legislative bargaining under weighted voting - nyu.edu · al. (1993). other cooperative solution...

Legislative Bargaining under Weighted Voting

By JAMES M SNYDER JR MICHAEL M TING AND STEPHEN ANSOLABEHERE

Organizations often distribute resources through weighted voting We analyze thissetting using a noncooperative bargaining game based on the Baron-Ferejohn(1989) model Unlike analyses derived from cooperative game theory we find thateach voterrsquos expected payoff is proportional to her voting weight An exceptionoccurs when many high-weight voters exist as low-weight voters may expectdisproportionately high payoffs due to proposal power The model also predictsthat ex post the coalition formateur (the party chosen to form a coalition) willreceive a disproportionately high payoff Using data from coalition governmentsfrom 1946 to 2001 we find strong evidence of such formateur effects (JEL D7 D72)

Collective decision-making frequently in-volves situations in which actors have differentnumbers of votes Some institutions assign un-equal voting weights explicitly Examples in-clude important political bodies such as theInternational Monetary Fund the EuropeanUnion (EU) Council of Ministers and the USElectoral College as well as many economiccartels such as the International Coffee Coun-cil Choice of weights is a subject of ongoingcontroversy in such bodies A wider class ofproblems involving weighted voting ariseswhen blocs of votes are assembled and casttogether Important examples are the formationof coalition governments voting in legislatureswith unified factions and shareholder voting incorporations

How does the distribution of votes affect whogets what Most theoretical and applied analy-ses of weighted voting employ power indicessuch as the Shapley-Shubik value the Banzhafindex and the Deegan-Packel index1 Theseindices often produce highly nonlinear relationsbetween expected outcomes and votes Typi-cally players with the largest weights receivedisproportionately high expected payoffs Manytheorists see these nonlinearities as natural re-flections of the subtleties of power For exam-ple William F Lucas (1978 p 184) writes ldquoItis fallacious to expect that onersquos voting power isdirectly proportional to the number of votes hecan deliver Power is not a trivial function ofonersquos strength as measured by his number ofvotes Simple additive or division arguments arenot sufficient but more complicated relationsare necessary to understand the real distribution

Snyder Department of Economics Massachusetts In-stitute of Technology 77 Massachusetts Ave CambridgeMA 02139 (e-mail millettmitedu) Ting Department ofPolitical Science Columbia University IAB Floor 7 420 W118 St New York NY 10027 (e-mail mmt2033columbiaedu) Ansolabehere Department of Political Science MIT77 Massachusetts Ave Cambridge MA 02139 (e-mailsdamitedu) We thank John Duggan for detailed com-ments and corrections on an earlier draft We also thank twoanonymous referees Massimo Morelli George TsebelisMilada Vachudova and seminar participants at New YorkUniversity the Midwest Political Science Association an-nual meeting and the Social Choice and Welfare meetingsfor helpful comments James Snyder and Michael Tinggratefully acknowledge the financial support of NationalScience Foundation Grant SES-0079035 Earlier drafts ofthis paper were written while Michael Ting was at theDepartment of Political Science at the University of NorthCarolina at Chapel Hill and Center for Basic Research in theSocial Sciences at Harvard University and he thanks themfor their support

1 See Lloyd S Shapley (1953) Shapley and Martin Shu-bik (1954) John F Banzhaf III (1965) and John DeeganJr and Edward W Packel (1978) for definitions of theseindices Applications include Banzhaf (1968) GuillermoOwen (1975) Samuel Merrill III (1978) Jacob S Dreyerand Andrew Schotter (1980) Manfred J Holler (1982)Robert H Bates and Da-Hsiang Donald Lien (1985) Am-non Rapoport and Esther Golan (1985) George Rabinowitzand Stuart Elaine MacDonald (1986) Kaare Strom et al(1994) Thomas Konig and Thomas Brauninger (1996)Emilio Calvo and J Javier Lasaga (1997) and Sven Berg etal (1993) Other cooperative solution concepts applied toweighted voting games include bargaining sets bargainingaspirations the kernel and the competitive solution Seefor example Norman Schofield (1976 1978 1982 1987)Richard D McKelvey et al (1978) Elaine Bennett (1983a1983b) and Holler (1987)

981

of influencerdquo2 These arguments have had asignificant impact on the field of political econ-omy and debates about the design of constitu-tions For example a large amount of literaturehas emerged employing power indices to studythe voting weights in the government of theEuropean Union3

While widely used power indices and the ar-guments they justify run contrary to an importantintuition Elementary microeconomic theoryteaches that in competitive situations perfect sub-stitutes have the same price4 In a political settingin which votes might be traded or transferred inthe formation of coalitions one might expect thesame logic to apply If a player has k votes thenthat player should command a price for thosevotes equal to the total price of k players that eachhave one vote In terms of expected payoffs theplayer with k votes should expect to have a payoffk times as great as the payoff expected by a playerwith one vote If ldquoexpected payoffrdquo can be used asa measure of ldquopowerrdquo then the player with k votesshould also expect to have k times as much poweras the player with one vote5

In this paper we present a straightforwardmodel of divide-the-dollar politics which cap-tures the simple price-theoretic intuition Weshow that the noncooperative bargaining modelof David P Baron and John A Ferejohn (1989)leads naturally to the result that expected pay-offs are proportional to voting weights6 Whilecooperative game theory concepts apply only tovoting power this model captures both votingpower and proposal power7 In the Baron-Ferejohn model a randomly drawn legislatormakes a proposalmdasha division of the dollarmdashwhich is then put to a vote Proposers seek tooffer as little of the dollar as possible to othersbecause they keep the residual for themselves

As in most noncooperative bargaining mod-els we must make an assumption about theprobability that each legislator is chosen tomake a proposal We analyze two natural casesFirst legislatorsrsquo proposal probabilities may beproportional to their voting weights This isconsistent with the empirical pattern found inthe formation of coalition governments8 Sec-ond legislatorsrsquo proposal probabilities may all

2 Another example is Steven J Brams and Paul J Affuso(1985 p 138) ldquoA measure like Banzhafrsquos is not only aneminently reasonable indicator of a crucial aspect of votingpowermdashthe ability of a member to change an outcome bychanging its votemdashbut also highlights the fact that size (asreflected by voting weights) and voting power may bearlittle relationship to each otherrdquo

3 See Brams and Affuso (1985) Berg et al (1993) KaisaHerne and Hannu Nurmi (1993) Madeline O Hosli (19931995) Mika Widgren (1994 2000) R J Johnston (1995)Jan-Erik Lane and Reinert Maeland (1995 1996) Lane et al(1995 1996) Matthias Bruckner and Torsten Peters (1996)Anthony L Teasdale (1996) George Tsebelis and GeoffreyGarrett (1996) Ulrich Bindseil and Cordula Hantke (1997)Dan S Felsenthal and Moshe Machover (1997 2000 2001)Annick Laruelle and Widgren (1998) Konig and Brauninger(1998) Garrett and Tsebelis (1999a 1999b 2001) Holler andWidgren (1999) Nurmi and Tommi Meskanen (1999) andMatthias Sutter (2000a 2000b)

4 Power indices are criticized on other grounds as wellSome critics emphasize the paradoxical flavor of the predic-tions that power indices yield (Brams and Affuso 1985 Holler1987 Felsenthal and Machover 1998 Garrett and Tsebelis1999a 1999b 2001) A further well-known problem is that theseindices capture only the influence that comes from having votesand ignore important sources of power especially the ability tomake proposals (James G March 1955) Moreover theseindices typically do not have a basis in non-cooperative gametheory an exception is Faruk Gul (1989) who provides anon-cooperative justification of the Shapley value

5 Theorists working on this problem commonly equatepower and expected payoffs There is some debate overwhether the definition of power should also include the

ability to change the outcome even though the action doesnot result in an increase in and may even lower the payofffor the pivotal actor eg William H Riker (1964 p 344)In this paper we use the terms power and expected payoffsinterchangeably

6 The Baron-Ferejohn model extends the Ariel Rubin-stein (1982) bargaining model to the case of three or moreplayers operating under simple majority rule This is themost widely used model of legislative bargaining and hasbeen used extensively to study various aspects of distribu-tive politics and government institutions See Joseph Har-rington (1989 1990a 1990b) Baron (1991 1996 1998)McKelvey and Raymond Riezman (1992) Baron and EhudKalai (1993) Randall Calvert and Nathan Dietz (1996)Eyal Winter (1996) V V Chari et al (1997) Daniel Dier-meier and Timothy J Feddersen (1998) Jeffrey S Banks andJohn Duggan (2000) William LeBlanc et al (2000) Nolan MMcCarty (2000a 2000b) Hulya Eraslan (2002) Peter Norman(2002) and Ansolabehere et al (2003) Other non-cooperativemodels of n-person bargaining include Reinhardt Selten(1981) Kenneth Binmore (1987) Gul (1989) Kalyan Chat-terjee et al (1993) and Benny Moldovanu and Winter(1995) Sergiu Hart and Andreu Mas-Colell (1996) andAkira Okada (1996) To our knowledge none of these hasbeen applied to the study of legislative politics

7 Massimo Morelli (1999) develops a model of ldquodemandbargainingrdquo that also captures these features and arrives atproportional payoffs albeit in a different bargaining envi-ronment We discuss his model and results further below

8 Diermeier and Antonio Merlo (2004) estimate that par-tiesrsquo ldquorecognition probabilitiesrdquo depend strongly on theirvoting weights as well as which party formed the previousgovernment

982 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

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be equal This is the most common assumptionin the theoretical literature It is also consistentwith an institution such as the EUrsquos Council ofMinisters in which countries have unequal vot-ing weights butmdashat least theoreticallymdashequalopportunities for making proposals

In the first case we find that each playerrsquosexpected payoff (continuation value) is equal tohis share of the total voting weight The intu-ition underlying this result is straightforwardand follows from a simple substitution argu-ment Suppose type-1 players have a continua-tion value of 2 and a voting weight of 1 whiletype-2 players have a continuation value of 5and a voting weight of 2 Then rational propos-ers seeking to minimize the costs of the coali-tions they construct will never include type-2players in their coalitions (except perhaps be-cause of ldquointegerrdquo issues) Proposers will sub-stitute type-1 players for type-2 playerswhenever possible since two type-1 playershave the same total voting weight as one type-2player but a total cost that is four-fifths asmuch

In the second case there are two types ofequilibria depending on the distribution of vot-ing weights The first is an ldquointeriorrdquo equilib-rium in which each playerrsquos expected payoff isequal to her share of the total voting weightmdashjust as in the case with proportional recognitionprobabilities This occurs when the distributionof voting weights is not too skewed

The second type is a ldquocornerrdquo equilibrium Inthis case the players can be divided into twodistinct groups defined by some cutoff weightt0 Players with voting weights less than t0 haveexpected payoffs that are greater than theirshares of the total voting weights while playerswith voting weights greater than (or equal to) t0have expected payoffs that are less than theirshares of the total voting weight All playerswith voting weights greater than (or equal to) t0have expected payoffs that are equal to some times their voting weight with 1 Thereason weak players have expected payoffs thatare greater than their shares of the voting weightis their proposal power By assumption this isassigned equally to all players The corner equi-libria occur when the weakest players are soweak that even if no other proposers ever includethem as coalition partners their proposal poweralone is enough to yield an expected payoffgreater than their share of the voting weight

The logic in the noncooperative model differssharply from that of the power indices Powerindices are based on the idea that all orderings(or winning coalitions or minimal-winning co-alitions) are equally likely to form regardless ofhow expensive or cheap they are Under com-petitive bargaining expensive coalitions willform rarely or not at all and cheap coalitionswill form quite often

The competitive bargaining logic producesseveral key predictions that differ from powerindices First as noted at an interior equilib-rium the expected payoffs of players dependlinearly on the number of votes Second in thecorner equilibria it is the weaker players whoreceive expected payoffs greater than theirweight By contrast power indices typicallyassign disproportionate power to players withhigher voting weights Third even ldquodummyrdquoplayers can have positive expected payoffs be-cause all players have some chance of beingchosen as proposer By contrast power indicesassign zero power to any player with too fewvotes to be pivotal in at least one coalition

The results also suggest that the linear pay-offs of the interior equilibrium are closely re-lated to low proposal power Supermajoritarianrequirements reduce proposal power and help toproduce the interior equilibrium Setting recog-nition probabilities to be proportional to votingweights also restores linearity in the payoffsWe therefore conjecture that institutional fea-tures such as an open rule will similarly makelinear payoffs more likely

In addition to the theoretical analysis we testpredictions of the model using data on coalitiongovernments in parliaments from 1946 to 2001An extensive literature examines the empiricalrelationship between partiesrsquo shares of parlia-mentary seats and their shares of cabinet postsin coalition governments to study theoreticalconjectures about bargaining generally A crit-ical weakness of this literature is that it focuseson the effects of seat shares on shares of cabinetposts but the theoretically relevant concept isshares of voting weights We correct this prob-lem and offer the first empirical study of therelationship between partiesrsquo shares of votingweights and their shares of cabinet governmentposts Consistent with the Baron-Ferejohnframework we find that the party recognized toform the government receives a share of postsmuch larger than its voting weight

983VOL 95 NO 4 SNYDER ET AL LEGISLATIVE BARGAINING UNDER WEIGHTED VOTING

Our theoretical results build on an importantliterature that uses noncooperative game theoryto study distributional politics under varyinginstitutional settings such as differential votingweights unicameral and bicameral legislaturesand supermajority rules Baron and Ferejohn(1989) develop the basic closed rule bargainingmodel in a unicameral legislature when all play-ers have equal voting weight Banks and Dug-gan (2000) provide an existence result but nota characterization for a generalized Baron-Ferejohn model that encompasses weightedvoting A series of papers characterizes the out-comes of the Baron-Ferejohn model underdifferent institutional arrangements includingveto-players and unanimity rules Winter (1996)and McCarty (2000a) study variants of theBaron-Ferejohn model with veto players andChari et al (1997) and McCarty (2000b) incor-porate an executive veto Merlo and CharlesWilson (1995) study the equilibria of a relatedgame under unanimity rule Baron (1998) andDiermeier and Feddersen (1998) compare leg-islatures with and without a ldquovote of confi-dencerdquo procedure McKelvey and Riezman(1992) analyze the effects of seniority rule An-solabehere et al (2003) characterize the equi-libria in a bicameral setting (also see Diermeierand Myerson 1999) LeBlanc et al (2000)show how different institutional arrangementsaffect the level of public investments Baron(1996) Calvert and Dietz (1996) Diermeier andMerlo (2000) and Matthew Jackson and BoazMoselle (2002) analyze variants of the Baron-Ferejohn model with externalities andor spatialpreferences To our knowledge no previousanalysis provides a characterization of the gen-eralized model under weighted voting with ma-jority or supermajority voting That is thecontribution of the current paper

I Model and Results

A The Model

Our model is based on the closed-ruledivide-the-dollar game studied by Baron andFerejohn (1989) Players in the game belong toa set N of legislators where N n 3Legislators have preferences over the funds al-located to their district with each legislator i 1 2 n receiving utility ui(x) xi from adivision of the dollar x (x1 x2 xn) X

where X xxi 0 i yeni1n xi 1 Each

legislator is of an integral type t 1 2 Twhere each type has a unique voting weight and1 T n Let T denote the set of types and lett(i) denote the type of legislator i The numberof type-t legislators is nt where nt is positiveand integer-valued so yent1

T nt n We denotea type-t legislatorrsquos voting weight wt and as-sume that (a) weights are positive integers and(b) wt wt1 for all t T For convenience wearrange the legislators so that legislators nt1 1 nt1 nt each have weight wt wheren0 0 Thus legislators are numbered in in-creasing order of weight

The legislature uses a generalized majorityrule For any coalition of legislators C N letnt(C) represent the number of type t legislatorscontained within Let w(C) yent1

T nt(C)wt rep-resent its total voting weight and let w w(N) yent1

T ntwt be the combined weight of alllegislators A coalition C is winning if and onlyif w(C) w where w (w wT w w2) isthe proportion of the total voting weight re-quired for passage Let W denote the set ofwinning coalitions

Weighted voting introduces a significanttechnical complication to this framework Aweighted voting game is homogeneous if allminimal winning coalitions have exactly thesame total voting weight All (strong) gameswith up to five players are homogeneous Forlarger populations of players however gamesare sometimes nonhomogeneous and roundingproblems may arise For example in a six-member majority-rule legislature all minimumwinning coalitions are of size or ldquoweightrdquo 4under unweighted voting But if votes areweighted (9 5 5 3 2 2) and 15 of 26 votes arerequired for victory then some minimum win-ning coalitions have a total weight of 15 (coa-litions consisting of players with weights 5 5 3and 2) while others have 16 or more Further-more the weight-9 player cannot be in a mini-mum-weight minimum-winning coalitionsince any winning coalition that includes herhas a weight of at least 16

Nonhomogeneity affects the general charac-terization of proposal strategies As the exampleillustrates some types of proposers may find itnecessary or optimal to choose coalitions thatare not of the smallest winning voting weight

We address this problem by examining thebehavior of ldquoreplicatedrdquo voting games That is


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we examine equilibrium strategies as the num-ber of players of each type is multiplied bysome positive integer r The basic gamedescribed above has r 1 and a game with rreplications has rn players a total weight of rwand a threshold for victory of rw We show thatthe effect of nonhomogeneity becomes small asr increases thus allowing us to derive somegeneral results We also show however thatthese results hold at r 1 for homogeneousgames and that for most ldquosmallrdquo games theeffect of nonhomogeneity is small even whenr 1

The sequence of the game is as follows Ineach period Nature randomly recognizes a pro-poser from N Recognition probabilities areequal among legislators of a given type anddraws are iid across periods Thus in eachperiod a type-t legislatorrsquos recognition probabil-ity is pt As noted in the introduction we willexamine two cases one in which recognitionprobabilities are uniform and one in which theyare proportional to voting weights The pro-poser proposes a division of the dollar from XAll legislators then vote for or against the pro-posal If the proposal receives weight rw insupport then the dollar is divided and the gameends If the proposal is rejected then a newproposer is randomly drawn and the game con-tinues We study the infinite-horizon game withno discounting The game can be treated as asequence of identical subgames where eachsubgame begins with Naturersquos move to draw aproposer

The solution concept used here is stationaryequilibrium Strategies under this concept arestationary thus each player uses history-independent strategies at all proposal-makingstages and voting strategies depend only on thecurrent proposal We therefore omit reference totime periods and game histories to conserve onnotation A proposal strategy for legislator i isthen i (X) where (X) is the set of prob-ability distributions over X A voting strategyfor legislator i is simply a mapping from theoffered amount to a vote i X3 0 1 where1 is a vote in favor

In a stationary equilibrium strategies mustsatisfy sequential rationality and weak domi-nance Stationary equilibria therefore have thefollowing properties Each player has a contin-uation value which gives her expected payoffprior to Naturersquos draw of a proposer By sta-

tionarity this value is time-invariant A playerwill vote for a proposal if it is at least hercontinuation value and against it otherwiseFinally for any legislator i recognized as pro-poser if the set of proposals X that provides herwith at least her continuation value and will beaccepted as non-empty then i must propose anelement of arg maxxXui(x)9

B Results

In this section we prove the existence of andcharacterize a stationary equilibrium to the leg-islative bargaining with weighted voting Spe-cifically there exists a stationary equilibrium inwhich the first legislator recognized makes aproposal that wins approval and ends the gameThese equilibria are symmetric in the sense thatany two players with the same voting weighthave the same continuation value And for suf-ficiently large games the legislatorsrsquo relativeprices are proportionate to their voting weights

Existence follows by establishing our modelas a special case of Banks and Duggan (2000)

PROPOSITION 1 A stationary no-delay equi-librium exists

PROOFAll proofs are in the Appendix

An important feature of these stationary equi-libria is their symmetry Types of legislators aredefined by their voting weight Any two legis-lators of the same type bring the same numberof votes to any potential coalition and theyhave the same probability of being chosen toform a coalition in the future As a result sta-tionary equilibria are ldquosymmetricrdquo in the fol-lowing sense

COMMENT 1 In a stationary equilibrium thecontinuation values for players of the same typeare equal

The result implies that at the beginning ofeach subgame players of each type t will havethe same continuation value vt This property

9 If X is empty then i proposes an allocation that will berejected and receives her continuation value in expectationThis possibility does not arise in the games we study


allows us to narrow the set of proposals thatmay occur in equilibrium At a stationary equi-librium the proposer must offer at least vt to atype-t player in order to obtain that playerrsquossupport Since proposers wish to minimize theiroffers every legislator must be offered either vtor 0 in equilibrium For each coalition C letv(C) yent1

T vtnt(C) be the total ldquocostrdquo of C Fora proposing legislator i of type t let v t minCiCCWv(Ci) be the minimum totalpayment proposed to coalition partners besidesherself The proposer retains the surplus andthus receives 1 v t

The result also implies that in equilibriumthe ex ante expected payoffs will be identicalamong legislators of a given type The ex postdistribution will heavily favor the proposerhowever because the proposer is the residualclaimant Since most empirical research exam-ines data that are the outcome of the coalitionprocess (rather than the expected outcome) theex post distribution will be of considerable usein testing this model The ex ante share is whatone would expect across a large number ofreplications of a given situation and it is ofconsiderable importance in thinking about thefairness of any given distribution of votes andvoting rule

What share of the overall value can coali-tion partners expect to receive in the station-ary equilibria A key intuition behind ourresults is the idea of a legislatorrsquos relativeldquopricerdquo The price a type-t coalition partnercan command equals that playerrsquos continua-tion value vt divided by his or her share ofthe voting weight in the rth replication iewt(rw) That is the legislatorrsquos relative priceis t vtrwwt for a type-t legislator If alegislator is relatively expensive to include ina coalition then the continuation value forthat legislator is high relative to the share ofvotes they would bring to a coalition imply-ing a price of t 1

Our results require a general expression forvt Let qt be the probability that a type-tlegislator is chosen as a coalition partnergiven that she is not the proposer Stationarityimplies that

(1) vt pt 1 v t 1 pt qtvt

Rearranging we can write a type-t legislatorrsquoscontinuation value as

(2) vt pt 1 vt

1 1 pt qt

The exact characterization of the continua-tion value depends on the nature of the recog-nition probabilities We begin by consideringthe case in which recognition probabilities areproportional to voting weight ie pt wt(rw)

PROPOSITION 2 Suppose pt wt(rw) for allt There exists a finite r1 such that if r r1 thenin any stationary equilibrium vt wt(rw) (iet 1) for all t

The proof of Proposition 2 works by estab-lishing the impossibility of a legislator type forwhich t 1 This is because on average anysuch cheap legislator must be chosen as a coa-lition partner with probability too high to keepher expected payoff below her share of thevoting weight wt(rw)

Two features of the equilibria identified inProposition 2 are worth noting First the equi-librium payoffs are unique and thus the station-ary equilibrium is unique up to differences inthe probabilities of choosing coalition partnersthat do not affect the playersrsquo expected payoffsSecond in equilibrium for all t we can solve for

v t and qt

(3) v t w w vt rw wt rw

(4) qt rw wt

rw wt

Note that each typersquos probability of being in-cluded in a coalition is slightly decreasing in wtbut the probabilities are all approximately equalfor large r

We now turn to the second case where rec-ognition probabilities are constant across play-ers ie pt 1(rn) This case correspondsmost closely to the original model of Baron andFerejohn (1989) and is also useful for illustrat-ing the effects of different distributions of pro-posal power

There are two subcases which depend onthe distribution of voting weight across typesof legislators In the first an ldquointeriorrdquo equi-librium arises and prices are linear in weightsIn the second ldquocorneringrdquo is possible andsome prices are non-linear In both a type-t


playerrsquos continuation value is given by sub-stituting pt into (2)

(5) vt 1 v t

rn rn 1qt

The following proposition characterizes thestationary equilibrium for the interior subcaseRecall that w1 mintwt In any stationaryequilibrium the expected payoff (continuationvalue) of each legislator is proportional to hisvoting weight at an interior solution

PROPOSITION 3 Suppose pt 1(rn) for allt Also suppose n (w w )w1 There exists afinite r2 such that if r r2 then in any station-ary equilibrium vt wt(rw) (ie t 1) forall t

As with Proposition 2 this distribution ofpayoffs is unique Note also that in the case ofsimple majority rule (w w2) the condition inProposition 3 becomes n w(2w1) This im-plies that proportional payoffs will occur whenthere are many low-weight players in thepopulation

We may compare the result here with Prop-osition 2 by solving (2) to obtain an approxi-mate expression for qt

(6) qt 1 rw w

rn 1wt

Clearly qt is increasing in wt Thus in contrastwith (4) one of the main reasons that types withhigher voting weights receive higher expectedpayoffs in equilibrium is that proposers are morelikely to choose them as coalition partners

The next proposition characterizes the sta-tionary equilibrium in the subcase where theconditions of Proposition 3 are not satisfied Inthis subcase linear payoffs cannot be sustainedin equilibrium and the equilibrium payoffs areat a ldquocornerrdquo Types with the smallest votingweights have expected payoffs greater than theirrelative weight while those with the largestweights receive expected payoffs lower thantheir relative weight Additionally those withrelatively large voting weight have expectedpayoffs that are proportional to voting weightsalthough it is unclear whether proportionality

holds for those with relatively small votingweight

PROPOSITION 4 Suppose pt 1rn for all tAlso suppose n (w w )w1 There exists afinite r3 such that if r r3 then in any station-ary equilibrium there is a unique type t0 maxtn (w w )wt and a number mint 1 such that vt wt(rw) for all t t0 and vt wt(rw) for all t t0

The proof of Proposition 4 also establishesthat if n (w w )wt0

then type t0 receives adisproportionately large share This condition isautomatically satisfied if for example (w w )wt0

is non-integral If n (w w )wt0

however then type t0 may belong to the set oftypes receiving proportional payoffs

It is worth noting the role played by ldquocheaprdquoand ldquoexpensiverdquo types in equilibrium Cheaptypes must constitute at least w w of the totalvoting weight for otherwise each cheap legis-lator would almost always be chosen as a coa-lition partner Using (5) this can be shown to beimpossible This implies that coalitions can beconstructed using cheap legislators almost ex-clusively and thus qt 0 for all expensivetypes10 Expensive legislators are therefore al-most never included in winning coalitions

C Discussion

In the corner equilibrium of Proposition 4the high voting weight types are ldquounderpaidrdquorelative to their voting weight while low-weight types are ldquooverpaidrdquo This is the oppo-site of what tends to happen for many of thepower indices

The intuition behind this result is that equalproposal probabilities disproportionately benefitvoters with low weights A corner equilibriumoccurs when the expected payoff to some low-weight voter is greater than his share of theweight even when no other proposers everchoose him as a coalition partner The highpayoff that occurs in the event that he is aproposer determines his entire expected payoff

10 The equilibrium payoffs are ldquoapproximatelyrdquo uniquein the sense that the variations from substituting qt 0 into(5) for types t t0 diminish to zero as r 3


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This suggests that if recognition probabilitiesmore closely approximated voting weights thenlinearity in expected payoffs and voting weightswould be restored When recognition probabil-ities coincide exactly with voting weights Prop-osition 2 predicts payoffs that are proportionalto voting weights

Note also that corner equilibria are morelikely to occur when the threshold for victory wis low since lower values of w imply greaterbenefits to being proposer Thus we expectlinear payoffs when w is high that is when thecollective choice rule is supermajoritarian Anopen rule also makes proposing less valuablesince the proposer must offer higher payoffs tohis coalition partners and must often build su-permajorities to reduce the probability ofcounter-proposals (see Baron and Ferejohn1989) Thus we also expect that linear payoffswill be more likely under an open rule

Finally a comparative static result followsimmediately Except for types that ldquocornerrdquo asmall increase in voting weight always in-creases a playerrsquos expected payoff but a smallincrease in proposal probability does not Thetypes that corner have low voting weights Forthese types a small increase in voting weightcarries no benefit since these types will stillldquoneverrdquo be included in a coalition A smallincrease in proposal probability however in-creases the expected payoff for the types thatcorner

II Examples of Small Legislatures

The propositions above apply to large legis-latures The basic logic underlying these resultshowever holds for relatively small legislaturesas well

Before proceeding we must revisit the matterof voting weights Each finite-weighted votinggame can be represented by many different vec-tors of actual votesmdashthat is there are differentdistributions of votes that produce the exactsame set of winning coalitions Consider twolegislatures with 101 seats In one party A has50 seats party B has 50 seats and party C has1 seat In the other party A has 34 seats partyB has 34 seats and party C has 33 seats Thesehave the same set of minimum winning coali-tions (AB AC BC) What vector of weightsshould we choose to characterize this game

Isbell (1956) shows that if a game has ahomogeneous representation then this repre-sentation is unique and the voting weights areminimum integer weights Nonhomogeneousgames do not always have a unique representa-tion in minimum integer weights but mostgames with small numbers of players do andeven when multiple representations exist thedifferences between them are usually minorThus in what follows we will use minimuminteger weights11

A A Comparison of Indices for SmallLegislatures

The Shapley-Shubik and Banzhaf indicesrepresent what players are expected to receiveassuming random formation of coalitions Theexpected payoffs from the Baron-Ferejohncompetitive bargaining model are analogous inthat they reflect what players expect to receiveaveraging over all possible proposers Here wecompare the expected payoffs from competitivebargaining and the Shapley-Shubik and Ban-zhaf indices for legislators with 3 4 5 and 6members In all cases we assume all legislatorshave equal proposal probabilities

The following example illustrates the generalapproach to finding the unique equilibrium pay-offs (brute force)

Example Consider the 5-player game withweights (3 1 1 1 1) and w 4 (w 7 so asimple majority of the weight is needed to passa proposal) This corresponds to row 2 of Ta-ble 1 Call the player with a voting weight of 3legislator A and call the players with weights of1 type-B legislators Let vA and vB be the cor-responding continuation values We show thatall stationary equilibria yield vA 3vB (vA 37 and vB 17)

Suppose vA 3vB Then legislator A is rela-tively expensive and all type-B proposers avoidher whenever possible When legislator A is theproposer she offers vB to one type-B legislator

11 Aaron B Strauss (2003) offers an algorithm for com-puting the minimum integer voting weights of any weightedvoting game In all applications examined in this papereither the derived minimum integer weights were unique orthe differences in the weights were inconsequential


and keeps 1 vB for herself When any type-Blegislator is the proposer that legislator offersvB to the three other type-B legislators andkeeps 1 3vB for herself Thus vA and vBsatisfy vA 1frasl5 (1 vB) and vB 1frasl5 (1 3vB) 1frasl5 (1frasl4 )vB 3frasl5 vB (The first term in theexpression for vB is for the event that the giventype-B legislator is the proposer The secondterm is for the event that legislator A is theproposer And the third term is for the eventthat one of the other three type-B legislators isthe proposer) Solving these two equationsyields vA 319 and vB 419 But this con-tradicts the assumption that vA 3vB so therecan be no equilibria with vA 3vB

Next suppose vA 3vB Then type-B legis-lators are relatively expensive and proposersavoid them whenever possible When legislatorA is the proposer she offers vB to one type-Blegislator and keeps 1 vB for herself Whenany type-B legislator is the proposer that legis-lator offers vA to legislator A and keeps 1 vAfor herself Thus vA and vB satisfy vA 1frasl5 (1 vB) 4frasl5 vA and vB 1frasl5 (1 vA) 1frasl5 (1frasl4 )vB (The first term in the expression forvA is for the event that legislator A is the pro-poser and the second term is for the event that atype-B legislator is the proposer The first termin the expression for vB is for the event that thegiven type-B legislator is the proposer and thesecond term is for the event that legislator A isthe proposer) Solving these two equationsyields vA 1 and vB 0 But this contradictsthe assumption that vA 3vB so there can be noequilibria with vA 3vB

The only remaining possibility is that vA 3vB in which case proposers do not favor eithertype We can construct an equilibrium in whichthis occurs as follows When legislator A is theproposer she offers vB to 1 type-B legislator andkeeps 1 vB for herself When any type-Blegislator is the proposer that legislator offersvA to legislator A with a probability of 34 andshe offers vB to the three other type-B legislatorswith a probability of 14 in either case shekeeps 1 vA 1 3vB for herself Then vAand vB satisfy vA 1frasl5 (1 vB) 4frasl5 (3frasl4) vAand vB 1frasl5 (1 vA) 1frasl5 (1frasl4)vB 3frasl5 (1frasl4 )vBSolving these two equations yields vA 37 andvB 17

Table 1 compares the Shapley-Shubik indexBanzhaf index and expected payoffs under the

competitive bargaining game for all strongfour-player and five-player weighted votinggames and a large sample of six-playergames12 The table shows that the results de-rived in the previous section hold for thesesmaller games In all but two cases the com-petitive bargaining model yields expected pay-offs equal to the playersrsquo shares of the votingweights The exceptions occur in the last tworows of the table where a cornering equilibriumis reached and in these cases the differencebetween the expected payoff and proportional-ity is quite small

The right-hand side of the table shows thedifferences between the power indices and theexpected payoffs of the competitive bargaininggame In many cases the differences are slightbut in some cases they are largemdashsee eg thefive-player game with weights (3 1 1 1 1) andthe six-player game with weights (4 1 1 1 11) and (4 3 3 1 1 1) In all cases the expectedpayoffs of the players with the largest weightsare lower than their Shapley-Shubik and Ban-zhaf indices (except in one case where they arethe same) In all but one case the expectedpayoffs of the players with the smallest weightsare lower than their power indices

Finally we should note that the basic logicdoes not necessarily require that the smallestweight be equal to 1 For example consider theseven-player game (3 3 3 2 2 2 2) This is anonhomogeneous game in its minimal integerform It is straightforward to show that anystationary equilibrium produces expected pay-offs of (3frasl17 3frasl17 3frasl17 2frasl17 2frasl17 2frasl17 2frasl17 )

B An Application Council of Ministers ofthe European Union

A prominent example of the application ofcooperative game theory solutions is the designof the government of the European Union Eachmember country has a single vote in the EUCouncil of Ministers but to compensate fordifferences in country size the votes have dif-ferent weights An extensive body of literatureanalyzes the distribution of power in the Coun-cil using cooperative game theory solution

12 A simple game is strong if the complement of a losingcoalition is always winningmdashso there are no blockingcoalitions


concepts13 Choice of voting weights has be-come increasingly controversial as the EuropeanUnion has expanded the number of countriesand grown in authority In the most recent roundof reform proposed at the Nice Summit inDecember 2000 the delegations came armedwith computer programs to gauge whether they

would be winners or losers under the Niceaccord14

13 See note 3 above

14 ldquoNice Summit who hopes to join the EU and whenthe issues the objectives dilemmas and resultsrdquo SundayTimes of London December 10 2000 page 1GN ldquoEverydelegation including Britainrsquos took special software toNice to work out majorities blocking thresholds and fair-ness ratiosrdquomdashthe building blocks for power indices IanBlack ldquoEU tries to figure out what it decided at Nicerdquo TheGuardian December 22 2000 Viewed on-line through the

TABLE 1mdashPOWER INDICES VERSUS EXPECTED PAYOFFS

Voting weights

Power indices and expected payoffsRow 1 Shapley-Shubik

Row 2 BanzhafRow 3 Expected payoffs Power indices relative to expected payoffs

2 1 1 1 0500 0167 0167 0167 125 084 084 0840500 0167 0167 0167 125 084 084 0840400 0200 0200 0200

3 1 1 1 1 0600 0100 0100 0100 0100 140 070 070 070 0700636 0091 0091 0091 0091 148 064 064 064 0640429 0143 0143 0143 0143

2 2 1 1 1 0300 0300 0133 0133 0133 105 105 093 093 0930286 0286 0143 0143 0143 100 100 100 100 1000286 0286 0143 0143 0143

3 2 2 1 1 0400 0233 0233 0067 0067 120 105 105 060 0600385 0231 0231 0077 0077 116 104 104 069 0690333 0222 0222 0111 0111

2 1 1 1 1 1 0333 0133 0133 0133 0133 0133 117 093 093 093 093 0930333 0133 0133 0133 0133 0133 117 093 093 093 093 0930286 0143 0143 0143 0143 0143

4 1 1 1 1 1 0666 0067 0067 0067 0067 0067 150 060 060 060 060 0600750 0050 0050 0050 0050 0050 169 045 045 045 045 0450444 0111 0111 0111 0111 0111

3 2 1 1 1 1 0400 0200 0100 0100 0100 0100 120 090 090 090 090 0900393 0179 0107 0107 0107 0107 118 081 096 096 096 0960333 0222 0111 0111 0111 0111

2 2 2 1 1 1a 0233 0233 0233 0100 0100 0100 105 105 105 090 090 0900233 0233 0233 0100 0100 0100 105 105 105 090 090 0900222 0222 0222 0111 0111 0111

3 3 2 1 1 1 0300 0300 0200 0067 0067 0067 110 110 110 074 074 0740286 0286 0214 0071 0071 0071 105 105 118 078 078 0780273 0273 0182 0091 0091 0091

4 2 2 1 1 1 0467 0167 0167 0067 0067 0067 171 092 092 074 074 0740462 0154 0154 0077 0077 0077 169 085 085 085 085 0850364 0182 0182 0091 0091 0091

3 2 2 2 1 1a 0300 0167 0167 0167 0100 0100 110 092 092 092 110 1100300 0167 0167 0167 0100 0100 110 092 092 092 110 1100273 0182 0182 0182 0091 0091

4 3 3 1 1 1b 0367 0267 0267 0033 0033 0033 122 119 119 040 040 0400346 0269 0269 0039 0039 0039 115 120 120 047 047 0470300 0225 0225 0083 0083 0083

3 3 2 2 2 1ab 0234 0234 0167 0167 0167 0033 103 103 110 110 110 0360234 0234 0167 0167 0167 0033 103 103 110 110 110 0360227 0227 0152 0152 0152 0091

a Non-homogeneous gameb Corner-solution in legislative bargaining game


Table 2 compares the Banzhaf index withexpected payoffs in the bargaining game for twoperiods in the history of the European Commu-nity15 In the original European Community thedistribution of votes was France 4 Germany 4Italy 4 Belgium 2 the Netherlands 2 andLuxembourg 1 with 12 of 17 votes required topass a measure The corresponding minimuminteger weights which are the relevant weightsfor the analysis of the game are (2 2 2 1 1 0)with six of eight votes required to pass a mea-sure As the table shows the value of the Ban-zhaf index for Luxembourg is 0 because as aldquodummyrdquo player it could never be pivotal in anyminimum winning coalition16 Belgium and theNetherlands have more power than their voteweights (0118) or than their minimum integerweights (0125) By contrast the competitive

bargaining model predicts a nearly proportionalpattern of expected payoffs to Council mem-bers The expected payoffs are not strictly pro-portional because we reach a corner equilibrium(with a corresponding 2021) Note thatalthough the dummy player receives much morethan its Banzhaf value of zero its expectedpayoff is slightly less than its vote share in theoriginal game

Expansion of the European Community in1973 added the United Kingdom Denmark andIreland The new system gave 10 votes each toFrance Germany Italy and the United King-dom 5 votes each to Belgium and the Nether-lands 3 votes each to Denmark and Ireland and2 votes to Luxembourg At least 41 of 58 voteswere required to pass a measure Luxembourg isno longer a dummy but its Banzhaf index of0016 is much lower than its vote share of0034 By contrast the competitive bargainingmodel gives Luxembourg an expected valueof 003517 Note that the competitive bargainingpayoffs are nearly linear with respect to theoriginal voting weights Also the addition ofother countries decreases each original mem-berrsquos expected payoff

Dow-Jones database May 2 2002 Quentin Peel ldquoCom-ment amp Analysis Europersquos guaranteed gridlockmdashTheabsurdly complex voting system of the Nice Treatyrdquo Fi-nancial Times July 9 2001 Viewed on-line through theDow-Jones database May 2 2002

15 Brams and Affuso (1985) Lane and Maeland (1995)Felsenthal and Machover (1997 2000) and others havecalculated the power indices for the Council in each of theseperiods

16 Luxembourg also has zero power when recognitionprobabilities are proportional to the minimum integerweights since it is then never recognized and never in-cluded in a coalition

17 The minimum integer weights are (6 6 6 6 3 3 22 1) and the quota is 25 votes The equilibrium is at acorner

TABLE 2mdashPOWER INDICES VERSUS EXPECTED PAYOFFS IN THE EUROPEAN COMMUNITY (EUROPEAN UNION)

Original EuropeanCommunity1957ndash1973

FranceGermany

ItalyBelgium

Netherlands Luxembourg Quota

Voting weightsNormalizeda

4 2 1 12170235 0118 0059 07051

Minimum integer weightsNormalizeda

2 1 0 680250 0125 0000 0751

Banzhaf indices 0238 0143 0000 mdashExpected payoffs 0238 0119 0048 mdash

Expanded EuropeanCommunity1973ndash1981

FranceGermanyItaly UK

BelgiumNetherlands

DenmarkIreland Luxembourg Quota


10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141

Banzhaf indices 0167 0091 0066 0016 mdashExpected payoffs 0170 0085 0057 0035 mdash

a These are also expected payoffs in bargaining games with proportional recognition probabilities


These examples illustrate a paradoxical fea-ture of the power indices Luxembourgrsquos Ban-zhaf value grew from zero to 0016 eventhough its vote share shrank from 117 to 258This might reflect the subtle nature of power orit might reflect problems with the Banzhaf in-dex such as those noted by Holler (1982 1987)and Garrett and Tsebelis (2001) Our noncoop-erative bargaining model with an endogenousagenda apparently does not have this featureLuxembourgrsquos vote share fell and its power fellIt is ultimately an empirical matter whether thismodel better captures the essence of collectivedecision making in the European Union

III Evidence from Coalition Governments

An extensive amount of literature uses coali-tion governments as a proving ground forcooperative and noncooperative bargainingmodels18 In this work the players are the par-ties in a parliament That is each partyrsquos mem-bers are assumed to vote as a bloc and thereforeeach party can be treated as a single player in aweighted voting game The ldquodollarrdquo to be di-vided is the collection of cabinet positions in thegovernment typically 10 to 30 posts

In this section we study the relationship be-tween voting weights and cabinet posts to testbasic predictions of the model analyzed aboveWe study coalition governments from 1946 to2001 in Australia Austria Belgium DenmarkFinland Germany Iceland Ireland Italy Lux-embourg the Netherlands Norway Portugaland Sweden This is approximately the same setof countries and governments used in previousstudies19

One key prediction of the model above isthat the party that is recognized to form acoalitionmdashthe formateurmdashwill receive ashare of the cabinet posts that is much largerthan its share of the voting weight In theprevious section we considered the ex anteshare of the value that each player is predictedto receive Actual coalitions reflect the expost division conditional on a specific partybeing chosen formateur

The three-party legislatures illustrate thedifference between the ex ante and ex postdivisions and lend considerable support forthe model In our sample 61 parliaments hadexactly three non-dummy parties in the par-liament and therefore a (1 1 1) distributionof voting weights In all of these cases aminimal winning (ie two-party) coalitiongovernment formed On average the forma-teur in these cases received 65 percent and thepartner received 35 percent of the cabinetposts The prediction of the Baron-Ferejohnmodel is 667 percent for the formateur and333 percent for the partner Thus the forma-teur took almost exactly the predicted shareThe strong formateur effect indicates that theBaron-Ferejohn model of legislative bargain-ing captures an essential feature of coalitionformation

Previous empirical studies claim to find littleor no evidence of a formateur effect Browneand Franklin (1973) study the relationship be-tween a partyrsquos share of cabinet posts and itsshare of parliamentary seats in the governmentThey find a nearly linear pattern a result thatsome scholars have called ldquoGamsonrsquos Lawrdquo20

The parties that form the governments theyobserve are larger and tend to receive less thanan equal share of posts based on their seatshares Warwick and Druckman (2001) regresseach partyrsquos share of posts on its share ofseats along with an indicator of whether thatparty formed the government They find a neg-ative formateur effect Some theorist have in-terpreted these results to imply that there is nobargaining advantage to being chosen to pro-

18 See Eric Browne and Mark N Franklin (1973)Browne and John P Frendreis (1980) Schofield andMichael Laver (1985) and Paul V Warwick and James NDruckman (2001) See also Laver and W Ben Hunt (1992)

19 For example Warwick and Druckman (2001) studyall of these countries except Australia and Portugal overthe period 1946 ndash1989 Data on party seat shares andgovernment cabinet post allocations are from the follow-ing sources Browne and Dreijmanis (1982) Muller andStrom (2000) and the European Journal of PoliticalResearch ldquoPolitical Data Yearbookrdquo special issues1992ndash2001 Following previous researchers we includeall coalition governments including minority govern-ments and non-minimal-winning governments except afew cases where one party had an outright majority butstill formed a coalition government These were in the

period immediately following World War II in the formeraxis powers Austria Italy and West Germany plus afew in Australia

20 William A Gamson (1961) predicted such a relation-ship based on an informal argument


pose a government and as evidence againstBaron-Ferejohnndashtype models21

A significant problem with these empiricalstudies is that they use seat shares rather thanvoting weights as the independent variable Froma game-theoretic point of view voting weights aremore relevant for making predictionsmdashvirtuallyall bargaining results are expressed in terms ofvoting weights not seat shares In games withlarge numbers of players the difference betweenvoting weight shares and seat shares is smallbut most parliamentary games involve only ahandful of players and in these cases the dif-ferences are important In our data for example50 percent of the parliaments have six or fewerparties The correlation between seat shares andvoting weights is positive but not extraordinar-ily high (088) Parties with quite different seatshares often have the same voting weight Inaddition there is a tendency for larger parties tohave voting weights smaller than their share ofseats22

Using voting weights as the independentvariable leads to completely different conclu-sions about the importance of the formateurThe first column of Table 3 presents a regres-sion of a partyrsquos share of cabinet posts on apartyrsquos voting weight plus a dummy variablefor whether that party is formateur This columnis similar to the specifications in previous re-gression analyses such as those by Browne andFranklin (1973) and Warwick and Druckman(2001) but uses voting weights instead of seatshares The coefficient on Formateur Dummy is015 with a standard error of 0046 Thus onaverage the party that forms the governmentreceives 15 percent more of the cabinet poststhan one would expect from its voting weightalone23 Note that in all specifications we clusterthe observations by country This corrects thestandard errors for serial dependence of partieswithin governments and also for serial depen-dence within countries over time

21 For example Morelli writes ldquoAll the models based onBaron and Ferejohn (1989) yield a disproportionate payoffshare for the proposal maker regardless of the distribution ofseats and hence they are not consistent with the basicempirical findings In contrast the demand bargaininggame introduced here performs very wellrdquo (Morelli 1999p 810)

22 Take for example the Austrian government of 1986In the parliament the Social Democrats held 90 seats thePeoplersquos Party 81 seats and the Freedom Party 12 seats

The shares of seats for the Social Democrats PeoplersquosParty and Freedom Party were 049 044 and 007 respec-tively Their shares of the voting weights in the parliamentby contrast were 033 033 and 033

23 The estimates are nearly the same when we dropminority governments Note also that if we replicate theexisting regression analyses with our data using seat sharesinstead of voting weights then we find similar results tothose in Browne and Franklin (1973) Browne and Frendreis(1980) and Warwick and Druckman (2001) The estimatedcoefficient on the Formateur dummy is small and statisti-cally insignificant (coeff 0009 se 0020)

TABLE 3mdashVOTING WEIGHTS AND THE ALLOCATION OF CABINET POSTS IN PARLIAMENTARY

GOVERNMENTS 1946ndash2001(Dep var share of cabinet posts)

Unweighted Weighted

(1) (2) (1) (2)

Formateur dummy 015 mdash 024 mdash(0046) (0041)

Share of voting weight in Parliament 111 mdash 101 mdash(0130) (0120)

Formateur predicted payoff mdash 066 mdash 073(0036) (0034)

Partner predicted payoff mdash 104 mdash 090(0200) (0179)

Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682

Notes Clustered standard errors in parentheses where each cluster is a country Statisticallysignificant at the 001 level


Mik Laver
Highlight

The formateur effect in column 1 providesstrong evidence that the parties chosen to forma coalition typically receive more than theirvoting weight Even this coefficient probablyunderestimates the value of being formateurhowever because some cabinet ministries aremore valuable than others and the formateurtypically controls a disproportionate share ofthe more valuable posts Laver and Hunt (1992)find that parties uniformly rank the prime min-ister as the most valuable post followed in mostcases by finance and foreign affairs As War-wick and Druckman (2001) point out the par-ties that form governments virtually always takethe office of prime minister and they also typ-ically take finance or foreign affairs or bothFollowing Warwick and Druckman (2001) weexperimented with various ways of assigninghigher payoffs to the more important ministriesIn all cases we find that the estimated value ofbeing formateur increases Column 3 show theresults when we assign a value of 3 to primeminister 2 to finance and foreign affairs re-spectively and 1 to all other ministries Theestimated coefficient on Formateur Dummy in-creases to 02424

We can push the analysis one step fartherTaking the bargaining model literally implies aspecific functional form Suppose the total vot-ing weight in the parliament is w and the for-mateurrsquos voting weight is wf If the formateurforms a government with a total voting weightof wg then it is predicted to receive a payoff of(w wg wf)w If a party with voting weightwi is included as a partner in the coalition thenit is predicted to receive wiw We can there-fore regress a partyrsquos share of cabinet posts ontwo variables (a) an indicator of whether theparty is formateur times (w wg wf)wwhich we call Formateur Predicted Payoff and(b) an indicator of whether the party is notformateur times wiw which we call PartnerPredicted Payoff If the bargaining model thatwe have analyzed is exactly right then a regres-sion of a partyrsquos share of cabinet posts on thesetwo variables should have an intercept of 0

and the coefficients on both variables shouldequal 1

Column 2 in the table presents the coefficientestimates for this specification The exact spec-ification clearly fails The coefficient on Forma-teur Predicted Payoff is approximately 066Thus although formateurs get more than theirldquofair sharerdquo they receive somewhat less thanpredicted by the specific game form that wehave analyzed Interestingly the coefficient onPartner Predicted Payoff is quite close to itspredicted value of 1 however the intercept issignificantly different from 0 Column 4 pre-sents the regression estimates using the 3-2-1weighting described above The coefficient onFormateur Predicted Payoff rises to 073 and thecoefficient on Partner Predicted Payoff falls to09025

There is a variety of reasons not to expect aperfect fit First as noted above we are unsureof the value of different ministerial posts Sharptests of specific models therefore require addi-tional and accurate information about the valueof government positions Second there may berestrictions on the sorts of coalitions that canform owing to particular ideological battles Forexample postwar European governments al-most never include the Communist Party Insome countries the ldquoeffectiverdquo bargaining en-vironment may be a restricted variant of the fullgame in which certain parties are excluded Inthese case the playersrsquo ldquoeffectiverdquo votingweights and proposal probabilities are differentfrom those of the full game Third the randomrecognition assumption of the model may bewrong In some countries such as Austria thelargest party is always recognized first26

Fourth the model assumes a closed rule (iethe formateur makes offers that cannot beamended) an open rule may be more appropri-

24 It is difficult to measure the actual value of differentministries this is an important subject for further empiricalinquiry Several previous analyses test the linearity predic-tion as well by including squared terms or step-functionterms Without an accurate measure of the value of differentposts however these tests are probably biased

25 We tried a variety of weighting schemes Givinghigher values to Prime Minister Finance and ForeignAffairs we typically find that there is no statisticallysignificant difference between the slopes on FormateurPredicted Payoff and Partner Predicted Payoff but thatboth slopes are below 1 and the constant is alwayspositive

26 On the other hand Diermeier and Merlo (2004) pro-vide evidence that the assumption of random recognitionwith probabilities proportional to seat shares provides agood first approximation to the frequency with which par-ties act as formateur


ate These represent important areas for furtherempirical and theoretical investigation27

Overall however the patterns of coalitionformation in European governments providestrong evidence for the two main qualitativepredictions of the Baron-Ferejohn approachThe party recognized to form a coalition hasadded leverage and thus gets more out of agiven situation than its vote share Moreoverthe ldquopricerdquo to the formateur of including aparty in a coalition is linear in its votingweight

IV Discussion

In this paper we characterize the equilibriumof the Baron-Ferejohn model under weightedvoting The analytical advantage of this nonco-operative approach is that it endogenizes theldquocostrdquo of coalition partners As a result themodel predicts that any actor with k votes hasthe same price or expected value in vote tradingas k actors with one vote each This predictiondiffers from that of conventional power indiceswhich have been the main approach to the studyof weighted voting

We also find empirical support for the generalapproach embodied in this model using data oncoalition government formation from 1946 to2001 The model predicts that the player chosento form a coalition will receive disproportion-ately more than his or her voting weight in theex post distribution of payoffs We conduct thefirst statistical analysis of the relationship be-

tween voting weights and seat shares and find alarge advantage to being formateur To the ex-tent that the estimated relationships deviatefrom the modelrsquos predictions small parties re-ceive higher payoffs than predicted and forma-teurs appear not to extract the full advantage oftheir privileged positions These more detailedfindings are suggestive but tentative due to ourinability to measure accurately the value of dif-ferent cabinet posts Further empirical researchis clearly required

The model is clearly quite stylized Severalextensions and further generalizations meritspecial attention First our results suggest thatproportional payoffs are sustainable by a widerange of proposal probabilities A more generalmodel is needed however to examine the in-teraction between proposal power and votingpower Second as a game of pure distribution itexcludes situations in which playersrsquo prefer-ences are correlated For example players mayhave spatial or ideological preferences Anyplayer ldquocloserdquo to the formateur then receivesadditional benefits from the formateur havingbeen chosen Spatial neighbors then may becheaper partners Third the model does notconsider a number of common institutional fea-tures The game employs a closed rule allowingno amendments to a proposal The open ruleintroduces analytical complications that haveyet to be solved Also in most contexts theformation of the agenda is often not random butis shaped by political parties or legislative com-mittees We conjecture that any model in whichthere is a residual claimant such as the forma-teur will have similar predictions In particularthe price-theoretic substitution argument shouldhold quite generally

APPENDIX

PROOF OF PROPOSITION 1Existence follows from Theorem 1 of Banks and Duggan (2000) It is sufficient to verify

that our game satisfies the Limited Shared Weak Preference (LSWP) property of Banks andDuggan

For any legislator i and outcome of x X let Ri(x) and Pi(x) denote the weak and strict uppercontour sets of x respectively For any coalition C let RC(x) iCRi(x) and PC(x) iCPi(x)denote the set of alternatives weakly and strongly preferred to x by all members of C respectivelyLet PC(x) be the closure of PC(x) LSWP holds if for all C and x RC(x) 1 implies RC(x) PC(x)

To show this note that RC(x) 1 iff yeniC xi 1 Thus for all such x and C PC(x) x

27 Of course another problem is that the model predictsminimal winning coalitions but we do not always observethem Also we assumed an ldquointeriorrdquo equilibrium for all cases


Xxi xi i C 13 A Since X is a compact subset of n the closure of PC(x) is PC(x) x Xxi xi i C Hence PC(x) RC(x) which establishes the result

PROOF OF COMMENT 1We begin by defining some useful notation Let i (i 1 n) represent the continuation value

for legislator i Let i be the probability that legislator i is chosen as a coalition partner given thatshe is not the proposer Then i pt(1 i) (1 pt)ii where i is the minimum total paymentproposed to coalition partners besides herself Rearranging i (1 i)[1pt (1pt 1)i]

Suppose that legislators j and k are of type t and j k Also without loss of generality let k minit(i) i Then j k may be written as (1 j)[1pt (1pt 1)j] (1 k)[1pt (1pt 1)k] For this to be true either (i) j k or (ii) j k (or both) We show that neitherof these holds addressing them in order

The probability of being chosen as a coalition partner can be represented as the sum of theprobabilities of being chosen partner by each of the other players weighted by the recognitionprobability Let legislator i be of type t and let i

m be the probability that i is chosen as acoalition partner given that legislator m 13 i is the proposer Then i [1(1 pt)] yenm13ipt(m)i

m We show that each element of these sums is such that jm k

m Consider firstthe situations where the proposer is m 13 j k The condition j k implies that in anycoalition formed by m legislator j will be included only if legislator k is also includedbecause for the same number of votes the proposer would choose the lower cost partnerTherefore j

m km for any proposer m 13 j k Second consider the situations in which j or

k is the proposer (these are the remaining probabilities in the calculation of j and k) It issufficient to show j

k kj If k

j 1 then the result is established So suppose kj 1 Then

there exists a cheapest set of coalition partners Cj for proposer j such that k Cj and Cj j W Since proposer k may select Cj as coalition partners and Cj k W sheincludes legislator j as a coalition partner only if there exists some set of coalition partners Ck j satisfying Ck k W and yeni Cj

i yeni Cki But then j k implies the existence

of a set of coalition partners for j Ck (Ck j) k satisfying yeni Cki yeni Ck

i Sinceyeni Ck

i yeni Cji we have yeni Ck

i yeniCji and so no such Ck exists Thus legislator

j cannot belong to any least cost set of coalition partners for proposer k ie jk 0

establishing the result Therefore j kTurning to (ii) let j k If there exists a minimum-cost set of coalition partners Cj for

proposer j such that k Cj and Cj j W then Cj is available to proposer k so k jOtherwise the coalition partners Cjk are available to proposer k and k j j k Thus k j or equivalently (1 j) (1 k) This implies (1 j)z (1 k)z for z z 1 only if z z 1 Using the result of (i) this implies (1 j)[1pt (1pt 1)j] (1 k)[1pt (1pt 1)k] only if j k 1 Now note that j 1 implies j 1 j orequivalently j yeniCj

i 1 Since w w wT we have Cj j 13 N and thus there existssome legislator l such that l 0 If l is recognized as proposer then since N j W we havel 1 j and hence l (1 l)(1pt(l )) j(1pt(l )) 0 contradiction Thus j 13 1 Weconclude that j k cannot hold so j k

Therefore j k implies j k and j k a contradiction

PROOFS OF PROPOSITIONS 2 3 AND 4To prove Propositions 2 3 and 4 we employ three lemmas The lemmas require some additional

notation Recall that the relative price of a type-t legislator is t vtrwwt Let TL t Tt minss t 13 1 denote the (possibly empty) set of ldquocheapestrdquo types Where convenient we willuse t and t to denote distinct types where t t Finally we use y and y to denote the lowestinteger greater than or equal to and highest integer lower than or equal to y respectively

LEMMA 1 In a stationary equilibrium for any 13 0 there exists a finite r13 such that for any t T and r r13 v t (rw wt)(rw) 13


PROOFLet tjj1

T represent an ordering of types such that ta tb

for any a b For a given r andproposer i of type t construct a coalition Ct i as follows Begin with Ct A and add legislatorsaccording to the following algorithm

1 If w(Ct) wt rw then Ct is complete2 Otherwise add a legislator (not i) of type tm where m min jntj

(Ct i) rntj and return

to step (1)

By this procedure w(Ct) rw wt and v(Ct) w(Ct)(rw wt) To see why the latter statementholds suppose otherwise Then v(NCt i) w(NCt i)(rw wt) This implies thatthere exists a type-t legislator in NCt i and type-t legislator in Ct such that t t whichcontradicts the construction of Ct

Clearly v(Ct) v t and thus w(Ct)(rw wt) v t Also by the algorithm above w(Ct) exceedsrw wt by less than wT where wT is the weight of the most expensive type So w(Ct) rw wt wT 1 Then v t (rw wt)(rw) 13 if

(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13

for r sufficiently large and some 131 132 0 It is then sufficient to verify that

(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132

Simplifying (A2) is satisfied for r r (wt(2w2131))(w131 w (w w131)2 4w2131) and(A3) is satisfied for r r (wtw) (wT 1)(132w) Thus letting r13 maxr r establishesthe result

LEMMA 2 In a stationary equilibrium there exists a finite rc such that for any r rc yentTLwtnt w

PROOFWe show that yentTL

wtnt w for r sufficiently large implies that qt3 1 for any t TL whichin turn implies t TL an obvious contradiction

If yentTLwtnt w then a least-cost winning coalition C must contain at least rwT 1

legislators of types not in TL not including the proposer C must then contain at least (rwT 1)(T 1) legislators of some type t TL This implies that for any k

(A4) if r wTk 1T 1 then nt C k

Now consider how many legislators of any type t TL can be excluded from C If nt(C) wtand nt(NC) wt then the proposer can replace wt type-t legislators in C with wt type-tlegislators in NC and achieve voting weight w(C) at strictly lower cost This contradicts theassumption that C is a least-cost winning coalition By (A4) nt(C) wt when r r wT(wT 1)(T 1) Thus r r implies nt(NC) wt and hence nt(NC) wT for all t TL

Suppose r r Then for any proposal from any proposer the probability that a legislator of typet TL is included in the proposed coalition is at least 1 wT(rnt 1) if the proposer is of typet or at least 1 wT(rnt) otherwise Therefore qt 1 wT(rnt 1) for all t TL This impliesthat for any (0 1) if r r maxr (1 wT)nt then qt 1


By (2) the type-t legislatorrsquos continuation value is vt [pt(1 v t)][1 (1 pt)qt] ByLemma 1 an upper bound on v t is [(rw wt)(rw)] 13 Combining results for any 13 0 and 0 if r rc maxr13 r then vt [pt(1 (rw wt)rw 13)][1 (1 pt)(1 )] This isgreater than wt(rw) if

pt 1 rw wt rw 13

1 1 pt 1

wt

rw

pt rw rw wt rw13 wt pt 1 pt

(A5) ptrw w w13

wt 1 pt

Let 13 1 w w let satisfy (A5) and choose r and r13 accordingly Then for all r rc we havevt wtw contradicting t TL

LEMMA 3 In a stationary equilibrium if r rc as defined in Lemma 2 then qt 2TwT(rnt) forall t TL

PROOFWe first derive a relationship between the number of legislators of any type t TL that can

belong to any least-cost winning coalition C and the number of legislators of some type t (t TL)that can belong to NC Suppose nt(C) type-t legislators are in C By Lemma 2 there exists rc suchthat for r rc yentTL

wtnt(NC) nt(C)wt Thus for some type t arg maxtTLnt(NC)

nt(NC) nt(C)wtyentTLwt

We now derive an upper bound on nt(C) Note that if nt(C) wt and nt(NC) wt then theproposer can exchange wt type-t legislators in C for wt type-t legislators in NC and achievevoting weight w(C) at strictly lower cost This contradicts the assumption that C is a least-costwinning coalition Thus either nt(C) wt or wt nt(C)wtyentTL

wt which simplifies to 1 nt(C)yentTL

wt or nt(C) 2 yentTLwt Thus nt(C) maxwt 2 yentTL

wt 2TwT in anyleast-cost winning coalition

Aggregating over all proposers and least-cost winning coalitions this implies qt 2TwT(rnt)

PROOF OF PROPOSITION 2Suppose that vt t for all types t TL Note that this implies TL 13 T for otherwise yent ntvt

1 an obvious contradiction We show that for r sufficiently large no such TL existsWe begin by deriving a lower bound on vt in two parts First Lemma 1 provides a lower bound

on 1 v t for any 13 0 there exists r13 such that if r r13 then 1 v t (rw rw wt)(rw) 13

Second we derive a lower bound on qt for some t TL Every proposer must assemble acoalition C with a weight (not including herself) of at least rw wT Thus

tTL

rqtntCwt tTL

rqtntCwt rw wT

Simplifying and applying Lemma 3 yields

tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt

for r rc Thus for any 0 and for all r r wT(1 2T yentTLnt(C)wt) we have yentTL

qtnt(C)wt w Thus for r maxrc r there exists some t TL such that qt (w


)yentTLnt(C)wt Since yentTL

nt(C)wt w this implies that for sufficiently small and r maxrc r qt ww for some w w

Substituting into (2) vt wt(rw) implies

pt rw rw wt rw 13

1 1 pt ww

wt

rw

Letting pt wt(rw) and simplifying yields

wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13

The left-hand side of (A6) is negative while for 13 sufficiently small the right-hand side is positiveand thus (A6) cannot hold Thus for suitably chosen 13 and r r1 maxrc r13 r we havevt wtw a contradiction

PROOF OF PROPOSITION 3Since yent rntvt 1 vt wt(rw) for all t if t 1 for all t It is therefore sufficient to provide

conditions under which there is no type such that t 1Note that t 1 for all t TL and therefore we check only t TL By Lemma 3 qt 2TwT(rnt)

for any t TL and all r rc By Lemma 2 and the definition of TL v t (rw wt)(rw) where mint Thus using (5) for r rc and t TL we obtain the following upper bound on vt

vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt

This implies the following upper bound on t

t w

w wt n wt r n 1r2TwT wt rnt

Thus t 1 for t TL only if

(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r

Simple manipulation reveals that [wt (2TwTwtnt)(n 1r)]r for type t and any 0 ifr rt() [2nTwT nt (2nTwT nt)

2 8ntTwTwt](2nt)wt Thus there exists t 0such that if r rt(t) then (A7) is satisfied only by n (w w )wt

This implies that for any r r2 maxrc maxtrt(t) (A7) holds for type t only if n (w w )wt Since w1 mintwt (A7) cannot hold for any type t if

(A8) n w w

w1

Thus when r r2 and (A8) holds there does not exist any type such that t 1 and hence t 1 for all t


PROOF OF PROPOSITION 4We first establish sufficient and necessary conditions for an equilibrium in which t 1 and hence

TL 13 T and then show uniqueness of the cutoff type t0 Let mint Also assume throughoutthat r rc

We begin by deriving a lower bound on vt and use it to derive a sufficient condition for t TL This is done by substituting an upper bound on v t and letting qt 0 in (5) Toderive an upper bound on v t note that a type-t proposer must build a coalition with partnersC such that w(C) rw wt By Lemma 2 there exists a set of coalition partners C such thatw(C) rw wt wT and C it(i) TL The proposer can then complete the coalitionby adding a legislator of type T Comment 1 implies that vT 1(rnT) and thus v t (rw wt)(rw) 1(rnT)

Substituting into (5) yields the following lower bound on vt

(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn

This implies the following lower bound on t

t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt

(A11) n w w wnT wt r

wt

Note that if (w w )wt is integral for some type t then (A11) is satisfied by n (w w )wt if(wnT wt)r wt or r w(nTwt) 1 Otherwise if (w w )wt is not integral for some t thenlet t (w w )wt (w w )wt Now (A11) is satisfied by n (w w )wt if (wnT wt)r t or r (wnT wt)t

We therefore conclude that if r r maxw(nTw1) 1 maxt(wnT wt)t then

(A12) n w w

wt

is sufficient to ensure that for any t t 1 and thus t TL and TL 13 T Since w1 mintwt (A12)will hold for some t if n (w w )w1 Note that if (A12) holds for some type t then it must alsohold for any t t

Next consider necessary conditions for t TL By the proof of Proposition 3 if r r2 then t 1 for t TL only if

(A13) n w w

wt

Putting the results together suppose that r r3 maxr2 r rc and let t0 maxt(A13)holds Clearly t0 is unique Then in a stationary equilibrium by (A13) t 1 (t TL)for t t0 and by (A12) t 1 (t TL) for t t0 If additionally t0 satisfies (A12) thent0

1


REFERENCES

Ansolabehere Stephen Snyder James M Jrand Ting Michael M ldquoBargaining in Bicam-eral Legislatures When and Why DoesMalapportionment Matterrdquo American Polit-ical Science Review 2003 97(3) pp 471ndash81

Banks Jeffrey S and Duggan John ldquoA Bargain-ing Model of Collective Choicerdquo AmericanPolitical Science Review 2000 94(1) pp73ndash88

Banzhaf John F III ldquoWeighted Voting DoesnrsquotWork A Mathematical Analysisrdquo RutgersLaw Review 1965 19(2) pp 317ndash43

Banzhaf John F III ldquoOne Man 3312 Votes AMathematical Analysis of the Electoral Col-legerdquo Villanova Law Review 1968 13(2)pp 304ndash32

Baron David P ldquoA Spatial Bargaining Theoryof Government Formation in ParliamentarySystemsrdquo American Political Science Re-view 1991 85(1) pp 137ndash64

Baron David P ldquoA Dynamic Theory of Collec-tive Goods Programsrdquo American PoliticalScience Review 1996 90(2) pp 316ndash30

Baron David P ldquoComparative Dynamics of Par-liamentary Governmentsrdquo American PoliticalScience Review 1998 92(3) pp 593ndash609

Baron David P and Ferejohn John A ldquoBargain-ing in Legislaturesrdquo American Political Sci-ence Review 1989 83(4) pp 1181ndash1206

Baron David and Kalai Ehud ldquoThe SimplestEquilibrium of a Majority-Rule DivisionGamerdquo Journal of Economic Theory 199361(2) pp 290ndash301

Bates Robert H and Lien Da-Hsiang DonaldldquoOn the Operations of the International Cof-fee Agreementrdquo International Organization1985 39(3) pp 553ndash59

Bennett Elaine ldquoThe Aspiration Approach toPredicting Coalition Formation and PayoffDistributions in Sidepayment Gamesrdquo Inter-national Journal of Game Theory 1983a12(1) pp 1ndash28

Bennett Elaine ldquoCharacterization Results forAspirations in Games with SidepaymentsrdquoMathematical Social Sciences 1983b 4(3)pp 229ndash41

Berg Sven Maeland Reinert Stenlund Hansand Lane Jan-Erik ldquoPolitics Economics andthe Measurement of Powerrdquo ScandinavianPolitical Studies 1993 16(3) pp 251ndash68

Bindseil Ulrich and Hantke Cordula ldquoThePower Distribution in Decision Makingamong EU Member Statesrdquo European Jour-nal of Political Economy 1997 13(1) pp171ndash85

Binmore Ken ldquoPerfect Equilibria in BargainingModelsrdquo in Kenneth Binmore and ParthaDasgupta eds The economics of bargainingOxford Basil Blackwell 1987 pp 77ndash105

Brams Steven J and Affuso Paul J ldquoNew Par-adoxes of Voting Power on the EC Council ofMinistersrdquo Electoral Studies 1985 4(2) pp135ndash39

Browne Eric C and Dreijmanis John Govern-ment coalitions in western democracies NewYork Longman Inc 1982

Browne Eric C and Franklin Mark N ldquoAspectsof Coalition Payoffs in European Parliamen-tary Democraciesrdquo American Political Sci-ence Review 1973 67(2) pp 453ndash69

Browne Eric C and Frendreis John P ldquoAllocat-ing Coalition Payoffs by Conventional NormAn Assessment of the Evidence from CabinetCoalition Situationsrdquo American Journal ofPolitical Science 1980 24(4) pp 753ndash68

Bruckner Matthias and Peters Torsten ldquoFurtherEvidence on EU Voting Powerrdquo Journal ofTheoretical Politics 1996 8(3) pp 415ndash19

Calvert Randall and Dietz Nathan ldquoLegislativeCoalitions in a Bargaining Model with Exter-nalitiesrdquo University of Rochester Wallis In-stitute Working Papers No 16 1996

Calvo Emilio and Lasaga J Javier ldquoProbabilis-tic Graphs and Power Indices An Applica-tion to the Spanish Parliamentrdquo Journal ofTheoretical Politics 1997 9(4) pp 477ndash501

Chari V Jones Larry E and Marimon RamonldquoThe Economics of Split-Ticket Voting inRepresentative Democraciesrdquo AmericanEconomic Review 1997 87(5) pp 957ndash76

Chatterjee Kalyan Dutta Bhaskar Ray Debrajand Sengupta Kunal ldquoA NoncooperativeTheory of Coalitional Bargainingrdquo Review ofEconomic Studies 1993 60(2) pp 463ndash77

Deegan John Jr and Packel Edward W ldquoANew Index of Power for Simple n-PersonGamesrdquo International Journal of Game The-ory 1978 7(2) pp 113ndash23

Diermeier Daniel and Feddersen Timothy JldquoCohesion in Legislatures and the Vote ofConfidence Procedurerdquo American PoliticalScience Review 1998 92(3) pp 611ndash21

Diermeier Daniel and Merlo Antonio ldquoGovern-


ment Turnover in Parliamentary Democra-ciesrdquo Journal of Economic Theory 200094(1) pp 46ndash79

Diermeier Daniel and Merlo Antonio ldquoAn Em-pirical Investigation of Coalitional Bargain-ing Proceduresrdquo Journal of PublicEconomics 2004 88(3ndash4) pp 783ndash97

Diermeier Daniel and Myerson Roger B ldquoBicam-eralism and Its Consequences for the InternalOrganization of Legislaturesrdquo American Eco-nomic Review 1999 89(5) pp 1182ndash96

Dreyer Jacob S and Schotter Andrew ldquoPowerRelationships in the International MonetaryFund The Consequences of Quota ChangesrdquoReview of Economics and Statistics 198062(1) pp 97ndash106

Eraslan Hulya ldquoUniqueness of Stationary Equi-librium Payoffs in the Baron-FerejohnModelrdquo Journal of Economic Theory 2002103(1) pp 11ndash30

Felsenthal Dan S and Machover Moshe ldquoTheWeighted Voting Rule in the EUrsquos Council ofMinisters 1958ndash1995 Intentions and Out-comesrdquo Electoral Studies 1997 16(1) pp33ndash47

Felsenthal Dan S and Machover Moshe Themeasurement of power theory and practiceproblems and paradoxes Cheltenham Ed-ward Elgar 1998

Felsenthal Dan S and Machover Moshe ldquoEn-largement of the EU and Weighted Voting inIts Council of Ministers 1958ndash1995rdquo Lon-don School of Economics and Political Sci-ence Centre for the Philosophy of theNatural and Social Sciences Voting PowerReport VPP 0100 2000

Felsenthal Dan S and Machover Moshe ldquoMythsand Meanings of Voting Power Commentson a Symposiumrdquo Journal of TheoreticalPolitics 2001 13(1) pp 81ndash97

Gamson William A ldquoA Theory of CoalitionFormationrdquo American Sociological Review1961 26(3) pp 373ndash82

Garrett Geoffrey and Tsebelis George ldquoMoreReasons to Resist the Temptation of PowerIndices in the EUrdquo Journal of TheoreticalPolitics 1999a 11(3) pp 331ndash38

Garrett Geoffrey and Tsebelis George ldquoWhyResist the Temptation of Power Indices in theEUrdquo Journal of Theoretical Politics 1999b11(3) pp 291ndash308

Garrett Geoffrey and Tsebelis George ldquoEvenMore Reasons to Resist the Temptation of

Power Indices in the EUrdquo Journal of Theo-retical Politics 2001 13(1) pp 99ndash105

Gul Faruk ldquoBargaining Foundations of Shap-ley Valuerdquo Econometrica 1989 57(1) pp81ndash95

Harrington Joseph E Jr ldquoThe AdvantageousNature of Risk Aversion in a Three-PlayerBargaining Game Where Acceptance of aProposal Requires a Simple Majorityrdquo Eco-nomics Letters 1989 30(3) pp 195ndash200

Harrington Joseph E Jr ldquoThe Power of theProposal Maker in a Model of EndogenousAgenda Formationrdquo Public Choice 1990a64(1) pp 1ndash20

Harrington Joseph E Jr ldquoThe Role of RiskPreferences in Bargaining When Acceptanceof a Proposal Requires Less Than UnanimousApprovalrdquo Journal of Risk and Uncertainty1990b 3(2) pp 135ndash54

Hart Sergiu and Mas-Colell Andreu ldquoBargain-ing and Valuerdquo Econometrica 1996 64(2)pp 357ndash80

Herne Kaisa and Nurmi Hannu ldquoThe Distribu-tion of A Priori Voting Power in the ECCouncil of Ministers and European Parlia-mentrdquo Scandinavian Political Studies 199316(3) pp 269ndash84

Holler Manfred J ldquoForming Coalitions andMeasuring Powerrdquo Political Studies 198230(2) pp 262ndash71

Holler Manfred J ldquoParadox Proof DecisionRules in Weighted Votingrdquo in Manfred JHoller ed The logic of multiparty systemsDordrecht The Netherlands Martinus-Nijhoff 1987 pp 425ndash36

Holler Manfred J and Widgren Mika ldquoWhyPower Indices for Assessing EuropeanDecision-Makingrdquo Journal of TheoreticalPolitics 1999 11(3) pp 321ndash30

Hosli Madeleine O ldquoAdmission of EuropeanFree Trade Association States to the Euro-pean Community Effects on Voting Powerin the European Community Council of Min-istersrdquo International Organization 199347(4) pp 629ndash43

Hosli Madeleine O ldquoThe Balance betweenSmall and Large Effects of a Double Major-ity System on Voting Power in the EuropeanUnionrdquo International Studies Quarterly1995 39(3) pp 352ndash70

Isbell John R ldquoA Class of Majority GamesrdquoQuarterly Journal of Mathematics OxfordSeries 1956 7(27) pp 183ndash87


Jackson Matthew O and Moselle Boaz ldquoCoali-tion and Party Formation in a LegislativeVoting Gamerdquo Journal of Economic Theory2002 103(1) pp 49ndash87

Johnston R J ldquoThe Conflict over QualifiedMajority Voting in the European UnionCouncil of Ministers An Analysis of the UKNegotiating Stance Using Power IndicesrdquoBritish Journal of Political Science 199525(2) pp 245ndash54

Konig Thomas and Brauninger Thomas ldquoPowerand Political Coordination in American andGerman Multi-Chamber Legislationrdquo Jour-nal of Theoretical Politics 1996 8(3) pp331ndash60

Konig Thomas and Brauninger Thomas ldquoTheInclusiveness of European Decision RulesrdquoJournal of Theoretical Politics 1998 10(1)pp 125ndash42

Lane Jan-Erik and Maeland Reinert ldquoVotingPower under the EU Constitutionrdquo Journalof Theoretical Politics 1995 7(2) pp 223ndash30

Lane Jan-Erik and Maeland Reinert ldquoConstitu-tional Power in the EU Reply to MatthiasBruckner and Torsten Petersrdquo Journal ofTheoretical Politics 1996 8(4) pp 547ndash52

Lane Jan-Erik Maeland Reinert and BergSven ldquoThe EU Parliament Seats States andPolitical Partiesrdquo Journal of Theoretical Pol-itics 1995 7(3) pp 395ndash400

Lane Jan-Erik Maeland Reinert and BergSven ldquoVoting Power under the EU Constitu-tionrdquo in Svein S Andersen and Kjell AEliassen eds The European Union Howdemocratic is it London SAGE Publica-tions 1996 pp 165ndash86

Laruelle Annick and Widgren Mika ldquoIs the Al-location of Voting Power among EU StatesFairrdquo Public Choice 1998 94(3ndash4) pp317ndash39

Laver Michael and Hunt W Ben Policy andparty competition New York Routledge1992

LeBlanc William Snyder James M Jr and Tri-pathi Micky ldquoMajority-Rule Bargaining andthe Under Provision of Public InvestmentGoodsrdquo Journal of Public Economics 200075(1) pp 21ndash47

Lucas William F ldquoMeasuring Power inWeighted Voting Systemsrdquo in Steven JBrams William F Lucas and Philip D Straf-fin Jr eds Political and related models

New York Springer-Verlag 1978 pp 183ndash238

March James G ldquoAn Introduction to the Theoryand Measurement of Influencerdquo American Po-litical Science Review 1955 49(2) pp 431ndash51

McCarty Nolan M ldquoPresidential Pork Execu-tive Veto Power and Distributive PoliticsrdquoAmerican Political Science Review 2000a94(1) pp 117ndash29

McCarty Nolan M ldquoProposal Rights VetoRights and Political Bargainingrdquo AmericanJournal of Political Science 2000b 44(3)pp 506ndash22

McKelvey Richard D Ordeshook Peter C andWiner Mark D ldquoThe Competitive Solutionfor n-Person Games without TransferableUtility with an Application to CommitteeGamesrdquo American Political Science Review1978 72(2) pp 599ndash615

McKelvey Richard D and Riezman RaymondldquoSeniority in Legislaturesrdquo American Politi-cal Science Review 1992 86(4) pp 951ndash65

Merlo Antonio and Wilson Charles A ldquoA Sto-chastic Model of Sequential Bargaining withComplete Informationrdquo Econometrica 199563(2) pp 371ndash99

Merrill Samuel III ldquoCitizen Voting Power un-der the Electoral College A StochasticModel Based on State Voting PatternsrdquoSIAM Journal on Applied Mathematics1978 34(2) pp 376ndash90

Moldovanu Benny and Winter Eyal ldquoOrder In-dependent Equilibriardquo Games and EconomicBehavior 1995 9(1) pp 21ndash34

Morelli Massimo ldquoDemand Competition andPolicy Compromise in Legislative Bargain-ingrdquo American Political Science Review1999 93(4) pp 809ndash20

Muller Wolfgang C and Strom Kaare Coalitiongovernments in western Europe Oxford Ox-ford University Press 2000

Norman Peter ldquoLegislative Bargaining and Co-alition Formationrdquo Journal of EconomicTheory 2002 102(2) pp 322ndash53

Nurmi Hannu and Meskanen Tommi ldquoA PrioriPower Measures and the Institutions of theEuropean Unionrdquo European Journal of Po-litical Research 1999 35(2) pp 161ndash79

Okada Akira ldquoA Noncooperative CoalitionalBargaining Game with Random ProposersrdquoGames and Economic Behavior 1996 16(1)pp 97ndash108

Owen Guillermo ldquoEvaluation of a Presidential


Election Gamerdquo American Political ScienceReview 1975 69(3) pp 947ndash53

Rabinowitz George and MacDonald StuartElaine ldquoThe Power of the States in US Pres-idential Electionsrdquo American Political Sci-ence Review 1986 80(1) pp 65ndash87

Rapoport Amnon and Golan Esther ldquoAssess-ment of Political Power in the Israeli Knes-setrdquo American Political Science Review1985 79(3) pp 673ndash92

Riker William H ldquoSome Ambiguities in theNotion of Powerrdquo Econometrica 196458(2) pp 341ndash49

Rubinstein Ariel ldquoPerfect Equilibrium in a Bar-gaining Modelrdquo Econometrica 1982 50(1)pp 97ndash109

Schofield Norman ldquoThe Kernel and Payoffs inEuropean Government Coalitionsrdquo PublicChoice 1976 26(26) pp 29ndash49

Schofield Norman ldquoGeneralized BargainingSets for Cooperative Gamesrdquo InternationalJournal of Game Theory 1978 7(2) pp183ndash99

Schofield Norman ldquoBargaining Set Theory andStability in Coalition Governmentsrdquo Mathe-matical Social Sciences 1982 3(1) pp 9ndash32

Schofield Norman ldquoBargaining in WeightedMajority Voting Games with an Applicationto Portfolio Distributionrdquo in Manfred JHoller ed The logic of multiparty systemsDordrecht The Netherlands Martinus-Nijhoff 1987 pp 51ndash65

Schofield Norman and Laver Michael ldquoBargain-ing Theory and Portfolio Payoffs in EuropeanCoalition Governments 1945ndash1983rdquo BritishJournal of Political Science 1985 15(2) pp143ndash64

Selten Reinhardt ldquoA Noncooperative Model ofCharacteristic Function Bargainingrdquo in Rob-ert Aumann John Harsanyi Werner Hilde-brand Michael Maschler Micha A PerlsJoachim Rosenmuller Reinhardt SeltenMartin Shubik and Gerald L Thompsoneds Essays in game theory and mathemati-cal economics in honor of Oskar Morgen-stern Manheim Bibliographiches Institut1981 pp 131ndash51

Shapley Lloyd S ldquoA Value for n-Person

Gamesrdquo in Harold W Kuhn and Albert WTucker eds Contributions to the theory ofgames Annals of mathematical studies Vol28 Princeton Princeton University Press1953 pp 307ndash17

Shapley Lloyd S and Shubik Martin ldquoAMethod for Evaluating the Distribution ofPower in a Committee Systemrdquo AmericanPolitical Science Review 1954 48(3) pp787ndash92

Strauss Aaron B ldquoAn Algorithm to CalculateMinimum Integer Weights for Arbitrary Vot-ing Gamesrdquo Unpublished Paper 2003

Strom Kaare Budge Ian and Laver Michael JldquoConstraints on Cabinet Formation in Parlia-mentary Democraciesrdquo American Journal ofPolitical Science 1994 38(2) pp 303ndash35

Sutter Matthias ldquoFair Allocation and Re-Weighting of Votes and Voting Power in theEU before and after the Next EnlargementrdquoJournal of Theoretical Politics 2000a 12(4)pp 433ndash49

Sutter Matthias ldquoFlexible Integration EMUand Relative Voting Power in the EUrdquo Pub-lic Choice 2000b 104(1ndash2) pp 41ndash62

Teasdale Anthony L ldquoThe Politics of MajorityVoting in Europerdquo Political Quarterly 199667(2) pp 101ndash15

Tsebelis George and Garrett Geoffrey ldquoAgendaSetting Power Power Indices and DecisionMaking in the European Unionrdquo Interna-tional Review of Law and Economics 199616(3) pp 345ndash61

Warwick Paul V and Druckman James NldquoPortfolio Salience and the Proportionality ofPayoffs in Coalition Governmentsrdquo BritishJournal of Political Science 2001 31(4) pp627ndash49

Widgren Mika ldquoVoting Power in the EC Deci-sion Making and the Consequences of TwoDifferent Enlargementsrdquo European Eco-nomic Review 1994 38(5) pp 1153ndash70

Widgren Mika ldquoA Note on Matthias SutterrdquoJournal of Theoretical Politics 2000 12(4)pp 451ndash54

Winter Eyal ldquoVoting and Vetoingrdquo AmericanPolitical Science Review 1996 90(4) pp813ndash23


of influencerdquo2 These arguments have had asignificant impact on the field of political econ-omy and debates about the design of constitu-tions For example a large amount of literaturehas emerged employing power indices to studythe voting weights in the government of theEuropean Union3

While widely used power indices and the ar-guments they justify run contrary to an importantintuition Elementary microeconomic theoryteaches that in competitive situations perfect sub-stitutes have the same price4 In a political settingin which votes might be traded or transferred inthe formation of coalitions one might expect thesame logic to apply If a player has k votes thenthat player should command a price for thosevotes equal to the total price of k players that eachhave one vote In terms of expected payoffs theplayer with k votes should expect to have a payoffk times as great as the payoff expected by a playerwith one vote If ldquoexpected payoffrdquo can be used asa measure of ldquopowerrdquo then the player with k votesshould also expect to have k times as much poweras the player with one vote5

In this paper we present a straightforwardmodel of divide-the-dollar politics which cap-tures the simple price-theoretic intuition Weshow that the noncooperative bargaining modelof David P Baron and John A Ferejohn (1989)leads naturally to the result that expected pay-offs are proportional to voting weights6 Whilecooperative game theory concepts apply only tovoting power this model captures both votingpower and proposal power7 In the Baron-Ferejohn model a randomly drawn legislatormakes a proposalmdasha division of the dollarmdashwhich is then put to a vote Proposers seek tooffer as little of the dollar as possible to othersbecause they keep the residual for themselves

As in most noncooperative bargaining mod-els we must make an assumption about theprobability that each legislator is chosen tomake a proposal We analyze two natural casesFirst legislatorsrsquo proposal probabilities may beproportional to their voting weights This isconsistent with the empirical pattern found inthe formation of coalition governments8 Sec-ond legislatorsrsquo proposal probabilities may all

2 Another example is Steven J Brams and Paul J Affuso(1985 p 138) ldquoA measure like Banzhafrsquos is not only aneminently reasonable indicator of a crucial aspect of votingpowermdashthe ability of a member to change an outcome bychanging its votemdashbut also highlights the fact that size (asreflected by voting weights) and voting power may bearlittle relationship to each otherrdquo

3 See Brams and Affuso (1985) Berg et al (1993) KaisaHerne and Hannu Nurmi (1993) Madeline O Hosli (19931995) Mika Widgren (1994 2000) R J Johnston (1995)Jan-Erik Lane and Reinert Maeland (1995 1996) Lane et al(1995 1996) Matthias Bruckner and Torsten Peters (1996)Anthony L Teasdale (1996) George Tsebelis and GeoffreyGarrett (1996) Ulrich Bindseil and Cordula Hantke (1997)Dan S Felsenthal and Moshe Machover (1997 2000 2001)Annick Laruelle and Widgren (1998) Konig and Brauninger(1998) Garrett and Tsebelis (1999a 1999b 2001) Holler andWidgren (1999) Nurmi and Tommi Meskanen (1999) andMatthias Sutter (2000a 2000b)

4 Power indices are criticized on other grounds as wellSome critics emphasize the paradoxical flavor of the predic-tions that power indices yield (Brams and Affuso 1985 Holler1987 Felsenthal and Machover 1998 Garrett and Tsebelis1999a 1999b 2001) A further well-known problem is that theseindices capture only the influence that comes from having votesand ignore important sources of power especially the ability tomake proposals (James G March 1955) Moreover theseindices typically do not have a basis in non-cooperative gametheory an exception is Faruk Gul (1989) who provides anon-cooperative justification of the Shapley value

5 Theorists working on this problem commonly equatepower and expected payoffs There is some debate overwhether the definition of power should also include the

ability to change the outcome even though the action doesnot result in an increase in and may even lower the payofffor the pivotal actor eg William H Riker (1964 p 344)In this paper we use the terms power and expected payoffsinterchangeably

6 The Baron-Ferejohn model extends the Ariel Rubin-stein (1982) bargaining model to the case of three or moreplayers operating under simple majority rule This is themost widely used model of legislative bargaining and hasbeen used extensively to study various aspects of distribu-tive politics and government institutions See Joseph Har-rington (1989 1990a 1990b) Baron (1991 1996 1998)McKelvey and Raymond Riezman (1992) Baron and EhudKalai (1993) Randall Calvert and Nathan Dietz (1996)Eyal Winter (1996) V V Chari et al (1997) Daniel Dier-meier and Timothy J Feddersen (1998) Jeffrey S Banks andJohn Duggan (2000) William LeBlanc et al (2000) Nolan MMcCarty (2000a 2000b) Hulya Eraslan (2002) Peter Norman(2002) and Ansolabehere et al (2003) Other non-cooperativemodels of n-person bargaining include Reinhardt Selten(1981) Kenneth Binmore (1987) Gul (1989) Kalyan Chat-terjee et al (1993) and Benny Moldovanu and Winter(1995) Sergiu Hart and Andreu Mas-Colell (1996) andAkira Okada (1996) To our knowledge none of these hasbeen applied to the study of legislative politics

7 Massimo Morelli (1999) develops a model of ldquodemandbargainingrdquo that also captures these features and arrives atproportional payoffs albeit in a different bargaining envi-ronment We discuss his model and results further below

8 Diermeier and Antonio Merlo (2004) estimate that par-tiesrsquo ldquorecognition probabilitiesrdquo depend strongly on theirvoting weights as well as which party formed the previousgovernment


Mik Laver
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I Model and Results

A The Model












Mik Laver
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Mik Laver
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Mik Laver
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B Results

















(2) vt pt 1 vt

1 1 pt qt





v t and qt


(4) qt rw wt

rw wt






(5) vt 1 v t

rn rn 1qt





(6) qt 1 rw w

rn 1wt










C Discussion





Mik Laver
Highlight
Mik Laver
Highlight
Mik Laver
Highlight




























13 See note 3 above



Voting weights



2 1 1 1 0500 0167 0167 0167 125 084 084 0840500 0167 0167 0167 125 084 084 0840400 0200 0200 0200

3 1 1 1 1 0600 0100 0100 0100 0100 140 070 070 070 0700636 0091 0091 0091 0091 148 064 064 064 0640429 0143 0143 0143 0143

2 2 1 1 1 0300 0300 0133 0133 0133 105 105 093 093 0930286 0286 0143 0143 0143 100 100 100 100 1000286 0286 0143 0143 0143

3 2 2 1 1 0400 0233 0233 0067 0067 120 105 105 060 0600385 0231 0231 0077 0077 116 104 104 069 0690333 0222 0222 0111 0111

2 1 1 1 1 1 0333 0133 0133 0133 0133 0133 117 093 093 093 093 0930333 0133 0133 0133 0133 0133 117 093 093 093 093 0930286 0143 0143 0143 0143 0143

4 1 1 1 1 1 0666 0067 0067 0067 0067 0067 150 060 060 060 060 0600750 0050 0050 0050 0050 0050 169 045 045 045 045 0450444 0111 0111 0111 0111 0111

3 2 1 1 1 1 0400 0200 0100 0100 0100 0100 120 090 090 090 090 0900393 0179 0107 0107 0107 0107 118 081 096 096 096 0960333 0222 0111 0111 0111 0111

2 2 2 1 1 1a 0233 0233 0233 0100 0100 0100 105 105 105 090 090 0900233 0233 0233 0100 0100 0100 105 105 105 090 090 0900222 0222 0222 0111 0111 0111

3 3 2 1 1 1 0300 0300 0200 0067 0067 0067 110 110 110 074 074 0740286 0286 0214 0071 0071 0071 105 105 118 078 078 0780273 0273 0182 0091 0091 0091

4 2 2 1 1 1 0467 0167 0167 0067 0067 0067 171 092 092 074 074 0740462 0154 0154 0077 0077 0077 169 085 085 085 085 0850364 0182 0182 0091 0091 0091

3 2 2 2 1 1a 0300 0167 0167 0167 0100 0100 110 092 092 092 110 1100300 0167 0167 0167 0100 0100 110 092 092 092 110 1100273 0182 0182 0182 0091 0091

4 3 3 1 1 1b 0367 0267 0267 0033 0033 0033 122 119 119 040 040 0400346 0269 0269 0039 0039 0039 115 120 120 047 047 0470300 0225 0225 0083 0083 0083

3 3 2 2 2 1ab 0234 0234 0167 0167 0167 0033 103 103 110 110 110 0360234 0234 0167 0167 0167 0033 103 103 110 110 110 0360227 0227 0152 0152 0152 0091












FranceGermany

ItalyBelgium



4 2 1 12170235 0118 0059 07051


2 1 0 680250 0125 0000 0751




BelgiumNetherlands



10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141


























Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES























































































































I Model and Results

A The Model












Mik Laver
Highlight
Mik Laver
Highlight
Mik Laver
Highlight






B Results

















(2) vt pt 1 vt

1 1 pt qt





v t and qt


(4) qt rw wt

rw wt






(5) vt 1 v t

rn rn 1qt





(6) qt 1 rw w

rn 1wt










C Discussion





Mik Laver
Highlight
Mik Laver
Highlight
Mik Laver
Highlight




























13 See note 3 above



Voting weights



2 1 1 1 0500 0167 0167 0167 125 084 084 0840500 0167 0167 0167 125 084 084 0840400 0200 0200 0200

3 1 1 1 1 0600 0100 0100 0100 0100 140 070 070 070 0700636 0091 0091 0091 0091 148 064 064 064 0640429 0143 0143 0143 0143

2 2 1 1 1 0300 0300 0133 0133 0133 105 105 093 093 0930286 0286 0143 0143 0143 100 100 100 100 1000286 0286 0143 0143 0143

3 2 2 1 1 0400 0233 0233 0067 0067 120 105 105 060 0600385 0231 0231 0077 0077 116 104 104 069 0690333 0222 0222 0111 0111

2 1 1 1 1 1 0333 0133 0133 0133 0133 0133 117 093 093 093 093 0930333 0133 0133 0133 0133 0133 117 093 093 093 093 0930286 0143 0143 0143 0143 0143

4 1 1 1 1 1 0666 0067 0067 0067 0067 0067 150 060 060 060 060 0600750 0050 0050 0050 0050 0050 169 045 045 045 045 0450444 0111 0111 0111 0111 0111

3 2 1 1 1 1 0400 0200 0100 0100 0100 0100 120 090 090 090 090 0900393 0179 0107 0107 0107 0107 118 081 096 096 096 0960333 0222 0111 0111 0111 0111

2 2 2 1 1 1a 0233 0233 0233 0100 0100 0100 105 105 105 090 090 0900233 0233 0233 0100 0100 0100 105 105 105 090 090 0900222 0222 0222 0111 0111 0111

3 3 2 1 1 1 0300 0300 0200 0067 0067 0067 110 110 110 074 074 0740286 0286 0214 0071 0071 0071 105 105 118 078 078 0780273 0273 0182 0091 0091 0091

4 2 2 1 1 1 0467 0167 0167 0067 0067 0067 171 092 092 074 074 0740462 0154 0154 0077 0077 0077 169 085 085 085 085 0850364 0182 0182 0091 0091 0091

3 2 2 2 1 1a 0300 0167 0167 0167 0100 0100 110 092 092 092 110 1100300 0167 0167 0167 0100 0100 110 092 092 092 110 1100273 0182 0182 0182 0091 0091

4 3 3 1 1 1b 0367 0267 0267 0033 0033 0033 122 119 119 040 040 0400346 0269 0269 0039 0039 0039 115 120 120 047 047 0470300 0225 0225 0083 0083 0083

3 3 2 2 2 1ab 0234 0234 0167 0167 0167 0033 103 103 110 110 110 0360234 0234 0167 0167 0167 0033 103 103 110 110 110 0360227 0227 0152 0152 0152 0091












FranceGermany

ItalyBelgium



4 2 1 12170235 0118 0059 07051


2 1 0 680250 0125 0000 0751




BelgiumNetherlands



10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141


























Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES


















































































































B Results

















(2) vt pt 1 vt

1 1 pt qt





v t and qt


(4) qt rw wt

rw wt






(5) vt 1 v t

rn rn 1qt





(6) qt 1 rw w

rn 1wt










C Discussion





Mik Laver
Highlight
Mik Laver
Highlight
Mik Laver
Highlight




























13 See note 3 above



Voting weights



2 1 1 1 0500 0167 0167 0167 125 084 084 0840500 0167 0167 0167 125 084 084 0840400 0200 0200 0200

3 1 1 1 1 0600 0100 0100 0100 0100 140 070 070 070 0700636 0091 0091 0091 0091 148 064 064 064 0640429 0143 0143 0143 0143

2 2 1 1 1 0300 0300 0133 0133 0133 105 105 093 093 0930286 0286 0143 0143 0143 100 100 100 100 1000286 0286 0143 0143 0143

3 2 2 1 1 0400 0233 0233 0067 0067 120 105 105 060 0600385 0231 0231 0077 0077 116 104 104 069 0690333 0222 0222 0111 0111

2 1 1 1 1 1 0333 0133 0133 0133 0133 0133 117 093 093 093 093 0930333 0133 0133 0133 0133 0133 117 093 093 093 093 0930286 0143 0143 0143 0143 0143

4 1 1 1 1 1 0666 0067 0067 0067 0067 0067 150 060 060 060 060 0600750 0050 0050 0050 0050 0050 169 045 045 045 045 0450444 0111 0111 0111 0111 0111

3 2 1 1 1 1 0400 0200 0100 0100 0100 0100 120 090 090 090 090 0900393 0179 0107 0107 0107 0107 118 081 096 096 096 0960333 0222 0111 0111 0111 0111

2 2 2 1 1 1a 0233 0233 0233 0100 0100 0100 105 105 105 090 090 0900233 0233 0233 0100 0100 0100 105 105 105 090 090 0900222 0222 0222 0111 0111 0111

3 3 2 1 1 1 0300 0300 0200 0067 0067 0067 110 110 110 074 074 0740286 0286 0214 0071 0071 0071 105 105 118 078 078 0780273 0273 0182 0091 0091 0091

4 2 2 1 1 1 0467 0167 0167 0067 0067 0067 171 092 092 074 074 0740462 0154 0154 0077 0077 0077 169 085 085 085 085 0850364 0182 0182 0091 0091 0091

3 2 2 2 1 1a 0300 0167 0167 0167 0100 0100 110 092 092 092 110 1100300 0167 0167 0167 0100 0100 110 092 092 092 110 1100273 0182 0182 0182 0091 0091

4 3 3 1 1 1b 0367 0267 0267 0033 0033 0033 122 119 119 040 040 0400346 0269 0269 0039 0039 0039 115 120 120 047 047 0470300 0225 0225 0083 0083 0083

3 3 2 2 2 1ab 0234 0234 0167 0167 0167 0033 103 103 110 110 110 0360234 0234 0167 0167 0167 0033 103 103 110 110 110 0360227 0227 0152 0152 0152 0091












FranceGermany

ItalyBelgium



4 2 1 12170235 0118 0059 07051


2 1 0 680250 0125 0000 0751




BelgiumNetherlands



10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141


























Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES




















































































































(2) vt pt 1 vt

1 1 pt qt





v t and qt


(4) qt rw wt

rw wt






(5) vt 1 v t

rn rn 1qt





(6) qt 1 rw w

rn 1wt










C Discussion





Mik Laver
Highlight
Mik Laver
Highlight
Mik Laver
Highlight




























13 See note 3 above



Voting weights



2 1 1 1 0500 0167 0167 0167 125 084 084 0840500 0167 0167 0167 125 084 084 0840400 0200 0200 0200

3 1 1 1 1 0600 0100 0100 0100 0100 140 070 070 070 0700636 0091 0091 0091 0091 148 064 064 064 0640429 0143 0143 0143 0143

2 2 1 1 1 0300 0300 0133 0133 0133 105 105 093 093 0930286 0286 0143 0143 0143 100 100 100 100 1000286 0286 0143 0143 0143

3 2 2 1 1 0400 0233 0233 0067 0067 120 105 105 060 0600385 0231 0231 0077 0077 116 104 104 069 0690333 0222 0222 0111 0111

2 1 1 1 1 1 0333 0133 0133 0133 0133 0133 117 093 093 093 093 0930333 0133 0133 0133 0133 0133 117 093 093 093 093 0930286 0143 0143 0143 0143 0143

4 1 1 1 1 1 0666 0067 0067 0067 0067 0067 150 060 060 060 060 0600750 0050 0050 0050 0050 0050 169 045 045 045 045 0450444 0111 0111 0111 0111 0111

3 2 1 1 1 1 0400 0200 0100 0100 0100 0100 120 090 090 090 090 0900393 0179 0107 0107 0107 0107 118 081 096 096 096 0960333 0222 0111 0111 0111 0111

2 2 2 1 1 1a 0233 0233 0233 0100 0100 0100 105 105 105 090 090 0900233 0233 0233 0100 0100 0100 105 105 105 090 090 0900222 0222 0222 0111 0111 0111

3 3 2 1 1 1 0300 0300 0200 0067 0067 0067 110 110 110 074 074 0740286 0286 0214 0071 0071 0071 105 105 118 078 078 0780273 0273 0182 0091 0091 0091

4 2 2 1 1 1 0467 0167 0167 0067 0067 0067 171 092 092 074 074 0740462 0154 0154 0077 0077 0077 169 085 085 085 085 0850364 0182 0182 0091 0091 0091

3 2 2 2 1 1a 0300 0167 0167 0167 0100 0100 110 092 092 092 110 1100300 0167 0167 0167 0100 0100 110 092 092 092 110 1100273 0182 0182 0182 0091 0091

4 3 3 1 1 1b 0367 0267 0267 0033 0033 0033 122 119 119 040 040 0400346 0269 0269 0039 0039 0039 115 120 120 047 047 0470300 0225 0225 0083 0083 0083

3 3 2 2 2 1ab 0234 0234 0167 0167 0167 0033 103 103 110 110 110 0360234 0234 0167 0167 0167 0033 103 103 110 110 110 0360227 0227 0152 0152 0152 0091












FranceGermany

ItalyBelgium



4 2 1 12170235 0118 0059 07051


2 1 0 680250 0125 0000 0751




BelgiumNetherlands



10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141


























Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES














































































































(5) vt 1 v t

rn rn 1qt





(6) qt 1 rw w

rn 1wt










C Discussion





Mik Laver
Highlight
Mik Laver
Highlight
Mik Laver
Highlight




























13 See note 3 above



Voting weights



2 1 1 1 0500 0167 0167 0167 125 084 084 0840500 0167 0167 0167 125 084 084 0840400 0200 0200 0200

3 1 1 1 1 0600 0100 0100 0100 0100 140 070 070 070 0700636 0091 0091 0091 0091 148 064 064 064 0640429 0143 0143 0143 0143

2 2 1 1 1 0300 0300 0133 0133 0133 105 105 093 093 0930286 0286 0143 0143 0143 100 100 100 100 1000286 0286 0143 0143 0143

3 2 2 1 1 0400 0233 0233 0067 0067 120 105 105 060 0600385 0231 0231 0077 0077 116 104 104 069 0690333 0222 0222 0111 0111

2 1 1 1 1 1 0333 0133 0133 0133 0133 0133 117 093 093 093 093 0930333 0133 0133 0133 0133 0133 117 093 093 093 093 0930286 0143 0143 0143 0143 0143

4 1 1 1 1 1 0666 0067 0067 0067 0067 0067 150 060 060 060 060 0600750 0050 0050 0050 0050 0050 169 045 045 045 045 0450444 0111 0111 0111 0111 0111

3 2 1 1 1 1 0400 0200 0100 0100 0100 0100 120 090 090 090 090 0900393 0179 0107 0107 0107 0107 118 081 096 096 096 0960333 0222 0111 0111 0111 0111

2 2 2 1 1 1a 0233 0233 0233 0100 0100 0100 105 105 105 090 090 0900233 0233 0233 0100 0100 0100 105 105 105 090 090 0900222 0222 0222 0111 0111 0111

3 3 2 1 1 1 0300 0300 0200 0067 0067 0067 110 110 110 074 074 0740286 0286 0214 0071 0071 0071 105 105 118 078 078 0780273 0273 0182 0091 0091 0091

4 2 2 1 1 1 0467 0167 0167 0067 0067 0067 171 092 092 074 074 0740462 0154 0154 0077 0077 0077 169 085 085 085 085 0850364 0182 0182 0091 0091 0091

3 2 2 2 1 1a 0300 0167 0167 0167 0100 0100 110 092 092 092 110 1100300 0167 0167 0167 0100 0100 110 092 092 092 110 1100273 0182 0182 0182 0091 0091

4 3 3 1 1 1b 0367 0267 0267 0033 0033 0033 122 119 119 040 040 0400346 0269 0269 0039 0039 0039 115 120 120 047 047 0470300 0225 0225 0083 0083 0083

3 3 2 2 2 1ab 0234 0234 0167 0167 0167 0033 103 103 110 110 110 0360234 0234 0167 0167 0167 0033 103 103 110 110 110 0360227 0227 0152 0152 0152 0091












FranceGermany

ItalyBelgium



4 2 1 12170235 0118 0059 07051


2 1 0 680250 0125 0000 0751




BelgiumNetherlands



10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141


























Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES








































































































































13 See note 3 above



Voting weights



2 1 1 1 0500 0167 0167 0167 125 084 084 0840500 0167 0167 0167 125 084 084 0840400 0200 0200 0200

3 1 1 1 1 0600 0100 0100 0100 0100 140 070 070 070 0700636 0091 0091 0091 0091 148 064 064 064 0640429 0143 0143 0143 0143

2 2 1 1 1 0300 0300 0133 0133 0133 105 105 093 093 0930286 0286 0143 0143 0143 100 100 100 100 1000286 0286 0143 0143 0143

3 2 2 1 1 0400 0233 0233 0067 0067 120 105 105 060 0600385 0231 0231 0077 0077 116 104 104 069 0690333 0222 0222 0111 0111

2 1 1 1 1 1 0333 0133 0133 0133 0133 0133 117 093 093 093 093 0930333 0133 0133 0133 0133 0133 117 093 093 093 093 0930286 0143 0143 0143 0143 0143

4 1 1 1 1 1 0666 0067 0067 0067 0067 0067 150 060 060 060 060 0600750 0050 0050 0050 0050 0050 169 045 045 045 045 0450444 0111 0111 0111 0111 0111

3 2 1 1 1 1 0400 0200 0100 0100 0100 0100 120 090 090 090 090 0900393 0179 0107 0107 0107 0107 118 081 096 096 096 0960333 0222 0111 0111 0111 0111

2 2 2 1 1 1a 0233 0233 0233 0100 0100 0100 105 105 105 090 090 0900233 0233 0233 0100 0100 0100 105 105 105 090 090 0900222 0222 0222 0111 0111 0111

3 3 2 1 1 1 0300 0300 0200 0067 0067 0067 110 110 110 074 074 0740286 0286 0214 0071 0071 0071 105 105 118 078 078 0780273 0273 0182 0091 0091 0091

4 2 2 1 1 1 0467 0167 0167 0067 0067 0067 171 092 092 074 074 0740462 0154 0154 0077 0077 0077 169 085 085 085 085 0850364 0182 0182 0091 0091 0091

3 2 2 2 1 1a 0300 0167 0167 0167 0100 0100 110 092 092 092 110 1100300 0167 0167 0167 0100 0100 110 092 092 092 110 1100273 0182 0182 0182 0091 0091

4 3 3 1 1 1b 0367 0267 0267 0033 0033 0033 122 119 119 040 040 0400346 0269 0269 0039 0039 0039 115 120 120 047 047 0470300 0225 0225 0083 0083 0083

3 3 2 2 2 1ab 0234 0234 0167 0167 0167 0033 103 103 110 110 110 0360234 0234 0167 0167 0167 0033 103 103 110 110 110 0360227 0227 0152 0152 0152 0091












FranceGermany

ItalyBelgium



4 2 1 12170235 0118 0059 07051


2 1 0 680250 0125 0000 0751




BelgiumNetherlands



10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141


























Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES






















































































































FranceGermany

ItalyBelgium



4 2 1 12170235 0118 0059 07051


2 1 0 680250 0125 0000 0751




BelgiumNetherlands



10 5 3 2 41580172 0085 0052 0034 07071


6 3 2 1 25350171 0086 0057 0029 07141


























Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES



































































































































Unweighted Weighted

(1) (2) (1) (2)





Constant 007 008 006 008(0020) (0028) (0017) (0026)

R2 072 069 081 078 Observations 682 682 682 682



Mik Laver
Highlight













IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES

























































































































IV Discussion





APPENDIX

















k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES























































































































k kj If k





i Sinceyeni Ck












PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES













































































































PROOFLet tjj1





to step (1)



(A1)rw wt wT 1

rw wt

rw wt

rw 131 132 13


(A2)rw wt

rw wt

rw wt

rw 131

(A3)wT 1

rw wt 132












pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES














































































































pt 1 rw wt rw 13

1 1 pt 1

wt

rw


(A5) ptrw w w13

wt 1 pt
















tTL

rqtntCwt tTL

rqtntCwt rw wT


tTL

qtntCwt w wT

r

2TwT

r tTL

ntCwt







pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES
















































































































pt rw rw wt rw 13

1 1 pt ww

wt

rw


wt r w

rw 13

1 pt w

w

(A6)wt

r w

w 1 w w w13





vt t wt

rw

1 t rw wt rw

rn rn 12TwT rnt


t w



(A7) w w wt n 1

r wt 2TwT wt

ntn

1

r




(A8) n w w

w1







(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES

















































































































(A9) vt t wt

rw

1 t rw wt rw 1rnT

rn


t w wrnT

w wt n wt r

Thus t 1 if

(A10) w w wt n 1

r w

nT wt


wt



(A12) n w w

wt



(A13) n w w

wt


1


REFERENCES













































































































REFERENCES













































































































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