The Convergence Coefficient across Political Systems

Abstract

Formal work on the electoral model often suggests that parties or candidates should locate themselves at the electoral mean. Recent research has found no evidence of such convergence. In order to explain nonconvergence, the stochastic electoral model is extended by including estimates of electoral valence. We introduce the notion of a convergence coefficient, 𝑐. It has been shown that high values of 𝑐 imply that there is a significant centrifugal tendency acting on parties. We used electoral surveys to construct a stochastic valence model of the the elections in various countries. We find that the convergence coefficient varies across elections in a country, across countries with similar regimes, and across political regimes. In some countries, the centripetal tendency leads parties to converge to the electoral mean. In others the centrifugal tendency dominates and some parties locate far from the electoral mean. In particular, for countries with proportional electoral systems, namely, Israel, Turkey, and Poland, the centrifugal tendency is very high. In the majoritarian polities of the United States and Great Britain, the centrifugal tendency is very low. In anocracies, the autocrat imposes limitations on how far from the origin the opposition parties can move.

1. Introduction

Work on modeling elections has often assumed that the policy space was restricted to one dimension or that there were at most two parties [1, 2]. The extensive formal literature on electoral competition has typically been based on the assumption that parties or candidates adopt positions in order to win and has inferred that parties will converge to the electoral median, under deterministic voting in one dimension, or to the electoral mean in stochastic models.

In this paper we offer a formal stochastic model of elections that emphasizes the importance of the idea of valence and use this notion to provide an explanation of why vote maximizing political leaders in some countries will not adopt convergent policy positions at the mean of the electoral distribution. In the standard spatial model, candidate positions matter to voters. However, as Stokes [3, 4] has emphasized, the nonpolicy evaluations, or valences, of candidates by the electorate are just as important. (See also Clarke et al.   [5, 6], Scotto et al. [7], and Clarke et al. [8].)

The main objective of this paper is to examine whether parties locate close to or far from the electoral mean (the electoral mean is the mean of voters ideal policies dimension by dimension) of various countries. We use Schofield’s [9] stochastic electoral model as a unifying framework that allows us to compare parties positions across different political systems. In this model, parties respond to their partisan constituencies after taking into account the anticipated electoral outcome and the positions of other parties. Voters decisions depend on parties’ locations and on the party’s valence, the voters’ overall common evaluation of the ability of a party leader to provide good governance. Using this model we examine whether parties converge to electoral mean in several elections in various countries under different political systems and use convergence, or the lack thereof, to classify political systems.

To examine whether parties converge to the electoral mean in each country in a particular election, we test whether any party has an incentive to stay or move away from the electoral mean to increase its vote share. In formal voting theory, it is usual to define a “Nash equilibrium” as a vector of party positions with the property that no party may make a unilateral move so as to increase its vote share. We use a variant of this concept, that is a “local Nash equilibrium” (LNE) where we consider only marginal moves from the position. One of the standard results in formal theory is the mean voter theorem, where the “Nash equilibrium” of a spatial voting game under vote maximization is one where all parties position themselves at the electoral mean. (For variants of the theorem see Enelow and Hinich [10–12]). We call such a vector the electoral mean.

To study each party’s best response to the electoral situation they face, we use the results presented in Schofield [9]. Schofield identifies a convergence coefficient, denoted 𝑐, whose value determines whether vote maximizing parties converge or not to the electoral mean. This coefficient depends on various parameters of the model. In particular it depends on the competence valences of the party leaders. Using 𝑗∈𝑃=1,…,𝑝 to denote the parties, the valence of party 𝑗, 𝜆_𝑗, essentially measures the electoral perception of the “quality” of 𝑗, the voters’ overall common evaluation of the ability of 𝑗’s leader to provide good governance. The valence terms, 𝝀=(𝜆₁,…,𝜆_𝑝), are assumed to be independent of the party’s positions and can be estimated as the intercept term in the appropriate stochastic model of the voter utility function. As Sanders et al. [13] comment, valence theory is based on the assumption that “voters maximize their utilities by choosing the party that is best able to deliver policy success.” These valence terms measure the bias in favor of one another of the party leaders [14].

The convergence coefficient, 𝑐, also depends on the weight that voters give to the policy differences they have with the various parties 𝛽. Lastly, 𝑐 depends on the variance/covariance matrix of the electoral distribution, 𝜎². By its construction, 𝑐≡𝑐(𝝀,𝛽,𝜎²) is dimensionless and thus independent of the units of measurement of the various parameters. We use the convergence coefficient to compare results across elections and countries and to classify political systems.

The convergence coefficient is a summary measure that provides an estimate of the centrifugal or centripetal forces acting on the parties. The Valence Theorem, presented in Section 2 (see Schofield [9] for the proof of this result) shows that if the policy space is two-dimensional and if 𝑐(𝝀,𝛽,𝜎²)<1, then the sufficient condition for convergence to the mean has been met and the “local Nash Equilibrium” (LNE) (the set of such local Nash Equilibria contains the set of Nash Equilibria) is one where all parties locate at the electoral mean. On the other hand, if 𝑤 is the dimension of the policy space and 𝑐(𝝀,𝛽,𝜎²)≥𝑤, then the LNE, if it exists, will be one where at least one party will have an incentive to diverge from the electoral mean in order to maximize its vote share. Thus, the necessary condition for convergence to the mean is that 𝑐(𝝀,𝛽,𝜎²)<𝑤.

In essence, a high empirical convergence coefficient of an election is a convenient measure of the electoral incentive of a small, or low valence, party to move away from the electoral mean to its core constituency position. We can interpret a high value of the convergence coefficient as a measure of the centrifugal tendency exerted on parties pulling them away from the electoral mean. The convergence coefficient is therefore a convenient, simple, and intuitive way to examine whether parties will have an incentive to locate close to, or far from, the electoral mean. We will show that there is a strong connection between the values of the convergence coefficient and the nature of the political system under which parties operate.

We used preelection polls to study elections in several countries operating under different political regimes. The factor analysis done on preelection surveys showed that for all elections the policy space was two-dimensional, except in Azerbaijan were it was one-dimensional. The position of voters along this two-dimensional space were then estimated and their voting intentions used to estimate party positions. We then ran a multinomial logit (MNL) model for the election using the estimated party and voter positions. The intercept of the MNL model gives the valences of each party/leader. Following Schofield’s [9] formal model, we rank parties according to their valence. Using these MNL estimates we calculate the convergence coefficient of the election and examine whether the party with the lowest valence has an incentive to locate close to or far from the electoral origin.

When comparing the convergence coefficients across countries we observe that in countries with proportional representation the convergence coefficient is high and that in countries with plurality systems or in anocracies it is low. Thus, suggesting that we can use the valence theorem and its associated convergence result to classify electoral systems.

The convergence coefficients for the 2005 and 2010 elections in the UK were not significantly different from 1, meeting the necessary condition for convergence to the mean. For the 2000, 2004, and 2008 US presidential elections, the convergence coefficient is significantly below 1 in 2000 and 2004 thus meeting the sufficient and thus necessary condition for convergence and not significantly different from 1 in 2008, only meeting the necessary condition for convergence. We suggest that the centrifugal tendency in the majoritarian polities like the United States and the United Kingdom is very low.

In contrast, the convergence coefficient gives an indication that the centrifugal tendency in Israel, Poland, and Turkey is very high. In these proportional representation systems with highly fragmented polities the convergence coefficients are significantly greater than 2 failing to meet the necessary condition for convergence to the electoral mean.

In the anocracies of Georgia, Russia, and Azerbaijan, where the President/autocrat dominates and controls who can run in legislative elections, the convergence coefficient is not significantly different from the dimension of the policy space (2 for Georgia and Russia and 1 for Azerbaijan), failing the necessary condition for convergence. While the analysis Georgia and Azerbaijan show that not all parties converge to the mean, in Russia it is likely that they did. Thus, in Russia opposition parties found it difficult to diverge from the mean. Note that convergence in anocracies may not generate a stable equilibrium as any change in the valence of the autocrat and the opposition may cause parties to diverge from the mean and may even lead to popular uprising that bring about changes in the governing parties such as in Georgia in previous elections or in the Arab revolutions.

We can also classify polities using the effective vote number and the effective seat number. (Fragmentation can be identified with the effective number. That is, let 𝐻_𝑣 (the Herfindahl index) be the sum of the squares of the relative vote shares and let 𝑒𝑛𝑣=𝐻_𝑣⁻¹ be the effective number of party vote strength. In the same way we can define ens as the effective number of party seat strength using shares of seats. See Laakso and Taagepera [15].) We examine how these two measures of fragmentation relate to the convergence coefficient for the polities we consider. The effective vote or seat numbers give an indication of the difficulty inherent in interparty negotiation over government. These two measures do not, however, address the fundamental aspect of democracy, namely, the electoral preferences for policy. Since convergence involves both preferences, in terms of the electoral covariance matrix and the effect of the electoral system, we argue that the Valence Theorem and the associated convergence coefficient allow for a more comprehensive way of classifying polities and political systems precisely because it is derived from the fundamental characteristics of the electorate. That is, while we can use the effective vote and seat number to identify which polities are fragmented, the Valence Theorem and the convergence result can help us understand why parties locate close to or far from the electoral mean and how, under some circumstances, these considerations lead to political fragmentation.

The next section presents Schofield’s [9] stochastic formal model of elections and implications it has for convergence to the mean. Section 3.1 applies the model to the elections to two plurality polities: The United States and the United Kingdom. In Section 3.2 we apply the model to polities using proportional electoral systems, namely, Israel, Turkey, and Poland. Section 3.3 considers the convergence coefficients for three “anocracies:” Azerbaijan, Georgia, and Russia. Comparisons between different fragmentation measures and the convergence coefficient are examined in Section 4. Section 5 concludes the paper. In the appendix we estimate the confidence intervals for the convergence coefficient as well as determining whether the low valence party has an incentive to deviate from the electoral mean.

2. The Spatial Voting Model with Valence

Recent research on modelling elections has followed earlier work by Stokes [3, 4] and emphasized the notion of valence of political candidates. As Sanders et al. [13] comment, valence theory extends the spatial or Downsian model of elections by considering not just the policy positions of parties but also the parties’ rival attractions in terms of their perceived ability to handle the most serious problems that face the country. Thus, voters maximize their utilities by choosing the party that they think is best able to deliver policy success.

Schofield and Sened [16] have also argued that Valence relates to voters’ judgments about positively or negatively evaluated conditions which they associate with particular parties or candidates. These judgements could refer to party leaders’ competence, integrity, moral stance or “charisma” over issues such as the ability to deal with the economy and politics.

Valence theory has led to a considerable theoretical literature on voting based on the assumption that valence plays an important role in the relationship between party positioning and the votes that parties receive. (Ansolabare and Snyder [17], Groseclose [18], Aragones and Palfrey [19, 20], Schofield [21], Schofield et al. [22], Miller and Schofield [23], Schofield and Miller [24], Peress [25]) Empirical work, based on multinomial logit (MNL) methods, has also shown the importance of electoral judgements in analyses of elections in the United States and the United Kingdom. (Clarke et al. [8, 26–28], Schofield [29], Schofield et al. [30, 31], Scotto et al. [7]) These empirical models of elections have a “probabilistic” component. That is, they all assume that “voter utility” is partly “Downsian” in the sense that it is based on the distance between party positions and voter preferred positions and partly due to valence. The estimates of a party’s valence is assumed to be subject to a “stochastic error.” In this paper we use the same methodology.

The pure “Downsian” spatial model of voting tends to predict that parties will converge to the center of the electoral distribution [10–12]. However, when valence is included, the prediction is very different. To see this suppose there are two parties, A and B, and both choose the same position at the electoral center, but A has much higher valence than B. This higher valence indicates that voters have a bias towards party A and as a consequence more voters will choose A over B. The question for B is whether it can gain votes by moving away from the center. It should be obvious that the optimal position of both A and B will depend on the various estimated parameters of the model. To answer this question we now present the details of the spatial model.

2.1. The Theoretical Model

To find the optimal party positions to the anticipated electoral outcome we use a Downsian vote model that has a valence component as presented in Schofield [9]. Let the set of parties be denoted by 𝑃=1,…,𝑝. The positions of the 𝑝 parties (We will use candidate, party and agents interchangeably throughout the paper.) in 𝑋⊆ℝ^𝑤 where 𝑤 is the dimension of the policy space it is given by the vector 𝐳=𝑧₁,…,𝑧_𝑗,…,𝑧_𝑝∈𝑋^𝑝. (1) Denote voter 𝑖’s ideal policy be 𝑥_𝑖∈𝑋 and her utility by 𝑢_𝑖(𝑥_𝑖,𝑧)=(𝑢_𝑖1(𝑥_𝑖,𝑧₁),…,𝑢_𝑖𝑝(𝑥_𝑖,𝑧_𝑝)), where 𝑢_𝑖𝑗𝑥_𝑖,𝑧_𝑗=𝜆_𝑗−𝛽𝑥_𝑖−𝑧_𝑗²+𝜖_𝑗=𝑢^∗_𝑖𝑗𝑥_𝑖,𝑧_𝑗+𝜖_𝑗. (2) Here 𝑢^∗_𝑖𝑗(𝑥_𝑖,𝑧_𝑗) is the observable component of the utility voter 𝑖 derived from party 𝑗. The competence valence of candidate 𝑗 is 𝜆_𝑗, and the competence valence vector 𝝀=(𝜆₁,𝜆₂,…,𝜆_𝑝) is such that 𝜆_𝑝≥𝜆_𝑝−1≥⋯≥𝜆₂≥𝜆₁, so that party 1 has the lowest valence. Note that 𝜆_𝑗 is the same for all voters and provides an estimate of the “quality” of party 𝑗 or its ability to govern. The term ‖𝑥_𝑖−𝑧_𝑗‖ is simply the Euclidean distance between voter 𝑖’s position 𝑥_𝑖 and candidate 𝑗’s position 𝑧_𝑗. The coefficient 𝛽 is the weight given to this policy difference. The error vector 𝝐=(𝜖₁,…,𝜖_𝑗,…,𝜖_𝑝) has a Type I extreme value distribution, where the variance of 𝜖_𝑗 is fixed at 𝜋²/6. Note that 𝛽 has dimension 1/𝐿², where 𝐿 is whatever unit of measurement used in 𝑋.

Since voter behavior is modeled by a probability vector, the probability that voter 𝑖 chooses party 𝑗 when parties position themselves at 𝐳 is 𝜌_𝑖𝑗(𝐳)=Pr𝑢_𝑖𝑗𝑥_𝑖,𝑧_𝑗>𝑢_𝑖𝑙𝑥_𝑖,𝑧_𝑙𝜖,∀𝑙≠𝑗=Pr_𝑙−𝜖_𝑗<𝑢^∗_𝑖𝑗𝑥_𝑖,𝑧_𝑗−𝑢^∗_𝑖𝑙𝑥_𝑖,𝑧_𝑗,∀𝑙≠𝑗. (3) Here Pr stands for the probability operator generated by the distribution assumption on 𝝐. Thus, the probability that 𝑖 votes for 𝑗 is given by the probability that 𝑢_𝑖𝑗(𝑥_𝑖,𝑧_𝑗)>𝑢_𝑖𝑗(𝑥_𝑖,𝑧_𝑙) for all 𝑙≠𝑗∈𝑃, that is, that 𝑖 gets a higher utility from 𝑗 than from any other party.

Train [32] showed that when the error vector 𝝐 has a Type I extreme value distribution, the probability 𝜌_𝑖𝑗(𝑧) has a Multinomial Logit (MNL) specification and can be estimated. Thus, for each voter 𝑖 and party 𝑗, the probability that voter 𝑖 chooses party 𝑗 at the vector 𝐳 is given by 𝜌_𝑖𝑗(𝐳)=exp𝑢^∗_𝑖𝑗𝑥_𝑖,𝑧_𝑗∑^𝑝_𝑘=1exp𝑢^∗_𝑖𝑘𝑥_𝑖,𝑧_𝑘. (4)

Voters decisions are stochastic in this framework. (See, for example, the models of McKelvey and Patty [14]. Note that there is a problem with the independence of irrelevant alternatives assumption (IIA) which can be avoided using a probit model [33]. However, Quinn et al. [34] have shown that probit and logit models tend to give very similar results. Indeed the results given here for the logit model carry through for probit, though they are less elegant.) Even though parties cannot perfectly anticipate how voters will vote, they can estimate the expected vote share of party 𝑗 as the average of these probabilities as follows: 𝑉_𝑗(𝐳)=1𝑛_𝑖∈𝑁𝜌_𝑖𝑗(𝐳). (5) We assume a party’s objective is to find the position that maximizes its expected vote share, as desired by “Downsian” opportunists. On the other hand, the party may desire to adopt a position that is preferred by the base of the party supporters, namely, the “guardians” of the party, as suggested by Roemer [35].

We assume that parties can estimate how their vote shares would change if they marginally move their policy position. The Local Nash Equilibrium (LNE) is that vector 𝐳 of party positions such that no party may shift position by a small amount to increase its vote share. More formally a LNE is a vector 𝐳=(𝑧₁,…,𝑧_𝑗,…,𝑧_𝑝) such that each vote share 𝑉_𝑗(𝐳) is weakly locally maximized at the position 𝑧_𝑗. To avoid problems with zero eigenvalues we also define a SLNE to be a vector that strictly locally maximizes 𝑉_𝑗(𝐳).

Using the estimated MNL coefficients we simulate these models and then relate any vector of party positions, 𝐳, to a vector of vote share functions 𝑉(𝐳)=(𝑉₁(𝐳),…,𝑉_𝑝(𝐳)), predicted by the particular model with 𝑝 parties. Moreover, we can examine whether in equilibrium parties position themselves at the electoral mean. (The electoral mean or origin is the mean of all voters’ positions, (1/𝑛)∑𝑥_𝑖 normalized to zero, so that (1/𝑛)∑𝑥_𝑖=0.) We call this vector the electoral mean.

Given the vector of policy position 𝐳, and since the probability that voter 𝑖 votes for party 𝑗 is given by (4), the impact of a marginal change in 𝑗’s position on the probability that 𝑖 votes for 𝑗 is then 𝑑𝜌_𝑖𝑗(𝐳)𝑑𝑧_{𝑗𝐳−𝑗}=2𝛽𝜌_𝑖𝑗1−𝜌_𝑖𝑗𝑥_𝑖−𝑧_𝑗, (6) where 𝐳_−𝑗 indicates that we are holding the positions of all parties but 𝑗 is fixed. The effect that 𝑗’s change in position has on the probability that 𝑖 votes for 𝑗 depends on the weight given to the policy differences with parties, 𝛽; on how likely is 𝑖 to vote for 𝑗, 𝜌_𝑖𝑗, and for any other party, (1−𝜌_𝑖𝑗) and on how far apart 𝑖’s ideal policy is from 𝑗’s, (𝑥_𝑖−𝑧_𝑗).

From (5), party 𝑗 adjusts its position to maximize its expected vote share, that is, 𝑗’s first order condition is 𝑑𝑉_𝑗(𝐳)𝑑𝑧_{𝑗𝑧−𝑗}=1𝑛_𝑖∈𝑁𝑑𝜌_𝑖𝑗𝑑𝑧_𝑗=1𝑛_𝑖∈𝑁2𝛽𝜌_𝑖𝑗1−𝜌_𝑖𝑗𝑥_𝑖−𝑧_𝑗=0, (7) where the third term follows after substituting in (6). The FOC for party 𝑗 in (7) is satisfied when _𝑖∈𝑁𝜌_𝑖𝑗1−𝜌_𝑖𝑗𝑥_𝑖−𝑧_𝑗=0 (8) so that the candidate for party 𝑗’s vote maximizing policy (See Schofield [36] for the proof.) is 𝑧^𝐶_𝑗=_𝑖∈𝑁𝛼_𝑖𝑗𝑥_𝑖,where𝛼_𝑖𝑗≡𝜌_𝑖𝑗1−𝜌_𝑖𝑗∑_𝑖∈𝑁𝜌_𝑖𝑗1−𝜌_𝑖𝑗, (9) where 𝛼_𝑖𝑗 represents the weight that party 𝑗 gives to voter 𝑖 when choosing its candidate vote maximizing policy. This weight depends on how likely is 𝑖 to vote for 𝑗, 𝜌_𝑖𝑗, and for any other party, (1−𝜌_𝑖𝑗) relative to all voters. (For example, if all voters are equally likely to vote for 𝑗, say with probability 𝑣, then the weight party 𝑗 gives to voter 𝑖 in its vote maximizing policy is 1/𝑛; that is, the weight 𝑗 gives each voter is just the inverse of the population size.) Note that 𝛼_𝑖𝑗 may be nonmonotonic in 𝜌_𝑖𝑗. To see this exclude voter 𝑖 from the denominator of 𝛼_𝑖𝑗. When ∑_{𝑎∈𝑁−𝑖}𝜌_𝑎𝑗(1−𝜌_𝑎𝑗)<2/3 then 𝛼_𝑖𝑗(𝜌_𝑖𝑗=0)<𝛼_𝑖𝑗(𝜌_𝑖𝑗=1)<𝛼_𝑖𝑗(𝜌_𝑖𝑗=1/2). Thus, if 𝑖 will for sure vote for 𝑗, 𝑖 receives a lower weight in 𝑗’s candidate position than a voter who will only vote for 𝑗 with probability 1/2 (an “undecided” voter). Party 𝑗 caters then to “undecided” voters by giving them a higher weight in 𝑗’s policy weight and thus a higher weight on its position. This is the most common case. When ∑_{𝑎∈𝑁−𝑖}𝜌_𝑎𝑗(1−𝜌_𝑎𝑗)>2/3, then 𝛼_𝑖𝑗 increases in 𝜌_𝑖𝑗. If 𝑗 expects a large enough vote share (excluding voter 𝑖), it gives a core supporter (a voter who votes for sure for 𝑗) a higher weight in its policy position than it gives other voters as there is no risk of doing so. The weights 𝛼_𝑖𝑗 are endogenously determined in the model.

Note that since voter 𝑖’s utility depends on how far 𝑖 is from party 𝑗, the probability that 𝑖 votes for 𝑗 given in (4) and the expected vote share of the party given in (5) are influenced by the voters and parties positions in the policy space. That is, in the empirical models estimated below, the positions of voters and parties in the policy space, together with the valence estimates, influence voters electoral choices.

Recall that we are interested in finding whether parties converge to or diverge from the electoral mean. Suppose that all parties locate at the same position, 𝑧_𝑘=𝑧 for all 𝑘∈𝑃. Thus, from (2) we see that 𝑢^∗_𝑖𝑘𝑥_𝑖,𝑧−𝑢^∗_𝑖𝑗𝑥_𝑖,𝑧=𝜆_𝑘−𝜆_𝑗, (10) so the probability that 𝑖 votes for 𝑗 in (4) is given by 𝜌_𝑖𝑗(𝐳)=1∑^𝑝_𝑘=1exp𝑢^∗_𝑖𝑘𝑥_𝑖,𝑧_𝑘−𝑢^∗_𝑖𝑗𝑥_𝑖,𝑧_𝑗=^𝑝_𝑘=1exp𝜆_𝑘−𝜆_𝑗⁻¹. (11) Clearly, in this case, 𝜌_𝑖𝑗(𝐳)=𝜌_𝑗(𝐳) is independent of voter 𝑖’s ideal point. Thus, from (9), the weight given by 𝑗 to each voter is also independent of voter 𝑖’s position and given by 𝛼_𝑗≡𝜌_𝑗1−𝜌_𝑗∑_𝑖∈𝑁𝜌_𝑗1−𝜌_𝑗=1𝑛, (12) so that 𝑗 gives each voter equal weight in its policy position. In this case, from (9), 𝑗’s candidate position is 𝑧^𝐶_𝑗=1𝑛_𝑖∈𝑁𝑥_𝑖, (13) that is, 𝑗’s candidate position is to locate at the electoral mean which we have placed at the electoral origin. Let 𝐳₀=(0,…,0) be the vector of party positions when all parties are at the electoral mean.

Moreover, as (11) indicates when parties locate at the mean 𝐳₀, only valence differences between parties matter in voters’ choices. The probability that a generic voter votes for party 1 (the party with the lowest valence) is 𝜌₁≡𝜌₁𝐳₀=^𝑝_𝑘=1exp𝜆_𝑘−𝜆₁⁻¹. (14)

Using this spatial model, Schofield [9] proved a Valence Theorem determining whether vote maximizing parties locate at the mean. The theorem showed that the spatial model is characterized by a convergence coefficient given by 𝑐≡𝑐𝝀,𝛽,𝜎²=2𝛽1−2𝜌₁𝜎². (15) The convergence coefficient depends on 𝛽, the weight given to policy differences; on 𝜌₁, the probability that a generic voter votes for the lowest valence party at the vector 𝐳₀ and on 𝜎², the electoral variance given by 𝜎²≡trace(∇), (16)where ∇ is the symmetric 𝑤×𝑤   electoral covariance matrix. (∇ is simply a description of the distribution of voter preferred points taken about the electoral mean.)

The convergence coefficient increases in 𝛽 and 𝜎² (and on its product 𝛽𝜎²) and decreases in 𝜌₁. As (14) indicates 𝜌₁ decreases if the valence differences between party 1 and the other parties increases, that is, when the difference between 𝜆₁ and {𝜆₂,…,𝜆_𝑝} increases.

The Valence Theorem allows us to characterize polities according to the value of their convergence coefficient. The theorem states that when the sufficient condition for convergence to the electoral mean is met, that is, when 𝑐<1, the LNE is one where all parties adopt the same position at the mean of the electoral distribution. A necessary condition, for convergence to the electoral mean is that 𝑐<𝑤, where 𝑤 is the dimension of the policy space. If 𝑐≥𝑤, then there may exist a nonconvergent LNE. Note that in this case, there may indeed be no LNE. However, there will exist a mixed strategy Nash equilibrium (MNE). In either of these two cases we expect at least one party will diverge from the electoral mean.

Note that 𝑐 is dimensionless, because 𝛽𝜎² has no dimension. In a sense 𝛽𝜎² is a measure of the polarization of the preferences of the electorate. Moreover, 𝜌₁ in (14) is a function of the distribution of beliefs about the competence of party leaders, which is a function of the difference (𝜆_𝑘−𝜆₁).

When some parties have a low valence, so the probability that a generic voter votes for party 1 (with the lowest valence when all parties locate at the origin), 𝜌₁ in (14) will tend to be small because the valence differences between party 1 and the other parties is sufficiently large. Thus, vote maximizing parties will not all converge to the electoral mean. In this case 𝑐 will be close to 2𝛽𝜎². If 2𝛽𝜎² is large because, for example, the electoral variance is large, then 𝑐 will be large, suggesting 𝑐>𝑤. In this case, the low valence party has an incentive to move away from the origin to increase its vote share. This implies the existence of a centrifugal force pulling some parties away from the origin.

Thus, for 𝛽𝜎² sufficiently large so that 𝑐≥𝑤, we expect parties to diverge from the electoral center. Indeed, we expect those parties that exhibit the lowest valence to move further away from the electoral center, implying that the centrifugal force on parties will be significant. Thus, in fragmented polities with a polarized electorate, the nature of the equilibrium tends to maintain this centrifugal characteristic.

On the contrary, in a polity where there are no very small or low valence parties, 𝜌₁ will tend to 1/2 and so 𝑐 will be small. In a polity with small 𝛽𝜎² and with low valence differences, so that 𝑐<1, we expect all parties to converge to the center. In this case, we expect this centripetal tendency to be maintained.

The convergence coefficient is a way of characterizing the Hessian (the 𝑤 by 𝑤  second derivatives of the vote share function) of party 1 with the lowest valence. The Hessian of the vote share function of party 1 is given by the characteristic matrix 𝐶₁=2𝛽1−2𝜌₁∇−𝐼. (17) Here 𝐼 is a 𝑤 by 𝑤 identity matrix and the other terms are as before. The eigenvalues of 𝐶₁ determine whether the vote share function of party 1 will be at a maximum, minimum, or at a saddlepoint at the electoral mean. If 𝐶₁ shows that party 1 is at a minimum or at a saddlepoint at the mean then party 1 has an incentive to locate away from the mean to increase its vote share. When all parties are at the mean and 𝑐<1, then all eigenvalues of the Hessian of the vote share function of the lowest valence party are negative indicating that the vote share function is at a maximum. The LNE must then be at the electoral mean.

For an arbitrary dimension, 𝑤, if 𝑐(𝝀,𝛽,𝜎²)≤1 in (15), then trace(𝐶₁)<0. In the two-dimensional case, if 𝑐(𝝀,𝛽,𝜎²)<1, then det(𝐶₁) must be positive, implying that both eigenvalues of 𝐶₁ are negative. It then follows that all {𝐶_𝑗} have negative eigenvalues, giving a SLNE and thus an LNE at the electoral mean. (This result follows from the application of the triangle inequality to the determinant. A parallel result can be obtained in more than two dimensions.)

The Valence Theorem asserts that if 𝑐(𝝀,𝛽,𝜎²)>𝑤 then the party with the lowest valence has an incentive to move away from the electoral mean to increase its vote share. When this is the case then other low valence parties may also find it advantageous to vacate the center. The value of the convergence coefficient, together with the analysis of the Hessians of the low valence parties, allows us to identify which parties have an incentive to move away from the electoral mean. The convergence coefficient then gives an easy and intuitive way to identify whether a low valence party should vacate the electoral mean.

In the next section, we estimate the convergence coefficient of various elections in different countries.

3. MNL Models of the Elections of Various Countries

We use the framework of the spatial model presented in Section 2 as a unifying methodology that allows us to study convergence across elections, countries, and political regimes. The Valence Theorem leads to the convergence coefficient of the election, a summary statistic that determines whether parties converge to or diverge from the electoral mean. Using this formal multinomial (MNL) spatial model, we now estimate the convergence coefficient for the elections in various countries. For each MNL estimation we choose a baseline party and normalize its coefficients to zero, then estimate the coefficients of all other parties relative to those of the base party. Using these coefficients we estimate the convergence coefficient and the characteristic matrix of the low valence parties to determine whether these parties converge to or diverge from the electoral mean in each election for each country. (These elections were studied in depth elsewhere. In this paper, we present only the calculations leading to the convergence coefficient and estimate the confidence intervals for the convergence coefficients that were not provided in earlier work.)

We study convergence under three political regimes (plurality, proportional representation, and anocracy) and group countries according to the similarities of their political regimes. Under plurality rule, we examine elections in two Anglo-Saxon countries: the US and the UK; under proportional representation we study Israel, Turkey, and Poland; and under anocracy, Georgia, Russia, and Azerbaijan. Since we use the same unifying methodology for all countries we present the methodology for the first elections in detail then condense the analysis to its basic components for the remaining countries. For each country we give a general description of the analysis and direct the reader to the full analysis of each election in the detailed country paper. We summarize the results across countries in various tables.

3.1. Convergence in Plurality Systems

We begin our analysis by examining the United States and the United Kingdom. Elections in these countries are carried out under plurality rule. We show that the electoral system in these countries produces relatively low convergence coefficients. (Relative to the convergence coefficient of other countries included in this study. In Section 4 we discuss how the values of the convergence coefficient are related to the political systems under which the countries operate.)

3.1.1. The 2000, 2004, and 2008 Elections in the United States

We construct stochastic models of the 2000, 2004, and 2008 US presidential elections using survey data taken from the American National Election Surveys (ANES). The factor analysis done on ten survey questions taken from the ANES (See Schofield et al. [30, 31] for the list of survey questions and the factor loadings and the full analysis of the US elections.) led us to conclude that voters preferences can be represented along the economic (𝐸=𝑥-axis) and social (𝑆=𝑦-axis) dimensions for all three elections. Voters located on the left of the economic axis are pro-redistribution. The social axis is determined by attitudes to abortion and gays. We interpreted greater values along this axis to mean more support for certain civil rights. Using the factor loadings we estimated each voter’s position in these two dimensions. Figures 1, 2, and 3 give a smoothing of the estimated voter distribution of the 2000, 2004, and 2008 elections, respectively.

Figure 1 

Distribution of voter ideal points and candidate positions in the 2000 US election.

Figure 2 

Distribution of voter ideal points and candidate positions in the 2004 US election.

Figure 3 

Distribution of voter ideal points and candidate positions in the 2008 US election.

Voters’ ideal points in the 2000 US election are characterized by the following electoral covariance matrix: ∇^US₂₀₀₀=𝜎²_𝐸=0.58𝜎_𝐸𝑆𝜎=−0.20_𝐸𝑆=−0.20𝜎²_𝑆=0.59. (18) The trace of electoral covariance matrix is 𝜎²_US2000≡trace(∇²⁰⁰⁰_US)=𝜎²_𝐸+𝜎²_𝑆=1.17. Given the negative covariance between these two dimensions, 𝜎_𝐸𝑆=−0.20, the correlation between these two factors is −0.344.

Using the spatial model presented in Section 2, we estimated the MNL model of the 2000 election. The coefficients for the US 2000 shown in Table 1 are 𝜆^US2000_rep=−0.43,𝜆^US2000_dem≡0.0,𝛽^US₂₀₀₀=0.82. (19) Bush’s competence valence, 𝜆^US2000_rep=−0.43, measures the common perception that voters in the sample have on Bush’s ability to govern and represents the nonpolicy component in the voter’s utility function in (2). As seen in Table 1, for the 2000 election Bush has a statistically significant lower valence than Gore, the democratic (baseline) candidate. Bush’s negative valence is an indication that voters regarded him as less able to govern than Gore, once policy differences are taken into account.

Table 1 

MNL spatial model for countries with plurality systems.

To find the convergence coefficient for this election, we assume that all parties locate at the electoral mean so that parties differ only in their valence terms (see Section 2). We can use (14) and the coefficients in (19) to estimate the probability that a typical US voter chooses to vote for the low valence Republican candidate, when both Bush and Gore locate at origin, 𝐳₀; that is, 𝜌^US2000_rep=²_𝑘=1exp𝜆^US_𝑘²⁰⁰⁰−𝜆^US2000_rep⁻¹=1+exp(0.43)⁻¹=0.40. (20) We found the estimate for 𝜌^US2000_rep using the MNL valence estimates. Note that since the central estimates of 𝝀=(𝜆₁,…,𝜆_𝑝) given by the MNL regressions depend on the sample of voters surveyed then so does 𝜌₁. Thus, to make inferences from empirical models we need the 95% confidence bounds of 𝜌₁. In the introduction of the appendix we derive the methodology used to find the confidence bounds of 𝜌₁. The bounds of 𝜌₁ are calculated in Appendix A.1.

The results indicate that in the 2000 election, both candidates found it in their best interest to locate at the electoral mean. To see this, we compute the convergence coefficient using (15) and the electoral covariance matrix in (18) ∇²⁰⁰⁰_US to determine whether the two parties converge to, or diverge from, the electoral mean.

Using (19) and (20) we have that 2𝛽^US₂₀₀₀(1−2𝜌^US2000_rep)=2×0.82×0.2=0.328 and from (18) the trace is 𝜎²_US2000=1.17 so that using (15) the convergence coefficient for 2000 US election is 𝑐²⁰⁰⁰_US≡2𝛽^US₂₀₀₀1−2𝜌^US2000_rep𝜎²_US2000=0.328×1.17=0.384. (21) Appendix A.1 shows that 𝑐²⁰⁰⁰_US is significantly less than 1 implying that 𝑐²⁰⁰⁰_US meets the sufficient and thus necessary condition for convergence to the electoral mean given in Section 2.

To check whether Bush, the low valence candidate, has an incentive to stay at the electoral origin, 𝐳₀, that is, whether Bush’s vote share function is at a maximum at 𝐳₀, we use the Hessian or characteristic matrix (of second order conditions) of Bush’s vote share function using (17) at 𝐳₀ as follows: 𝐶^US2000_rep=2𝛽^US₂₀₀₀1−2𝜌^US2000_rep∇^US₂₀₀₀=.−𝐼=0.3280.58−0.20−0.200.59−𝐼−0.81−0.06−0.06−0.81 (22) Because the characteristic matrix for Bush 𝐶^US2000_rep is estimated using the MNL coefficients of the 2000 US sample, 𝐶^US2000_rep depends on the sample of voters surveyed. The confidence bounds on 𝐶^US2000_rep in Appendix A.1 suggest that if Bush positions himself at the electoral origin, then with probability exceeding 95%, his vote share function would be at a maximum. We infer that, with probability exceeding 95%, the origin is an LNE for the spatial model for the 2000 US election. The valence differences between Bush and Gore are not large enough to cause either of them to move from the origin. The unique local Nash equilibrium was one where both candidates converge to the electoral origin in order to maximize their vote shares.

All the components needed to derive the convergence coefficient for 2000 US election and its confidence bounds are summarized in Table 2.

Table 2 

The convergence coefficient in plurality systems.

Bush faced Kerry as the democratic candidate in the 2004 US election. The distribution of voters in 2004 gives the following electoral covariance matrix along the economic and social dimensions: ∇^US₂₀₀₄=𝜎²_𝐸=0.58𝜎_𝐸𝑆𝜎=−0.177_𝐸𝑆=−0.177𝜎²_𝑆=0.59. (23) While the covariance between economic and social axes differs, the trace 𝜎²_US2004=trace(∇²⁰⁰⁴_US)=𝜎²_𝐸+𝜎²_𝑆=1.17 is similar to that in the 2000 US election.

From Table 1, the MNL estimates of the spatial model for the 2004 US election are 𝜆^US2004_rep=−0.43,𝜆^US2004_dem≡0.0,𝛽^US₂₀₀₄=0.95. (24) Bush has a significantly lower valence (𝜆^US2004_rep=−0.43) than Kerry (𝜆^US2004_dem≡0.0), the baseline candidate.

From (14) the probability that a US voter chooses Bush, the low valence candidate, when both Bush and Kerry are at the electoral origin, 𝐳₀, is 𝜌^US2004_rep=²_𝑘=1exp𝜆^US_𝑘²⁰⁰⁴−𝜆^US2004_rep⁻¹=[1+exp(0.43)]⁻¹=0.40. (25) The confidence bounds for 𝜌^US2004_rep are given in Appendix A.1. Since Bush’s valence, relative to that of his opponent, was similar in the two elections, it is not surprising that the probability of voting Republican is similar in the two elections, compare (20) and (25). From (15), 2𝛽^US₂₀₀₄(1−2𝜌^US2004_rep)=2×0.95×0.2=0.38 and 𝜎²_US2004=1.17, so that the convergence coefficient of the 2004 election is 𝑐²⁰⁰⁴_US=2𝛽^US₂₀₀₄1−2𝜌^US2004_rep𝜎²_US2004=0.38×1.19=0.45. (26) Since 𝑐²⁰⁰⁴_US=0.45 is significantly less than 1 (see Appendix A.1), the sufficient condition for convergence given in Section 2 is met. Moreover, from (17) Bush’s characteristic matrix is 𝐶^US2004_rep=2𝛽^US₂₀₀₄1−2𝜌^US2004_rep∇^US₂₀₀₄=.−𝐼=0.380.53−0.18−0.180.66−𝐼−0.80−0.06−0.06−0.75 (27) If Bush positions himself at the electoral origin, then with probability exceeding 95% (see Appendix A.1), his vote share function would be at a maximum. Bush, the low valence candidate, has then no incentive to move from the origin, 𝐳₀. With probability exceeding 95%, the mean is an LNE for model of the 2004 US election.

Our analysis suggests that Obama’s victory over McCain in the 2008 US election was the result of an overall shift in the relative valences of the Democratic and Republican candidates as compared to those of the candidates in the 2000 and 2004 elections. The electoral covariance matrix for the sample in 2008 along the economic and social dimensions is ∇^US₂₀₀₈=𝜎²_𝐸=0.80𝜎_𝐸𝑆𝜎=−0.127_𝐸𝑆=−0.127𝜎²_𝑆=0.83. (28) Relative to the two previous elections the “variance” of the electoral distribution 𝜎²_US2008=trace(∇^US₂₀₀₈)=𝜎²_𝐸+𝜎²_𝑆=1.63 increased, while the covariance between these dimensions 𝜎_𝐸𝑆=−0.127 decreased.

The MNL estimates of the spatial model given in Table 1 for the 2008 US election are 𝜆^US2008_rep=−0.84,𝜆^US2008_dem≡0.0,𝛽^US₂₀₀₈=0.85. (29) Obama, the baseline candidate, has a significantly higher valence than McCain.

From (14) the probability that a voter chooses McCain, when both candidates are at the origin, 𝐳₀, is 𝜌^US2008_rep=²_𝑘=1exp𝜆^US_𝑘²⁰⁰⁸−𝜆^US2008_rep⁻¹=1+exp(0.84)⁻¹=0.30. (30) From (15), 2𝛽²⁰⁰⁸_US(1−2𝜌^US2008_dem)=2×0.85×0.4=0.68, and 𝜎²_US2008=1.63, so the convergence coefficient is 𝑐²⁰⁰⁸_US=2𝛽^US₂₀₀₈1−2𝜌^US2008_dem𝜎²_US2008=0.68×1.63=1.11. (31) Appendix A.1 shows that 𝑐²⁰⁰⁸_US=1.11 is significantly greater than 1 and significantly less than 2. The Valence Theorem then states that the necessary but not the sufficient condition for convergence has been met. To check whether the low valence candidate, McCain, has an incentive to move from the electoral mean, we examine McCain’s characteristic matrix using (17) to get 𝐶^US2008_rep=2𝛽^US₂₀₀₈1−2𝜌^US2008_rep∇^US₂₀₀₈=.−𝐼=0.680.80−0.127−0.1270.83−𝐼−0.46−0.086−0.086−0.44 (32) With probability exceeding 95% (see Appendix A.1) McCain’s vote share function is at a maximum when he locates at the origin, and thus has no incentive to move. Thus, with probability exceeding 95%, the electoral origin is an LNE for the spatial model for the 2008 US election.

In conclusion, Table 2 illustrates that the convergence coefficient varies across elections in the same country even when there are only two parties. This is to be expected as from (15) the convergence coefficient depends on the “variance” of the electoral distribution, 𝜎²=trace(∇); on the weight voters give to differences with party’s policies, 𝛽; and on the probability that a voter chooses the party with the lowest valence, 𝜌₁. The electoral distributions of the 2000 and 2004 are quite similar, as can be seen by comparing (18) and (23). Voters’ preferences had however substantially changed by 2008, see (28). The electoral variance along both axes increased relative to 2000 and 2004. While the 2000 and 2004 convergence coefficients are indistinguishable from each other, the 2008 coefficient is significantly different from that in 2000 and 2004. In spite of these differences, candidates in all three elections had no incentive to move from the origin.

3.1.2. The 2005 and 2010 Elections in Great Britain

We study the 2005 and 2010 elections in the UK using the British Election Study (BES). (The full analysis of the 2005 and 2010 elections in Great Britain can be found in Schofield et al. [37].) The factor analysis conducted on the questions of the two surveys led us to conclude that the same two dimensions mattered in voter choices in the two elections. The first factor deals with issues on “EU membership,” “Immigrants,” “Asylum seekers,” and “Terrorism.” A voter who feels strongly about nationalism has a high value in the nationalism dimension (Nat=𝑥-axis). Items such as “tax/spend,” “free market,” “international monetary transfer,” “international companies,” and “worry about job loss overseas” have strong influence in the economic (𝐸=𝑦-axis) dimension with higher values indicating a promarket attitude. Figures 4 and 5 present the smoothed electoral distribution obtained from these analyses for the 2005 and 2010 elections.

Figure 4 

Electoral distribution and estimated party positions in Britain in 2005.

Figure 5 

Voter and party positions in Britain in 2010.

The electoral covariance matrix for the 2005 UK election is ∇^UK₂₀₀₅=𝜎²_Nat=1.646𝜎_Nat𝐸𝜎=0.00_𝐸Nat=0.067𝜎²_𝐸=3.961, (33) where 𝜎²_UK2005≡trace(∇^UK₂₀₀₅)=𝜎²_Nat+𝜎²_𝐸=5.607.

From Table 1, the MNL estimates of the spatial model for the 2005 UK are 𝜆^UK2005_Lab=0.52,𝜆^UK2005_Con𝜆=0.27,_UK2005Lib≡0.0,𝛽^UK₂₀₀₅=0.15. (34) Both the Labour (Lab) and the Conservative (Con) parties had a significantly higher valence than the Liberal Democrats (Lib), the baseline party.

From (14), the probability that a voter chooses the Liberal Democratic Party, the lowest valence party, when all parties locate at the origin, 𝐳₀, is 𝜌^UK2005_Lib=³_𝑘=1exp𝜆^UK_𝑘²⁰⁰⁵−𝜆^UK2005_Lib⁻¹=1+exp(0.52)+exp(0.27)⁻¹=0.25. (35)

Given that 2𝛽^UK₂₀₀₅(1−2𝜌^UK2005_Lib)=2×0.15×0.5=0.15 and since 𝜎²_UK2005=5.607 in (33), from (15) the convergence coefficient, in Table 2, is 𝑐²⁰⁰⁵_UK=2𝛽^UK₂₀₀₅1−2𝜌^UK2005_Lib𝜎²_UK2005=0.15×5.607=0.84. (36) Appendix A.1 shows that 𝑐²⁰⁰⁵_UK is significantly less than 1 and thus meets the sufficient and necessary conditions for convergence given in Section 2. From (17) the characteristic matrix of the Liberal Democratic Party is 𝐶^2005UK_Lib=2𝛽^UK₂₀₀₅1−2𝜌^UK2005_Lib∇^UK₂₀₀₅=.−𝐼=0.151.6460.00.0673.961−𝐼−0.750.00.01−0.406 (37) From the 95% confidence bounds in Appendix A.1, we conclude that if the LibDem locates at the origin, it is maximizing its vote share and has no incentive to vacate the center. Thus, with probability exceeding 95%, the origin is an LNE for the 2005 UK election.

The electoral covariance matrix for the 2010 UK election is ∇^UK₂₀₁₀=𝜎²_Nat=0.601𝜎_Nat𝐸𝜎=0.067_𝐸Nat=0.067𝜎²_𝐸=0.861, (38) where 𝜎²_UK2010≡trace(∇^UK₂₀₁₀)=1.462, lower than in 2005.

From Table 1, the MNL estimates of the spatial model of the 2010 election are 𝜆^UK2010_Lab=−0.04,𝜆^UK2010_Con𝜆=0.17,_UK2010Lib≡0.0,𝛽^UK₂₀₁₀=0.86. (39) Given the great popular discontent with Gordon Brown, the Labour leader, heading into the 2010 election, it is not surprising to find that both Conservatives and Liberal Democrats (the base party) had significantly higher valences than Labour.

From (14) the probability that a voter chooses Labour, when all parties locate at the origin, 𝐳₀, is 𝜌^UK2010_Lab=³_𝑘=1exp𝜆^UK_𝑘²⁰¹⁰−𝜆^UK2010_Lab⁻¹=1+exp(0.21)+exp(0.04)⁻¹=0.319. (40) Since 2𝛽^UK₂₀₁₀(1−2𝜌^UK2010_Lab)=2×0.86×0.362=0.622 and 𝜎²_UK2010=1.462 in (38), from (15) the convergence coefficient, in Table 2, is 𝑐²⁰¹⁰_UK=2𝛽^UK₂₀₁₀1−2𝜌²⁰¹⁰_Lab𝜎²_UK2010=0.622×1.462=0.91. (41) The convergence coefficient 𝑐²⁰¹⁰_UK=0.91 is significantly less than 1 (see Appendix A.1), meeting the sufficient and thus necessary condition for convergence. From (17), Labour’s characteristic matrix is 𝐶^UK2010_Lab=2𝛽^UK₂₀₁₀1−2𝜌^UK2010_Lab∇^UK₂₀₁₀=.−𝐼=0.6220.6010.0670.0670.861−𝐼−0.630.0420.042−0.46 (42) If Labour, the low valence party, locates at the origin, then with probability exceeding 95%, its vote share function is at a maximum (see Appendix A.1) giving it no incentive to move from the mean. Thus, with probability exceeding 95%, the electoral origin is an LNE for the 2010 UK election.

The major shift in voters’ preferences between the two elections led to very different electoral outcomes as evidenced by the electoral covariance matrices in (33) and (38). Voter dissatisfaction with the governing Labour leader led to a dramatic decrease in his competence valence and on the probability of voting Labour. Even though the electoral variance fell in 2010 relative to 2005, the increase in the convergence coefficient meant that this lower variance was more than compensated by the lower probability of voting Labour in 2010. The analysis for the UK elections shows that the convergence coefficient reflects not only changes in the electoral distribution but also changes in voters’ valence preferences as the convergence coefficient of the 2005 election is substantially lower than the one for the 2010 election.

The analysis of these two Anglo-Saxon countries illustrate that even under plurality rule the convergence coefficient varies from election to election and from country to country. The analysis for the 2010 UK election highlights that candidates’ valences matter and that parties understand how their valence affects their electoral prospects and may adjust their positions to increase their votes. This section illustrates that under plurality, the convergence coefficient has low values that generally satisfy the necessary condition for convergence to the mean and is thus below the dimension of the policy space.

3.2. Convergence in Proportional Systems?

We now estimate the convergence coefficients for three parliamentary countries using proportional representation: Israel, Turkey, and Poland. As is well known, these countries are characterized by multiparty elections in which generally no party wins a legislative majority leading then to coalitions governments. This section shows that these countries are characterized by very high convergence coefficients.

3.2.1. The 1996 Election in Israel

In the 1996, as in previous elections, Israel had approximately nineteen parties attaining seats in the Knesset. (These include parties on the left, on the center, on the right, as well as religious parties. On the left there is Labor, Merets, Democrat, Communists and Balad; those on the center include Olim, Third Way, Center, Shinui; those on the right Likud, Gesher, Tsomet and Yisrael. The religious parties are Shas, Yahadut, NRP, Moledet, and Techiya.) There were small parties with 2 seats to moderately large parties such as Likud and Labor whose seat strengths lie in the range 19 to 44, out of a total of 120 Knesset seats. Since Likud and Labour compete for dominance of coalition government, these large parties must maximize their seat strength. Moreover, Israel uses a highly proportional electoral system with close correspondence between seat and vote shares. Thus one can consider vote shares as the maximand and for these parties.

Schofield et al. [30] performed a factor analysis of the surveys conducted by Arian and Shamir [38] for the 1996 Israeli election. The two dimensions identified by the factor analysis were Security (𝑆=𝑥-axis) and Religion (𝑅=𝑦-axis). “Security” refers to attitudes toward peace initiatives; “religion” to the significance of religious considerations in government policy. A voter on the left of the security axis is interpreted as supporting negotiations with the PLO, while higher values on the religious axis indicates support for the importance of the Jewish faith in Israel. The distribution of voters is shown in Figure 6.

Figure 6 

Party positions and voter distribution in Israel in the 1996 election.

Voter distribution along these two axes gives the following covariance matrix: ∇^I_!996=𝜎²_𝑆=1.00𝜎_𝑆𝑅𝜎=0.591_𝑅𝑆=0.591𝜎²_𝑅=0.732, (43) giving a “variance” of 𝜎²_I1996≡trace(∇^I_!996)=1.732.

Only the seven largest parties are included in the MNL estimation. These include Likud, Labor, NRP, Moledat, Third Way (TW), and Shas with Meretz being the base party. From Table 2, the MNL coefficients for the 1996 election in Israel (I) are 𝜆^I1996_Lik=0.78,𝜆^I1996_Lab𝜆=0.999,_I1996NRP=−0.626,𝜆^I1996_MO𝜆=−1.259,_I1996TW≡−2.291,𝜆^I1996_Shas𝜆=−2.023,_I1996Merezt≡0.0,𝛽^I₁₉₉₆=1.207. (44) The 𝛽-coefficient and the valence estimates for all parties are significantly nonzero. The two largest parties, Likud and Labour, have significantly higher valences than the other smaller parties with Third Way (TW) having the smallest valence.

From (14), the probability that an Israeli votes for TW, when all parties locate at the mean is 𝜌^I1996_TW=⁷_𝑘=1exp𝜆^I_𝑗¹⁹⁹⁶−𝜆^I1996_TW⁻¹=1+𝑒^3.071+𝑒^3.29+𝑒^1.665+𝑒^1.032+𝑒^0.268+𝑒^2.291−1≃0.014. (45)

Given that 2𝛽^I₁₉₉₆(1−2𝜌^I1996_TW)=2×1.207×0.972=2.346 and since 𝜎²_I1996=1.732 from (43), then using (15) we compute the convergence coefficient for Israel, in Table 4, as 𝑐^I₁₉₉₆=2𝛽^I₁₉₉₆1−2𝜌^I1996_TW𝜎²_I1996=2.346×1.732=4.06. (46)

The 95% confidence intervals for 𝑐^I₁₉₉₆=4.06 in Appendix A.2 confirm that the necessary condition is not satisfied as 𝑐^I₁₉₉₆=4.06 is significantly higher than 2, the dimension of the policy space. Moreover, at the electoral mean the vote share function of Third Way is not at a maximum since its Hessian from (17) 𝐶^I1996_TW=2𝛽^I₁₉₉₆1−2𝜌^I1996_TW∇^I_!996=−𝐼=2.3461.000.5910.5910.732−𝐼1.3461.3861.3860.717 (47) shows that if TW locates at the mean its vote share function is at a saddlepoint since 𝐶^I1996_TW has one positive (2.453) and one negative (−0.39) eigenvalue. Appendix A.2 confirms that 𝐶^I1996_TW has one negative and one positive eigenvalue at both its lower and upper bounds. Thus, with a high degree of certainty TW deviates from the mean to maximize its votes and the electoral mean is not a LNE for the 1996 Israeli election.

3.2.2. The 1999 and 2002 Elections in Turkey

We used factor analysis of electoral survey data of Veri Arastima for TUSES to study the 1999 and 2002 Turkish elections. (See Schofield et al. [39] for details of the estimation.) The analysis indicates that voters made decisions in a two-dimensional space during the two elections. Voters who support secularism or “Kemalism” are placed on the left of the Religious (𝑅=𝑥) axis and those supporting Turkish nationalism (𝑁=𝑦) to the north. Figures 7 and 8 give the distribution of voters along these two dimensions surveyed in these two elections.

Figure 7 

Party positions and voter distribution in the 1999 Turkish election.

Figure 8 

Party positions and voter distribution in Turkey in 2002.

Minor differences between these two figures include the disappearance of the Virtue Party (FP) which was banned by the Constitutional Court in 2001 and the change of the name of the pro-Kurdish party from HADEP to DEHAP. (For simplicity, the pro-Kurdish party is denoted HADEP in the various figures and tables. Notice that the HADEP position in Figures 8 and 9 is interpreted as secular and nonnationalistic.) The most important change is the emergence of the new Justice and Development Party (AKP) in 2002, essentially substituting for the outlawed Virtue Party.

Figure 9 

Voter distribution and party-positions in Poland in 1997.

The parties included in the analysis of the 1999 election are the Democratic Left Party (DSP), the National Action party (MHP), the Vitue Party (VP), the Motherland Party (ANAP), the True Path Party (DYP), the Republican People’s Party (CHP), and the People’s Democratic Party (HADEP). A DSP minority government formed, supported by ANAP and DYP. This only lasted about 4 months and was replaced by a DSP-ANAP-MHP coalition, indicating the difficulty of negotiating a coalition compromise across the disparate policy positions of the coalition members.

In the 1999 election, the electoral covariance matrix along the Religious (𝑅) and Nationalism (𝑁) axes is ∇^T_!999=𝜎²_𝑅=1.20𝜎_𝑅𝑁𝜎=0.78_𝑁𝑅=0.78𝜎²_𝑁=1.14, (48) with 𝜎²_T1999≡trace(∇^T_!999)=2.34.

Using DYP as the base party, from Table 3, the 1999 MNL coefficients are 𝜆^T1999_FP=−0.16,𝜆^T1999_MHP𝜆=0.66,_T1999DYP≡0.0,𝜆^T1999_HADEP𝜆=−0.071,_T1999ANAP=0.34,𝜆^T1999_CHP𝜆≡0.73,_T1999DSP=0.72,𝛽^T₁₉₉₉=0.38. (49) The 𝛽-coefficient and the valence estimates of DSP and MHP and CHP are significantly nonzero. The probability that a Turkish voter chooses FP with lowest valence in 1999, when all parties locate at the mean, 𝜌^T1999_FP in (14), is 𝜌^T1999_FP=⁷_𝑘=1exp𝜆^T_𝑗¹⁹⁹⁹−𝜆^T1999_FP⁻¹=1+𝑒^0.82+𝑒^0.16+𝑒^0.09+𝑒^0.5+𝑒^0.89+𝑒^0.88−1≃0.08. (50)

Table 3 

MNL spatial model for countries with proportional systems.

Table 4 

The convergence coefficient in proportional systems.

Given that 2𝛽^T₁₉₉₉(1−2𝜌^T1999_FP)=2×0.38×0.84=0.64 and since 𝜎²_T1999=2.34 in (48), then using (15), Turkey’s convergence coefficient in 1999, in Table 4, is 𝑐^T₁₉₉₉=2𝛽^T₁₉₉₉1−2𝜌^T1999_FP𝜎²_T1999=0.64×2.34=1.49. (51) The convergence coefficient is significantly higher that 1 and significantly lower than 2 (see Appendix A.2). From (17) FP’s Hessian at the origin is 𝐶^T1999_FP=2𝛽^T₁₉₉₉1−2𝜌^T1999_FP∇^T_!999=.−𝐼=0.641.200.780.781.14−𝐼−0.240.4480.448−0.27 (52) When at the electoral origin, FP’s characteristic function shows that its vote share function is at a saddlepoint as the eigenvalues of 𝐶^T1999_FP are −0.74 with minor eigenvector (+1−1.116) and +0.23 with major eigenvector (+1,+0.896). Moreover, as seen in Appendix A.2, the 95% confidence bounds show that at the lower bound of 𝐶^T1999_FP FP has no incentive to move but it does at the upper bound. Since FP wants to move at the central estimate of 𝐶^T1999_FP in (52) it is probable that in general FP wants to move away from the mean to increase its vote share. Moreover, since the convergence coefficient is significantly greater than 2, then with a high degree confidence, the electoral mean cannot be a LNE for Turkey in 1999.

The electoral covariance matrix of the 2002 Turkish election is ∇^T₂₀₀₂=𝜎²_𝑅=1.18𝜎_𝑅𝑁𝜎=0.74_𝑁𝑅=0.74𝜎²_𝑁=1.15, (53) with 𝜎²_T2002=trace(∇^T₂₀₀₂)=2.33.

Note that the covariance matrix of 1999 in (48) and that of 2002 in (53) suggest few changes in the distribution of voters between these two election. Figures 8 and 9 suggest that there were few changes in party positions between these two elections. The basis of support for the AKP may be regarded as similar to that of the banned FP, suggesting that the leader of this party changed the party’s position on the religion axis, adopting a much less radical position. One would think of this as generating political stability in Turkey. Yet, between 1999 and 2002, Turkey experienced two severe economic crises and in 2002, a 10% electoral cut-off rule was instituted. The crises and the cut-off rule changed the political landscape in Turkey. In the 2002 election, seven parties obtained less than 10% of the vote and won no seats. The AKP won 34% of the vote, and due to the cut-off rule, obtained a majority of the seats (363 out of 550).

Our analysis reflects this change in the political landscape. Using DYP as the base party, from Table 3, the 2002 MNL coefficients are 𝜆^T2002_ANAP=−0.31,𝜆^T2002_MHP𝜆=−0.12,_T2002DYP≡0.0,𝜆^T2002_HADEP𝜆=0.43,_T2002AKP=0.78,𝜆^T2002_CHP≡1.33,𝛽^T₂₀₀₂=1.52. (54) The 𝛽-coefficient and the valences of AKP and CHP are significantly nonzero with ANAP having the lowest valence. The probability of voting ANAP, when parties locate at the mean, 𝜌^T20029_ANAP in (14), is 𝜌^T2002_ANAP=⁶_𝑘=1exp𝜆^T_𝑗²⁰⁰²−𝜆^T2002_ANAP⁻¹=1+𝑒^0.19+𝑒^0.31+𝑒^0.74+𝑒^1.09+𝑒^1.164−1≃0.08. (55)

Given that 2𝛽^T₂₀₀₂(1−2𝜌^T2002_ANAP)=2×1.52×0.84=2.55 and since 𝜎²_T2002=2.33 from (53), then using (15) we find that the 2002 convergence coefficient for Turkey, in Table 4, is 𝑐^T₂₀₀₂=2𝛽^T₂₀₀₂1−2𝜌^T20029_ANAP𝜎²_T2002=2.55×2.33=5.94. (56) The political changes induced by the cut-off rule led to a higher convergence coefficient in 2002 relative to 1999 (increasing from a low of 𝑐^T₁₉₉₉=1.49 in (51) to a high 𝑐^T₂₀₀₂=5.94 in (56)). An indication that a more fractionalized polity emerged from this reform. The convergence coefficient of the 2002 election is significantly above 2, the dimension of the policy space (see Appendix A.2) giving ANAP an incentive to locate far from the mean. ANAP’s characteristic matrix using (17) is 𝐶^T2002_ANAP=2𝛽^T₂₀₀₂1−2𝜌^T2002_ANAP∇^T₂₀₀₂=.−𝐼=2.551.180.740.741.15−𝐼2.011.881.881.93 (57) When at the origin, 𝐶^T2002_ANAP indicates that ANAP is minimizing its vote share since its eigenvalues are both positive (0.090 and 3.850). This together with the 95% confidence bounds in Appendix A.2 implies that there is a high probability that ANAP will vacate the center and that the mean is not an LNE for Turkey in 2002.

3.2.3. The 1997 Polish Election

In the election held in Poland in 1997 (In this election Poland used an open-list proportional representation electoral system with a threshold of 5% nationwide vote for parties and 8% for electoral coalitions. Votes are translated into seats using the D’Hondt method.) the following five parties won seats in the Sejm (lower house). The left-wing excommunist Democratic Left Alliance (SLD) and the agrarian Polish Peoples’ Party (PSL), both of which have been the most frequent governing parties in the postcommunist period. The Freedom Union (UW) and the Solidarity Election Action (AWS) had grown out of the Solidarity movement. AWS combined various mostly right wing and Christian groups under one label, while UW was formed based on the liberal wing of Solidarity. The remaining party is the Movement for Reconstruction of Poland (ROP).

Applying factor analysis to questions from the Polish National Election Survey an economic and a social value dimensions were identified (see [40]). The economic dimension is influenced by issues such as privatization versus state ownership of enterprises, fighting unemployment versus keeping inflation and government expenditure under control, proportional versus flat income tax, support versus opposition to state subsidies to agriculture, and state versus individual social responsibility. The separation of church and state versus the influence of church over politics, complete decommunization versus equal rights for former nomenclature, and abortion rights regardless of situation versus no such rights regardless of situation are the most influential issues in this social values dimension. The distribution of voters along these dimensions is seen in Figure 9. (See Schofield et al. [40] for details of the estimation.)

The covariance matrix for the 1997 Polish (P) election is ∇^P₁₉₉₇=𝜎²_𝐸=1.00𝜎_𝐸𝑆𝜎=0.0_𝑆𝐸=0.0𝜎²_𝑆=1.00, (58) with variance 𝜎²_P1997=trace(∇^P₁₉₉₇)=2.00.

From Table 3, the MNL coefficients for the 1997 election are 𝜆^P1997_UPR=−2.3,𝜆^P1997_UP𝜆=−0.56,_P1997ROP≡0.0,𝜆^P1997_PSL𝜆=0.07,_P1997UW≡0.73,𝜆^P1997_SLD𝜆=1.40,_P1997AWS=1.92,𝛽^P₁₉₉₇=1.74. (59) The 𝛽-coefficient and valence estimates for all parties except UP and PSL are significantly nonzero. The probability of voting UPR with lowest valence, in 1997, when parties locate at the mean, 𝜌^P1997_TW in (14), is 𝜌^P1997_UPR=⁶_𝑘=1exp𝜆^P_𝑗¹⁹⁹⁷−𝜆^P1997_UPR⁻¹=1+𝑒^0.048+𝑒^3.08+𝑒^4.27+𝑒^3.77+𝑒^2.42−1≃0.01. (60)

Given that 2𝛽^P₁₉₉₇(1−2𝜌^P1997_UPR)=2×1.74×0.98=3.41 and since 𝜎²_P1997=2 from (58), then using (15) the convergence coefficient for Poland, in Table 4, is 𝑐^P₁₉₉₇=2𝛽^P₁₉₉₇1−2𝜌^P1997_UPR𝜎²_P1997=3.41×2=6.82. (61) Appendix A.2 shows that 𝑐^P₁₉₉₇=6.82 is significantly greater than 2 and thus fails the necessary condition for convergence to the mean. UPR’s Hessian from (17) is 𝐶^P1997_UPR=2𝛽^P₁₉₉₇1−2𝜌^P1997_UPR∇^P₁₉₉₇=.−𝐼=3.411.00.00.01.0−𝐼2.410.00.02.41 (62) The trace (= 3.82), the determinant (= 5.80), and the eigenvalues of 𝐶^I_UPR(2.41,1.41) are positive. The 95% confidence bound of 𝐶^I_UPR in Appendix A.2 also shows positive eigenvalues at the lower and upper bounds of 𝐶^I_UPR. Thus, with a high degree of certainty UPR locates far from the origin to maximize its votes and the electoral mean is not a LNE for 1997 Polish election.

Summarizing, in this section we examined three countries that use proportional representation. Their convergence coefficients are significantly higher than 2, the dimension of the policy space and are also much higher than that of the US and the UK. A high convergence coefficient signals then a high degree of political fractionalization in these multi-party parliamentary democracies.

3.3. Convergence in Anocracies

We now study elections in Georgia, Russia, and Azerbaijan. In these partial democracies or anocracies, (The term “partial democracy” has been applied to new democracies lacking the full array of democratic institutions present in western democracies (see [41].)) the President/autocrat holds regular presidential and legislative elections while exerting undue influence on the elections. Anocracies lack important democratic institutions such as freedom of the press. Autocrats hold regular elections in an attempt to give their regime legitimacy. The autocrat “buys” legitimacy by rewarding their supporters and opposition members with well-paid legislative positions and give legislators the ability to influence policies. Opposition parties participate in elections to become known political entities. This allows them to regularly communicate with voters. Their objective is to oust the autocrat either in a future election or through popular uprisings. We assume that opposition parties maximize their vote share even when understanding that there is little chance of ousting the autocrat in the election.

3.3.1. The 2008 Georgian Election

We use the postelection survey conducted by GORBI-GALLUP International from March 19 through April 3, 2008, to built a formal model of the 2008 election in Georgia (see [42]). The factor analysis done on the survey questions determined that there were two dimensions describing voters’ attitudes towards democracy and the west. One dimension is strongly related with the respondents’ attitude toward the US, the EU and NATO with larger values in the West (𝑊=𝑦-axis) dimension implying a stronger anti-western attitude. Along the democracy (𝐷=𝑥-axis) dimension larger values are associated with negative judgements on the current state of democratic institutions in Georgia, coupled with a demand for more democracy. The electoral distribution along these two dimensions is given in Figure 10. The points (S, G, P, N) in Figure 10 represent the estimated positions of the four candidates: Saakashvili (S), Gachechiladze (G), Patarkatsishvili (P), and Natelashvili (N). (See Schofield et al. [39] for details of the estimation.)

Figure 10 

Voter distribution and candidate positions in the 2008 Georgian election.

The 2008 electoral covariance matrix in the Democracy (𝐷) and West (𝑊) axes is ∇^G₂₀₀₈=𝜎²_𝐷=0.82𝜎_𝐷𝑊𝜎=0.03_𝑊𝐷=0.03𝜎²_𝑊=0.91 (63) with 𝜎²_G2008≡trace(∇^G₂₀₀₈)=1.73.

From Table 5, the MNL estimates of the 2008 election with Natelashvili as the base candidate are 𝜆^G2008_S=2.56,𝜆^G2008_G=1.50,𝜆^G2008_P𝜆=0.53,_G2008N≡0.0,𝛽^G₂₀₀₈=0.78. (64) All coefficients are significantly nonzero showing Natelashvili as having the lowest valence.

Table 5 

MNL spatial model in anocracies.

The probability that a Georgian votes for Natelashvili, when all candidates locate at the mean, is 𝜌^G2008_N=⁴_𝑘=1exp𝜆^G_𝑗²⁰⁰⁸−𝜆^G2008_N⁻¹=1+𝑒^2.56+𝑒^1.50+𝑒^0.53−1≃0.05. (65)

Given that 2𝛽^G₂₀₀₈(1−2𝜌^G2008_N)=2×0.78×0.9=1.4 and since 𝜎²_G2008=1.73 from (63), then using (15) Georgia’s the convergence coefficient, in Table 6, is 𝑐^G₂₀₀₈=2𝛽^G20081−2𝜌^G2008_N𝜎²_G2008=1.4×1.73=2.42. (66)

Table 6 

The convergence coefficient in anocracies.

As shown in Appendix A.3, 𝑐^G₂₀₀₈ is not significantly different from 2 and thus fails the necessary condition for convergence to the mean. Natelashvili’s Hessian or characteristic matrix, from (17), is 𝐶^G2008_N=2𝛽^G₂₀₀₈1−2𝜌^G2008_N∇^G₂₀₀₈=.−𝐼=1.40.820.030.030.91−𝐼0.150.040.040.28 (67) Since the eigenvalues of 𝐶^G2008_N are both positive (+0.139,+0.291), Natelashvili’s vote share function is at a minimum when he is at the mean and has an incentive to move to increase his vote share. This together with the analysis of the 95% confidence intervals of 𝐶^G2008_N in Appendix A.3 shows that with a high degree of certainty Natelashvili will locate far from the mean. This is not surprising since Georgians managed to induce three major changes in government through mass protests prior to this election. Thus, with a high degree of certainty Natelashvili locates far from the origin in this election and the electoral mean cannot be an LNE for the 2008 Georgian election.

3.3.2. The 2007 Russian Election

The analysis of the 2007 Russian election concentrates on four parties: the pro-Kremlin United Russia party (ER), Liberal Democratic Party (LDPR), Communist Party (CPRF), and Fair Russia (SR). Voters’ ideological preferences were measured according to two questions taken from the survey conducted by VCIOM (Russian Public Opinion Research Center) in May 2007 (see [43]). The first dimension gives a measure of voters general (dis)satisfaction (𝐷=𝑥-axis). High values in this dimension correspond to negative feelings toward “justice,” “labor” and, to a lesser extent, “order,” “state,” “stability,” and “equality.” Also, those with high values of the first axis tend to feel neutral toward order, elite, West, and non-Russians. The second dimension measures the voter’s degree of economic liberalism (𝐸=𝑦-axis). High values correspond to positive feelings to “freedom,” “business,” “capitalism,” “well-being,” “success,” and “progress,” and to negative feelings toward “communism,” “socialism,” “USSR,” and related concepts. The distribution of voter preferences along these two dimensions can be seen in Figure 11. (See Schofield and Zakharov [43] for details of the estimation.)

Figure 11 

Party positions and voters distribution in the 2007 Russian election.

The 2007 electoral covariance matrix along the (dis) satisfaction (𝐷) and economic liberalism (𝐸) axes is ∇^R₂₀₀₇=𝜎²_𝐷=2.95𝜎_𝐷𝐸𝜎=0.13_𝐸𝐷=0.13𝜎²_𝐸=2.95, (68) with 𝜎²_R2007≡trace(∇^R₂₀₀₇)=5.9.

From Table 5, the MNL estimates of the spatial model for Russia are 𝜆^R2007_SR=−0.4,𝜆^R2007_𝐸𝑅≡0,𝜆^R2007_LDPR𝜆=0.153,_R2007CPRF=1.971,𝛽^R₂₀₀₇=0.181. (69) Distance and all valences, except for that of the LDPR party, are significantly nonzero. When parties locate at the mean, the probability that a Russian votes for Fair Russia (SR) with lowest valence, from (14) is 𝜌^R2007_SR=⁴_𝑘=1exp𝜆^R_𝑗²⁰⁰⁷−𝜆^R2007_SR⁻¹=1+𝑒^0.4+𝑒^0.553+𝑒^2.371−1≃0.07. (70)

Given that 2𝛽^R₂₀₀₇(1−2𝜌^R2007_SR)=2×0.181×0.86=0.31 and since 𝜎²_R2007=5.9 from (68), then using (15) Russia’s convergence coefficient, in Table 6, is 𝑐^R₂₀₀₇=2𝛽^R₂₀₀₇1−2𝜌^R2007_SR𝜎²_R2007=0.31×5.9=1.83. (71) Since 𝑐^R₂₀₀₇ is not significantly different from 2 (see Appendix A.3), the necessary condition for convergence is not met. The characteristic matrix or Hessian of Fair Russia (SR) from (17) is 𝐶^R2007_SR=2𝛽^R₂₀₀₇1−2𝜌^R2007_SR∇^R₂₀₀₇=.−𝐼=0.312.950.130.132.95−𝐼−0.0860.040.04−0.086 (72) The eigenvalues are both negative (−0.126,−0.046), implying that at this central estimate Fair Russia is maximizing its vote share and thus has no incentive to vacate the origin. This conclusion holds at the lower 95% bound of 𝐶^R2007_SR in Appendix A.3. However, at the upper bound of 𝐶^R2007_SR Fair Russia is minimizing its vote share. It seems then that with the Russian President and his party exerting much influence over the election and Putin being so popular that Fair Russia is more likely to remain at the origin. (This result however highlights that unexpected political events could prompt Fair Russia to move from the origin.) It is then likely that the electoral mean is a LNE for the 2007 Russian election.

3.3.3. The 2010 Election in Azerbaijan

In the 2010 election in Azerbaijan, 2,500 candidates filed application to run in the election, but only 690 were given permission by the electoral commission. The parties that competed in the election were the Yeni Azerbaijan Party (the party of the President, YAP), Civic Solidarity Party (VHP), Motherland Party (AVP), Azerbaijan Popular Front Party (AXCP), and Musavat (MP). Various small parties formed political blocks.

President Ilham Aliyev’s ruling Yeni Azerbaijan Party took a majority of 72 out of 125 seats. Nominally independent candidates, who were aligned with the government, received 38 seats, and 10 small opposition or quasiopposition parties took 10 seats. The Democratic Reforms party, Great Creation, the Movement for National Rebirth, Umid, Civic Welfare, Adalet (Justice), and the Popular Front of United Azerbaijan most of which were represented in the previous parliament, won one seat a piece. Civic Solidarity retained its 3 seats and Ana Vaten kept the 2 seats they had in the previous legislature. For the first time, not a single candidate from the opposition Azerbaijan Popular Front (AXCP) or Musavat were elected.

We organized a small preelection survey of 2010 election in Azerbaijan allowing us to construct a model of the election (see [42]). For VHP and AVP, the estimation of their party positions was very sensitive to inclusion or exclusion of one respondent. Thus, we used only the small subset of 149 voters who completed the factor analysis questions and intended to vote for YAP or the AXCP+MP coalition.

The factor analysis showed that voters were only concerned with one dimension: the “demand for democracy” with higher values being associated with voters who had a negative evaluation of the current democratic situation in Azerbaijan, who did not think that free opinion is allowed, had a low degree of trust in key national political institutions, and expected that the 2010 parliamentary election would be undemocratic. Figure 12 shows the distribution of voters and the party positions at the mean of their supporters. (See [42] for details of the estimation.) In this one dimensional model the variance is 𝜎²_A2010≡trace∇²⁰¹⁰_G=0.93. (73)

Figure 12 

Voter distribution and activist positions in the 2010 Azerbaijani election.

The binomial logit estimates for the 2010 election with AXCP-MP as the base party, in Table 5, are 𝜆^A2010_YAP=1.30,𝜆^A2010_AXCP-MP≡0.0,𝛽^A₂₀₁₀=1.34. (74) All coefficients are significantly nonzero with AXCP-MP having the lowest valence. If these two parties locate at the mean, the probability that an Azerbaijani votes AXCP-MP from (14) is 𝜌^A2010_AXCP-MP=²_𝑘=1exp𝜆^A_𝑗²⁰¹⁰−𝜆^A2010_AXCP-MP⁻¹=1+𝑒^1.3−1≃0.21. (75)

Given that 2𝛽^A₂₀₁₀(1−2𝜌^A2010_AXCP-MP)=2×1.34×0.58=1.554 and since 𝜎²_A2010=0.93 from (73), then using (15) the convergence coefficient for Azerbaijan, in Table 6, is 𝑐^A₂₀₁₀=2𝛽^A₂₀₁₀1−2𝜌^A2010_AXCP-MP𝜎²_A2010=1.554×0.93=1.445. (76) Given that 𝑐^A₂₀₁₀ is not significantly different from 1, the dimension of the policy space (see Appendix A.3) and the necessary condition for convergence is not met. The one dimensional Hessian of AXCP-MP from (17) is 𝐶^A2010_AXCP-MP=2𝛽^A₂₀₁₀1−2𝜌^A2010_AXCP-MP𝜎²_A2010−𝐼=1.554×0.93−1=0.445. (77) Clearly, 𝐶^A2010_AXCP-MP has a single positive eigenvalue indicating the AXCP+MP is minimizing its vote share at the origin. The 95% bounds of 𝐶^A2010_AXCP-MP in Appendix A.3 shows that this matrix has positive eigenvalues at the lower and upper bounds of the confidence interval. Thus, with a high degree of certainty AXCP+MP will deviate from the origin and the electoral mean is not a LNE for the 2010 election in Azerbaijan.

This section illustrates that for the three anocracies that we consider the convergence coefficient does not satisfy the necessary condition for convergence to the mean. That is, these convergence coefficients are not significantly different from the dimension of the policy space. As a consequence, parties are at a knife-edge equilibrium. Under some conditions, parties converge to the mean, under others they diverge. Which equilibrium materializes depends on how popular or unpopular the President/autocrat and his party are and so depends on the valence of all parties and on how dispersed voters are in the policy space. Thus any change in valence can substantially affect party positions.

4. Convergence across Political Systems

In the previous sections we used the unifying framework of Schofield’s [9] stochastic electoral model outlined in Section 2 to study whether parties locate near or far from the electoral mean for countries with plurality and proportional representation systems and in anocracies. Using this framework we estimated the convergence coefficient for various elections in different countries. We will now use this dimensionless coefficient to compare convergence to the electoral mean across elections, countries, and political systems. We can then illustrate the use of the convergence coefficient to classify political systems. Table 7 presents a summary of the convergence coefficients across elections, countries, and political systems that we now discuss.

Table 7 

Convergence and fragmentation.

As Table 7 indicates the two countries using plurality systems (the US and the UK) studied in Section 3.1 meet the conditions for convergence to the mean. Thus, suggesting that plurality rule imposes a strong centripetal tendency that keeps parties close to the mean. Our analysis suggests that in countries with plurality systems the convergence coefficient will be low at or below the dimension of the policy space.

Of the anocratic countries that we studied in Section 3.3, Georgia seems to have the highest convergence coefficient, 𝑐^G₂₀₀₈=2.42 in (66) which is not different from 2, suggesting that parties can diverge from the mean. (Note that prior to 2008 Georgians had already brought about three major political changes through mass popular revolt. This rebellious “tradition” may give opposition candidates the ability to position themselves away from the mean.) The convergence coefficient of all three anocracies was not significantly different than the dimension of the policy space [2 for Georgia and Russia and 1 for Azerbaijan: 𝑐^G₂₀₀₈=2.42 given in (66), 𝑐^Ru₂₀₀₇=1.83 in (71), and 𝑐^A₂₀₁₀=1.44 in (76)]. These results suggest that convergence in anocracies is fragile and depends on the distribution of voters’ preferences as well as on the valences of the autocrat and the opposition parties.

The countries with proportional systems studied in Section 3.2 have convergence coefficients that are significantly above their two-dimensional policy space signalling the lack of convergence of small valence parties to the electoral mean (from Table 7, Israel’s 𝑐^I₁₉₉₆=4.06 in (46), Turkey’s 𝑐^T₁₉₉₉=1.49 in (51) in 1999, and 𝑐^T₂₀₀₂=5.94 in (56) in 2002 and Poland’s 𝑐^P₁₉₉₇=6.82 in (61)). Having no possibility of forming government, these small parties maximize their vote shares by locating closer to their core supporters. Elections lead to multiparty legislatures producing a highly fragmented party system where coalition governments are the norm. Note that changes to the electoral process in Turkey between 1999 and 2002 forced parties to move from locating close to the mean in 1999 to diverging towards their partisan constituencies so as to increase their vote shares in 2002. These results suggest that in countries with proportional systems, with highly fragmented political parties, divergence from the mean is the norm.

We can explain the lack of convergence to the mean in proportional systems with multiparty (>3) legislatures by noting that the convergence coefficient 𝑐≡𝑐(𝝀,𝛽,𝜎²)=2𝛽[1−2𝜌₁]𝜎² in (15) depends on fundamental characteristics of the electorate. These characteristics include the weight given by voters to the distance to the parties’ positions, 𝛽; the electoral variance, 𝜎² in (16); and the probability that a voter chooses the lowest valence party, 𝜌₁ in (14). Thus, in countries with many parties, the smallest low valence parties have little chance of receiving much support, a low 𝜌₁. If, in addition, voters care a lot about policy differences (a high 𝛽) and if the electorate is very dispersed (a high 𝜎²), then small parties will have an incentive to move towards their core supporters and away from the mean. That is, in highly fragmented polities where voters and correspondingly parties are very dispersed, we observe high convergence coefficients.

In essence, Schofield’s [9] Valence theorem gives a simple summary statistic, the convergence coefficient, that measures the degree of fragmentation, or lack thereof, in each polity. Poland is an extreme case of this fragmentation and correspondingly has a very high convergence coefficient (see Table 7).

The are other measures of political fragmentation in the literature. The effective number of party vote strength (env) used by Laakso and Taagepera [15] serves to measure how many dominant parties there are in a polity a given election. To find the env, let the Herfindahl index of the election be given by 𝐻_𝑣=^𝑝_𝑗=1𝑣²_𝑗, (78) where 𝑣_𝑗 is the vote share of party 𝑗 for 𝑗=1,…,𝑝. This Herfindahl index 𝐻_𝑣 gives a measure of the party size in an election and measures how competitive the election was. Laakso and Taagepera’s effective number of party vote strength is then the inverse of 𝐻_𝑣; that is,𝑒𝑛𝑣=𝐻_𝑣⁻¹. (79) In the same way we can define the effective number of party seat strength (𝑒𝑛𝑠) using seat shares instead of vote shares giving us a measure of the strength of parties in a legislature.

We calculate the 𝑒𝑛𝑣 and 𝑒𝑛𝑠 for each election we consider (see Table 7) using all the parties that obtained votes in each election and exclude parties that ran in the election but that got no votes. We now compare the level of fragmentation given by the 𝑒𝑛𝑣 and 𝑒𝑛𝑠 with that given by the convergence coefficient for each country and each election under the three political systems that we studied.

We first examine countries with plurality rule. In Table 7 we see that for the US, the 𝑒𝑛𝑣 and the 𝑒𝑛𝑠 at the Presidential and House levels are closely aligned. There is little variation between the 𝑒𝑛𝑣 and 𝑒𝑛𝑣 indices in the three elections. According to these indices there is essentially no change in political fragmentation across these three elections. The convergence coefficient however rises in 2008 relative to 2000 and 2004 indicating that in 2008 the dispersion among voters was higher than in the previous two elections. For the US, the convergence coefficient provides more information than do 𝑒𝑛𝑣 or 𝑒𝑛𝑣. For the UK, the convergence coefficient shows that the electorate was more dispersed in 2010 than in 2005 (see Tables 2 and 7). This dispersion led to the first minority government since 1974 which resulted in higher effective number of parties as measured by the 𝑒𝑛𝑣 and 𝑒𝑛𝑣. All three measures, 𝑐, 𝑒𝑛𝑣, and 𝑒𝑛𝑠, indicate that the United Kingdom became more fragmented in 2010. Thus, in the countries using plurality, the convergence coefficient tends to provide more information than the 𝑒𝑛𝑣 and 𝑒𝑛𝑠 numbers do as the convergence coefficient takes into account the degree of dispersion among the electorate and the valence of parties.

Polities with high convergence coefficients (Israel, Turkey in 2002 and Poland in Table 7) had a large number of parties competing in these elections. The greater the number of parties obtaining votes, and thus effectively competing in the election, led to large 𝑒𝑛𝑣 values. These elections produced highly fragmented legislatures leading to very high 𝑒𝑛𝑠 values. Having a large number of effective parties competing in the election and greater effective number of parties in the legislature does not necessarily translate into a higher convergence coefficient. The convergence coefficient is lower for Israel with a larger number of effective parties (higher 𝑒𝑛𝑣 and 𝑒𝑛𝑠) than for Poland with fewer parties. Changes in the Turkish electoral system between 1999 and 2002 in which a minimum cut-off rule has instituted led to a high 𝑒𝑛𝑣 but a low 𝑒𝑛𝑠. Small parties were however able to gain enough votes leading to a high convergence coefficient, an indication that these parties would disperse themselves in the policy space. The 𝑒𝑛𝑣 and 𝑒𝑛𝑠 values of the 2002 Turkish election show high party fragmentation but no legislative fragmentation. This shows that these three measures of fragmentation provide different information about a particular election.

The convergence coefficient suggests that a way of interpreting the arguments of Duverger [44] and Riker [45] on the effects of proportional electoral methods on electoral outcomes: the strong centrifugal tendency pulling all parties away from the electoral mean towards their core constituency. This tendency will be particularly strong for small, or low valence, parties. In particular, even small parties in such a polity can assign a nonnegligible probability to becoming a member of a coalition government, and it is this phenomenon that maintains the fragmentation of the party system. For example, in Poland no party can obtain a majority and parties and coalitions regularly form and dissolve. In general the convergence coefficients in Poland were of the order of 6.0 in the elections in the 1990’s.

For countries using proportional representation, while the 𝑒𝑛𝑣 and 𝑒𝑛𝑠 give a measure of electoral and legislative dispersion, the convergence coefficient provides a measure that summarizes dispersion across voters and parties in the policy space.

In the anocratic countries studied, the convergence coefficient seems in line with the 𝑒𝑛𝑣 in presidential elections but going in the opposite direction in parliamentary elections (see Table 7). In these countries, the convergence coefficient does not meet the necessary condition for convergence to the mean. These countries that we study show that parties could either converge to or diverge from the mean under anocracy as the equilibrium is fragile. Changes in valences, for example, of the autocrat or in voters’ preferences, can lead small valence opposition parties to diverge from the mean and to mount popular uprisings as happened in previous elections in Georgia or in recent Arab uprisings.

The convergence coefficient reflects information that the 𝑒𝑛𝑣 and 𝑒𝑛𝑠 cannot capture as it reflects the preferences of the electorate through the policy weight, 𝛽; the perceived ability of parties or candidates to govern as captured by their valences 𝝀=(𝜆₁,…,𝜆_𝑝); and the dispersion of voters’ preferences in the policy space, 𝜎². All of which are not taken into account in the 𝑒𝑛𝑣 and 𝑒𝑛𝑠. Moreover, 𝑒𝑛𝑣 and 𝑒𝑛𝑠 have nothing to say about the dispersion in parties’ positions relative to the mean.

The analysis carried out in this section suggests that there is an inverse relationship between the degree of fractionalization in a polity and the convergence coefficient. By our interpretation of the nature of the convergence coefficient, the convergence effect in presidential elections in the United States is stronger than in parliamentary elections in Great Britain. That is, our results suggest that democratic presidential systems have fewer parties and a low convergence coefficient. Parliamentary democracies operating under plurality rule tend to have more parties than presidential democracies and a somewhat higher convergence coefficient. Parliamentary democracies operating under proportional representation tend to have multiparty legislatures and high convergence coefficients. Anocratic countries tend to have multiple parties competing in the election but low convergence coefficients as opposition parties remain close to the electoral mean when Presidents/autocrats have high valences and diverge when they do not.

5. Conclusion

In this paper, Schofield’s [9] Valence Theorem together with multinomial logit models of elections are used as a unifying framework to compare the convergence properties of parties across elections, countries, and political systems. We found evidence to support the hypothesis that in countries with proportional representation parties located away from the electoral mean.

We relate the convergence coefficient to the effective number of parties according to both vote (𝑒𝑛𝑣) and seat (𝑒𝑛𝑠) shares and showed how the characteristics of the electorate and the political regime under which parties operate. Then, compare the convergence coefficient to the fractionalization measures provided by the 𝑒𝑛𝑣 and 𝑒𝑛𝑠. The advantage of the convergence coefficient is that it is a summary statistic that incorporates the preferences of voters, the valence of parties, and the dispersion of voters and parties in the policy space.

Appendix

A. Confidence Intervals

Schofield’s [9] Valence Theorem, presented in Section 2, perfectly predicts whether parties converge to or diverge from the electoral origin. Convergence or divergence depends on the value of the convergence coefficient 𝑐≡2𝛽[1−2𝜌₁]𝜎² in (15) and on the Characteristic matrix of party 1 with lowest valence, 𝐶₁=2𝛽(1−2𝜌₁)∇−𝐼 in (17). Both 𝑐 and 𝐶₁ depend on 𝛽 and on 𝜌₁=[∑^𝑝_𝑘=1exp(𝜆_𝑘−𝜆₁)]⁻¹ in (14).

The central estimate of 𝛽 and of 𝝀=(𝜆₁,…,𝜆_𝑝) given by the MNL regressions depend on the sample of voters surveyed as do 𝜌₁, 𝑐, and 𝐶₁. Thus, to make inferences from empirical models we need the 95% confidence bounds of these estimates. Using these bounds we assert with some degree of certainty whether parties converge to or diverge from the electoral mean or if there is a knife-edge unstable equilibrium.

To build these bounds, we could perform simulations of the election. For each simulation we could generate the value of 𝛽, 𝝀=(𝜆₁,…,𝜆_𝑝), 𝜌₁, 𝑐 and 𝐶₁. Repeating the simulation many times would generate their distribution from which we could derive their 95% confidence bounds. Note that 𝑐 and 𝐶₁ increase in 𝛽 and decrease in 𝜌₁. So that given the electoral covariance matrix ∇ and variance/trace 𝜎² in (16) of an election, when in a simulation 𝛽 has a low value and 𝜌₁ a high one, the values of 𝑐 and 𝐶₁ are low with the opposite being true when 𝛽 is high and 𝜌₁ is low. Since we have not performed simulations for the elections in this study, we use these features of 𝑐 and 𝐶₁ to generate our confidence bounds.

Let 𝐿 identify the lower and 𝑈 the upper bounds of the 95% confidence intervals of any estimate. The MNL estimation for an election gives the confidence bounds of 𝛽 and 𝜆₁, (𝛽^𝐿,𝛽^𝑈) and [𝜆^𝐿₁,𝜆^𝑈₁]. To estimate the bounds on 𝜌₁ in (14), [𝜌^𝐿₁,𝜌^𝑈₁], we use the bounds on 𝜆₁ and Taylor’s Theorem, which asserts that 𝜌₁𝜆₁±ℎ=𝜌₁𝜆₁±ℎ𝑑𝜌₁𝑑𝜆₁=𝜌₁𝜆₁±ℎ𝜌₁𝜆₁1−𝜌₁𝜆₁=𝜌₁𝜆₁1±ℎ1−𝜌₁𝜆₁=𝜌^𝐿₁,𝜌^𝑈₁. (A.1)

Using (15) and the bounds on 𝛽 and 𝜌₁ we build the confidence intervals for the convergence coefficient 𝑐 as follows. In (15) use 𝛽^𝐿 and 𝜌^𝑈₁ to get the lower bound of 𝑐, 𝑐^𝐿, and use 𝛽^𝑈 and 𝜌^𝐿₁ for the upper bound of 𝑐, 𝑐^𝑈. The 95% confidence interval of the convergence coefficient is then 𝑐^𝐿,𝑐^𝑈=2𝛽^𝐿1−2𝜌^𝑈₁𝜎²,2𝛽^𝑈1−2𝜌^𝐿₁𝜎². (A.2)

Following a similar procedure we estimate the bounds for 𝐶₁ using (17) and the corresponding bounds of 𝛽 and 𝜌₁ to get the bounds for the Hessian of the lowest valence party 𝐶^𝐿₁,𝐶^𝑈₁=2𝛽^𝐿1−2𝜌^𝑈₁∇−𝐼,2𝛽^𝑈1−2𝜌^𝐿₁∇−𝐼. (A.3) Clearly, the bounds for 𝑐 and 𝐶₁ must be similar to those generated by repeated simulations.

Using these procedures, we now derive the 95% confidence intervals for the central estimates of 𝜌₁, 𝑐, and 𝐶₁ for each of the elections studied (see summary in Tables 2, 4, and 6). We first derive the detail of the confidence bounds for the 2000 US election, then in less detail those of other elections. Table 7 gives the values needed to derive the confidence intervals for the convergence coefficient of the election.

A.1. Convergence in Plurality Systems

A.1.1. Confidence Bounds for the 2000, 2004, and 2008 US Elections

US 2000 Election. From Table 1, the 95% confidence interval for 𝛽^US₂₀₀₀=0.82 are [𝛽^US𝐿₂₀₀₀,𝛽^US𝑈₂₀₀₀]=[0.82±1.96×0.06]=[0.71,0.93]. Using (A.1), the bounds for 𝜌^US2000_rep=0.4 in (20) are [𝜌^US2000𝐿_rep,𝜌^US2000𝑈_rep]=[0.35,0.44]. Using these bounds and (18), the bounds for the convergence coefficient for the 2000 US election in (21) from (A.2) are 𝑐^US𝐿₂₀₀₀,𝑐^US𝑈₂₀₀₀=[2(),2(=[].0.71)(1−2×0.44)(1.170.93)(1−2×0.35)(1.17)]0.20,0.65 (A.4) With 95% confidence, the convergence coefficient is below 1 meeting the sufficient and thus necessary condition for convergence to the mean. The bounds on Bush’s characteristic matrix in (22) from (A.3) are 𝐶^US2000𝐿_rep,𝐶^US2000𝑈_rep=2()2()=,.0.71)(1−2×0.440.58−0.20−0.200.59−𝐼,0.93)(1−2×0.350.58−0.20−0.200.59−𝐼−0.90−0.03−0.03−0.90−0.68−0.11−0.11−0.67 (A.5) Since the eigenvalues of the lower and upper bounds of 𝐶^US2000_rep are negative [𝐶^US2000𝐿_rep=(−0.87,−0.93),𝐶^US2000𝑈_Bush=(−0.79,−0.57)], with 95% confidence Bush’s vote share is at a maximum when all parties locate at the mean. Thus, with a high degree of certainty the origin is a LNE for the 2000 US election.

US 2004 Election. From Table 1, the 95% confidence bounds of 𝛽^US₂₀₀₄=0.95 is [𝛽^US𝐿₂₀₀₄,𝛽^US𝑈₂₀₀₄]=[0.95±1.96×0.07]=[0.82,1.08]. Using (A.1), the bounds of 𝜌^US2004_rep=0.4 in (25) are [𝜌^US2004𝐿_rep,𝜌^US2004𝑈_rep]=[0.35,0.44]. The bounds for 𝑐^US₂₀₀₄=0.38 in (21) from (A.2) and for the characteristic matrix of Bush 𝐶²⁰⁰⁴_rep in (27) from (A.3) are 𝑐^US𝐿₂₀₀₄,𝑐^US𝑈₂₀₀₄=[2(),2(=[],𝐶0.82)(1−2×0.44)(1.171.08)(1−2×0.35)(1.17)]0.23,0.76_{US2004𝐿rep},𝐶^US2004𝑈_rep=2()2()=,.0.82)(1−2×0.440.58−0.18−0.180.59−𝐼,1.08)(1−2×0.350.58−0.18−0.180.59−𝐼−0.89−0.04−0.04−0.88−0.62−0.12−0.12−0.62 (A.6) The convergence coefficient is significantly below 1. Bush maximizes his vote share when located at the origin since the eigenvalues of the lower and upper bounds of 𝐶^US2004_rep are negative [𝐶^US2004𝐿_rep=(−0.87,−0.93),𝐶^US2004𝑈_rep=(−0.79,−0.57)]. Thus, with 95% confidence Bush does not want to move from the mean implying that with a great certainty the origin is a LNE for the 2004 US election.

US 2008 Election. From Table 1 the bounds of 𝛽^US₂₀₀₈=0.85 are [𝛽^US𝐿₂₀₀₈,𝛽^US𝑈₂₀₀₈]=[0.85±1.96×0.06]=[0.73,0.97]. Using (A.1), those of 𝜌^US2008_rep in (30) are [𝜌^US2008𝐿_rep,𝜌^US2080𝑈_rep]=[0.26,0.35]. So that the bounds for c^US₂₀₀₈=1.1 in (31) from (A.2) and for McCain’s characteristic matrix C^US2008_rep in (32) from (A.3) are 𝑐^US𝐿₂₀₀₈,𝑐^US𝑈₂₀₀₈=[2(),2(=[],𝐶0.73)(1−2×0.35)(1.630.97)(1−2×0.26)(1.63)]0.71,1.52_{US2008𝐿rep},𝐶^US2008𝑈_rep=2()2()=,.0.73)(1−2×0.350.80−0.13−0.130.83−𝐼,0.97)(1−2×0.260.80−0.13−0.130.83−𝐼−0.65−0.06−0.06−0.64−0.26−0.12−0.12−0.23 (A.7) The convergence coefficient is not statistically different from 1 and thus meets the necessary but not the sufficient condition for convergence. Since the eigenvalues of the lower and upper bounds of 𝐶^US2008_rep are negative[𝐶^US2008𝐿_rep=(−0.75,−0.59),𝐶^US2008𝑈_rep=(−0.37,−0.12)], then with 95% confidence McCain stays at the origin. With a high degree of certainty the mean is an LNE for the 2008 US election.

A.1.2. Confidence Bounds for the 2005 and 2010 UK Elections

UK 2005 Election. From Table 1, the bounds of 𝛽^UK₂₀₀₅=0.15 are [𝛽^UK𝐿₂₀₀₅,𝛽^UK𝑈₂₀₀₅]=[0.15±1.96×0.01]=[0.13,0.17]. Using (A.1), those for 𝜌^UK2005_lib in (35) are [𝜌^UK2005𝐿_lib,𝜌^UK2005𝑈_lib]=[0.18,0.32], so that those for 𝑐^UK₂₀₀₅ in (36) from (A.2) and for the Liberal Democrats’ characteristic matrix 𝐶^UK2005_lib in (37) from (A.3) are 𝑐^UK𝐿₂₀₀₅,𝑐^UK𝑈₂₀₀₅=[2(),2(=[],𝐶0.13)(1−2×0.32)(5.610.17)(1−2×0.18)(5.61)]0.51,1.25_{UK2005𝐿lib},𝐶^UK2005𝑈_lib=2()2()=,.0.13)(1−2×0.321.650.000.003.96−𝐼,0.17)(1−2×0.181.650.000.003.96−𝐼−0.850.000.00−0.64−0.630.000.00−0.12 (A.8) With 𝑐^UK₂₀₀₅ not significantly different from 1, the necessary but not the sufficient condition for convergence to the mean has been met. The eigenvalues of the bounds on 𝐶^UK2005_lib are negative, [𝐶^UK2005𝐿_lib=(−0.85,−0.64),𝐶^UK2005𝑈_lib=(−0.37,−0.12)]. With 95% confidence the LibDem locate at the origin and the mean is an LNE of the 2005 UK election.

UK 2010 Election. From Table 1, the bounds of 𝛽^UK₂₀₁₀=0.86 are [𝛽^UK𝐿₂₀₁₀,𝛽^UK𝑈₂₀₁₀]=[0.86±1.96×0.02]=[0.81,0.90]. Using (A.1), those for 𝜌^UK2010_lab in (40) are [𝜌^UK2010𝐿_lab,𝜌^UK2010𝑈_lab]=[0.29,0.32]. So that those for 𝑐²⁰¹⁰_UK in (41) from (A.2) and for Labour’s characteristic matrix 𝐶^UK2010_lab in (42) from (A.3) are 𝑐^2010𝐿_UK,𝑐^2010𝑈_UK=[2(),2(=[],𝐶0.81)(1−2×0.32)(1.460.90)(1−2×0.29)(1.46)]0.86,1.10_{UK2010𝐿lib},𝐶^UK2010𝑈_lib=2()2()=,.0.81)(1−2×0.320.600.070.070.86−𝐼,0.90)(1−2×0.290.600.070.070.86−𝐼−0.650.040.04−0.49−0.550.050.05−0.35 (A.9) The convergence coefficient meets the necessary but not the sufficient condition for convergence to the mean as is not significantly different from 1. The eigenvalues of the bounds of 𝐶^UK2010_lib are negative, [𝐶^UK2010𝐿_lab=(−0.66,−0.48),𝐶^UK2015𝑈_lab=(−0.56,−0.34)]. Thus, with 95% confidence Labour does not want to move from the origin and the origin is an LNE of the model of the 2010 UK election.

A.2. Convergence in Proportional Systems

A.2.1. Confidence Bounds for the 1996 Israeli Election

From Table 3, the bounds of 𝛽^I₁₉₉₆=1.207 are [𝛽^I𝐿₁₉₉₆,𝛽^I𝑈₁₉₉₆]=[1.207±1.96×0.065]=[1.076,1.338]. Using (A.1), those for 𝜌^I1996_TW in (45) are [𝜌^I1996𝐿_TW,𝜌^I1996𝑈_TW]=[0.006,0.034], implying that those of 𝑐^I₁₉₉₆ in (46) from (A.2) and for the TW’s characteristic matrix 𝐶^I1996_TW in (47) from (A.3) are 𝑐^I𝐿₁₉₉₆,𝑐^I𝑈₁₉₉₆=[2(),2(=[],𝐶1.076)(1−2×0.034)(1.7321.338)(1−2×0.006)(1.732)]3.474,4.579_I1996𝐿TW,𝐶^I1996𝑈_TW=2()2()=,.1.076)(1−2×0.0341.000.5910.5910.732−𝐼,1.338)(1−2×0.0061.000.5910.5910.732−𝐼1.0061.1851.1850.4681.6441.5631.5630.935 (A.10) Since 𝑐^I₁₉₉₆ is significantly greater than 2, the necessary condition for convergence to the electoral mean is not met. The lower and upper bounds of 𝐶^I1996_TW have one negative and one positive eigenvalue, [𝐶^I1996𝐿_𝑇𝑊=(−0.48,1.95),𝐶^I1996𝑈_TW=(−0.313,2.892)], TW is at a saddle point at both bounds. Thus, with 95% confidence TW locates away from the origin and the origin fails to be a LNE for the 1996 Israeli election.

A.2.2. Confidence Bounds for the 1999 and 2002 Turkish Elections

1999 Turkish Election. From Table 3, the bounds of 𝛽^T₁₉₉₉=0.375 are [𝛽^T𝐿₁₉₉₉,𝛽^T𝑈₁₉₉₉]=[0.375±1.96×0.088]=[0.203,0.547]. Using (A.1), those for 𝜌^T1999_FP in (50) are [𝜌^T1999𝐿_FP,𝜌^T1999𝑈_FP]=[0.046,0.145], so that those of 𝑐^T₁₉₉₉ in (51) from (A.2) and for the FP’s characteristic matrix 𝐶^T1999_FP in (52) from (A.3) are 𝑐^T𝐿₁₉₉₉,𝑐^T𝑈₁₉₉₉=[2(),2(=[],𝐶0.203)(1−2×0.145)(2.340.547)(1−2×0.046)(2.34)]0.675,2.234_T1999𝐿FP,𝐶^T1999𝑈_FP=2()2()=,.0.203)(1−2×0.1451.200.780.781.14−𝐼,0.547)(1−2×0.0461.200.780.781.14−𝐼−0.6540.2250.225−0.6710.1920.7750.7750.132 (A.11) Since 𝑐^T₁₉₉₉ is significantly greater than 2, the necessary condition for convergence to the mean is not met. 𝐶^T1999𝐿_FP has two negative eigenvalues, [𝐶^T1999𝐿_FP=(−0.888,−0.437)] indicating that at the lower bound FP has no incentive to move from the origin. However, 𝐶^T1999𝑈_FP has one negative and one positive eigenvalue 𝐶^T1999𝑈_FP=(−0.614,0.938); thus, FP is at a saddlepoint at the upper bound and wants to move from the mean. At the central estimate of 𝐶^T1999_FP given in (52) FP is also at a saddlepoint. It is more probable that FP wants to move and that the electoral mean is not a LNE of 1999 Turkish election.

2002 Turkish Election. From Table 3, the bounds of 𝛽^T₂₀₀₂=1.52 are [𝛽^T𝐿₂₀₀₂,𝛽^T𝑈₂₀₀₂]=[1.52±1.96×0.12]=[1.285,1.755]. Using (A.1), those for 𝜌^T2002_ANAP in (55) are [𝜌^T2002𝐿_ANAP,𝜌^T2002𝑈_ANAP]=[0.038,0.133], implying that those of 𝑐^T₂₀₀₂ in (56) from (A.2) and for the ANAP’s characteristic matrix 𝐶^T2002_ANAP in (57) from (A.3) are 𝑐^T𝐿₂₀₀₂,𝑐^T𝑈₂₀₀₂=[2(),2(=[],𝐶1.285)(1−2×0.133)(2.331.755)(1−2×0.038)(2.33)]4.338,7.438_{T2002𝐿ANAP},𝐶^T2002𝑈_ANAP=2()2()=,.1.285)(1−2×0.1331.180.740.741.15−𝐼,1.755)(1−2×0.0381.180.740.741.15−𝐼−0.6600.2130.213−0.6690.1720.7350.7350.142 (A.12) Since 𝑐^T₂₀₀₂ is significantly greater than 2, the necessary condition for convergence to the mean has not been met. The eigenvalues of 𝐶^T2002𝐿_ANAP are all negative, 𝐶^T2002𝐿_ANAP=(−0.878,−0.451) so that at the lower bound ANAP remain at the mean. However, at 𝐶^T2002𝑈_ANAP there is one negative and one positive eigenvalue 𝐶^T2002𝑈_ANAP=(−0.578,0.892), ANAP is at a saddlepoint and wants to move. At the central estimate of 𝐶^T2002_ANAP in (57) the eigenvalues are both positive and ANAP is minimizing its vote share. There is a high likelihood that ANAP wants to move from the origin and that the electoral mean is not a LNE of 2002 Turkish election.

A.2.3. Confidence Bounds for the 1997 Polish Election

From Table 3, the bounds of 𝛽^P₁₉₉₇=1.739 are [𝛽^P𝐿₁₉₉₇,𝛽^P𝑈₁₉₉₇]=[1.739±1.96×0.12]=[1.512,1.966]. Using (A.1), those for 𝜌^P1997_UPR in (60) are [𝜌^P𝐿₁₉₉₇,𝜌^P𝑈₁₉₉₇]=[0.002,0.022], so that those of 𝑐^P₁₉₉₇ in (61) from (A.2) and for the UPR’s characteristic matrix 𝐶^P1997_UPR in (62) from (A.3) are 𝑐^P𝐿₁₉₉₇,𝑐^P𝑈₁₉₉₇=[2(2),2(2=[],𝐶1.512)(1−2×0.022)(1.966)(1−2×0.002)()]5.782,7.833_P𝐿1997,𝐶^P𝑈₁₉₉₇=2()2()=,.1.512)(1−2×0.0221001−𝐼,1.966)(1−2×0.0021001−𝐼1.8910.0000.0001.8912.9160.0000.0002.916 (A.13) With 𝑐^P₁₉₉₇ significantly greater than 2, the necessary condition for convergence to the mean is not met. The eigenvalues of the bounds of 𝐶^P₁₉₉₇ are positive, [𝐶^P1997𝐿_UPR=(1.891,1.891),𝐶^P1997𝐿_UPR=(2.916,2.916)] as are those of the central estimate of 𝐶^P₁₉₉₇ in (62). Thus, with a high probability UPR will not locate at the mean and the electoral mean is not a LNE of 1997 Polish election.

A.3. Convergence in Anocracies

A.3.1. Confidence Bounds for the 2008 Georgian Election

From Table 5, the bounds of 𝛽^G₂₀₀₈=0.78 are [𝛽^G𝐿₂₀₀₈,𝛽^G𝑈₂₀₀₈]=[0.78±1.96×0.06]=[0.66,0.89]. Using (A.1), those for 𝜌^G2008_N=0.05 in (65) are [𝜌^G200𝐿8_N,𝜌^G2008𝑈_N]=[0.03,0.07]. So that those of 𝑐^G₂₀₀₈ in (66) from (A.2) and for Natelashvili’s characteristic matrix 𝐶^G2008_N in (67) from (A.3) are 𝑐^G𝐿₂₀₀₈,𝑐^G𝑈₂₀₀₈=[2(),2(=[],𝐶0.66)(1−2×0.07)(1.730.89)(1−2×0.03)(1.73)]1.99,2.89_G2008𝐿N,𝐶^G2008𝑈_N=2()2()=,.0.66)(1−2×0.070.820.030.030.91−𝐼,0.89)(1−2×0.030.820.030.030.91−𝐼−0.060.030.030.050.370.050.050.52 (A.14) Since 𝑐^G₂₀₀₈ is not statistically different from 2, the necessary condition for convergence is not met. The lower bound of 𝐶^G2008_N has one negative and one positive eigenvalue [𝐶^G2008𝐿_N=(−0.068,0.058)], so that at the lower bound Natelashvili’s vote share function is at a saddlepoint. The upper bound has two positive eigenvalues [𝐶^G200𝑈_N=(0.355,0.535)] so that at the upper bound Natelashvili is minimizing his vote share. At the central estimate of 𝐶^G2008_N in (67) Natelashvili is also minimizing his vote share. Thus, with a high probability Natelashvili diverges from the mean and the mean cannot be a LNE of the 2008 Georgian election.

A.3.2. Confidence Bounds for the 2007 Russian Election

From Table 5, the bounds of 𝛽^R₂₀₀₇=0.181 are [𝛽^R𝐿₂₀₀₇,𝛽^R𝑈₂₀₀₇]=[0.18±1.96×0.01]=[0.15,0.20]. Using (A.1), those for 𝜌^R2007_SR=0.07 in (70) are [𝜌^R2007L_SR,𝜌^R2007𝑈_SR]=[0.04,0.12]. So that those of 𝑐^R₂₀₀₇ in (71) from (A.2) and for SR’s characteristic matrix 𝐶^R2007_SR in (72) from (A.3) are 𝑐^R𝐿₂₀₀₇,𝑐^R𝑈₂₀₀₇=[2(),2(=[],𝐶0.15)(1−2×0.12)(5.90.15)(1−2×0.04)(5.9)]1.35,2.28_R2007𝐿SR,𝐶^R2007𝑈_SR=2()2()=,.0.15)(1−2×0.122.950.130.132.95−𝐼,0.2)(1−2×0.042.950.130.132.95−𝐼−0.330.030.03−0.330.140.050.050.14 (A.15) With 𝑐^R₂₀₀₇ not significantly different from 2, the necessary for convergence is not met. The lower bound of 𝐶^R2007_SR has two negative eigenvalues, [𝐶^R2007𝐿_SR=(−0.30,−0.36)] implying that at lower bound SR’s vote share is at a maximum and SR stays at the origin. However, at the upper bound there are two positive eigenvalues, [𝐶^R2007𝑈_SR=(0.09,0.19)]. Thus at the upper bound SR’s vote share is at minimum and SR wants to move. At the central estimate of 𝐶^R2007_SR in (72) SR also has two negative eigenvalues suggesting that SR wants to remain at the origin. So it seems more likely that SR will stay at the origin and that the mean is a LNE of the 2007 Russian election.

A.3.3. Confidence Bounds for the 2010 Azerbaijani Election

From Table 5 the bounds for 𝛽^A₂₀₁₀=1.34 are [𝛽^A𝐿₂₀₁₀,𝛽^A𝑈₂₀₁₀]=[1.34±1.96×0.29]=[0.77,1.91]. Using (A.1), those for 𝜌^A2010_AXCP-MP=0.21 in (75) are [𝜌^A2010𝐿_AXCP-MP,𝜌^A2010𝑈_AXCP-MP]=[0.08,0.47]. So that those of 𝑐^A₂₀₁₀ in (76) from (A.2) and for AXCP-MP’s characteristic matrix 𝐶^A2010_AXCP-MP in (77) from (A.3) are 𝑐^A𝐿₂₀₁₀,𝑐^A𝑈₂₀₁₀=[2(),2(=[],𝐶0.77)(1−2×0.47)(0.931.91)(1−2×0.08)(0.93)]0.085,2.984_{A2010𝐿AXCP-MP},𝐶^A2010𝑈_AXCP-MP=[2()2()]=[].0.77)(1−2×0.47)(0.445−1,1.91)(1−2×0.08)(0.445−10.037,1.428 (A.16) With 𝑐^A₂₀₁₀ not significantly different from 1, the dimension of the policy space, the necessary and the sufficient (in this case the same) conditions for convergence are not met. This one-dimensional characteristic matrix has positive eigenvalues at the lower and upper bounds as does the central estimate of 𝐶^A2010_AXCP-MP=0.445 in (77). It is then very likely that AXCP-MP locates far from the origin and that the electoral mean is not an LNE for the 2010 election in Azerbaijan.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Prepared for presentation at the Journées Louis-André Gérard-Varet, 24-28 June, Marseille, and for presentation at the joint LSE-WashU workshop on Comparative political economy, September, 2013. This paper is based on work supported by NSF grant 0715929 and a Weidenbaum Center grant. Earlier versions were completed while Gallego was a visitor at the Center and later while Schofield was the Glenn Campbell and Rita Ricardo-Campbell National Fellow at the Hoover Institution, Stanford.