An Integrated Computational Model of Multiparty Electoral Competition

Statstcal Scence 2002, Vol. 17, No. 4, 405 419 Insttute of Mathematcal Statstcs, 2002 An Integrated Computatonal Model of Multparty Electoral Competton Kevn M. Qunn and Andrew D. Martn Abstract. Most theoretc models of multparty electoral competton make the assumpton that party leaders are motvated to maxmze ther vote share or seat share. In pluralty-rule systems ths s a sensble assumpton. However, n proportonal representaton systems, ths assumpton s questonable snce the ablty to make publc polcy s not strctly ncreasng n vote shares or seat shares. We present a theoretc model n whch party leaders choose electoral declaratons wth an eye toward the expected polcy outcome of the coalton barganng game nduced by the party declaratons and the partes belefs about ctzens votng behavor. To test ths model, we turn to data from the 1989 Dutch parlamentary electon. We use Markov chan Monte Carlo methods to estmate the partes belefs about mass votng behavor and to average over measurement uncertanty and mssng data. Due to the complexty of the partes objectve functons and the uncertanty n objectve functon estmates, equlbra are found numercally. Unlke prevous models of multparty electoral competton, the equlbrum results are consstent wth the emprcal declaratons of the four major Dutch partes. Key words and phrases: Monte Carlo method, votng behavor, electoral strategy, coalton formaton. 1. INTRODUCTION Democratc poltcs s fundamentally about allocatng resources among competng nterests. Those who wn enjoy the perqustes of offce and the publc polcy benefts that governng entals. Those who lose must prepare to compete n the next electon. In the Unted States and other essentally two-party systems, understandng who wns and who loses s smple; one party gets a majorty of the vote and wns, whle the other party fals to do so and loses. In such systems, both partes are nterested n maxmzng the number of votes they receve snce recevng the most votes mples more or less complete control over both government Kevn M. Qunn s Assstant Professor n the Department of Poltcal Scence and Center for Statstcs and Socal Scences at Unversty of Washngton, Seattle, Washngton 98195-4320 (e-mal: qunn@stat. washngton.edu). Andrew D. Martn s Assstant Professor n the Department of Poltcal Scence at Washngton Unversty, St. Lous, Mssour 63130 (e-mal: admartn@artsc.wustl.edu). perqustes and polcy. In parlamentary democraces wth more than two partes the most common form of democratc government across the globe wnnng and losng s not so clear-cut. In these systems, rarely do partes garner a majorty of votes. Typcally, two or more partes must jon together n a governng coalton. Snce the ultmate mx of perqustes and polcy s a result of barganng between coalton partners, no party s lkely to get ts most preferred outcome. Thus, wnnng and losng s defned n terms of dong as well as possble condtonal on the strateges of the other partes. Partes must be concerned wth gettng votes, but they also must consder ther prospects of sharng resources wth other partes n a governng coalton (Laver and Shepsle, 1996). To understand legslatve poltcs and publc polcy outputs n multparty systems, t s thus necessary to jontly understand votng behavor, electoral strategy and coalton formaton. Much has been wrtten about each stage of the process separately, but lttle has been done to emprcally jon these lteratures n a prncpled fashon. Ths s lkely due to the complex nterdepen- 405

406 K. M. QUINN AND A. D. MARTIN dences of these stages, as well as the problem of accountng for uncertanty at all levels of the emprcal model. In ths artcle we develop a smulaton-based methodology that allows us to deal wth ths complexty. In ths text we offer a method that allows us to test the predctons of one partcular model of coalton formaton n multparty democraces. The advantage of ths approach over an analytc one s that by ncorporatng stochastc behavor nto the theory, we te emprcal analyss drectly to the theoretcal model. Further, we gauge the predctons of models that would be otherwse ntractable. The dsadvantage of the approach s that the generalty of the theoretcal model s sacrfced, although the method can be used to consder other explanatons of coalton formaton. Ths artcle s organzed as follows. In Secton 2, we revew current understandng of electoral competton n multparty democraces. We dscuss the 1989 Dutch parlamentary electon the applcaton n ths artcle n Secton 3. Secton 4 contans our model and focuses on vote maxmzaton and the polcy motvatons of poltcal partes. Ths secton dscusses how to ntegrate our estmates of key model parameters nto the modelng framework and how to solve for equlbra numercally. In Secton 5, we report the fndngs from the operatonalzaton of the model wthn the context of the Netherlands n 1989. Here we account for uncertanty n model nputs, present estmates of voter utlty functons and apply the results from the statstcal model to estmate equlbrum party strateges. The fnal secton concludes wth a dscusson of the methodologcal and substantve mplcatons of ths research, and some speculatve thoughts as to how these results relate to the more tradtonal, qualtatve lterature on partes and party systems. 2. UNDERSTANDING MULTIPARTY ELECTORAL COMPETITION Understandng why poltcal partes put forth a gven set of electoral platforms to an electorate s of profound theoretcal mportance. (We use the terms electoral platforms and electoraldeclaratonsnterchangeably throughout the artcle.) Knowledge of ths process nforms our understandng of such key ssues as representaton, realgnment and the stablty of electoral systems. Indeed, n pure multparty parlamentary systems, understandng coalton formaton means understandng legslatve outputs and publc polcy formaton. Over the past 40 years a great deal has been learned about what equlbrum electoral declaratons look lke n pluralty-rule systems of varous sorts (Downs, 1957; Davs and Hnch, 1966; Enelow and Hnch, 1984, 1990; Coughln, 1992). Ths lterature traces ts ntellectual roots to the spatal votng model (Hotellng, 1929; Downs, 1957; Black, 1958). The spatal votng model s predcated on the assumpton that the polcy content of poltcal ssues can be represented n a (typcally) low-dmensonal Eucldean space. Ths space s typcally called a polcy space or an ssue space. For llustraton, consder the smplest one-dmensonal case, wth a parlament choosng the ncome tax rate. Currently the tax rate s zero. The choce of tax rate can be thought of as pckng a locaton on [0, 1] or, after sutable transformaton, pckng a locaton on the real number lne. It s mportant to note that the polcy space need not be defned n terms of monetary consequences; other ssues such as socal polcy can be thought of n the same fashon, wth ponts on the lne representng the lberalness of the polcy. Hgher dmensons arse wth multple ssues, whch are typcally correlated, thus yeldng a low-dmensonal space. The key assumpton s that the polces relevant to decson makng map nto ths space. To understand ndvdual decson makng, the two relevant factors are the polcy postons and the deal ponts of the actors. In the prevous llustraton, the locaton of the status quo s zero and each proposal can be represented by a pont on the lne. The spatal model asserts that each actor has an deal pont, or preferred polcy poston, n the space. Ths deal pont corresponds to the polcy that the ndvdual would enact f they had dctatoral power. Wth the polcy locatons and the deal ponts, the fnal assumpton one has to make to understand behavor s the utlty functon of the actors. The assumpton of symmetrc, quadratc utltes (Enelow and Hnch, 1984) s typcal, although others use dfferent functonal forms (Coughln, 1992). These functons take the polcy locatons and deal ponts as arguments, and produce a personal utlty to the decson maker. The model predcts that ndvduals wll vote for the optons from whch they derve greatest utlty. The spatal model has been appled n essentally two ways. Some have used the model to understand votng by the mass publc (Downs, 1957). Here canddates standng for electon can be thought of as the polcy locatons n the space, and the voters can be thought of as the decson makers. By assumng a specfc random

MULTIPARTY ELECTORAL COMPETITION 407 utlty functon, one can use models of multnomal choce to statstcally model votng behavor (e.g., Qunn, Martn and Whtford, 1999). The spatal model has also been appled to the study of commttee decson makng (Shepsle, 1979). In ths case, the legslators are the decson makers, who are votng over proposals n the polcy space. One queston of great mportance when studyng multparty systems s, What electoral declaratons do partes make? In other words, what locaton n ths Eucldean space wll partes present to voters n a gven electon? What makes ths queston nterestng s the fact that these choces are nterdependent. That s, t s not just the locaton of a sngle party that determnes voter choce, but rather t s the locatons of every party that determne how voters choose. Moreover, the choces the partes make depend crucally on ther goals. To understand such an nterdependent choce settng, we make use of the game theoretc concept of pure strategy Nash equlbrum (Nash, 1950). Ths equlbrum concept suggests that each party wll choose a poston such that t s a best response to the best responses of all opposng partes. Such an equlbrum has the property that t s defecton-proof; no party can unlaterally mprove ther standng by adoptng an alternatve strategy. Wth some exceptons (Wttman, 1983; Shepsle and Cohen, 1990), the man fndngs from the spatal votng lterature are that partes n pluralty-rule systems wll generally tend to take farly centrst postons n equlbrum. These Downsan convergence results hold even for multparty systems wth hgh-dmensonal ssue spaces as long as canddates are motvated by wnnng and are suffcently unsure of voter behavor (Enelow and Hnch, 1989; Coughln, 1992). Whle scholars have ganed a good understandng of electoral competton n two-party systems (and pluralty-rule systems more generally), the same cannot be sad for multparty proportonal rule (PR) systems. Ths s partcularly nterestng snce a very large proporton of the world s democraces feature some form of PR. The reasons for the dscrepancy between our knowledge of electoral strategy n pluralty-rule systems and PR systems are manfold. Nonetheless, one of the key reasons has to do wth the lnkage between electoral declaratons and polcy outcomes n the two systems. In multparty PR systems, polcy s ultmately decded by a coalton government composed of a number of the partes who have ganed seats n parlament. As Austen-Smth and Banks (1988) showed, the power to determne polcy s not monotoncally ncreasng n vote shares or seat shares n certan types of PR systems. Ths mples that a theoretc model of electoral competton n a multparty PR system requres a model of cabnet formaton that s lnked to the electoral declaratons put forth by the partes. Because of the complexty of ths lnkage, few scholars have undertaken such an emprcal or theoretcal modelng enterprse. Rather than formulatng such a model of multparty electoral competton, many researchers mplctly or explctly assume that partes n PR systems choose electoral declaratons to maxmze vote share or seat share. Ths has led several researchers to remark upon the apparent dscrepancy between the predctons derved from theores of party strategy based on vote maxmzaton and estmates of the actual electoral declaratons put forth by partes (Dalton, 1985; Lsthaug, Macdonald and Rabnowtz, 1990; Rabnowtz, Macdonald and Lsthaug, 1991; Iversen, 1994). Others contend that polcy outputs enter the decson calculus of the partes and thus explan the dvergence n electoral declaratons (Laver and Shepsle, 1996). Our purpose n ths artcle s to compare the Nash equlbrum predctons of vote-maxmzng and polcy-motvated partes. We put forth a general modelng strategy that can be used to evaluate equlbrum electoral strateges under a varety of party motvatons. Snce the relatonshp between electoral declaratons and polcy outcomes s extremely dffcult to deal wth analytcally n anythng other than the smplest of cases, we calculate equlbra numercally. Our approach allows us to ncorporate uncertanty at all stages -measurement, estmaton and postestmaton equlbrum computaton nto our analyss. One of the dangers of such a computatonal approach s the possblty that results may depend crtcally on the values of key parameters that have been set by the researcher. We attempt to mnmze ths problem by estmatng model parameters from a representatve natonal survey of the Dutch electorate. Whle such an approach decreases the generalty of our results, t does ncrease the valdty of our results for the system under study. Our emprcal focus s on the Netherlands a naton wth an essentally pure PR system n the 1989 electon. To foreshadow, the results suggest that the electoral declaratons actually chosen by the four major Dutch partes n 1989 are much more consstent wth polcy-seekng behavor than wth vote-maxmzng behavor. 3. THE 1989 DUTCH PARLIAMENTARY ELECTION Operatonalzng our computatonal model requres us to obtan estmates of voter deal ponts, voter

408 K. M. QUINN AND A. D. MARTIN perceptons of party platforms at the tme of the survey and voter utlty functons. To do ths we examne data from the Netherlands n 1989. The Netherlands features a very pure form of PR n whch there s a sngle natonal dstrct and an extremely low electoral threshold (0.67%). (In other words, all partes recevng two thrds of 1% of the natonal vote wll gan at least one seat n Parlament.) Such features suggest that the ncentves for voters to engage n sophstcated votng (contrary to our assumptons) are relatvely weak compared to other PR systems. Further, the fact that the Netherlands has a sngle natonal dstrct means that dstrct-specfc attrbutes of canddates are constant across the whole naton. Ths mples that a natonally representatve sample survey has the potental to provde the nformaton about votng behavor that we desre. Whle a number of Dutch partes have acheved representaton n the lower house of parlament n past years, we focus on the four major partes: the PvdA, the CDA, the VVD and the D66. The PvdA (Partj van de Arbed) s a tradtonal socal democratc party wth left-wng economc stands and moderate to lberal socal postons. The CDA (Chrsten Democratsch Appèl) s a Chrstan democratc party that was formed from the three major relgous partes n 1980. The CDA tends to take relatvely conservatve postons on socal ssues (partcularly aborton and euthanasa) and moderate to slghtly rghtst postons on economc ssues. The VVD (Volkspartj voor Vrjhed en Democrate) s a tradtonal European lberal party that takes conservatve economc postons and moderate postons on socal polcy. Fnally, the D66 (Democraten 66) s a relatve newcomer to the party system that has fought for reform of the electoral system and a lberal socal agenda. It tends to take moderate postons on economc ssues. We do not explctly model the remanng Dutch partes. Instead, we assume that they each receve a constant percentage of the vote and vote aganst all possble cabnets n the cabnet formaton stage of the model. (Ths s not an entrely unreasonable assumpton gven Dutch poltcal hstory pror to 1989, and especally gven the atttudes of the major party leaders toward the ffth largest party Groen Lnks.) Our data come from the 1989 Dutch Parlamentary Electon Study (Anker and Oppenhus, 1993). Ths s the eghth natonal electon study to be conducted n the Netherlands, and was desgned to study the September 6, 1989 electon for the Second Chamber of the Dutch Parlament. Ths data set contans N = 1784 ndvduals who reported votng for one of the four major Dutch partes. It also ncluded a battery of demographc varables, as well as a battery of fve ssue questons (on aborton, euthanasa, ncome equalzaton, nuclear power and nuclear weapons), where the voters were asked to report ther own preferences as well as ther perceptons of the postons of the four major partes. 4. A MODEL OF PARLIAMENTARY ELECTORAL COMPETITION We can thnk about the process by whch electoral declaratons lead to polcy outputs as beng smlar to the process n Fgure 1. At the outset of electon, partes make electoral polcy declaratons. These declaratons, typcally formalzed n party manfestos, locate the partes n the polcy space. Gven those declaratons, voters cast ballots for the partes. The expectaton s that voters choose partes close to them n the polcy space. After the electon, vote totals are translated nto seat shares by an exogenous rule. After seats are allocated, the partes enter a barganng game, where they ultmately vote over cabnets, portfolo allocatons and other perqustes. Ths game results n a cabnet, typcally consstng of more than one party, that forms publc polcy. Note that a far amount of knowledge exsts regardng each of the fve steps ndvdually. An equlbrum n ths game conssts of a set of party polcy platforms and strateges for formng coalton governments such that no party has an ncentve to unlaterally change ts polcy platform or ts rule for formng a coalton government. In what follows we assume equlbrum behavor n the later stages of the game and focus on fndng equlbrum polcy declaratons. The sequental structure of ths process FIG.1. Schematc model of parlamentary electoral competton.

MULTIPARTY ELECTORAL COMPETITION 409 also suggests that equlbrum electoral declaratons can be found n the frst node by workng backward through the process (Selten, 1965). In the remander of ths secton we sequentally operatonalze the process sketched above. We then go on to show how equlbrum electoral declaratons can be calculated under the assumpton of polcy-motvated partes and under the assumpton of vote-maxmzng partes. 4.1 Calculatng Emprcal Electoral Declaratons and Ctzen Polcy Preferences The key to the spatal votng model s locatng voters and partes n the same ssue space. To measure voter deal ponts we employ a factor analytc measurement model of the responses to the fve ssue questons n the survey. (See Qunn, Martn and Whtford, 1999, for a justfcaton of ths procedure.) An example of one of the ssue questons n the Dutch Parlamentary Electon Study (Anker and Oppenhus, 1993) s the followng: Aborton: Some people thnk that the government should forbd aborton n all crcumstances; other people thnk that every woman should have the rght to decde whether she wants an aborton. Of course, there are also people wth an ntermedate opnon. Suppose we place the people who thnk that aborton should be forbdden under all crcumstances at the begnnng of ths lne (at number 1) and the people who thnk that every woman has the rght to decde for herself at the end of ths lne (at number 7). Frst I shall ask you to ndcate the poston of a number of poltcal partes on ths lne. If you do not know what poston a party has on ths ssue, please do not hestate to tell me. Where would you place the CDA on ths lne? And the PvdA? And the VVD? And D66? And where would you place yourself on ths lne? In addton to ths queston on aborton, ctzens were asked smlarly constructed questons on nuclear power, state attempts to reduce ncome nequalty, euthanasa and the deployment of nuclear weapons. As n all survey research, tem nonresponse s a serous problem. Here we deal wth mssng data through multple mputaton. We use a multvarate Normal model to characterze the jont dstrbuton of the raw survey data. In addton to the demographc covarates lsted n Table 1 and the responses to the ssue questons, measures of poltcal knowledge were ncluded n the mputaton model to help predct mssng responses. We employed the software of Kng, Honaker, Joseph and Scheve (2001), whose approach s based on the work of Lttle and Rubn (1987) and Schafer (1997). The prmary dfference s that the algorthm generates mputatons va mportance samplng from the posteror dstrbuton rather than usng data augmentaton or the expectaton-maxmzaton algorthm. We treat the vote choce response as a nomnal varable. We created and stored fve mputed data sets. Let = 1,...,N ndex ctzens and j = 1,...,J ndex partes. Let w represent the vector of ctzen s responses to the fve ssue questons and let represent the vector of ctzen s perceptons of party j on the fve ssues. Our goal s to use the data on these ssue questons to estmate ctzen s deal pont θ n the two-dmensonal ssue space X. We assume that the underlyng Dutch ssue space s twodmensonal, wth one dmenson reflectng underlyng w (j) TABLE 1 Socal-structural covarates used to estmate Dutch voter utlty functons, 1989 Varable Relgosty Catholc Socal class Unon member Income Educaton Age Rural Urban Descrpton Relgosty dummy (0 = not relgous, 1 = relgous) Catholc dummy (0 = not Catholc, 1 = Catholc) Socal class scale (0 = lowest socal class to 4 = hghest socal class) Unon dummy (0 = not unon member, 1 = unon member) Income scale (0 = lowest to 6 = hghest) Educaton scale (0 = lowest to 4 = hghest) Age categores (0 = lowest to 12 = hghest) Rural dummy (0 = not rural resdence, 1 = rural resdence) Urban dummy (0 = not urban resdence, 1 = urban resdence)

410 K. M. QUINN AND A. D. MARTIN economc preferences and the other reflectng preferences on socal polcy. Ths s consstent wth much of the lterature on Dutch votng behavor (see Qunn, Martn and Whtford, 1999, for a revew). We would also lke to make nferences about ctzen s percepton of party j s electoral platform (denoted ψ (j) )n the two-dmensonal ssue space as well as the mean percepton across the populaton of party j s electoral platform (denoted ψ (j) ) n the two-dmensonal ssue space. The measurement model used to estmate voter deal ponts s gven by (1) w = θ + ε, where ε N (0, ) and s a dagonal matrx wth the rth dagonal element gven by σrr 2. We assume the followng prors: r N ( r, r ), σrr 2 IG(ν/2,δ/2) and θ N (0, ),where r denotes the nonzero ] elements of the rth row of and =. (Some of the elements of are set to 0 for [ 1 ρ ρ 1 reasons of dentfcaton.) Let N (a, b) denote a normal dstrbuton wth mean a and varance b, and let IG(a, b) denote an nverse gamma dstrbuton wth shape a and scale b. We assume a unform hyperpror for ρ. It s assumed that s R K and θ s K 1, where R s the number of manfest ssue responses (n ths case, R = 5) and K s the number of latent ssue dmensons (n ths case, K = 2). In ths applcaton, we set r to be a vector of zeros, r = 10, 000I, ν = 2andδ = 1. Ths measurement model s ftted va Markov chan Monte Carlo (MCMC). We can sample from the dstrbuton of the average voter percepton of the party declaratons for all partes denoted ( ψ) by vewng t as a determnstc functon of the model parameters. Appendx A descrbes the algorthm employed and how the dstrbuton of ( ψ) s calculated wthn the MCMC algorthm. It s mportant to note that our MCMC samplng produces a seres of draws that are approxmately from p(θ, ψ D ms,d obs ). We ft the model to each of the fve mputed data sets. To llustrate the fndngs from our measurement model, Fgure 2 dsplays the mean perceptons of the party electoral declaratons and the underlyng densty of voter deal ponts based on a sngle draw from the posteror dstrbuton of these parameters. Ths comports well wth our qualtatve understandng of the Dutch party system. Table 2 presents the results from our measurement model averaged across the fve mputed data sets. FIG. 2. Emprcal party postons of Dutch partes overlad on a hghest densty plot of voter deal ponts, 1989. Runnng from lghtest to darkest, the hghest densty contours represent the smallest regons of the polcy space that contan 95, 75, 50, and 1% of the underlyng densty of voter deal ponts. 4.2 Calculatng Seat Strengths Gven Electoral Declaratons and Ctzen Polcy Preferences The next step of the process s to use the measures of voter and party locaton obtaned above to estmate voter utlty functons. Wth ths nformaton we can then make nferences about the lkely vote share and seat shares of partes for a partcular confguraton of party electoral declaratons. We assume that voters vote expressvely n the electon accordng to a probablstc rule. That s, they tend to vote myopcally for the party that puts forth the electoral declaraton they most prefer regardless of the polcy consequences. To operatonalze ths, we employ a multnomal logt (MNL) model. The MNL model can be motvated by a random utlty assumpton. Let z j denote the utlty to voter of votng for party j. Assume (2) z j = x j β + ε j, where ε j follows a log-webull dstrbuton (McFadden, 1974) and { 1, f zj = max(z y j = ), 0, otherwse.

MULTIPARTY ELECTORAL COMPETITION 411 TABLE 2 Summary of the posteror densty of a factor model of the Dutch ssue space, 1989 (results are averaged over the fve mputed data sets) Posteror Posteror Posteror 95% BCI Parameter mean medan std. dev. Lower Upper Economc factor λ aborton 0.000 0.000 0.000 0.000 0.000 λ nuclear plants 0.595 0.595 0.034 0.662 0.528 λ equalze ncome 0.551 0.551 0.033 0.616 0.484 λ euthanasa 0.000 0.000 0.000 0.000 0.000 λ nuclear weapons 0.519 0.518 0.036 0.588 0.450 Socal factor λ aborton 0.730 0.730 0.036 0.800 0.661 λ nuclear plants 0.000 0.000 0.000 0.000 0.000 λ equalze ncome 0.000 0.000 0.000 0.000 0.000 λ euthanasa 0.798 0.798 0.038 0.871 0.722 λ nuclear weapons 0.134 0.134 0.030 0.191 0.074 ρ economc, relgous 0.221 0.220 0.029 0.163 0.279 N 1754 Total Gbbs teratons per data set 5000 Number of factor models 5 Then the MNL samplng densty s gven by where N J f(y β) = Pr(y j = 1 β) y j, =1 j=1 exp(x (3) j Pr(y j = 1 β) = β) p k=1 exp(x k β). In what follows we employ mproper, unform prors for the elements of β. Hstorcally, socologcal factors played the domnant role n explanng voter choce n the Netherlands. As denomnatonal and class lnes began to soften n the late 1960s and early 1970s, ssue votng became ncreasngly mportant. Recent work (Alvarez and Nagler, 1998; Qunn, Martn and Whtford, 1999) suggests that socologcal factors and ssue preferences play an mportant, jont role n determnng voter behavor. For these reasons we choose to nclude both socologcal varables and ssue varables n our emprcal model of votng behavor. The specfc covarates used n the analyss are presented n Table 1. To capture the voter ssue preferences we nclude the squared dstance between each voter s deal polcy θ and the mean perceved party locaton of each party ψ (j) as a choce-specfc covarate for j = 1,...,J. Snce the MNL covarates X depend on draws from p(θ, ψ D ms,d obs ),wetakefvedraws from p(θ, ψ D ms,d obs ) for each of the fve mputed data sets and then ft a MNL model to each of these 25 data sets. We use the Metropols algorthm to sample from the posteror densty of ths model. Note that ths allows us to obtan a sample approxmately from p(β D ms,d obs,θ, ψ). Table 3 summarzes the results from the 25 MNL models. To llustrate the substantve mplcatons of changes n party electoral declaratons, we graph the expected vote share of a gven party aganst potental polcy locatons of the party n the two-dmensonal polcy space, condtonal on the other three partes remanng fxed at ther estmated actual locatons n the polcy space. The vote shares are computed usng the coeffcent estmates from the vote choce model. Fgures 3 6 dsplay the mpact changes n party electoral declaratons on votng behavor. From these fgures we can see that the vote shares of the two largest partes CDA and PvdA are qute senstve to changes n electoral declaratons, yet each of these two partes can count on mantanng about 6 10% of the vote regardless of the electoral poston taken wthn the [ 2, 2] square. On the other hand, the vote shares of the smaller partes VVD and D66 are less senstve to the partes polcy postons. It s also nterestng to note that no party could gan a smple majorty of the vote gven the emprcal locatons of the other partes.

412 K. M. QUINN AND A. D. MARTIN TABLE 3 Results from the 25 MNL models Posteror Posteror Posteror 95% BCI Parameter mean medan std. dev. Lower Upper Spatal dstance 0.295 0.294 0.029 0.354 0.241 Relgosty PvdA 0.101 0.101 0.220 0.324 0.530 VVD 0.406 0.402 0.271 0.106 0.956 CDA 1.430 1.434 0.268 0.903 1.958 Catholc PvdA 0.401 0.401 0.185 0.033 0.764 VVD 0.280 0.298 0.236 0.277 0.704 CDA 0.979 0.983 0.195 0.581 1.353 Class PvdA 0.639 0.635 0.114 0.866 0.425 VVD 0.038 0.033 0.116 0.282 0.176 CDA 0.316 0.311 0.104 0.533 0.125 Unon member PvdA 1.008 0.996 0.196 0.664 1.433 VVD 0.090 0.115 0.311 0.650 0.562 CDA 0.176 0.159 0.250 0.258 0.710 Income PvdA 0.094 0.093 0.060 0.215 0.022 VVD 0.174 0.177 0.069 0.031 0.302 CDA 0.054 0.055 0.059 0.064 0.163 Educaton PvdA 0.313 0.316 0.106 0.520 0.103 VVD 0.189 0.191 0.105 0.387 0.038 CDA 0.155 0.160 0.111 0.367 0.084 Age PvdA 0.107 0.107 0.038 0.034 0.181 VVD 0.102 0.099 0.048 0.015 0.201 CDA 0.151 0.151 0.042 0.072 0.234 Rural PvdA 0.162 0.178 0.306 0.724 0.488 VVD 0.112 0.113 0.302 0.491 0.686 CDA 0.027 0.020 0.309 0.551 0.627 Urban PvdA 0.122 0.116 0.220 0.561 0.319 VVD 0.156 0.161 0.260 0.616 0.454 CDA 0.112 0.107 0.261 0.613 0.393 Constant PvdA 2.736 2.729 0.334 2.094 3.400 VVD 0.042 0.036 0.427 0.891 0.761 CDA 0.746 0.735 0.408 0.035 1.579 FIG. 3. Expected CDA vote share at varous electoral declaratons condtonal on emprcal locatons of the other Dutch partes, 1989. FIG. 4. Expected PvdA vote share at varous electoral declaratons condtonal on emprcal locatons of the other Dutch partes, 1989.

MULTIPARTY ELECTORAL COMPETITION 413 FIG. 5. Expected VVD vote share at varous electoral declaratons condtonal on emprcal locatons of the other Dutch partes, 1989. FIG. 6. Expected D66 vote share at varous electoral declaratons condtonal on emprcal locatons of the other Dutch partes, 1989. Wth knowledge of the voters utlty functons [emboded n p(β D ms,d obs,θ, ψ)] we can go on to make nferences about the vote shares and seat shares that would accrue to partes f they were to put forth a dfferent set of electoral declaratons. Calculatng vote shares for partes s smple because t only requres pluggng new values of the covarates nto equaton (3) and calculatng the votng probabltes for the voters. A determnstc rule maps votes nto parlamentary seats. We assume the d Hondt rule (see Cox, 1997) translates ctzens votes n the parlamentary electon nto seats, whch s the actual nsttutonal rule used n the Netherlands. 4.3 Calculatng Equlbrum Cabnets and Polcy Outcomes Gven Electoral Declaratons and Seat Strengths Wth knowledge of the polcy postons and seat shares of the partes, we can go on to examne the formaton of governments and publc polcy n the Parlament. To model cabnet formaton and polcy enactments we use a verson of Laver and Shepsle s (1996) model of cabnet formaton. The Laver Shepsle model assumes the exstence of a K-dmensonal polcy space X R K. Partes j = 1,...,J are assumed to have fxed polcy postons ψ j X. As noted above, n the Dutch applcaton, the set of partes s assumed to consst of PvdA, CDA, VVD and D66. It s assumed that partes are motvated by publc polcy and attempt to produce publc polcy that s most consstent wth ther polcy postons. More concretely, we assume that the utlty party j attaches to a partcular polcy outcome s gven by the negatve squared dstance between ψ j and the polcy outcome n queston. In addton to a polcy poston, each party s assumed to have a nonnegatve seat strength. Ths s smply the number of seats the party n queston controls n the parlament. Gven the set of partes and the partes polcy postons, utlty functons and seat strengths, the Laver Shepsle model works as follows. (See Chapters 3 and 4 of Laver and Shepsle, 1996, for a more detaled dscusson of ther model.) Stage I of the model conssts of a party from the set of partes beng chosen to propose a cabnet. A cabnet conssts of an allocaton of the varous mnstry portfolos (.e., the Fnance Mnster, the Home Affars Mnster, etc.) to the partes. Followng the actual parlamentary rules used n the Dutch Parlament, we assume that the party wth the largest seat strength s allowed to make the frst proposal. We make the smplfyng assumpton that poltcal competton s prmarly over two mnstres the Fnance Mnstry and the Mnstry of Home Affars. (Laver and Shepsle, 1996, page 153, presented evdence from a survey of country experts that suggests ths assumpton s reasonable.) Each mnstry corresponds wth one of the ssue dmensons n the polcy space. In Stage II of the model, partes controllng mnstres under the Stage I proposal are allowed to veto the proposed cabnet by refusng to accept the portfolos they were allocated. If the cabnet s vetoed, we go to Stage I and a new party s allowed to propose a cabnet. If no party vetoes a proposed cabnet we move to Stage III of the model. In Stage III, the Parlament as a whole votes for ether the proposed cabnet or for the status quo cabnet (the last cabnet approved by a majorty of the Parlament). For our purposes, the game stops when a parlamentary majorty approves a cabnet. Laver and Shepsle assumed that a cabnet produces publc polcy dmenson by dmenson. In other words, the party that controls a partcular mnstry sets polcy on the dmenson that corresponds wth that mnstry. For nstance, n the Dutch case we assume the polcy space s two-dmensonal. The frst dmenson corresponds to economc polcy controlled by the Fnance Mnstry and the second dmenson corresponds to socal polcy controlled by the Mnstry of Home Affars.

414 K. M. QUINN AND A. D. MARTIN A cabnet may consst of CDA controllng the Fnance Mnstry and PvdA controllng the Mnstry of Home Affars. In ths case, CDA would set economc polcy to ts preferred mx of economc polces (gven by the frst element of ψ CDA ) and PvdA would set socal polcy to ts preferred mx of socal polces (gven by the second element of ψ PvdA ). An equlbrum cabnet n the Laver Shepsle model s an allocaton of mnsteral portfolos that s majorty preferred to all other possble allocatons of mnsteral portfolos. It s mportant to remember that partes are assumed to be motvated not by controllng portfolos but by the polcy outcomes that result from a partcular cabnet. Thus a party may well prefer not to be a member of a governng coalton f stayng n opposton produces publc polcy that s more n lne wth ts polcy preferences. Equlbrum cabnets can be found by searchng through the set of cabnets that are majorty preferred to the status quo cabnet (the cabnet n place at the tme of the electon). To summarze, once we know the partes polcy declaratons and the assocated seat strengths, we can fnd equlbrum cabnets and polcy outcomes usng the Laver and Shepsle (1996) model. 4.4 Fndng Equlbrum Polcy Declaratons Suppose that party j s lookng down the game tree and s choosng ts electoral declaraton ψ j.partyj s goal s to fnd the value of ψ j that maxmzes ts utlty gven ts belefs about the values of the other quanttes n the model (ψ j,d ms,d obs,θ, ψ,β). We say that a collecton of party electoral declaratons ψ s an equlbrum confguraton of electoral declaratons f no party can unlaterally choose a dfferent electoral declaraton and mprove ts payoff. To search for equlbrum confguratons of electoral declaratons we need to defne the partes utlty functons and show how to calculate expected utlty. In what follows, we let u j (ψ j ψ j,d ms,d obs,θ, ψ,β) denote the utlty j receves from puttng forth electoral declaraton ψ j gven partcular values of ψ j,d ms, D obs,θ, ψ and β. We look at two possbltes for u j (ψ j ψ j,d ms, D obs,θ, ψ,β). Frst, that partes are motvated by maxmzng vote share. Under ths assumpton, the utlty that party j gets from electoral declaraton ψ j s smply the fracton of the votes j would expect to receve gven ths polcy declaraton and the values of the other model parameters. The second type of party motvaton we explore s that partes are motvated by publc polcy n a fashon consstent wth the Laver and Shepsle (1996) model. As noted above, gven values of all the model parameters, t s possble to fnd the polcy outcome of the ultmate coalton barganng game nduced by the electoral declaratons, voter deal ponts and the voter utlty functons. We assume that the payoff party j gets from a partcular polcy outcome s the negatve squared dstance between the spatal locaton of the polcy outcome and the party s true deal pont (whch we take to be ψ (j) ). Whle t s not possble to easly wrte down ths functonal relatonshp between polcy declaratons and utlty, t s possble to evaluate ths utlty functon at partcular values of the model parameters. Ths wll prove mportant when calculatng expected utlty below. Snce D ms,θ, ψ and β are not known wth certanty, we need to average over these quanttes to calculate the expected utlty of an electoral declaraton. If we knew the jont dstrbuton of the quanttes gven the observed data we could stll calculate expected utlty accordng to U j (ψ j ψ j,d obs ) (4) = u j (ψ j ψ j,d ms,d obs,θ, ψ,β) p(d ms,θ, ψ,β D obs )dd ms dθ d ψdβ. Rather than try to deal wth the rght hand sde of Equaton (4) drectly, we use the rules of condtonal probablty and proceed to factor p(d ms,θ, ψ,β D obs ) nto smaller chunks and construct a Monte Carlo estmate of U j (ψ j ψ j,d obs ). Ths works as follows. For l = 1,...,L: Draw D (l) ms from p(d ms D obs ). Draw (θ (l), ψ (l) ) from p(θ, ψ D obs,d (l) ms ). Draw β (l) from p(β D obs,d (l) ms,θ(l), ψ (l) ). Calculate (5) Û j (ψ j ψ j,d obs ) = 1 L L l=1 u j (ψ j ψ j,d (l) ms,d obs,θ (l), ψ (l),β (l) ). Such an estmate can be made arbtrarly accurate by ncreasng the number of random draws L. As noted above, draws from p(d ms D obs ), p(θ, ψ D obs,d (l) ms ) and p(β D obs,d (l) ms,θ(l), ψ (l) ) are avalable from MCMC samplng. As a practcal matter, we saved fve draws of D ms from p(d ms D obs ), fve draws of (θ, ψ) from p(θ, ψ D obs,d ms ) for each of the

MULTIPARTY ELECTORAL COMPETITION 415 fve prevous draws of D (l) ms and fve draws from p(β D obs,d ms,θ, ψ) for each of the 25 prevous draws of (θ, ψ). Because of the ntractablty of the partes expected utlty functons t s not feasble to characterze equlbrum declaratons analytcally. Instead equlbra are found va an teratve tâtonnement process. Ths works as follows: 1. Randomly assgn startng declaratons. 2. Randomly permute partes. (a) Party 1 maxmzes expected utlty condtonal on the other partes electoral declaratons. (b) Party 2 maxmzes expected utlty condtonal on the other partes electoral declaratons. (c) Smlarly for remanng partes. 3. (a) If no movement n the last two permutatons, return current confguraton of electoral declaratons. These are supported n equlbrum. (b) Else go to step 2. Whle such an algorthm s not guaranteed to fnd all potental equlbrum declaratons, the declaratons that t returns (f any) wll n fact be part of a pure strategy Nash equlbrum. The advantage of focusng on declaratons that are arrved at through such an adaptve process s that they are consstent wth a very smple model of human reasonng n whch each player mentally works through such a process of adaptve optmzaton untl a steady state s reached. Ratonal players would then play the approprate equlbrum strateges (see Fudenberg and Trole, 1991, Secton 1.2.5, for a more detaled dscusson). 5.1 Vote Maxmzaton 5. EQUILIBRIUM RESULTS We begn by examnng the case n whch the partes are motvated solely by vote maxmzaton. As noted above, the assumpton of vote-maxmzng partes s farly common n the substantve lterature on multparty electoral competton as well as some of the ratonal choce lterature. As we show below, ths assumpton s not supported by the 1989 Dutch data. Wth vote-maxmzng partes, u j (ψ j ψ j,d ms, D obs,θ, ψ,β) s smply the vote share that party j can expect from choosng ψ j gven ψ j,d ms,d obs,θ, ψ and β. We numercally found constellatons of electoral declaratons that are supported n Nash equlbrum gven the vote-maxmzng specfcaton of party utlty. Fgure 7 dsplays these equlbrum platforms. FIG. 7. Equlbrum electoral declaratons wth vote-maxmzng partes, 1989. As we expect, the vote-maxmzng equlbrum declaratons are very centrst and located very near the mean of the voter dstrbuton. (For smlar, theoretc results, see, Ln, Enelow and Dorussen, 1999, and Erkson and Romero, 1990.) The mnor departures from the mean voter poston are due to the ncluson of the socologcal varables n the voter utlty functons. A cursory comparson of Fgures 7 and 2 reveals that the equlbrum declaratons wth vote-maxmzng partes are qute dfferent from the emprcal party declaratons. Based on our 1989 data, the four major Dutch partes dd not take electoral postons that are consstent wth vote-maxmzng behavor and equlbrum play. 5.2 Polcy-Seekng Partes Next we assume that partes put forth electoral declaratons to nfluence the polcy outcome of the mpendng cabnet formaton game. To calculate u j (ψ j ψ j, D ms,d obs,θ, ψ,β) n ths case, we calculate the vote shares that are mpled by ψ,d ms,d obs,θ, ψ and β; allocate seats accordng to the d Hondt method; calculate the equlbrum cabnet and assocated polcy output based on the Laver and Shepsle (1996) model; and then calculate the negatve squared dstance between ths polcy outcome and party j s deal polcy. Averagng over all 125 draws of the model parameters gves us an estmate of the expected utlty of ψ j to

416 K. M. QUINN AND A. D. MARTIN FIG.9. 1989. Expected utlty to CDA of varous electoral declaratons, FIG. 8. Equlbrum electoral declaratons wth polcy-seekng partes, 1989. party j gven the other party declaratons and the observed data. Once agan we calculate Nash equlbra numercally based upon ths specfcaton of utlty. Fgure 8 dsplays the equlbrum party declaratons n whch partes are motvated by polcy goals. Ths fgure reveals a confguraton of party postons that s much more n lne wth the emprcal locatons depcted n Fgure 2. To provde some ntuton as to why we see such dvergence n electoral declaratons wth polcy-seekng partes we examne the CDA s expected utlty functon condtonal on the equlbrum declaratons of the other partes and the observed data. Fgure 9 depcts ths functon. Frst, note that ths s not an easy functon to maxmze. Because of the low dmensonalty of the search space, we employed a very smple grd search on the [ 2, 2] square. [Before mplementng the grd search, we expermented wth varous dervatvefree optmzaton algorthms, ncludng the Nelder and Mead (1965), smplex method. Due to the complexty of the objectve functon, as llustrated n Fgure 9, these methods dd not converge. In all of the Nash fndng grd searches, we multstarted them from varous ponts n the space, and have acheved the same equlbrum from each run. We have also used varous grd szes for the grd search. Of course, ths n no way guarantees that the equlbrum s unque, but we do know that t s an equlbrum.] More mportantly, Fgure 9 provdes mportant nsght nto why polcyseekng partes wll tend to dverge from the center of the voter dstrbuton. At ts equlbrum declaraton, the CDA s able to control economc polcy but not socal polcy there wll always be a majorty n parlament who are wllng to block cabnets n whch CDA can enact ts rather extreme (relatve to the other partes) preferred socal polces. The only way that CDA can gan control over socal polcy s to move south of the PvdA s poston. However, snce we assume that partes must try to enact polcy consstent wth ther electoral declaratons, the CDA would actually be forced to enact socal polcy that s more lberal than the socal polcy pursued by the PvdA n equlbrum. Ths s why we see the dp n the CDA s expected utlty as t takes centrst postons. If the CDA were to take extremely lberal socal postons (correspondng to negatve values on the socal dmenson), t once agan s unable to mplement socal polcy and consequently ts utlty s roughly the same as when t chooses extreme, conservatve postons. We also make a prelmnary attempt to examne the extent to whch these equlbrum results are dependent upon the salence of polcy ssues estmated from the 1989 data. As noted prevously, ssue votng has become ncreasngly prevalent n the Dutch electorate n recent years. Would we expect to see a dfferent constellaton of party declaratons n equlbrum f voters attached ncreasng amounts of weght to ssue concerns? To examne ths we computed a polcy-maxmzng equlbrum n whch the MNL coeffcent on spatal dstance was set to be equal to the values drawn from the MNL posteror mnus 1.00. Ths mples that voters are behavng n a much more determnstc

MULTIPARTY ELECTORAL COMPETITION 417 manner. The equlbrum results (not shown here) are nearly dentcal to those from the emprcally calbrated polcy-seekng model. Ths suggests that the polcy-seekng equlbra may be relatvely robust to modest changes n the model parameters. As a conjecture, we speculate that ths model may help elucdate some of the mcrofoundatons underlyng the observatons of scholars such as Lpset and Rokkan (1967) regardng the relatve stass n most party systems. It may well be the case that once an electoral equlbrum s reached, t may take very large changes n ether voter preferences or the preferences of party leaders to result n notceable shfts n the polcy declaratons of partes. Further, ths model suggests that the reason many European partes n PR systems have not completely muted ther programmatc appeals and become true catchall partes has less to do wth deologcal consstency as suggested by scholars such as Krchhemer (1966) and more to do wth the nonmonotonc relatonshp between seats and poltcal power. Fnally, t must be remembered that the real reason we see electoral dvergence n our model s because partes are assumed to have dverse polcy preferences. To understand the reasons for ths, t s useful to return to the more tradtonal lterature on party recrutment. 6. CONCLUSION The results presented here strongly suggest that to understand electoral competton n multparty democraces, t s not enough to know voter preferences and the dmensonalty of the ssue space. Instead, expectatons of postelectoral coalton barganng very lkely exert a substantal effect on the electoral declaratons of poltcal partes. Ths leaves us wth the theoretcal challenge of developng models of multparty electoral poltcs that are based on rcher assumptons about party motvatons. In large part, ths theoretcal enterprse revolves around the correct specfcaton of the partes objectve functons. The specfcaton of these objectve functons presupposes an understandng of (1) how electoral declaratons translate nto seat strengths, (2) how electoral declaratons and seat strengths translate nto governng coaltons, and (3) how electoral declaratons, seat strengths and coalton structures translate nto polcy outcomes and government perqustes. The approach that we have adopted s but one of many ways to specfy partes objectve functons. The power of the computatonal approach presented n ths artcle s that t does not requre the ablty to wrte down the partes objectve functons n closed form. Instead, all t requres s that one can evaluate the objectve functons at any pont n the polcy space. Whle computatonal solutons lack the generalty of analytcal results, t s our belef that n areas such as multparty electoral competton, where the phenomena under study are both extremely complex and where smple theoretcal models seem at odds wth emprcal realty, a combnaton of emprcal analyss and computatonal modelng offers great promse. By comparng the computed results of specfc models aganst emprcal observatons, we can gan a better understandng of the strengths and weaknesses of the underlyng theoretcal models wth the hope of contnually refnng and generalzng such models. Further, just as numercal methods are often used to solve appled general equlbrum models n economcs to better understand the polcy mpact of varous types of government nterventon, the modelng strategy sketched here can be used to gan nsght nto the normatve consequences of demographc and nsttutonal changes. For nstance, t may be nterestng to examne the effect of the Dutch populaton becomng ncreasngly secular. Would such a change result n dfferent coalton governments or has the hstory of the Dutch party system put the CDA n a poston where t can contnue to be a major player n coalton poltcs despte the loss n votes that would come from the ncreasng secularzaton of poltcs? Usng our modelng strategy t s sensble to entertan a counterfactual change n the composton of the votng publc snce (wth nonstrategc voters) ths should not affect votng behavor. Addtonally, t would be possble to explore the mplcatons of ncludng the addton of another party nto the poltcal sphere (after makng strong assumptons about the effect of socologcal characterstcs), changes n the dstrbuton of voter preferences and the case when voters choose purely on the ssues. It s worth emphaszng that when conductng such counterfactual exercses one must be careful to entertan counterfactuals that are consstent wth assumptons under whch the model parameters were estmated. For nstance, t s not possble for the model sketched here to answer the queston of what would happen to equlbrum electoral declaratons f an external shock such as a war or economc crss gave poltcal eltes the opportunty to engage n heresthetc maneuvers that changed the dmensonalty and components of the ssue space. Such heresthetcal changes would yeld an entrely dfferent spatal representaton, whch would be mpossble to explore gven our model.

418 K. M. QUINN AND A. D. MARTIN Addtonally, t would be mpossble to explore the possblty of polcy-mnded sophstcated votng. Smlarly, relaxng the ndependence of rrelevant alternatve assumptons makes explorng the mplcatons of party change mpossble. Substantvely, ths modelng strategy has the potental to clarfy the mcrofoundatons underlyng more tradtonal qualtatve analyses of European party systems. It s mportant to recognze that studyng party declaratons, votng behavor and coalton formaton ndependently cannot provde answers to fundamental questons about multparty democraces. To do so, t s vtal to ntegrate these separate levels of analyss nto a sngle theoretcal and emprcal model. The methodology we provde here allows us to do so. APPENDIX: MCMC ALGORITHM FOR THE MEASUREMENT MODEL The algorthm used to ft the measurement model s the followng: 1. For = 1,...,N,drawθ from N ( ˆθ,V θ ),where V θ = ( + 1 ) 1 and ˆθ = V θ ( 1 w ). 2. For r = 1,...,R, draw r from N ( ˆ r,v r ), where V r = ( r + σrr 2 r r ) 1, ˆ r = V r ( 1 r r + σrr 2 r w r ) and r denotes the columns of that correspond to r. 3. For r = 1,...,R,drawσrr 2 from IG[(α + n)/2, (β + e r )/2],wheree r = (w r r ) (w r r ). 4. Draw ρ va the Metropols step: Let ρ (c) denote the current value of ρ. Draw ρ (p) from N (ρ (c),δ),whereδ s a user defned tunng parameter. Set ρ = ρ (p) wth probablty { mn 1, p(ρ(p) w,, ) p(ρ (c) w,, ) else set ρ = ρ (p), where p(ρ w,, ) = f(w,,ρ)p( )p( )p(ρ). Iteratng through ths sequence of draws yelds a seres of draws from the jont posteror of (,,,ρ).wth the excepton of the Metropols step to sample the factor correlaton coeffcent, ths algorthm s very smlar to that n Lopes and West (1999). We can sample from the jont dstrbuton of the voter deal ponts (θ) and the average voter percepton of the party declaratons at the tme of the survey ( ψ) by vewng t as a determnstc functon of the model parameters. We can do so from p(θ, ψ D obs,d ms ) by nsertng the followng steps nto the MCMC samplng scheme: } ; For = 1,...,N and j = 1,...,J,drawψ (j) from N ( ˆψ (j),v (j) ψ ), wherev (j) ψ = ( + 1 ) 1 and ˆψ (j) = V (j) ψ ( 1 w (j) ). Calculate ψ (j) = n 1 n=1 ψ (j). Ths assumes that the voters use the same perceptual process to map party declaratons from the fvedmensonal ssue space to the two-dmensonal predctve space as they use to map ther own preferences from the hgher dmensonal space to the lower dmensonal space. ACKNOWLEDGMENTS An earler verson of ths paper was presented at the 2000 Annual Meetng of the Amercan Poltcal Scence Assocaton, Washngton, DC. We thank Larry Bartels, Sd Chb, Garrett Glasgow, Ed Greenberg, Brad Jones, Gary Kng, Norman Schofeld and Andy Whtford for helpful comments and dscussons. Ths work s based on research supported by NSF Grants SES 01-35855 to Washngton Unversty and SES 01-36676 to the Unversty of Washngton. Addtonal fnancal support was provded by the Center n Poltcal Economy at Washngton Unversty and the Center for Statstcs and the Socal Scences wth funds from the Unversty Intatves Fund at the Unversty of Washngton. REFERENCES ALVAREZ, R. M. and NAGLER, J. (1998). When poltcs and models collde: Estmatng models of multparty electons. Amercan Journal of Poltcal Scence 42 55 96. ANKER, H. and OPPENHUIS, E. V. (1993). Dutch Parlamentary electon study, 1989 (computer fle). Dutch Electoral Research Foundaton and Netherlands Central Bureau of Statstcs, Amsterdam. AUSTEN-SMITH, D. andbanks, J. (1988). Electons, coaltons, and legslatve outcomes. Amercan Poltcal Scence Revew 82 405 422. BLACK, D. (1958). The Theory of Commttees and Electons. Cambrdge Unv. Press. COUGHLIN, P. J. (1992). Probablstc Votng Theory. Cambrdge Unv. Press. COX, G. W. (1997). Makng Votes Count: Strategc Coordnaton n the World s Electoral Systems. Cambrdge Unv. Press. DALTON, R. J. (1985). Poltcal partes and poltcal representaton: Party supporters and party eltes n nne natons. Comparatve Poltcal Studes 18 267 299. DAVIS, O. and HINICH, M. (1966). A mathematcal model of polcy formaton n a democratc socety. In Mathematcal Applcatons n Poltcal Scence (J.L.Bernd,ed.)2 175 208. Southern Methodst Unv. Press, Dallas.