The Essays on Collusion Paul Johnson - Bienvenue au site ...€¦ · The Essays on Collusion Paul...
Transcript of The Essays on Collusion Paul Johnson - Bienvenue au site ...€¦ · The Essays on Collusion Paul...
Université de Montréal
T h e Essays on Collusion
Paul Johnson
Département des science8 économiques
M t 6 des arts et des sciences
Thése présentée b la Facuit6 des études supérieumi
as mede l'obtention du paie de
PhiloaophiaeDoctor (Ph.D.)
en sciencee ecOnomiques
The author has granted a non- L'auteur a accord6 une licence non exclusive licence allowhg the exclusive permettant B la National Li'brary of Canada to Biblioth6que natioaale du Canada de
copies of this thesis in microform, paper or ekctronic formats.
The author retains ownership of the copyright in this thesis. Neither the thesis nor suôstantial extracts fiom it may be printed or othawise reproduced without the auihor's pamission.
vendre des copies de cette thése sous la forme de microfiche/n]m, de reproduction sin papier ou sur format ekcmnique.
L'auteur conserve la propriété du droit d'auteur qui pro@@ cette thèse. Ni la these ni des extraits sabstantiels de celle-ci ne doiveni être imprimés ou autrement reproduits sans son autorisation.
Cette thèse intitulée:
présentée par:
Paul Johnson
a été dvaluée par un jury composé das personnes suivantes:
Gérard Gaudet, pxésidëfit-rapporteur Jacques Rosert, dzrecteai: de recherche Yves Sprumont, membre du jury Davfd Martfmort, examinateur externe tlnfversfté de Pau - France Jean Roy, reprhentant du doyen H.E.C.
Liste des articles et auteurs
Collusion in a Mode1 of Repeated Auctions
Paul Johnson and Jacques Robert
On Cartel Stability
Paul Johnson
Search, Matching and Moral Hazard
Paul Johnson
This thesis defina collusion broadly as play in a repcated eQme which diflm
6rom play in a one h o t game. The analysis of colluirion is an important part of
many branches of economics. In industrial organization, for example, if collusion
rsere not present then ae could restnct investigation to the study of oligopohtic
and competitive markets. A n o h important part of modem economic theory is
the analysis of situations whm them d t s soma kind of priviieged information.
Game theorists would say that in euch a scenario th- d t s incomplete infor-
mation. The predominant theme of the thme essays composing thia thesis is the
study of repeated gamea under incomplete information.
Typically, repeated F e analyses have assumeci the preiience of complete in-
formation. However, many exampl= of repeated interaction must be treated in
an incomplete idormation scenario. Auctions, the subject of the first eesay, are
the most natural example of thb. The goal of thîs study is to understand how
biddae collude in auctions. The main innovation iii an expliut traatment of the
repeated nature of the ppme to e n d o p h the threats neeemary to support non
competitive behavior. This andysis yields severd testable implicatiom about the
behavior of colluding agents in auctions which are not apparent h m the few
mdeis which have atudied collusion in auctions fiom a static point of view.
Nearly every m d constructed to study collusion m a h pfedictions whkh are
at dds with accepteci stylized facts. The maet prominent of th- paradoxes is the
atabiity which theoretid m d e b p d c t yet exnpiricai and dlegoricai ewidenœ
rejects. This is the subject of the second m. hcomplete idormation takes the
fonn of h m players evaluate friture pqmfb. This is an important detail becam
futur% pa@b are the only twl a cartel c m use to enforce the play of collusive
equilibna The developed model impaieri that patience is private information and
heterogeneous and develop pdctiom which co11trast Btmngly with accepteci
models of colidon. The pTedictions of the model are supporteci with discussion
of some empincai evidence on castels.
The third essay develops a model which can have sociologica as weli as ec+
nomic applications, in that it studies how rational agents form surpius creating
partnemhips in a repeated, incomplete information environment. Previous work
has assumed an exogenous production technology which partners use to mate this
surplus. Furthemore an agent's type, which affects surplus creation, has always
been assumed to be obsemble. This esaay studies matching with an endogenous
production technology, in the sense that the surplus is a function of the levei of
collusion which cm be supporteci. CoUusion c m be supported to va,rying degree
based upon the type (patience, or discount factor) of each agent in the partner-
ship. A special attention is turned to contmsting the implications of the model in
the presence of complete as opposed to incomplete information.
Introduction
The predominant theme of the three emays compoeing this th& is the study
of collusion under inmmplete information. CoilUWon is defined as eqgilibrium
shtegies to the repeated game which diffa fkom the repetition of the equilibrium
strategies to the stage p n e . The anal@ of coilusion is an important part of
many branches of economics. In industrial organization, for example, if coiiusion
were not present then we could restrict investigation to the study of oligopolistic
and competitive markete. Incomplete information is a way which game theorists
study situations where al1 information is not common knowledge. Its applications
are ubiquitous in modern game theoretic economics with applications as diveme
as the study of oligopoly and the cold war.
The dust essay studies coilusion in a set of markets which were the sub ject of
the f h t systematic investigation into the economics of incomplete information:
Vichy's (1961) seminal study of auctions. With few exceptions, the auction
literature has maintained the assumption that agents participate in s one shot
game: an auction is held and the participante part company never to meet again.
This is the point of departure for this work as a repeated environment is explicitly
developed. The repeateà nature of such an auction immediately suggests the
p068ibility of bidder collusion. Recently there has been Borne pmgress made into
undenstanding the behavior of rational agents coUuding in auctions, but this mrk
has abstracted away h m a repeated game analysis. For interna1 coIlslsfency
t h ie raason enmgh to study collusion in an exp1icitly mpeated environment.
Studying collusion in explicitly repeated auctiofls is aiso of intemt h m a more
pragmatic point of view. Firstly, the predictions generated by a static mode1 do
not al- carry o~ to a dyiiamic modal (in a way to be d e dear bdow).
Secondly, a dynemic model yialds richer results than its static couterpart. And
thitdly, an entire dase of results on auctioneer behavior emerges which dom not
arise in a static model.
One of the eaiiiest examinations of coiiusion in auctions w a ~ done by Cornanor
and Schsaherman (1976) who studied a puadcm in auction behaviot: that of iden-
tical bids subrnittd by biddue (under uaual hypothm identical bids should be
observed with zero probability). M&ee and McMilian (1992) were able to m l v e
this parada by showing that if biddem were unable to efktuate sidepayments
amongst themselves, perhaps because of a "papa trailn which miibss c d y d e
tection very likely, then the optimal response of a cartel would be for every bidder
to bid the r e s e m price and let the auctioneer act eii s randomization device. Of
course if the auctioneu nera to anard the good in a pdetermirred way, Say to the
bidder who's name was &st dph8betically, then a little bit more coordination by
the cartel is required. The key insight hm is that a cartel must =ounce &ciency
in oràer to ovacome incentive compatibility problems bmught about by the &
tence of privately held information. Though they only examineci a static model,
McAfiee and McMïlian ammeci the exbtmce of scmie sort of trigger-strate@ which
makea adherence to the &el more ptostable than defiction.
.Collusion in a Modal of Repeated Auctiond' studies collusion in auctions h m
an explicitly npeated game point of view. Thus inacad of assumjng the &me
of some land of tri- strategy, tri- strategies are explicitly comtmcted. An
immediate ontcome is that one can &y give an c~zunp1e of a series of auctiom
whue McAIæ and McMiUan's (1982) strategy of bidding the merve price is not
supportable. Thus, added to the familiar incentive campatibiity constrahts is a
new dynsmic "tri@ conetraint (or in the teRninology uscd by Abreu, Peute,
and S t d e t t i (1986), Wmissibilityn). Raughly, the intuition is that a bidder
can value an object vary highly today-so much sa that she L prepared to faa any
d b 1 e pttniE;hmat in mturn for p-g the item tday. The inefIiciency of
the M&ee and McMü1an Vatn bidding fuacfion is the driving force behind this
behavior. The obvious question is that if for certain Senes of auctions bidding the
reserve price is not an admissible scheme, then what is? and what is the optima
scheme? E'urthemore, what c m be said about cartel behavior across tirne: can
incomplete information cause some kind of instability in the cartel? Does the role
of the auctioneer dinu fiom that of a static model when his main weapon in
combatting collusion is the resarve price? This assay Q an attempt to anmm the
above questions.
The essay uses techniques k s t developed by Abrau, Pearce, and Stacchetti
(1986) to endogenize punishmenta. These punjshments difter h m the usual re
peated game analyses in that an auction ie a game of incomplete information. The
main results are that a cartel permits a finite number of bidding leveb, and that
collusion ie found to be stable acms time. Perhap the ma& intetestiag insights
are the number of predictions which eriila due to the explicitly dynamic nature
of the model. For example auctioneer profits can be dected by the fkequency at
wbicb auctions aie held.
Unfortunattelp for cartels, coiiusion L i tarel~ if ever, long 1-g. Thb lairt
observation is troubiing for economists, since m a t repeated game mdel s of col-
lusion p d c t puléct cartel stab'ity over the. For a liteiature which ha8 been in
existence since the t h e of Fkiedman (1971), aiid has been the subject of sa much
mearch d o r t this is problematie. The second -y, 'On Cartel Stabiüty" o f f i
a nconciliation of theory with the accepteù stylized fact that cartels are generally
unstable.
One of the moiit commonly made wnmptions in the literatnn on colîusion
is that one playet is just as patient as anothm pîayer-piayem &are a cornmon
discount fisctort. Hsrrington (1989) is an interasting exception to this tendency,
in that he perniits coliuding finns to have àifKerent discount rates. Despite fhis
heterogeneity, tbese diflaing discount rates remmin pufectly known. In this essay
a mode1 where discount facto113 en privste information is developed. Not only are
dimunt factors private idormation, but they are abject to change as the game
P~gresses*
Suppose that discount fBCtors (patience) are unobeservab1e and change as play-
ers play a repeated game. A rnaxhhing cartel is fsced with a tradeofE on one
hand, the more profitable the collusion is the more diiBailt it is to aupport, thus
a s m d dmp in a discount factor cause8 mtly ddceton; on the other hand, the
1- profitable collusion is the 1- difficult it is to support, thus mail changes in
discount ratai are udikeiy to deetabilize the cartel resulting in mutually profitable
coopezation. if puniahmenta wem h p l e as in Abreu (1988), a razor dge d t
d d be obtaiired, as any .gent with a diecount fwtor belon a certain l d aonld
d d i ta the mnuimum extent p d t t e d by the structure of the m e , wherea%
agents with discount fkctom above a certain Id muid respect cartei des. This
emay a h to characterize the efforts of an optimiaiig cartel (a concept which is
esnhilly debed) when con)onted with mch a scenario.
Agents àraw iid discount f&om in e w y peziod. 1 argue that something akin
to defection and punjshment arises on the equiiibrium path (the mdation prin-
aple is invoked throughout, so d d ' o n where agents &port their discount
factora does not occur). The moet &ibg d t of t h section is the structure
which optimai collusive schemes must take: u d i h Abteu's (1988) simple strategy
profiles and iinlihe Abreu, Pearee, and Stacchetti's (1986) and Abren, Pea,rce, and
Staeehetti's (1990) bang-bang results, a Upunishmentn must be made to fit the
"crime". In other words, future pay& depend non trivially on current action, so
minor "infractions" are foiiowed by play which i more profitable than the play
following a major Yiafractionn of cartel des. The essay concludea with a brief
discussion of some empirical evidence which appears to be in line with the pre
dictions of the model. This entaile a diScuswon of B~bezat's (1989) study of the
International Steel Cartel which existeci in Europe between the fht and second
mr1d war. Hi8 ckussion implied that member nations of the cartel were eubject
to different internal pmame~ leading to non respect of quotas. In response, pun-
inhments were stnictured accordingiy. Bertrand's (1999) study of firms' impUcit
contracts with w m k is also discussed.
The finai essay of the thesis exmines a mdei which can have economic as weii
as mciologicai implications. The esmy deveiops an extension of Bechri (1973)
mariage model with the search fictions used by Smith (1997). Theee models
study h m rational agents c h m partnem in order to gain accem to a ptoductian
technology quiring the inputs of two agents. M d a g e was the motivating ex-
ampie of Becker (1973) but euch a modal cari be applied to many other situations.
This fiterature aims to characterize the conditions under which agents of a certain
type ody accept partneni of M a r type, i r when the matching is assortative.
useasch, bfafI?hing and Mord Hazard1' examines Bedtd8 model in two difcerent
idormational gcenarios when much is d y and the production technology is
endogenous ta the model.
To ses how the production technology is endogenized, comîder the meaning
and implications of type in a hPo sided matching modal. Oftentimes type is not
terribly relevant. For instance if a fVm hireg a contractor to accomplish a specific
task, tben ail details about the service: tams of delivery and payment may be
perfectly measurable and domable in a contnct. Thus the type of the parties
is supduous since the tramaction taLes place, in essence, on a spot marka.
In longer term relationships, type may be more relevant. For instance, if the
same fim hires a permanent employee, then type becomes more relevant. It is
probably important to the h n to have employees who are willing worlc hard at
their specified ta&. Similarly it is probably important to a worker to work in an
agreeable work environment and receive dequate compensation. The finn and
the worker may have information on a potential match (through derences or
reputation), and this may be of help in the matching choie. Altematively, the
finn and wotker may have unreliable or insuffident information on which to mdre
a decision (a new h m without a reputation, or a ~ 0 t h with no pnor arpaiience
or who was self employed). In any aie the pdtab ' i ty of a match k a fuaction
of the willingaess (or ab'ity) to work overtime on a pmject, ot the awardhg of
perfomance through pay inmeases ammg many othv things.
The previous discmion mgues that type n d not neceRRnrily be job qecific
productivity, eàucation or other factors which dint11 enter into the production
function of the mariage modei studied by &cber. Alternativelly, type could be
a cornmitment to do houeehold chores, visit in-laws or not hold grudges aRer
arguments. This suggests that type enteni into the joint production hinction in a
qecial way, aa one codd aimp teneg on a pmmisa to do dishes or tsks out the
gazbage. This pogsibility of non fulfilment of a promise (implicit or atplicit) hints
that this type of scenario can be modeled as a repeated game where cooperative
(collusive) outcomea may ocm. This is the approach which tbis essay takes.
There are two main innovations in this -y. First is the aforementioned
endogenization of the production technology. The technology employed by Becker
(1973) or Smith (1997) does not ailow for the pdbi l i ty of agents' Udetecting".
The other innovation is the treatment of a matching model under incomplete
information. The incomplete information is akin to the incomplete information
in the second essay, in that it is the patience of the agents (thus theh ability to
mllude) which can vary. Additionally, evolutionary game theory is used to justify
the choice of an equilibrium criteriun which deets a single equilibrium. This
is necmsaxy since aven a concept ul strong aa sequential eqdibrium does not
permit the modales to study under what conditions amrtative matching o b t a i ~ .
Attention is focused on h m the ab'ity of agents to d d ~ t h m agreements affects
matching, as w d l as contnwthg the eqpilibrium srising nnda the complete and
incomplete infornationai scenarics.
The main insight of the emmy is that the fdbility of amortative matchhg is
c l d y linked to the ease with which diffbent le* of coiinsion can be supporkd.
Contrast this wîth the hdinga of Becker (1973) and Smith (1997). Becker fin&
that assortative m a t h g o b t h unda a h a i t any condition, whereas Smith
requires a log nipermoduiarity condition on the production technology. For ex-
ample, unmme that there are two types of agents. The patient type can support
the play of a highly profitable form of coilusion. The impatient type csn only
mpport the play of a lemer profitable fomi of coliusim. If the patient player hàs
it Udifû~ult'' to support the veqr profitable form of collusion, then assortative
matching may be eschewed in favor of an equilibrium where playars are inWei-
ent between matching with a patient or impatient partner. The imposition of
incomplete information adds another layer of complexity to the prablem, and one
of the main pais of the essay is to contrast mrtative matchhg unda incomplete
aiid mmplete information scenarios.
Le thème principale de cette th&, compoaée de trois essais, est 1'4tude de la
collusion en information ineornpl&te. La collusion se définit comme des etrat6gies
déquilibre dans le cadra d'un jeu r6pété qui sont différente8 de la répétition des
stxat6gies d'équilibres du jeu d'une seule étape. L'analyse de la collusion est im-
portante pour beaucoup de branches de l'économie. En organisation industrielle
par exemple, s'il n'existait pas de possibilit6 de collusion, on pourrait restreindre la
recherche B 1'4 tude des marchés oligopolistiques et concurrentiels. L'information
incomplbte représente l'environnement utilisé par les théoriciens des jeux pour
6tudier des modèles dans lesquels l'information n'est pas de connaissance mm-
mune. Sa applications sont omniprésentes dans la science économique moderne de
la théorie des jeux parce qu'elle est devenue un élhent standard d'un grand nom-
bre de modhlm, avec des applications diverses telles que l'analyse des oligopoles
ou de la guerre froide.
Dans le premier essai, on 6tudie la collusion dans un ensemble de marchés qui a
été l'objet des premières recherches systématiques dans l'économie de l'iafomation
incomplhte : les ventes aux enchères de Mude "originale" de Vickrey (1961).
Sauf dans un nombre M t 6 d'exceptions, la littérature sur les enchères a main-
tenu l'hypothèse que les agents participent & un jeu d'une sede période : une
vente aux enchères a lieu et les pdcipants se séparent ensuite pour ne plus
jamais ee rencontrer. C'est le point de ddpart de ce travail dans lequel un envi-
ronnement est d6veloppé explicitement. La répétition dkne telle vente aux
enchèms su- immédiatement la ps ib i i t6 de collusion entre les enchérisseurs
(les achetenre). Ii y a eu récemment quelques pro- dans la compréhension
du comportement des agents rationnels coopérant dans les ventes aux enchères,
mais ces travaux ne se sont pas attachés B l'analyse en jeux répétés. Pour des
raisons de cohérence interne, il est donc raisonnable d'hdier le collusion dans un
enWomement explicitement répété.
L'btude de la collusion dans des ventes aux enchères explicitement répétées a
également de l'intérêt d'un point de vue plus pragmatique. D'abord, les prédictions
générées par un modèle statique ne se transposent pas toujours h un modèîe dy-
namique (d'une fwon qu'on explicitera plus bas). Ermite, un modèle dynamique
apporte des résultats plus riches que son équivalent statique. Enfin, il émerge
des modèles dynamiques une ciasse entière de r h i t a t s sur le comportement des
vendeurs, qui ne peuvent être obtenus dans les modèIes statiques.
L'une des premières recherches sur la collusion dans les ventes aux enchères
a bt6 dectuée par Comanor et Schdermen (1976) qui ont dtudii un pazadoxe
dans le comportement d'enchère : les mises identiques soumises par les achetews
(sous les hypothèses habitueIIes, des mises identiques devraient être obsenréea avec
probabilite zéro). McAfee et MeMillan ont pu résoudre ce paradoxe en montrant
que si les acheteurs étaient incapables d'effectuer des papent8 de tranderta entre
eux, par exemple B cauae de la pOggibilit6 de retracer tout accord M t , qui rendrait
la detection tda probable, dors la réponse optimale d'un cartel serait que chape
acheteur mise le prix de réserve et laisse le vendeur agir de fwon B rendre le rérniltat
aldatoire. Bien sûr, si le vendeur accorde le bien d o n un processne préd6temin8,
par exemple B I'aeheteur dont le nom anive en premier dans l'ordre alphabétique,
le cartel a besoin d'un peu plus de coordination. L'intuition ci4 ici est que le cartel
doit renoncer B I'efEcacité de fwon B surmonter les p roblba incitatifs introduits
par l'existence d'information privée. Bien qu'ils se soient contentb d'examiner
un modèle statique, McAfiee et McMillan ont supposé l'existence d'une forme
de stratégie de la gachette qui rand l'adhésion au cartel plus profitable que la
défection.
"Collusion in a Mode1 of Repeated Auctions" (Collusion dans un modèle
d'enchères r8ptStées) Btudie la collusion dans les ventes sw enchères du point
de vue dtun jeu explicitement répété. Ainsi, au lieu de suppaeer l'existence d'une
forme de stratdgie de la gachette, ces stratégies sont expiicitememt construites.
Un résultat immédiat est que l'on peut fdement donner un exemple d'une série
d'enchhw dans lesquelles la strathgie de McAfee et McMillan de miser le prix de
réserve n'est pas soutenable. Par codquent, en plus des habituelles contraintes
de compatibilith avec les incitations, on a une contrainte de la gachette dynamique
nouvelle (ou, seion la terminologie d9Abreu, Pearce et Stacchetti (1986)' une con-
trainte d'"admissibilit6"). En gros, l'intuition est qu'un acheteur peut donner
beaucoup de valeur B un objet aujourd'hui, si bien qu'il est prêt B faire face h toute
punition crédible du moment qu'il peut le posseder tout de suite. L'inefficacith de
la fonction de mise Uplate" de McAfee et McMillan est la force qui sous-tend ce
comportement. La question &ridente est que si, pour certaines suites de miws, le
prix de réserve n'est pas un schéma admissible, dom qu'est-ce qui l'est? Et qnel
est le schha optimal? De plus, que peut on dire du comportement du d e i dans
le tempe? L'idormation incampl&te peut-elle cik une sorte d'instabilit 4 dans le
cartel? Le rôle du vendeur esMl d i f f h t de son rôle dam un modèîe statique où
sa meiUeure arme pour combattre Is coliusioa &ide dans le prix de &me? Cet
essai propose des réponses B ces questions.
Cet essai utilise des techniques d'abord déveioppées psr Abreu, Pearce et S tx-
chetti pour endogénéiser les punitions. Ces punitions difIérent de celles qu'on
trouve habituellement dans les jeux répétés en cela qu'une enchhe est un jeu B
information incompléte. Les résultats prinapaux sont qu'un cartel autorise un
nombre fini de niveaux de mises et que la collueion est stable dans le temps.
Le phénomène le plus intéressant at sans doute le nombre de prédictions qui
émergent grâce B la nature explicitement dynamique du modèle. Par exemple, les
profits du vendeur peuvent être aEect6s par la fréquence avec laquelle les ventes
aux enchères sont tenua.
Mallieusement pour l a ca,rtels, la collusion dure rarement longtemps. Cette
observation est troublante pour les économistes parce que la plupart des modèîes
de collusion en jeu rép6t4 prMbent une partaite stabilitd du cartel dans le temps.
Pour une littérature qui existe depuis F'riedman (197l) et qui a fait l'objet de tant
d'efforts de recherche, c'est problématique. Le second essai, intitul4 "ûn Cartel
Stabiity" (Sur la stabilith des cartels), &oncilie la thhrie avec le fsit stylisé que
les cartels sont généralement instables.
L'une des h y p o t h k les plus communes dans la littérature sur la collusion
est qye les joueurs ont le même degré de patience-qu'ile ont le même taux
d'escompte. Harrington (1989) eet une exception intéressante i cette tendance;
ü permet en &et aux firme8 en coliusion d'avoir des taux d'escompte diErenta*
Md@ cette h6thgénéit6, Ies d i f fb t s taux d'escompte restent -tement
connus. Dans cet essai, on imp- une h6térogenéit4 et une inobservsbilit6 eur les
tao* d'escomptes des agents.
Supposons que les facteurs d'escompte (les de@ de patience) sont inobserv-
ables et varient B mesure que les joueurs jouent le jeu répét6. Un cartei mrurimis*
teur fait fme B un arbitrage : d'un côt4, plus la co11usion est profitable et plus
elle est difncile soutenir; ainsi, une petite chute du taux d'escompte entzaine des
défections coûteuses. De l'autre côte, moins la collusion est profitable, moins il est
dinide de la soutenir, ainsi de petites variations dans les taux d'escomptes sont
peu susceptibles de déstabiliser un cartel rMtan t d'une collusion mutuellement
bénéfique. Si les punitions &aient simples, comme dans Abreu (1988), on pourrait
obtenir comme réeultat que les agents qui ont un facteur d'escompte en dessous
d'un certain niveau font défaut autant qu'il est pennis par la structure du jeu
alors que les agents qui ont un faeteur d'escompte supérieur B un certain niveau
respectent les règles du cartel. Cet essai vise B caractériser les efforts d'un cartel
optimisateur (un concept qui est défini avec précaution) qui est confxont6 B un tel
scénario*
Supposons que les facteurs d'escompte des agents sont tirés B chaque période
d'une distribution iid. Je soutiens qu'un phénomène relié B la défection et aux
punitions émerge sur le sentier d'équilibre (on invoque le principe de révélation
tout au long du travail si bien que la forme de défiition par laquelle les agents
mentent sur leurs véritables f a e u s d'escompte n'a jamais lieu). Le résultat le
plus édifiant de cette secti~n est la structure que doivent prendre les s c b h
optimaux de dusion : au contraire des profils simp1es de stratégie de Abren
(1988) et des résultats %mg-bang" de Abreu, Pesrce et Stacchetti (1986 et 1990),
ch- Upmitionn doit étm B la mesun du "crime". En d'autres termes, les
gaina futurs dependent de façon non triviale de l'action courante si bien que des
"inhctions" mineures sont suivie d'un jeu qui est plus profitable que le jeu
suivant les "inüactions" majeures aux régles du cartel. L'essai est conclu par une
brbve discussion des &idences empiriques qui sont dans la lignée des prédictions du
modbe. Ceci entraîne une dkusion de l'étude de Barbezat (1989) sur le cartel
international de l'acier qui existait en Europe entre la première et la seconde
guerre mondiale. Son analyse indique que les nations membres du cartel étaient
sujettes & des pressions intemes menant au non respect des quotas. En r6ponse,
les punitions étaient structurées de fqon appropri&.
Le dernier essai de la thése htudie un modèle qui peut avoir des implications s+
ciologiques aussi bien qu'économiques. L'essai dhveloppe une extension du modèle
de mariage de Becker (1973) avec les coûts de recherche utilisés par Smith (1997).
Ces modèles étudient comment les agents rationnels choisissent leurs partenaires
de façon B avoir acch B une technologie de production requ6rant les inputs des
deux agents. Le mariage &tait l'axample utilisé par Becker mais un tel modèle peut
être appliqué B beaucoup d'autres situations. Cette littérature vise B caxactériser
les conditions sous lesquelles des agents d'un certain type acceptent seulement
des partenaires d'un type similairet c'est-8-dire quand l'appariement est assorti.
"Sean&, Matchhg and M d Hazard" (Recherche, Appariement et Aiéaa Moral)
6tudie le modèle de Becker daas deux scénario8 infomationnels Mbts quand
la techerche est coûteuse et la technologie de production aidogène au modèle.
Pour voir comment la techn01agie de production est endogénéisée, considérons
la signification et les implications du type dans deux modéles d'appariement par-
allèles. Souvent, le type n'est pae vraiment important. Par exemple, si une &-me
engage un contracteur pour accomplir une tâche spécifique, les détails du ser-
vice (termes de la livraison et paiement) peuvent être padaitement mesurables
et exécutables dans un contrat. Alors, le type des parties est s u p d u puisque la
transaction a lieu, essentiellement, sur un m8tch8 courant. Dans des relations de
plus long terme, le type peut être plus important. Par exemple, si la même &me
engage un employé permanent, le type devient plus d6terminant. Il est proba-
blement important pour la firme d'avoir des employ6s prêts B travaiiler fort sur
leurs tâches spéaBques. De la même façon, il est probablement important pour
un travailleur d'avoir un environnement de travail agréable et de percevoir une
compensation adéquate. La h e et le travailleur peuvent avoir de l'information
sur un appariement potentiel ( B travers des références ou la dputation) et ceci
peut aider au choix d'appariement. D'un autre côté, la firme et le travailleur peu-
vent avoir, pour prendre une d a o n , de l'information non fiable ou insuffisante
(une nouvelle &me sans réputation ou un travaiileur sans expérience préalable ou
qui était son propre employe). Dans tous les cas, la profitabfit4 de l'appariement
est une fonction, entre autres choses, de la volonté (ou de la capacitd) B t r a d e r
en temps supplémentaire sur le projet ou de la rémunération de la performance
par des augmentations de salaire.
La d i s d o n précédente fait valoir que le type ne représente pas nécasahnent
la productivït6 speafiqne B l'emploi, l'éducation ou d'autres facteurs qui entrent
directement dans la faction de production du m u e de mariage &di& par
Becker. Le type peut être un engagement hrire des cornées ménagères, B visiter
la belle-Me ou B ne pae g d e r de rancune aprb mie dispute. Ceei su&m
que le type entre dans la fonction de production jointe d'une fapon particulière
car l'on pourrait toujours renier une promesse de faire la M e ou de sortir
lea poubelles. Cette p088ibilit6 de ne pas remplir ses engagements (implicites
ou explicites) suggère que ce type de scénario peut être modéM comme un jeu
répét6 où peuvent spparaitre des revenus de la coopération (de la collusion). C'est
l'approche que prend cet essai.
Il y a deux grandes innovations dans cet essai. D'abord, il y a 19endog4n&ation
mentiorm6e plus haut de la technologie de production. La technologie employée
par Becker ou Smith ne permet pas la "défection" des agents. L'autre inno-
vation est le traitement d'un modèle d'appariement en information incompl&e.
L'information ineomplbte est ici reliée B l'information incompl&te qu'on a dans le
second essai en cela que c9est la patience des agents (donc leur inclination B la col-
lusion) qui peut varier. De plus, on utilise la thbrie des jeux évolutionniste pour
justifier le choix d'un critère d'équilibre qui sélectionne un seul équilibre. Ceci est
nécessaire parce que même un concept aussi puissant que l'équilibre séqyentiel ne
permet pas au mod6lisateur d'4tudier sous q u e b conditions on obtient un a p
pariement essorti. L'attention se concentre sur la fqon dont la capacité des agents
B violer letus engagements &te l'appariement aussi bien que eur la comparaison
des &@libres émergeant de scénarios en information compléte et incomplhte!.
L'apport principal de cet essai est que la réalisabiit6 de l'appariement assorti
est 4troitement bée B la f W t 6 avec îaqyelle d i fbn t s niveaux de d u s i o n peu-
vent être supportés. Comparons ceci avec les rMta te de Beclrer et de Smith.
Becha trouve qne l'appariement assorti peut être obtenu sous presque n'importe
qgeiles conditions dom que Smith reqniert une condition de log supermodnlrvit6
sur la technologie de production. Par exemple, suppo8on8 qu'il y a deux types
d'agents. Le type patient peut supporter le jeu d b e forme hautement profitable
de collueion. Le type impatient peut seulement supporter le jeu d'une forme de
collusion moins profitable. Si le joueur patient trouve "difficile" de supporter
la forme de couusion tds profitable, le matching assorti peut être contoumé en
faveur d'un équilibre où les joueurs sont indifférents entre s'apparier avec un paxte
naire patient ou impatient. L'imposition de l'information incompl&e ajoute un
degr6 de complexité au problème et l'un des buts importants de l'essai est de
comparer l'appariement assorti sous Mérents scénaxias en information compl4te
et incompl&e.
Table of contents
Introduction iii
Coiiwion in a Mode1 of Repeated Auctiom 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 AnExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Assumptions and Notation . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Static Representation of Repeated Auctions . . . . . . . . . . . . 11
1.5 Optimd Collusive Rules . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 AuctionearBehavior . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7 Conclusion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 IntfOduction 47
. . . . . . . . . . . . . . . . . . . . . . 2.2 Assumptions and Notation 50
. . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Static Repreeentation 54
2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Wts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
. . . . . . . . . . . . . 2.6 Tnterpretatim of Changing Discount Ratcs 7ï
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Con&sion 74
Search. Matching and Moral Hasard 75
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 76
3.2 Mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 E'ullInformation 86
. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 RivateInformation 102
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
List of figures
Acknowledgment s
1 am indebted ta a pst many people and organizatiom who have hdped me
during my studiea at the Univemit6 de Montréal.
F i d y 1 muid like ta thank the Centre de Recherche sur les Thsports (Cm)
and the Centre de Développement en Ekonomie (CRDE) for pmviding top notch
fru?ilities which 1 uaed for four ye-. The CKT supported me t h u g h my k t two
yeam through a series of eeholarships. The CRDE supported me through travel
assistance to numemus conferences and most importaatly through inviting guest
speahs to the department which greatly benefitted me. 1 was also mpported
through the Départment des Sciences Economiques et the Univeraith de Montrhd
for my k t three years. 1 would a b Hce to thanlt my advisor, Jacques Robert, for
supporthg me through my l u t two yem through grants h m the Social Sciences
and Humanities Researeh Council of Canada (SSHRC).
1 benefitted h m my interaction with my fdow students through a series of
student seminars organizeà by Lars Viuber and later by Stefim Ambec. With-
out Karine Gobert, who graciously translated the Résumh, 1 think 1 would have
required ewther yesr to get this section readable.
The single largest source of help thughout my writing this thesis has corne
fimm my advisor Jacques Robert. The nret -y of t h thesis stems h m joint
mtk beheai us. His contribution did not end tben as he prwided iwaluable
input and critichm for di aqpccte of the th&. In addition to bis intellectual
contribution he provided me with hancing so that 1 new had to worry about
fiom where fun& f a coderence8 or living would corne-
Lastly 1 would like to thsnk my bxdy. My parents have always b a n wiihg
to support me throughout any endeam which 1 chose to underth. 1 consider
myself very lucky. Duriiig the writing of this theh my family gmw to include
my fiancée habelle and her firmily Gilles, N i d + Phiiippe, Maryiise and M e .
Isabelle was essential for my sanity throughout the past three yeams, and our
numerous trips to Joliette to nBit her femily Iet me forget about economics for at
leaat a little while.
Collusion in a model of Repeated Auctions
Paul Johnson and Jacques Robert
A model of h t pria sealeci bid auctions is developed where bidders meet repest eàly wbile independently drawing private vaiuations in each pesiod. Attention is focused on symmetric collusive bidding equiiibria whsn sidepayments are not ai- lowed. Via an approach introduad by Abreu, Pesree and Stacchetti endogenous piinifihments are characterized and 4 in the construction of optimal collusive bidding strate*. This analysis Mers h m mal repested game treatments due to the presence of incomplete idormation. Optimal co11usive bidding strategies are generdy inefficient and have a bang-baug nature which implies that ddection is never observeci. Auctioneer responses are elso studied, which in this explicitIy dynamic setting give rise to iaeights not apparent in a static formulation.
Achiowledgxnemts: The authore benefitteà from the helpful comments of Yims Spnunont ea d M patticipanl at the 1998 Canadiaa Ekonomic Theory C o n f i c e and the 1998 North American Summer Meetings of the Eeonometric Society. ûf course emm remah our m. Support uns provideci by Canada's SSHRC, the Département de Scienas Econcuniques at the U n i d t é de Montréal and by the Centre de Recherche sur las lhnrporb at the Univemit6 de Mon-.
1.1 Introduction
It is weli recognized that auctiam account for a signibnt lare of mirrant ece
nomic aetivity* Fkom a pragmatic point of view, this is pcdbly why auctiom
have been the focus of so much rrsaarch dor t for nearly forty y m . With few
exceptions, the literature hie maintained the amumption that agents participate
in a one shot game: an auction is held and the participants part Company nevu
to meet again. This i the point of departure for this work, and is justifieci by
the fact that in most auctions there is a core of long nui biddersl. The repeated
nature of mch an auction immediataly mggeste the pogiibility of bidder coliusion.
RecentIy there has bcen mme progre~e maâe into undemtmding the behavk of
rational agents colluding in auctions, but this work has abtracteci away from a
repeated game analysis. For intemai consistency this is reason enough to study
coilusion in an explicitly rapeated environment. Studying coilusion in auctions in
a repeatd setting is pertinent for several additional reamns. FirstIy, the predic-
tiom generated by a static model do not always carry over to a dynamic model (in
a way to be made dear Mm). Secondy, a dynamic model yieldP richer resuits
than its static counterpart. And thirdly, an entire daes of resdts on auctioneer
behavior mages which does not atiae in a static model.
One of the earliest examinations of coUusion in auctions was done by Cornanor
and Schankeman (1976) who esmineci rotating bid schemes (the moat famous
ca~+study is perbapa the "ph88e&of-themoon achemen detailed ôy Smith (1961)).
They aiso studied a p d o a in auction behavior: that of identicai M& mbmit-
tod by b i d d d . McAfee and McMiil~n (1992) were able to resolve this puadm
by showing that if bidders were unable to efktuate sidepayments amon@ them-
selves, perhap because of a "paper Wn which makes c d y deteetion vuy W y ,
then the optimal response of a cartel would be for every bidder to bid the resave
pria and let the suctioneer act as a randomization device. Of coiuee if the auc-
tioneer were to award the good in a pndeteTmined way, say to the bidder who'e
name was fi& dphabeticaily, then a little bit more coordination by the ring is
required. The key insight here is that a ring must renomce efficiency in order
to overcome incentive compatibility problems brought about by the existence of
pnvatdy held information. Though they only examineci a static model, McAfiee
and McMillan assumai the existence of soma sort of trigger-strategy which makes
adherence to the cartel more profitable than d e f i o n .
When the restriction on sidepayments is lifted, the andysis changes mther
drasticaily. As Graham and Marshali (1987) show, second price and En- auc-
tions are susaptible to a mechanism cailed a pre-auction hocLout which not ody
succeeds in winning the objcct at the reserve price (when the cartel is al1 in&
siva), but always awBTds the object to a buyer who values it the most. Efficiency
is retained. In their paper, M d e e and McMillan (1992) dm study 'strong car-
tels". These carteis can eflaetuate tmu&ers amongst themselves and can exclude
non serious bidders. They a h propose a mechaiiiasn which is efficient and wias
the item at the rerienn priœ. Agdo, th- mdels abrrtraet away fiom repated
play which muet be used to jw obedience to mch mechanham.
Ridnun (1971) was the bt to forma,îize the YoUT theorem showing that
repeated partnerships enable plagers to coordinate to equilibria which Psnto dom-
inate any single stage Nash m b r i u m . Uaul anaiysis has absfracted away h m
any uncertaiaty, but if a modd of auctim L to be studied one must keep in mind
that uncertainty ia the raison d'êtm for auctions. In tact it PB thi8 incomplete in-
formation in an auction which changes the ruinlyshi of repeated play, since players
(in a riease) play a difterent game in each period. The @8ct information assump
tion has been relaxed by Grien and Porta (1984) a~ naU as by Abm, Pearce,
and Stcrechetti (1986) and Abreu, Pearce, and S t d e t t i (IWO), mainiy thn,ugh
the study of Cournot oligopoly whm esch periodb pria gim an imperféct signal
about (private information) h-epednc quaiititie The uncertainty in auctions
is somewhat different h m the uncertainty deecribed above. In Coumot eampe-
tition an Uauction-like" uncertainty would be more d9n to firms holding private
idormation about stochastic coab, as oppased to the demand side uncertsinty
studied by Green and Porta and Abreu, Peam and Stacchetti. Neverthelem,
the approach introduad by Abreu, Pearce and Stacchetti proves to be particnlarly
usefùl in Btudying tepeated auctions.
An immediate outcome of studying expliatly repeated auctions is that one
ean easily give an m p l e of a series of auctions where McAfee and McMillan's
strategy of bidding the reserve price is not supportable. Thus, added to the
familiar incentive compatibility comtdnts ie a new dynamic Wggd' constraint
(or in the terminology used by Abreu, Pearce and Stacchetti, Ulllm;.a;bii;tf').
Roughly, the intuition is tbst a bidder can d u e an object vsly highly today-
a0 much m that ha is prepued to faea rny credible puniRhment in ietum for
poesessing the item t d a y The inefficiency of the MeAae and McMi11an %t"
bidding fiinction is the dnving f- behind t h behavior. Section 1.2 gives a
simpIe example of this phenornenon. The obviow qyestion is that if for eerhin
d e s of auctions biddiag the n e v ~ price is not an ulminnible scheme, then what
is? and what is the optimal scheme? Furfhermore, what can be said about cartel
behavior amas tune: cm incomplete information cause! some h d of instability
in the cartel? Does the role of the auctioneer ciSm h m that of a static mode1
where hia main weapon in combatting collusion is the resarva price? This paper,
therefore, is an attempt to answar the a b m questio~s.
Section 1.3 presents the model to be studied through a d e s of BSSUrnptions.
Section 1.4 shows that the method of A b m , Pearce and S t h e t t i csn be used
in auction games. Some of the major pointa of the papa are stated in mtion 1.5
where optimal coilusive bidder behavior is studied. The role of the auctionea is
presented in section 1.6 whue several implications mise only due to the repeated
nature of the scenario. Section 1.7 b n d y concludes. Many of the proofs can be
found in the appendix.
To motivate the loilowing sections, this section pnsents a simple ewmple. Con-
aider two bidders who meet repeatedly in a first pnce d e d bid auction, discount
ing future earnings with a cornmon discount rate 6 E (O, l), drawing independent
vaiuatioas, denoted u, each paiod h m a d o m àistribution over [O, 11. FoUow-
hg McAfee and McMillan, aippoae they decide to use a form of tacit collusion
to in- theh expected profits whem each bids a m (the nserw pnce) in each
peziod. The auctioneer is BBSUrned to r a n d o h equaliy in the case of a draw.
Suppose that this aUusion @es thst ueh playv use Nash stratagies forever if
a winning bid above aaro is ever obssrvad. In order for rny bidda type to pder
coiiuding it is nemsary and aufiicient that the highest valuation type prefm to
adhere to cartel rules:
The sufficiency follol~s b m the fact that the incentive to cheat for a player with
duation one is mater than for any other vaiuation. The left hmd Qde repre-
sents the gains to obeying the collusive d e for an agent with valuation one, and
the nght haad aide represents the gains ta delecting fiom the coflU8ive d e for
an agent with valuation one. However, for say 6 < 617 this coilusive rule wiii
eventually provoke defiction (the arimirrrrib'ity constraint is not satisfied). The
inefnciency of the coiiusive d e drives a high vaiuation agent to prefer cheating.
The obvioas question becomes what is the form of collusion which should be used
to guarantee the participants the highest discounted expected payofb? Can Borne
kind of dedection be permitted? How should ddectors be puninhed?
1.3 Assumptions and Notation
A set N = {1,2,. . . ,n} of ex-ante identical potentid buym compete in an in-
finitely repeated auction game with discounting. Detiüed assumptio~ can be
found belm.
Tlic Stage Game
(Aesumption 1). Each potential buyer ie -ante symmetric, drawing an
independent private value for tha object in question h m a common continuou8
distribution hc t ion F(-) with strict1y positive continuody diffanatiable density,
f (B), defined on a compact support [O, a]. For mu0118 to be made dear later agmme
1 Fu that the h ~ a r d function, H(v) := '*, is strictly decreasing in v.' F'urthermore
F and f are common knowledge.
(Assumption 2). Each potential buyer is ririk neutmi.
(Aesumption 3). The auctioneer allocates objects via a k t price auction,
randomizing equally among the ninnm in the case of a draw. The auctioneer
publicly announces the amount of the winning bid, but does not announce any
other idonnation (including the identity of the winner).
(Assunption 4). Sidepayments ara not permitted betwean potential buyas.
(Aesumption 5). The der's reserve price is nomalized to zero.
(Assumption 6). Unicity of equilibrium to the ~tage game is ammeci. ' The Repeuted Game
Let p(t) : [O, a] x [O, cl x .!7! ?!?. x [O, C] + [O, C] be a bidding lunction at time
t. This bidding huiction sench currant duation as weU as aU previous winning
bids, (bw(t) E [O, cl) into the bidding space [O, cl. Let P = x$(t) be the strategy tE
set of amilable bidding functions.
(Assumption 7). Playve discount future shge profite with a c m o n d i e
count rate 6 E (O, 1).
(Aesumption 8). Vaiuatiom ara drawn independentiy and identidy rerociri
time.
(Aesnmption 9). The 4 restricta itaeif to symmetric and undominated
(in the sense of Pareto) collusive schemes.
(Assumption 10). The equïübrium concept useà is that of Rubinstein's
(1979) (eubgame) pdect equilibrium.
Assumption 7 above permits the restriction that each potentid buyer mbmit
bids in a compact i n t d [O, cl. This can be done by taking c to be defined as
the hieest possible payoff one can zeceive in the entire gama, Le. c = Ae-
sumption 9 can be done away with by impoeing the Nash bargainhg axioms. T b
would not be a departme h m the mainstream of the literature eina modeler8
mally choase such a focal point when conhnkd with 8 continuum of possible
payadl vectors. However, pronding a criterium for the selection of equilibria in
repeated games is outside the scope of this paper. Our objective is simply to study
the best symmetric colluive strategy* Remark that under the informational struc-
ture imposeci by Assumption 3 a cartel must, md can, be all encompassing. This
is because a cartel cannot distinguish cheating by a member or non-member, and
it can ctedibly menace any deviation wîth a piinishment. Findy, note that due
to the assumptions of ex-ante eymmetry end continuone distribution funetion, a
theorem from Miigrom and Weber (1982) can ùe used to show the existence of
a symmetnc bidding eqyilibrium where the bidding huictions are increasing in
valuation.
1.4 Stat ic Representation of Repeated Auctions
With thme essumptions and notations in mind, we non proceed to uialyze the
repeated game as a single stiye m e . Abmu, P m , and Stacchetti (1986) were
the k t to use a generalization of the technique8 of dynamic programming to
analyze noneooperative games. This pennits a vesy condent representation of
repeated games as much more tractable single stage gamm. The intuition is the
foîlowing. The single stage repmtation of a npeated game 'Yactorized' the
gains acauing to each playa into two parts. The k t part is the gain fiom the
m e n t etage game and the second part is expected, biture gains which can be a
function of any observable action taken in the current penod. So this represen-
tation needs two elements: a payoff hction for the stage game and a function
d d b i n g friture gaim. Proposition 1 dates that an equiiibrium to the single
stage game which satbîm a W generationn criterium is an equilibrim to the
repeated game. Proposition 2 states that any perfect equilibrium to the repeated
game can be represented as a Nash equilibrium to the single stage game. Thus
the single stage representation is equivalent to the d y n d c p e . Before stating
these propositions some definitions are pmsented.
Definition 1 Let W be a bouodeà Bord subeet of R Let /3 : [O, a] + [O, c]
snd U : [O,c] + W be Bod memarable hctions and cdl (/3, U) a coliusive
mechanim. (8, Cr) is d d nAminsible with feepat to W in fm all b 6 @([O, a]):
x(v, b) is the expecteà payoff of an agent with valuation u who bids b given that
the otha agents foUow strate@ 80 U(bw) represeats the apected gains of ali
agents, given that the ninning bid ia bw. The operator pi.. ( 0 1 b) ie the conditionai
expectation of fiitwe gains, h m the pempective of an agent having bid b where
aU other playexs follow stratew Note that this conditional expeckition is nell
d&ed due to the assuxnptions on (Boral) measumbility of @ and U, aa well as
the boundedness of W.
Furthetmore dehe u(u; #3, U) := n(v, p(v)) + (U(bw) 1 @(v)). Note that
an ex-ante value can be attacheci to u(v; P, U), denoted as &u(v; @, U), by taking
the expectation over the valuation. Findy a set valued function B (W) is defineci.
DMtion 2 For any (Borel) W C R deîme:
Dehition 3 A bounded (Borel) W such that W c B(W) is d e d self generat-
Propositions 1 and 2 are! non comllaries to the propositions on self-generation
and factorbation in Abreu, P a c e and Stacchetti, since we are asmred of the
existence of the conàitiond expectation. The d e r ia referred to Abm, Pearce,
and Stacchetti (1986) for the prooh. In the foîlai*ing take V to be the set of
pawfi.
Proposition 2 V = B(V).
1.5 Optimal Collusive Ruks
The g d of thie section is to characterize optimal coiiusive schemas with the help
of the Abreu, Peam and Stacchetti iItatic game. M t s ara p m t d in taro
mbsecti0118. Subsection 1.5 prese~ts pmpositiom which are used in Subsection
1.5 to present the problem of a rnahizbg cartel succinctiy. Many of the proofs
appear in the appendix
Punishmeats and Rewards
In explicitly repeated auetions there are hm types of constraints which need to
be satisfied. One corresponds to the mai seif deetion constraiat: if B is the
coUu8ive bidding d e , then type v prefers bidding @(u) to bidding P(ur). The
other constraint wiil be refmed to as admisibility. This constraint stateri that
if /3 is the bidding d e , type u prefm bidding in /3([O, a]), as o p p d to bidding
outside P([O, a]). These two constraints are quaiitatively dinarant becauae in the
latter case, a defection can be detateci.
' Q p i d y the incentive compatibility constrainta force the mdnninms to be
atmctured in such a way that types d f select. In the probIem at band, the mech-
anism must indeed be so stnrctured since vaiuatiops are not observable. Consider
a collusive mechanism {p(-) , WC)) whem P O the bidding fiinction mapping types
into biàs and whara U ( P ) is the future expecfed wfb when the ninning bid is
P. Let Q(v; 8) denote the probability that a plqyu bidding D(u) w i ~ the auction
@ven that @ is the bidding hction u d by waryone else. Since the auctioneer
anards the object to the highest bidda, and in the case of draws randomizes,
Q(u; /3) can be WRtten explicitly. Obvioudy if u is in an h t d where @ suggests
submitting an increasing bid, then Q(u; 8) = F(U)~-'. Othertffise if B is constant
Incentive compatibility can thus be nrittea:
Using the logic of Myarson (1981) ne know that incentive compatibility is
equivalent to demanding that
We can thus obtain a more convenient chamcterization of incentive compatibility
which lets us expIicitly d v e fm the bidding hction in krms of type conditiond
It is important to note that the bidding hction nü] g e n d y be c o m p d of
(positively) sloped and flat regions. h m McATee and McMillan (1992) ne know
that a perfectly flat bidding scheme is a Yolk" type neult which speaks to the
case when 6 is arbitrarily close to ualty.
If we are to study collusion in auctions from an explicitly repeated point of
view, as aras mggesteci in the Introduction and in Section 1.2, another constraint
muat be i m p d : achissibility. The admiwnbility constraints wi l l be relevant
whenevar /3 contains flat regions. The followhg lemms fwnishes some preliminary
results. Its proof c m be found in the appendix.
Lemma 1 Let denote the lowest expected profit d b l y attainable. An in-
centive compatible mechaniSm ID(*), II(*)) ie ahissible if and only if for al1 v* at
the highest edp of a flat bidding range na have:
[u' - P (v' ) JQ (v' ; P) + +(@ (v' ) ) F (ue)"-l + s la U(P(S))~F(~I)"-L 2 tP
The above b similsr to the example introduceà in Section 1.2, in that we compare
gains to deviation with gaim to cornpliance. Tb proof consists primariïy of
showing the existence of discontinuities in the biddhg hction, s h d g that the
impdtion of the hatshest @bIe puninhment is dnays beneficid. Finaüy we
p m that Sali u* at the highest edge of a flet bidding range raspect dmissibiity
then dl types do.
Using the incentive compatib'ity constsaiut (1.1) and the above lemma, we
can rewrite 8àmissibility ais:
Note that when the bidàing function k strictly increasing admissibility is auto-
matically satisfled.
Defie R as an incentive compatible and ruiminnible mechanism which yields
the highest level of expected utility Ü. Define M as an incentive compatible and
admissible mechenism whieh yielde the lowest level of expected utility Ü. h m
section 5 of Abreu, Pearce, and Stacchetti (1986) the wt of perfect payons is
compact, so we are ssrmred that W and are weU defineci. The reniainder of this
section aims to chamterize 5Y[ and
In order to satisfy admissib'ity, d b l e punishments mwt be adable. Propo-
sition 3 shom that the hmhest p d b 1 e puniffhment gim the same payofb as
the single stage Nash equiiibrium. Its p m f can be found in the appenàix.
Proposition 3 No petfèct a- a n giue h a h n the puyoff ~ o c i u t e d wüh
the stage gume Nosh cquüi6rium repcotcd ad inJinitum.
Thk pmpition k an important step in endogaiiising pnnishmenfs as it charac-
terizes the gains to optimal (in the sense of Abreu (1986)) puidmentS. The proof
relie8 heavily on the tact that H(u) is demathg in v dong with the k t that the
oniy information d e & by the auctioneer is the winning bid (Assumption 3).
kieqyality (1.2) leade us to an important r d t on collusion in repeated auc-
t ions.
Proposition 4 An optimal collwiue mechanian, = {&), CI(*)), mwt d i b i t
the bang- bang pmperty :
U(b) = Ü for almoat e u q b E @([O, a]).
Proof of Proposition 4 Let {Q(*; /9), U(*)) be an optimal colluWve mechaaiSm,
generating Ü, for which the bang-bang property does not hold. Therefore, there
exists an intend, [VI, va), such that U(B(u)) c Ü for aiI v E [uI , w]. In thi8
case we show that there gtiets an alternative incentive compatible, admhsib1e
mechanhm which generates higher rents for all types. This mechanism consista of
raising U(P(u)) by some emall amount AU whenever v E [ui, s]. Denote this new
continuation huiction by W. AU fiat bidding regions above q are cut into two
ngions othenrise, the pmbab'ity of winning ia heid constant for dl other types.
b m equation (1.1) , if we iniacrcriurcr continuation payofb in [vi, q] and wish to
praserve typa conditionai pmbabiilitiea of winning an auction (Q(b, p)), it is neag
s ~ r y to change bide in order to retain incentive compatibility. Bide of types 10-
than ul are unadkted ôy such a change. Studying inequality (1.2) immediately
s h m that typa in [vl, -1 nrill sa* nAmiwPibility. The variation in the rents to
vcyaregivenby:
which is strictly positive for ail AU > O.
Consider the intermis b, ü'] whem a !, 2 and for which the bidding bc t ion
is flat. Find teClfTISively a fi where the fdowing holds:
Create a new incentive compatible bidding function (using the new continuation
bc t ion ü'), dinerent h m f l only in that O(*; b) exhibits discontinuities at
aii Ci but constant on ail the b, &] and [Ci , vil. We now have Au(v) = (U-Û~)[Q(V~,Ü,)-Q(~,V~)] 2 O for aU v E [û4,üi]. Addi-
tiondy for ad v E [ut, fi] ne have Au@) = [u -QI [Q(s, 4) -Q(a, Fi)] +Au(%) 2
O. Hence the variation in rents for dl types is non negative by construction. It m
mains to verify that the new mechanism is admissible for all u > y if the original
mechanhm was.
Firet consider checkhg adminsibiîîty in [ûi)~i]. With the original mechsnimi
we have
Under the new mechanisxn we! have that:
Since u(ûi; B., U*) = U(ûi; p, U), and Qh, wr) < Q(Q, ût) I) have that:
So the admissibility constraint, which can be written:
is sure to be satiefied, since the original mechanism was assumeci to be admissible
and Q(&; P', U.) > Q(%, B, U) and (vi - iB (&)) < (Fi - Nui)) . Finally, check admissibility in h,Çi]. Notice that as AU a p p d e s zero,
Cr( must approach y,. It fo11ows that for a AU enfsciently dose to zero, the
addssibility constraint wil i be respecteci for 4. O
Consider the MMee and McMiiian Bat bidding scheme. In this instance, the
incentive compatibiiity constraint is trivial &ce there is ody one 4'advised'9 bid
(the mservation pria). However theh static formulation abstracted away from
admïssibility as was ailuded to in the ~ l i n p l e of section 1.2. Pmpogition 4 states
that an optimd wliusive scheme shouid be sfnictured 80 that any bid in fl([O, O])
be treated as mpecting cartel niles, and eiiy bid not in @([O, a]) should be treated
as dtlfection6.
It follm àirectiy h m Pmpdtion 4 that the bidding fuaction taks on the
f d a r forai:
It is also easy to veiify that any fiat section in the bidding hct ion must be pre-
ceded and foiiowed by discontinuities. Suppose that a flat section somewhere in
the bidding function &arts at vaiuatim v8 > O and en& at vsluation va* < a.
Since the probability of a bidder with valuation v* winning is discretely greater
than the probability of a bidder with valuation v* - E for any E > 0, the de-
nominetor of the above equation increases discretely at va. Therefo~e /3(v8) mwt
inaease W e t e l y as weii. The argwnent that a flat section must be followed by
s diacontinuity is precieely the same.
Coiiusive Schemes as O ptimization Problems
Define the operator 9 in the following mannex:
subject to:
The andnint is the ulmiriihiity ccmtdnt where the bid hm been mWtuted
out ushg equation (1.3) and w h m the variable K can be seen to tdoe the place
of bp - a. @(K) in equation (1.4) represents the diffierence in the per period
rents betwean dusion ushg bidding bction and playhg the Nash stage game
equiiibrium strategy. Note that if
then the coilusive scheme piving payofi of U'(K) every period is admissible. This
amounts to cornpuhg K with the hture payons generated by K.
The following lemma regroups s e v d preliminary results. Its proof is rdegated
to the appendix.
Lemma 2 i) O(.) is non decmasing in its argument.
ii) *(O) = O.
iii) For any K 2 o[i - (l in)] , i t (K) = Jt H(v)[! - F(U)"-~]~F(V) .
iv) There is a hi te 0 such that iY(K) BK for all K.
The above lemma contains some immediatdy interpretable rrsalts. The fmt
that 9 ( ~ ) L non dscreaeing in ita srgument, m e w that gains to collusion are
non decreasiog in 6. And in k t , when playare am perfectly myopic (6 = O) then
coilusion gains and the gains to the Nash stage game equiiibrium strategy are
eqnd. This of course foUm h m the fact that *(O) = O. The interpmtation of
part iii U st r a i g h ~ o d to intapret and is best ciasli as ir W t of McAke and
McMillm (1992) who only consider the incentive compatibility const~aint. The
interpretation of condition iv wili be postpond until the statement of proposition
5.
It is passible to prova that O(K) subject to constmint (1.5) ie continuous
by appeeling to the Theonm of the Muimum, however this does not imply that
collusive profits are continuow in 6. Consider Fi- 1.1 and 1.2 which plot B(K)
against K. The m h a l admissible and incentive compatible coIIusiw payofi am
@en by the intersection of e(K) and the line with dope 9. This follows becam
whenever IY(K) 2 YK we know that the future qec ted collusive rents are large
enough to satisty edmissibility required to do rce this collusion. As in Figure 1.1,
if O(K) is concave in K, then collusive gains am indeed continuous in K. However
if q(K) ie not giobdy concave, as ie the case in Figure 1.2 then a mail change
in d muld generate a large change in the intersection of O(K) with 9. The
relationship b e t m the concavity of i t (K) and the parameters of the problem is
a complex one and we have no mason to believe that O(K) is giobally concave for
ail distributions satisfying Assumption 1. Neverthelm we can state the following
proposition.
Proposition 5 i) For dl 6 tlrm ezW an optimal collusive scheme.
ii) The= cMts a d > O mch that if 6 < 8 M collusion t possible.
Proof of Proposition 6 i) Let 9' = mp{sup ha H(u) [Q(u; 8) - F(t~)*-~]dF(u)} K b
eubjaet to constiraint (1.5). Let Kr and j3' be the argument8 which obtain the sup.
Amme that 9' is not aâmi&ble, Le. Ao(K.) < K.. But th- exists mxne
sequence K, + K8 auch that @(K,) 2 K, for ail n. If the sequeme {K,) is
non monotone or monotone d d g , thm ne have a contradiction since 8(K)
i non decreasing ia K. SO assume that Km t K.. SiDca Kn t IP, thara axiete
a sequence en J, O mch that K. + En = Km. Mdring a mbstitution we obtsin
&P(K*) < Kn + En. But for n 8Uf6iciently lm*, wê have &Y (K.) c K, which
is a contradiction since K. 2 K. and O h nondecreasing*
ü) Rom Lemma 2 we how that there exists a finite number 6 such that
O(K) BK for all K. For a collusive scheme to be admissible and incentive
compatible rire require that &o(K) 2 K. Putting # = assures us that for
di 6 < 6, this cannot be the case.
The next proposition n m dom the clam of optimal collueive bidding b c -
tions.
Proposition 6 ~f Ü > then an optirnd collusive bidding finctiun a n cuntuin
no continuow incnose in the bUlding findion.
The proof of proposition 6 can be fond in the appendk The implications of
this proposition are strong as it implies that an optimally collucihg cartel nill uee
a bidding function which has a finite range. For example all bidders with type
in [w , vil bid 4, ail bidders with type in [VI, q] bid bi etc. Athey, Bapell, and
Sanchirico (1Q98) find a M a r r d t for collusion in a Coumot oligopoly with
incomplete information about cod.
1.6 Auctioneer Behavior
The tractability of the mode1 used up to now has corne at the expense of many
s i m p m g assumptions. This eiurent section argues that despite the simplicity
of the modd, anaiysis of coiiusion in an expiicitly rapaateà environment can lead
to interegting and robust insights m to how an auctioneer can bat st~cture a
series of auctions to resist bidder callusion. This d o n has t h subsections,
each presenting a Mer& way that auctioneers could design auctions to lessen
theh losses due to co11USion. The following lemma is a preIiminary nsult used in
each of the following suktions and ita pmof can be found in the appendix.
Lemma 3 If an optimal coilusive bidding fiuiction exhibits one or more discon-
tinuities, then there exists at Iecist one type indifbnt betneen adhering to and
ddecting h m preeeribed behavior.
Reserve Prices and Bidding Ceilings
The use of reservation prieeg to combat collusion hm been studied by Graham and
Marshall (1987) as well as McAfee and McMillan (1992). This andysis ie strength-
e n d by the observation that auctionaare in fact do uaa rawrve pri- to increase
their profita in the f w of collusion? In McAfee and MdWlan (1992), since the
cartel colludes so that 9 types are a w e â the good with quai probability, the
objective of the wctioneer k to dve the folloning-
where r denotes the raierva price and the seller's valuation.
in the more g m d a w studied in this papa, the ta& of hding an optimal
raierve price is a more damting one. This is principaiiy due to the fwt that
an optimal coilusive d e depanda on the ~ese~lltion pria in a non trivial way
(as was the carie studied by McAfee and McMillsn (1992)). The objective of the
auctioneer is:
where the bidding function not only depends on the valuation but ah30 on the
reserve price. It is an easy exercice to extend the iaalysis of subsection 1.5 to
accommodate a non zero m e price. Tbis is neces8ary to do in order to d e
rive an optimal collusive bidding hinction for a given reserve price. Assuming
dinerentiabüity and concavity of the problem, ne are looking for r* such that:
In McAfee and McMillan (19Q2), @(v, r ) h equal to one. In a more traditional non
F T collusive setting (for example Myemon (1981)) g ( u , r ) L eqtaal to In a
generai coliU8ive setting, it is not p d b l e to aasign a d u e to %(us r). This ie due
to the ambiguous a&ct of e naenia prica on the nAmiwribiity constraint: a change
in the naave prie a f k b the rdmiaib0iQ constraint h u g h LI; additionally a
change in the leeave @ce modifie8 a colIasive biddiiy function by changing type
conditional pmbab'itim of winriing the auction. Hence, the impact of the resave
pnce on the ability of the cartel to cdude is ambiguorni, a fortiori so is its impact
on the equiîibnum bidcüng schedule.
A tool to inmase auctioneer profits which is only apparent fkom sa explicitly
repeated context is a bidding ceiling. Consider the example pnanteà in section
1.2 but with a bidding ceiling of 1/4. Single stage expected Nash profits non
inmase and bemme 31/182. Thdore in orda for there to be no type of bidder
who does not want to defèct h m the efrategy of bidding zero it is neceesary and
d c i e n t that:
When the le& hand side represents the gains to obeying the collusive d e for an
agent with valuation one, and the ri@ hand side rep~esents the gains to defect-
ing fkom the collusive d e for an agent with vahation one. This equation is oniy
satisfied for 6 larger than 6/7. That bidding ceilings can poeitivdy effect auction-
eer p d t s is perhaps surprising, but the intuition is quite straightfomard if one
thinks in terms of a repeated context: lowering the bidding ceiiing makes future
punishments I e s severe, and a less severe piininhment cm support a 1- profitable
fom of collusion. The foilowing propmition gives #mie generai conditions as to
when l o d g a bidding ceilhg can be usefui.
Proposition 7 Suppose that the cunnt p i i a &hg, 8, b &*ctly gmter &un
the higheut bid pnscrikd bg the optt'md colltu~iw bidding findion. Z%en auction-
ccr p m j f b can be ah+@ M m d by a lowering of the ôid enling.
Proof of Proposition 7 Assume, without losg of gendty , that the bidding
cailing is less thsn or e q d to the highest single etage Nash bid. N a m a r k
that since is atnctiy greater than the highest prescribed bid, 6 can be 10- by
E > O whiie stiU preserving this inequaiity. Furthermore since iowering 6 by any
amount in- the single-stage Na& expected payofh, any indifferent typa wiïl
now stnetly pder defecting h m the cartel to obedience. T h d o r e , the cartel
must remedy this problem by decreasing the length of certain Vat spots" on the
bidding hct ion or adding more àiscontinuities. Either reqonse results in 10-
profits to the cartel and higher profit to the auctioneer. If the opthal bidàing
function contains no diseontinuities, then there may not be an indifferent agent
(lemma 3). Howevu, by chooeing the bidding ceiling such that the Nash single
etage payof and the collusive (bidding zero) single stage payoff mer by lem than
nœL then at least one type strictly preféns defection. This new bidding ceiîing a n s
obviously leads to a higher average winning bid. O
Stated differently, the abow that a necessary condition for an auctioneer
to mâitimiae ptofib ie to have the bidding ceiling quel to the higheet p&bed
bid of an optimal collusive biddiiig fuaction. This is a etrong r d t since it implies
thst auctionani al- profit fimm the impogition of a ceiîing. Unfortunately,
while bidding c- and mmve pricee may be usenil in incteaaing auctioneer
pap& the following proposition abm that they an not d u e n t for msicimiaing
auctimeer profits. The intuition ir simiiar to that of proposition 7 in that one ciu,
outlaw 0 t h portions of the range of bids in order to weaiœn future pmhhments.
Proporition 8 Contmlling the bidding d i n g and mcruc prie is not mcient
for an uuctioneer to maCrnize profa
Proof of Ptoposition 8 Suppme that the auctioneer hm rnaxhnhed hie profit
with mect to the bidding cailing and floor.' h m lemma 3 and proposition 7
there d s t s at hast one type indifterent between detecting h m and adhering to
cartel rules. Denote the prescribed bids of this bidding function {bi, . . . , bc), mch
that a, > a-, for dl à. Propa3ition 7 implies that 6 = bk. Non outlaw bidding in
an interval just belm bk: (bk -6, bc). Single stage Nash payofb must increase when
any i n t e portion of the biddhg range is outlaned. Since there exists a type
indifferent between adhering and defécting under the old system, this type now
strictly prefm defecting. Thedore, in order for cartel profits not to d m , the
cartel must corne up with a more profitable collusive scheme* By assuming that
the cartel had chosen an optimal collusive bidding fpnction, the only way for the
cartel to improve payoffs is to permit bidding at bt - IZ The lrey is recognizing
that since E cm be taken to bearbitrerily smdl the admimibility constraint wili
not be binding between bidding bk - r and bk. Consider three possibiliües: 1)
replace b by bk - E, 2) replace 44 by b - E, 3) add a new bidding level at bk - E.
In each cluie if a cartel were abk to increase its pmfitabiüty this muid imply that
it had originally ch- a non optimd coilusive bidding hinction-a contradiction.
The surprising d t s on biddiiy ceilings is qnik intuitive, yet to our knowledge
bidding ceilin@ do not exist.
The inefficiency of the McAfa and McMiilan coiiusive scheme has been retained
to a certain extent; thus one would expect to O- identical bids. Due to
the presence of thme identical biàs the choice of tiebreaking mie used by the
auctioneer becorna important h u g h its dtéct on the admîssibilitg constraint.
Consider the example in section 1.2. Aesuming a randomisation on the part of
the auctioneer in case of ties (a post randomization) na see that the McAfiee and
McMillan scheme is enforceable only when 6 2 617. Now consider what happene if
the auctioneer, pior to bids being aubmitted, publicly designates a winner in case
of equality in the highest bid (a ante randomization). Assuaze the designation
ie made by a fiip of a coin. In this cese the neceesary and GUfficient condition for
obdence to this scheme is the foliowing:
The diffmce hem is the first term on the Ieft hand side. This term refiects that
a player with s valuation of one has been desipated as the loser in case of a tie.
The gains to defecting b m cartel rules have i n d ~rir is reffected by solving for
6 in the abcm inequality to reveai r5 2 12/13. The above ugument is formalized
in the fdowing pmpoeition.
Proposition 8 Biddet pmjb un& a port riindomhtton un ohwys ut tead as
laqe as under cz ante mndomiaatton.
P d of Pmpodition 9 Consider the optimai d w i v e d e m e {& 6, . . . ), and
suppose u E [$, uh,]. The nAminnibility constraint under a post raadomization
is:
Now consider the admissibility corutraint under ex ante randomization:
Cornparhg Q(ui+, , vé) with F(u;)"-' implies that one can use lemma 3 to anive at
the following conclusions. If there exista one or more àiscontinuities in the bidding
function then ex ante randomizstion &ctly domhates ex ps t randomization. If
there exists no discontiouity then ex ante mdomization weakly dominates a
post randomization. O
The above axgwnent implies that a castel would always pder to face ex poet nui-
domiaetion. The existence of rotating bidding schemes may be interprated to be
the cartel's naction to an uUn8~~0mmodating" auctionasl. Tbat the enect on auc-
tioneer profita of such a b p l e change in tbe randomization d e is unambiguous,
it is suprising that one does obeava auctioneers who use ex past randomization.
ORentimes esrvices can be procured through aucti011s occurirg with a regular
fkquency. One might think of a garbage collection contract behg amdeci by a
municipaiity through auction averp year. For the praient purposes, the important
part of this acenario is that the auction occm every year. Once again consider the
example of section 1.2, but now cicnnune that the auctioneer d s 2 items et every
other period-the auctionesr bundles item. In thjs c a ~ a necessary and d u e n t
condition for obedience to the McAfee and McMillan scheme is the foiiowhg
inequaiity :
Solving for d one obtsins 6 2 One observes two effects hm. The first
is the reduction in the effective discount rate h m 6 to &? The second is the
increase in the gains to cheating. Again we formalize the above intuition into the
following proposition.
Proposition 10 Bidder projib an aiways decraosing in the number of i t m bun-
dled together.
Pmof of Proposition 10 Fot simplicity consida bundling hro items together,
and compare the correspondhg nAminnibility constraint with the AAmisffib'ity con-
straint when there is no bundling. Again consida the d u s i v e scheme {v:, u;, . . . ), and suppase v O [u;, &]. One can write down the admjssibility constdnts and
of the dccrease in the discount k t m is e t ra igh t fod . Usiiig lemmcr 3 one can
reeson dong the same lines as in the pmvious pmpdtion. If the bidding bc t ion
has st leset one discontinuity then domination by bundiing L strict. If the bidding
hinction contninn no discontinuity then the domination t weak. O
It is important to etrese that this lest v e n t hm ignored important considersr
tions mch as my corrts a municipality may incw in making a long term commit-
ment to a single d c e eupplier.
1.7 Conclusion
Collusion without mdepayrnents in explicitly repeated auctions hm been studied.
It is the repeated nature of the modei which dinkrentiatai it fmm earlier work.
Techniques Brst developed by Abmi, Pesrce and Stachetti were used to endog-
enize prlniRhments. Tbese punishments Mer h m the usual tngger strategies in
that an auction ia a game of incomplete information. CollmCon is chaxacterized by
its stability and inafnciency. Collusion is dectuated by submitting identical bids,
or ueing a rotating bidding schemepredictions which are commonly nsed car-
tel tactics. Studying an explicitly rapeated p e leacb to several insights about
optima auctioneer behavior not apparent when studying a static game. The
main hding of the papa, that &eh ara obligeci to uee a randomization nile to
avacome incentive compatib'ity problems, L supportecl by the observation that
identicai bids and rotating bid schemea are obsarpcd.
1.8 Appendix
Proof of Lemm 1 S~ppaile /3 to be e~oe t~n t on [ui,uj]. Fht m a r k that
Q(u;@) is discoatiauous at uj, and that in particufar Q(uj;>B) < F(u3)'? S u p
pose that the condition in the statameat of the lemma is violateci and that @
iû continuous at v$. TbeD Uw(uj + E ) ) 2 f a dl E > O, th- Bgsts a
small E such that bidders of type vj pnfm to bid B(uj + E), violating the inca-
tive compatibility constraints. Now suppose that P is dieoontinuous et %. Rom
equation (1.1) one can v e that B is aleo discontinuous at vj. Therefora th-
exists a b $ @([O, a]) which is arbitrady cl- to B([ui, ~$1) ;vat which discretdy
inmases the pmbabüity of winning. A (credibIe) punidment mwt be adable
to dissuade eueh a deviation. This punishment can be made as sevare as possible
without harming cartel gains, in so far as it is only used ofF the equilibrinm path,
i.e. bw p([O, a]).
To prove the only if part, notice that aAmiwnbiIity implies that dl types respect
the above inequality, and in pdcular vj respects the abova inequality.
To prove the if part, define B+ = lin$(v). Similady dehe 8- = W ( U ) . *f
Since is diecontinuou8 at uj m have th& 8+ > Bo. Any bid in (Bo, B+] wil l
win the auction with probab'ity F(vj)"". The inequality in the statement of
the lemma implim that v3 prefers foliowing bis ecpdibrium rtntegy rather than
deviating. Fix u c vg. By monotonicity of pref;erences, u a h prefenr biddhg /3-
tban bidding in (p, 8+]. A simiiar argument shows that dl u > uj prefii bidding
B+ to bidding in (Bo, Pi]. O
Ptoof of Proposition S Notice &t that Assumption 3 implies that puni&-
ments m o t be bidder epecifie-all players must receive the same expected payoff
h m piininhment.
Gains to a player with valuation v fmm the collUBive mechaniam (B, LI) are:
which can be used to calculate ex-ante rents:
Consider infinite, or unrelenting, punishments. Let VI, CI- ) be an incentive
compatible collusive mechaaism which n h i x h s (1.6) :
min : Io H(v)Q(v;fi)dF(u) + b l 0 &@I(u))@(~)~-~ 81 #L
Since payofb are constrained to be in V, the perfect equiiibrium set, and V =
B(V), there atiets another collusive couple, (b, 4) giving the saxne pagofb ae
loi& (u' (b) 1 h(u))]dF(v). Working iterativeiy in this manner one obt- that
the payoff to thie ptinishment k
00
Incentive mmpatibiity m c p h O(.; p) to be non decreasing. h m Lemma 4 in
the appendix the above expreeslon ie minimized when we nile out constant bidding
regions, i.e. when the bidding d e s eta c o ~ e d to induce efficiency so that
Q(u; B) = F(V)"-~. The single stage Nash strategy accomplishes this task while
satisfying incentive compatibility- Note that when bidding d e s are co~mtrained
to induce efficiency payofi an ncceesarily equal to the Nash equilibrium stage
g=e payofts-
The next step is to show that participants alaayrs bid with probability one.
Suppose a mechanism which randomly select8 agents not to bid. The identity of
the winning bidder is unobsuvable, so fbtura gains can only be a function of the
winning bid. Thedore for a certain agent with a given type to be inMete~t
between bidding and not bidding, aîi agents with l m valuation must stnctly
pder not bidding and all agents with higher valuation must strictly prefér bidding.
Consider a meehanism that in the flrst round outlaws bidding b&w T > 0, and let
p~ be the future expected gains if nobody bids. Admjssibility of the mechaniem
And since F(r)n-l > (1 - F(8))F(s)nw1 for dl r in [O, r] nns have that:
This pmvious expression represents using the N d bidding stmtegy in the i n t d
[OJ]. Thdore we have contradicteci the suppdtion that outlawhg bidding in
a certain region dominated ueing the Nash bidding etrategy. O
Proof of Lemma 2 i) Fix K8 > K. Obvioudy any Q(.; p) satisfying constraint
(1.5) for K will satisfy (1.5) for P.
ii) When K = O the l& hand side of (1.5) is zero, so in order for the con-
straint to be respectai the right hand side must be lets thsn or equd to zero. By
proposition 3, the only bidding fiuiction satisrying this requirement is the single
stsge Nash equilibrium bidding fhction.
iii) Incentive compatibility impliea that due to informationai ssymmetries col-
lusive gains can be no higher thur those finsa using a scheme awaràing the good
to alI types with equal probability (McAfiee and MeMiilan (1992) Theorem 1).
When such a scheme is used, a necemary and d c i e n t condition for ali types to
respect rilmiaribiity b for type a to mecfi admhibiiity. For type a to respect
admhibility implies K 2 o[l - (lfn)].
iv) Suppose Q is the function dbveà h m the bidhg fisnction which solvar
(1.4) subject to (1.5). Let Q advise bidding a constant amount on [vol vJ, [ri, us], . . . [vm-L, c
The above equality ia obtained from the fact that
Rom iaequallty (1.5) we have that
n F v b a where 7 = max. : *&. Note that 7 is finite. SuWtuting this into equation
(1.7) we obtain:
This pmves the ela8tence of a finite number, such that O(K) 5 BK. O
The proof of propdtion 6 mabes reference to the fdowing lemma which we
state and prove before the proof of proposition 6.
Lemma 4 For any O < ui < ut < o it is the case that:
Proof of Lemma 4 Mer an integration by parts the above inequality cm be
written:
Since Ht(v) < O, it d c e s to show:
Since it is the case that F(u6) < F(v) < F(ul), F(u) can be written:
F(u) = W(u4) +(1 -A)F(ul) for some A E (0,l).
Therefore aRer substitution md a reamrrigamant it is BUfficient to show that:
Which is always satMed eince F(v)" is a convex function of F(u). O
Proof of Proposition 4 Suppose an optimal collusive bidding function wbich ie
mntinuousl y increaeiag for some interval [vo , vil c [O, a] . To show a contradiction,
modify the bidding d e so that it awatds the object to types in [uo, vl] with eqpd
probability and respects incentive compatibiiity. There are two uuies which wili
be considerd separately: 1) Type vl is weakiy prefém the old scheme. 2) Type
ul strictly prefers the new &me. In each treatrnent incentive compatibiiity is
wified while using the original value v, then it is show11 that the new bidding
hct ion generates a higher Ü.
Cue 1 Suppore that aRer invoking 4, type ul is just as welloE as under the
old bidding firnction. Therefore bidding levels b1 and q can k maintainecl and ail
incentive compatib'ity constrainta am safisfieci. Now apply lemm 4 to see that
the expected value to collusion bas been i n d . Suppom that type VI strictiy
prefers the old scherne. Then if bl and b are held constant, the type inc i i f f i t
between bidding & and bl wil i be located to the left of vl. Now raj, 4 and
b> mch that the in-t type between bidding 6 and bl remaias ul and such
that the indiffe~ent type betwem bidding 4 and mains u2. Notice that this
change in the bidding function has not changeci the probabiiities that any type
greater than ul wim the auction and aii incentive compatibiiity and dmhsibility
constraints are mpected. Now apply lemma 4.
Carie 2 Suppase that type ul atrictly prefm the new scheme. Then the type
indinerent between bidding and moves to the right of VI . Call this new
indinerent type v:. Incite u: to move baclc to ul by decreasing the continuation
payoff if the winni~g bid is Y. Such a change produces a more profitable bidding
function by lemma 4, but may not produce higher mts. Invoke Proposition 4 to be
assured of the existence of a collusive mechankm with a bidding fnnction wbicb is
never continuously i n d g and having the banPbang property. This bangobang
mechaniam uses a bidding function which is even more profitable. Fhthermore,
such a bang-bang mech& generates fi = J: H(v)Q(v; /P)dF(u), where is
the newest bidding hction. So sinœ the bang-bang mechanhm produces rents
which are generated by a bidding b c t i o n which domhates the original scheme,
the propdtion is p d . O
Proof of Lemma 3 Suppase thm does not exist an indift-t type undu rrn
optirrui collU8ive biddiqg de. CalI this d e @ and demte the single stagp profits
it generates by $: H(v)Q(v; p)dF(v). hie to the lack of an inmkxent type, there
aasts q > O aich tbat /F is incentive compatible if it satisfies Jt Ip(v) - Pt(v) (du c
q. Now we cari choose a (finite) sequence of bidding hctiom mch that
for all k E {1,2, . . . , K - 1): 1) ' IPk(v) - /3k+1(~) ldv < 7, 2) PK(u) = O for di
u, 3) ,ûL = @, and 4) &'H(u)(Q(v; Bk+1) - Q ( u ; p ) ) d ~ ( u ) > O. We are d of
4) because H(v) was assumed strictly decreasing, and we! are agnired that K > 1
because there is at least one discontinuity in the bidding fuaction. Obviody this
sequence contains at le& two bidding fnnctions which are incentive compatible
and at leaat one which g im higher single stage payos than p, contradicting the
hypothesis on the optimality of p. Ci
Figure 1.1: Graph of g(K)
Figure 1.3: A bidding fuaction
1 1 n ~ i t 5 d i f i i d t o @ ~ u i ~ p h d a p d y ~ r h o t ~ c t i o ~
'Under urud hypdmen identid bi& ahouid be obrerved with m probability.
=In a reœnt prpar Athey, Bagwell. .iid sancwco (lm) have eJcplond tht bue*
'This L the cme fbr mat cornmon dkibutiom. See Bagnoii anci Barlptrom (1989) for more
details*
6Q~otinghm W i n (1992): "...bPr symmtdc Bntprice auctiona there is eume p ~ p t i o n
that the symmetric equIiibrfum is the unique equiiibnum."
'Note th contrary to Abmu, Peazœ, and Stacchetti (1986) the bang-bang property is
detined m that it iavokea the hamht punirkm- aii d. This P not a nsceawy condition for
Pmpaition 4, but a ddent one. Thb ia b u e r ler than muimrl punidment may d c e
to enmue obedianœ to a biddhg mie bidding zero regarcUeerr of duation. To eee that
a muOmd punishment ir ~uf8aent. notice that deviation wiü k o k a n d wîth probability zero
thua wiiS not a Furthenaore, e harsher puniahment am always support a scheme
supportable by a we&r ptlnfnhment. Note that if the admiasibiiity constraint h binding, then
the W e s t pdble p-ta m u t ba u d aa d.
the dimadon in d o n ïL of Greham and Mazshail(1987).
'It oui be shown thrt p d t s move continuoudy in the bidding alltig and mserve pr ia
Sice we un consider a compact region ovet which ta dect the reaerve pria. we am assud of
the existence of a mwimum.
On Castel Stability
Paul Johneon
This paper studies collusion when discount rates diffa among cartel members, with the god of estabiishing a model where cartel instability arises. The me+ lation principle is invokeà hughout, thus it is not iiterally true that defection in the sense of misreporthg type aMss, though it ie srgued that equilibrium strate@ csn be interpreted to suggast Uundercutting" behavior. The model has players drawing iid discount faetors in every period. The main implication is that the baog-bang d t 8 of Abreu, Pearce and Stacchetti, as well as the aim- ple punishment results of Abreu no longer hold. In other words, players choose actions baseà on type and each action invokes a unique streaxn of hture payofi. The model'a ptedictions are &O compard with the sme empirical evidence on collusion.
Acknowledgareiitr: This d nns supporteci by the SSHRC. 1 wodd lilte to th.alt Jacques Robert for extmnely useM critichm and comments. 1 muid aiso like to thank neminar participants at the summer lm9 meetingt of the Socidt6 Canadienne d'Eeonomique and the Canadian bnomics Association. Of comq any remaining amas mwt remciin mine done..
Introduction
Co11usion and cartels, as economisfs dehe the wrds, take many forma. A cartel
of countries can collude to prevent ovu exploitation of fi& stocks, or s cartel of
firrrm can cdude to raise prias. Despite the difhnt fornu that coilusion can
take, d cartels shue certain properties. &t, collusion is an sttempt at a Pareto
improvement (at least for the membem of the cartel). If countrie8 sign no treaty
to protect fish stodrs, then the end &t is that each country catcha fewer fuh
than if a treaty had been signed and respecteci. Second, cartel members alwap
have some motivation not to respect a collusive agreement. Firms competing é, la
Bertrand and having agreed to fix pria above marginal coet, codd inmase short
term profits by undercutting other cartel membem. Unfomuiately for cartels, this
latter &et invariably seems to dominate the former. Collusion is almost never
stable: colluding biddem chisel one another (aee Smith (1961)), member countnes
of international cartels do not respect quotas (see Barbezat (l989)), flrms do not
respect price guidelines ( s a Gmeeman (1996)) and firms do not respect agreements
with workers (see Bertrand (1999)).
This lest observation is troubling for economists, since most repeated game
models of coilusion predict @ect cartel stability over t h e . For a literature
that has been in existence for cnm 25 yeam and the mbject of so much d
effolt, tbis is problematic. Whiîe some mqy be qnidt to point out the widely
h m mrk of Porter (1983) as well as Green and Portet (l984)', tbeh is not
a w h d y sathQing m01utio~ to this paradooc for trro reamns. Fi, th& modal
d i e s aiüreiy on the im@& nature of information at every stage of the game.
However, there is much evidence that cartels main unstable in gasnea of perfect
monitoring. Secondly, detection or "undmutting" never a h in their model. In
fact the ravueionary periods should not even be considered to ba piinishnients
since it ia common knowledge that each pl- has ptoduced the cartel suggested
quantity. The p d t m sMU being unsolveû, this papa offers a reconciiiation of
theory nith the rccepteà stylized fact that cartels are generaliy unstable.
One of the m a t commody made lvgumptions in the literatnre on collusion
is that one player is jwt as patient as anotha player-playem &are a common
discount rate. Two interesthg examples are Harrington (1989) and Lehrer and
Paumer (1999). Harrington (1989) permits colluding &ms to have differeot dig
count rates. Lehrer and Pauzner (1999) study hm the set of feseible p a p e
change when discount rates are hetmgeneous. Despite the heterogeneity in both
these papenr, these difiering discount rates remain perfectly known. This is the
point of departwe of this paper-not only is a possible heterogeneity i m p d , but
this heterogeneity is 88sumed to be private idormation. This adds a more corn-
plicated structure to the game, but it would seem that thjs is more nslistic than
the ellnplifying symmetry and perfect information aewmptionse
This introduction has hinted that with the introduction of prioste informa-
tion, a modal emergs whieh rapücates an important stylized fa& of eollueion.
The revefation principle i d extedvely, and as mch it is not tme that the
end result ha8 agents '<ddecting'' by misre!porfing theh types (disco~t rates*)
Rather, agents can be thought of as tnrthfully nporting th& levels of patience
and are advised to taha actions which could be interpreted as a "defecfion" or a
<pitninhmentP Pethaps the nord Uuadercutting" is be8t suited in this eenre, sina
it c- none of the connotation of disegacmient which 'cheating" or Udef'ti~n'9
do. This undercutting sesult flm dinctly h m opthizing behavk an agent un-
demats if and only if unde~cutting is betta for her than not undercutting. Thue
is soma evidence that cartels reco@e that undamitting may ba inevitable due
to the presence of private information. The International Steel Caztel detaüed in
Barbezat (1989) is a good example of t h type of behavk. This csse study nül
be discussed in more detail in e t i o n 2.5.
The model developed in this papa assumes agents draw iid discount factors
in every period. This assumption need not be interpreted literally as section
2.6 argues that instead of supposing that preferencers change in every period, we
can equivalentiy assume that expectations about future market conditions could
change. In this model, I argue, something dixi to dektion and punidment arises
on the equilibrium path (of course the caveat about the revelation princip1e must
be lrept in mind). The maat striking r d t of this essumption is the etnicture
which optimal colluaive schemes must take' unlike Fkiedman's (197l) unrelenthg
punishments, Abreu7s (1988) simple strategy profiles and unlilre Abreu, Pearce,
and Stacchetti's (1986) and Abm, Pearce, and Stacchetti's (1990) bang-bang
results, "punishmentsn mut be maàe to fit the "crimev'. In other wods future
payoffs depend non trivialiy on currant action. There is soma evidence that cartel
punishments are neither "bangObmg" not unrelenthg rn in Medman (1971).
Section 2.2 fomally intduces the game to be studied thmugh s e d as
sumptions and mme notation. section 2.3 ad8pts the method of Abreu, Peame,
and Staccbetti (1986) for the preaent p n r p o ~ ~ . Section 2 4 presenta tno exam-
pies. Section 2.5 addressee the prob1em of cartel behavior when discount rates
are private information, and may change with the. Section 2.6 considem mme
applications.
2.2 Assurnptions and Notation
This paper stuclies a clam of gamee eatisfying certain proparties giving a flavor of
the prisoner'a dilemma: cooperatiw play nsulta in payoffs which Pareto dominate
noncooperative play. Roughly, these propaties etcrte that the more profitable is a
cartel, then the p a t e r are gains to detection and the greater are losses if others
defect. Many gamm of interest to aiialyze in a repeated context satisfy these
properties. The most natural example is collusion in Cournot (quantity) competi-
tion. Broadly spealing, the game to be studied is sy~~lrnetric and is one of perfect
information. Eiirthermore it is repeated infinitely often. The distinguishing feic
ture of this mode1 is that discount rates @daences) are ex post ssymmetric and
private information. An important restriction is that the modal imposes sym-
metry on the strategies dong the lines of Abreu, Pearce, and Stacchetti (1986).
This restriction requires that aRer any hhtory, aU players play the same strategy.
This, admittedly, entails a lm of generality since, in general, optimal symmetric
collusion Is not optimd (asyxnmetrie) co~usion. Nevertheleas, symmetry in etrat-
egy provides rigdicaat hpli6cation and does not change the fhct that changing
discount faetors caa produce cartel instability.
Denote the stage game by ï' = (SI,. . .,S,,,rI,. . .,n,). When N = (1, ..., n}
is the set of p lam and Si is the pura strategy sprca of pIayer i. Play ocnuir
&mdt8~eousiy in the stage game. The riait neutrai ptatèrencee of player i are
meaeurtd by the bounded payoff fiuiction : S + II2 Symmetry L i m p d by
the fdowing sesumptions:
Al. Si =Si V i , j E N.
A2. ri(si,. . .,Si, . .. ,S j , . . . ,am) = ~ ~ ( 8 ~ ~ . . . , 8jt . . . ,&, . . . ,am) E N.
With the ammption of symmetry? attention can be mtricted to srbitrary player
i E N. The following two technical ammptions wii l be needed:
A3. Si is a compact i n t d of the real numbeis.
A4. s is once continuody dinerentiable.
Note that due to A3, for any si, 4 E Sr such that si > 4, there exists an sf E S1
such that s+ > sr > el. This important fact wi i l be r e f d to as the 'mntinuity
property' of the game.
1t will be convenient to relabei elements of the action space in the f011owbg man-
ner :
We now assume that profits are strictly deczeasing in one's own actions and strictly
inmasing in the actions of another player:
AS.
It follm that there ia a unique beet mpontse for plqysr i to any 8-6; d this beat
mponse a*. We can condude that ifss 2 4 then there is a stronger temptation
to d e k t f hm playing SC than from p l m g 4:
Note that, in essence, a mplete and transitive orda is being ~snmed to arist
on the strategy space. A5 givee ur stnct dominance of an action (s,.) which
eneures d e n c e and unicity of a Nash equilibrium to r. In order to amid caees
where the stage game 8chiew a Psreto optimum the following is essiimed:
A6. For any interior point s of S we have:
It fouonte that 8- := {st 1 rj(8i, 8i, . . . , si) 1 9(4,4, . . . , 4) V s', E Si) i~
the unique most profitable collusion which satisfies anonymity. The follonring
assumption permits a considerable simplification of the and@ since it, dong
with A3 and A4, implim that the Nash stage game equilibrium strategy is an
optimal punishment in the sense of Abmu (1988).
A7. r(sh) ecpals the opportunity CO& of playing the gbme. Normaiize both
to zero.
The repeated gme is the infinite replication of the stage gamee Gains wil l be
discounteci by ployer i with discount fictor & E [a, b] c [O, 1). DiScaunt fwtors
are dram according to:
AB. At the beglnning of every period each player draum a &cotant br-
ter with the atomlesa probability meseute f defined on the meamrable speea
([a, b], %([a, b])). Where !B([a, b]) denofes the the Bonl subsets of [a, 4. This is a highiy TeStTictive assurnption, neverthelem it can be aeai to be nec-
emxuy to obtain tractable mults. Suppoae, momentdy, that discount fact01~8
are h w n according ta a Markov pmce88. Ekpectatio~ about othed discount
fwtoni is an important part of any strategy but &ce it may not be the case that
actions and discount factors are one to one in the strategy, the recwsive nature
of the problem is lost. In other words, one may only be able to infer a (non
singleton) set of discount factorri fkom any given action. This being the case, con-
ditional probabüities depend on the entire history of actions in a non trivial -y.
Though perhaps surprishg at fmt sight, A9 can be interpreted in ways which
do not assume that preferenca change. Such a discussion wii i be delayed until
section 2.6.
Rubinstein's (1979) concept of (subgame) @ect quilibrium wiii be used as
the equiiibrium concept to the repeated game. Sina niture gains are discounted,
this implies that a strategy is an equilibnum i f a d only if thare exkts no profitable
one shot deviation (Abreu (l988), Proposition 1). Fbthermore, we wil l only study
symmetric strategy profiles (see Abm, Pearce, and Stacchetti (1986)). This does
imply a los8 of gaeraiity, yet it is mgned in mtion 2.4 that the main conclusions
of the mode1 are robust to a webing of this mtriction.
2.3 Static Representation
The method of Abreu, Pearce, and Stiicchetti (1986) is a eonvenient tool to analyze
(subgame) pariect m e h i c strategies in repeated gzunes. The restriction to
symmetric stratepie8 is important thce .Ru any history the strategy of playsr
i is required to be the ssma as the sttategy for player j for a,iî i, j E N. Thair
method entails Y&ctonzing" the gains scctning to each player into two partai so the
game can be analyzed as a static game. The first part ie the gain h m the nvrant
stage game and the second part is expected, fbtwe gains which can be a huiction
of any obgervable action t a h in the crurent period. So th& representation needs
two elements: a pa- function for the stage gama and a function describing
hiture gaias. In Abrau, Pearce, and Staechetti (1986) Proposition 1 states that
'an equilibrium to the new static game which satisfies a "seIf generatiodY criterhm
is an equilibrium to the repeated game. hoposition 2 states that any perf't
equilibrium to the npeated game can be represented as a Nash equilibrium to the
new static game* Thus the static representation ie equivalent to the dynamic game.
The following clasely mimics the prcilentation in A b m , Pearce and Stacchetti to
modify theu static repnsantation to meet out needs.
Define C to be the set of Borel meanvable fiinctions fimm [a, b] into Sc: C :=
{O 1 O : [a, b] + Sr). Dehe n(W) to be the set of Borel meatmable hctions
h m S into a bounded set W c IL. Q(W) := {u 1 u : S + W). Due to the
Lestriction made on m e t r i c sfrategies (O$, uy) = (of, uI) for aiî i and je When
then is no danger of confusion 1 shaîi write 6 to denote x 6( and 4 6 ) to denote EN
x O(&). For any pair (a, u) E C x n(W) write: iEN
Note that ni(o(6)) and u(o(6)) denote gains to the atage game and future ex-
pected geins respectiveiy far an arbitrary player and for a given vector of discout
rata 6. Note that these expectatiom are weU defined because compoeitions of
Borel measurable functions are Bord measur~b1e, as weU as the fact that r and
W are bounded.
D a t i o n 1 For any W c R, (q u) E C x n(W) ie calleci admissible with respect
to W iff the foliowing two conditions are satisfied:
Dcanition 2 For W C R, B(W) := {& (O; u) 1 (u, u) admissible wrt W).
Thus for e w y tu E B(W) th- exista at le& one pair (o,u) admissible
with respect to W with d u e W. Ch- one such pair, a : B(W) + S and
u : B(W) + n(W). Thus, ifwedehe W c R to be selfgeneratingif W c B(W),
propdtions 1 and 2 h m (Abreu, Pearce, and Stacchetti 1986) f o h . Ewe dafine
V to be the set of paf& equiîibrium expected payofb, then the construction in
APS assures us that th- exkits a sttategy with vdnt w E V which does not
depend on (privately observable) past paetunt fators.
Since W is the mge of u, m are dmianàhg that for (almat) any dnw of
discount factors, d = x Ji, nia have that u(u(6)) 5 &(a; u). It is important #EN
to note that the left haod side of the previous inequality does not m d y hold
in expectation but for any conceivable vector of discount futors. If this were
only true in axpectation, we would not be chaclriiig the entire range of u. Thw
then might be realiza,tions of 6 such that u(o(6)) > & (u; u), which would violate
&generation, and thus the static game wouId not be equident to the repeated
grne*
This section pretmts thme examples. The k t goal is to illwtrate how the meth-
ode of section 2.3 can be applied to the problem at hand. Secondly, these examples
show that chmghg discount fsctors can engender undercutting dong the equilib
rium path.
A Bimpie two playv symmetric p e d be used. This example is coI18istent
with A5 in that eamtidy it is a prisoner's dilemma with a continuum of actiom.
Action spaces an Si = S2 = [O, 11, and payofb are given by:
r r 2 si 'y.
Let d = 1/2 with pmbab'ility 112 and 6 = O with probability 112. Let 0(1/2) =
57
r and @(O) = 0. We can calculate p e ~ period gains: n(o(1/2)) = (s - $)/2 and
n(u(0)) = (8 + #)/2. Let x be the continuation value if both types play a(l/2),
and let y be the antinuation d u e if one type plays ~(112) and the other plays
m. If both players play zero, let the continuation value e q d m. Since na
are restricting study to symmetric strategie8, payoffip must be the same for both
play- irrespective of previous asymmetric actions. The problem then can be
writ ten:
Subject to the foliowhg constraints:
The f h t constraint is the incentive compatibiity constraint for type 112, and the
second constraint is self-grneration. These conetraînts yield:
Thus the program b d v e d for s - 118, z - 119 and 119. It may be surpr%hg
that z = y, &ce one might thirik that the incentive constraints for type 112 may
not be satMed. H m , ittypc l/2 were to play action O, then zero future gains
would occur with probabiity 112. If one player were to play 8 then zero future
gains muid never occur.
Now 1 j e why o(0) = O and why the continuation d u e when both players
play zero ia zero. Charly, in any equilibrium, type O m u t play 8 = O, since she
places no weight on future gains and s = O k a dominant strategy to the stage
game. Letting the continuation value when both players play uai be positive
is not optimal sime it do- not change and maltee the incentive compatibility
constraint for type 112 more dificuit to respect.
The second example shows that more than tao values can be taken by the
continuation hinction. Suppara d = 112 with probabüity 0.576 and 6 = O with
complementaq probability. Again let 8 be the action taken by type 112 (type
zero wil l play action zero), and let z be the continuation d u e if both players play
s and let y be the continuation value if one player play s and the 0 t h plays zero
(O wiil be obtained if both types play zero). The problem can be mtten similady
to the first exampie and optimal d u e s can be calculateci to be s E 0.179246,
x Y 0.139005 and y 0.104008. Note that the continuation value is clearly not
bang-bang, nor is the '$unishment9' d e m e simple.
Consider, briefiy, the pasibility thst asymmetric strategies are wed. If cf =
112 with probability p and zero with probabiîity 1 - p, then ne ciui write down
the optimization problem as foliows:
w is continuation gains to a playa who haa played s m long as her partner hee
played S. z is continuation gains to a player who has played r so long as ber
partner hm played zero. y is continuation gains to a pl- who h a p l a d zero
so long as her pcutna has played S. z is continuation gains to a piayer who hae
pla@ zero iio long as h a putner hai played zero. The comtraiats eaii be writtan:
- Ideaüy, we muid iike to set w = z = U and y = z = O. However, it is necessary to
check if th- exists an asymmetric equilibrium strategy which gives one player
and the other playar zero. If there does not exbt such an asymmetric equilibrium,
then it L necaeeary to have x < W . in any case, dlowing aeymmetric strategies
does not change the implication that defection occm on the equiiibrium path.
This section aims to show that the d t s of Abreu, P m , and Stacchetti (1986)
on the optimality of l<bsng-bmf eqyiîibria as w d as the results of Abreu (1988)
on the opümelity of %mpleW pnnishments do not cury over to a context in which
defection arises. Locdy, it is u y e d that optimd collurion, in general, mwt be
%ore type mealing" than bang-bang oollwion.
Fiiet we Wnte dom the optimization pmblem faced by the cartel at time 1:
- U = max : &(u;u)
bC-)mW
Subject to, for almarit every b!, ai1~08t every vector 6 := (4,. . . ,a), and aU 7 E C:
The Grst maetraint is rrlmicaPibility and the seeond ie self grneration. It is impor-
tant to stress that for the second inequdity u(u(6)) r a p m b friture gains if a
vector, 6, of discount rates were drawn. Note that u(u(6)) does not represent an
expectation.
Lat (oL,ul) mlve the above pmblem. Baseci on the discount factors drawn
at time 1, the cartel wi l l propose another admisgible and seKgenerating couple,
(#, u2) whoae ex-ante value is lem thbn or quai to the ex-ante value of (el, ul)
and greeter than zero: O 5 &(g; u2) &(ul; ul). This fol lm since u1 promises
an element of a self generating set for ahont e w y dran of diecount factors (Le.
u' promises an element of the i n t d [0,U]).
In the foIlowing, it wi l l be convenient to rinite n(o(bi)) = Er, [ri(@(Jt, iLc))];
similady u(a(Jr)) = &-,[~(0(6~,b-~))]. The k t reault cornes directly nom the
admissibility constraint.
Proposition 1 r(o(*)) Q demain9 in 6, and u(u(.)) ir incmaâng in 6.
Proof of Proposition 1 Fix Jr > 6:. Rom aciminnibility na have:
Thezrefo]:e, since di > 4, w have that u(u(bi)) 2 u(o(4)). Additionally, WE! have
%(@(a;)) r *(@(Ji))*
Using these monotonicity pmpextiea, one can obtain a chatacterization of col-
ldve rulea when discount rata change. First notice that both the expressions:
exist almoat evetywhere since each is bounded and monotone in 6. Using this fut
we can state the following proposition whose proof is an elemaitary application
of inequaiitiea (2.1):
Proposition 2 For any admUsible coupte (O, u), the following dation hotde for
almost euety di:
-+(0(6,)) = Jiu (O(&)).
Using this result we can modifjr the program of an optimal synimetric cartel
in the first pdod and write it a bit more explicitly:
Subject to, for a l m a every Ji, and aima every vector 6 := (di1,. . . ,&):
The kst constraint is the W b i l i t y constraint. Define the folowing: T*(u) :=
max : & ri(&, ~ ( 6 ~ ~ ) ) f (&). Thus, the second constraint states that playing 8i
- a one period b a t response to o (which is amin) is not profitable. Due to the
monotonicity of a and u, we only need check this for the least patient type. The
third constraint is self grneration.
In general, optimal collusion wi l i require the division of the space of dismunt
factors into three distinct intervals. Any type in the first interval, [O, 6& will gain
re(o) in the curent period and will imke continuation gains of zero. Any type
in the second intend, [di, &], wîîl have cnrrent gains strictly and continuody
decraasing in discount factor, and continuation g a h continuously inueaeing in
discount factor. Any type in the third intemal, [&, 1), wi l l have constant cumnt
gains and continuation gains expal to the ex ante expected value of optimal col-
lusion. It is important to notice that any particder example of a game satisfying
the 88sumptions of section 2.2, could have a > cfL, or a > 4. The former case
opouid g in that nobody plays ro(g) and the allusion never completaly falls apart.
The latter case would mult in a coUusion where evay type ie adviseci to play the
same action and the collusion is stable. It can be seen that such a coliusion would
entaii the ph@g of s* = (s 1 r(s) + a& = n(s,)), where E6 = 1: df(d6).
The following jusfifies the existena of the Brst region [O, hl.
Proposition 3 Let a = O. lf (n, u) & an optimal admissibfe couple, then o(0) =
8- and u(u(0)) = O.
Ptoof of Proposition 3 It ahould be dear that any type zero agent must play
SM,, since ie a etrictly dominant action to the stage gune and no weight
is placed on future gains. It remaina to show that u(c(0)) = O. Suppose that
u(o(0)) > O. On one hand, this would not change the actions of the type zero
agent and not iaenaee Ü since m y future gains to a type zero agent would be
weighted by zero. On the other hand this would cause crll type eufsciently clw to
6 = O to decre88e their actions. A6 eseuree us that such an eventudity decrem
the profitability of coliusion, which confradicts the fact that (a, u) was assurneci
optimal. O
The property discussed above has a coroliaxy which demonstrates the existence
of some Ji, euch that evuy type in the interval [O, JI] play 8~ and receives zero
hiture payonS.
Proposition 4 Let o = O and uuppose h t (o, u) b an optimal admissible couple.
men, them h t s a tVpc Ji 2 O which plaga 8- and d v u rem &tu= payoff~.
Ptoof of Proposition 4 Rom propdtion 3 na have n(a(0)) = T* (o) and u(a(0)) =
O. By incentive compatibiity ne have 2'(4 2 n(o(6)) + Ou(o(6)) for .II 6. Ra
caii that ~(o(6)) = -6u(o(b)). if the previous inequaiity L Wct then there
wiîi exist > O such that for al1 6 < 4 incentive compatibiity requlles that
r(o(6)) = n*(a) and u(u(6)) = 0. [1
Non consider types in the uppa range of the support of 6. It is claimecl that
in this range, diflrkmnt types may take the same action. We wodd lüra every
type to play as high an action ~ E I posgSble, WU re~uires appropriate recompen-
&ion thrwgh incred htm gains. Howem, tbis is not always passible as the
following example demomtrates.
Consider the game pmmted in section 2.4 whaa 6 c m *al 112 or 1/4 with
equal probabiity. Note that the m a t profitable symmetric colludon a player with
type 114 can support is the playing of 0.4 because Eb = 318. This follows because
a(l + &&) = s + f is solved for 0.4. By a similar argument note that a player
with type 112 can support the playing of 0.8. Letting sl and s2 be the actions
taken by 112 and 114 respectively, and letting z be the continuation gains if both
players play si, y be the continuation gains if one player plays $1 and the 0 t h
play y, and r be the continuation gains if both players play 32, one can pose the
problem in the following way:
Subject to:
The mnstraints ara the incentive compatibility conskdnts for typa 112 and type
1/4, the comtnht stating that 114 preEais phmg q to playhg 0, and seif gens-
ation. This program can be dved to show that the optimal symmefric collusion
advises both types to play action 0.4 and promises future payos of 0.64. Note
that despite the tact that discount factors are nibject to change, collwnon is per-
fectly stable and a bang-bang continuation function, or nimple pnniRhment, is
SuffiCient to support optimal coUusion.
T b example shows that bunchjng can occur 'at the top". The intuition is the
following. in order that 112 be motivated to inmase 81, she must be appropriately
rscompensed through an inmase in z and y (proposition 2). But when z and y
increase, it is necessary that Ü, the d u e of collusion, inuease by at least as much
in order to respect self genmtion. But Ü is endogenous to the problem a d as
euch, the increase in Ü brought about by the inmase in si may not be large
enough to ensure dmbibility. Thus the tension between the aiimissibility and
self generation constraints may provolce bunching near the moet patient type.
The taio previous cases have explorecl the extremes of the possible support of
the discount f8ctor. Now we study the case whem di p d b l e discount firctors do
not fell into one of these trro categorie8. Let these intermediate discount f e o r s
be in the i n t d [&, 6d. It follows that for Qpe~ in [O, Ji] or in [b, l), co11usion
is not type revealing. The following propoeition shows that for types in [&, 611,
coUusion is patectly type revealurg.
Prapodtion 6 Optimal collksion mwt n v d dmud any type not in [O, br] U
[&, 1)-
Proof of Proposition 5 Suppm (q u) admissible with nspect to a bounded
self generating B d set W. M e r m a r e amme to the con- that O is not
M y type reveaîing (o"(sr) := {Ji 1 o(&) = ai) L not a hgleton) on an interval
such that 6 > a a d 8 < 6. Thare must be a àiscontinuity in both %(a(*))
and u(o(*)) at 6. Due to the fact that f is amumed atodeas and 7 t is BBlPUrned
conthuous such a discontinuity in r(a(8)) must be due to a âiacontiwity in a(*).
Pi& d E (& 8) and define a new strategy, cf, to be identical to o eve~ywhere
except on the intend [b, q. On tbis intaval dehe o' by:
Remark that we can write (2.4) due to the continuity property as weU as due to
the fect that the admissibility of (a, u) Mplies that a@+ r ) > u(8 for aii e > 0.
We can write (2.3) if a(h) # s-. This h paranteeci since b < 1.
Now it ie ne~es8ary to flad a continuation function, u', such that (gm,u')
is admissible with respect to a seIf generating set. Define u' to be equal to u
everynhere except when the discout factor îie~ in the intaval [%,a. In this
interval define u* by:
This is Suflbicient to show that [é 8) and [8, self aled due to the f ~ t that future
payofb ase inaeaaing in the discount rate. Now it nmdns to show that 8+ E
prefè~~ n p o h g + E aa opposed to 8 for d E > O. Eiom the uim;ani€aility of
(o, u) we have:
Showing the desired seIf seiection means vuifying the folîowing:
Due to the continuity property, 0°(6) can be ch- so that 1 r(om(8)) -r(c(8)) [<
Q for ali 111 > O. Thus, oe(8) can be ch- such that for any > O we have
1 u*(o'(~)) -u(o(6)) I< Q by equation (2.5). Using (2.5) and (2.6) it is suflicient
to show that for any fixad > 0:
This c m seen to be d e d since:
Since [&al phy a hi&- action this mwt remit in more profitable colîUBion
due to A6. Sinœ ;d < b then &generation MOWB triviaiîy since the con- huîi
of the range of u equais the convex hdi of the range of u'. Cl
It is northwhiie pointing out why the above argument does not work for [O, di]
or [&, 1). Foi d E [&, 1) in order to respect W b i l i t y it is necessary to increase
the range of the continuation function. As ariul argued eaziier, this may not respect
self generation. For 6 E [O, Ji] there need not be a diseontinuity in a(.) at Ji. This
follows since typai below Ji are co118tfained to play 8& (see propdtion 4).
The structure of optimal co11usion which is detailed a b m is important for the
following reason. If collusion is to be (partially) type revealing then two diffetent
types may take different actions, And by incentive compatibility, this implies that
the leas patient is an agent then the higher wi l l be her cumnt payoff, and the Iower
d l be her future payoffi. In other words, impatient agents will "undercut" today
and be "punished" in the h twe (due to the restriction on symrnetric strategies,
di players must be punished equally). This is the behavior that 1 argue cm be
interpreted as cartel instabiiity.
For 6 E [O, 4) u (&, l), continuity of r(a(b)) and u(u(6)) follows trivially (they
are constant). The foilowing proposiüon shm that continuity of a and u holds
in generai for ali 6 E [O, 1).
Proposition 0 If (o, u) is an optimal admkible cdIwiue dmtw then, bath o
and u am continww in 6.
Proof oi Proposition 6 Suppose that o is an optima collusive action strategy
which ha^ a point of ciismntinuity at 6. Let the limit h m the left of O(&) be a ( b )
and let the Mt h m the right ofu(6) be a+(@. h m nAminnibilitywe knaw that
a (6) c a+ (a). Adm;wpibility a h implies that 6 is just indinkrent between p1aMg
a ( d ) and &(ô). Now advise a new action sw and a new continuation hction
defineci in the fo11owing mamer. Cho- such that n(o+(b)) - %(se) = A <
r(o+(J)) - r(r(6)). Increase the expecteù cootiauation gai^ to a player having
pl@ S. by A/d (the convex hull of the range of u wiil m a i n unchanged). Now
type d ie indifcete~t betrPeen choosing my of the three actions: a-(&), 2, a+(&).
We csn advise similar new action8 for all types in a neighborhood lese than 6. Such
a change WU alter the distribution of actions, and tu such may cause a type greater
thaa 6 to wish to play a*. If tbis is the case then we how that 6 strictly prefers
playing so to playing ~ ~ ( 6 ) or O+ (9, thus we can lower continuation gains untild is
again indifferent between the three actions. Such a decrease in continuation gains
will dissuade any type p a t e r than 6 ftom playing S.. Thus we have i n d
the actions of certain types, which, by A6, ensures higha pmfitabiity of collusion
contdcting the suppoeition of optimal discontinuoue collusion. O
As a corollary to the above proposition we can use propoeition 2 and integrate
by parts to obtain the following characterization for optimal symmetric collusion:
6r n(@(Si)) + biu(0(6.)) = n(o(tz)) + au(o(a)) + / U(U (8))dû VJi ' b i ~ [a, b]
a
It is worthwhile pointhg out that the above propositions can be taken together
to enable us to say what an optimd collusive mecbanhm is, and what it ie not.
Speaficaliy, the above namnn the choice of optima collusive strategies to one of
four pdbiities: o(m) L constant for dl d E [a? b]; a(-) is constant for all 6 E [a, JI]
and i n d g for dl d E [&, 61; o(b) is incnepine for aU d E [a, &] and constant
for dl 6 E [b, q; or oc) is constant for a d E [a, Ji], increasing for aii 6 E [JI, &]
and constant for all6 E [b, b]. This des out case13 where, for example, u(.) is
increasing then constant then increathg.
The above studied optimal colluBian. But the i~t8bii i ty of such collusion
prevents a cartel h m dways dering optimal gains. Consider the meiumre auo
case where in the b t round ali plsyers àraw the 1- p d b l e discount factor.
Unless it is the case that a > 61, the eneuing p a y a must be very low and, thus,
cannot be optimd. So the mecbanhm at time t, (ut, ut), dong wi th the d m of
discount factors at time t, cho- the mechanirim rrt time t + 1. At p d o d one,
since the cartel is not constrained by past commitments, it may use the optimal
mechanim. Mer types are reporteci, with positive probabiity, it wil l be the case
that the cartel p m m h hture gains which are not Pareto optimal. Thus, future
mechanisme may be 'lesss" type revealing, dbcontinuous and thus not optimal.
Now consider h m to replicate any payoff u in the interior of V, the set of
perfect payofk. v is equal to a weighted aim of currsnt and future payons. h tu re
payoffs are weighted by the expectation of the discount factor, E(b) = J: bf (ds).
Consider any current and future payofb whicb yield u. It is easy to see that a
decrease in current p a y a by E(6) unïts dong with an inmase in future p a p e
by one unit yielàs u. We cari aiways inmase and decreaee current and future
payofb IBO long as they are not extrema points of V. T b implies that th- are
an infinite number of way to mete out noo-extmmal continuation payofb.
There mstg eome evidence giving support to the ldnd of cartel behavior de-
scribed in thir section. Barbezat (1989) &dies the International Steel Cartel
which comprfsed Cermany, Ehœ, Bdgim and Luxembourg starüng in 1926.
He reports 'For an over quota pmduction betnean 7.5 siid 28 percent they paid
one doilas; between 28 and 30.8 percent tno dollam; and beyond 30.8 percent . . . four doiiare." (footnote 9.) This L precidy the type of punishment impIied by
the preceding propdtion. The &al ssemed to recognize that member countries
may have been subject to intemal pnesnree whieh sjgnificantly changeci incentives.
The degree of tbese internal premm may, of CO-, have Mned acrms countries
and across time (Germany's war reparations, for example). The above propoei-
tion says tbat the &el may be able to do better by letting types sort themselves.
Note also that these puniahments were neither d e n t i n g as in Rieclman (1971),
nor were they optimal in the sense of Abreu (1988).
2.6 Interpretation of Changing Discount Rates
The previous sections have ass iund discount rates to be the only heterogeneity
between agents. This was imposed for several reasons. F M , it is fairly straightfor-
nard to incorporate diking discount rates into a frirly tractable modei. Second,
the commonly made asmnption of symmetry and p d e e t Monnation can be ar-
gued to be u~veatistic. Third, and m a t important, patience pmvides a ünlesge
betwe!en fbture r e d and current actions. Despite the plausibility of heteme
neow patience, no rigomus arplanation for this hetmgeneity hre been advanced.
Since the ovemhelming majority of the literattue hm foerised on the synimetric
case, it wouid be usehi to be able to provide a more rigomus explaairtion for this
deputun. HCVtiDgbn (1989) argues th& either iramm in the capital mar-
kat or d i f f d g de- of agency pmblems among &mcr muid 8ccount for such
a heterogeneity. These are piausible explansticma and one muid imagine th-
factoni changing ow time- Howemr they are explanafions which reiy on f ~ t o n r
entinly outside the =ope of b model and, as such, cannot be ruisesged. The
goai of this sutsection, therefore, b to p d d e not only a plausible but a ( m d y )
emdogenow explmation of a heterogeneity which plryii the m e role as patience.
Any privately obsaved variable which provides a hicage between future n
wax& and cment &ions d d be used to derive results s idar to th- of
the previous sections. As a simple example, consider a model where patience is
syxunetric and common knowledge, but w h m payofb are nibjact to permanent
multiplicative shocka. M e r m o r e , suppose agents have di.fEiit (private) expec-
tations about the size of these shoclcs. h this case the relevant payofE to agent i
would be ?ri(s) + bpiz(i(8, bpi) - Urhere pi is a parameter refiecting i's expectations
about future profitability in an industry. The parameter pi bas now taken the
place of in the previous analysis. AB another example suppasa that in every
puiod each agent must pay dollars worth of intenst on loans ta aeditors. If
agent i is unable to mhburse kt in any paiod, then he must declan bankniptcy
and future play stops for him. If current profits are greater than or equd to k,
then agent i has no incentive to cheat b m a weil constructed collusive scheme ae
mming stabiity o f ~ u n t rates and expectations on friture poyoL. Now assume
that the unount agent i must reimburse in any peziod chmges in an unvdiable
way. Obviously, a pl- might not have a choiœ but to defect h n a p d e d des .
This one time defection codd have dewtating dbcb on the participants since
not only do fimu d m h m being cheated, but a p d b l y pdnlul puniahment
strategy must be phyeà.
Recent wrk by Bertrand (1999) has empincally examineci relationships ba
tween firms and t~~rkem. M.cccod and Malcomon (1989) show that repetition
a d 8Ufficiently large! discount f~ctors permits firms and norhers to achieve Pamto
improving equilibria which an unobtainable in a static game. Bertrand (1999)
hds that fimu f&g cornpetition h m foreign industries an more M y to renege
on agreements with th& worlers: "Cornpetitive premms, by lonering expectad
ruture mts and shortenhg the corporate horizon, ducen the cost amciated
with reneging on an implicit contract (page 9).
In these two examples the discount rate hetmgeneity has merely bean n+
placed by heterogeneity in another variable which links future mmrdis to current
actions. However, this nould seem to be a more aconomically interdng per-
spective because it demonstrates how expectations about observable panmeters
change cumnt behavior. C o n t e this with the approach of using unobservable
preferences . It m u t be s t 4 that the variable ehanging expectati0118 about future pay-
ofls must be private information, since if this wem not the case then a cartel would
be able to restnicture the collusive scheme. The above malyses lets us make pre-
dictions on when pnce wars should occur. If it can be suppased that expectatiom
play a major mle in an indnstry, then it can be pasited that price rpan shouid
be more frcquent when expectations are more variant. ThPs it is necessary that
expectations systematically change in variance dong the busine88 cycle. Similarly,
if it can be mppused that indebtednem is a mqBor factor in an indwtxy, then price
should be more fmpent when indebtedness is hi&.
2.7 Conclusion
This papa hm p m t e d a model where colIuding optimiaing agent8 d b i t un-
dercutting behavior which was intetpnted as cartel instability. This inetability is
brought about as a direct d t of allowing disc30unt factors, or patience, to vary
acroerr t h e and acrorn, agents. It was cugued that the bmg-bang remiits of A b ~ u ,
Pearce, and Stacchetti (1986) as weli aa the h p l e puniRhment d t s of Abreu
(1988) do not cany over to such a sanario. Instead punishments, in optimal
collusion must be made to "fit the &en, and as such actions reveai type. The
interpntation of changing dismunt factors n a d not rely on changes in preferences,
since it was argued that heterogeneous private expectations about future market
profitability, or credit crunches may resuit in simiiar resulte. The model rnakes
predictions which evidence fkom Barbezat (1989) and Bertrand (1999) appear to
back up.
Notes
Search, Matching and Mord Hazard
Paul Johnson
This papa re-examines Becker's (1973) marriage mode1 with -ch fictions dong the lines of Smith (1997). There an two main Innovations. First, ne interpret surplus creation to be the malt of Pareto impmving collusion. Thw, partie8 in a relationship ean choose to ex- a moral hazard variable or "ddect." Collusion can be of difiking pdtability, so type is patience (discount factor). Since players am fkee to march for new pmtners in every pariod, usud enforcement mechsnisms (trigger strategies) c m o t be used. Mead, players are interpreted to buni a SUfjicient qumtity of monay upon meeting a nen p m e r BO that defection fiom couusion is rendered unprofitable. The second innovation is the ensrnination of the search eqgilibrium under the asmmption that type is unobservable. An interesthg implication of type unobservability is the appeaxance of "def'tion" equiiibria.
Acknow1edgmenta: This niiearch was supportad by the SSHRC. 1 nould like to thank Jaccpues Robd for much iadghthrl criticisni and chmusion. This paper iIso bandtted fmm the commeete of seminat participants at the mi~f0e~011omic theory Sanmu at Penn State University. Enom mmt remah mine done.
Becker (1973) studied a matehing model where type ia icliosynaatic, production
is exogenow to the model and search le fkictionleas. Smith (1997) and Shinier and
Smith (1997) studied Beck's matchhg modal from the perspective whem type is
etill idiosynuatic, and production is stii l exogenow but where th- exist search
fictions. The goal of this literature is to study under what conditions partners,
in equilibrium, are of eimilar type - the matching is a8ortative. This literatue
has studied the formation of long run relationships, yet has not permitted the set
of equiiibrium payoffi to in- due to repeated play. The current paper off=
a production process susceptible to Pareto improving collusion while retaining a
search friction.
Consider the meaning and implications of type in a t.Ro sided matching model.
Oftentimes type is not t d b l y relevant. For instance if a h hires a contractor to
accompiish a specific task, then al1 detaile about the service: terms of delivery and
payment may be pdectly measurable and enforaable in a contract. Thus the type
of the parties is mperfiuous since the transaction t h place, in essence, on s spot
muket. in longer tem relationships, type rnay be more relevant. For instance, if
the same firm himi a permanent employeel, then the relationship may be subject
to more moral hazard on both sida due to the incompletmess of a contract. It
is probably important to the hm to have e m p l m who are willing work hard
at theh specified tada?. Similazlyt it is pmhbly important to a noriter to aork
in an agreeable work environment and receive adequate compensation. The Bna
and the worlar may have infornation on a potentid match (through derences
or reputation), and this mqy be of help in the matching chaice. Altematively, the
h and worker may have nnnliable or insufiiâent idormation on which to make
a decision (a new h wîthout a reputation, ot a worker with no pnor experience
or who was self employed). In any case, the pmfitability of a match is a fiinction
of the willinpem (or ability) ta work overtime on a project, or the awmding of
performance through pay increases among many 0th things. The h - w o r k e r
exampIe seem.s to be pertinent to the amortative matching literature since many
economieto, including Kremer and Maskin (1996)~ note aad rationalize a ment
trend of higher intr&firm wage correlation.
The previoua discussion argues that type need not necessarily be job s p d c
productivity, education or other factors which dirrcUy enter into the production
function of Becker's (1973) marriaga model. instead, type is the ability of parties
to overcome the p-ce of opportunities to take admatage of one another. This
paper interprets a relationship subject to moral hazard as a relationship suscep
tible to Pareto improving collusion. For instance, a marriage igords both parties
opportunitiea to renege on promises (implicit or stplicit) to visit in-Iaws or share
h o d o l d chores. However, the repeated nature of a relationship between spouses
may permit the creation of a much larger surplus than betwcen casual acquain-
tances. The interpretation of murfrrge as Pareto improving collusion has been
made by Lundberg and Pollak (1993) who andyze -g nithin muriages.
They use a noncooperative sqiiillbrium M a thnat point, whemas 0th- have
useci divorce as a b a t point.
On a more abetrsct led, thjs papa pmp- a seatch modd whera type is
heterogeneous and ricarch is coetiy. Moreover, the m e far which parhem are
searehed has a flavor of the prisoner's dilemxna in that more profitable outcames
occur if people cm somehow collude. This motivates the interpretation of type as
patience, or discount IBCtor* Smith (1997) and Shimer and Smith (1997) explore
a s d mode1 where type is heterogeneous in a non fransfef8bIe and trader-
able utility p a d i p reepectively* Thh paper studies the non trandkrable utility
p d g m . Their work demonstrates that the existence of a seareb quilibriuxn is
fer from trivial to show with a continuum of types. In a game with a production
technology refiecting the ability of agents to coilude (woperate), the regulazity
conditions of these two papers are not met4. It is for this reason that a con-
siderable simplification is made in d g that th- exkt only two types of
agents.
The very fa t that agents search implies that they are fhe to choase with whom
to play the game. Thus the traditionai repeatd game andysis which implicitly
866umes that players have no other outside opportmities cannot be applied hem.
Thus, there needs to be a way WM collusive outcornes cm be eupported whexi
players are not obliged to remain with a single partner and receive a puainhment.
The main obstade to onmorne is the ability of players to defect while changing
partners in every period. Such a etrategy can be rend& unprofitable by giving
players the abiiity to (verifiably) bum a quantity of money bdore playing with
a new partner BO that chaagiag partnm becornes pmhiiitively expensive. This
is the main thrust of Carmidd and MacLeod (1997) who use the givïng of
intrinsically duelem giRs in lieu of b-g money. Th& papa leaves no room
for the d a n g e of money since the net eftect of nich would be nil. Hmver, it
is shown that under certain circumntan~e8 having the ab%@ to transfer rnoney
is a Pareto implovement in this context. Thes is a papa iining wolutiomry
game theory to d e the equilibrium concept, thus we can think of this as a uNo
PILISSitesn @ereafter NP) npoinment. It is important to note that this constmint
is pdcular to coIIusion games-in Smith (1997), Kremer and Maskin (1996) or
Beclm (1973) no aich defection is pdb le .
Ghoah and Ray (1996) study the abïlity of agents to collude in a search model
where diseount factom and hietories ue unobservable. They amme e h e d p m
portion of players having a discount factor strictly gnster than zero and the
remahder have a discount factor of zero, so (non) assortative matching is not
an issue. In their paper, newly matched gents initially cooperate to a certain
extent a f t e ~ which type is revealed and fidl cooperation ensues. Their equilibrium
is partidarly interesthg since it is robust to b i i a t d deviation, Le. no matched
pair can deviate by forgoing the unprofitable ucourtship" at the beginning of a
relationdip. In general, this remit does not carry over to the current papa since
it is money buming in j h r e dationshipe which mdtes collusion supportable.
However, the intuestecl reader l o d d pay spsiai attention to siibseetion 3.4 to
eee how G h d and Ray's (1996) notion of bilaterai rationaiity carry o v u in this
context.
If matching is to be 88SQ)rtative then th- iir a positive pmbabiîity that any p e
tentiai partnet wiU be rejected - a waiting the, or aogenous much friction. This
is one reason why matching may faü to be amortative. The gaxne studied in this
papa aEorâs another explmation as to why matehing may be amrhtive. Since
di profitable nlatiannhips are subject to morai hazard, the NP conshaht must
be respecteci in every match. But sinœ the profitability of a Mtch is pdtinly
dated to the extent of mord hazard prment, a more profitable match mdtes the
NP constraint more difficult to satis&. Ailegorical apdenœ might lead one to
believe that the winning of a parhier's trust and confidence is a more difficuît
obstacle than the physid meeting of a prospective partner. M e r m o r e , more
pmfound relationships maUy requin! a coUttBhip which is more t h g than do
more ephemd relatiodips.
It WU be assumed that aoy player's history ie private information. This is to
avoid the possibility of a trigger stratqgy of the type "due to play with Bnyone
who has ever cheated" could support coopertttive play. 1 argue that it may be
unwise to permit the use of such a strong tooi in thh case that history mre
o m b l e . For example, long term relationships may be kept mostly implicit to
increase flexibility between a worker and km or between a h u s b d and niTe.
Due to the implicit nature of the contract it rnay be a very Micuit ta& to
assess blame. For example, divorce proceedhg~ sometima degenerate due to the
parties' diBering interpretation of the impliat marriage contract; or a worker may
choose to quit a job becauee ehe did not receive a raise, however h m the hm's
perspective the raise may not have been meriteci.
The eqgiiibrium concept usad in the search modal is important ae it quite
drastically chuigee certain conclusions. Obviously a stronger concept than Nash
equilibrium WU be needed, since t h strategy %je& every potentid partnef is a
Nash eqiiilbrium, This natwally le& to mmhhg perfect bayesiœaa eqdibrium
(or a variant Me Krep and Wilson's (1982) sequeatid equilïbrium '). However,
1 show that even a stmng refinement U e @kt bayesian eqtdibrium Ieaâs to
unpalatable results; eaamtially the modei b e s di abüity to p d c t when amor-
tative matching arises. The appmach 1 tdre is one bassd on psyodl dominance.
A ke!y component to which is the n d t y to have unnnimous consent to fom a
partnefShip.
The paper stuclies tao idormational sccnMa: one where type is observable
and the other whem type is u~~obpetvable. This b a departure fhm the matching
literature which hm only treated the uise whera type is observabie. Section 3.3
ammes that type is obsemable. Intuitiveiy, this is equivaient to 9 information
on a prospective partner behg adable. Section 3.4 assumes that type is unob-
servable. htuitivdy, this is equivalerit to partial or no information (hpertect or
incomplete information) available on a prospective partna. The remainder of the
pape inciudes section 3.2 which presents the model to be useà and section 3.5
which bridy codudee.
The model used throughout this papa sime to be as simple as poseible. The game
at the heart of thia model is inspiml by the weU known prisoner's dilemma with
the addition of dowing for diiEerent Yevelsn of collusion.
THE PLAYERS: There exist13 a continuum of players of finite mam. AU
players have prefe~ences wbich aie ~epreseatable by a utility fiinction of the form:
where 6 is a diecount futor and 4 is the quaafity "utils" wnsumed in paiod t.
Playem are risk neutrai so utils will be intapreted as monetary sums. Opportunity
costs are zero, so an mmatched player receives a surplus of zero. A proportion,
p > O, of playas hava discount rate a, rnd a proportion, 1 - p , of players have a
dismunt rate Ji. Note that p wili be, in part, chosen by the type of equilibrium.
The proportion of type &, players is common knowledge. M e r reetzictions wil i
be pleced on C& and Ji below.
THE GAME: The game to be played will be denoted J?. l? is a two person
symmetnc prisoner's dilemma game with three pure strategies available to each
player. ïï : S x S + R denotee the payoff fuaction sending the set of s t r a t e
gies (S) into the mals. Like Smith (1997) tbie papa assumes that utiiity is non
transfmble. There exista a unique single stage Naah equilibrium in pure strate
@es. Denote the action corresponding to this pure strategy by s* and normalize
ïI(s*, a*) = O to be the same as the opportunity cost of playing the game. The
two remciining pure strategies wil l be denoted by and st. se is a W o r m best
tesponse to both sh and sr. The followhg shorthaad wii l be usefbi. n(se) wiU
denote payofb when both playem play se. r(eh) ririll denote payofb when both
players play ah. n(sl) is dehed similarly. **(a) wili denote the papff to play-
h g a* when one's opponent play8 q. r*(&) ie defineci sbdady. The notation
z(sn, 8') wil l denote the payofb to playing 8b when one's partner haa played se.
THE REPEATED GAME The mpeated game combts of the infinite reg
etition of the stage p e . PayoS and discount rates are such that:
Intuiti~eiy~ i' is m d y a prisoner's dilemma, where action sh is an equilibrium
to the infinteiy npeated game when the diacount factor is a, but not when the
discount fcrctor is 4. Action 4 is an equilibrium to the repeateà game when the
discount factor is either a or ifi. Note that since the opportunity cmt of playhg
the game is equd to the payofb to the single stage Nash equilibria, reliance on an
melenthg Nash equilibrium puniiphment ie optimal in the seme of Abreu (1986).
It is easy to give an example of such a game. If L = 2/3 and 61 = 113 then such
a game might resemble the following:
The rd interest of the mode1 ie a eesreh structure which is i m p d upon the
p1ayeis playing r. Play begh whem every playv is unmatched. The extensive
form of the aearching game is as follm.
1. Seaichirig plam an matcbed et =dom with a plaw h m the unmltched
popdation. Historia are piinte I n l h ~ t i i o n .
2. if so desirod, players may d a b l y burn a quantity of money. Money may
as0 be t r a n s f d to one's parfner.
3. Players observe, or not, theh putna's type and choarie to play I', or not,
for one period with theh ptutner. Unanimity is required for players to play
4. if r ia played, actions are observed and payons mceived.
5. Players decide whether to accept playhg next period with their current
partner, or setuch for a new paitner whose type h unhiown.
Note that in this scenario search has no dinct caets and the game may be played
with a new partner in ewry period. The dietinction between type being obsemble
is an important one, aad will be studiad in the remahder of the paper.
POPULATION DEMOGRAPHICS: An important component of any
matching mode1 is population demographics. For inetance, if a matching qui-
librium is 888ortative and no new players enter into the population then after the
tirst period of matching p = 112. If a matching equilibrium is non assortative and
no new players enter the population, then eRet the Bret period all players wil l
hava been matched. This latter case drastically changes the implications of the
following sections since plsycn' histories are non obauvable, thua buming money
is no longer necmesry. To awid this difncdty, 1 unnune that in every period s
mam, p, of n m agents are barn aiid enter the popdation unmatched of which a
fmd proportion, p, are of type a. It shouid be obvions that if the eqdibrium rsen non uiiwrtative, then p = 8.
It remaine to study the am where matchhg is 888ortative. We thedore have the
foiîowing propoeition:
Proposition 1 If tlie cguilàbnum is tasuortutive, thm tue hate the following:
Proof of Proposition 1 If @ = ) it is obvious that p = 4. We are looking for a
state in whkh the total mase of agents does not change, nor does the proportion,
p, of type 6h agents. If the totd mam is constant then we mu& have p = eO>? + (1 - p)l), where s is the total mass of all agents in the economy. If the proportion
of type Jh agents L constant, then we mnst have:
substituting out 8 in equation (3.2) yields = 9&-+. U8ing the quadratic
formula to solve for p yields:
negative, thetefore the numerator must be negrrtive &o. This implies that the
3.3 Full Information
In this section the simplest Monnational structure ie impoeed: type (discount
fwtor) is obsenable. Understanding the NP constaint is criticel to undelcgtend-
hg the following results. This constraint requires that no player can profitably
u n i l a t d y deviate fiom a wuch equiiibrium by cheating on a new partner in ev-
ery period. Primarüy, this coflstdnt is satisfied through the d a b l e burning of
a quantity of money. It WU be shown that transfiig money a h will be useful
under certain conditions. Note that money buming is only used to aatise the NP
constraints and not as a signal since type is obeervable; this is a major difference
between this and the foilowing saction. R d that a quantity T of money can be
verifiably bunieci only before the type of one's p-er is known.
There are three equilibria of interest to this game. The ht is an aaeortative
equilibrium where a type matches only with a similar type. Non assortative qui-
libria have all types automatically matching with any other type. Non assortative
equilibria corne in two flavom. 4 is always obliged to play sl with any type, hm-
ever 4 could play either q or sh ifmatched wîth another type &,. Thme equiiibria
are examhed in the s u W o n s belm.
Strategiee are c o m p d of three eiements. The b t element wiU be a type
dependent quantity of money to burn. The second dement wii l be an action
(element of {ah, &,$)) dependent on type as weii aa the quantity of money bnnied
by both agents. The third dement lrill be a decision (dependent on the quantity
of money burned a d the btory of actions played) to remah with one'~ cnrrent
p-er or search for a new one! Note that thjs third eiement of a strategy enables
matches to be endd if a partner d d ' . An eqdibrium wiii be a strategy which
sathfies the No P d t e a (NP) comîmint, u weli as the hdividual Rationaiity
(IR) constraint for both types of agent. The IR constraint is standad. The NP
constraint will be more formally pre~e~ted in each of the following subsections
which treat the sesortative and non assortative cases separately.
AB aUuded to in the introduction, the eqdibrium concept is important. For
instance, consider the strategy "bm nothing and play se". Such a strategy ia
cleariy a Nash equilibrium. More intemtingiy, BU& a &rat= is aiso a pedect
bayesian equilibrium. This motivates the ut3e of the payoff dominance concept to
r&e equilibria. The following subsections show that this equiiibrium concept
gives the mode1 much more predictive power than it otherwise would have.
Assortat ive
if it is the case that the search equiiibrium is assortative (in the sense that type 6,
is only accepteci by type q), there may be fmo qnantities of money to be bumed.
The case where ody one quantity of money is bumed is discusged later. Let T;
be the quantity of money burned by type bh and kt qA be the qumtity of money
burned by type 4 in an amrtative equilibrinm. ~t and TiA mmt sath& the
follcmhg No Parasites (NP) comtmints:
Or, more conveniently:
An equilibrium concept even a~ strong as Mect bayesian eqdibrium dom not
delete strategies of the type where Tt + a and TA + a (e > O) must be bumed
in order that agents play together. This is the k t instance when the p a w
dominance criterhm is employed. Clesly, burning Tt or TA sasatisfying the NP
constrsints with strict inequaiity is dominated by the bnrning of T: or GA satis-
fying the NP coflstraints with equality. Hereafter Tt and qA ;A be understood
to denote the 10- values respecthg (3.3) and (3.4) respectively.
It may well be the case that Tt or be negstive. If this is the case then
the physid fiction of hding a match dominatm the W i o n due to the ability
of playem to defect. This is undesirable since players muid bave to %umn a
negative qnantity of mone' It is obvioudy not acceptable to have agents îiterally
burniiig negative quantities d money, but the use of transfii un save the same
purpost. The f011dg pmpdtion gives the condition unda which transterring
mmey can mabe the NP wI18frBYlfs k sathhi nith qpality.
Proposition 2 IfpTt + (1 - p)qA l;A O, Bm the NP c o n s h i n t ~ con k satiafied
with cquulity thmugh the we of tmwfer.
Proof of Proposition 2 If both Tt and qA are greater than zero, then the
inequaüty in the statement of the proposition is vaiid, and t r d m need not
be used. Suppose that Tt < O and qA > O, and instead of burning money, br
t d m money to a prospective partner whüe bh neither bums nor exchanges
money. Denote the t d b r by t and the amount of money bumed by b. In ordu
that 4'8 NP constraint be satisfied we nquire that gt 2 qA ;Ai the inequaüty is
strict, then simply set b > O to sa* the NP coflstraint with equality). In orda
that bh's NP constmint be satMed with equality na require that (1 -p ) t 2 -T,A,
or (1 - p)? +PT: 2 O. Supposing that Th > O and qA < O leads to exactly the
same condition. 13
Note that when Tt c O and TiA T;' O, then the NP constraints camot be saüsfied
with equelity. Furthemore, when p ia dose to zero or unity, tr8518fers can always
be used to satisfy the NP constraints with equality. Thus, we obtain:
The amount of money to be bumed cannot be too high so as to make playing
the game unptoatable. We thus have the foliowing individual ratimality (IR)
It is s t r a i g h t f " ~ to check that (3.1) puantees tbst the IR and NP oonstraints
are internally consistent.
Non assortative
A non essortetive search equilibrium is one where both typm accept playing with
both types. It will dways be the case that 4, when p a M with & or 61, wii l play
$1. Howeve~, if bh were palled with bh eitha 8h or st muld be played. This is an
important distinction. 1 wiU caii the type of equiiibrium where bh plap q with
a and 4 "non eseortative"; %ybridn wil i nfa to the equiîibriwn where bh plays
4 with bh and SI with 61. Since type is 8s~wned to be obrwrvable, no signal need
be sent. In a non assortative search equilibrium money must still be bmed to
s a t e the NP coI16kaints. Note that since type is observable, we may permit
each type to b u m a different amount of money, despite the fact that each type
may be equaily as "productive." In a hybnd equiiibrium it is wefid to recall that
money is required to be burned belon one's opponent's type ie obeennd, thdore
bh must bum a single qnantity of money regardle88 of her partner.
f f i e TFA and TiNA to be the mount of money bumed in a non assortative
Since plays q with a and 4 then har NP constraint is:
Now consider a hybnd equilibrium. These two equiiibria are identic
the perspective of 4, so we wiIi have qNA = qH. Since & plays sh when paireci
with cfh then Jhts NP constaaint fl Mer fiom T [ ~ above. Cali the new amount
of money to be bunieci Tk. Ti must be such that prefas not defécting once
matcbed with type bn:
which yields:
Similarly Tt muet be such that 4 prefers not ddécting once matched with type
Since the right hand aide of inequality (3.9) may be 1- than or greater than the
right hand side of inequality (3.10), T! must sa* both constdnfs. The NP
con8traints for 4 are identical in a hybnd equilibrium to the NP wnstraints in a
non amortathe eqdibrium.
Note that in non mrtative and hybnd equiiibrium it wili never be useful to
use traders to bum negative quantitities of money. The intuition ie that since a
partner ie found at the beginning of play, the fiction due to the ability of players
to ddect wil i always dominate the friction due to the tirne to look for a new
partner.
PayoE dominance requires that qNA, TfA and Ti satisfy the NP coI1StT8iPts
with eqnality. Perfeet bayesian equilibrium does not select the most profitable
equilibrium between non assortative and hybrid equilibria for a. M n e TFA and
Tt as above. We have that Tk > ~r". Due to this, thara exist beliafs entaihg
that the ody rationai strategy b d on these beliefk is to buni TFA and play
si with both 61 and &. Thus burning TtA and playing si is a patfect bayesian
equilibrium, even if buming Tk and playing sh with bringp higher expected
payofi-
Cornparison
in a fkictiodess world where bh and 6i have the ab- to commit to plaging 8b
and si respectiveiy, assortative matching is the Brst best. This section uses the
payoff dominana criterium to pndict whether matehing wiîl be assortative or not
when there am s e d fictions and moral hazard.
If ne ignore the NP constraints, type 6 aiways pderrr a non amrtative e<iui-
librium since in either equilibrim a match is equaiiy as profitable and in a non
assortative equiiibrium search time Q mmmud O . . The following proposition illus-
trates how the NP constraints change thb conclusion. Additionally, the propoei-
tion illustrates the importance of treating difterent types differently even if they
are equally as productive.
Proposition 3 If > O or PT: + (1 - p)qA 2 O, then typc 4 b indiffmnt
between hybrid, non ossortatiue and assortative epuüibriu.
Proof of Proposition 3 It ie clear that 4 is indifterent bett~een a hybrîd and
non 888ortative eqyiiibrium. Therefore, we show that di is inMérent between a
non mrtative and mrtative equiiibrium. The money that 6 burns in atpecta-
tion in an 888ortative ecpiîibrium mwis the money that 4 bunie in expectation
in a non ~sortative equilibrium is equal ta:
The expected payoff to Ji in a non ammrtative apniubrium minus the expected
Aithough the a b m proof is entirely algebraic, the propoeition is intuitive. When
the NP constraint is just binding, agents are just îndifkrent between cheating and
playing the equilibrium strategy (cooperating). Suppose Ji hm found a match, w
that in a non assortative or assortative eqdibrium her gains to obedience are the
same. Therefore gains to defection are equal in the two aws:
Thus, the value of behg unmatched 5 equal in the two cases.
We can a h conclude that if 6r'~ NP constraint m o t be satisfied with equd-
ity, then non assortative matching is strictly preferred to asmrtative matching.
This, dong with proposition 3, assures us that br would dways be willing to ac-
cept 4 as a partner. Thdore , since a match requires consent of both p&ners,
the type of equiIibrium wii l be dependent on the preferences of a. The foliowing
lemmas study &s preferences as to Merent types of equiîibrium as a function of
the parameters of the game. Together, these lemmas lead to the main mult of
the section*
Lemma 1 Suppom Tt > O or PT': + (1 -p)Tt 1 O. Then whenever *(ah) - li.(sl) < *-, 6, is inMererit b e h e n a hybnd and non aasortative M b -
rium. Similad, whenever %*(ah) - Z(Q) > ?*, is indifférent betrieui a
hybrid and assortative equilibrium.
Proof of Lemma 1 F i notice that Tk 1: detemined by (3.10) rather than
(3.9) ia
Simplification of the above tena yielcb the promiseà resuit.
Now suppose that no (q) -n'(sr) < W. We know that T: is detenninec
by (3.10). &, is indifférent between hybrid and assortative equilibria ifl:
Substitution for T t A and Tt aud simplification shows this to dwap be the case.
Now mppoee that re(ah) - ~ ' ( 8 ) > ?** Assortative eqyiiibria wiîi be
wil l gin the same gains as a hybnd equilibna iff:
By the previow 1- we know that Tt is defined t h u g h (3.6). mer mbsti-
tuüng in far TI and na see that Q is al~lp~ys indifkent between hybnd and
The following lemma studiee the ielationship betmsn aaaortative and non
asmrtative matchhg*
Lemma 2 Suppoee T: > O or PT, + (1 - P)T: 2. Then whenevar n8(sh) - r8(sl) < i~~ p d i m amrtative to non assortative mat*g. ~imilar~y,
whenever r0(sh) - Ir .(si) > **, bh p d m non cuisortative to assortative
matching*
Proof of Lemma 2 So long as dhls NP constraints bind, rents to h m aseor-
tative rnatching can be readily caiculated to be:
h t s to 6i, fiom non amortative matching can be caiculated to be:
Cornparison of these two expressions yields the pmmised d t . O
Thus using the above lemmas, when no(q) -n'(si) c 6, ordm the
types of eqpilibrium in the followhg mamer:
asmrtative + non amorfative - hybrid.
non amortath + issortatim - hybnd.
Note that hybnd equilibria do not mur.
Obviously n8(sb) - n'(si) 5 uw is the lray property of the gazne nhich
detenaines if matching is assortative or non assortative. This property b k s the
diflerence in the gaine to defection and the difl'ce in the gains to collusion in an
intarpretable way. Intuitively, if additional coliusion yields only a mail in~~ease
in the profitability of the dationship yet a laqe iacrease in the gains to exercis-
hg the moral hazard variable, then non amrtative matching wi l l ensue. Becker's
(1973) rnmhge mode1 predicts that in a nontransfbmble utîlity scenario assor-
tative matching will mise for any (monotone increasing) production teehnology.
Smith (1997) shows that when players are impatient and se& is lengthy, eearch
is asmrtative when the production technology is log-aipermodular. Therefore, for
the case when production is subject to coUusion players cho- the equïübrium
basai on the with which different levels of coilusion can be mpported.
The above has given conditions undu which eqdibrium is mrtative or
non amrtative so long as Tt > O or pTt + (1 - p)qA ;A O* Since whenever
fl(sk) - f l (q) > de, equilibrium is non amrtative, it remninn to study
the equiiibrium when n'(ah) - n'(al) ww tuad bn9s ~iiortative NP con-
atxaint does not bind.
If the NP cormtdnte nare jwt biiding, thai, for a, the rents h m an ut
mrtative eqailibrium minus the rents fkom a non amorfative ecpilibrium -al
**- 1- h - ( T * ( S ~ ) - ~ ~ ( 8 ~ ) ) . F a 8üppo8e TL < 0 a d qA 2 0 ~d flt + (1 - p ) c O In this case a burns no money but meives a trader h m 6,.
The discounted expectrd d u e of the lasg due to the NP constraint not b'iding
can be esldated to be: Thus bh ptefere an 1188ortative equjlibrium
whenever:
Now suppaee that T: < O and qA c O. In thie case dh bnnis no money and
receives no t d w h m 4. The discounteci expected value of the loss due to the
NP constraint not binding can be calculatecl to be: ,&]. Thus prefm an
assortative equilibrium whenever:
We can finaliy state the main proposition of this section.
Proposition 4 Matching d l & assortativc whenevec
Similady, matchhg d l k non ~ssortatiue whenevec
Untü this section, ne have ignod the dects of population demographics on
equilibrium. Since the choice of equüibrium (to a certain extent) chooses the
proportion of patient players, such ~EE&s should be studied. Uf course, such
effects may be trivial in that the combination of 8, p and the type of equi l ibn~
may cause one type of equilibrium to always be played. The following example
shows that this need not always be the case.
h a i l that if the equilibrium is non assortatiw, then p = p. If the equiiibriwn
îs assortative and p + 112 then p = we Now suppose the following:
b b P 4. for p = .S., we have Tt < 0, and pTt + (1 - *)TA < O,
1. assures us that if the equiiibrium ie assortative, then p < B. 2. amres w
that i f p = B thenTt >O. 3. amre~wthatifT~ >~,thentheeqdibrium
LOO
is amrtative (pmpooition 4). 4. siiauras us that for a certain vahie of p a's NP
constraint cannot biid. 5. aseurar us that for this value of p, type da p d e m non
asaortative matching.
5. requires riome jdcat ion . The value to of non ~wrtative matching can
be caiculated to be ?-. h 1- it The d u e to 6h of amrtative matching when
Th = O is: *. 5. L eimply the combination of these two expressions.
Consider the environment in period one. Since p = p, we are essured that
T[ > O (by 2.). By 3. the equilibrium ia essortative. In period two, p changes
e, eo the non negativity constraint on Tt binds by 4, thus mch that p = w-l
T: = O. Now we use 5. to see that non assortative rnatching D p r e f d to
assortative matching. Of course in m o d 3, p = p and the cycle repeats itself.
So in general, in every odd numbered period thete is assortative matching and in
every even numbered period there is non asmrtative matching.
However, if such an environment nen to extst then the amount of money to
be burned (Th and q, presumably) is miscalculateci. In this case then would be
dinerent quantities of money to bum in odd and even periods. Let the superscript
o denote an assortative (odd) pezîod, and the superscript e denote a non 888or-
tative (even) period. Thus, ne can use the NP constrsints, the non negativity
comtraints, and the pay& dominance cRtetium to write the folowing expiicitly:
The NP constrainta assure us that no agent has an incentive to defect in
any given period. Ho-, it may be the cam that an agent be prepareà to
"sit out" a period in anticipation of the next. The dgebra involveà in checking
such an ewntuality is nimbersoma, nevertheles the following lemmes provide an
alternative method of analyzing this case.
Lemma 3 If > O, then br obtaîns idaiticai payoffi in odd and even rounds.
Proof of Lemma 3 The lemma can be proveà in a fashion similar to the earlier I
indinerence propositions. We know that ia dmys greater than zero, so if
> 0, c& is inHetent between defiting and coopsrsting in the two casea. Since
coopuating gains are equal in the two am, gains to ddeetion must be equal in
the triro cases. Thus, gains to being iinrnatched are v a l in the two cases. O
The previow lemma i m p b that ifv > O, then 4 wiil always pnda participating
aa oppposC do aitting out" (thce fiitm gains are dlscounted). Additiondy, if
P = O, then br strictly p d u s non sisorfative (even) paiods to 888ortative (odd)
mods, thus, b1 may pdbr to dt out durhg as8ortative periods.
The following lemma studiea how bh acts in euch a ecenario.
Lemma 4 & must eventudly b d it preferable to participate in the economy.
Roof of Lemma 4 Suppoae not. Thedore in evezy period only 6( participates.
(4 must participate Bnce any auch period would be equivaient to non asaortative
matching.) Thus the mass of type agents approacbes infinity, and ee audi p
approaches unity and Th approaches &[rr8(q) - *(a)] which is a contradiction
since by sssumption 2, a would gain strictly positive payofb by participating. O
Thus, a may prefer to eit out for ~everal periods, but must eventuaily rwnter
the economy. In this case, cycling has a period of greater than two.
3.4 Private Information
In this section type is sssumed unobservable. The NP constraints must be re
tained as agents am s t i l l able to change partners at will. The p m m c e of private
information does, however, require the introduction of incentive compatibility con-
straints: type & must prefer sending signal (burning) Th rather than 9, and type
must pd' sending mgnal (burning) Tc to sending signal Th. These constraints
WU be referred to by ICn and ICI respectiveIy. Th~bus an ecpilibrium must aatisfy
ICh and ICI in addition to the NP and IR constraînts. In thb context money
b d g not only serves to sa* a NP cormtmht, but dm serves as a signal
dong the lines of Camerer (1988). Note that ICI mcphs that Ji prefer burning
8 to b d g Tb and playing a*.
Thsn ue four eqtdibria of interest which a r k in this idonnationai set-
asaortative, non mrtative, hybrid and "defiionW. The notions of assortative,
non aseortative and hybrid equilibria are drin to thme deveiopd in the previow
section. Defection equilibna have both types buining the eune amount of money,
but playing dïfE@rent actions. This type of equiiibrium is ody possible when type
is unobservable. Each of the ciifferant types of equilibria WU be d i s c e in a
series of subaections.
Assort at ive Equiiibria
Assortative equiIibria under ineomplete information are similar to 888ortative equi-
libria under full information. The constrainta for eaeh type can be nritten in the
foliowing mamer. For type a:
For type 4:
These are the fnmilinr IR, NP and new IC comtnhts, mpectively of each type.
The IR and NP comhabts are identicai to the IR and NP constraints of section
3.3. The NP constraints must raepect the constaaint that no negative quantities
of money be bumed.
The only jU8tification needed is to arplain why the mndmht where bums
qA and then defects, is not induded. Similarly the constraint where Ji biinis
T' and then play (faithfully) 8s is not induded either. To see why, consider
bUTIIiIIg v. We have that tbere is a unique d E (O, 1) such that:
For any discount factors above d the le& hand side is strictly greater than the
right hand side. Since ne how that the left hand side is naalrly greater than the
right hand side for Ji, we am assured that the inequality is ~trict for a. A War
mgurnent shom why 4 wili never faithfully play 4.
Let us now reconsider G h d and Ray's (1996) notion of bilateral rationality.
While the eqiiilibrium concept of this paper rules out unilateral deviations, newly
matched agents may have an incentive to mach an agreement wherein no money is
burned. Such a bilateral deviation might be profitable since it is only expectations
about fiture market conditions whkh deter deviation. Ghosh and Ray's (1996)
work show that type unobgervability L essential to d e v e m eq@ibrium which
satisfies bilateral rationdity. It is a h esseotial that a b i l a t d propasal offering
to fongo the biirning of money be lem profitable than bplpiI1g money. This is
trne if thue is a dcient ly large proportion of qpts ready to defeet and cause
dciently large laescrs once auch a p m p d L acœpted. In an BBBOrtative eqyb
librium w h m type b unobsentable, hiu, no incentive to biaterally renegotiate
Non amorfative
Non -dative equilibria need not s a t e incentive compatibiiity c o ~ t s , nev-
ertheless the anal@ is changexi under private information since only one quantity
of money can be bumed. This foi lm since if th- nue two distinct quantities of
money to burn, it muid dways be optimal to burn the lesser quantity of money.
Thus there wilI only be one NP constraint:
which will be satisfied with equality if the payoff dominance criteriun is need.
Additionally, there rrrill only be one IR cowtmbt: -TtA + a(sl)/(l - 4) 2 O.
This is always satisfied so long as (3.1 1) holds with equality due to (3.1).
Hybrid Equiiibria
A hybrid equiiibrium when type is unobservsble is eunila to a hybrid equilibrium
when type is observable: Jn signais ha type by burning a pester amount of
money than 4, but if she is unluelty enough to have b w n a parher who hm not
sigaalled thet she is of type bh, then ahe dom not s e a d the next period in the
hope of hdiiig a patient putner. Of coiirse a hybrid ecpilibrium wil l involve Jr
burning money as in a non assortative equiîibrium, thus ne rnay write qNA and
TB intercbarigeably. Let qNA be the signai sent by Jr and let Tk be the signal sent by 6''. Unob-
Note that the above exdudes the p d b i i t y that 6' ddects upon meeting another
6,; implicitly the constrsint:
is m e d satisfied. But this is exactiy the NP constraint wbich defines
ICh ~e~ui res that tîh pder busning T: to burning TA:
Defection Equilibria
A defection equilibrium has both & and 6, buming an identical quantity of money,
TD, and upon meeting a new partner bh and Ji play Mèrent actions. Th- ara
t h types of defdon equilibria which an of intanst:
Defation 1: & only plays an with a permanent partner*
Defation 2 ody play si with a peimanent partner*
Defation t Jh plays both sh and al with a permanent partna.
Thae dehitions do not epedt to the behavior of 6, nor to the initial actiom
talcen by prospective partneus. For example, a defection 1 eqaillbrium codd have
bn piaying q or 84 upon meeting a new parber, whereas Ji couid piay s*. hr-
themore, dFi, aRer meeting somebody who has played se, couid chouse to piay q
permanently. Initiai actions wiii Muence the qpantity of money to be bumed by
eech type. However, notiœ that in a type 2 defection equiiibrium m u t play d
at every round. For instance, in a d e f i o n 1 equilibrium whae &, dways plays
4 and Ji always play8 s*, the d u e which & d g n s to being unmatched deeream
thce there is a probabiity (1 - p) that she WU receive payoff8 of r(sr, se). Rec-
ognizing this, Jh wiî l need to burn legs money upon meeting a new partner to
respect the NP constrainta. The next proposition uses this insight to amive et sa
important d t :
Proposition 5 If the NP conshint bànàa in a defdion 1 cpuüibrhm, then 4
b indi#mnt between thb equilibrium and an wortative equilibnum whm fypc b
obsemed.
If the NP conshint binàa in a defdon 8 equilibnum, then bh kt indÿlmnt
between this equàlibrium and a non assortutive equüibrium when Eype is obscrved.
If the NP conatmkt binds in a defedion 8 eguüibriwn, then cih is àndiferent
ktween thb equilibrium and a hgbrid quiiib6um when typc b okmed.
Proof of Proposition 5 For each eqyilibrium, if the NP constraint binds, then
is just indifferent batneen staying with a long tam partner and def'ting* Thus
the vaiuea to being unattacheci in a defection 1, 2 or 3 equilibnum m m be just
quai to the d u e of behg unattached in 8n 8%80rtativey non amortative or hybrid
equilibrium mspectively. 0
A comliary to the a h pmpo&ion is that in a d e f i o n equiiibrium the amount
of money bmed is lem than the amount of money bmed in a hill information
equiiibrium, Also note that defiifion equilibria mdDe &, strictly worse off when
NP daes not bind.
It is easy to give a h p l e , albeit samewhat trivial, example of a deteetion
equiiibrium where bh plays ah and 6i plays a*. Suppose the following:
1. p is arbitrarily dose to unity,
Rom 1, both NP constraint~ wi l i be binding in an assortative equilibrium (using
t d e r s ) . a always p r e f i playhg sh to 81 so long as 3. holds. The incentive
compatibility constraint for 61 is not typical, as it says that prefers to mas-
querade as & rather than playing an esriortative equilibn~m. This iip given by
condition 2. Since the mass of 4 types hm been assumai to be arbitrarily small,
Jn wodd tolerate an infestation rather than raise Th or play 81. A W a r example
can be given where 4, plays sr. When type is observable a defection equilibrium
d d not Mse as d d aimply refuse to play with 6,.
In general, there an thna &aga in the construction of a defection equiiibrium.
Fit the NP conetraint for a is useà to d&e the quatity of money which is
burned in quilibrium. This amount wil l be denoted p. Second, the incentive
compatibiiity constraint must be s a W d for type 5. This constraint states that
4 prefers bnrning to buamhg anothsr amount which may &t in another
type of equiiibrium. Third* must prefer a ddection quiiibrium to raising Td
by a quantity which would render bc98 masquerade unprofitable. Similady, &
m u t pdiw a d d ' n equilibrium to a non aesortative or hybrid squilibriwn.
Note that if the pafect bayeth equiiibrium concept b to be useà, a ddection
equiiibrium could ariet for a larger set of economies.
Implications of Type Unobservability
As was mention4 in the introduction of this section, money brvning in this in-
formational conte not only caves to hilâll an NP comtraint but also as a way
of communicating idormation. This is an important disfindion with section 3.3
where type was o m b l e . This mbsection hm for goaI the study of the implica-
tions of the introduction of the incentive compatibility constraints on equilibrium.
The first r d t veTifies that pmpoeition 3 holds when type is mobgervable.
Proposition 8 men type is unobserwble, so long as Ji b NP co~tmint O bind-
ing, then die U indvemt bettueen hvbrul, non tusortutive and asaortatàue equi-
libria and pnfm defatton equitibna.
Proof of Proposition 6 We nad that the introduction of the IC con-
straints does not change propdtion 3. The d y rrqy in which this muid be
untme w d d be if qA, qNA or Tf wem changed. None can be dacrsased thce
an NP constraint muid be vio1ated. Suppae or incrcwar (due to a viole
tion of IC,). Then 4 muid etrîctly p d m a non assortative eqdibrium. But dl
couid refum to incraiiis or rad a non amdative eqyilibriimn m l d eneue,
contradicting the payoff dominance criteRum.
To ria that 3 p d t n a dektion quiiibrium, simply d the ICI constraint
in the d a t i o n of a defection equilibrium. This consbaint rsqnirea tbat &6i p&a
a deféction equiiibrium to an assortetive equiiibrium. O
Note that if 4)s NP consttdiit is not binding then 4 strictly prafae a non sg
sortative or hybrid equilibrium to an iusortative eqoilibrium. Thus, ne condude
that, like when type was obsenisble, bb selecta the type of equilibrium.
An important coro11ary to the above proposition ie that the introduction of the
incentive compatib'ity coI16traints can never force to be i n d . Suppose!
that ICh was violated in an assortative equilibrium at the values of Te and qA dehed by the NP constraints- The only way to satisfy ICh is to raise qA (loaiering
T', is not passible, since that would violate an NP constraint). Thus if were
to be increased, then would be strictly wom on than in a non assortetive
equilibrium (kom the above propdtion). But by d u h g to incnsse TI, br can
guarantee that a non amrtative equilibrium e m e . Thus 4 wodd never increase
TA. The same 81:gument holds for qH.
In the previous section we were able to otder a's preferences over equilibria
in an intuitive way so long as the NP constraint was binding (i.e. so long as the
physical fnction of sead did not dominate the mord h a z d fiction of &).
The 88sumption that type is unobeervabIe now affords another mason for the NP
mnstraint not to bind - the IC constr8ints. Thus like in the pmviow section the
anaiysis is complicated when the NP constraints do not bind, neverthel- certain
recuits can be obtained.
Proposition 7 examines the impact of incentive compatib'ity m long as the
NP constraints for the ddccton equiiibria binds. Proposition 8 8 a t a m i n e s the
choice betnesn an 888ortative and defectjon 1 equîiibrium, provided that ICI is
violated and that the NP constrcrint does not bind for the debection 1 equilibrium.
hpoeition 9 examines the choiœ betmeen a non assortative and defaction 2
equilibrium, provided that the NP constdnt does not bind for the ddmtion 2
equilibriwn.
Proposition 7 If % NP co~tmànt bind M a d e f ' o n 1 eguilibràum, then &,
con dwaya guarontcc hmdf the uume payoff as un ossortatiue typc obaewabZe
equüàbtium*
Similady, f bn i NP oonstroint biwb in a defcetion 9 equüibrium, then Jn
ma dways guamntee herself the sume puyof aa a non assortatiue ope obaentable
equitibrium.
Proof of Proposition 7 This foUows dhctly from proposition 5. O
Stated dinerently, when the NP constraints biid for the delection equilibria, in-
centive compatibility does not a f t i bh9s payoffi* Note that although bn may
guarantee herself the sama expmteâ payofk as in an msortative full information
equilibrium, the equilibrium is quite difilemt. Additionaily, whm the NP con-
atraints bind a deiection 1 or 2 eqyilibnuxn is always predared to a delection 3
equiiibrium.
Proposition 8 Suppo8e ICI 6 violüted in an aaortotiue equilibriurn and the NP
anahint fur o defection 1 equdibn'um doeu not b i d . Then a defation 1 equüàb-
Proof of Proposition 8 If Ici is violatexi in an assortative equilibrium then,
Thus, gains to an essortative equilibrium will be equal to gains to a full information
assortative equilibrium minus -&. If the NP conetraint for a ddection 1 equilibriwn does not bind, then Td =
~t + r(sh, s*) is lem than zero. (Note that hem we hava sssumed that 61 plays
s*, it might be the case that 61 play si and remains with a type br agent. If this
is the case, the necessary change should be obvious.) Thus gaine to a ddection 1
equilibrium d l be equal to the gaine to a full information assortative equilibrium
Comparing the loesee in the two cases leads to the promised m d t . O
Proposition 8 Suppose NP dues not bind /or o derdon 8 equilibriurn. m e n a
defation 8 e~uilibrium d l be p n f e d to a n o n twortatiue equdibnum @,
Proof of Pmpositlon 9 Gaina to a non asaortative ecpdibnum wben type P
unobservable can be calcdated to be:
Asguming that the NP constrdnt does not bind for a defection 2 equilibrium,
gains are equal to:
Combining the two expressions, canceiîjng out and posing qNA = &[r8 (si) - r(sc)] le& to the promiseci result. O
The most stRking implication of considering the mode1 whexi type is unobserv-
able is the appearmce of defecüon equilibna Typicai analyses of collueion have
shared the implication that defection should never be obriemd since a cartel couid
dways structure coiiusion in mch a way as to dimude non respect of an a-
ment. The current analysjs diffeis in that cartels (which in this csse are fomed
of only tm agents) fom endogenousiy. This d t s in a traie off: to increase the
profitability of a match with a patient type it is necesaary to decrease the prof-
itability of meeting an impatient type. Defection equiübria are a key ingredient
to the folioning proposition.
Proposition IO 6'' naiuea hiohet puyoffi den Lypc ir o b r d k , Pahem 4
d u e s hi* poyoff when type b unoQmabZe 80 long oi her NP cowfminb
a n binding.
Proof of Proposition 10 The Grst part of the proof shows that bh is mme ofE
when typm are unobemble. Suppoae, to the con-, that & is strictly bettet
OB when typa L unobaemb1e. Recell that choix! the type of equilibrium whm
type na8 obearvable. Thdore , if the resulting equiiibrium is amrtative, non
assortative or hybrid, thm the only -y for bh to be better off iil for the mount
of money sbe bunis to decre88e. This is not passible since a violation of an NP
consbaint would enme. The remaining poss'bility is that a defection equiiibrium
be played. But proposition 5 d us that was aiways at leest as weU off
in an assortative or non assortative full information M b r i u m as in a defection
equilibrium.
Next, we examine the d a r e of type cfr. Since we have assumeci that bi9e NP
constraints are binding? 4 is always indiftmnt betnreeD amortative, non assorts
tive and hybrid equilibria when type is otservable. Men type is unobservable a
violation of ICh wil l not affect the amount of money that Jr burns. If ICl is vie
lated in a hybrid or aesortative equilibrinm, then wouid rehisa to raise Ti, since
a non assortative eqdibrium muid obtain, leaving him just as d off. The ICI
corutraint in deféction equilibria ensures w that bc prefm a defection equilibrium
to an assortative equilibrium. Th, since ddection equiiibria only oecur wheri
type ie unobservable, m crn condude that if a defection eqdibriwn obtains, thm
6' is better off. 0
Perhaps an interesüng neult un be obtdned by noticing that the sœnario
where type is unobservsble m w Pareto dominate the sœnario where type is o b
m b l e . We know that bh is indiffemt betneen a defection and mrtative
equilibrium. 6 prefe118 a deféction eqdibrium to a non 8880it8tive equiiibrium in:
The intuition es to why ari incomplete informational scenario may dominate a
complete information ecenario is that the adion set of a when type is unobserv-
able is containeci in the action aet of 4, when type is observable. Namely, when
type is unobservable, 4 may not rehiee to play with 61 before both have played.
3.5 Conclusion
This papa mexamined Becker's (1973) mamiage model with search Mctions.
Tm major changes to the made1 were studied. First the production technology
was endogenized to reffect the pdbil ity for Pareto improving collusion. The
partnership is faced with the obstacle of agents defecting in e m y period and
matching with a new partner. This obstacle is amcorne by making meeting new
partnem prohibitivdy expensive. Second was the ammination of the model unda
incomplete infomation: unobservrrbiiity of type. An eqiiilibrium concept stronger
than pafect bayesian eqdibrium ie nemary to select a single equilbrium. This
paper uses a prpffdoniannce concept. The goal of the papa was to atudy when
oniy like types match together (assortative matching) a d when types match with
differmt typeil (na mrtative matcbing). The key condition was argned to be
the "easen wi th which tha di&crant de- of coUusion wem supporteci. The ad-
dition Ot imcompIete infofl~lllction pedts the existence of defection equiübria and
diminishe the payofB of patient types whiie incremhg the payofb of impatient
types*
uorkers.
'Sec Shimer and Smifh (1987) footaate 5.
'The p e to be etudied in not a Wte game abce the mount of money to be burned b
an element of the nrl nomben. This paai a urhnicai o W e ahce the notions of punct
bayesian and ee~uential equilibrium ate d&ed for Orte gmm.
'II a p4yer w b k to Wect her pattnet before p-, without lom of mty, assume
thrt b a h p l q m play a', and th- teceive rro m.
Bibliography
AEREU, D. (1986): "Extremd huiïibria of O l i g o p o ~ c Supergames? Journd of Economic Thcory, 39,191-225.
(1988): "On the Theory of Mnitely Repeated Games with Disconnting," Ewnometricu, 66,383-396.
ABREU, D., D. PEARCE, AND E. STACCHETTI (1986): Wptimal Cartel Qui- libria with h p d a t Monitoring>1) Joumd of Economic !lheory, 39,251-269.
(1990): Toward a Theory of Discountd Repeated Cames with Imperfect Monitoring," Economehicu, 58,1041-1063.
ATIIEY, S ., K. B AOWEU, AND C. S ANCH~R~CO (1998): "COUusion under Private Information ," Mimeo-MIT.
BAGNOU, M., AND T. BERGSTROM (1989): "Log-Concave Probability and its Applications," Discussion paper 8923; University of Michigan.
BARBEZAT, D. (1989): "Cooperation and Rivalry in the International S t d Car- tel, 19261933," Journal of Econornic Hbtory, 49,435-447.
BECKER, G. (1973): "A Theory of Mamage, Part 1," Journal of Pol i t id Emn- omy, 81, 813-846.
BERTRAND, M. (1999): the Invisible Handshake to the Invisible Hand? How Import Campetition Changes the Ernp1oymemt Relatiomhip," NBER Working Paper 6900.
C m I C. (1988): ''Gh as Economic Signais and Social Signas," Amerùm Journd of Soeiology, M, s1-214.
CARMICWAEL, H. L., AND W. B. MACLEOD (1997): Wif't Giviog and The EvoIution of Cooperation," Intemotid Economic RGuiew, 38,485-509.
COMANOR, W. S., MD M. SCBAN- (1976): UIdentid Bide and C e e l Behaviot? Bell J o u d of Econumb, ?,2ûl-î3.
FWEDMAN, J. W. (1971): *A Nonmopetatiw Equilibri~m for Supergames," Review of Economic Studies, 28,l-12.
G ~ o s s , P., AND D. RAY (1996): "COoperati011 in Cor~lmunity interation With- out Information Flows," Rewew of Economic Studim, 63,491-519.
GRAHAM, D. A., AND R. C. MARSAU (1987): 'Co11u8ive Bidder Behavior at SingleObje& Second-Mce and English Auctiom: J o u d of Polü id Econ- oniy, 95,1217-1239.
GREEN, E. J., AM, R. H. PO- (1984): 'fNoncooperative Collusion under Imperfect Price Information," EmomeinEq 52,87-100.
GROSSMAN, P. 2. (1996): The DynaSOics of a Stable Cartel: The Raiiroad Express 1851-1913," Econornic Inquiry, 34,220-236.
HARRINGTON, J. E. (1989): UCoiiusion Among Asymmetric F i - The Case of Different Discount Facto# Intemationai Journal of IndwMai Organization, 7,289-307.
KREE~ER, M., AND E. M A S K ~ (1996): w~ge hequahy by Segregation and SU," NBER Working Paper 5718.
LEIIRER, E., AND A. PAUZNER (1999): "Repeated Games with Dine~ential Time Pde~ences," Econornetricq 67,393-412.
LUNDBERG, S., AND R. A. POLLAK (1993): "Separate Sphem Bargaining and the Marriage Market," Journal of Politid Eoonomy, 101,988-1010.
MACLEOD, W. B., AND J. MALCOIUSON (1989): UImplicit Contracts, incentive Compatibiity and Involuntary Unemp10yment," Economdrica, 57,447-80.
MC-, R. P., AND J. MCMUAN (1992): UBidding Rings," The Amhaan Eceanomic Review, 82,579499.
MIL~R~M, P. R., AND R. J. WEBER (1982): "A ThCOry of Auctions a d Corn- petitive Bidding," Eoonometvh, 50, 10094122.
PORTER, R H. (1983): *Optimal Cartel TEgger-Price Strategies," Joumd of Eoonomic Theory, 29,313-338.
RUBINSTEIN, A. (1979): '<Epdibrium in Supagamm with the Owt.lgng Cn- terian," Jawnd of Ec01u)mic Tkorg, 21,l-9.
SMITH, L. (1997): "The M d e i with Surch Fkictions," MIT Mimeo.
SMITH, Rb A. (1961): The h d b l e Electrid Congphcy," Fortune, 63.
VICKRBY, W. (1961): "Coutempecdation, Auctiom and Cornpetitive Seaied Tenders," Journd of Finance, 16, û-37.
WILSON, R. (1992): USttstegic Anaiyais of Auctions," in Bandbook of Garne Theory with Economic Applicotio~, d. by R J. Aumann, and S. Hart, vol. 1, pp. 227-279. Elsevier Science Publishem B.V.