Electoral Rule Disproportionality and Platform Polarization

Transcription

1 Electoral Rule Disproportionality and Platform Polarization Konstantinos Matakos Orestis Troumpounis Dimitrios Xefteris December 19, 2013 Abstract Despite what common perception dictates, theoretical literature lacks a complete formal argument regarding the relationship between the electoral rule disproportionality and platform polarization. In that direction, we consider a unidimensional electoral competition model with mainly (but not necessarily purely) policy-motivated parties. We first show that in equilibrium platform polarization (measured by any of the popular measures of polarization) is decreasing in the level of the electoral rule disproportionality. We then argue that the number of parties has a(n) positive (ambiguous) effect on platform polarization when polarization is measured by the distance between the two most distant platforms (the Dalton index of polarization). By constructing a large database we provide empirical evidence in support of our theoretical findings: electoral rule disproportionality is a major determinant of polarization while the number of competing parties has the predicted explanatory power only if it is measured by the distance between the two most distant platforms. Keywords: proportional representation; disproportional electoral systems; polarization; policy-motivated parties; number of parties; Duvergerian predictions London School of Economics and Political Science, Government Department, Connaught Hall CON.5.10, London WC2A 2AE, UK; k.matakos@lse.ac.uk Department of Economics, Lancaster University, The Management School, Bailrigg, LA1 4YX, UK, o.troumpounis@lancaster.ac.uk Department of Economics, University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus, xefteris.dimitrios@ucy.ac.cy 1

2 1 Introduction Party-system (platform) polarization is known to have a significant effect on a wide array of policy issues ranging from the political stability (Sartori, 1976) and cabinet-survival (Warwick, 1994) to the effectiveness of policy-making (Tsebelis, 2002) and political representation (Abney et al., 2007; Huber and Powell, 1994). In addition to those dimensions, polarization has a strong negative effect on economic performance and growth (Frye, 2002) as well as on the risk of social tension and violent conflict within and between nation states (Esteban and Schneider, 2008). So far, many alternative theories were developed to identify the determinants of polarization. For example, some attribute it to informational or media-related factors (e.g., Grosser and Palfrey 2013; DellaVigna and Kaplan 2007) while others to diverging policy preferences of candidates (e.g., Calvert 1985; Roemer 1994). In this paper we focus on one institutional (electoral rule) and one structural (number of parties) determinant of polarization and ask the following questions; How does the electoral rule disproportionality determine how polarized a party-system is when parties are mainly (but not necessarily purely) policy motivated? 1 How does the nature of political competition (number of competing parties) affect polarization? Finally, does empirical evidence support the predictions of the theory? This paper aims at answering these questions by the means of a conventional formal model of electoral competition in a unidimensional policy space. In light of contradicting results of recent empirical studies regarding the effect of the electoral rule disproportionality on platform polarization (e.g., Curini and Hino 2012; Dow 2011, 2001; Calvo and Hellwig 2011; Andrews and Money 2009; Ezrow 2008; Dalton 2008; Budge and McDonald 2006) the development of robust theoretical predictions becomes extremely relevant. In the first part of this paper we do precisely this: we formally develop two distinct theoretical predictions. We show existence of a unique equilibrium 1 In the main text we present our results considering only purely office motivated parties. In the appendix we show that our results are robust to partially office motivated parties. 2

3 in which the degree of platform polarization is; a) decreasing in the level of electoral rule disproportionality independently of how polarization is measured; and b) increasing in the number of competing parties only if polarization is measured by the distance between the two most distant platforms (MDP). If polarization is measured by the Dalton index (DI) then the effect of the number of competing parties on platform polarization is ambiguous. 2 In the second part of the paper we provide empirical evidence in support of these theoretical findings. Overall, our paper provides a unified framework for studying both theoretically and empirically the effect of electoral rule disproportionality on polarization. Our paper is, of course, not the first to formally explore the institutional and structural determinants of platform polarization since other models also describe how electoral equilibria are affected by changes in the institutional and structural elements of the political environment. For example, Cox (1990) argued in favor of a negative link between the disproportionality of the electoral rule and platform polarization, by analyzing a model with purely office-motivated parties. More recently, Calvo and Hellwig (2011) applied an alternative approach based on a probabilistic voting model (a là Adams et al. 2005) that also led to formal arguments in favor of the negative relationship between the electoral rule disproportionality and platform polarization. Finally, Iaryczower and Mattozzi (2004) by the means of a citizen-candidate model show that certain equilibria of a proportional system lead to larger polarization than certain equilibria of plurality rule elections. We believe though that the features of an equilibrium of a formal model can better support an empirically testable hypothesis if this equilibrium a) is proven to exist, b) is essentially unique and c) relies on standard assumptions regarding the interests of all participating agents (parties and voters). To our knowledge the present formal analysis 2 In the appendix, we consider a variation of the model with endogenous party-entry that includes an entry stage in the game and we show that in line with the Duvergerian predictions (Duverger, 1954) the number of competing parties is decreasing in the level of electoral rule disproportionality. Our results therefore point at a second indirect channel through which disproportionality affects polarization: as the number of parties is decreasing in the level of electoral rule disproportionality, the centripetal forces of disproportionality are further amplified. 3

4 is the first to combine all these three features. Cox (1990) clearly stresses that his results only tell what will happen if there is an equilibrium; they do not guarantee that an equilibrium will exist. The approach by Calvo and Hellwig (2011) on the other hand guarantees equilibrium existence but it relies on non-conventional assumptions about voters preferences: voters do not care only about parties proposed platforms but also about the asymmetry between a party s seatand vote-share. Finally, given the nature of a citizen candidate model that they employ, Iaryczower and Mattozzi (2004) can not provide straightforward comparisons across electoral systems due to the well-known problem of multiplicity of equilibria (for a discussion see e.g., Dhillon and Lockwood 2002). By introducing policy motives (a là Calvert 1985; Wittman 1977) this paper actually surmounts the complexities of establishing existence of an equilibrium in a framework with standard voters preferences. At the same time our model provides a unique equilibrium prediction that allows us to explore and empirically test the link through which electoral rule disproportionality and the structure of the party-system affect polarization. 3 Let us mention here that a common link among Cox (1990); Iaryczower and Mattozzi (2004); Calvo and Hellwig (2011) and our paper is the analysis of a continuum of disproportionalities essentially considering any rule that lies between a purely proportional and a first past the post (FPTP) electoral rule. Such continuum not only allows the analysis of different electoral systems in an abstract way but also permits the comparison of different electoral rules that belong in the same family but whose disproportionality may vary significantly (e.g. in the family of proportional representation (PR) systems the electoral rule in Italy is much more disproportional than in Netherlands). 4 3 This route to overcome problems of possible non-existence of equilibria in electoral competition models is not novel in the literature. Groseclose (2001), for example, used this framework to deal with the non-existence of pure strategy equilibria in competition of two office-motivated candidates of unequal valence. 4 For important pairwise comparisons between FPTP and proportional systems see for example Lizzeri and Persico (2001) on public good provision, Morelli (2004) on party formation, Austen-Smith (2000) on redistribution, Persson et al. (2003) on corruption. For a pairwise comparison between plurality and runoff elections see Osborne and Slivinski (1996). Finally, Myerson (1993a,b) offers pairwise comparisons between PR, approval voting, FPTP and Borda rule mainly on the issues of corruption and campaign promises. 4

5 In order to fully grasp why electoral disproportionality acts as a centripetal force, hence resulting in low levels of polarization, we first model a two-party election. The leftist party has its preferred policy at the extreme left of the policy space while the rightist party has its preferred policy at the extreme right. 5 First, parties announce their platforms. Second, voters observe the announced platforms and vote for the party that proposed the platform closer to their ideal policy. Finally, a policy is implemented according to the parliamentary-mean model (Ortuño-Ortín, 1997; Llavador, 2006; De Sinopoli and Iannantuoni, 2007; Merrill and Adams, 2007). 6 In such model the implemented policy is a weighted average of parties announced platforms where each party s weight is determined by its seat-share in parliament. Hence, the policy outcome is a function of the announced platforms, the electoral outcome, and the disproportionality of the electoral system. According to our results, parties platforms in equilibrium converge towards the ideal policy of the median voter as the electoral system becomes more disproportional (in favor of the winner). The intuition is clear. On one hand, a move towards the median harms a party since it proposes a platform further away from its ideal policy. On the other hand, as parties move towards the median they increase their vote-share and hence their weight in the implemented policy. As the disproportionality of the electoral system increases, proposing a moderate platform may be worthwhile since the incentives to obtain some extra votes are amplified. This result generates the first empirically testable hypothesis of our model: an increase in electoral rule disproportionality decreases platform polarization. We conduct the same analysis with three parties (a leftist, a centrist and a rightist one) and we show that in equilibrium platform polarization not only moves as in the twoparty case (higher electoral rule disproportionality leads to less platform polarization) 5 Our results do not hinge on this since our main predictions are robust to parties with non-extreme policy platforms when we impose some mild assumptions on the distribution of voters (the ones required in proposition 2). 6 We borrow the name of this model from Merrill and Adams (2007). An alternative way of modeling PR elections is the dominant party model where each party implements its proposed platform with probability equal to its vote (or seat) share (Merrill and Adams, 2007; Faravelli and Sanchez-Pages, 2012; Iaryczower and Mattozzi, 2004). Merrill and Adams (2007) provide a comparison between the two ways of modeling PR systems and their effect on equilibrium proposed platforms. In our model both ways of modeling are equivalent and provide the same results and intuition if parties are risk neutral. We will actually further interpret our results through the dominant party model at a later stage. 5

6 but is also slightly higher compared to the two-party case when measured by the MDP. Hence, ceteris paribus and under an MDP measure, a larger number of competing parties increases platform polarization. On the other hand, our theory shows that the Dalton index (DI) does not provide a clear global prediction regarding the effect of the number of parties on the degree of platform polarization. This contrast between the MDP and the DI generates the second empirically testable hypothesis: the number of parties affects positively the degree of platform polarization when polarization is measured by the MDP while the effect is ambiguous (a point missed by all previous empirical literature) when polarization is measured by the DI. Our empirical analysis departs from recent literature in terms of the estimation methodology and survey design in the following aspects. First, by combining three different sources of data (Comparative Political Data Set-I, Comparative Manifesto Project and the Carey archive) we construct a balanced panel of 23 advanced democracies (OECD states) with a large number of electoral observations over a 50-year-long period (more than 300 observations). Second, our study simultaneously considers: a) both continuous and categorical measures of the electoral rule (dis)proportionality; b) two different measures of platform polarization (the MDP and the DI as detailed above); and c) employs both random and fixed effects estimators. Our empirical findings provide strong support in favor of our main theoretical predictions; the level of platform polarization is decreasing in the level of electoral disproportionality. Furthermore, as our theory predicts the number of competing parties has a significant positive impact on platform polarization only when we consider the MDP and an insignificant (and possibly ambiguous) effect when the DI is employed. The rest of the paper is organized as follows: In section two we present our theoretical model. In section three we provide our main set of results while in section four we perform our empirical analysis. We conclude in section five. Formal proofs, descriptive statistics and a version of the model with endogenously determined number of competing parties are incorporated in the appendix. 6

7 2 The Model We construct a formal model in line with the parliamentary-mean model (e.g. Ortuño- Ortín 1997; Llavador 2006; De Sinopoli and Iannantuoni 2007; Merrill and Adams 2007; Casamatta and De Donder 2005). 7 The innovation of our benchmark model of electoral competition is that the implemented policy does not only depend on parties vote-shares and proposed platforms but also on the (dis)proportionality of the electoral system. 8 We first consider two parties (j = L, R) that compete in an election and announce their platforms. Voters observe these platforms and vote for one of the two parties. Given parties vote-shares (V L and V R ), the announced platforms (p L and p R ), and the (dis)proportionality of the electoral system (n) a policy ˆp is implemented. The policy space is assumed to be continuous, one-dimensional, and represented by the interval Π = [0, 1]. We assume that there is a continuum of voters. Voters are distributed on the policy space according to a continuous and twice-differentiable probability measure F on Π. Let π i Π denote the ideal policy of individual i. We assume that each voter cares about the remoteness (but not the direction) of the proposed platform p j from his ideal policy π i. Specifically, individual i s preferences are represented by a utility function u : Π R, where u is a continuous, strictly decreasing function with respect to the distance between p j and π i. This utility function is symmetric, that is for all π i and for all x such that π i + x Π, π i x Π then it holds that u(π i + x; π i ) = u(π i x; π i ). Voters support the party that proposes the platform closer to their ideal point. Formally, a voter i with ideal policy π i votes for party j if u(p j ; π i ) > u(p j ; π i ) and splits his vote if u(p L ; π i ) = u(p R ; π i ). We denote as π the ideal policy of the indifferent voter, that is the ideal policy of the voter for whom it holds that u(p L ; π) = u(p R ; π). Given that preferences 7 In existing approaches, the implemented policy is assumed to be a convex combination of the proposed platforms weighted by a function that depends solely on parties vote-shares. As Ortuño-Ortín (1997) states Clearly, the specific function used should [also] depend on many institutional and cultural factors. In this paper we consider the effect of one of the most important institutional factors, namely the disproportionality of the electoral system. 8 Merrill and Adams (2007) also consider that the implemented policy depends on parties parliament seat-shares but they analyze strategic behavior of parties only under a purely proportional rule. That is, in their case the parliament seat-share of a party coincides with its vote-share. 7

8 are assumed to be symmetric, the location of the indifferent voter is always half distance between the platforms proposed by the two parties. Formally, π = (p L + p R )/2 and therefore parties vote-shares are given by F ( π) = F ( p L+p R ), if p 2 L < p R V L (p L, p R ) = 1, if p 2 L = p R 1 F ( π) = 1 F ( p L+p R ), if p 2 L > p R and V R (p L, p R ) = 1 V L (p L, p R ). Parties are policy motivated. Their payoffs depend on the implemented policy rather than on an exogenous given office value for winning the election. Each party j has an ideal policy π j Π. We assume that parties have an ideal policy at the extremes of the policy line, that is, π L = 0 and π R = 1 and that party s j preferences over policies are the same as the preferences of a voter with the same ideal policy. Formally, party s j payoffs when policy ˆp is implemented are defined as U j (ˆp) = u(ˆp; π j ). Let us mention that our results are robust to parties with non-extreme policy preferences if the conditions on the distribution of individuals required in proposition 2 hold. The implemented policy is a function of parties power in the parliament (parties seat-shares are denoted by (S L, S R )) and parties proposed platforms (p L, p R ). Notice that parties seat-shares are of course a function of parties vote-shares (V L, V R ) which ultimately are a function of the proposed platforms and the disproportionality of the electoral system denoted by n, where n 1. 9 We formally define the implemented policy function as: ˆp(p L, p R, n) = S L (p L, p R, n) p L + S R (p L, p R, n) p R 9 The analysis could incorporate the case of degressive proportionality where n < 1 and the electoral system is disproportional in favor of the loser (Koriyama et al., 2013; Bracco, 2013). In order to apply the term disproportionality as it is conventionally understood, that is in favor of the winner, we assume throughout the paper that n 1. This helps in the clarity and intuitive understanding of our results. 8

9 This function captures the post electoral compromise between parties platforms depending on parties parliamentary power. 10 The way parties vote-shares (V L and V R ) translate into seat-shares (S L and S R ) in the parliament depending on the disproportionality of the electoral system (n) follows Theil (1969): 11. S L S R = ( ) n VL V R Through the above formula and n = 1 one captures a purely proportional representation system where no distortions are present. Letting n = 3 the seat allocation is based on the famous cube law which is used in the literature as a good approximation of the distortions created in favor of the winner in FPTP elections. In general as n increases the electoral system is more disproportional in favor of the winner of the election. Since we know that S L + S R = 1 and that S L /S R = (V L /V R ) n we can rewrite the seat-shares as follows: S L = VL n/(v L n + V R n) and S R = VR n/(v L n + V R n ). Therefore the implemented policy function can be rewritten as: ˆp(p L, p R, n) = V L (p L, p R ) n V L (p L, p R ) n + V R (p L, p R ) p V R (p L, p R ) n n L + V L (p L, p R ) n + V R (p L, p R ) p n R In existing models of pure proportional representation (e.g. Ortuño-Ortín 1997; Llavador 2006; Merrill and Adams 2007) the associated weights to parties platforms are assumed to be proportional to parties vote-shares (captured by our function by assuming n = 1). As depicted in figure 1, by allowing n to take values larger than one, and for a given electoral outcome, we increase the weight put on the policy proposed by the winner of the election. Notice that the weights determined through this specific functional form 10 For two-party compromise models under pure PR elections where S L = V L and S R = V R see Llavador (2006); Ortuño-Ortín (1997). For multiparty models see De Sinopoli and Iannantuoni (2007, 2008); Gerber and Ortuño-Ortín (1998); Austen-Smith and Banks (1988); Merrill and Adams (2007). 11 Taagepera (1986) offers a further analysis of the above formula and empirical estimations of parameter n. For an overview of measures of bias and the proportionality in the relationship between vote-shares and seat-shares see Grofman (1983). Gallagher (1991) provides an analysis of indices and empirical measures of disproportionality. For recent applications of this formula see Calvo and Hellwig (2011); Ergun (2013); King (1990). 9

10 Figure 1: The weight of a party s proposal (i.e. its seat-share) as a function of its voteshare for the cases where n = 1, n = 3, and n. are identical to the seminal contest success function introduced by Tullock (1980, pg ). Remember though that since we assume that n 1 the weight function is convex for values smaller than one-half, and concave for values larger than one-half. Hence, in the electoral contest winning the election makes a difference. If n then parties actually compete in a winner-take-all election where the implemented policy converges to the winner s proposed platform. As in most spatial models of this type individuals vote for one of the parties once they observe the announced platforms. Hence, parties are the actual players of the game. Parties strategically announce their platforms, which are enough to determine the voters behavior and hence the outcome of the game and the corresponding payoffs. The equilibrium concept we apply is Nash equilibrium in pure strategies. 3 Results For clarity of presentation we first analyze the case of uniformly distributed voters and then we provide our general result. 10

11 Figure 2: The incentives for policy convergence. Figure 2a and 2b if n = 1, 2c if n = Uniformly distributed voters Consider first a pure PR system (where n = 1) depicted in Figure 2a. Let parties platforms coincide with their ideal policies, hence they propose platforms at the the extremes of the policy space (p L = π L = 0 and p R = π R = 1). The voter who is located at one-half is indifferent between the two proposed platforms ( π = 0.5). Voters on the left of the indifferent voter support the platform of the left party, while voters on the right of the indifferent voter support the right party. Given that voters are uniformly distributed each party obtains 50% of the votes. Combining this vote-share with the proportional system (n = 1) each party s platform has an equal (50%) weight on the implemented policy. Hence, the implemented policy is one-half (which coincides with both the median and the ideal policy of the indifferent voter). Let us first exploit the possible incentives for any of the two parties to deviate from this strategy and converge towards the median (Figure 2b). Let for example party L announce platform p L = 0.2. Now the indifferent voter moves further to the right (now π = 0.6). Hence, party L obtains 60% of the votes while party R obtains 40% of the votes. When the implemented policy is determined party L now has a larger weight than 11

12 before (now 60% compared to 50%), but at the same time proposes a policy further from its ideal point (p L = 0.2 compared to p L = 0). Therefore the implemented policy is ˆp = 0.52 and hence party L has no incentives to deviate from its initial strategy since by doing so the implemented policy moves further from its ideal point and is worse off. Consider now that the electoral system is more disproportional, hence it favors the winner of the election (let for example n = 3) as depicted in Figure 2c. As before, if party L announces platform 0.2 rather than zero, it obtains 60% of the vote-share. Because of the disproportionality such vote-share translates to a disproportionally high weight of 77% on its proposed platform. In contrast to the case when n = 1 such weight now compensates the loss because of proposing a moderate platform. It turns out that the implemented policy is Hence, party L has incentives to converge towards the median since it brings the implemented policy closer to its ideal point. The general mechanism which provides incentives to parties to propose less differentiated platforms when the degree of disproportionality of the electoral rule increases must have been made clear by now. Analyzing a uniform distribution of voters, though, does not only formally prove the negative direction of this relationship but further guarantees a closed form solution regarding parties equilibrium strategies. That is, it enhances our understanding on quantitative aspects of the relationship between platform polarization and electoral rule disproportionality other than its direction. Proposition 1. Let π i U[0, 1]. Then: (i) There exists a unique equilibrium (p L, p R ) = ( n 1, n+1 2n 2n ) (ii) The distance between p R The unique equilibrium (p L, p R and p L is decreasing in n (iii) ˆp = π = 0.5. ) of two party competition for different values of electoral disproportionality n [1, 20] and a graphical interpretation of proposition 1 is depicted in figure 3. If n = 1 (i.e. no distortions are present) both parties platforms coincide with their ideal policies, hence they stick at the the extremes of the policy space (p L = π L = 0 and p R when n = 3 is (p L, p R = π R = 1). It is easy to calculate that the unique equilibrium ) = (1/3, 2/3). In the extreme case of a winner-take-all election 12

13 (n ) where the winner can perfectly implement his platform we obtain the standard result of full convergence to the median. 3.2 The General Result Let us first note that our game is guaranteed to have Nash equilibrium for any admissible distribution function F. This is true because the equilibrium existence conditions of Glicksberg (1952) are trivially satisfied in our setup (compactness of the strategy space and continuity of the payoff function in own strategies). The following result generalizes the case of a uniform distribution to a large family of cumulative distribution functions F. 12 Proposition 2. Let m denote the median of any c.d.f. F such that F (x) n /[F (x) n + (1 F (x)) n ] is a log-concave function. Then: (i) There exists a unique equilibrium (p L (n), p R (n)) where p L (n) < p R (n) (ii) If for some n 0 we have that 0 < p L (n 0) < p R (n 0) < 1 then ˆp(n 0 ) = π(n 0 ) = m, p R (n 0) p L (n 0) = 1/[n 0 f(m)] and for every n 1 and n 2 such that n 0 < n 1 < n 2 we have p L (n 1) p R (n 1) > p L (n 2) p R (n 2) (iii) p L (n) p R (n) when n (iv) If m = 0.5 then p L = 1 p R. For any degree of disproportionality there exists a unique equilibrium where the leftist party proposes a policy at the left of the median and the rightist party proposes a policy at the right of the median voter. In equilibrium the implemented policy coincides with the median voter s ideal point while both parties propose platforms that diverge from the median by the same distance. Despite our analysis possibly allowing an asymmetric distribution of voters ideal points, as we show, parties propose platforms that are symmetric with respect to the median no matter how skewed the distribution is. The asymmetry in terms of the distribution is of course reflected in parties payoffs. 12 Notice that the assumption of log-concavity is usually imposed on the distribution of voters ideal points F (for example in Llavador 2006; Ortuño-Ortín 1997). However, here we require that the weight function S J is log-concave. In order to relate the log-concavity of S J with the primitives of the model it can be shown that S J is log-concave for a large family of F s. A general example guaranteeing the log-concavity of S J is when voters are distributed according to any unimodal beta distribution (that is π i Beta(α, β) with α 1, β 1). 13

14 From a comparative perspective, as the disproportionality of the electoral system increases parties have incentives to moderate their policies and converge towards the median. Full convergence to the median is predicted in the case of winner-take-all elections (n ). The intuition behind parties incentives to moderate or not their platforms is similar to the case of uniformly distributed voters. Notice that our general result fully backups the recent empirical results of Curini and Hino (2012) regarding the effect of the amount of independent voters on platform polarization. The equilibrium condition, p R (n 0) p L (n 0) = 1/[n 0 f(m)], clearly shows that polarization (p R (n 0) p L (n 0)) is decreasing in the density of moderate voters that can be a proxy of the share of independent voters in the electorate (Curini and Hino, 2012; Green, 2007; Lin et al., 1999). 3.3 The Three Parties Case We now consider the case of a three-party electoral race (a leftist, a centrist and a rightist party). 13 It is straightforward that the complexity of the analysis increases several orders in magnitude when we increase the cardinality of the set of players from two to three. For example, the game is no longer strictly competitive as when only two-parties compete and, hence, the equilibrium characterization cannot follow from a standard combination of the popular properties that strictly competitive games have (see proof of Proposition 2). Hence, to guarantee tractability some further assumptions are in order. We consider that a) the ideal policy of the centrist party is at one-half (that is π C = 0.5) and the leftist and the rightist party are modeled as in the two-party scenario analyzed above (π L = 0 and π R = 1) and b) an equilibrium in this case is a pure-strategy Nash equilibrium such that the distribution of policy proposals is symmetric about the center of the policy space. Since now three parties compete in the election each party s seat-share is given by the 13 The result trivially extends to k parties where two parties have ideal policies at the two extremes (that is at 0 and 1) and k 2 parties have an ideal policy at

15 following expression (Theil, 1969; Taagepera, 1986): S J = V n J V n L + V n C + V n R and therefore the implemented policy function for the three-party model is accordingly defined as: V L (p L, p R ) n ˆp(p L, p C, p R, n) = V L (p L, p R ) n + V C (p L, p R ) n + V R (p L, p R ) p n L V C (p L, p R ) n + V L (p L, p R ) n + V C (p L, p R ) n + V R (p L, p R ) p n C V R (p L, p R ) n + V L (p L, p R ) n + V C (p L, p R ) n + V R (p L, p R ) p n R The following result holds: 14 Proposition 3. Let π i U[0, 1]. Then: (i) There exists a unique equilibrium (p L ( n 1, 0.5, n+3 ) (ii) The distance between p 2(n+1) 2(n+1) R 0.5., p C, p R ) = and p L is decreasing in n (iii) ˆp = π = As we observe the centrist party proposes a platform equal to its ideal policy (p C = π C = 0.5). The two other parties differentiate and propose more extreme platforms. The extent to what parties differentiate depends on the level of disproportionality (see figure 3 depicting the proposed platforms (p L, p R ) for different values of electoral disproportionality n [1, 20]). The more proportional the electoral system is, the higher are the centrifugal forces, and hence polarization increases. As it was the case for our general result and two-party competition electoral proportionality acts as a centrifugal force. Clearly, if parties compete in a winner-take-all election (n ) then the standard result of convergence to the median applies (that is (p L, p C, p R ) (0.5, 0.5, 0.5)). Note that the qualitative implications of the above proposition directly extend to a general class of distributions (at least for the class of log-concave distributions) and are not restricted to 14 Without providing a formal definition of voters strategies it is necessary to mention that each voter i supports party j that proposes the closest platform to his ideal policy. If a voter is indifferent between two or even three platforms then he randomizes his vote. 15

16 the uniform case. Notice that in all results presented above, including the one allowing for an asymmetric distribution of voters, the implemented policy coincides with the one of the median voter which implies that the citizens welfare is independent of the precise value of n. Here we would like to stress that this invariance of social welfare to electoral changes is only apparent and a consequence of our choice to interpret our model as a parliamentary mean model where the implemented policy is a convex combination of the proposed platforms. If alternatively one chooses to interpret our model as a dominant party model (Merrill and Adams, 2007; Faravelli and Sanchez-Pages, 2012; Iaryczower and Mattozzi, 2004) - this is possible by interpreting parties seat shares as the probabilities that each of the parties can implement its proposed platform - then the equilibrium results presented in our formal analysis remain unaffected conditional on parties being risk neutral. Now it becomes clear that even if the expected policy outcome is still invariant in n, the variance of the policy outcome is decreasing in n (both in the two and in the three parties case) and, hence, social welfare is increasing in n. Hence, a welfare analysis of such an interpretation of our model shows that if voters do not care only about the first moment but also have some preferences regarding the second moment and are risk-averse they would tend to be better off under disproportional electoral systems. 3.4 Measures of Polarization Having characterized the equilibrium under different theoretical specifications and having made clear the interaction of the centripetal and the centrifugal forces of the political environment we now turn attention to the concept of polarization. It is not our intention to provide arguments for or against any definition of polarization. We merely utilize our theoretical findings in order to demonstrate that the choice of a measure of polarization in an empirical study can, to a great extent, affect the significance level (and even the direction) of the results. The empirical literature (see table A.I) has mainly considered two measures of po- 16

17 Figure 3: Left: The effect of electoral disproportionality n [1, 20] on proposed platforms for two (p L, p R ) and three-party competition (p L, p R, p C ) as characterized in propositions 1 and 3. Right: Polarization measured by the Dalton index in the case of two and three party competition for n [1, 20]. larization. The first one is the one that we have used so far in our theoretical analysis. According to tis measure polarization is nothing else but the distance between the two most distant platforms (MDP). The second measure is the Dalton index (DI) that is formally defined as follows: DI = j ( ) 2 pj p V j 0.5 where p denotes the weighted mean of parties proposed platforms (each party j is weighted by its vote-share V j ), p j is the platform proposed by party j while the difference between the two is normalized by the mid-point ideology position (in our case 0.5). The index takes value zero when all parties converge to a single position and one when parties are equally split between the two extreme positions Here the index takes value from zero to one since vote shares in our theory have been defined in the zero one interval. In our empirical part and in our data vote shares are given by percentages and therefore the DI simply takes values from zero to ten. 17

18 Figure 3 summarizes our findings. On the left we plot the results presented in propositions 1 and 3 and as one can see the following relationship holds: Remark 1. Polarization (measured by the MDP) in a three-party election is larger than in a two-party election (p R p L p R p L with the equality holding for n = 1). First notice that polarization (measured by MDP) is clearly decreasing in the level of disproportionality both in the two and the three parties case. Nevertheless, in the three parties case, the centrifugal force identified in the proportionality of the electoral system is now amplified as the number of competing parties increases. Comparing the distance between the most distant platforms of these two cases we observe that in the unique equilibrium of the three-party election this distance is larger than in the two-party election (for any n > 1 it holds that p L < p L and that p R > p R ). The presence of a third party makes competition for centrist voters tougher and hence parties have less incentives to moderate their policies in return for a slightly larger share of moderate votes. As far as the DI is concerned, notice first that, similar to the MDP, the DI is as well decreasing in the level of disproportionality both in the two and in the three parties case. Nevertheless, given that the DI is affected by parties vote-shares the relationship between polarization (measured by the DI) and the number of parties for a given level of disproportionality is not any longer that clear (see figure 3). Remark 2. Polarization (measured by the DI) in a three-party election is larger (smaller) than in a two-party election when n is large (small). When n is small the DI for the two parties case takes very high values given that the two parties are located at the extremes of the policy space and each obtains half of the votes. For the same small values of n the DI in the three parties case is smaller than the DI of the two parties case because of the presence of the centrist party. When n is small the centrist party obtains a very large vote-share and this drives polarization (measured by DI) of the three-parties case to low levels. As n increases, though, the leftist and the rightist parties platforms converge to the centre. This makes the vote-share of the 18

19 centrist party, and thereafter the influence of the centrist party in determining the level of polarization (measured by the DI), reduce rapidly and, essentially, disappear (when n takes very large values). Hence, when n is large it should be the case that the DI should converge to the MDP which, as we argued above, is larger in the three parties case than in the two parties case. Clearly the previous remarks have important implications regarding the use of the MDP or the DI in the empirical applications. If our model describes sufficiently well the nature of actual political competition then we should expect to find empirical evidence which suggests that a) platform polarization (measured either by the MDP and or by the DI) is decreasing in the level of electoral rule disproportionality and b) the number of parties has a clear positive effect on platform polarization only if polarization is measured by the MDP; when it is measured by the DI the direction of the effect should be ambiguous. The ambiguity of such result not only is persistent but the effect of the number of parties on the DI is further inconclusive if we were to extend our model to more than three parties. 4 Empirical Analysis As outlined above, our theoretical analysis makes two clear predictions regarding the determinants of platform polarization that can tested against the data. Thus, we formulate and test the following two hypotheses: (H.1) Electoral System Hypothesis (Propositions 1 and 2): Platform polarization (measured either by the DI or the MDP index) is decreasing in the disproportionality of the electoral rule n. (H.2) Number of Parties Hypothesis (Remark 1): Platform polarization (measured by the MDP index) is increasing in the number of competing parties. Both hypotheses have been explored by a number of related studies in the past yielding inconclusive empirical findings. While most approaches fail to garner enough support for 19

20 H1 (e.g., Budge and McDonald 2006; Ezrow 2008; Dalton 2008) few of them provide some (conditional) evidence in favor of either H1 (e.g., Calvo and Hellwig 2011; Dow 2011) or H2 (Andrews and Money, 2009). 16 Curini and Hino (2012) find support for two additional institutional hypotheses: the cabinet-parties conditional hypothesis and the electoral spill-over hypothesis while Calvo and Hellwig (2011) find conditional support for H1, but only for the large (governing) parties. 17 Our empirical analysis shows that the main reason why most existing studies find weak (or even no) support for the electoral-system hypothesis is that most of these studies utilize small and unbalanced data sets over a short time frame (see Table A.I). As a result, most of these studies do not include enough observations of electoral rule transitions per country that could be used to estimate the within-country effect of disproportionality on polarization. 18 On top of it, by including very few observations for each country - in many instances only one - even the cross-country effects, that several of those papers are in fact estimating, are based on a single observation per country. That is, they take into account only a snapshot of cross-country polarization levels and it is therefore impossible for them to disentangle the variation in polarization that is related to electoral rules from country-specific trends or other country intrinsic characteristics (e.g. time or country specific shocks that might have occurred during this limited period of observation). Therefore, by considering an enlarged and balanced panel, that includes on average 16 For a complete comparative presentation of the empirical literature, and their findings see Table A.I in the appendix. 17 We also include the variables introduced by Curini and Hino (2012) as additional controls in our regressions but we do not find a significant effect of spill-overs while some specifications provide partial support regarding the cabinet- parties conditional hypothesis. Nevertheless, Curini and Hino (2012) also test a simpler version of their model that includes only H.1 and H.2 and is therefore directly comparable to our model presented in columns 1 to 3 in Table I. The fact that we find a statistically significant effect of disproportionality on polarization (when we estimate their model using our data) while they do not is therefore clear evidence that our different predictions cannot be attributed to these two additional hypotheses. Calvo and Hellwig (2011) follow a different empirical approach and predict that the disproportionality of the electoral rule acts as a centripetal force but only for dominant parties, while it is a centrifugal force for smaller ones that are predicted to occupy more extreme positions. We, on the other hand, predict unconditional support for H.1, irrespective of party size. 18 This is particularly the case for studies that use observations from the CSES-2 and -3 mass surveys (e.g., Dalton 2008; Dow 2011; Curini and Hino 2012) and for studies which - despite the use of the CMP data - aggregate their data at the country level or limit attention to a short time frame (e.g. Budge and McDonald 2006; Ezrow 2008). 20

21 thirteen observations for each country, our work is an improvement on both fronts: not only we introduce some within country variation in the electoral rule disproportionality, but we also improve significantly the cross-country comparison, thus obtaining a more accurate picture of the effects of interest. 4.1 Data Description and Measurement We construct a balanced panel that combines electoral, political, institutional, socioeconomic, and demographic data for more than 300 elections from 23 OECD countries during the period from 1960 to 2006 by combining three different sources (Carey, 2012; Armingeon et al., 2012; Volkens et al., 2012). We describe our data and main variables in this section and provide the summary statistics in Table A.II of the appendix. 19 The Dependent Variable: Our dependent variable, platform polarization, is constructed using data from the Volkens et al. (2012) Comparative Manifesto Project (CMP) dataset compiled by the Berlin Centre for Social Research (WZB). The latter records the position of the platforms proposed by hundreds of political parties since 1946 in a unideminsional ideology (left-right) space. In line with our theoretical model we consider a unidimensional policy space where zero stands for extreme left and one for extreme right. 20 In order to maintain consistency with our theoretical model and also existing literature we measure polarization in two different ways: First, as the distance between the two most distant platforms (MDP) (e.g., Budge and McDonald 2006; Andrews and Money 2009). Second, following Dalton (2008), 21 we formally define the Dalton index (DI) of platform polarization for country i, in election year t exactly as in Section 3.4, substituting the platform proposed by party j, p j, with the ideological (platform) position of that party in election year t at country i, weighted by the party s vote-share. 19 More information on the data and estimation codes can be found in the appendix. 20 Technically, the CMP provides parties positions on a -100 (extreme left) to 100 (extreme right) scale. We perform an affine, monotonic, order preserving transformation of the index by adding 100 to each party s position then, dividing the sum by 200. Other studies that use the CMP to measure platform polarization are: Budge and McDonald (2006); Ezrow (2008); Andrews and Money (2009). 21 Curini and Hino (2012) also use the Dalton index while Ezrow (2008) and Dow (2011) use a very close analogue that incorporates all parties positions weighted by their vote-shares. 21

22 Nevertheless, let us now stress that as in the empirical analysis we measure vote shares in percentage terms (0-100), the index takes value zero when all parties converge to a single position and ten (instead of one) when parties are equally split between the two extreme positions. 22 In general, our findings are robust to the use of either of the two measures of polarization with one notable exception: as demonstrated in section 3.4, the effect of the number of parties on polarization, while always positive when polarization is measured by the MDP index, is ambiguous when the Dalton index is used (see figure 3). We present our estimates using both measures of polarization and we show that our main results are consistent with the findings of our theory and the two hypotheses outlined above. The Main Explanatory Variables: Disproportionality (n) and the Number of Competing Parties. Our key explanatory variable is the measure of the electoral rule disproportionality (parameter n) as defined in our theoretical section. By combining data from two different sources (Armingeon et al., 2012; Carey, 2012) we estimate the disproportionality parameter n for twenty-one countries in our sample by applying the formula proposed by Taagepera (1986): n = [log(v )/log(s)] (1/M) where V is the total number of voters, S is the total number of parliamentary seats and M is the average electoral district magnitude. Estimating the disproportionality parameter permits a tight fit between our model s predictions and our empirical estimation. An additional advantage of using this variable, given that the electoral rule is a sluggish institutional variable, is that it is continuous and exhibits some within-country variation, therefore allowing for both within- and cross-country comparisons. Nevertheless, in order to insulate our results, we also repeat our estimates using two alternative measures of disproportionality, that provide qualitatively identical results. The first one is a binary 22 Obviously, this transformation is trivial and does not affect our results in any way. 22

23 variable that takes the value of one whenever the FPTP rule with Single-Member Districts (SMD) is applied and zero otherwise. The second one is the natural log of the mean electoral district magnitude conveying the idea that a larger district magnitude reduces the effective threshold required for a party to occupy a parliamentary seat, hence, making the electoral system more proportional (Taagepera, 1986; Carey and Hix, 2011). The use of these three alternative measures not only increases the robustness of our findings but also allows us to address any concerns related to limited within country variation and endogeneity. On one hand, the binary variable records only a radical change from PR to a FPTP rule and vice versa. Such radical changes occurred only in three countries (Greece, Italy and New Zealand). Since these changes are not frequent, it is less likely that our analysis suffers from an endogeneity problem, especially since our dependent variable varies relatively more than any of our independent ones. On the other hand, the two continuous versions of our independent variable can address to some extent the problem of limited within country variation since both the disproportionality parameter n and the district magnitude vary in more than half of the countries of our sample. 23 It is worth noting that under all three specifications our results are robust, statistically significant and identical in the direction of the effect. Finally, following the literature (e.g., Andrews and Money 2009; Curini and Hino 2012), we test the number of competing parties hypothesis (H.2) using the Effective Number of Parties (ENP) index and its natural logarithm as our independent variables. 24 Nevertheless, our results are robust when we control for the actual number of parties Empirical Estimation Since empirical evidence (e.g., Gallagher 1991) and theoretical literature (e.g., Duverger 1954) suggest that electoral rules may also affect polarization through the structure of 23 Our estimates are robust to the use of alternative measures of electoral rule disproportionality such the natural log of median district magnitude. Additional estimates are available upon request by the authors. 24 Laakso and Taagepera (1979) define the effective number of political parties as 1/ j (V j) These results are available upon request by the authors. 23