ou comes,
he Y-equilib ium no ion is mo e pe missi e han pai wise s abili y o he
J-equilib ium no ions. In pa icula , i always exis s.
P oposi ion 5.4: Y-equilib ium and Pa e o Op imali y
The se o Y-equilib ium pai ings =The se o Pa e o-op imal pai ings.
P oo :
Gi en any pai ing (xi), de ine L((xi)) o be he se o all double ons {i,j}
such ha xi%ijand xj%ji. No ice ha o e e y i, he double on {i,xi}
is in L((xi)).
Le ‹Y,(xi)›be a Y-equilib ium and (yi)be a pai ing ha Pa e o-
domina es (xi). Ob iously, L((xi)) ⊇Y. Clea ly, L((yi)) ⊇L((xi)). In
ac , he inclusion is s ic since a leas one agen , say j, is s ic ly
be e o in (yi)and he e o e he pai {j,yj}is in L((yi)) −L((xi)). The

122 Chap e 5. A Compa ison o Game Theo y
uple ‹L((yi)),(yi)›is a pa a-Y-equilib ium wi h a la ge se o pe missible
pai s, which con adic s ‹Y,(xi)›being a Y-equilib ium.
On he o he hand, le (xi)be a Pa e o-op imal pai ing. We now show
ha he uple ‹L((xi)),(xi)›is a Y-equilib ium. I no , hen he e is a pa a-
Y-equilib ium ‹Z,(yi)›wi h Z⊃L((xi)). All agen s a e weakly be e o in
(yi) han in (xi). The se Zcon ains a leas one pai {i,j}which is no
in L((xi)). Wi hou loss o gene ali y, suppose ha jixi. In ha case,
yi%ijixiand he e o e, (yi)Pa e o-domina es (xi).
Example: The Common- anking Two-sided Ma ching Economy
E e y mixed pai ing is Pa e o-op imal and he e o e, is a Y-equilib ium
pai ing. The Y-equilib ium pai ing {i1↔j1,i2↔j2,...}is suppo ed
by a maximally es ic ed pe missible se which con ains only he
equilib ium ma ches (and all he ma ches be ween any wo agen s om
he same side).
5.4 P es ige by Pa ne : S a us Equilib ium
We now u n o he s a us equilib ium concep discussed in Chap e 3
( e e ed o as an S-equilib ium in Rich e and Rubins ein (2024)). Ha mony
is es ablished by a s a us o de ing o he agen s ha blocks an agen om
app oaching ce ain o he agen s. In a s a us equilib ium, agen s a e pai ed
up, and no agen can app oach any o he agen who has a highe s a us han his
cu en pa ne . The only agen s he has he cou age o app oach a e hose wi h
a (weakly) lowe s a us han his own pa ne . An equilib ium is ha monious in
ha no agen can ind a di e en pa ne who is bo h app oachable and mo e
desi able.
5.4 P es ige by Pa ne : S a us Equilib ium 123
De ini ion: S a us Equilib ium
A s a us equilib ium is a uple ‹P,(xi)›whe e (xi)is a pai ing and Pis
a weak o de ing o he agen s such ha , o e e y agen i, he e is no j
such ha jixiand xiPj .
S a us equilib ium pai ings ha e he p ope ies ha a leas one agen ( he
agen wi h he highes - anked pa ne ) ge s a pa ne whom he mos p e e s,
and i p e e ences a e s ic , hen a mos one agen ge s his leas -p e e ed
pa ne (i can only be he agen wi h he lowes - anked pa ne ).
We now es ablish some ela ionships be ween he s a us equilib ium
concep and he J2-equilib ium, Pa e o op imali y, and pai wise s abili y.
P oposi ion 5.5: S a us Equilib ium P ope ies
(i) E e y s a us equilib ium pai ing is a J2-equilib ium pai ing and
he e o e, is Pa e o-op imal.
(ii) The no ions o s a us equilib ium and pai wise-s abili y a e dis inc ;
i is possible o ei he no ion o exis when he o he one does no .
P oo :
(i) Le ‹P,(xi)›be a s a us equilib ium and b eak ies so ha Pis s ic .
De ine a powe anking by anking agen s acco ding o he s a us o
hei pa ne s: iji xiPx j. The uple ‹,(xi)›is a J2-equilib ium: I
jixi o some i,j, hen i mus be ha j Pxisince ‹,(xi)›is a s a us
equilib ium. Bu hen xji, and iis p e en ed om app oaching jby
he powe o j’s pa ne . By P oposi ion 5.2, he pai ing is also Pa e o-
op imal.
(ii) In he economy depic ed in Table 5.1, he e is no pai wise-s able
pai ing, bu he o de ing 1P2P3P4 suppo s he pai ings {1↔3,2↔4}
and {1↔4,2↔3}as s a us equilib ia.
124 Chap e 5. A Compa ison o Game Theo y
In he economy depic ed in Table 5.2, he highligh ed pai ing
{1↔2,3↔4}is pai wise s able, bu he e is no s a us equilib ium: The
highligh ed pai ing is no a s a us equilib ium pai ing because no agen
ge s his i s -bes . Nei he a e he o he wo pai ings because in each o
hem, wo agen s ge hei las choice.
Example: The Common- anking Two-sided Ma ching Economy
By P oposi ion 5.5, only mixed pai ings can be s a us equilib ium
pai ings. In ac , any mixed pai ing is a s a us equilib ium pai ing wi h
any anking P ha sa is ies i1Pi2P...Pin/2and j1Pj2P...Pjn/2( o any
agen i, e e y agen ha idesi es mo e han his pa ne has a highe
s a us han i’s pa ne ).
5.5 P es ige by Sel : Ini ial S a us Equilib ium
We adap he ini ial s a us equilib ium concep o he ma ching economy by
aking an agen ’s ini ial s a us o be himsel . Recall ha his concep belongs o
he choice g oup o solu ion concep s (see Sec ion 0.4). E e y agen chooses
his pa ne , bu he s a us anking only allows an agen o app oach agen s
wi h he same s a us o lowe . In equilib ium, he s a us anking is such ha
he indi idual choices o m a pai ing ( ha is, i ichooses j, hen jchooses i).
Fo mally, a candida e o an ini ial s a us equilib ium is a uple ‹P,(xi)›whe e
iPj is in e p e ed as “i’s s a us is a leas as high as j’s” and (xi)is a pai ing. This
adap ed ini ial s a us equilib ium concep is e e ed o as a C-equilib ium in
Rich e and Rubins ein (2024).
De ini ion: Ini ial S a us Equilib ium
An ini ial s a us equilib ium is a uple ‹P,(xi)›such ha , o e e y agen
i, his pa ne xiis i’s mos -p e e ed pa ne in {j∈N|iPj }.
5.5 P es ige by Sel : Ini ial S a us Equilib ium 125
Ob iously, any wo ma ched agen s in an ini ial s a us equilib ium mus ha e
he same s a us. The e o e, e e y ini ial s a us equilib ium is also a s a us
equilib ium and, by P oposi ion 5.5, i s pai ing is Pa e o-op imal.
O pa icula in e es is he ela ionship be ween he ini ial s a us equilib-
ium and he J1-equilib ium. I ‹P,(xi)›is an ini ial s a us equilib ium, hen
‹,(xi)›is a J1-equilib ium whe e is any s ic ie-b eaking o P( ha is, ij
only i iPj ).
Howe e , unlike he ini ial s a us equilib ium concep , a J1-equilib ium
does no equi e ha an agen be weakly s onge han his pa ne . Many
pai ings can be a J1-equilib ium pai ing bu no an ini ial s a us equilib ium
pai ing. Fo example, conside he wo-sided ma ching economy wi h
N1={1,3},N2={2,4}, and a agedy: 1 lo es 2, 2 lo es 3, 3 lo es 4, and 4 lo es
1. Bo h mixed pai ings a e J1-equilib ium pai ings. One is he i s -bes o N1’s
membe s. I is suppo ed by any powe ela ion ha anks N1’s membe s abo e
N2’s membe s. The o he is he opposi e. Nei he is an ini ial s a us equilib ium
pai ing (sho ly, we will see why).
We need an addi ional concep . We say ha he ma ching economy is pai -
ankable i he se o agen s Ncan be pa i ioned in o double ons I1,...,In/2,
such ha each agen in Iqp e e s his pa ne in he double on o any membe
o Iq+1∪∙∙∙∪In/2. In o he wo ds, he agen s can be pa i ioned in o a sequence
o double ons whe e e e y agen ’s pa ne is his bes choice om hose who a e
no ahead o him.
Pai - ankabili y is a s ong p ope y o a ma ching economy which eme ges
in some na u al se ings. Two classical amilies o pai - ankable ma ching
economies a e:
(i) Agen s li e in a me ic space and ank pa ne s by hei closeness. The
i s double on can consis o he wo closes agen s, and each subsequen
double on consis s o he wo closes among hose emaining.
(ii) Agen s a e posi ioned on a line, and each has single-peaked p e e ences
o e he o he agen s wi h a peak a one o his neighbou s. This implies ha an
ex eme agen op- anks his only neighbou and, o any se o agen s, he e a e
132 Chap e 5. A Compa ison o Game Theo y
The di e ence be ween he Nash and Deb eu o mula ions is pu ely
seman ic: e e y playe is ei he no in e es ed in mo ing om a non-c isis
p o ile o a c isis p o ile (in he Nash o mula ion) o is no e en allowed o
do so (in Deb eu’s o mula ion).
All p o iles in which a ba e majo i y o exac ly τ= (n+1)/2 agen s choose
he same posi ion (wha e e i is), while he es choose hei peaks, a e non-
c isis Nash equilib ia. These equilib ia can be ex emely unna u al in ha he
coali ion which suppo s hem does no ha e any hing o do wi h he posi ion
being suppo ed. In pa icula , he e a e non-c isis Nash equilib ia o any
o e all posi ion, e en ex eme ones ha a e ou side o [L,R], and he agen s
suppo ing he o e all posi ion need no be hose whose peaks a e closes o i .
We will now see ha he e a e no o he non-c isis Nash equilib ia.
P oposi ion 5.8: Nash Equilib ium in he Vo ing Game
I n≥5, hen he se o non-c isis Nash equilib ia in he o ing game
consis s o all p o iles o which he e is a posi ion chosen by exac ly τ
agen s while he es choose hei peaks.
P oo :
These a e Nash equilib ia: No agen a he majo i y posi ion can de ia e
p o i ably since, i he did so, hen a c isis would ensue because his
o me posi ion would no longe be a majo i y posi ion and nei he
would his new posi ion (all o he agen s a e choosing hei peaks which
a e dis inc , so any new posi ion would ha e a mos wo agen s, bu
n≥5). All o he agen s a e a hei i s -bes , hey choose hei peak,
and no c isis occu s. The e o e, hey do no wan o de ia e.
To see ha he e a e no o he non-c isis Nash equilib ia, conside a
Nash equilib ium in which a leas τagen s choose a common posi ion
. An agen who does no choose is no c i ical in main aining ha mony
and he e o e, mus be a his peak. I s ic ly mo e han τagen s choose
, hen a leas one o hem is no a his peak and could de ia e p o i ably.

5.11 Compa ing ou App oaches wi h Nash Equilib ium 133
Commen : In Rich e and Rubins ein (2021), we conduc ed simila compa -
isons and eached simila conclusions ega ding o he condi ions o “holding
he g oup oge he ”:
(i) a consensus among a supe majo i y o agen s,
(ii) all posi ions a e su icien ly close o he median posi ion, o
(iii) all posi ions a e su icien ly close o he a e age posi ion.
5.11 Compa ing ou App oaches wi h Nash Equilib ium
The abo e analysis cla i ies he signi ican di e ences be ween he con ex Y-
equilib ium, he biased p e e ences equilib ium, and he Nash equilib ium
o he abo e poli ical game. Fo he con ex Y-equilib ium concep , Mis he
only o e all posi ion. Fo he biased p e e ences equilib ium concep , ypically
only he ex eme posi ions, −1 and 1, a e o e all posi ions. In con as , o he
Nash equilib ium concep , all posi ions, e en hose ou side he ange [L,R], a e
o e all posi ions. Fu he mo e, a con ex Y-equilib ium is “mono onic” in he
sense ha i agen i’s ideal posi ion is o he le o j’s hen his chosen posi ion
is weakly o he le o j’s. In con as , he e a e always non-mono onic Nash
equilib ia. The biased p e e ences equilib ium case is less clea : he exis ence
o a non-mono onic equilib ium depends on he unde lying u ili y unc ions.
No ice ha he Nash equilib ia equi e a high deg ee o coo dina ion
be ween he agen s. In con as , he Y-equilib ium and biased p e e ences
equilib ium concep s only equi e ha agen s know ei he he social es ic-
ions o biases, bu no he beha iou o o he s. This is like he ma ke place
whe e indi iduals only need o know p ices, bu no o he agen s’ ac ions.
Le us emphasise: we a e no saying ha he s anda d game- heo e ical
app oach is “w ong”, no do we insis ha he Y-equilib ium o biased
p e e ences equilib ium app oaches a e “ igh ”. Ra he , and as al eady
men ioned, we a e sugges ing ha he eade no au oma ically apply Nash-
equilib ium-like concep s bu ins ead conside s al e na i e solu ion concep s
in he spi i o hose desc ibed in his book.
Re e ences
Abdulkadı o˘
glu, A ıla, and Tay un Sönmez (1998), “Random se ial dic a o ship
and he co e om andom endowmen s in house alloca ion p oblems.”
Econome ica, 66, 689–701. [18]
Alcalde, José (1994), “Exchange-p oo ness o di o ce-p oo ness? S abili y in
one-sided ma ching ma ke s.” Economic Design, 1, 275–287. [126]
A akan, Alp, Michael Rich e , and Ma an Tsu (2023), “E icien sea ch,
ma ching, in es men s.” mimeo.[108]
Bagwell, Kyle (2007), “The economic analysis o ad e ising.” Handbook o
Indus ial O ganiza ion, 3, 1701–1844. [92]
Balasko, Y es (1979), “Budge -cons ained Pa e o-e icien alloca ions.” Jou nal
o Economic Theo y, 21, 359–379. [101]
Becke , Ga y S., and Ke in M. Mu phy (1988), “A heo y o a ional addic ion.”
Jou nal o Poli ical Economy, 96, 675–700. [92]
Benassy, Jean-Pascal (1986), Mac oeconomics: An In oduc ion o he Non-
Wal asian App oach. Else ie , New Yo k, NY. [100]
Bi kho , Ga e (1946), “T es obse aciones sob e el algeb a lineal ( h ee
obse a ions on linea algeb a).” Uni e sidad Nacional de Tucumán Re is a
Se ie A, 5, 147–151. [109]
Bowles, Samuel, and He be Gin is (1992), “Powe and weal h in a compe i i e
capi alis economy.” Philosophy and Public A ai s, 21, 324–353. [13]
Buchanan, James M. (1965), “An economic heo y o clubs.” Economica, 32, 1–
14. [6]
136 Re e ences
Budish, E ic, Yeon-Koo Che, Fuhi o Kojima, and Paul Milg om (2013), “De-
signing andom alloca ion mechanisms: Theo y and applica ions.” Ame ican
Economic Re iew, 103, 585–623. [109]
Deb eu, Ge a d (1952), “A social equilib ium exis ence heo em.” P oceedings
o he Na ional Academy o Sciences, 38, 886 – 893. [131]
Downs, An hony (1957), An Economic Theo y o Democ acy. Ha pe . [128]
Edelman, Paul H., and Robe E. Jamison (1985), “The heo y o con ex
geome ies.” Geome iae Dedica a, 19, 247–270. [72]
Edgewo h, F ancis Ysid o (1881), Ma hema ical Psychics: An Essay on he
Applica ion o Ma hema ics o he Mo al Sciences, olume 10. CK Paul. [5]
Foley, Duncan Ka l (1966), Resou ce alloca ion and he public sec o . Yale
Uni e si y. [43,55]
Gale, Da id (1984), “Equilib ium in a disc e e exchange economy wi h money.”
In e na ional Jou nal o Game Theo y, 13, 61–64. [105]
Gale, Da id, and Lloyd S. Shapley (1962), “College admissions and he s abili y
o ma iage.” The Ame ican Ma hema ical Mon hly, 69, 9–15. [27,111,113]
Gale, Da id, and Ma ilda So omayo (1985), “Some ema ks on he s able
ma ching p oblem.” Disc e e Applied Ma hema ics, 11, 223–232. [27]
Gibbs, Lau a (2002), Aesop’s Fables. Ox o d Wo ld’s Classics, Ox o d Uni e si y
P ess. [91]
G ossman, He schel I. (1995), “Robin hood and he edis ibu ion o p ope y
income.” Eu opean Jou nal o Poli ical Economy, 11, 399–410. [13]
Hi shlei e , Jack (1994), “The da k side o he o ce: Wes e n economic
associa ion in e na ional 1993 p esiden ial add ess.” Economic Inqui y, 32,
1–10. [13]
Re e ences 137
Ho elling, Ha old (1929), “S abili y in compe i ion.” The Economic Jou nal, 39,
41–57. [128]
Keiding, Hans (1981), “Exis ence o budge cons ained Pa e o e icien
alloca ions.” Jou nal o Economic Theo y, 24, 393–397. [101]
Malin aud, Edmond (1972), Lec u es on Mic oeconomic Theo y. Ad anced
ex books in economics, No h-Holland Publishing Company. [47]
Osbo ne, Ma in, and A iel Rubins ein (2023), Models in Mic oeconomic
Theo y, Expanded second edi ion. Open Book Publishe s. [23]
Piccione, Michele, and A iel Rubins ein (2007), “Equilib ium in he jungle.” The
Economic Jou nal, 117, 883–896. [14,33,34]
Rich e , Michael, and A iel Rubins ein (2015), “Back o undamen als: Equilib-
ium in abs ac economies.” Ame ican Economic Re iew, 105, 2570–94. [65,
82]
Rich e , Michael, and A iel Rubins ein (2019), “Con ex p e e ences: A new
de ini ion.” Theo e ical Economics, 14, 1169–1183. [71,72,75,84]
Rich e , Michael, and A iel Rubins ein (2020), “The pe missible and he
o bidden.” Jou nal o Economic Theo y, 188, 105042. [38,61]
Rich e , Michael, and A iel Rubins ein (2021), “Holding a g oup oge he : Non-
game heo y e sus game heo y.” The Economic Jou nal, 131, 2629–2641.
[112,133]
Rich e , Michael, and A iel Rubins ein (2024), “Unila e al s abili y in ma ching
p oblems.” Jou nal o Economic Theo y, 216, 105780. [111,122,124]
Rubins ein, A iel (2005), Lec u e No es in Mic oeconomic Theo y. P ince on, NJ:
P ince on Uni e si y. [46]
Rubins ein, A iel (2012), Economic Fables. Open Book Publishe s. [1]

138 Re e ences
Rubins ein, A iel, and Ashe Wolinksy (2022), “Biased p e e ences equilib ium.”
Economics and Philosophy, 38, 24–33. [91]
Rubins ein, A iel, and Kemal Yıldız (2022), “An é ude in modeling he
de inabili y o equilib ium.” Re iew o Economic Design, 26, 543–552. [27,
115]
Shapley, Lloyd, and He be Sca (1974), “On co es and indi isibili y.” Jou nal
o Ma hema ical Economics, 1, 23–37. [5,14,21,22,69,88,104]
Shapley, Lloyd S., and Ma in Shubik (1971), “The assignmen game I: The
co e.” In e na ional Jou nal o Game Theo y, 1, 111–130. [105,106,107,108]
Sp umon , Y es (1991), “The di ision p oblem wi h single-peaked p e e ences:
A cha ac e iza ion o he uni o m alloca ion ule.” Econome ica, 59, 509–
519. [6]
Tóbiás, Á on (2022), “Equilib ium non-exis ence in gene alized games.” Games
and Economic Beha io , 135, 327–337. [131]
Va ian, Hal (1974), “Equi y, en y, and e iciency.” Jou nal o Economic Theo y,
9, 63–91. [43,55]
Abou he Team
Alessand a Tosi was he managing edi o o his book.
Jenni e Mo ia y copyedi ed he book.
The co e image was c ea ed by A iel Rubins ein om a concep by Michael Rich e
and A iel Rubins ein. The co e o his book was p oduced by Jee anjo Kau Nagpal
in InDesign using Fon in and Calib i on s.
Michael Rich e and A iel Rubins ein ypese he book in LaTeX and p oduced he
pape back and ha dback edi ions.
Came on C aig p oduced he PDF and HTML edi ions. Con e sion was pe o med
wi h open sou ce so wa e eely a ailable on ou Gi Hub page a h ps://gi hub.com/
OpenBookPublishe s.
This book need no end he e…
Sha e
All ou books—including he one you ha e jus ead—a e ee o access online so ha
s uden s, esea che s and membe s o he public who can’ a o d a p in ed edi ion will ha e
access o he same ideas. This i le will be accessed online by hund eds o eade s each mon h
ac oss he globe: why no sha e he link so ha someone you know is one o hem?
This book and addi ional con en is a ailable a :
h ps://doi.o g/10.11647/OBP.0404
Dona e
Open Book Publishe s is an awa d-winning, schola -led, no - o -p o i p ess making knowledge
eely a ailable one book a a ime. We don’ cha ge au ho s o publish wi h us: ins ead, ou
wo k is suppo ed by ou lib a y membe s and by dona ions om people who belie e ha
esea ch shouldn’ be locked behind paywalls.
Why no join hem in eeing knowledge by suppo ing us:
h ps://www.openbookpublishe s.com/suppo -us
Follow @OpenBookPublish
Read mo e a he Open Book Publishe s