CMPO Working Paper Series No. 00/21
Passive Industry Interests in a Large Polity:
A Large Poisson Game Approach to
Endogenous Lobby Formation
Clare Leaver
and
Miltos Makris
Centre for Market and Public Organisation,
and Department of Economics, University of Bristol
May 2000
Abstract
In this paper we offer a theory of lobby formation that is consistent with passive industry groups. Our
focus is on the variable `political influence' costs associated with direct transfers and the decision to
contribute towards these costs in a large polity. Modelling the latter decision as a Large Poisson Game,
we show that it can never be an equilibrium for more than one lobby to attempt to solicit policy
favours. Rather a single group that does not oppose the legislator's prior preference dominates the
policy process. Direct corollaries are that `money never changes hands' and lobbying is purely
counteractive.
JEL Classification: D72
Keywords: lobbying, collective action, common agency, Large Poisson Games
Acknowledgements
We are grateful to David Austen-Smith, Stephen Coate, Paul Grout, Ian Jewitt, Jean-Jaques Laffont and
participants in the Young Economists' Conference at Oxford and Departmental and Economic Theory
group seminars at the University of Bristol for helpful comments prior to this draft. The usual
disclaimer applies. This project falls under the Political Imperatives in Welfare Services Reform
programme funded by the Leverhulme Trust.
Address for Correspondence
CMPO
University of Bristol
8 Woodland Road
Bristol
BS8 1TN
[email protected]
[email protected]
CMPO is funded by the Leverhulme Trust.
1. Introduction
Forany particularpolicy issue itis often easy toidentify a numberofdi¤ erent
interests groups, orsub-sets ofcitizens likely toshare a common policy goal. It
is acommon observation, however, thatsuch groups seem todi¤ erin thedegree
towhich they participate, collectively, in the politicalprocess in an attemptto
furtherthese goals. M oreover, in a numberofcases, lobby group competition
appears tobe absentaltogether, with justone‘voice’appearingtodominatethe
lobbyingprocess.
T hemostobviousexampleofsuchaphenomenonisthesubjectofitsownliterature, namely‘regulatory capture’. Inspired by Stigler’s (19 7 1) re-interpretation
ofU S regulatory policy, this literature suggests thatindustry groups lack competition tosuch an extentthattheycan e¤ ectively ‘capture’legislators and thus
ensure thatregulatory policy is supplied exclusively in theirfavour. T he dominanceofasinglelobbyhas alsobeennotedin tradepolicy. ForexampleKrueger
(19 9 0) points tothe factthatindustrialusers such as softdrink manufacturers,
as wellas ordinaryconsumers, playedasurprisinglypassiveroleinthedebatebetweensugargrowersandtheU S governmentovertheissueofsugarpricesupports.
Similarly, inacross-sectionalstudyof435 4-digitindustriesG awande(19 9 7 ) notes
thatthere is little evidence tosuggestthata protectionistlobby faced e¤ ective
competition from eitherapro-exportordownstream group, letaloneconsumers,
when lobbyingfornon-tari¤ barriers on Japaneseimports.
N aturallythis asymmetryhas notgoneunnoticedbyeconomists andpolitical
scientists. R ather, theuniversalresponsehasbeentobuildonO lson’s(19 65) work
on collective action in which itis suggested thatcertain groups failtoorganise,
and thus participate, collectively, in policy debates, because atsome level the
…xed costs of group formation exceed the expected rewards. R eceived theory
therefore suggests a taxonomy ofgroup formation driven by variations in group
size and permembernetbene…ts. T hatis, ifa group sharinga common policy
goalcontains atleastone memberforwhom the bene…ts from politicalaction
exceeds theassociatedcosts, even ifbornealone, itis suggestedthatthegroup is
privileged and willalways …nd a‘voice’in thepolicyprocess. A nalogously, ifthe
costs associatedwithformationexceedthetotalamountbywhichagroup stands
togain from politicalparticipation, collective action is sub-optimal. In such an
eventthegroup is thoughttobelatent: always failingtoparticipateinthepolicy
process.
In the remaining case, however, when aggregate, butno individual, bene…t
2
exceeds the associated …xed costs, the outcome is deemed to be dependanton
the leveloffree-ridingand thus ‘indeterminate’. T hatis, since the provision of
a favourable policy outcome has all the characteristics of a pure public good,
depending on the size of the group, members face incentives to take an easy,
ifnotfree, ride. T hus to ‘form’, and hence be intermediate, a group mustbe
su¢ciently smalltolimitthe possibility ofeasy / free-ridingand thereby ensure
thatacoalition is willingtobearthe …xed costs ofpoliticalparticipation. O nce
group size becomes large it is conjectured that actions will not be considered
pivotaland thus thateasy/ free-ridingwillensurethegroup remains latent.
T akingtheregulatoryexampleabove, this theoryoflobbyformation does indeedo¤ eraplausibleexplanation as towhygroups representingregulatedindustries lacklobbyingcompetition. Firstly, industrygroups oftenhaveapre-existing
structure (trade, shareholderorunion associations) and are smallerin size, implying thatthe …xed organisationalcosts they face are likely to be lowerthan
those oftheirobvious competitors, consumers. T his …xed costdi¤ erentialobviously increases the likelihood that: a) the industry group faces an aggregate, if
notper-member, netgainfrom politicalparticipationandb) theconsumergroup
faces an aggregatenetloss thatensures itis always latent. Secondly, in theevent
thatbothgroups faceanaggregatenetgain, thelargersizeoftheconsumergroup
increasestheprobabilitythatfree-ridingwillprohibittheformationofaconsumer
butnotofaproducerlobby. In otherwords group sizeand permembernetbene…ts combinetosuggestthatthe consumergroup willprovetobelatentbutthe
industrygroup intermediate.
H owever, whilstthe above logic fares wellas an explanation ofpatterns of
regulatorygroup formation, ithasfarlesspredictivepowerinotherpolicyspheres.
In particular the examples cited above pose two worrying questions. Firstly,
as D ixit(19 9 6) has noted, O lsonian theories actually supportthe existence of
an active industrialsugarusers group as wellas a farming lobby; like wise for
pro-exportand downstream industrylobbies in addition toactive protectionists.
P utverysimplyallthesegroups represent‘industryinterests’, implyingthatthe
playing …eld should level, both in terms ofthe likelihood of an aggregate net
gain and any problems stemmingfrom free-riding. O fcourse itis possible that
su¢ciently greatdi¤ erences remain toexplain the observed lack ofcompetition
but, exante atleast, O lsonianlogicseemstohavelostitsbite. Secondlyandmore
worryingly, inadditiontothelackofcompetitionfornon-tari¤ barriers, G awande
(19 9 7 ) observedthat“lobbyingbyexport-orientedindustries seems tobedirected
3
atinstruments thatfacilitateexports ratherthan anti-protectionism”1 . Ifrobust
this resultsurelyquestions thevalidityofthewholeO lsonian approach. For, ifa
group has …nanced the …xed costs associated with collective participation in one
policydimension whywould itremain silentin another?
T heaboveexamples thereforesuggestthatreceivedtheoryo¤ ers us aninsight
intothe issue ofpassive consumergroups, butcrucially notpassive industry interests. T hus the obvious question, and hence the subjectofthis paper, is why
dosomeindustry interestgroups participateinthepolicyprocess whilstothers of
similarsize and permembernetbene…ts donot;alternatively, as commentators
haveoften asked, ‘wherewas Coca-Colain thedebateoversugarprices?’
W e attemptto shed lighton the empiricalpuzzle posed by ‘passive industry groups’ by re-formulating current thinking in two ways. Firstly, we focus
attention away from the exogenous asymmetries thatdrive O lsonian theories of
group formation. T hatis, sinceweseektoexplain whyvariation in participation
arises even when ‘the playing…eld is level’in O lsonian terms, we remove exogenous asymmetry by normalising…xed organisationalcosts tozero2 . In contrast
toexistingmodels, then, ourmodelingapproachfocuses oneachindustrygroup’s
abilitytoraisevariable politicalin‡uencecosts, and in particularthosecosts associated with lobbyingby virtue ofdirecttransfers topolicy makers. Similarly,
ratherthan lookingtoexogenous di¤ erences in potentialmembership toexplain
variations information across intermediate groups, weallowforthepossibilityof
groups ofapproximatelyequalsize.
Secondly, we note thatthe totalnumberofstakeholding …rms, orpotential
benefactors, across allindustrygroups is actuallyoftenlarge. Forinstanceinthe
tradeexampleabovepotentialcontributors toany group spanned allpro-export,
downstream andprotectionist…rmsintheU S.T hus, whilstanyoneindustrygroup
mightnotbe considered large, the population, orratherthe expected numberof
all…rms withapolicystake, almostcertainlywillbe. D rivenbythis observation,
weinvestigatethedecisiontocontributetowards variablepoliticalin‡uencecosts
byadoptingrecentdevelopments in games with alargenumberofplayers.
W hen modellinginteractions in alargepolity, ones notices thatthe standard
assumption thatevery playertakes otherplayers’behaviouras given and known
1
G awande(19 9 7 ), pp. 6
5.
N ote the suggestion here is simply that, across the groups we wish toconsider, …xed costs
mightbe similar. In normalisingthese costs tozerothe implicitsuggestion is eitherthatsuch
organisationalcosts have been borne foralternative purposes (e.g. union formation), orthata
su¢cientresidualremains from otheractivities tocoverfurthercosts (e.g. hiringprofessional
lobbyists).
2
4
seems implausible. O r, as M yerson (19 9 8 ) puts it, in games with a very large
number of players ‘it is unrealistic to assume thatevery player knows all the
other players in the game; instead, a more realistic modelshould admitsome
uncertainty aboutthe number ofplayers in the game.’ In factM yerson (19 9 8 )
goes even further, proposing the use ofso-called L arge P oisson G ames as more
realisticmodelingdevices ofsuch situations. W e therefore followhis suggestion
and modelthe information structure underlying the contribution decision as a
L argeP oisson G ame.
In essence we showthat, in the absence ofex ante asymmetries, adoptinga
L argeP oissonG ameapproachtoendogenouslobbyformationdoesindeedo¤ eran
insightintothe phenomenon ofpassiveindustry groups. D rawingon established
results from the literature on policy menu-auctions we demonstrate thatitcan
never be an equilibrium formore than one industry lobby toattempttosolicit
policy favours. M ore intuitively, we showthatwhen the totalnumberofpotentialcontributors across allgroups is very large the free-riderproblem becomes
insurmountable;no group is abletoraise su¢cientcontributions too¤ erstrictly
positivetransfers. H owever, ratherthan eliminatingin‡uenceseekingaltogether,
asO lsonconjectured, thisleavesthepoliticalmarketplaceopentogroupsthat‘do
notneed topay’. In otherwords tolonelobbies thatdonotneed tocompensate
legislators fortheirpolicychoices. Inthis sense, then, industrylobbyingbecomes
purely counteractive; a single group may choose to enterthe policy process to
ensurethatparticipation becomes acostlyactivityforits competitors.
T heremainderofthepaperis organisedas follows. T henextsectiondiscusses
the relationship between ourapproach and othermodels ofthelobbyingprocess
in more detail. Section 3 describes the details ofourmodel. Sections 4 and 5
presentresults from the common agencypolicy selection gameand the standard
/ largeP oisson contribution games respectivelyand Section 6concludes.
2. R elated L iterature
T his paperrelates directly tothree strands ofliterature: a)‘intra-group’models
ofthestrategicinteractionbetweenthemembers ofanyonegroup overindividual
contributionstowardstheprovisionofcollectivelobbyingactivity;b) ‘inter-group’
models ofthe strategicinteraction between agiven numberoflobbies, each able
to …nance the costs ofpoliticalparticipation; c) fully endogenous models that
seektoderive simultaneously an equilibrium policy choice and equilibrium level
oflobbygroup participation.
5
In all of the ‘intra-group’ models the …nal policy choice is seen as a publicgood. Such models therefore focus on the extenttowhich free-ridingcauses
under-provisionofthispublicgood.3 Ineverycasetherelationship betweengroup
resources and the supply ofthe publicgood, orpolicy favour, is determined exogenously and thus allsuch models lack a micro-founded theory ofendogenous
policyselection.
T he…rstandcertainlythemostin‡uentialcontributioninthis…eldwas O lson
(19 6
5). A bstracting from the e¤ ectofgroup competition on individualcontributions, O lson suggested that(own) group sizeplays an importantrole in group
formation. In particularO lson conjectured thatindividuals in su¢ciently small
non-privileged groups could regard theiractions as pivotaland thus thatcollective action was a possibility; whilst“by contrastin a large group in which no
single individual’s contribution makes aperceptible di¤ erence...itis certain that
a collective good willnot be provided [voluntarily]
”4. In one sense, then, our
papercon…rms O lson’s conjecture thatfree-ridingmay inhibitgroup formation.
H owever, ourpointofdepartureis tohighlightthattheendogeneityofthepolicy
choiceactuallyimplies thatasinglelobbymaystillform even ifallcontributions
areconsidered non-pivotal.
A lso relevant in this contextare ‘inter-group’ models ofendogenous policy
selection thatabstractfrom collective action problems and therefore treatthe
numberandidentityofcompetinglobbies as given. T wobasicapproaches canbe
distinguished. T he …rstsetofmodels - the common agency approach - can be
seen as an attempttodescribe lobbyingunder(policy maker) certainty. Followingpioneering work by B ernheim &W hinston (19 8 6
), such models assume that
competing groups simultaneously o¤ erthe policy makera ‘menu’ ofbids in an
attempttosway policyin theirfavour.5 In contrast, in thesecond setofmodels
- thestrategicinformation transmission approach - participants faceuncertainty
3
Importantcontributions tothe literature on the private provision ofpublic goods include
O lson (19 6
5), Chamberlain (19 7 4), P alfrey & R osenthal(19 8 5), B ergstrom, B lume & V arian
(19 86) and B agnoli & L ipman (19 89 ). For interesting applications to lobbying activities in
particularsee Stigler(19 7 4), A usten-Smith (19 8 1) and M agee, B rock& Y oung(19 8 9 ).
4
O lson (19 6
5) pp.44.
5
T he application ofthe menu-auction / common agency approach to models ofeconomic
in‡uenceseekingwaspopularisedbyG rossman& H elpman(19 9 4, 19 9 6) inapplicationstotrade
policy and electoralcompetition respectively. O fthe burgenoing literature thathas followed
D ixitetal(19 9 7 ) provideaveryusefulequilibrium characterisationtheorem;aresultwhich has
been used by B esley & Coate (19 9 7 a) tointroduce lobby action intoa modelofrepresentative
democracy.
6
overpolicy outcomes. Consequently interestgroups become competing‘experts’
or‘advocates’;incurringinformationacquisitioncosts tosubsequentlyo¤ erpolicy
advice6. Clearly, given thatourintention in this paperis toabstractfrom …xed
costs, we restrictattention to the …rst- common agency - approach to policy
selection.
T hestrategicinformationliteraturedoes howeverhavesomeinterestingimplications forgroup formation. In particular, abstractingfrom anycollectiveaction
problems, A usten-Smith & W right(19 9 2) notethat“legislators areoftenlobbied
byjustoneoftwocompetinggroups”7 and hence they concludethat“groups do
notalways compete by clashingdirectly overasingle legislator’s vote”8 . In contrasttoourresults, however, theprediction is thatsuch lobbyingwillnotalways
becounteractive. Inparticular, thesuggestioninthatpaperisthatitisthegroup
thatdisagrees withthelegislator’s votingpredisposition thatwilltypicallylobby
alone.
T he…nalclass ofmodels thatare relevantin this contextarethosethatseek
toendogeniseboththeprocess ofpolicyselectionandlobbygroup formation. T o
dateonlyM itra(19 9 9 ) has attemptedtocombineafullymicro-founded modelof
policyselectionwithanindividualcontributiongame9 . InfactM itra’smodelmay
be seen as mutually compatible with ourown. T hatis, whilstboth approaches
precede the benchmark common agency modelwith an individualcontribution
game, the focus ofM itra’s work is on the ability ofdi¤ erentinterestgroups to
coverthe…xed ratherthan variablecosts associated with lobbyingactivities.
B y assuming, …rstly thatpre-play communication resolves O lson’s ‘indeterminacy’when aggregate, butnoindividual, bene…texceeds the costs associated
withpoliticalparticipation, andsecondlythat“oncethelobbyisformed, thelobby
machinerycan enforceperfectcoordination amongmembers ofthatgroup in the
collectionofpoliticalcontributions”10 M itraensuresthatplayerscancrediblycom6
See in particularA usten-Smith & W right(19 9 2). Contributions in a similarvein include
A usten-Smith (19 9 3), L ohmann (19 9 8 ), D ewatripont& T irole (19 9 9 ) and Krishna & M organ
(19 9 9 ).
7
A usten-Smith & W right(19 9 2) pp. 230.
8
A usten-Smith & W right(19 9 2) pp. 245.
9
Since the …rstversion ofthis paperwas written a numberofotherpapers have addressed
the issue ofendogenous lobby formation. Contributions include D amania& Fredrikson (2000)
whoalsomodelthe policy selection stage as a menu-auction butfocus on the implications for
free-ridingin arepeatedgame;andFelli & M erlo(2000) whoendogeniselobbyformation inthe
contextofthecitizen candidatemodeldeveloped byB esley& Coate(19 9 7 a).
10
M itra(19 9 9 ) pp. 1122.
7
mittomakingcontributions. T hus, byensuringthatitis onlythosegroups that
facean aggregatenetloss thatremain latent, M itrais abletoestablish aunique
equilibrium: only thosegroups with …xed costs less than an endogenously determinedcriticalvalueform. T hus, sothisstorygoes, pro-exportersanddownstream
…rms failed tolobbyovernon-tari¤ barriers and industrialsugarusers keptsilent
oversugarprices, because their…xed costs ultimately proved toohigh. Itwould
appear, then, thatthe relative merits ofa variable ratherthan a …xed costapproach toindustry group participationis an empiricalquestionand is onethatis
lefttofutureresearch.
3. T he M odel
O urpolity is populated by N agents11 each endowed with y units ofa private
good. Each agentis of‘type’t2 T = f1;2 g, where tdenotes the preference for
apublicpolicy g 2 G = f1;2 g:L etr(t) > 0 denote the …xed proportion ofeach
typein thepolity. W edenotethegross utilityofarepresentativeagentoftypet
byU (g;t;y) = ut(g)+ y; whereut(g) = H t> 0 ift= g andut(g) = 0 ift6
= g:W e
assumethatr(t); ut(g); G and thetypesetT areallcommon knowledge.
W ede…ne‘lobbyingactivity’as anattempttoin‡uencethechoiceofg 2 G by
virtueofnon-negativetransferstothepolicymakingauthority. W ewillconsidera
group tohaveformedalobbyifandonlyifsu¢ cientcontributions of y havebeen
raisedfrom individualagentsforthelobbytodemonstrateanabilitytoo¤ erstrictly
positive transfers tothe policymaker. T henumberofsuch lobbies is given byL,
whereL ·2 andisendogenouslydetermined. T oavoidunnecessarycomplications
weassumethepre-existenceoftwo‘manifestos’, indexed byt= f1;2 g; such that
itis common knowledge thata lobby with manifestotwill, by virtue ofdirect
transfers tothepolicymaker, seektosecurethepolicychoicefavoured byagents
oftypet(i.e. g = t), regardless oftheactualdistributionofitsbenefactors’policy
preferences.
W e denotethenumberofcontributions ofy toanylobby tby x(yt) and thus
its gross utility by U (g;t;y;x(yt)) = x(yt)ut(g) + x(yt)y:T hus, given the above
de…nition oflobby formation, we consideragroup tohaveformed alobby ifand
onlyifx(yt) > 0 :D enotingthesetoflobbies byL, thereareclearlyfourpossible
lobbyingcon…gurations: nolobbies, L= f;g; ifx(yt) = 0 8 t= 1;2 ; only lobby
11
Since ourfocus is competition between industry groups the appropriate unitofanalysis is
the individual…rm. W e usetheterm agenttore‡ectthegeneralityofourapproach.
8
1, L= f1g; ifx(y1 ) > 0 and x(y2 ) = 0 ;only lobby 2, L= f2 g; ifx(y1 ) = 0 and
x(y2 ) > 0 ; orlobbyingcompetition L= f1;2 g ifx(yt) > 0 8t:
T he…nalplayerinourmodelisapolicymakerresponsibleforthepolicychoice
g 2 G = f1;2 g:Intakingthis decisionweassumethatthepolicymakerconsiders
boththeutilityshederives from thepolicyoutcomeitself12 , up (g), as wellas any
conditionaltransfers from the L lobbies, which we denote by fB t(g)gg2G. T he
policymaker’s objectivefunction is thereforegiven by
¼ p = up (g) +
X
B t(g);
(3.1)
t2L
where, withoutany loss ofgenerality, we assume up (1) = H p ¸ up (2 ) = 0 , implyingthatthe policy makershares a preference ordering with lobby 1.13 Consequently, in the absence oftransfers, we assume thatthe policy makeralways
implements g = 1:
In lightoftheabove, itshould be clearthattheremainingaction spaces and
pay-o¤ s are as follows. Every lobby t2 L(with resources x(yt)y available for
in‡uenceseeking) mustdecideon ascheduleofpayments thatitwillo¤ ertothe
policymakerinanattempttosecureapolicychoiceg = t. W edenotethis choice
by B t(g) 2 B t = [0 ; x(yt)y] 8 g 2 G, where the bounds on the feasible setB t
re‡ectthefactthat, byassumption, nolobbycan o¤ ermorethan ithas received
from individualagents ordemand payments from the policy maker. L obby t’s
objectivefunction is thereforegiven by
¼ t = U (g;t;y;x(yt)) ¡B t(g):
(3.2)
Each individualagentmustdecide between contributingtheirendowmenty
tolobby1; lobby2 and notcontributingatall. W edenotethis choicebyc2 C =
fy1;y2 ;;g; the resultingoutcome, oraction pro…le, by the vectorx= fx(c
)gc2C
14
andthesetofpossibleactionpro…lesbyZ (C ) , wherexdeterminesL. A ssuming
thatagents areaware thatthelobbies willreturn an equalshareofany ‘unused’
12
Such preferences couldeasilybeexplainedbyvirtueoftherepresentativedemocracymodel
ofB esley& Coate(19 9 7 a).
13
Inanattempttopreservegeneralitywemakenoassumptions concerningrelativepreference
intensity between any ofthe players. In particularwe considera policy makerwith nopolicy
preferences (i.e. Hp = 0 )as a specialcase. In whatfollows we willmake extensive use ofthe
notion ofa lobby opposing the policy maker’s preferences. N ote this refers only to the case
whereHp > 0 :
14
T oaidtheexpositionoftheL argeP oissonG amewefollowM yerson’s (19 9 7 , 19 9 8) notation
throughtoutourdiscussion ofthecontribution stage game.
9
contributions, orx(yt)y¡B t(g), totheirbenefactors15 , theobjectivefunction for
arepresentativeagentoftypetis given by,
¼ t= ut(g(c
;x¡ )) + y¡Á(c
;x¡ );
where
8
>
>
<
Á (c
;x¡ ) = >
>
:
(3.3)
9
ifc= y1 >
>
=
ifc= y2 >;
>
;
0 ifc= ;
1
B (g(c
;x¡ ))
x(y1 ) 1
1
B (g(c
;x¡ ))
x(y2 ) 2
x¡ is the action pro…le of the other players and Z (C )¡ is the set of possible
correspondingaction pro…les.
A tthis pointwealsoimposethestabilitycondition
y ¸ m ax[H t];
t2f1;2 g
(3.4)
which simply says that endowments are such that every individual agent can
contribute an amount greaterthan orequalto their‘willingness to pay’ fora
policy favour. Itis ofinterestto note thatsuch an assumption willalways be
necessarytoruleoutnon-existenceofequilibriain menu-auctions when payment
schedules arebounded from aboveas wellas below.
From aboveitshould beclearthattherearetwostages, orsub-games, tothe
lobbying process. T he …rstis the contribution stage in which every individual
agent decides which lobby, if any, to contribute to. T he second is the policy
selection stage in which every lobby, that has formed in the …rst stage game,
selects a payment schedule in an attemptto in‡uence the policy choice. O ur
aim, then, is to establish equilibrium lobbying con…gurations Lo16.T hatis, the
numberand identity ofindustry groups able toraise su¢cientcontributions to
demonstrate an ability to o¤ erequilibrium directtransfers ffB to(g)gg2Gg. W e
establish such equilibriabyanalysingthetwostages in reverseorder.
15
O bviouslythenotionthatlobbiesreturn‘unused’contributionsisnotuniversallyapplicable,
although ithas been usedbyB agnoli & L ipman (19 8 9 ) andFelli & M erlo(2000) amongothers.
N ote thatourrefund assumption is formally equivalenttothe casewhere, exante, benefactors
(credibly) committomakingacontribution and thus, expost, onlymakethenecessaryequilibrium payment. R ealworld examples ofsuch amechanism abound. Forinstance, closetohome
a scheme called on the residents ofKingsdown, B ristol, topledge a willingness topay towards
acampaign againstparkingrestrictions.
16
T hroughoutwe usethe supercriptotodenote an equilibrium value.
10
4. P olicySelection Stage G ame
W e follow B ernheim and W hinston (19 86
), and numerous others, in modelling
in‡uence seekingas atwostage menu-auction. W e assume then thatthe policy
selection gamebegins with each ‘formed’lobbyt2 Lchoosingapaymentscheduletomaximisegroup utility, takingas given similaroptimisingbehaviourbythe
otherlobbies and anticipatingan optimisingpolicychoice. Itis completed when,
inlightofallthese‘simultaneousandnon-cooperativeo¤ ers’, thepolicymakerselects apolicyg o thatmaximisesherutility. A nequilibrium ofthis policyselection
game is therefore any pair[go;ffB to(g)gg2Ggt2L (x)] thatsatis…es the requirement
that: a) foreachlobbyt, fB to(g)gg2G is optimal(maximising3.2 above) giventhe
o¤ ers fB so(g)gs2L(x)=ftg ofthe otherL ¡t lobbies17 and the policy choice induced
byallL o¤ ers and b) go is optimal(maximising3.1) given allsuch L o¤ ers.
Since, forthepurposes ofthecontributionstage, werequirethepolicychoices
and actualpayments made in all such equilibria we proceed as follows. U sing
the equilibrium characterisation theorem forcommon agency games from D ixit,
G rossman and H elpman (19 9 7 ) we …rstexamine the two cases when only one
lobbyforms beforeproceedingtothetwolobbyscenario.
A dapted toourmodelthecharacterisation theorem yields:
P roposition 4.1. (D ixit, G rossman, H elpman19 9 7 ) A vectorofpaymentschedules ffB io(g)gg2Ggt2L (x) and a policy choice goconstitute an equilibrium ofthe
policyselection gameforanyL 2 f1;2 g ifandonlyif:
(a) forallt2 L:
B to(g) 2 B t 8g 2 f1;2 g;
(b)
go = arg m ax [up (g) +
g2f1;2 g
X
t2L
B to(g)] 8g 2 f1;2 g;
(4.1)
(4.2)
(c) foreveryt2 L, [go ;B to(go)] solves thefollowingproblem:
max
x(yt)ut(g) + x(yt)y¡B t
(g;B t)2G£Bt
17
N otethatL
¡t = # L=ftg:
11
(4.3)
subjectto
up (g) +
X
s2L=ft]
0
B so(g) + B t ¸ 0m ax [up (g ) +
g 2f1;2 g
X
0
B to(g )]
(4.4)
s2L=ft]
Explanation18 : Condition (a) simply corresponds to the requirementthat
paymentschedules mustbefeasible, whilecondition(b) follows from optimisation
bythepolicymaker. Condition (c) is morecomplicatedandresults from viewing
each lobby’s ‘bestresponse’calculation as aconstrained maximisation problem.
In essence, each lobby tis aware thatifthe policy makeris tochoose apar0
ticularpolicy g she musthave noincentive todeviate toheroutside option g ;
0
where g is the policy she would choose iflobby to¤ ered nothingand the other
L ¡t lobbies maintained theirequilibrium paymentschedules fB so(g)gg2G. H ence
condition4.4. T herefore, iflobbytsatis…es this constraint, itknows thatite¤ ectivelyhas thepolicymaker‘on side’andthus thatitcan proposeapolicygo and
correspondingpaymentB to(go) thatmaximises its own utility. H ence condition
4.3.
4.1. Equilibriawith O ne L obby
From proposition 4.1 wehave,
P roposition 4.2. IfL= f1g thengo = 1 is theuniqueequilibrium policychoice
and B 1o (1) = 0 is theuniqueequilibrium payment.
Explanation: Sincepreferences arecommon knowledgelobby1 knows that,
intheabsenceofcompetitionfrom anothergroup, thepolicymakerwillsetgo = 1
evenifitmakes azerocontribution. Inotherwords, regardless ofits size, lobby1
…nds itcan‘freeride’onthebackofsharingapreferenceorderingwiththepolicy
maker.
P roposition 4.3. IfL= f2 g then:
(a) Ifthe policymakerhas ‘stronger’policypreferences than lobby2 (i.e. H p >
x(y2 )H 2 > 0 ); theng o = 1 is theuniqueequilibrium policychoiceandB 2o(1) = 0
18
W eo¤ erverbalexplanations inplaceofformalproofs inthetext. Suchproofs maybefound
in theappendix.
12
is theuniqueequilibrium payment;
(b) Ifthepolicymakerandlobby2 have‘equal’policypreferences(i.e. x(y2 )H
H p > 0 ); theneithergo = 2 and B 2o(2 ) = H p org o = 1 and B 2o(1) = 0 ;
2
=
(c) Ifthepolicymakerhas‘weaker’policypreferencesthanlobby2 (i.e. x(y2 )H 2 >
H p ¸0 ), then g o = 2 is the uniqueequilibrium policy choiceand B 2o(2 ) = H p is
theuniqueequilibrium payment.
Explanation: Sincepreferences arecommon knowledgelobby2 knows that,
intheabsenceofcompetitionfrom anothergroup, apolicymakerwillsetg = 1 if
itmakes azerocontribution. Ittherefore alsoknows thattosecure its preferred
outcome itmusto¤ ersuch apolicy makeran amountthatatleastcompensates
herforany utility lostby choosingg = 2 ratherthan g = 1; thatis H p . Clearly,
in equilibrium, this paymentwillonlybemadeifitis lowerthan orequaltothe
lobby’s ‘willingness topay’forthepolicyfavour, orratherH p ·x(y2 )H 2 . Should
thegroup provetoosmalland/orpreferenceintensitytoo‘weak’theequilibrium
paymentwillclearlybezeroand g = 1.
4.2. Equilibriawith T woL obbies
A sB esleyandCoate(19 9 7 b) note, whilstadiscretepolicychoiceensuresaunique
equilibrium policy choice, the equilibrium characterisation theorem given above
fails totie down equilibrium paymentschedules when the numberofprincipals
is greaterthan one. T hus, since in this contextwe mustspecify every policy
choice and equilibrium paymentthatany agentmightexpectwhen makingher
contribution decision, weinvokethe‘truthfulness re…nement’forcommonagency
games developed byB ernheim and W hinston (19 8 6
)19 . A de…nition ofaglobally
truthfulpaymentscheduleis given below:
D e…nition 4.4. (D ixit, G rossman and H elpman (19 9 7 )). A feasible payment
schedule Bet 2 B t forlobbytis said tobetruthfulrelativetotheconstantuot if
19
Bet(g) = m ax[0 ; m inf(x(yt)ut(g) ¡uot); x(yt)yg] 8g 2 f1;2 g;
Itis nowstandard tojustify the use ofthe‘truthfulness’re…nementby noting: …rstly that
truthfulstrategies are extremely simple and thus possibly ‘focal’; and secondly thattruthful
equilibria are both e¢ cient (maximising the sum ofgross utilities) and coalition-proof. See
B ernheim & W hinston (19 8 6
). H owever, truthfulstrategies very often resultin complex payments schedules. A n alternative set ofstrategies - so-called naturalstrategies - thatdo not
exhibitthis problem have been suggested by Kirchsteiger& P rat(19 9 9 ). N evertheless, in the
caseofabinary policychoice, truthfuland naturalequilibriacoincide.
13
whereuot = x(yt)ut(go ) ¡B to(go) orequilibrium utilitynetofanypayment20 .
Itshould thereforebeclearthatreplacingcondition 4.1 with
B to(g) = Bet(g) 8g 2 f1;2 g
in proposition 4.1 above yields atheorem (hereafterproposition 4.10) which may
be used to characterise allthe truthful equilibria ofthis policy selection game
when L = 2 .
W e…rstestablish twousefullemmas.
L emma4.5. Inanytruthfulequilibrium Beto(t) ¸ Beto(s) witht6
= s 2 f1;2 g:
Explanation: N o lobby willo¤ ermore forits worstoutcome than forits
bestoutcome. Itshould be clearfrom above that‘globaltruthfulness’ implies
thatlobbies o¤ ertheirtrue (net) willingness topayforanypolicychoicerelative
tothe equilibrium policy choice. T herefore, ifa lobby anticipates thatits best
(worst) option willbe chosen in equilibrium, itcannoto¤ ermore (less) forthe
remainingoption than itwillpay in this equilibrium. H ence Be1o(1) ¸ Be1o(2 ) and
Be2o(2 ) ¸ Be2o(1):
L emma4.6
. Inanytruthfulequilibrium ofthepolicyselectiongame, whenL =
2 ; Beto(s) = 0 witht6
= s 2 f1;2 g:
Explanation: N olobby willo¤ erapositive paymentforits worstoutcome.
Since preferences are common knowledge, lobby 2 knows thatthe policy maker
willsetg = 1 ifitmakes a zerocontribution and thus thatitmustprovide the
policy makerwith atleastthe utility she receives underthis outside option if
g = 2 is evertobe chosen. Clearly positive payments forg = 1 (lobby 2’s worst
outcome) simply serve to increase the paymentneeded to realise g = 2 (lobby
2’s bestoutcome). T he same logicapplies iflobby 1 anticipates thatthe policy
makerwillsetg = 2 when itmakes azerocontribution.
Incontrast, iflobby1 anticipates thatthepolicymakerwillsetg = 1 whenit
makes azerocontribution, thenitknows thatitcane¤ ectively‘freeride’ando¤ er
azeropaymenteven forits bestoutcome (g = 1). T hus, since allpayments are
N otethatequilibrium payments aretruthfulby construction and thus B to(go)´ Bet(go):A
proofofsuch ‘localtruthfulness’is given in D ixit, G rossman & H elpman (19 9 7 ).
20
14
‘truthful’, weknowlobby1 mustalsoo¤ erazeropaymentforits worstoutcome
(g = 2 ).
W e are nowin a position toestablish ourcentralproposition forthe policy
selection stage game concerning possible equilibrium policy choices and correspondingequilibrium payments:
P roposition 4.7 . IfL= f1;2 g then:
(a) Ifthe policy makerhas ‘stronger’policy preferences than lobby 2, orifthe
policymakerandlobby2 have‘equal’policypreferences(i.e. H p ¸x(y2 )H 2 > 0 );
theng o = 1 is theuniqueequilibrium policychoiceand Be1o(1) = 0 ; Be2o(1) = 0 are
theuniqueequilibrium payments;
(b) Ifthepolicymakerhas‘weaker’policypreferencesthanlobby2 (i.e. x(y2 )H
H p ¸0 ), then:
2
>
(i) Iflobby1 is more ‘powerful’21 (i.e. x(y1 )H 1 > x(y2 )H 2 ¡H p ); then go = 1 is
the unique equilibrium policy choice and Be1o(1) = x(y2 )H 2 ¡H p ; Be2o (1) = 0 are
theuniqueequilibrium payments;
(ii) Iflobby 2 is more ‘powerful’(i.e. x(y1 )H 1 < x(y2 )H 2 ¡H p ); then go = 2 is
the unique equilibrium policy choice and Be1o(2 ) = 0 ; Be2o(2 ) = x(y1)H 1 + H p are
theuniqueequilibrium payments;
(iii) Ifthelobbies areequally‘powerful’(i.e. x(y1 )H 1 = x(y2 )H 2 ¡H p ); then
eithergo = 1 and Be1o(1) = x(y2 )H 2 ¡H p (y1 ); Be2o (1) = 0 ,
orgo = 2 and Be1o(2 ) = 0 ; Be2o (2 ) = x(y1 )H 1 + H p aretheequilibrium payments.
Explanation: T he intuition behind this resultis as follows. Considerthe
case ofapolicy makerwith ‘stronger’policy preferences than lobby 2 and begin
bysupposingthatgo = 2 :Sincepreferencesarecommonknowledgelobby2 knows
thatwhen go = 2 lobby 1 willhave o¤ ered the policy makerits true willingness
to pay (x(y1 )H 1 > 0 ) forg = 1. L obby 2 also knows thatifito¤ ers nothing
forg = 2 the policy makerwillchoose g = 1; derivingautility ofH p + x(y1 )H 1
underthis ‘outsideoption’:U singthis informationlobby2 thereforeknows thatit
musto¤ erthepolicymakerx(y1 )H 1 + H p ifgo = 2 is indeedtobeanequilibrium
policy choice. Clearly lobby 2 willonly be prepared tomake this paymentifit
is lowerthan orequalto the group’s bene…tfrom the policy favour, implying
21
Inthis contextalobby’s ‘power’referstoits willingnesstopaywithanadjustmenttore‡ect
the policy maker’s preferences.
15
x(y2 )H 2 ¸x(y1 )H 1 + H p :H owever, the factthatlobby 1 has formed is evidence
thatx(y1)H 1 > 0 andthus wecanbecertainthatlobby2 willnotbepreparedto
make the necessarypaymentsinceH p > x(y2 )H 2 :Ittherefore follows thatg = 2
cannotbeanequilibrium policychoicewhenthepolicymakerhas‘stronger’policy
preferences than lobby2.
O uralternativeis thus tosupposego = 1:A gainsincepreferencesarecommon
knowledge lobby 1 knows thatwhen go = 1 lobby 2 willhave o¤ ered the policy
makerits truewillingness topay(x(y2 )H 2 ) forg = 2 . L obby1 alsoknows thatif
ito¤ ers nothingforg = 1 the policy makerwillchoose g = 1, derivinga utility
ofH p underthis ‘outsideoption’. U singthis informationlobby1 thereforeknows
thatitcan o¤ erthe policy makernothingand stillsecure an equilibrium policy
choiceofg o = 1 since H p > x(y2 )H 2 . Finallywecan be certain thatlobby 1 will
beprepared tomakethenecessarypaymentsincex(y1 )H 1 > 0 when lobby1 has
formed.
Ittherefore follows thatgo = 1 is the unique equilibrium policy choice and
o
e
B 1 (1) = 0 ; Be2o(1) = 0 aretheuniqueequilibrium paymentswhenthepolicymaker
has ‘stronger’policypreferences thanlobby2. N otethattheintuitionbehindthe
remainingresults is exactlyanalogous.
Following this logic we therefore see thatthe policy selection game, forany
givennumberoflobbies, has auniqueequilibrium ineverycaseexceptx(y2 )H 2 =
x(y1)H 1 + H p . For the purpose ofthe contribution stage game, we make the
(purely) simplifyingassumption thatin such an eventagents anticipate thatthe
equilibrium withthepolicymakerchoosingher‘default’policy(g = 1)willprevail.
Finally itwillprove usefultoemphasise the followingresults which, in part,
drive the main result of our paper. T hat is, we have shown that any lobby
thatdoes notoppose the preference orderingofthe policy makerwillnever pay
tosecure apolicy favour, unless itcompetes forthatfavourwith anothergroup
exhibitingstrongerpreferencesthanthoseofthepolicymaker. Incontrastalobby
thatopposes the policy maker’s preferences willalways face a costtopersuade
theauthoritytoimplementits preferred policy.
5. Contribution Stage G ame
A rmedwith theresults ofthepolicyselectiongamewecannowmovetothe…rst
stage ofthe lobbying process, orcontribution stage, in which every individual
agentdecides whethertocontribute theendowmenty tolobby 1; lobby 2 ornot
to contribute atall. In otherwords, having established the equilibrium policy
16
choicego and payments fB to(go )gt2L(x) foreveryaction pro…lex2 Z (C ); wenow
analyse the contribution decision c2 C = fy1 ;y2 ;;g:A ssumingallagents choose
ctomaximise3.3above, wemayestablishequilibrium actionpro…les xo andthus
completeourcharacterisation ofallpossibleequilibrium lobbyingcon…gurations:
Instandardvoluntary-contributiongames itis assumedthatthetotalnumber
ofplayers N in any game is common knowledge, and thus by implication that
every playertakes otheragents’ behaviour(i.e. x¡ ) as given and known when
contemplatingherbestresponse. In any modern polity, however, the totalnumberofpotentialbenefactors across all groups willoften be large - an assertion
certainly borneoutin the trade examples cited in theIntroduction. A s aresult,
theassumptionthatplayersconditiontheirstrategiesonx¡ beginstolookunrealistic. W ethereforefollowM yerson’s(19 9 8) suggestionandmodelthecontribution
stagegameasaL argeP oissonG ame. A consequenceofthisapproach, asM yerson
(19 9 7 , 19 9 8) has shown, is thatplayers form ‘beliefs’overx¡ whencontemplating
theirbestresponseandthus weestablishanequilibrium actionpro…lexo whenit
is indeed optimalforeveryplayertobehaveaccordingtothese‘beliefs’.22
N evertheless, itwillaid theunderstandingoftheL arge P oisson Contribution
G ametobearinmindtheimplications ofmodellingthecontributiongameunder
theconventionalassumptionthatN is commonknowledge. T henextsub-section
therefore brie‡y discusses the standard voluntary-contribution game adapted to
ourmodel.
5.1. Standard V oluntary-Contribution G ame
W hen N is common knowledge, players are aware ofevery possible elementx¡
ofthe setZ (C )¡ . Each playertherefore resolves theircontribution decision by
calculating which elements ofC = fy1;y2 ;;g constitute a bestresponse to any
action pro…lex¡ . Formally, each playersolves theproblem
h
i
m ax ¼ t(c
;x¡ ) ´ut(g(c
;x¡ )) + y¡Á(c
;x¡ ) 8x¡ 2 Z (C )¡ :
c
2C
(5.1)
In such agame we assertthatany action pro…le xfrom which noplayerhas an
incentivetodeviateifnootherplayerdeviates is anequilibrium actionpro…lexo:
In such aframework, itcan be shown aftersome tedious butstandard calculations, thatatleastone pure strategy N ash equilibrium exists foreach possible
22
Inthis respect, ourapproach tolobbyformation echoes thatofA usten-Smith(19 8 1). H owever, inthatpaper, ‘beliefs’overx¡ wereimposedapriori andthus lackedtherequirementthat
thesebeliefs should be consistentwith equilibrium.
17
lobbycon…guration. Intuitively, this resultrests on thefactthatagents havetwo
possible motives fordeviation. Firstly, ifthe equilibrium policy choice is ‘uno
favourable’, orgo 6
= t, thereis an incentivetodeviatetoanotheraction c6
= c
if
o
this alternativechoiceis pivotalinsecuringg = tandresults inahighernetpayo
o¤ givenÁ (c
;xo¡ ):Secondly, iftheequilibrium choiceis ‘costly’, orÁ(c
;xo¡ ) > 0 ;
o
thereis an incentivetodeviate toanotheraction c6
= c ifthis alternativechoice
is less costly and results inahighernetpayo¤ givenut(go):N otethatthesecond
motiveincludes theincentivetofreerideas aspecialcase;thatis, whentheequio
librium policychoiceis ‘favourable’, org = t, butanotheractionc6
= c
results in
o¡
Á(c
;x ) = 0 :
W ith this logic in mind itshould be clearthatthe results ofthe policy selection game allowus to distinguish between two classes ofequilibria. O n the
one hand, there are those in which agents perceive an alternative action to be
pivotalyetresultin a lowernetpay-o¤ . A lternatively, there are those in which
equilibrium actions areneitherpivotalnorcostly. In eithercase noagenthas an
incentive tosave on costs by free-riding. N ote thatin the …rstclass ofequilibria the con…guration willalways feature lobby 2. Intuitively, iflobby 2 has not
formed, contributions tolobby1 cannotbepivotalsincelobby1 does notoppose
the preference ordering ofthe policy maker. B y the same logic, in the second
class ofequilibriathe con…guration willneverresultin strictly positive transfers
to the policy maker. T he above results clearly echo O lson’s notion thatifthe
totalnumberofplayers is common knowledge and actions possibly pivotal, the
contribution stage game willhave multiple equilibria- in particularone ofthese
industrylobbies “mightform butitmightnot”23.
In solvingthe standard contribution game, however, we notice thatmany of
theequilibriaaresupportedbyconditions thatseem somewhat‘implausible’ina
gamewherethetotalnumberofpotentialcontributors is large. Intuitively, would
an agentreally be willing to believe thatthe policy makercould have ‘strong’
preferences or that his contribution might be pivotal if he expected the total
numberofothercontributors tobe very large? Ifnotthen clearly the …rstclass
ofequilibriawilldisappear.
Inthenextsectionwethereforemodelthecontributionstageas aL argeP oissonG ame, allowingus toexploreO lson’s notionthatinlargegroups “itis certain
thatacollective good willnotbe provided [voluntarily]
”24. In e¤ ectwe see that
O lsonwas correcttoconjecturethatinlargegames free-ridingwilllimitthenum23
See O lson (19 6
5) pp. 43.
O lson (19 6
5) pp. 44.
24
18
berofgroupsthatform. H owever, armedwithanendogenous policychoicewesee
thatthis e¤ ectwillnotbesymmetricacross groups as, one, butnotboth groups
will failto form. T hus, by demonstrating thatourlarge P oisson contribution
game exhibits only the latter- non-pivotaland non-costly - class ofequilibria,
we establish thatalone industrygroup can bepartofaunique equilibrium in a
non-trivialway.
5.2. L arge P oisson V oluntary-Contribution G ame
For the reasons given above we now relax the assumption thatN is common
knowledge. M ore speci…cally we assume thatthe totalnumberofplayers is a
random variable. H owever, as M yerson (19 9 7 ) shows, the use ofa P oisson distribution allows such games tobe simpli…ed considerably. W e therefore assume
hereafterthatN is aP oisson random variable(P R V ) with meann. N otethatby
thedecomposition property oftheP oisson distribution (seeM yerson (19 9 7 )) the
totalnumberofplayers with typetis aalsoaP RV , with mean nr(t).
Introducingsuchuncertaintyimpliesthatplayerscannolongeremployastrategypro…lethatassigns astrategytoeach speci…cindividualin thegame, simply
because they are notaware ofwhothey allare. Instead we assume thatplayers
use the only information they dohave, namely the type setT. W e followM yerson(19 9 8) indescribingthestrategicbehaviourofplayers inthis P oissongamein
terms ofa distributionalstrategy ¿, de…ned as any probability distribution over
the setC £T such thatthe marginaldistribution on T is equaltor:N ote that
any ¿(c
;t) should be interpreted as the ‘beliefs’ players choose tohold overthe
likelybehaviourofanytype, orrathertheperceived probabilitythatarandomly
sampledplayerwillhavetypetandchooseactionc
:25 G ivenr; any¿ is alsoassociated with aunique strategy function ¾ (cjt), orconditionalprobability thata
randomlysampled playerwillchoosecifheis oftypet.
Sincethetotalnumberofplayers is aP RV withmeann; itfollows that, when
theplayers behaveaccordingto¿; thenumberofplayers ofanytypethatchoose
any action cin C is a P R V with mean n¿(c
;T) ´ n¿(c
)26, and thus thatthe
25
“...each playeris only aware ofthe possible types ofthe otherplayers, and soplayers can
only form perceptions abouthowthe strategic behaviourofotherplayers is likely to depend
on their types...T hat is, going to a modelof population uncerainty requires us to specify a
probability distribution overactions foreach type ofplayer, ratherthan foreach individual”,
M yerson (19 9 7 ) pp. 4,5.
26
A ny ¿ alsoinduces amarginalprobability distribution on C and thus ¿(c)= ¿(c;T ):H ereafterweuse¿(c)todenotethemarginalprobability ofany action.
19
expected action pro…leinsuchagameis n¿ ´fn¿(c
)gc2C :T herefore, sinceitcan
beshownthatthenumberofplayers inaP oissongamewhochoosetheactioncis
independentofthenumberofplayers whochooseallotheractions27 , weestablish
that, foranyxin Z (C ), theprobabilitythatxis theaction pro…leoftheplayers
in thegameis
"
#
Y e ¡n ¿(c)(n¿(c
))x(c)
P(xjn ¿) =
:
(5.3)
x(c
)!
c
2C
T hen, usingthe factthatany playerin the P oisson game assesses the same
probabilities fortheaction pro…leoftheotherplayers in thegame(notcounting
himself)28 , wemayre-writetheoptimisation problem 5.1 as
2
h
i
4 E ¼ t(c
max
;x¡ ) ´
i
c2C
X
x¡ 2Z (C )¡
3
i
P(x¡ jn ¿) ut(g(c
;x¡ )) + y¡Á(c
;x¡ ) 5 : (5.4)
h
In otherwords each playerresolves his ‘contribution decision’fora game ofsize
n by formulating‘beliefs’¿ overthe likely behaviourofeach type and then calculatingwhichelements ofC = fy1 ;y2 ;;g maximisehis expectedutilitygiventhe
resultingexpected action pro…len¿.
Following M yerson (19 9 8), we letS(b;n¿) µ T denote the setofalltypes
for whom choosing action b 2 C would maximise 5.4 when n is the expected
numberofplayers and¿ is thedistributionalstrategy. T henwemayassertthata
distributionalstrategy ¿ is an equilibrium ofthe P oisson voluntary-contribution
game¤ de…ned byfT, n , r, C, E [¼ t(:)]g ifand onlyif
¿(b;S(b;n¿)) = ¿(b) 8b 2 C.
T hatis, if“alltheprobabilityofchoosingaction b comes from types forwhom b
is an optimalaction, when everyone else is expected tobehave accordingtothis
distributionalstrategy”29 . A lternativelyany¿, suchthatnotype hasanincentive
todeviatewhen the othertype does notdeviate, is an equilibrium distributional
strategy¿ o .
Inlightoftheabovediscussionweproceedbyattemptingtoestablishwhether,
in alargegameofsizen , anequilibrium distributionalstrategyexists tosupport
27
A proofofwhathas been termed the ‘independentactions’property ofP oisson games can
be found in M yerson (19 9 7 ).
28
T his resulthas beentermedthe‘environmentalequivalence’propertyofP oissongamesand,
again, aproofcan befound in M yerson (19 9 7 ).
29
M yerson (19 9 8), pp. 7 .
20
each lobbying con…guration. T hatis, we seek to verify the conditions, ifany,
underwhich: a) ¿(;;S(;;n ¿)) = ¿(;) = 1 which would supportLo = f;g; b)
¿(y1;S(y1 ;n ¿)) = ¿(y1) > 0 and ¿(y2 ;S(y2 ;n¿)) = ¿(y2 ) = 0 supportingLo =
f1g; c) ¿(y1 ;S(b;n ¿)) = ¿(y1 ) = 0 and ¿(y2 ;S(y2 ;n¿)) = ¿(y2 ) > 0 supporting
Lo = f2 g; and …nally ¿(y1 ;S(b;n¿)) = ¿(y1 ) > 0 and ¿(y2 ;S(y2 ;n ¿)) = ¿(y2 ) >
0 supportingLo = f1;2 g:
Itappears then, thatwemust…nd P(x¡ jn¿) foreveryx¡ in Z (C )¡ :FortunatelyM yerson (19 9 8) has shown thatthis is notactually thecase, provingthat
itis possibleto…ndasolutiontooptimisationproblems suchas 5.4, foranygiven
setof‘beliefs’, byfollowingatwostageprocedure30 .
T he …rststep is tosimplify the optimisation problem by partitioningZ (C )¡
intoevents, orpay-o¤ equivalentsub-sets ofindividualactionpro…les x¡ ; relevant
to each action c2 C. T aking c= y1 as an example, the results ofthe policy
selection suggestthattherearethreerelevantevents:31
(i) “T hepolicymakerhas ‘stronger’policypreferences than lobby2” :
W 1 = fx¡ 2 Z (C )¡ jH
¸x(y2 )H 2 g
p
implying
¼ t(y1 ;x¡ ) = ut(g(y1;x¡ )) + y¡Á(y1 ;x¡ )
= ut(1) + y:
(ii) “T he policy makerhas ‘weaker’policy preferences than lobby 2 and lobby
1 is more‘powerful”’:
W 2 = fx¡ 2 Z (C )¡ j(x(y1)¡ + 1)H 1 + H
p
¸x(y2 )H
2
> H p ;x(y2 ) > 0 g
implying
¼ t(y1 ;x¡ ) = ut(g(y1;x¡ )) + y¡Á(y1 ;x¡ )
Be1o(1)
= ut(1) + y¡
;
x(y1)¡ + 1
30
where Be1o (1) = x(y2 )H 2 ¡H p :
In the text we give only a brief outline of the steps invloved in solving a large P oisson
contributiongame. A fullerexplanantionandproofofoursubsequentpropositionmaybefound
in theappendix.
31
N otethatin this casex(y2 )¡ = x(y2 )and x(;)¡ = x(;):
21
(iii) “T he policy makerhas ‘weaker’policy preferences than lobby 2 and lobby
2 is more‘powerful”’:
W 3 = fx¡ 2 Z (C )¡ jx(y2 )H
2
> (x(y1 )¡ + 1)H 1 + H p ;x(y2 ) > 0 g
implying
¼ t(y1 ;x¡ ) = ut(g(y1;x¡ )) + y¡Á(y1 ;x¡ )
= ut(2 ) + y:
T hesecondstep is toestimatetheprobabilities oftheseevents. H erewemake
useoftheconceptofthemagnitude ¹ ofaneventW , orrathertherateatwhich
theprobabilityofthis eventgoes tozeroin the equilibriaoflargeP oisson games
wherethesizeparametern tendstoin…nity:T hatis, we…rstderivethemagnitude
ofeach eventusingM yerson’s “M agnitudeT heorem” which implies that, forany
eventW 2 Z (C )¡ ,
X
(P(W jn¿ n ))
¹ = n!
lim1 log
= n!
lim1 m¡ ax ¿ n (c
)ª
n
xn 2W c
2C
Ã
!
x¡
)
n (c
;
n¿ n (c
)
wherethefunction ª :<+ ! <is de…ned bytheequations
ª (µ) =
ª (0 ) =
µ(1 ¡log(µ)) ¡1 ·0 , 8µ > 0
lim ª (µ) = ¡1;
µ! 0
and theconvention that
if¿ n (c
) =
if¿ n (c
) =
0 and x¡
) = 0 then ¿ n (c
)ª [x¡
)=n¿ n (c
)]= 0
n (c
n (c
¡
¡
0 and xn (c
) > 0 then ¿ n (c
)ª [xn (c
)=n¿ n (c
)]= ¡1
is adopted.
Inessencethe“M agnitudeT heorem” tells us thattherateatwhichtheprobabilityofaneventW tends tozeroas n ! 1 is determinedbytherateatwhich
the probability ofits elements, orindividualaction pro…les x¡
end tozeroas
n; t
n ! 1 :M orespeci…cally, themagnitudeofanyeventW ; is equaltothegreatest
magnitude ofany vectorx¡
e that¹ cannotbe strictly positive since
n 2 W . N ot
the logofa probability is always less than orequaltozero. Further, this value
22
¹ is e¤ ectively determined by the di¤ erence between the actual numberofcontributions x¡
) andtheexpected numbern¿ n (c
), giventhesizeparametern and
n (c
beliefs ¿.
H avingestablished the magnitudeofeveryeventforagiven setof‘beliefs’¿,
the …naltaskis tosolve 5.4. H ere we may exploitthe factthatthe “M agnitude
T heorem” implies as a corollary that, forany two events W i and W j i 6
= jin
¡
Z (C ) ; if¹ i > ¹ jthen
lim
n! 1
P(W j jn¿ n )
= 0:
P(W i jn¿ n )
O rratherthat, as thesizeofthegamen tends toin…nityandforanygivensetof
‘beliefs’¿; players’perceptionoftheprobabilityofeventswithnegativemagnitude
becomes in…nitesimallysmallrelativetothoseevents withzeromagnitude. T hen,
repeatingthis procedure foreach setof‘beliefs’ we may verify whetheractions
thatareconsistentwith these‘beliefs’maximise5.4.32
T oprovethemain proposition wemakeuseofthefollowinglemma:
L emma5.1. A s n ! 1 :
(a) If¿ : ¿(yt) = 0 8 teach playeranticipates that: ifH p ¸ H 2 , the policy
makerwillalwaysimplementgo = 1 andthus thatnoactionwillbepivotal;
butifH p < H 2 ; thatthepolicymakerwillimplementgo = 2 ifand onlyif
c= y2 :
(b) If¿ : ¿(y1 ) < ¿(y2 )HH 12 each playeranticipates thatthe policy makerwill
always implementgo = 2 andthus thatnoactionwillbepivotal:
(c) If¿ : ¿(y1) > ¿(y2 ) = 0 or¿(y1 ) ¸ ¿(y2 )HH 21 and ¿(y2 ) > 0 each player
anticipates thatthe policy makerwillalways implementgo = 1 and thus
thatnoaction willbepivotal.
Explanation. W hen beliefs are such that no other player is expected to
contribute (¿(yt) = 0 8t) and c= y2 lobby 2 willform and actalone(L= f2 g):
32
M yerson(19 9 8 ) alsoproves theso-called“O ¤ setT heorem” whichis veryusefullforcomparingtheprobabilities ofevents thatdi¤ erbya…niteadditivetranslation. Infact, this theorem is
essentialforestimatingpivotprobabilities invotinggames. H owever, inourmodelweworkwith
events thatdonotdi¤ erbyaddinga…xedvectorofintegers andthus theuseofthe“M agnitude
T heorem” and its corollary su¢ces forourpurpose.
23
P roposition 4.3 therefore implies thatgo = 2 should be expected with certainty
onlyifH p < H 2 . In allothercases - i.e. ifc2 fy1;;g - lobby2 willnotform and
thus P roposition 4.2 implies thatanyplayerwillexpectgo = 1 with certainty.
W henbeliefsaresuchthatsomecontributionsareexpectedtobemade(¿(yt) 6
=
0 forsome t), then, given thatthe expected totalnumberofplayers is large, no
playeris willing to believe thathe is pivotal. M ore speci…cally if the relative
¿(y1 )
probability thata randomly selected playerwillcontribute to lobby 1, ¿(y
, is
2)
high enough each playeranticipates thatthepolicymakerwillalways implement
go = 1:Consequently contributions toeitherlobby areneverexpected tobepiv1)
otal. A nalogously if ¿(y
is lowenough players anticipate thatthe policy maker
¿(y2 )
will always implement go = 2 . A gain contributions to either lobby are never
expected tobepivotal.
W earenow…nallyinapositiontostateourcentralpropositionconcerningthe
endogenous formation oflobbygroups given an endogenous binarypolicychoice:
P roposition 5.2. IntheP oissonvoluntary-contributiongame¤ :
(a) Ifthe policy makerhas ‘stronger’ policy preferences than agents oftype 2,
orifthe policy makerand agents oftype 2 have equalpolicy preferences, (i.e.
H p ¸ H 2 > 0 ) then the setofequilibrium distributionalstrategies is such that
Lo = f;g andLo = f1g aretheonlypossibleequilibrium lobbyingcon…gurations.
(b) Ifthe policy makerhas positive but‘weaker’policy preferences than agents
oftype 2 (i.e. H 2 > H p > 0 ) then thesetofequilibrium distributionalstrategies
is such thatLo = f1g is theuniqueequilibrium lobbyingcon…guration.
(b) Ifthe policy makerhas nopolicy preferences (i.e. H p = 0 ) then the setof
equilibrium distributionalstrategies is such thatLo = f1g and Lo = f2 g arethe
onlypossibleequilibrium lobbyingcon…gurations.
Explanation: L= f2 g cannotbe an equilibrium when H p > 0 :T osee this,
…rstly note thatP roposition 4.3 implies that, even when acting alone, lobby 2
musto¤ erastrictlypositivetransferH p tosecurego = 2 :H owever, alsonotethat
L emma5.1 implies thatif¿(y2 ) > 0 contributions tolobby 2 are neverexpected
tobe pivotalwhen the expected totalnumberofpotentialcontributors is large.
Free-ridingis thereforecompleteand thus lobby2 fails toform.
L= f1;2 g cannotbeanequilibrium. T his follows from thefactthatP roposition 4.6implies thatallpotentialcontributors tolobby tanticipate thatstrictly
positive, orcostly, transfers willbe necessary to secure go = t; whilstL emma
24
5.1 implies thatwhen ¿(yt) > 0 8tcontributions towards these variable political
in‡uence costs are neverexpected to be pivotal. A ll agents therefore have an
incentivetofree-ride.
L= f;g cannotbe an equilibrium when H 2 > H p > 0 :P roposition 4.3 and
L emma 5.1 imply thatwhen ¿(yt) = 0 8tand H 2 > H p > 0 allagents oftype 2
havean incentive to‘deviate’from this distributionalstrategy and contributeto
lobby2 inordertosecureg o = 2 . Incontrast, L= f;g canbeanequilibrium when
H 2 ·H p ; sinceL emma5.1 implies thatnoactions areexpected tobepivotal.
Finally, L = f1g can always be an equilibrium, whilstL = f2 g can be an
equilibrium whenH p = 0 . Inbothcases P ropositions 4.2 and4.3 andL emma5.1
implythatnoplayerhas an incentivetodeviatefrom thecorrespondingdistributionalstrategysincecontributions areunnecessary tosecure thepreferred policy
choice. Inotherwords inthesecases theincentivetofree-ridehas disappeared.
In essence, then, industry lobbyingis purely counteractive. T osee this recall
from abovethatlargegamepopulationuncertaintyensures thatalobbyopposing
the policy maker’s preferences, willneverform. A n immediate corollary tothis
resultis thatin equilibrium lobbying bears no variable politicalin‡uence cost.
H owever, this does notmean thatlobbyingexists in anon-trivialway. In fact, if
alobbyforms, demonstratinganabilitytoo¤ erstrictlypositivetransfersalongthe
equilibrium path, itwillonlybewhenanopposinggroup posesacrediblethreat, or
ratheris privileged, on an o¤ -equilibrium path. In e¤ ect, then, an industrylobby
forms simply toensure thatits opponentfaces competition and thus acollective
action problem.
A sanillustration, considerthecaseH 2 > H p > 0 :Inthiscase, L= f;g cannot
beanequilibrium sincelobby2 poses acrediblethreattoform ifnocontributions
areexpectedtobemade. If, ontheotherhand, lobby1 is expectedtoform, then
lobby 2 willinevitably face costs to in‡uence policy and hence, given thatthe
expected size ofthe polityis large, an insurmountable collective action problem.
A ccordingly, lobby 1 forms toensure thatlobby 2 neverforms.33 In fact, in this
case, L= f1g is the unique lobbyingcon…guration. T he above consideration is
reinforcedifweconsiderwhathappens ifH 2 ·H p :Inthis case, lobby2 nolonger
poses a credible threattoform on any o¤ -equilibrium path and therefore either
L= f;g orL= f1g can bepartoftheequilibrium.
33
T he same intuition holds when Hp = 0 : N ote, however, thatin this case lobby 2 can also
deterlobby1 from forming. T hereforeeitherL= f2g orL= f1 g canbepartoftheequilibrium.
25
6
. Conclusion
G iventheempiricalobservationthatcertainindustrylobbygroupsappeartohave
played asurprisingly passive role in policy debates, this papero¤ ers atheory of
lobby formation that, in contrast to the O lsonian approach, is not driven by
asymmetries in …xed organisationalcosts orgroup size. R ather, welooktoother
factors toexplain patterns ofgroup formation, namely the relationship between
population size and the free-riderproblem associated with the procurement of
variable ‘politicalin‡uence’costs. M ore speci…cally, we build on O lson’s (19 6
5)
pointthatin large groups in which “no single individual’s contribution makes
a perceptible di¤ erence...itis certain thata collective good willnotbe provided
[voluntarily]
”. H owever, incontrasttoO lson, weexploretherelationship between
population sizeand free-ridingin thecontextoflobbygroup competition overan
endogenous policychoicedetermined bydirecttransfers.
Inessenceweshowthat, when thepolicychoiceis endogenous, exanteasymmetriesabsentandtheexpectedsizeofthepolityislarge, onlyoneoftwoindustry
groups willbe besetby free-riderproblems and thus failtoform. Furthermore,
we suggestthatifa group opposes the policy maker’s policy preference we can
be certain that, atleaston this policy issue, itwillalways remain passive. T he
cornerstoneofthis resultis simplythatin amenu-auction, agroup’s equilibrium
transfermustcoverthewelfarelossitimposes onothers participatinginthepoliticalprocess. T hus, ifagroup shares apreferenceorderingwith thepolicymaker
and lacks competition from anotherlobby, its equilibrium transferwillactually
be zero. B y implication then, in an environmentwhere allcontributions are expected to be non-pivotaland thus the incentive to free-ride on costly transfers
complete, we see that, iflobbyingexists, itwillonly come from the group that
shares apreferenceorderingwith thepolicymaker.
In short, by abstracting from the endogeneity ofthe policy choice, existing
theories oflobbyformationhaveoverlookedthefactthatfree-riderproblems need
notbesymmetric, even when groups are ofapproximately equalsize. R ather, in
lightofthisobservation, thispapershowsthatitispossibletoexplainvariationsin
industrygroup formation, withoutrecoursetoasymmetries in…xedorganisational
costs and group size. M oreimportantlyperhaps, weo¤ erareason whyG awande
(19 9 7 ) observed thatpro-exportgroups preferred todirecte¤ ortatinstruments
thatfacilitateexportsratherthancompetewithanti-protectionistgroupsoverthe
levelofnon-tari¤ barriers34. O urapproach suggests that, iffree-ridinglimits the
34
See G awande (19 9 7 ) pp. 6
5.
26
extenttowhichindividual…rmscontributetowardsthevariablepoliticalin‡uence
costs oflobbying, it is indeed rationalfor lobbies to specialise in areas where
policymaker’s eitherdonothaveapriorpreference orarealready ‘on-side’. W e
therefore avoid the worrying question posed earlier: ifa group has formed by
virtue of…nancingthe …xed costs associated with collective participation in one
policydimension whywould itremain silentin another?
Finally we note thata corollary ofourresults is that‘money neverchanges
hands’, invalidatingan often heard criticism ofthe common agency approach to
economic in‡uence seeking. B utthis, then, raises the question whatdolobbies
do? A common answer has been that lobbies exist, among other reasons, to
actively change the votingdisposition ofa legislatororpolicy maker. H owever,
ouranalysis shows that, in alarge polity, such groups willbe besetby free-rider
problems and failto form. W e therefore suggestthatallindustry lobbying by
virtue ofdirecttransfers, is purely counteractive; such lobbies do notexistto
changethedefaultpolicyoutcomebutrathertoensurethatothergroups cannot.
O uranalysis alsoo¤ ers some interestingperipheralresults. B y endogenising
industry lobby formation, we develop a modelthat- in contrasttothe existing
literature on policy formation in the presence oflobbies - allows us to predict
equilibriamoreprecisely. W earethereforeabletosuggestthatanon-benevolent
policymaker(i.e. up (g) = 0 8g) willalways choose asociallysub-optimalpolicy
choice. T hereasonbeingthatonegroup willalways lackrepresentationand thus
suchapolicymakerwillfailtomaximisethegrossutilities ofallgroupsinsociety.
Similarly, ourresultsimplythatthechoicebyasemi-benevolentpolicymaker(i.e.
up (g) > 0 forsomeg) willalways beintendedtomaximisesocialwelfare, sinceno
group willeverform tooppose this choice. O r, rather, a semi-benevolentpolicy
makerwillalways implementthe socially optimalchoice, even ifa group lacks
representation.
Finally, re‡ectingonthefactthatourresultsarepotentiallymutuallycompatiblewiththoseofM itra(19 9 9 ) weseetwodirections forfutureresearch. N amely,
both an empiricaland a theoreticalinvestigation ofthe importance ofvariable
politicalin‡uence costs, relative to…xed organisationalcosts, in explainingthe
variation in industry group participation.
7 . A ppendix
P roofofP roposition4.1. A formalproofofthis propositionis given in D ixit,
G rossman and H elpman (19 9 7 ).
27
In this form thecharacterisation theorem clearlyaids intuitiveunderstanding
ofthe common agency problem and is therefore presented as such in the text.
H oweverforthepurposeofprovingsubsequentpropositions itis usefultore-state
proposition 4.1. as follows.
P roposition 4.1.¤ A vector of payment schedules ffB to(g)gg2Ggt2L and a
policy choice go constitute an equilibrium ofthe policy selection game forany
L = f1;2 g ifand onlyif:
(a) forallt2 L:
(b)
up (go) +
(4.1* )
B to(g) 2 B t 8g 2 f1;2 g;
X
t2L
B to(go) ¸up (g) +
X
t2L
(4.2* )
B to(g) 8g 2 f1;2 g;
(c) foreveryt2 L, t6
= s:
(i)8g 2 f1;2 g
x(yt)ut(g o) + up (go ) +
X
B so(g o) ¸x(yt)ut(g) + up (g) +
s2L =ftg
X
B so (g) (4.3* )
s2L=ftg
(ii)
0
B to(go) = 0max [up (g ) +
g 2f1;2 g
X
0
B so(g )]¡[up (go) +
s2L=ftg
X
B so(go)]:
(4.4* )
s2L=ftg
Explanation: Condition (a) remains unchanged. Condition (b) simply reexpresses optimisationbythepolicymakerinamoreconvenientform. Inthecase
ofcondition (c), optimisation in4.3 bylobbytclearlyimplies thattheconstraint
willbindwithequality, hencecondition4.4¤;thatis, eachlobbyis awarethat, for
anyproposed policychoice, itis neverin its interesttoprovidethepolicymaker
with more than herreservation utility. Furtheritshould be clearthatcondition
4.3¤ follows directly from substitutingcondition 4.4¤ intothe objective function
in 4.3. M ore intuitively, the existence ofan ‘outside option’ on the partofthe
policy-makerimplies thatthelobbymusttakeintoaccountanyutilitythepolicy
makerderives from the policy choice itself, togetherwith any o¤ ers from other
lobbies, when contemplatingits actions.
28
P roofofP roposition 4.2. Suppose go = 2 :T hen when L= f1g condition
4.3¤ ofthecharacterisation theorem implies
0 ¸x(y1)H 1 + H p ;
which cannotbethecase. Sogo = 1 is theuniqueequilibrium policychoice.
L etting
0
0
gb1 = arg 0m ax [up (g ) + 0 ]
g 2f1;2 g
0
wehave gb1 = 1; and thus when L= f1g condition 4.4¤ implies B 1o (1) = 0 :
P roofofP roposition 4.3. P art(a): suppose go = 2 :T hen when L= f2 g
condition 4.3¤ ofthecharacterisation theorem implies
x(y2 )H
2
¸H p ;
which cannotbethecase. Sogo = 1 is theuniqueequilibrium policychoice.
L etting
0
0
gb2 = arg 0m ax [up (g ) + 0 ]
g 2f1;2 g
0
we have gb2 = 1; and thus when L= f2 g condition 4.4¤ implies B 2o(1) = 0 . P arts
(b) and (c) can beproved in an analogous way.
P roofofL emma4.5. T his follows immediatelyfrom D e…nition 4.4.
A tthis point, note thatby invokingthe ‘truthfulness’re…nementtoP roposition 4.1* weend up with
B to(g) 2 B~t(g) 8g 2 f1;2 g;
instead of4.1* .
P roofofL emma4.6
. L et,
0
0
0
gb1 = arg 0m ax [up (g ) + Be2o(g )]
g 2f1;2 g
and
0
0
0
gb2 = arg 0m ax [up (g ) + Be1o(g )]:
g 2f1;2 g
29
0
(4.1¤ )
0
L emma 4.5 implies that Be1o (1) ¸ Be1o(2 ) and thus gb2 = 1:H ence, we have from
4.4¤ that
X
up (go) +
Beto(go) = H p + Be1o (1) 8go 2 f1;2 g
(6
.1)
and
t2L
0
0
H p + Be1o(1) = up (gb1 ) + Be2o(gb1):
(6
.2)
Suppose thatgo = 1:T hen clearly 6.1 implies that Be2o(1) = 0 :A lternatively
supposethatgo = 2 . T hen 6
.1 and 4.2¤ implyrespectively,
X
t2L
and
X
t2L
Beto (2 ) = H p + Be1o (1)
Beto(2 ) ¸H p +
X
t2L
Beto (1);
0
togetherimplyingthatBe2o (1) ·0 :T hus, given 4.1¤ , Be2o(1) = 0 :
0
Since we have Be2o (1) = 0 , if gb1 = 1 then clearly 6
.2 implies Be1o(1) = 0 and
0
¤0 e o
thus, by L emma4.5 and condition 4.1 ; B 1 (2 ) = 0 :H owever, ifgb1 = 2 then 6.2
becomes
H p + Be1o(1) = Be2o(2 ):
(6
.3)
Supposego = 1; then Be2o(1) = 0 ; 4.2¤ and 6.3 togetherimply
Be2o(2 ) ¸ Be2o(2 ) + Be1o(2 )
0
and thus, given 4.1¤ , Be1o(2 ) = 0 :A lternativelysupposego = 2 , then 6
.1 becomes
Be2o (2 ) + Be1o(2 ) = H p + Be1o(1)
and thus, given 6
.3, Be1o(2 ) = 0 :
P roof ofP roposition 4.7 . P art(a): Suppose go = 2 :T hen when L =
f1;2 g condition 4.3¤ ofthecharacterisation theorem implies
x(y2 )H 2 + Be1o(2 ) ¸H p + Be1o(1):
(6
.4)
0
N ote thatL emma 4.5 implies Be1o(2 ) = 0 , which togetherwith condition 4.1¤ of
thecharacterisation theorem (i.e.‘truthfulness’) implies
Be1o (1) = m infx(y1 )H 1 ;x(y1 )yg:
30
T hen, given thestabilitycondition
y ¸ m ax[H t];
t2f1;2 g
wehave Be1o(1) = x(y1 )H 1 :T hen 6.4 clearlybecomes
x(y2 )H
or, sincex(y1 )H
1
2
¸H p + x(y1 )H
1
> 0 when L= f1;2 g,
x(y2 )H
2
> H p;
which cannotbethecase. Sogo = 1 is theuniqueequilibrium policychoice.
T hen from L emma 4.5 we have Be2o(1) = 0 ; which, togetherwith condition
0
4.1¤ of the characterisation theorem and the stability condition above, implies
Be2o(2 ) = x(y2 )H 2 :A gain let
0
0
0
gb1 = arg 0m ax [up (g ) + Be2o(g )]
g 2f1;2 g
and notethat
H
p
0
¸x(y2 )H 2 :
T hus gb1 = 1 andthus from condition4.4¤ ofthecharacterisationtheorem wehave
Be1o(1) = 0 :
P art (b)(i): Suppose go = 2 : T hen, in a similar way to that above, 6.4
becomes
x(y2 )H 2 ¸H p + x(y1 )H 1;
which cannotbe the case. So go = 1 is the unique equilibrium policy choice.
0
Similarly, then Be2o (2 ) = x(y2 )H 2 :D e…ning gb1 as in part(a), and noting
H
p
< x(y2 )H
2
0
wehavegb1 = 2 . T husclearlycondition4.4¤ ofthecharacterisationtheorem implies
Be1o(1) = x(y2 )H 2 ¡H p :H encethe result. P arts (ii) and (iii) can beproved in an
analogous way.
P roof of L emma 5.1. Following the procedure outlined in the text, we
beginbyusingthe“M agnitudeT heorem” toderivethemagnitudeofeveryevent
relevanttoeach c2 C.
31
(a) Events relevanttoc= y1 (notethatin this casex(y2 ) = x(y2 )¡ and x(;) =
x(;)¡ ):
(i) W 1 = f x¡ 2 Z (C )¡ jx(y2 ) · HH p2 g implying¼ t(y1;x¡ ) = ut(1) + y:
T o…ndthemagnitudeforthis eventwehavetosolvethefollowingmaximisation problem
"
max ¿ n (;)ª
x¡
n 2W 1
Ã
!
xn (;)
+ ¿ n (y1)ª
n ¿ n (;)
Ã
!
x(y1 )¡
+ ¿ n (y2 )ª
n¿ n (y1 )
Ã
!#
xn (y2 )
n¿ n (y2 )
:
(6
.5)
N otethatthevectorx 2 W 1 does notfeatureanyconstraints withrespect
tothefeasiblechoices forxn (;) and xn (y1 )¡ :T hus, given thatthefunction
ª reaches its maximum valueatª (1) = 0 ; weseethatxn (;) = n¿ n (;) and
xn (y1 )¡ = n ¿ n (y1 ) arepartofthesolution totheaboveproblem.
¡
D e…newith m n 1 themaximum valueoftheobjectivefunction in 6.5. T hen
itfollows directlythat
m
n1
=
(
0 ifn¿ n (y2 )H 2 ·H p
® n 1 ifn¿ n (y2 )H 2 > H p
³
)
;
´
2)
where® n 1 = ¿ n (y2 )ª n ¿x(y
< 0 withtheinequalityfollowingonn¿ n (y2 )H 2 >
n (y2 )
H p ¸ 0 ; xn (y2 )H 2 · H p and the properties ofª (). From the ‘M agnitude
T heorem’ we know thatthe magnitude ofW 1 , ¹ 1 , is the limitofm n1 as
n ! 1 :H ence, itis clearthat
¹1 =
(
0 if¿(y2 ) = 0
® 1 < 0 if¿(y2 ) > 0
)
;
where® 1 = n!
lim1 ® n 1 :
B y followinganalogous steps we …nd the magnitudes ofallremainingevents
as stated below.
(ii) W 2 = f x¡ 2 Z (C )¡ j(x(y1 )¡ + 1)H 1 + H p ¸ x(y2 )H 2 > H p g implying
B~ o (1)
¼ t(y1 ;x¡ ) = ut(1) + y¡ x(y11)¡ + 1 ; where B~1o(1) = x(y2 )H 2 ¡H p :
T hemagnitudeofthis eventis
¹2 =
(
0 if¿(y2 ) 2 (0 ;¿(y1 )HH 12 ]
® 2 < 0 otherwise
32
)
:
(iii) W 3 = fx¡ 2 Z (C )¡ jx(y2 )H
ut(2 ) + y:
2
> (x(y1 )¡ + 1)H 1 + H p gimplying¼ t(y1 ;x¡ ) =
T hemagnitudeofthis eventis
¹3 =
(
0 if¿(y2 ) > ¿(y1)HH 12
® 3 < 0 otherwise
)
:
N ote that the above events are pairwise adjoint and such that
Z (C )¡ . H ence
cases.
P3
S3
i= 1
i= 1
Wi =
P r(W i) = 1:T his holds analogously forthe following
(b) Events relevanttoc= y2 (notethatin this casex(y1 ) = x(y1 )¡ and x(;) =
x(;)¡ ) ifH 2 ·H p:
(i) W 4 = fx¡ 2 Z (C )¡ jx(y1)H 1 + H p ¸(x(y2 )¡ + 1)H 2 g implying¼ t(y2 ;x¡ ) =
ut(1) + y:
T hemagnitudeofthis eventis
¹4 =
(
0 if¿(y2 ) 2 [0 ;¿(y1 )HH 21 ]
® 4 < 0 otherwise
)
:
(ii) W 5 = fx¡ 2 Z (C )¡ j(x(y2 )¡ + 1)H 2 > x(y1)H 1 + H p g implying¼ t(y2 ;x¡ ) =
B o (2 )
ut(2 ) + y¡ x(y22 )¡ + 1 whereB 2o(2 ) = x(y1 )H 1 + H p :
T hemagnitudeofthis eventis
¹5 =
(
(c) Events relevanttoc= y2 ifH
0 if¿(y2 ) > ¿(y1 )HH 12
® 5 < 0 otherwise
2
)
:
> H p:
(i) W 4 a = fx¡ 2 Z (C )¡ jx(y1)H 1 + H
¼ t(y2 ;x¡ ) = ut(1) + y:
p
¸(x(y2 )¡ + 1)H 2 ; x(y1 ) > 0 g implying
T hemagnitudeofthis eventis
¹4 a =
(
0 if¿(y1 ) > 0 and ¿(y2 ) 2 [0 ;¿(y1 )HH 21 ]
® 4 a < 0 otherwise
33
)
:
(ii) W 5a = f x¡ 2 Z (C )¡ jx(y1 ) = 0 orx(y1 ) > 0 and (x(y2 )¡ + 1)H 2 >
B o (2 )
x(y1 )H 1 + H p ; x(y2 )¡ > 0 g implying¼ t(y2 ;x¡ ) = ut(2 )+ y¡ x(y22)¡ + 1 where
B 2o(2 ) = x(y1 )H 1 + H p :
T hemagnitudeofthis eventis
¹ 5a =
(
0 if¿(y1 ) = 0 or¿(y2 ) > ¿(y1 )HH 12
® 5a < 0 otherwise
)
:
(d) Events relevanttoc= ;(notethatin this casex(y1) = x(y1 )¡ and x(y2 ) =
x(y2 )¡ ):
(i) W 6 = f x¡ 2 Z (C )¡ jx(y1 )H 1 + H p ¸ x(y2 )H 2 g; implying ¼ t(;;x¡ ) =
ut(1) + y. T hemagnitudeofthis eventis
¹6 =
(
0 if¿(y2 ) 2 [0 ;¿(y1 )HH 21 ]
® 6 < 0 otherwise
(ii) W 7 = f x¡ 2 Z (C )¡ jx(y2 )H 2 > x(y1)H
ut(2 ) + y. T hemagnitudeofthis eventis
¹ 7=
(
1
)
:
+ H p g; implying ¼ t(;;x¡ ) =
0 if¿(y2 ) > ¿(y1)HH 12
® 7 < 0 otherwise
)
:
W e are nowready toprove L emma 5.1. W e proceed as follows. First, for
each possible con…guration ofbeliefs ¿ we compare the magnitudes ofthe
events relevanttoeachc2 C. T hen, usingthe‘corollary’tothe‘M agnitude
T heorem’, we determine the events the probability ofwhich willdisappear
tozeroas n ! 1 :
Itfollows, then, bysimpleinspection that:
(i) W hen ¿ : ¿(y1) = 0 and ¿(y2 ) = 0 and H p ¸ H 2 only ¹ 1 = ¹ 4 = ¹ 6 = 0 ;
with allremaining magnitudes less than zero:T hus the ‘corollary’ to the
‘M agnitudeT heorem’implies P(W 1 jn¿) = P(W 4 jn¿) = P(W 6 jn¿) = 1
as n ! 1 :A lternatively, ifH 2 > H p ; only ¹ 1 = ¹ 5 = ¹ 6 = 0 ; and thus
P(W 1 jn ¿) = P(W 5 jn ¿) = P(W 6 jn¿) = 1 as n ! 1 :
34
(ii) W hen ¿ :¿(y1 )_> 0 and ¿(y2 ) = 0 and H p ¸H 2 only¹ 1 = ¹ 4 = ¹ 6 = 0 and
thus P(W 1 jn ¿) = P(W 4 jn¿) = P(W 6 jn¿) = 1 as n ! 1 :A lternatively,
ifH 2 > H p ; only¹ 1 = ¹ 4 a = ¹ 6 = 0 ; and thus P(W 1 jn¿) = P(W 4 a jn¿) =
P(W 6 jn ¿) = 1 as n ! 1 :
(iii) W hen ¿ :if0 ·¿(y1 ) < ¿(y2 )HH 12 and H p ¸H 2 only ¹ 3 = ¹ 5 = ¹ 7 = 0 , and
thus P(W 3 jn ¿) = P(W 5 jn¿) = P(W 7 jn¿) = 1 as n ! 1 :A lternatively,
ifH 2 > H p ; only¹ 3 = ¹ 5a = ¹ 7 = 0 , and thus P(W 3 jn¿) = P(W 5a jn¿) =
P(W 7 jn ¿) = 1 as n ! 1 :
(iv) W hen ¿ : ¿(yt) > 0 8tand ¿(y1 ) ¸ ¿(y2 )HH 12 then: a) ifH p ¸ H 2 only
¹ 2 = ¹ 4 = ¹ 6 = 0 , and thus P(W 2 jn¿) = P(W 4 jn¿) = P(W 6 jn¿) = 1
as n ! 1 ; b) ifH p < H 2 only ¹ 2 = ¹ 4 a = ¹ 6 = 0 , and thus P(W 2 jn¿) =
P(W 4 a jn ¿) = P(W 6 jn ¿) = 1 as n ! 1 :
L emma5.1, as stated in thetext, follows directlyfrom theseprobabilities.
P roof of P roposition 5.2. T he distributionalstrategy ¿ : ¿(yt) > 0 8t
supportingL= f1;2 g; is never an equilibrium. Suppose not. T hen, given n and
¿; we require ¿(y1 ;S(y1 ;n ¿)) = ¿(y1 ) > 0 and ¿(y2 ;S(y2 ;n¿)) = ¿(y2 ) > 0 :N ote
L emma 5.1 implies thatunderthese beliefs, as n ! 1 , either¿(y1 ) ¸ ¿(y2 )HH 12
and P(W 2 jn ¿) = P(W 4 jn¿) = P(W 6 jn¿) = 1 or¿(y1) < ¿(y2 )HH 12 and
P(W 3 jn ¿) = P(W 5 jn ¿) = P(W 7 jn¿) = 1:T hus in the…rstcase5.4 becomes:
E[¼ t(c
;x¡ )]= ut(1) + y
forallc2 fy2 ;;g and
E[¼ t(c
;x¡ )]= ut(1) + y ¡
B 1o(1)
x(y1 )¡ + 1
when c= y1 :T herefore, given B 1o(1) = x(y2 )H 2 ¡H p > 0 ; c= y1 is a strictly
dominatedstrategyforalltandthus¾(y1 jt) = 0 8t:T heresultisS(y1 ;n¿) = f;g
and thus ¿(y1 ;S(y1;n ¿)) 6
= ¿(y1 ) > 0 ; the probability ofacontribution tolobby
1 cannot‘comefrom types forwhom choosingy1 is an optimalaction’. L ikewise,
in the second case, c= y2 is a strictly dominated strategy for alltand thus
¾ (y2 jt) = 0 8tand S(y2 ;n ¿) = f;g, again implying¿(y2 ;S(y2 ;n¿)) 6
= ¿(y2 ) > 0 :
o
A nalogously, itcan beshown thatwhen H p > 0 ; thefactthatB 2 (2 ) > 0 ; implies
c= y2 is astrictly dominated strategy foralltunderthe distributionalstrategy
35
¿ : ¿(y1 ) = 0 , ¿(y2 ) > 0 supportingL= f2 g:T hus again S(y2 ;n¿) = f;g and
¿(y2 ;S(y2 ;n ¿)) 6
= ¿(y2 ) > 0 :
T heexistenceofequilibriafortheremainingdistributionalstrategies supportingthelobbyingcon…gurations L= f;g, L= f1g and L= f2 g ifH p = 0 can be
veri…ed in an exactlyanalogous manner.
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