Strategic Electoral Rule Choice Under Uncertainty

Transcription

1 Strategic Electoral Rule Choice Under Uncertainty Konstantinos Matakos University of Rochester Dimitrios Xefteris University of Cyprus October, 01 Abstract We study electoral rule choice in a multi-party model with offi ce-motivated parties and electoral outcome uncertainty. We show that when all dominant parties (parties with positive probability of winning the elections) have suffi ciently good chances of winning, then they agree to change the PR-rule with a more majoritarian one in order to increase their chances of forming a single-party government. We identify the exact degree of disproportionality of the new rule and we prove that it is increasing in the expected vote share of the minority parties (parties with zero probability of winning). The necessary and suffi cient conditions for such collusion in favor of a majoritarian rule are: a) the high rents from a single-party government, b) suffi cient uncertainty over the electoral outcome and c) ideological proximity of the dominant parties. Under mild conditions, we show that a unique and dynamically stable electoral reform equilibrium exist. Our predictions are supported by empirical evidence. Keywords: electoral reform; majority premium; single-party government; uncertainty; strategic coordination; ideology JEL classification: D7, H10 The authors would like to thank the participants at the following conferences: ECORE 009 at ECARES (ULB), Brussels; ASSET 009 at Boğaziçi, İstanbul and the Public Economics Workshop at Warwick. We would also like to thank David Austen-Smith, François Maniquet and Christoph Vanberg for useful comments and suggestions. In particular, Dimitrios Xefteris would like to thank Enriqueta Aragonès and Josep Colomer for their comments and Konstantinos Matakos would also like to thank Bhaskar Dutta, Herakles Polemarchakis and Ben Lockwood for their supervision. W. Allen Wallis Institute of Political Economy, University of Rochester; kmatakos@z.rochester.edu Department of Economics, University of Cyprus; xefteris.dimitrios@ucy.ac.cy 1

2 1 Introduction The choice of the electoral rule is a strategic decision of major importance, made in every democracy, since it sets the rules of the game. In terms of policy implementation, more important than the electoral outcome itself, is the allocation of parliamentary seats in the legislature, according to the applied electoral rule. This is so, because the ability of any government, single-party or coalition one, to implement its policies critically depends on the size of its parliamentary majority. Hence, the number of parliamentary seats allocated to the winner is a significant determinant of political power (Blais, 1991). Under a Proportional Representation rule (hereinafter PR), a party that secures a very large vote share can sometimes fail to capture the absolute parliamentary majority, whereas, under a more majoritarian rule (such as First-Past-The-Post) it could have been easier for an offi ce-motivated party, simply by securing the parliamentary majority, to form a single-party government and enjoy the spoils of offi ce alone, or advance its policy agenda facing less opposition in the parliament (Blais, 1991; Tsebelis, 1999). It is exactly this feature of nonproportional electoral rules, to distort the allocation of parliamentary seats in favour of the largest parties, that generates incentives for strategic electoral rule choice. Of course, one can argue that non-proportional electoral rules are deemed to be unfair and as such, they are rejected by the political systems. Yet, a closer inspection reveals that in many countries electoral rules other than pure PR are actually applied in order to transform votes into parliamentary seats (Norris, 1997). Moreover, in the last forty years many majoritarian electoral rule reforms were enacted (Riera, 01). With these observations in mind, a set of interesting questions arise. Firstly, how strategic considerations and opportunistic incentives affect parties decisions vis-à-vis electoral rule change? Secondly, which electoral rule do parties choose, given their expected vote share in the forthcoming elections? Finally, what are the key determinants of electoral rule choice? Only strategic (opportunistic) calculations matter, or does ideology play an equally important role? To answer these questions, we develop a model of multi-party electoral competition with two, offi ce-motivated, dominant parties and uncertainty over the electoral outcome. In such a framework, we model electoral rule choice endogenously, as an optimal, strategic decision made by the

3 parties 1, through a parliamentary voting procedure, within the context of uncertain electoral competition. To put it more simply, in this paper we will try to explain why and how parties choose the rules of the game. We will also present some stylized empirical evidence to support its theoretical predictions. 1.1 Recent Literature Duverger s (1954) famous law postulates that it is the electoral rule, determined exogenously by some pre-existing constitutional arrangement, which is responsible for shaping the political landscape and the structure of the party-system. Recently, a number of scholars have gone in the opposite direction. Turning Duverger upside down, Colomer (005) presents and tests the hypothesis that it is, in fact, the number of parties that can explain the choice of electoral rules. He argues that existing political parties tend to choose electoral systems that allow them to [C]rystallize, consolidate and reinforce the current party system instead of changing it dramatically. He also concludes that political systems that are dominated by few parties tend to establish majoritarian electoral rules. Boix (1999) also suggests that the existing variation in electoral rules across parliamentary democracies is due to the strategic decisions that ruling parties make, anticipating the coordinating consequences of different electoral rules, in order to maximize their representation in the legislature. He finds that, if the electoral competition is less uncertain and the existing electoral rule serves the current ruling parties then, status-quo bias prevails and parties have limited incentives to modify the electoral rule. However, if the degree of uncertainty increases due to new voters, or change in their preferences, the ruling parties will consider changing the current electoral rule depending on two conditions: Firstly, the strength of the other parties, and secondly, the coordinating capabilities of the dominant parties. Similarly, if the new entrants are expected to be weak, a non-proportional rule is maintained regardless of the structure of the old political system. We note that, although 1 In line with Ansolabehere and Snyder (006) who consider that ruling parties control the flow of public funding, we consider that ruling parties influence electoral reforms as opposed to studies which assume that institutions are directly formed by the citizens (see for example Barbera and Jackson 004 and Jack and Lagunoff 006). Riker states it as follows: Simple majority rules favor bipartisanship whereas, proportional and simple majority two-round electoral rules favor multi-partism. 3

4 it is out of the scope of this paper to explicitly model new party entry, our theoretical model is, nonetheless, robust to alternative interpretations and can account for the effects of small party entry 3 on electoral rule choice. In the same vein, Benoit (004) studies the endogenous choice of electoral rules by parliamentary parties and develops a theoretical framework that classifies patterns of electoral rule change in various political systems. He derives conditions for endogenous electoral rule change by rational, seat-maximizing political parties. He predicts that electoral rule change occurs endogenously, when two conditions are met: First, the existence of a coalition of parties willing to agree on electoral rule reform, such that all of them are expected to score seat gains under the newly proposed rule. Second, the ability of those parties to muster enough votes in the current parliament in order to enact this change. Finally, in a paper closely related to ours, Ergun (010) studies the change of electoral rule from plurality to PR 4 assuming offi ce-motivated, rational, seat-maximizing parties. He finds that for any change to take place the following two conditions have to be satisfied: First, the government must be formed by a coalition. Second, the larger the number of parties and the more equitable the distribution of the spoils amongst them, the more likely the change to a PR rule is. That is, starting from the opposite direction, Ergun adopts the same counter-duvergerian approach that strategic motivations and party-system structure determine the choice of electoral rules. 1. Our Contribution Our paper, building on existing literature of endogenous electoral rule choice (Colomer, 005), formalizes the idea that it is parties that choose the electoral rule. It models explicitly how dominant parties 5 coordinate in order to increase their chances of forming a single-party government and consolidate the two-party system by eliminating the role that smaller parties would have other- 3 Palfrey (1984) considers a two-dominant-party model with small party entry. But Palfrey s results relate to the position that the two dominant parties occupy in anticipation of new party entry. Our analysis is distinct, since we focus on another strategic choice made by the two dominant parties: the electoral rule. 4 We model the reverse direction of the change from PR to a more majoritarian rule. Yet, as it will become evident, the choice of PR as the starting point in our model does not affect our results because we are able to prove uniqueness. 5 The term dominant party refers to a party that has a positive probability of winning the election, whereas the rest are minority parties with zero probability of coming first in the forthcoming election. 4

5 wise played in a coalition government. Moreover, we extend our understanding on electoral rule choice in two directions: Firstly, we explicitly derive the conditions that allow dominant parties to coordinate in adapting a more majoritarian electoral rule and we identify the exact degree of disproportionality that they introduce into the new rule. Moreover, under mild assumptions, we show that dominant parties agree on a unique (non-proportional) electoral reform proposal. Secondly, we identify the drivers of this strategic coordination: their desire for offi ce and electoral uncertainty. Rather than simply being rational seat-maximizers, parties desire more seats to serve another end: securing the absolute parliamentary majority, allowing them to form a single-party government and enjoy alone the spoils of offi ce. It is their strict preference for single-party governments, in conjunction with electoral uncertainty, that triggers collusion. The latter is a sine qua non condition for aligning dominant parties incentives, therefore, enabling strategic coordination. Our modelling approach introduces four new elements in the study of electoral rule choice that were absent from the traditional body of literature: We introduce offi ce-motivated parties, we model electoral rule change in the presence of electoral uncertainty, we introduce ideology as an additional explanatory factor of electoral rule choice, and finally, we consider a broader set of electoral rules as candidates for the electoral reform process. We discuss each one of them separately. Firstly, the formulation of parties preferences incorporates their desire to form a single-party government. Their goal to win as many parliamentary seats as possible serves a dual purpose. On the one hand, it increases their bargaining power, in the case of a coalition formation and their control over the legislative agenda, if in opposition 6. On the other hand, as their seat share rises above the parliamentary majority threshold, they gain the ability to form a single-party government and enjoy alone the perks of holding offi ce (both legislative and executive). Parliamentary seats serve this end and hence, the utility of an extra seat is less when a party commands the absolute majority in the Parliament. As a result, further seat gains have a smaller impact, once the prospect of forming a single-party government is guaranteed. Figure1 (Appendix B) depicts such a utility function. The first discontinuity captures parties received rents when in government (coalition), 6 For instance, more seats increase their ability to influence the agenda when in opposition (e.g. Tsebelis, 1999), or occupy more ministerial portfolios and pursue their policies with greater success within a coalition government. 5

6 the second jump reflects their preference for single-party governments, whereas, the change in slope (after one-half) captures the decreasing marginal utility of an extra seat. Our second element is the introduction of electoral uncertainty. Uncertainty plays a key role in our model since it is the second driving force (the other is their desire for single-party government) that allows dominant parties to collude. In particular, when there is suffi cient electoral uncertainty 7, dominant parties might have a common interest to coordinate and adapt a less proportional electoral rule. In other words, we expect that in party-systems with two dominant parties and competitive elections, the major players will collude in order to consolidate the bipartisan system. As a result, a more majoritarian rule will be chosen. In fact, electoral uncertainty is a necessary condition for strategic coordination among dominant parties. The intuition is very simple: without competitive elections, the sure loser will never agree to a more majoritarian electoral rule, since its chances of forming a single-party government are zero. Introducing parties with ideological preferences, in order to study how ideology interacts with strategic (opportunistic) considerations in the choice of electoral rules, is the third key element of our model. So far, the literature on electoral rule reform has solely focused on rational, seatmaximizing parties, omitting completely from the discussion one extra dimension affecting electoral rule choice: ideology. Therefore, in the last part of our paper we allow for parties having welldefined preferences over ideology in a left-right dimension. In such a framework, incentives for strategic coordination over adapting a more majoritarian electoral rule may come into conflict with ideological differences between the two dominant parties. For instance, when the two parties are ideologically very distant and diverge, it seems more plausible that a more proportional rule will be preferred over a majoritarian one. The intuition is that, by sticking to proportionality, dominant parties insure against the risk of a single-party government being formed by a diametrically opposed party. This is confirmed by some empirical examples (e.g. Italy), where extreme ideological divergence has blocked, until recently, any attempt to substitute the PR with a more majoritarian rule. The formal analysis of the effect of ideology on electoral reform choice indicates that our 7 In our context, the term suffi cient electoral uncertainty is meant to imply that elections are competitive and contested by both dominant parties (Andrews and Jackman, 005). 6

7 findings are dynamically stable. When parties are ideologically divergent we find that they adopt proportional representation and when parties compete under proportional representation they tend to adopt divergent political platforms. Moreover when parties are ideologically convergent we find that they adopt a majoritarian rule and when parties compete under a majoritarian rule they tend to select ideologically convergent platforms (Cox 1990 and Calvo and Hellwig 011 show that centripetal incentives are stronger in majoritarian systems while proportional systems are dominated by centrifugal incentives). Here, we need to clarify that in our set-up, parties in favor of electoral rule reform, need only have expected utility gains, not necessarily realized, ex-post, seat gains (e.g. Benoit, 004). This is so, because expected utility incorporates something more than mere seat gains: the change in the probability of forming a single-party government, for the two parties favoring the reform. Eventually, only one of them will form a single-party government. Yet, in the presence of electoral uncertainty and in the absence of extreme ideological differences, this is utility enhancing for both, in expected terms. The final element, is to consider a broader class of electoral rules (other than PR and FPTP). Following Sartori (1976) who claims that the most common distortions to the PR rule are the introduction of majority premia and of exclusion clauses 8, we allow our electoral rule reform proposals to take more generic forms. To capture all possible degrees of disproportionality (from pure PR to FPTP with a unique district) we introduce in the theoretical modeling the, so called, majority premium system (Sartori, 1976). This rule allocates a fraction of the seats according to PR while, the rest are given to the first party as a premium 9. Hence, by varying the amount of the premium we can simulate electoral rules with different degrees of disproportionality. 8 E.g. the 10% entry barrier in Turkey, the 5% in Germany and the 3% in Greece. 9 Greece introduced this electoral rule in 004 and was applied in three consecutive elections (007, 009 and 01). 7

8 The Model.1 Basic Set Up and Preferences Formally, we let N = {1,, 3} to be the set of parties participating in the electoral competition. All parties in N are assumed to be represented in the parliament. Each party i N holds a n proportion of seats s 0 i in the current parliament, such that s 0 i = 1. Moreover, parties have information about each other s expected vote share in the forthcoming elections. i=1 Party 3, is assumed to be a minority party. That is, its vote share shall never exceed that of Party 1 or Party. The expectations on future vote shares are formed by information that is commonly available to all Parties. The vote share of the minority Party 3 is assumed to be fixed at a level, whereas, the vote shares of the other two parties are subject to uncertainty 10. Formally, the vote share of Party 1 in the coming elections will be modeled as a random variable: v 1 Φ 1, where Φ 1 is a uniform distribution in [a 1, b 1 ] [0, 1]. It follows that, since v = 1 v 1, it must be the case that v [a, b ] where a = 1 b 1 and b = 1 a 1. These expectations need to satisfy n v i = 1 and < a i, i {1, }. That is, both b 1 and b are smaller than 1. As it will be evident in the analysis section, the way we chose to model uncertainty is a reduced form of assuming that the vote share distribution {v 1, v, } is a random draw from a multivariate distribution Z(v 1, v, ) under the restriction that < min{v 1, v }. That is, certainty about is not required for our results but improves significantly the readability of the formal parts of the paper. The share of seats that party i N occupies in the new parliament will be defined as s l i(v i ), where l will be the applied electoral rule. As stated before, we do not limit our attention to a i=1 single electoral rule. Rather, we want to consider the transition from PR to a wider range of possible electoral rules. In order to capture the big diversity of electoral rules existing in the world 10 Practically, our model can be viewed as a multi-party model in the following sense: can be interpreted as the sum of the vote shares of all other parties. As long as is less than the vote share of each of the two dominant parties, our minority party stands for the collection of all other small parties. 8

9 we will adopt the following mechanism l that is based on a variation of the PR with the use of a majority premium (Sartori, 1976). That is, l is the proportion of parliamentary seats allocated to the winning party as a bonus (premium). Obviously, l [0, 1] and, thereafter: s l i(v i ) = v i (1 l) if v i < 1 and s l i(v i ) = v i (1 l) + l if v i > 1. Given our assumptions, the third party will never be entitled to this bonus as a result of never winning the election. The above seat allocation mechanism allows us to capture a wide variety of electoral rules, from pure PR to mixed systems and FPTP 11. The first part is the proportional allocation of the seats minus the reserved premium, whereas the second part is the bonus given to the winner. The utility of a party i N will be defined as: u i (v 1, v,, l) = s l i(v i ), if s l i(v i ) 1/ min{s l 1(v 1 ), s l (v ), s l 3( )} s l i(v i ) + r 1, if 1/ min{s l 1(v 1 ), s l (v ), s l 3( )} s l i(v i ) 1/ gs l i(v i ) + r, if s l i(v i ) > 1/ where 0 r 1 < 1 + r 1 < r and g [0, 1]. The first part is the utility of party i when it stays in opposition and some other party forms a single-party government. In this case, i s utility depends only on the number of its seats. The second part is the utility of party i when no party can form a single-party government and as a result, a coalition government is formed. Thus, on top of its seat share party i receives extra rents r 1 from holding offi ce. Finally, the third part gives the utility of party i when it forms a single-party government. In this case, the payoff of the party depends, as before, on its seat share and on offi ce rents r (which are obviously higher than the rents i gets in a coalition government). Parties in this environment care about forming a single-party government ( 1 + r 1 < r ). Moreover, g [0, 1] implies that the utility from an extra seat when the party is not able to form a 11 To see this, consider the case of l = 0. Then, our electoral rule is pure PR, whereas in the case of l = 1, it transforms into a unique district FPTP system (the most disproportional electoral rule possible). For values of l between those two extremes all the other electoral rules can be simulated. Table 1 provides the exact calculations on how our proposed mechanism can replicate the electoral results in Greece (PR with majority premium). 9

10 single-party government (s l i(v i ) 1/), is generically higher than the utility from an extra seat when the party is able to form a single-party government 1. Solely for analytical and demonstration purposes, in the remainder we will assume that r 1 = g = 0 and r = 1. That is: s l i(v i ), if s l i(v i ) 1/ u i (v 1, v,, l) = 1, if s l i(v i ) > 1/ This simplification still captures the desire that parties have for single-party governments. Yet, it is open to two main criticisms: First, it implies that a party which has a majority in the parliament does not have any gain by increasing its seat share; and secondly, it suggests that when a party cannot form a single-party government then its utility solely depends on its vote share independently of whether it participates in a coalition government or if it stays in opposition. In Appendix A, we present results for the general case (0 r 1 < 1 + r 1 < r and g [0, 1]) and we show that they are in line with the simpler case that we exhaustively analyze in the paper. As it will become evident in the coming sections, more general assumptions regarding the values of r 1, r and g do not contradict any of the forces that are crucial for our results. It will become clear from the proofs and by the analysis of the general case (Appendix A) that it is not the constant nature of u i ( ) for s l i(v i ) > 1/ that drives our results but, the assumption that the gain from an extra seat should be less for a party that already has a majority in the parliament compared to that when this party is in a coalition government or in opposition. Finally, in Section 4 we will allow the utility of the parties to vary depending on whether they are in a coalition government or just in the opposition. This will be introduced as an ideological component in the utility function which is robust to alternative interpretations such as rent gains from participating in a coalition government as opposed to staying in opposition. To summarize, our preference structure is a rough but analytically convenient approximation of more general preference structures such that the marginal utility of an extra seat, for a party 1 A party derives utility from an extra seat in two ways: directly, and indirectly (by increasing its chances of forming a single-party government). Once its seat share exceeds the parliamentary majority threshold, the second channel disappears. Hence, the marginal utility of an extra seat might decrease. 10

11 has the majority in the parliament, is less than the marginal utility when a party is in a coalition government or in opposition.. The Game Structure After defining the preferences of the parties, we proceed with the structure of the electoral reform game. Formally, the game has three stages: (i) The current rule is l = 0 (pure PR) and the party with the largest share of seats in the current parliament (either party one or party two) shall bear the role of the Proposer of an electoral reform. That is, it will propose l [0, 1]. (ii) Parties vote on the proposal l. If the votes in favor of the reform surpass a given threshold W [0, 1] (given exogenously by the constitution) then, the electoral reform proposal is accepted and the forthcoming elections take place according to the new rule l. In the opposite case, that is, if the proposal does not gather the necessary parliamentary support W, the electoral reform is cancelled and future elections are conducted according to the proportional rule l = 0. (iii) Elections take place and each party, according to its vote share and the applied electoral rule, l or l = 0, takes its new seat share in the Parliament and computes its utility..3 Understanding the Proposer s Problem For simplicity, let us assume without any loss in generality, that the Proposer is always Party 1. That is, we assume that s 0 1 > s 0 > s 0 3. Since the electoral rule determines the seat allocation for the parties and thus, their utility, Party 1 will propose an electoral rule l [0, 1] such that, given the threshold W [0, 1], maximizes its expected utility. That is, if W s 0 1 Party 1 will propose l [0, 1] such that l arg max{eu 1 (v 1, v,, l)}. In other words, if the current seat share of the Proposer exceeds the required threshold for enacting an electoral reform, the Proposer faces an unconstrained maximization program. On the contrary, if W > s 0 1 the proposer needs the support of at least one of the two other parties in order to implement an electoral rule reform. An obvious, but nonetheless useful Lemma, demonstrates the behavior of the minority party in such cases. 11

12 Lemma 1 The minority party never consents to any electoral reform proposal. Proof. Since the minority party expects to receive the premium l with probability zero, it strictly prefers the PR rule (l = 0). Its expected utility from any electoral rule l 0 is: Eu 3 (v 1, v,, l) = (1 l), which is clearly decreasing in l. Given the above observation, Party 1 will have to secure Party s support in order to proceed with an electoral rule reform. That is, it has to propose l arg max{eu 1 (v 1, v,, l)} s.t. Eu (v 1, v,, l) Eu (v 1, v,, 0), facing this time a constrained maximization program (Incentive Compatibility Constraint 13 of Party ). In general, since we have assumed that is fixed and common knowledge and that v 1 = 1 v, the Proposer faces one source of uncertainty (information about v 1 is equivalent to information about v ). Thus, the proposer s expected utility is given by expression: Eu 1 (v 1, v,, l) = 1 (b 1 a 1 ) (1 )/ a 1 v 1 (1 l)dv 1 + 1/ l (1 )/ [v 1 (1 l) + l]dv 1 + and equivalently, Party s expected utility is: Eu (v 1, v,, l) = 1 (1 v 1/ l 3)/ (b a v ) (1 l)dv + [v (1 l) + l]dv + a (1 )/ b 1 1/ l b 1/ l dv 1 dv. Notice that there exist two critical vote shares. The first one, (1 )/, defines the necessary vote share for one of the two parties to be first (and get the premium l). The second, 1/ l, is the vote share that the first party needs in order to get the majority of seats in the parliament, given electoral rule l. Obviously, if (1 )/ 1/ l the first party will have the a majority in the parliament as long as it wins and thus, the expected utility of Party 1 will be: [ ] (1 v3 )/ Eu 1 (v 1, v,, l) = 1 b 1 (b 1 a 1 v ) 1 (1 l)dv 1 + dv 1 a 1 and for Party : [ (1 v3 )/ Eu (v 1, v,, l) = 1 (b a v ) (1 l)dv + a 13 Hereinafter Eu (v 1, v,, l) Eu (v 1, v,, 0) is referred to as the ICC. (1 )/ b (1 )/ dv ] 1

13 As 1/ l is decreasing in l, an increase in the majority premium (an increase in l) for the party that runs first in elections, not only increases its seat gains but, it also increases its chances of obtaining the majority of parliamentary seats and hence, forming a single-party government. It is this dual impact of the electoral rule on parties utility that makes the electoral reform process such an important strategic decision..4 Definitions We will classify the results given the following definitions. Definition 1 The electoral reform is possible if W 1 This is a direct implication of Lemma 1. The minority party never consents to an electoral reform. Thus, if a reform is to take place, W must be such that Party 3 cannot block it. Definition The electoral reform process is trivial if W s 0 1 When the Proposer (Party 1) has a large enough proportion of seats in the current Parliament, so as to be able to choose the electoral rule at will, we shall call the reform process is a trivial one, since it only depends on the preferences of the Proposer. Definition 3 The electoral competition is trivial if either a 1 > 1, or b 1 < 1 The above definition just describes the case that the probability of Party 1 running first in the coming elections is either 1 or 0. In such cases the winning party, which will also receive the premium l, is known with certainty. On the other hand, when electoral competition is non-trivial, both Parties 1 and have a positive probability of winning and thus, getting the majority premium l. Definition 4 The Proposer (Party 1) is the leading party if and only if: E(v 1 ) = a 1+b 1 > 1 If a party is expected to run first in the elections, then it shall be called the leading party. Since 1 is the threshold above which Party 1 is the winner, Party 1 is the leading party if and only if its expected vote share exceeds this threshold. Otherwise, Party is the leading one. 13

14 3 Results Since we want to study electoral rule choice under uncertainty, we focus on the case of nontrivial electoral competition, in order to ensure that there is enough uncertainty over the electoral outcome. We explore the case of trivial electoral competition in Appendix A. Always assuming that an electoral reform is possible (W 1 ) we can state the following results. Proposition 1 When the electoral competition is non-trivial then l {0, }. 14 This is the main result of this paper. In an environment of electoral uncertainty, the proportionality distortion l that the proposer might introduce into the electoral rule as a majority premium, will be such that it guarantees to the winner the majority of seats in the parliament. That is, if Party 1, alone or with the support of Party, sponsors an electoral rule reform, it will be such that it consolidates the status quo in favor of the two dominant parties. This result is a summary of the next two Propositions. Hence, we restrict further analysis in the following section in order to combine Propositions 1, and 3 together. We now present two Propositions that build on the previous result and state explicitly the necessary and suffi cient conditions for an electoral rule change to take place. Before doing so, for expositional ease, we define function f i (a i, b i, ) which measures the expected utility gains (or losses) for a party when there is a change in the electoral rule from l = 0 to l =. Definition 5 For i = {1, }, define f i (a i, b i, ) = (b i 1 )( b i 1 ) ( a i ) if b i < 1. Otherwise if b i 1 define f i(a i, b i, ) = + () )( 1 ( v a i +1 +a i )( 1 ). Observe that f i ( ) 0 implies that party i prefers l = to l = 0, which in turn, implies that the ICC for i is satisfied (Eu i (v 1, v,, ) Eu i (v 1, v,, 0)). That is, f i ( ) is the re-stated version of the ICC. Intuitively, f i ( ) 0, means that Party i has enough chances of winning the elections (see that whenever b i 1, it is easily checked that f i( ) > 0 is always satisfied). Proposition When the electoral competition is non-trivial but the electoral reform process is trivial (W s 0 1), then the Parliament departs from PR (l = 0) and adapts l = if and only if f 1 (a 1, b 1, ) All Proofs in Appendix B. 14

15 Proposition 3 When both the electoral competition and the electoral reform process are non-trivial then in order for the Parliament to depart from PR (l = 0) and adapt l = there are two sets of both necessary and suffi cient conditions: EITHER (i) the Proposer is the leading party and f (a, b, ) 0 is satisfied OR (ii) the Proposer is not the leading party and f 1 (a 1, b 1, ) 0. Corollary 1 In the symmetric case (a 1 = a and b 1 = b ), we always have l =. Proof. Note that, in the symmetric case, since a 1+b 1 = a +b = 1, it is again easily checked that f i ( ) > 0 for every b i, for i {1, }. Then, both conditions (i) and (ii) of Proposition 3 are trivially satisfied and the result follows immediately. In the remainder, we provide an idea of the proof, followed by a discussion for each case separately, since this is the main result of the paper. First of all, we note that formally the proof is derived from the maximization program of the proposer. As shown in Figures and 3 (Appendix B), Eu i (v 1, v,, l) for i = {1, } is convex with respect to l (and strictly convex for some values of l) for l [0, ] and decreasing for l (, 1]. Hence, the are exactly two values of l that are candidates for an optimum (denoted as l ): Either 0, or. That is, in equilibrium, the proposer chooses one of those two strategies (notice that in each case the equilibrium is unique). Moreover, when the electoral reform process is non-trivial, the Proposer faces a constrained maximization program. As a result, Party 1 chooses l in order to maximize its expected utility Eu 1 (v 1, v,, l), satisfying at the same time the ICC of Party (f (.) 0) 15. Next, we argue why only those two values of l are candidates for an equilibrium. Ideally, the proposer would prefer to propose the PR rule (l = 0), if she knows that she will run second and propose l =, otherwise. Yet, in our environment of electoral uncertainty, the Proposer has to compare the expected loss from proposing l = and running second (area ABCD in Fig. 4) 16, with the expected loss from winning elections but not being able to form a single-party government because it has proposed l = 0 (area DEFH in Fig. 4) 17. This statement, graphically depicted in Figure 4, is mathematically expressed by f i ( ), which measures the difference between areas 15 Recall that by Lemma 1 the minority party always prefers the PR rule (l = 0) and never agrees to accept any l Or equivalently, ABCD is the expected gain from proposing the PR (l = 0) and running second. 17 Or equivalently, DEFH is the expected gain from winning in the election and forming a single party government, as a result of proposing l = v3. 15

16 DEFH and ABCD. The same analysis applies for the receiver of the proposal (Party ) in deciding whether to accept it or not. Now, the intuition behind our result is more clear. Assuming that the electoral reform process is trivial, the proposer proposes l = whenever the expected benefit from forming a single-party government (as a result of its proposal) exceeds the expected loss of seats (and utility) whenever it runs second. Otherwise, it proposes l = 0. If the electoral reform is non-trivial, and the proposer is the leading party it always prefers to propose l =, since for Party 1 expected benefits always exceed expected losses 18. But, when the Proposer faces a constrained maximization program it has to satisfy the ICC of Party, which is analogous to the previous inequality for Party 1 (i.e. f (.) 0). Whenever the ICC of Party is satisfied, Party 1 proposes l =, which is always accepted. Otherwise, Party will reject any proposal l 0. As result, Party 1 proposes the PR (l = 0). The case where the Proposer is not the leading party is, in fact, the mirror image of this case. Party is now the leading party, and as a result, its ICC is satisfied. Hence, it always accepts a proposal of l =. But now, it is Party 1 that will propose l = whenever its expected gain exceeds its expected loss (f 1 (.) 0), despite it not being the leading party. Otherwise it proposes l = 0. This completes our argument. 3.1 Discussion and Empirical Evidence Our results, so far, highlight the following feature. Once expected gains from distorting the PR rule exceed expected losses, the two parties are faced with a strategic decision: What is the desired (optimal) level of distortion (premium l) to be introduced into the PR rule. Their dilemma is summarized in the following question: Given that the outcome is uncertain, which electoral rule guarantees me a single-party government, if I win, but at the same time minimizes the loss of parliamentary seats in case of defeat? We already know the answer: it is. An important implication of this is the fact that the equilibrium outcome is unique. Once the two dominant parties agree to depart from the PR rule, there is a unique value of l 0 that is proposed and 18 Being a leading party ( ai+bi > 1 v3 ) implies f i (.) > 0. But, the reverse is not true. To see this, notice that in the symmetric case ( a1+b1 = a+b = 1 v3 ) we still have f i (.) > 0 for i = 1,. Hence, by continuity of f i (.) it is still possible to have f i (.) > 0 even if i is not the leading party. 16

17 accepted (l = ). In fact, the value of l is such that it is the minimum required majority premium that always guarantees to the winner the prospect of a single-party government. This key feature of our model implies that offi ce-motivated parties, not only can agree to distort the PR rule in order to consolidate the two-party system, but they can also agree on a unique new electoral rule. On the other hand, if the two dominant parties agree to disagree, the PR rule (l = 0) is maintained. As a result, we either have a unique electoral reform equilibrium (l = ), or we do not have an electoral rule reform at all. Moreover, when dominant parties agree on electoral rule reform, a simple comparative statics analysis on the equilibrium value of the majority premium (l ( ) ) reveals that l is increasing in the electoral power of third parties ( minority party in the language of our model). This yields an interesting insight in the relationship between new-party entry and electoral rule reform. Although, as stressed in the introduction, our model does not aspire to explore party entry in this set up, the minority party can be viewed as the sum of many small parties, as long as they do not stand any chance of outperforming any of the two dominant ones ( < a i ). As such, our model can accommodate expected new-party entry, modeled as an increase in, as long as the new entrant is not expected to upset the dominance of the two other parties. Then, l > 0 implies that the entry of a new party, or an increase in the electoral power of smaller parties, causes the electoral rule to become more majoritarian. In fact, this point is made by Boix (1999). 19 These predictions are supported empirically by Riera (01), who finds that an increase in the Effective Number of Parties (ENP) 0 is positively associated with a majoritarian electoral reform. Since in our model we always have l > 0 and ENP > 0, it is clear that an increase in the expected vote-share of the smaller (minority) parties, meaning an increase in the ENP Index, leads to an increase in the majority premium l. As a result, our model predicts a positive relationship between ENP and the likelihood of a majoritarian electoral rule reform. In fact, we provide a rational choice explanation to the (Riera, 01) that defies conventional wisdom 19 Boix asserts that: [W]henever the new entrants are weak, a majoritarian electoral rule is maintained (or reinforced) regardless of the structure of the old political system. 0 1 The ENP Index is defined as. In our model, given that we always assume that < i 1 (vi) 3, we can compute the ENP Index for the symmetric case using expected vote-shares (E[v i ] = ai+bi = 1 v3 ) for i {1, }. Then: ENP 4(1 3v ENP = 3() +1. Clearly, = 3) > 0, for any v [3() +1] 3 < 1 3. Hence, in our model, an increase in will always result in an increase in the ENP. 17

18 of endogenous electoral rule choice (Colomer, 005) by offering a modified version of Colomer s statement, based on the number of dominant parties that matters the most. As long as the number of dominant parties remains unchanged, an increase in the number (or in the electoral power) of smaller (non-winning) parties causes the adaptation of more, not less, majoritarian electoral rules. Empirically, since l is measuring the degree of disproportionality of the electoral rule, our results imply that we should expect the distribution of applied electoral rules across parliamentary democracies (based on electoral rule disproportionality measured by the Gallagher Index 1 ) to be concentrated around two points: when the Index is equal to 0 (resulting from PR being applied), and when the Index is around 3 (when l = is applied). This is exactly what data from OECD countries suggest in Figure 5, where despite the rough approximation technique that we employ (assuming that all OECD countries have bipartisan systems where the two dominant parties are of equal size), we observe that the outcomes produced by electoral rules are concentrated around two main clusters: results generated by PR rules, and those generated by modified (more disproportional) rules, as our model predicts. This point clearly hints the idea that, for any given political environment, there exist a unique optimal electoral rule, from the perspective of dominant parties who care about consolidating the current structure of the party-system. In particular, in party-systems dominated by two major parties (such as the majority of OECD countries 3 ) we should observe some convergence in electoral rules, which is consistent with our model s prediction. And in fact, we do observe such a convergence (Fig. 6) when we plot the actual electoral rule disproportionality (measured by the Gallagher Index) against the predicted one, by applying our model s (optimal) electoral 1 A commonly used measure of electoral rule disproportionality is the Gallagher Index defined as: G(l) = 1 i (v i s l i ). If one assumes the symmetric case (i.e. systems with two dominant parties of almost equal size) and replaces the vote shares of dominant parties with their expected ones (i.e. Ev i = 1 v3 ), we can define: 3(v3) 4 +() 4() (recall that by Corollary 1 in the symmetric case we always have l = v3 ). Otherwise, G ( v3 ) = if l = 0, we can compute G (0) = 0. In our sample of OECD countries, the average vote-share of third parties (other than the two major ones) is 1.7%. Assuming that the two-dominant parties are of equal size (in fact in our sample this is roughly correct), one can compute the Gallagher Index for the symmetric case and obtain the value of From the list of OECD democracies we identify the following 16 countries that roughly fit this definition: Australia, Austria, Canada, France, Germany, Greece, Ireland, Japan, Luxembourg, N. Zealand, Norway, Portugal, Spain, Sweden, UK and USA. The remaining six cases are countries with fragmented party-systems which apply the PR rule. 18

19 rule l =, for the symmetric case 4. Despite our rough approximation, since it is true that in some countries the two dominant parties have not been historically perfectly symmetric in terms of vote-shares (e.g. Sweden, Japan), and despite some outliers, namely the countries that are influenced by the Westminster system or apply the FPTP rule (e.g. Australia, New Zealand, USA, Canada), the fit of our model s predicted Gallagher Index (under the assumptions stated before) against actual data is reassuring. In order to further check this claim, and identify whether empirical patterns of electoral rule change fit our model, we apply its predictions on actual electoral data from Greece. Greece underwent four major electoral rule reforms in the period under consideration, namely in 1988, 1991, 004 and 006. Table 1 presents the actual electoral results, the vote and seat share allocations, the actual ( l) majority premium 5 and the optimal one (l ). We also report the observed deviations between the actual and the predicted values. In fact, our model seems to perform quite well in predicting the direction of electoral rule change in Greece. As one can observe in Figures 7 and 8, from the actual electoral rule is more disproportional than the optimum. Yet, the first electoral reform (1988) introduces the PR rule which results in more proportionality than it desired by the dominant parties. As a result, another reform follows (1991) that makes again the rule more majoritarian. During the third electoral reform (004), which was first applied in the 007 elections, a 40-seat majority premium is introduced, quite close to the predicted optimum. Finally, Greek Parliament amended the rule in 006 introducing a 50-seat premium, in accordance with the theoretical predictions of the model. Moreover, with the exception of the 1989 elections, the observed gap between the actual and optimal majority premium is shrinking over time. Overall, we observe that the actual and predicted values are moving in the same direction and do not differ significantly from each other. That is, both dominant parties in Greece have been quite strategical in choosing the right electoral rule, in order to consolidate the bipartisan system and 4 Recall that, in symmetric bipartisan systems (such as most OECD countries) our model predicts that l = v3. By replacing this value to the Gallagher formula G (l ) we can get an expression of G as a function of (see footnote 0). Then, we can calibrate our model by replacing with the sum of vote-shares of the remaining (non-dominant) parties, within each country, to obtain a unique, country-specific value of the Gallagher Index that our model would have predicted, if our optimally chosen electoral rule (l ) were to be applied. 5 Our model can accomodate a wide range of potential electoral rules. One need only compute l = s1 ṽ1 1 ṽ 1 in order to find the actual majority premium l, which corresponds to the value of the premium when we allow the actual seat allocation to be replicated by our model. 19

20 avoid political power-sharing with smaller parties (single-party governments). 3. Technical Remarks On a final note, two clarifications are in order. First, with respect to the structure of the bargaining process. One might worry that the results presented so far critically depend on the fact that bargaining between the two dominant parties does not allow for counter-proposals (we adopt a take-it-or-leave-it bargaining protocol). Whereas, this statement would have been generically true in any other context, in this particular set up, enriching the bargaining process plays absolutely no role in driving the results. The reason for this is the convexity of the expected utility function (Eu i (l, v)), with respect to l (Fig. and 3). As stressed in Proposition 1, due to convexity of Eu i, there are only two candidate-values of l for an optimum (l = 0 and l = ). And it is also true that by assuming a non-trivial electoral reform process, one of the two parties can always guarantee its most preferred outcome (l = 0), in case there is no agreement between them. That is, it has a veto power, since it can block the electoral rule reform. Hence, the existence of an alternative bargaining process, where Parties could engage in consecutive counter-proposals is equivalent to our set-up and the results obtained under any such formulation would have been identical. Therefore, for simplicity, but without any loss in generality, we refrain from adapting a more complex bargaining protocol. Secondly, the same rationale applies when one considers our choice to study only one-sided transitions from the PR rule (l = 0) to more majoritarian ones. Since, as argued above (see also Fig. and 3), there are only two candidate values of l for an optimum, in equilibrium we must have either l = 0 or l =. Therefore, regardless of our starting point, be it the PR rule or a more majoritarian one, we should expect to end-up in the same place. Yet, starting from l = (or from any other l 0 for that matter) would have been much more unintuitive and more diffi cult to motivate as our chosen starting point. On the other hand, assuming that the PR is the status-quo rule resonates much better with actual data and is much more intuitive. Since we have proved that there exist a unique electoral reform equilibrium, other than the status-quo rule (in our case the PR), the choice of any other rule as a starting point would have yielded identical results. 0

21 4 Ideology and Dynamic Stability So far, our discussion has attempted to shed some light on how and why electoral rule change takes place in parliamentary democracies, as a result of strategic choice by the parties. Yet, we did not provide an account on how ideology might influence this choice. And although we do not aspire to conduct an exhaustive analysis on the role of ideology in electoral rule reform, we will provide some results that yield useful insights on the importance of ideology in electoral rule choice. So far, we have shown that the predictions of our model fit well the Greek election data. Nevertheless, there are some notable cases (e.g. Italy), where a two-dominant-party system failed to produce a more majoritarian electoral rule. Practically, from 1945 and until the proposed electoral reform of 1993, Italy used the PR rule. Nonetheless, political competition was dominated by two major parties (PCI and CD 6 ), especially during the 1970s, when they reached the peak of their electoral appeal. Then, one might expect that according to our predictions, the PR should have been substituted with a more majoritarian rule. We will show, in this extension of the basic model, that this did not happen due to the extreme ideological distance between the two dominant parties in Italy during the Cold War era. The intuition behind this Italian Paradox is that both parties utilized the PR rule as an implicit insurance mechanism against the risk of facing a single-party government formed by an ideologically polar-opposite opponent. That is, strategic incentives to collude were mitigated, or even cancelled off, due to extreme ideological divergence. As a result, the PR rule was sustained as an insurance device. Hence, when ideology comes into play, strategic incentives might be reversed. In the remainder of this section, we built into our model preferences over ideology and explore their effect on electoral rule choice. Nevertheless, we need to stress that the scope of this section is limited to the symmetric case, in order to motivate the Italian Paradox. Since we focus on the symmetric case, the two dominant parties are of equal electoral size, in expected terms 7, and also symmetrically positioned in terms of ideology. Hence, both have equal chances of winning the election. Moreover, we let [ 1, 1] be the ideology space. Then, each party 6 PCI is the Italian Communist Party and CD is the Christian-Democrat one. 7 That is, we have that a 1 = a and b 1 = b. This implies that: b i = 1 a i, which in turn implies that a i+b i = 1 v3 for i {1, }. 1

22 i N has its own ideology denoted by x i [ 1, 1], which will implement if it forms a single-party government. If not, then we assume that a grand-coalition is formed. As a result, each party influences the ideology of the coalition relative to its own seat share s l i(v i ). Symmetry, in terms of ideological distance, also implies that: x 1 = ε, x = ε and x 3 = 0, such that ε [0, 1]. That is, the Proposer (Party 1) is the left party, Party is the right party and the minority Party 3 is positioned in the centre, equidistant from the other two. The utility of a party i N shall be defined as: s l i(v i ) (x i π), if s l i(v i ) 1/ u i (v 1, v,, l) = 1 (x i π), if s l i(v i ) > 1/ Clearly, s l i(v i ) and l are defined as before. The extra term, (x i π), measures the disutility that each party receives when ideology other than its own is implemented. So, π(s l i(v i )) denotes the implemented ideology and clearly it depends on the electoral outcome (v i ), the electoral rule applied (l) and the resulting allocation of seats (s l i). A brief comment with respect to π and the formation of government is in order. Whenever we have a single-party government (s l i(v i ) > 1/), π = x i. But in case no party has the absolute parliamentary majority, the grand-coalition chooses its π according to the relative parliamentary strength of each one of its members 8. But given symmetry, it is equally likely that the major partner in any coalition government will be one of the two dominant parties. Hence, in expected terms, the ideology of the median will be implemented 9. That is, π = 0, which in our particular case happens to coincide with the ideology of the centrist party x 3. For demonstration simplicity we assume that ε = 1 (in the Appendix we provide equivalent results about any ε [0, 1]). That is, the two dominant parties occupy polar opposite positions in the ideological spectrum. Thus, the model exhibits maximum ideological divergence. It is interesting to compare the results of this special case with the results of the symmetric case of the standard version of our model, where we have implicitly assumed that ε = 0 (no ideological component present). 8 One can think of a coalition government distributing the ministerial portfolios to the parties based on their parliamentary strength. 9 We stress that this statement is not an assumption. It is trivially derived when one considers that the ideology of the median is the Condorcet winner.

23 Proposition 4 Assume symmetry and a non-trivial electoral reform process. Then, (i) when ε = 0 the optimal choice of electoral rule is l =, (ii) when ε = 1, the optimal choice is the PR rule (l = 0). This simple example demonstrates the role of ideology in the strategic choice of electoral rule. Whereas, under complete ideological convergence the two dominant parties were always able to collude and substitute the PR rule with a more majoritarian one (see Corollary 1 for the symmetric case), when the ideological distance between the two dominant parties is maximized, the incentives to collude disappear. On the contrary, they have an incentive to use the PR rule as an implicit insurance mechanism, in case they lose elections. And although they still care about forming a single-party government, their risk-aversion over the prospect of their polar-opposite ideological opponent doing the same forces them to stick with the PR rule. This could potentially explain why Italy never departed from the PR, especially during the 1970s, when ideological divergence between the communists and the christian-democrats was at its peak. Moreover, after the collapse of the iron curtain, the subsequent dissolution of the Italian Communist Party (PCI) resulted in a significant decrease in its electoral power. As a result, the ideological distance between the left and right shrunk and a more moderate party emerged from PCI, as the dominant party in the center-left. This occurred in the 199 elections. A proposal to adopt a more majoritarian electoral rule followed immediately afterwards (1993). Going back to the Greek Parliamentary election data, on Table 1 and Fig. 7 and 8, we can see yet another application of the electoral rule reform as insurance against an ideologically distant opponent. During the 1980s political competition was dominated by corruption allegations and high ideological divergence between the two dominant parties, the governing socialist PASOK and the conservative opposition ND. The governing party fearing the prospect of a single-party government being formed by the opposition, in the forthcoming elections, amended the electoral rule to pure PR in It did so to impede the formation of a single-party government and clearly, it achieved its objective (see Fig. 8). This is depicted as a steep dive in Figure 7. It is clear that during those electoral periods, the electoral rule was more proportional than before. We attribute this larger degree of proportionality to the increase in the degree of ideological divergence that was 3

24 observed back then. Hence, empirical evidence from both Greece and Italy seem to confirm our theoretical findings and intuition. On a final note, we need to stress that this attempt to account for ideology is far from being complete. In this paper, we have focused on the symmetric case, both in terms of electoral strength and in terms of ideological proximity. We also made particular assumptions about the coalition formation process. These assumptions might fit the particular cases of Greece 30 or Italy. However, our model does not explore what happens when symmetry is dropped. While acknowledging this limitation, we stress that the model s predictions, even in its simplest form, are in line with empirical observations and intuition. Moreover, it allows us to isolate the impact of ideology on electoral rule choice. Of course, more analysis is warranted on the impact of ideology on electoral rule choice. We leave this to future research. 4.1 Dynamic Stability The static nature of our model raises some concerns regarding the stability of the equilibrium electoral rules that we identify. We will briefly argue that the inclusion of an ideological component in the game as presented above guarantees the dynamic stability of the model. If parties know that the electoral rule that they will design will have a long life length then the uncertainty that we assume about the next elections can be easily interpreted as the distribution of the expected outcomes for the many elections that will take place under a certain electoral rule. Moreover, in this dynamic context, the symmetric scenario that we consider above (the scenario in which both parties are expected to win the next election with equal probability) can be interpreted as each of the major parties expecting to win half of the times during the period in which this electoral rule will be applied (data confirm that in most countries each of the two largest parties won approximately half of the elections in any arbitrary long period). If at the time of the electoral rule choice the ideologies of the two major parties diverge then proposition 4 dictates that they will choose PR and when parties compete under a PR rule they tend to adopt highly polarized political platforms (see for example Cox 1990 and Calvo and Hellwig 011). On the other hand, if at the 30 For example in Greece the two dominant parties the socialist left PASOK and the conservative ND are approximately of equal strength and have shared almost equal time in offi ce since

25 time of electoral rule choice the two major parties exhibit ideological proximity then they choose a majoritarian electoral rule (l = ) and when parties compete under majoritarian electoral rules they tend to adopt more convergent platforms (this holds especially for major parties; see Calvo and Hellwig 011). So what we observe is that we have two stable dynamic paths. PR leads to platform polarization and platform polarization leads to PR. Majoritarian systems lead to platform convergence and platform convergence leads to majoritarian systems. 5 Conclusions The main finding of our paper is the strategic coordination among dominant parties in adapting a more majoritarian than the PR electoral rule, in order to consolidate the two-party system and eliminate the role of smaller parties. Furthermore, our model identifies the three driving factors that allow this collusion between dominant parties, in the form of introducing a majority premium to the PR, to take place. The first key element is suffi cient electoral uncertainty. The second is their desire to form a single-party government. When elections are competitive and both dominant parties have enough chances of winning, they have incentives to cooperate in order to eliminate the impact of third parties, in the same spirit that big firms would like to drive smaller competitors out of the market. That is, dominant parties have incentives to consolidate their position in the party system, since the terms of political competition are obviously favorable to them under the status quo. But, for collusion to take place the two conditions mentioned above are indispensable. The desire for offi ce and single-party governments generates the incentive to distort the proportionality of the electoral rule, whereas electoral uncertainty allows for those incentives to be aligned, creating enough room for collusion. The third, and more subtle, factor is ideology. Driven by empirical observation, suggesting that are party-systems in which the two dominant parties do not cooperate to introduce a more majoritarian electoral rule, we identify ideological proximity between dominant parties as the final necessary condition for collusion. Hence, we introduce an ideology component, as an extra dimension, into the preferences of the parties. We show that, a third key condition was implicitly assumed in order for the two dominant parties to coordinate: ideological proximity. That is, when 5

26 ideological distance between the two dominant parties is large, our model predicts that dominant parties strategic incentives to agree on a more majoritarian rule are reversed. The fear that their polar-opposite ideological opponent will form a single-party government dominates over their desire for offi ce. Hence, they utilize the choice of the electoral rule as an insurance device against the risk of having an ideologically opposed single-party government. In the case of maximum ideological divergence, our key result states that both parties are better off by sticking to the PR rule. This can explain, to some extend, the prevalence of PR rule in some bipartisan systems and backs up the dynamic stability of our results. 6

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29 6 Appendix A 6.1 The General Case Assume that: u i (v 1, v,, l) = s l i(v i ), if s l i(v i ) 1/ s l 3( ) s l i(v i ) + r 1, if 1/ s l 3( ) s l i(v i ) 1/, gs l i(v i ) + r, if s l i(v i ) > 1/ where 0 r 1 < 1 + r 1 < r and g [0, 1] and that a 1 = a < 1 < b 1 = b (symmetric parties). Notice that s l i(v i ) = v i (1 l) when v i < (1 )/, s l i(v i ) = v i (1 l) + l when v i = (1 )/ (parties 1 and split the bonus when they tie) and that s l i(v i ) = v i (1 l) + l when v i > (1 )/. Therefore, the expected utility of party 1 (since parties 1 and are symmetric we derive the outcome of the game only by studying party 1) from an electoral rule l is given by: 1 Eu 1 (v 1, v,, l) = 1 () [ (b 1 a 1 v ) 1 (1 l)dv 1 + a 1 + 1/ l (1 )/ [v 1 (1 l) + r 1 + l]dv 1 + and from an electoral rule l > is given by: Eu 1 (v 1, v,, l) = 1 We compute: (b 1 a 1 ) [ (1 v3)/ b 1 1/ l a 1 v 1 (1 l)dv 1 + (1 v 3 )/ 1 () [v 1 (1 l) + r 1 ]dv 1 + [g(v 1 (1 l) + l) + r ]dv 1 ] b 1 (1 )/ [g(v 1 (1 l) + l) + r ]dv 1 ]. Eu 1 (v 1, v,, l = 0) = 1+4a 1 +g 4b 1 g 8r 1 +4r 8b 1 r 8(a 1 b 1 ) and Eu 1 (v 1, v,, l = ) = 1+4a 1 +g 4b 1 g+4l 4a 1 l+4gl 8b 1gl+4b 1 gl 3g +4r 8b 1 r 4 r 8(a 1 b 1 ). Without loss of generality we normalize r 1 = 1 and we find that Eu 1 (v 1, v,, l = ) > Eu 1 (v 1, v,, l = 0) holds if and only if: r > 7+8b 1 4b 1 +g 8b 1g+4b 1 g+15 8b 1 3g 4v

30 and that Eu 1 (v 1,v,,l) l > 0 for l holds if and only if: r > 1 (11 3g). 4 Moreover we observe that Eu 1(v 1,v,,l) l < 0 for l > occurs if and only if: g < 3+b b 1. That is, for suffi ciently high r (suffi ciently high rents from forming a single party government) and for suffi ciently low g (suffi ciently low gain from an extra seat when the party already has a majority of votes in the parliament) both parties maximize their expected utilities by applying the l = electoral rule. To conclude let us state that there are three more possibilities. For r suffi ciently low the most probable outcome is that parties will choose l = 0 (PR). There is a very small set of parameter values 31, given by: 7+8b 1 4b 1 +g 8b 1g+4b 1 g+4 8b 1 4v 3 4 < r < 7+8b 1 4b 1 +g 8b 1g+4b 1 g+6 8b 1 6g +7v 3 3g v 3 that leads parties to adopt an electoral rule l (0, l = 1 ) such that: 11a 1 11g 3a 1 g+b 1g 11b 1 g+3g 6b 1 g +3b 1 g +11 3g 4a 1 r +4gr 8b 1 gr +4b 1 gr 4 r (a 1 g+b 1g b 1 g+) Finally, in the extreme case where both r and g are very large, the first-past-the-post electoral rule (l = 1) may be selected by the parties.. 6. Insuffi cient Electoral Uncertainty When the electoral competition is trivial we can state the following results. Proposition 5 When both the electoral reform process and the electoral competition are trivial and the proposer is: (i)the leading party, then l max{0, 1/ a 1 1 a 1 }, (ii) not the leading party, then l = This claim is derived from parametric analysis using Mathematica. 30

31 Proof. Since the electoral reform process is trivial, Party 1 faces an unconstrained maximization program. Moreover, since the electoral competition is trivial, if it is the leading party, its expected utility is given by: Eu 1 (v 1, v,, l) = 1 1/ l [ (b 1 a 1 ) a 1 [v 1 (1 l) + l]dv 1 + b 1 1/ l dv 1 ]. Then, it proposes l such that it secures with certainty the majority of the seats in the new parliament. That is, it sets a 1 (1 l ) + l = 1/, which implies that s l 1 = 1/, a 1. Then, solving for l yields the result. If it is not the leading party, Eu 1 (v 1, v,, l) = 1 [ b 1 (b 1 a 1 v ) 1 (1 l)dv 1 ], which is strictly decreasing in a 1 l. Hence, l = 0. This completes the proof. This result can be viewed as the simplest case scenario. The idea behind this proposition is very simple. Since the electoral reform process is trivial, the proposer holds enough seats in the current parliament to enact any electoral rule reform, without the need to satisfy the ICC of Party. Hence, the proposer just chooses l in order to maximize expected utility. Given that the electoral competition is trivial, if the Proposer is not the leading party (i.e. sure loser) it proposes that the PR rule is not amended (l = 0). As a result, the electoral rule does not change. Otherwise, it proposes l max{0, 1/ a 1 1 a 1 }. That is, if Party 1 is the leading party (i.e. sure winner), the proposed proportionality distortion l will be such that it will guarantee the formation of a singleparty government by Party 1. The solution to this unconstrained maximization problem yields with certainty the majority of the seats in the parliament for every possible realization of a 1. Hence, by choosing this level of l the leading party ensures the highest possible level of utility. On the other hand, if it is not the leading party it can never get the premium l, making its utility strictly decreasing on l. Hence, any distortion to the PR rule is not desirable. Proposition 6 When the electoral competition is trivial but the electoral reform process is nontrivial, then l = 0. Proof. Since electoral competition is trivial one party is a sure loser and the other is a sure 1 winner. That is, for i = 1 or, Eu i (v i, l) = [ b 1 (b 1 a 1 v ) i (1 l)dv i ], which is strictly decreasing a 1 in l. Hence, one party always prefers l = 0 but the other, as shown in Proposition 1 prefers l max{0, 1/ a 1 1 a 1 }. Since the electoral reform process is non-trivial, the two parties have to agree on the electoral reform. So, the only equilibrium is l = 0. 31

32 The idea that drives the result is that in this case there is no room for collusion. The strategic incentives of the two dominant parties do not align, because there is no uncertainty over the outcome of the electoral competition. The leading party will always prefer a value of l > 0 but the other party will always reject this proposal because its utility is strictly decreasing in l. Since the electoral reform is non-trivial and requires the consent of both dominant parties, it is obvious why no electoral rule reform will ever be enacted by this parliament. That is, the status-quo is maintained and PR persists as the electoral rule (i.e. l = 0). 6.3 Arbitrary degree of ideological differentiation For ε [0, 1] the Proposer 3 (Party 1) faces the following maximization program 33 : 1 (b 1 a 1 ) 1 1/ l a 1 (1 v 3 )/ a 1 v 1 (1 l)dv 1 + ( ε ε) dv 1 max Eu 1 (v 1, v,, l) = l [0,1] 1/ l (1 )/ 1/ l 1 1/ l [v 1 (1 l) + l]dv 1 + ( ε 0) dv 1 s.t. Eu (v 1, v,, l) Eu (v 1, v,, 0). b 1 1/ l b 1 1/ l dv 1 ( ε ( ε)) dv 1 Note that the first three terms of the above expression are identical with the maximization program presented in Section. Moreover, we can simplify the second line of the expression that refers to the ideological component of the utility by applying the symmetry conditions. Clearly, the probability of Party 1 forming a single-party government and implementing its ideology is identical with that of Party doing the same. That is, 1 1/ l a 1 = b 1 1/ l. Then with probability 1 (b 1 1/ l ), no party succeeds in forming a single-party government. Thus, the ideology of the median is implemented. Hence we can rewrite the Proposer s expected utility as follows: Eu 1 (v 1, v,, l) = 1 (b 1 a 1 ) (1 v 3 )/ a 1 v 1 (1 l)dv 1 + 1/ l (1 )/ [v 1 (1 l) + l]dv 1 + b 1 1/ l dv 1 4ε (b 1 1/ l ) [1 (b 1 1/ l )]ε (b 1 1/ l )0 3 By symmetry Party faces an analogous program. 33 In this section we focus again on the non-trivial electoral reform process. 3

33 As before, a critical value for l is l =, since for every l (, 1] the above expression [ ] (1 v3 )/ collapses to 34 : Eu 1 (v 1, v,, l) = 1 b 1 (b 1 a 1 v ) 1 (1 l)dv 1 + dv 1 ε. Finally, we note a 1 (1 )/ that apart from imposing symmetry the rest of the model remains as specified in Section. Proposition 7 Assume the electoral reform process is non-trivial. Let ε [0, 1]. Then ε, and ε, satisfying ε > ε > 0 such that: (i) ε (ε, 1] we have l {0}; (ii) ε (ε, ε ] we have l {0, l l } and (iii) ε [0, ε ] we have l {0, }. Proof. First we note that we restrict attention to values of l [0, ] since for all other values Eu 1 (v 1, v,, l) is decreasing with respect to l, as shown in Proposition 4. For part (i) we need to show that Eu 1 (v 1, v,, l)/ l < 0, that is decreasing, for all ε [1, ε ). We compute: [ ] Eu 1 (v 1,v,,l) = 1 l (+a 1 ) l(+a 1 )+1/8+(+a 1 )/ ε l b 1 a 1. Clearly, Eu 1(v 1,v,,l) < 0 iff l (v () l 3 + a 1) l( + a 1) + 1/8 + (+a 1 ) ε < 0 which implies that for every l [0, be satisfied: ε > ] the following should (+a 1 ) = ε. Then the result follows, since there is a unique candidate for a maximum, namely l = 0, due to the fact that Eu 1 (v 1, v,, l) is decreasing for every l. For part (ii) we need to show that for every ε > ε, Eu 1 (v 1, v,, l) is concave in [0, That is we compute: ]. Eu 1 (v 1,v,,l) l = 1 b 1 a 1 [ 1 4 ε ] < 0 iff ε > 1/8 = ε. Hence, by concavity of Eu 1 (v 1, v,, l) there may exist an interior maximizer l [0, Eu 1 (.) is decreasing for l ( ) such that l <. Since, 1], and by an argument analogous to that of Proposition 1, we conclude that there are only two candidate values for an optimum, namely l {0, l }. If an interior maximizer does not exist, then we trivially have l = and we are in case (iii). For part (iii), we just note that for any other value of ε, such that ε ε, Eu 1 (v 1, v,, l) is convex in [0, ]. Therefore, as shown in the proof of Proposition 1, the two candidates for a maximum are the same, namely l {0, }. This completes the proof. The above result demonstrates how ideological divergence between parties affects the strategic choice of electoral rules. It is clear that when ideological divergence is high we end up with more proportional electoral rules. Moreover, for significantly large values of ε, PR is the unique choice of electoral rule that parties are willing to accept. These are exactly the situations where the 34 By symmetry, when l = v3, we have that: b 1 1/ l = b 1 1 v3 = b1 a1. 33

34 PR is used as an insurance against the prospect of facing a single-party government with polaropposite ideology. For intermediate values 35 of ideological proximity, we observe that dominant parties might still collude in choosing a less proportional rule. Yet, compared to the standard symmetric case where ideology was absent (ε = 0), the majoritarian rule that is chosen is less disproportional than l =, which was always chosen. Finally, for smaller values of ε (high ideological proximity), parties face an identical choice as before. But still, there are some values of ε [0, ε ] such that Eu i (v 1, v,, l = 0) > Eu i1 (v 1, v,, l = ) for i = {1, }. That is, even in this case the PR rule might be again chosen. Contrast this with the initial case, where under all circumstances l = was chosen, and one can get a clear idea on how ideology alters the incentives of dominant parties to collude. 7 Appendix B 7.1 Proofs Proposition 1. Given the structure of the game, to show that when the electoral competition is non-trivial then l {0, }, is quite easy. In the first part of the proof we shall demonstrate that, for both parties the exact bonus l that maximizes their expected utility is either 0 or. Then given this result, we will offer a trivial argument to show that if both parties maximize their expected utility with a bonus, this specific electoral reform takes place, and in case at least one maximizes its expected utility with l = 0 no electoral reform takes place. For the first part of the proof we need to prove that the expected utility of party 1 is convex in l [0, v ], strictly convex in a subset of [0, 3 ] and decreasing in (, 1]. The arguments are equivalent for party. Since the electoral competition is non-trivial, we have that b 1 > 1. If b 1 < 1/ then there exist ˆl (0, 1) s.t. 1/ ˆl (1 v3)/ 1 [ (b 1 a 1 ) a 1 v 1 ()dv 1 + b 1 (1 )/ 1 ˆl [v 1 ()+l]dv 1 ], for l (ˆl, = b 1. For l [0, ˆl] we have that Eu 1 (v 1, v,, l) = ] we have that Eu 1 (v 1, v,, l) = 35 Parametric analysis using Mathematica reveals that the region [ε, ε ] represents a very small fraction of parameter values; it is extremely unlikely that parties will agree on an electoral rule different to 0 or. 34

35 (1 v3)/ 1 [ (b 1 a 1 ) a 1 v 1 (1 l)dv 1 + Eu 1 (v 1, v,, l) = 1 1/ l (1 )/ (b 1 a 1 ) [ (1 v3)/ for l [0, ˆl], Eu 1 (v 1,v,,l) l if Eu 1(v 1,v,,l) l is convex in [0, [v 1 (1 l) + l]dv 1 + a 1 v 1 ()dv 1 + > 0 for l (ˆl, b 1 (1 )/ 0 for l [0, ˆl] then Eu 1(v 1,v,,l) l b 1 1/ l dv 1 ] and for l (, 1] we have that dv 1 ]. One may observe that Eu 1 (v 1,v,,l) = 0 l ], and Eu 1(v 1,v,,l) < 0 for l ( l, 1]. Moreover, > 0 for l (ˆl, ]. That is, Eu 1 (v 1, v,, l) v ], strictly convex in a subset of [0, 3 ] and decreasing in (, 1]. The only candidates for maximum are {0, 1/ l (1 )/ If b 1 > 1/, for l [0, l)dv 1 + [v 1 ()+l]dv 1 + b 1 (1 )/ b 1 1/ l }. ] we have that Eu 1 (v 1, v,, l) = 1 (b 1 a 1 ) [ (1 v3)/ dv 1 ] and for l (, 1] we have that Eu 1 (v 1, v,, l) = 1 a 1 v 1 (1 l)dv 1 + (b 1 a 1 ) [ (1 v3)/ dv 1 ]. Just, as before Eu 1 (v 1,v,,l) v > 0 for l [0, 3 l ], and Eu 1(v 1,v,,l) l a 1 v 1 (1 < 0 for l ( v, 1]. That is, Eu 1 (v 1, v,, l) is strictly convex in [0, 3 ] and decreasing in (, 1]. The only candidates for maximum are {0, }. This concludes the first part of the proof. If both parties maximize their expected utility with l = then party one proposes this electoral reform and party votes for it. If the optimal bonus for the proposing party 1 is l = 0 then it does not propose any electoral reform. And when l = 0 maximizes the expected utility of party then it always votes against any electoral reform. Proposition. We will only prove the result for values of b 1 < 1.First let us note that from Proposition 1 we have that when the electoral competition is non-trivial then there are only two possible candidate values for an optimum, that is either l = 0 or l =. When the electoral rule reform is trivial then this implies that the proposer faces an unconstrained maximization problem. Hence for a change of the electoral rule from l = 0 to l = it suffi ces to show that Eu 1 (v 1, v,, l = ) > Eu 1 (v 1, v,, l = 0).Assume this condition holds. Then this implies (1 v3)/ 1 that [ (b 1 a 1 v ) 1 (1 v b 1 (1 v3)/ 3 )dv 1 + dv 1 ] > 1 [ b 1 (b 1 a 1 v ) 1 dv 1 + v 1 dv 1 ]. After some a 1 a 1 algebraic manipulation this yields which implies b 1 (1 )/ (1 )/ b 1 (1 v 1 )dv 1 > ( dv 1 b 1 (1 )/ (1 )/ (1 v 3 )/ ) a 1 v 1 dv 1 > (1 )/ a 1 v 1 dv 1 (1 )/ (1 )/ a 1 v 1 (1 )dv 1, v 1 dv 1. It is then easily checked that expanding the 35

36 integrals yields the desired inequality (b 1 1 )( b 1 1 ) ( )( 1 +a 1 )( 1 a 1 ) which in turn implies f 1 (a 1, b 1, ) 0. This completes the argument. Reversing the argument assume that the above inequality holds true but no electoral rule change takes place, that is l = 0. But then simple algebra yields Eu 1 (v 1, v,, l = ) > Eu 1 (v 1, v,, l = 0) which implies l = a clear contradiction. Then the only if part follows. Proposition 3. We will be using an argument analogous to Proposition. Let us first note that, in this case, the duality of the conditions is due to the fact that the proposer faces the constrained version of the maximization problem (given that the electoral reform process requires the consent of both parties). That is for l = 0 to change to l = both conditions Eu 1 (v 1, v,, l = ) > Eu 1 (v 1, v,, l = 0) and Eu (v 1, v,, l = ) > Eu (v 1, v,, l = 0) need to be satisfied simultaneously. It can be easily checked by an analogous argument (as in Proposition 4), that the condition (b 1 1 )( b 1 1 ) ( )( 1 +a 1 )( 1 a 1 ), implying f 1 (a 1, b 1, ) 0, is both necessary and suffi cient condition for Party 1 to support l =. A directly analogous condition is also true for Party by symmetric nature of the problem. Hence we only need to check that the leading party condition implies the above inequality for the other party as well. Note that a 1 +b 1 > 1 implies b 1 1 > 1 a 1. In turn this implies that (b 1 1 )( b 1 1 ) > ( )( 1 + a 1 )( 1 a 1 ).Thus, f 1 (a 1, b 1, ) > 0. By analogy, f (a, b, ) > 0 when Party is the leading party and with a similar argument as in the proof of Proposition 4 it can be shown that this condition is both necessary and suffi cient. This completes the argument. Proposition 4. Firstly, symmetry implies that the electoral competition is non-trivial as well. Secondly, we note that for every l (, 1] the function Eu 1 (v 1, v,, l) is decreasing with respect to l, since Eu 1(v i,l) l [0, < 0. Hence, we can restrict our search for an optimum in the interval ] as we did before. For case (i) we only need to show that symmetry implies the condition of Proposition 3, since then the same reasoning applies. Furthermore, we need only show that the condition is satisfied only for Party 1, due to symmetry. Check that a 1+b 1 = a +b = 1 implies b 1 1 = 1 a 1. But then the condition becomes: ( b 1 1 ) ( )( 1 + a 1 ). Since ( ) < 1 and b a 1 for every a 1, b 1, we conclude that the condition is always satisfied and hence, l = l 1. For case (ii) Eu 1 becomes Eu 1 (v 1, v,, l) = v 1 (1 (b 1 a 1 ) [ (1 v3)/ a 1 36

37 l)dv 1 + 1/ l (1 )/ [v 1 (1 l) + l]dv 1 + b 1 1/ l dv 1 ε (b 1 1/ l ) ε ]. Then one can compute Eu 1(.) l = 1 [ l / l 3/8 (b 1 a 1 1 a ) () 1 v ] < 0 for every l [0, 3 ], since l / l 3/8 < 0. Hence, Eu () 1 (v 1, v,, l) is strictly decreasing for every l [0, 1] and as a result, the unique candidate for an optimum is l = 0. This completes the proof. 37

38 Figure 1: Utility u i (s l i) for Party i as a function of its seat-share s l i(v i ). 38

39 Figure : Expected Utility Eu i (l, v i ) as a function of l for i = 1, when l =. Figure 3: Expected Utility Eu i (l, v i ) as a function of l for i = 1, when l = 0. 39

40 Figure 4: Utility u i (v i, l) as a function of vote share for i = 1, and for different values of l. The Degree of Electoral Rule Disproportionality In OECD Countries from Density Electoral Rule Disproportionality (Gallagher Index) Source: Comparative Political Studies Data Set 1 Kernel density estimate Figure 5: The Distribution of Electoral Rules (according to their Disproportionality based on the Gallagher Index) across 3 OECD Countries (from ). 40

41 Figure 6: Predicted vs. Actual Electoral Rule Disproportionality (as measured by the Gallagher Index) in 16 OECD Countries with Bipartisan Party-systems (symmetric case) from Actual vs. Predicted Majority Premium (in Seats) Greek National Elections ( ) Parliamentary Seats Electoral Periods ( ) Actual Premium Premium Gap (Actual minus Predicted Premium) Predicted Premium Source: Own calculations and Greek Ministry of Interior Figure 7: The Model s predicted Majority Premium (l ) and its fit with actual Greek electoral data ( ) 41