What are the effects of entry of new extremist parties on the policy platforms of mainstream parties?

Abstract

We examine the consequences to policy-seeking, center-left and center-right parties under proportional representation following entry by extremist parties either at one or both ends of a unidimensional political spectrum. Assuming a symmetric, unimodal voter distribution, we show that the entry of a single extremist party on either the left or right drives both mainstream parties in the direction opposite to the extremist party. We argue that this three-party scenario is the most empirically relevant case in contemporary European politics. We also extend results of Casamatta and De Donder that project moderation of mainstream parties at equilibrium for PR elections with two extremist parties – one on each end – to a symmetric, unimodal voter distribution. In a setting not considered by these authors, we show that this moderating effect on the mainstream parties for a symmetric voter distribution is reversed if the voter distribution is sufficiently bimodal.

Keywords

Extremist parties Mainstream parties Party competition Policy-seeking Voter distribution

I. Introduction

Adams and Merrill (2006) trace the consequences of the entry into unidimensional political competition by a centrist party in a two-party situation with single-seat competition under plurality where there is a center-right party and a center-left party, both of which are policy-seeking, i.e., entry that changes a two-party game into a three-party game. While Duvergerian ideas of the dynamics of plurality-based competition (see Duverger, 1954; Fey, 1997; Palfrey, 1984) might suggest that the center-right and the center-left party would operate to squeeze out the center party by each moving further to the center, Adams and Merrill demonstrate the counterintuitive result that, if the major parties are both policy-seeking parties, they can be expected to propose policies that are much more divergent than what we should observe without the third party.

While new parties of many stripes may enter the political process, a common phenomenon today, especially in European polities, is the appearance of an extreme party that substantially cuts into the support of more centrist parties. Most commonly this extremist party is a populist party on the right (see Abou-Chadi and Krause, 2018; Eatwell, 2017; Kitschelt with McGann, 2017; Mudde, 2017; Widfeldt, 2014). We model this situation as a three-party game, with a center-right and a center-left party initially, followed by the entry of an extremist party – thus converting a two-party game into a three-party game. Such a scenario can be thought of as a natural complement to the Adams-Merrill (2006) set-up.

We first highlight the consequences of the entry of a single extremist party at one end of the political spectrum, since we believe this scenario has considerable empirical relevance, not just in proportional representation (PR) settings, which are the focus of this paper, like the Netherlands, but also in plurality settings, such as the UK, and in two-round elections, such as in France. But sometimes we have extremist parties entering the political space from both ends of the political spectrum – although usually in recent decades it is the right-wing party that is the greatest threat to the center. This possibility leads us to consider the four-party case, with a center-right and a center-left party at the beginning followed by entry of more extreme parties from both sides of the political spectrum. When there are two new extremist parties, one at each end of the spectrum, we extend findings of Casamatta and De Donder (2005) from log-concave voter ideal densities to any symmetric unimodal densities. Then we provide a new theorem that extends their work further, by examining what happens when the underlying voter distribution is bimodal.

We see our modeling as very relevant to events in the real world in terms of the decline in the vote share of mainstream parties throughout much of Europe (including the UK) and the response of established parties to new and more extreme ones that begin to garner substantial voter support. Our set-up allows us to consider both the empirically common case where it is a new (or renewed) right-wing party that is the greatest threat, and the case where the threat to the center comes from both ends of the spectrum.

1. Empirical Evidence about the Appearance of Extremist Parties and the Response to them of Mainstream Parties

Throughout Europe there are challenges to mainstream parties by more extremist parties. For example, since 2006, the Dutch Party for Freedom (PVV), which has taken anti-immigration views similar to those of the late Pim Fortuyn, has had parliamentary representation. After winning 24 seats in 2010 and joining, then withdrawing from, a coalition government with mainstream parties on the center right, this party dropped to 15 seats and out of government. In the UK, UKIP, a Eurosceptic and nationalist yet populist political party, has gone from nowhere to being the largest British party in terms of members of the European Parliament and, although winning only one MP in the House of Commons, received the third highest vote share in the last parliamentary election. Moreover, with a hard-exit Brexit ever more likely as of this writing (January 2019), the importance of UKIP was enhanced because of the role it played in successfully lobbying for Brexit and an anticipation (until internal divisions within UKIP surfaced) that they would make inroads into areas of traditional Labour support¹ as well in areas of traditional Conservative Party support. Similarly, in France, there is the potential for the Le Penistes, who offer a strong nationalist anti-immigration stance, to consistently become its second most important party. In these countries, the major challenge to mainstream parties from extremist parties seems to be coming from only one end of the political spectrum.²

Previous empirical work has looked at the reaction of center parties to the emergence of extremist parties on their flanks, but the conclusions have been mixed. Analyzing cross-national data, however, to test competing expectations is not that straightforward. On the one hand, there can be long-run time trends that operate largely independently of the structure of the party system. Alonso and Fonseca (2012) note that, in their study of 18 Western European countries over the period 1975–2005, mainstream parties of the left and right have trended in an anti-immigration direction, regardless of whether or not extreme right parties have been present. For example, in Austria, Germany, the Netherlands, and Sweden, the mainstream left moved in an anti-immigration direction before the emergence of significant far-right parties. On the other hand, some empirical research that focuses on immigration issues (see Bale et al., 2010, among others) suggests that the mainstream right has responded to the rise of radical right parties with a harder line on immigration. But both Alonso and Fonseca (2012) and Akkerman (2015) conclude that the degree of mainstream accommodation reported in this previous research is overstated.

2. Theoretical Expectations for Mainstream Response

The absence of clear empirical findings suggests the desirability of modeling the incentive structure of the center parties more formally, so as to develop better understanding of the factors that affect whether centrist party response to new extreme parties is one of accommodation or one of rejection. The modeling that is closest to our own is the ground-breaking work of Casamatta and De Donder (2005), who obtain explicit equilibria for the cases of two or four parties, under the assumptions that the voter distribution is log concave, but do not consider the case of three parties, when a single extreme party is present.³ While their work also includes the plurality case, which we mention briefly below (see note 11 in section II.1), as well as results for PR systems, we focus on extending this work in the PR context. We contribute in four ways: (a) most importantly for the potential empirical relevance of our modeling results, we obtain new results for the case of three parties when only one extreme party is present; (b) we prove the equilibrium results for any unimodal, symmetric voter distribution;⁴ (c) given two extremist parties, one on each flank, we demonstrate divergence of the mainstream parties at equilibrium for a sufficiently bimodal (polarized), symmetric distribution; and (d) we consider two different assumptions about how coalitions might form.

As in Casamatta and De Donder (2005) and Adams and Merrill (2006), we assume that the motivations of the two mainstream parties are policy-seeking and that party utility is a function of distance to the parliamentary mean. In this unidimensional setting, when parties are policy-seeking and the voter distribution is symmetric and unimodal, we show that entry of only one extremist party moves both mainstream parties in the opposite direction from that party. For symmetric, unimodal voter distributions, the entry of two extremist parties again, by contrast, drives both the center-right and the center-left party further toward the center. In one sense, this result is the opposite of what happens when a centrist party joins the competition under the Adams-Merrill (2006) model. In another sense, however, the findings here for the effect of extreme parties and the findings by Adams and Merrill (2006) for the effect of a centrist party are similar. In each case, the mainstream parties are driven away from the newly entering party or parties.⁵

We assume that, in the short run, the mainstream parties are free to move but that the position of the extreme party is fixed, which we believe realistically reflects the behavior of extreme parties, especially in their early years.⁶ For example, Jean-Marie Le Pen maintained a quite consistent stance on immigration and related topics for decades; it was only when his daughter Marine inherited the party that there was any real ideological movement – the renunciation of anything directly smacking of racism and anti-Semitism that accompanied her purge of her father from the party. But even she maintained the party’s anti-immigration stance. More generally, Adams et al. (2006), in a study of eight Western European democracies from 1976–1998, find that niche or ideologically-based parties typically did not respond to shifts in public opinion (or were punished at the polls if they did), while mainstream parties did respond to public opinion shifts, without paying electoral penalties.

However, any tendency of mainstream parties to move away from entering extreme parties can disappear or be reversed under alternative conditions, including bimodality of the voter distribution and/or discounting by the mainstream parties of the relevance of the extreme parties to the competition to join a governing coalition. The political climate in some countries may be such that particular parties, normally extremist parties, are not deemed ‘worthy‘ of participating in coalitions. Thus, as we suggest later, the policy locations of those parties may be less influential on the platform choices of mainstream parties than are the choices made by other mainstream parties. Moreover, the appearance of extreme parties may proceed hand in hand with increasing bimodality of the voter distribution, as a synergy develops between rejection of establishment norms in the electorate and appeals by extremist parties. Such effects can provide the conditions under which mainstream parties would respond to extreme party entry by moving to positions that are more, rather than less, extreme.⁷ We may summarize our findings in the following three statements:

1) Given either one or two extreme parties, if the voter distribution is symmetric and unimodal and mainstream parties account for all parties in assessing the expected policy outcome (i.e., place it at the parliamentary mean), they should move away from the extreme party(s).

2) If the voter distribution is symmetric and unimodal but mainstream parties disregard extreme parties in determining the expected policy outcome (i.e., place it at the mainstream party mean), the mainstream parties should stay put.

3) If the voter distribution is symmetric and sufficiently bimodal, however, we show that the mainstream parties should move to more extreme stands when extreme parties enter on their flanks, i.e., right-of-center mainstream parties should shift further right and left-of-center mainstream parties should shift further left.

II. Formalizing the Model

Employing a one-dimensional spatial model, represented by a line centered at 0, we investigate the changes in the equilibrium ideological locations of two mainstream parties, one a center-left party, one a center-right party, for scenarios in which new and more extreme parties are added to the party system. Initially there are two parties: a mainstream left party located (with notation chosen in anticipation of the introduction of further parties) at a point $x_{2}$ between −1 and 0, and a mainstream right party, located at position $x_{3}$ between 0 and +1, so that overall, $- 1 \leq x_{2} \leq 0 \leq x_{3} \leq 1$ . In the three-party model, an extreme party enters on the right,⁸ at some fixed location, $x_{4} \geq 1.0$ . In the four-party model, an additional extreme party enters on the left, at the fixed location $x_{1} \leq - 1.0$ . We assume that voters choose the party closest to their position. As noted earlier, in our theoretical work, we focus on the case where the electoral system is a proportional one, though our modeling framework has potential application to the plurality case as well.

1. Policy-seeking Motivations

In this paper we assume that the parties are policy-seeking, i.e., each party seeks to optimize its utility for the expected outcome. Policy-seeking assumptions were pioneered by Wittman (1973, 1977) and have been pursued by Calvert (1985), Londregan and Romer (1993), Groseclose (2001), and Adams, Merrill, and Grofman (2005), among many others, including Casamatta and De Donder (2005). We assume throughout that seat assignments under proportional representation (PR) faithfully reflect the parties’ vote shares, i.e., seat shares are exactly equivalent to vote shares. Furthermore, we assume that each party’s utility for a policy position is based on linear loss, i.e., party $j' s$ utility for position x is given by $- | q_{j} - x |$ , where $q_{j}$ is party $j' s$ preferred (ideal) position. Finally, we assume that parties faithfully maintain the same policy intentions before and after election and government formation.

Each party’s vote share is only one means toward the end of achieving a policy outcome. Because vote shares and participation in a government/governing coalition is not known in advance, the policy outcome is probabilistic. For the initial models discussed in this section, all parties are assumed to be potential members of a governing coalition. We consider two alternative interpretations of policy-seeking goals and show that, under certain assumptions, these two interpretations lead to the same mathematical formulation. Thus, whichever interpretation is more realistic, or whether some intermediate assumption would be most appropriate, the analyses and the conclusions would be the same.

Our first interpretation supposes that policy is a compromise among the proposals of all parties making up a parliament, weighted by their seat shares, without regard to whether a party is in government or opposition. Merrill and Adams (2007) comment that this parliamentary-mean model is perhaps most relevant to consensual democracies – such as Switzerland, Belgium, and the Netherlands – in which opposition parties’ prerogatives provide a degree of policy influence that may approach that of the governing parties.⁹ Furthermore, they cite empirical support by Warwick (2001), in analyses of ten democracies, that the parliamentary mean is a significant predictor of government policy declarations.

Accordingly, given N parties whose vote shares are $p_{i}, i = 1, . . ., N$ , we assume that party j seeks to minimize the distance between its ideal location and the weighted mean of the actual locations of all parties, where the weights are the vote shares of the respective parties. This weighted mean, denoted by $\bar{x}$ , where $\bar{x} = \sum_{i = 1}^{N} p_{i} x_{i}$ , is referred to as the parliamentary mean. Equivalently, party j seeks to maximize its utility $U_{j}$ for the parliamentary mean, where this utility is the negative of the distance between its ideal point and the parliamentary mean, i.e.,

U_{j} = - | q_{j} - \bar{x} |,

(1)

where $q_{j}, j = 1, . . ., N$ denote the ideal points (preferred policy positions) of the parties.¹⁰

Alternatively, as Merrill and Adams (2007) note, we may assume that following the election, one of the parties becomes primarily responsible for forming policy and that, a priori, the identity of this party is probabilistic, but related to the electoral strength of the parties. Under this dominant-party model in its pure form, the party awarded the opportunity to form a government (the formateur) might fully implement its preferred policies. Warwick (2001), in empirical analyses, suggests that the formateur does indeed derive policy advantages. Accordingly, instead of minimizing the distance to a parliamentary mean, participants might suppose that a single party would be expected to determine policy and that utility would depend on a lottery over the competing parties with weights proportional to the seat, and hence vote shares, i.e., $U_{j}' = - \sum_{i} p_{i} | q_{j} - x_{i} |$ . Thus the dominant-party model could reflect a PR system in which policy is expected to be set primarily by a single party, but in which the identity of this party is not known in advance.¹¹ If it is the case that the ideal points $q_{j}$ are more extreme than all party positions¹² (i.e., if both $q_{3} \geq x_{4}$ and $q_{4} \geq x_{4}$ , and similarly, $q_{2} \leq x_{1}$ and $q_{1} \leq x_{1}$ when one or both extreme parties are in the model), this definition of $U_{j}'$ is equivalent to that of $U_{j}$ given by equation (1) above for the parliamentary mean model.¹³

Since the formula for party utility for the outcome is the same for both the parliamentary-mean and the dominant-party models, provided the ideal points $q_{j}$ are more extreme than all party positions, this formula is also the same for an intermediate model that combines the aspects of both of these approaches (as long as the party ideal points are more extreme than all party positions).¹⁴ Such an intermediate model, in which policy is neither determined entirely by a single party nor arises as a complete compromise between all parties involved, appears more realistic than either pure model. With this robustness check in mind we proceed.

2. The Effects on Mainstream Parties of Entry of New, Extreme Parties at Fixed Locations, for a Normal Distribution of Voter Points

We illustrate, for a normal voter distribution, the dynamic effects of successive entry of extreme parties (at fixed locations) upon the Nash equilibrium¹⁵ strategies of the mainstream parties. We assume that the voter distribution has mean 0 and standard deviation 0.5.

We begin with only two (mainstream) parties – a center-left party and a center-right party. If, say, the center-right party moves right, it faces a trade-off because its position ( $x_{3}$ ) improves but its vote share ( $p_{3}$ ) decreases, so its contribution ( $p_{3} x_{3}$ ) to the parliamentary mean can either increase or decrease, depending on the trade-off.¹⁶

For the parliamentary mean model, under the assumption that the ideal points of the mainstream parties straddle $\bar{x}$ , i.e., $q_{2} \leq \bar{x} \leq q_{3}$ , Table 1 and Figure 1 indicate Nash equilibria for the mainstream parties as additional parties are added dynamically under the assumptions of normality. In the two-party scenario, the optimal (Nash equilibrium) positions for the mainstream parties ( $\pm 0.627$ ) are about 1.25 standard deviations from the mean voter position. When, in addition to two mainstream parties, an extreme party enters on the right (at the fixed position, +1.0), the mainstream party on the right moderates its position to 0.477, a little less than one standard deviation from the mean.

Table 1.

Policy-seeking Nash equilibrium for mainstream parties, with two-, three-, and four-party contests, assuming a normal electorate.

a. Parliamentary-mean model where the ideal points of the mainstream parties straddle the parliamentary mean ( $q_{2} \leq \bar{x} \leq q_{3}$ ).

	Extreme left party $x_{1}$	Mainstream left party $x_{2}$	Mainstream right party $x_{3}$	Extreme right party $x_{4}$
2 parties		–0.627 (50.0%)	0.627 (50.0%)
3 parties		–0.777 (38.2%)	0.477 (54.8%)	1 (7.0%)
4 parties	–1 (7.7%)	–0.427 (42.3%)	0.427 (42.3%)	1 (7.7%)

Note: Voters are normally distributed, with mean 0 and standard deviation 0.5. Extreme candidates, if present, are fixed at $x_{1} = - 1$ and $x_{4} + 1$ . These equilibria apply not only for the parliamentary-mean model when $q_{2} \leq \bar{x} \leq q_{3}$ , but also to the dominant-party model if $q_{2} \leq x_{1}$ and $q_{3} \geq x_{4}$ . For the four-party scenario, it is assumed that at equilibrium, $q_{2} = - q_{3}$ (this is the case for the two-party scenario by Proposition 1). Values in parentheses are the vote shares at equilibrium.

b. Dominant-party model when mainstream party ideal points are less extreme than the extreme parties, but more extreme than the two-party equilibrium locations.

	Extreme left party $x_{1}$	Mainstream left party $x_{2}$	Mainstream right party $x_{3}$	Extreme right party $x_{4}$
2 parties		–0.627 (50.0%)	0.627 (50.0%)
3 parties		–0.704 (43.9%)	0.550 (50.1%)	1 (6.1%)
4 parties	–1 (6.2%)	–0.540 (43.8%)	0.540 (43.8%)	1 (6.2%)

Note: Mainstream party ideal points are located at $\pm 0.8$ . Voters are normally distributed, with mean 0 and standard deviation 0.5. For the four-party scenario, it is assumed that at equilibrium, $q_{2} = - q_{3}$ . Values in parentheses are the vote shares at equilibrium.

Figure 1.

Effect of entrance of extreme parties on Nash equilibria of mainstream parties

This moderation can be intuitively explained because moderation draws vote share away from the center-left party (to the center-right party) as well as transfers vote share from the center-right party (to the extreme right party) – both of which move the parliamentary mean toward the center-right party’s ideal point. These effects are partly counterbalanced by the fact that the center-right’s more moderate position per se moves the parliamentary mean leftward. These effects suggest that, given a specific position for the center-left party, the center-right party achieves an interior equilibrium between the center point and the party’s ideal point, as illustrated for the parliamentary-mean model by the solid arrow in the example in Figure 2. That the net impact of all these considerations favors moderation at equilibrium for the center-right party is explained below.

Figure 2.

Utility maximization under parliamentary-mean and dominant-party models.

The mainstream left party, however, moves further to the left, in this case to location −0.777 (a little over 1.5 standard deviations from the mean). Thus, following the entry of an extreme party on the right, both mainstream parties move to the left, away from the extreme party. Roughly speaking, the center-left party, having lost vote share to the center-right party, attempts to make up its loss in utility by gaining desired policy position by moving left; the details are discussed below. Similarly, following entry of an extreme party on the left (but none on the right), both mainstream parties move right, again away from the extreme party.

Moving on to the four-party model, we note that adding an additional extreme left party to the three-party scenario obviates the need of the mainstream left party to move left and both mainstream parties draw in toward the center to symmetric locations at $\pm 0.427$ , each located less than a standard deviation from the mean (see Figure 1). Again, following the entry of extreme parties, each mainstream party moves away from its most proximate extreme party.

In the dominant-party model, the Nash equilibrium results are the same as above for the parliamentary-mean model, as long as we make the more stringent assumption that $q_{2} \leq x_{1}$ and $q_{3} \geq x_{4}$ . Suppose, instead, that in the dominant-party model the ideal locations of the mainstream parties are less extreme than the locations of the extreme parties (but still more extreme than their two-party equilibrium locations). Then as extreme parties enter, the equilibrium locations of the mainstream parties move in the same directions as for the parliamentary-mean model, but these movements are less pronounced, as indicated in Table 1.¹⁷

3. General Results for the Movements of the Mainstream Parties in Response to the Entry of Extreme Parties and Nash Equilibria

In this section we show that, in general, the response of mainstream parties to the entry of extreme party(s) is similar to that described above for a normal voter distribution and tabulated in Table 1.

As detailed above, voters are posited to choose the party closest to their position. We assume henceforth that voter ideal points have a distribution with mean equal to 0, specified by the density f and the cumulative distribution function F and that f is differentiable. More specifically, we assume that the outcome of an election is specified by the parliamentary mean (although, as we have indicated above, other assumptions lead to the same party utility functions under certain conditions). Denote by $m_{jk}$ the midpoint between party positions $x_{j}$ and $x_{k}$ , so that $m_{12} = (x_{1} + x_{2}) / 2$ , etc. Assuming proximity voting, in a three-party scenario with locations $x_{2}$ , $x_{3}$ , and $x_{4}$ , the vote shares of the parties are given by:

\begin{matrix} p_{2} = F (m_{23}) \\ p_{3} = F (m_{34}) - F (m_{23}) \\ p_{4} = 1 - F (m_{34}); \end{matrix}

(2a)

similarly, in a four-party scenario with locations $x_{1}$ , $x_{2}$ , $x_{3}$ , and $x_{4}$ , the vote shares are given by:

p_{1} = F (m_{12}), p_{2} = F (m_{23}) - F (m_{12}), p_{3} = F (m_{34}) - F (m_{23}), p_{4} = 1 - F (m_{34}) .

(2b)

We assume, for now, that the two extreme parties, if present, are fixed at $x_{1} = - 1$ and $x_{4} = + 1$ , respectively, and that these are the ideal points of these extreme parties (i.e., $q_{1} = x_{1}$ and $q_{4} = x_{4}$ ).

Our first proposition shows that in the base model with only two parties, a Nash equilibrium exists and is unique, and we specify the values of the equilibrium locations for each party.

PROPOSITION 1 (Nash equilibrium in a two-party model): In a model consisting of only the two mainstream parties, assume that these parties are policy-seeking based on the parliamentary mean, and that $q_{2} \leq \bar{x}$ and $q_{3} \geq \bar{x}$ . Assume further that the voter density, f, is symmetric and $f (0) > 0$ . Then a unique Nash equilibrium occurs for $x_{2} = - 1 / (2 f (0))$ and $x_{3} = + 1 / (2 f (0))$ .

PROOF:Casamatta and De Donder (2005)¹⁸ show that for a symmetric voter density, log concavity of $F (x)$ (i.e., $f (x) / F (x)$ is decreasing with x) is sufficient for a Nash equilibrium at $x_{2} = - 1 / (2 f (0))$ and $x_{3} = + 1 / (2 f (0))$ . In the Appendix, we show that symmetry alone is sufficient for this result to hold.

Definition: A symmetric probability density f is unimodal if $f (x)$ declines (or does not increase) as x recedes from the median on either side of that median.¹⁹

LEMMA 1: Assume that the two mainstream parties are policy-seeking based on the parliamentary mean, and that $q_{2} \leq \bar{x}$ and $q_{3} \geq \bar{x}$ . If there is one extreme party $x_{4}$ on the right (the three-party model), the rate of change of utility of each mainstream party with respect to that party’s location is given by

\begin{matrix} \frac{\partial U_{3}}{\partial x_{3}} = \int_{m_{23}}^{m_{34}} f (x) dx - [f (m_{23}) (x_{3} - m_{23}) + f (m_{34}) (m_{34} - x_{3})] \\ \frac{\partial U_{2}}{\partial x_{2}} = - \int_{- \infty}^{m_{23}} f (x) dx + f (m_{23}) (m_{23} - x_{2}) \end{matrix}

(3a)

If there are two extreme parties $x_{1}$ and $x_{4}$ , the first on the left and the latter on the right (the four-party model), the rate of change of utility of each mainstream party with respect to that party’s location is given by

\begin{matrix} \frac{\partial U_{3}}{\partial x_{3}} = \int_{m_{23}}^{m_{34}} f (x) dx - [f (m_{23}) (x_{3} - m_{23}) + f (m_{34}) (m_{34} - x_{3})] \\ \frac{\partial U_{2}}{\partial x_{2}} = - \int_{m_{12}}^{m_{23}} f (x) dx + [f (m_{12}) (x_{2} - m_{12}) + f (m_{23}) (m_{23} - x_{2})] \end{matrix} .

(3b)

Suppose further that the voter density, $f$ , is symmetric and unimodal. If $\partial U_{3} / \partial U_{3} \partial x_{3} \partial x_{3}$ is continuous, then, for either the three-party or four-party model and for any fixed value of $x_{2}$ , $U_{3}$ , reaches a maximum at some point $x_{3}'$ on the interval from 0 to $x_{4}$ and this point satisfies the recursive equation

x_{3} = \frac{p_{3} + m_{23} f (m_{23}) - m_{34} f (m_{34})}{f (m_{23}) - f (m_{34})} .

(4)

Similar recursive equations for maximum points hold under similar conditions for $x_{2}$ and $x_{3}$ in the three-party and four-party scenarios.

PROOF: See the Appendix.

Thus for example, in the three-party scenario (see Figure 3),

\partial U_{3} / \partial U_{3} \partial x_{3} \partial x_{3} = - A + B,

(5a)

where

$A = - \int_{m_{23}}^{x_{3}} f (x) dx + f (m_{23}) (x_{3} - m_{23})$ , and

$B = \int_{x_{3}}^{m_{34}} f (x) dx - f (m_{34}) (m_{34} - x_{3})$ .

Figure 3.

Three-party scenario depicting $\partial U_{3} / \partial U_{3} \partial x_{3} \partial x_{3}$ and $\partial U_{2} / \partial U_{2} \partial x_{2} \partial x_{2}$

Thus, A is the difference between the area of the rectangle between $m_{23}$ and $x_{3}$ and the integral of the probability density from $m_{23}$ to $x_{3}$ ; B is the difference between the integral of the probability density from $x_{3}$ to $m_{34}$ and the area of the rectangle between and $x_{3}$ and $m_{34}$ (see Figure 3).

By a similar argument (see Figure 3),

\partial U_{2} / \partial U_{2} \partial x_{2} \partial x_{2} = A' - B',

(5b)

where

$A' = - \int_{x_{2}}^{m_{23}} f (x) dx + f (m_{23}) (m_{23} - x_{2})$ , and

$B' = \int_{- \infty}^{x_{2}} f (x) dx$ .

Note that $\partial U_{3} / \partial U_{3} \partial x_{3} \partial x_{3}$ is independent of the location of $q_{3}$ , as long as $q_{3} \geq \bar{x}$ ; a similar statement holds for $\partial U_{2} / \partial U_{2} \partial x_{2} \partial x_{2}$ .

PROPOSITION 2 (Effects of the Extreme Party in a Three-Party Scenario): Assume the voter distribution is symmetric and unimodal, $f (0) > 0$ , and the two mainstream parties are policy-seeking based on the parliamentary mean, and that $q_{2} \leq \bar{x}$ and $q_{3} \geq \bar{x}$ . If a right-wing extreme party is added to this two-party scenario, both the mainstream parties move to the left. Similarly, if the extreme party is left wing, the mainstream parties move to the right.

PROOF: See the Appendix.

Intuitively, in the presence of the right-wing extreme party, any rightward movement by the center-right party draws vote share from the extreme right party, thus pushing the parliamentary mean leftward (because the center-right party exerts a weaker pull per vote on the parliamentary mean than does the extreme right party). At the same time such rightward movement by the center-right party would add vote share to the center-left party, so both of these effects by themselves decrease the center-right party’s utility.²⁰ But by moving right, the center-right party (Party 3) also gains utility because $x_{3}$ and hence $p_{3} x_{3}$ are increasing, which by itself would shift the parliamentary mean to the right, increasing the center-right party’s utility.²¹ These countervailing effects create a trade-off and lead to an equilibrium location between the center point and Party 3’s ideal point.²²

Why is the three-party equilibrium position for Party 3 to the left of the two-party equilibrium position? To see this, as noted above, when there are three parties, the shift of votes from the extreme-right party to the center-right party when the latter moves right pulls the parliamentary mean to the left, but with only two parties, this effect, of course, does not occur. The other two effects (losing votes to the center-left party and increase of $p_{3} x_{3}$ ) are the same in either a two-party or three-party scenario. Thus, rightward movement by the center-right party relative to its two-party equilibrium decreases its utility, while, conversely, moving left increases that utility. Hence, the equilibrium for the center-right party in a three-party scenario is to the left of its equilibrium in a two-party scenario.²³

Now we turn to the center-left party (Party 2). Why is the three-party equilibrium for this party also further to the left than in the two-party model? To see this, note that when the center-right party moves left (in reaction to the extreme right party), the cut-point $m_{23}$ between the two mainstream parties also moves left away from the point where the density peaks, so the center-left party can better afford to lose support to the center-right party than in the two-party scenario (i.e., it loses fewer votes by moving left). Since its benefit by gaining leftward position (the $p_{2} x_{2}$ term) is the same as in the two-party model, movement left is a net benefit, until a three-party equilibrium is reached.

PROPOSITION 3 (Effects of Extreme Parties in a Four-Party Scenario): Assume the voter distribution is symmetric and unimodal, $f (0) > 0$ , and the two mainstream parties are policy-seeking based on the parliamentary mean, and that $q_{2} \leq \bar{x}$ and $q_{3} \geq \bar{x}$ . If we add to this two-party situation two extreme parties at fixed positions, one to the left and one to the right, then both mainstream parties move inward, i.e., become less extreme.

PROOF: Casamatta and De Donder (2005) show that for a symmetric voter density, log concavity of $F (x)$ is sufficient for a four-party equilibrium (i.e., in which all four parties including the extreme parties are free to move) and provide a recursive formula for the party positions at equilibrium. The form of this equilibrium implies that the mainstream parties moderate their positions relative to their two-party equilibrium positions. Our proof of Proposition 3, for the slightly more general class of symmetric unimodal voter distributions, under the assumption that the positions of the extreme parties are fixed, is similar to that for Proposition 2 and is given in the Appendix.

Next, in a context not considered by Casamatta and De Donder (2005), we show that if the voter distribution is sufficiently bimodal, i.e., polarized instead of unimodal, the expectations are quite different: the mainstream parties may be motivated to diverge to extreme positions.

PROPOSITION 4 (Divergence of Equilibria for Bimodal Voter Distributions): Assume that the voter density f is symmetric and bimodal, $f (0) > 0$ , and that c denotes the least value greater than the right-hand mode for which $f (c) = f (0)$ (see the example in Figure 4). In the four-party model, if $x_{3}$ and $x_{2} (= - x_{3})$ constitute a Nash equilibrium, and $x_{1} < x_{2}$ , $x_{4} > x_{3}$ , then

x_{3} \geq c + \frac{h}{f (0)},

(6)

where $h = \int_{0}^{c} [f (x) - f (0)] dx$ is the area of (either) hump.

Figure 4.

Mixed normal voter distribution with extreme Nash equilibria

PROOF: See the Appendix.

Intuitively, the proposition says that in order for a Nash equilibrium to occur, the mainstream parties would need to be separated by a factor that increases with the size of the humps (that define each mode) and inversely related to the density at the center point of the scale. Thus, for example, if the voter distribution is the mixture of two equally weighted normal distributions with the same variance but means separated by three standard deviations,²⁴ and if $x_{3}$ and $x_{2} (= - x_{3})$ constitute a Nash equilibrium, then, by Proposition 4, $x_{3} / σ \geq 2.43 + 0.178 / 0.130 = 3.80$ , so $x_{3}$ would need to be nearly four standard deviations from the mean (see Figure 4). Accordingly, the mainstream parties would have to be exceedingly extreme to achieve equilibrium, so equilibrium under this circumstance would be correspondingly unlikely. In general, for bimodal distributions, the mainstream equilibrium locations rapidly diverge from one another as bimodality becomes more pronounced.

Finally, we investigate the existence and nature of Nash equilibria in the three- and four-party models for fixed extreme parties.

PROPOSITION 5 (Policy-seeking Nash Equilibrium): Assume that the voter density, f, is symmetric and unimodal, $f (0) > 0$ , and that $\partial U_{3} / \partial U_{3} \partial x_{3} \partial x_{3}$ is continuous as a function of $x_{2}$ and $x_{3}$ . Furthermore, if recursive application of equations (4) leads to sequences of values of $x_{3}'$ that converge to a limit $x_{3} *$ (and $x_{2} *$ is defined as $- x_{3} *$ ), then $(x_{2} *, x_{3} *)$ constitute a policy-seeking Nash equilibrium.²⁵

PROOF: See the Appendix.

4. Model Robustness: Effects of Alternative Assumptions on the Locations of Nash Equilibria

In this section, we note the effects of several alternative assumptions; details are in the Appendix. First, suppose that mainstream parties consider the extreme parties as pariahs that could not be considered for a coalition government under any circumstances, so that the two mainstream parties evaluate their utility of the election outcome entirely in terms of their own locations. Tests of this scenario for normal voter distributions indicate that the mainstream parties neither moderate nor do they become more extreme at equilibrium. Intuitively, discounting the contribution of an extreme right party to policy formation annuls any value that the extreme party might provide to compensate for the mainstream party’s loss of vote share to that extreme party.

Second, under an assumption that citizens located at relatively extreme locations abstain if no party is located near their position, we suggest by example that for plausible levels of abstention, it appears likely that mainstream parties would be motivated to moderate their strategies when extreme parties materialize, just as when there was no abstention.

Third, we find little effect on equilibrium positions when the extreme parties, as well as the mainstream parties, can change location.

Fourth, adding an independent office-seeking component exerts a strong moderating effect on optimal strategies of mainstream parties, drawing them toward the center point of the scale, whether or not extreme parties impinge. However, mainstream parties are drawn to complete convergence with a smaller admixture of vote share when extreme parties are present than when they are not. Finally, we explore in the Appendix the effects on equilibria of alternative values of the fixed positions of the extreme parties or if seat share is super-proportional relative to vote share.

III. Discussion

We have determined Nash equilibrium strategies for a pair of policy-seeking, mainstream parties under a PR system, on the flank of one of which, an extreme party enters. Under the assumption that the voter density is symmetric and unimodal and the policy outcome is interpreted as the parliamentary mean, we find that at equilibrium both mainstream parties move away from the new extreme party. If two extreme parties enter on each flank, we extend – to any symmetric, unimodal voter density – the result of Cassamata and De Donder (2005) that both mainstream parties moderate at equilibrium. Furthermore, we show that, under certain conditions, our results continue to hold when the policy outcome is determined by one dominant party.²⁶

On the other hand, we prove for the scenario with extreme parties on both ends of the spectrum that, if the voter density is significantly bimodal, any equilibria for the mainstream parties are rapidly divergent as polarization increases. Finally, examples suggest that if the mainstream parties discount extreme party potential to be part of government, their equilibrium positions are little affected by the entry of those extreme parties.

Alonso and Fonseca (2012), in their comparison of party systems with and without the presence of extreme right parties that take a strong anti-immigration position, find that Christian Democratic parties were less anti-immigration when extreme right parties were present than when they were not. This finding accords with our Proposition 2, which indicates that, under our assumptions, the introduction of an extreme party on the right should motivate the proximate mainstream right party to moderate its stand.

Mainstream party movement in the direction opposite to that of a burgeoning extreme party is suggested by three recent presidential elections: that of Alexander Van der Bellen as President of Austria over far-right Freedom Party candidate Norbert Hofer in December 2016, and in the elections of Jacques Chirac in 2002 and Emmanuel Macron in 2017 as President of France.

Florian Bieber of the University of Graz, Austria, in a New York Times op-ed (Bieber, 2016) suggests that the following lesson can be drawn from the 2016 Austrian presidential election:

Catering to the Freedom Party is likely to be a losing strategy for the centrists. One lesson from the presidential election is that presenting a real alternative is the way to defeat the far right. … [I]nstead of pandering to the far right, Mr. Van der Bellen presented himself as a clear alternative.²⁷

In the French presidential elections of 2002, the success of Jean-Marie Le Pen in entering the second round of the French presidential election led to a consensus repudiation of his politics by the left (denied a place on the second round because of fragmentation of the left-leaning vote), as well as by the right – in the person of Jacques Chirac. While in 2017, Emmanuel Macron, on the center right, successfully took a stance clearly in opposition to the Le Peniste attacks on the European Union and its immigration policies.

Supplemental Material

Supplemental_Material – Supplemental material for What are the Effects of Entry of New Extremist Parties on the Policy Platforms of Mainstream Parties?

Supplemental material, Supplemental_Material for What are the Effects of Entry of New Extremist Parties on the Policy Platforms of Mainstream Parties? by Samuel Merrill and Bernard Grofman in Journal of Theoretical Politics

Footnotes

Acknowledgements

We thank Jim Adams and Eri Bertsou for thoughtful and helpful comments on previous versions of this work.

Authors’ Note

A previous version of this paper was presented at the Annual Meeting of the American Political Science Association, Philadelphia, September 1–4, 2016.

Funding

Grofman’s work on this project was supported by the Jack W. Peltason (Bren Foundation) Chair at the University of California, Irvine.

Supplemental Material

The Appendix for this article is available online at http://www.socsci.uci.edu/~bgrofman/ or .

Notes

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