# Ideological Ambiguity and Political Spectrum

Hector Galindo-Silva
<EMAIL_ADDRESS>Department of Economics, Pontificia Universidad
Javeriana. I thank participants at several seminars, as well as the editor and
two anonymous referees for very helpful remarks and suggestions. Any remaining
errors are my own.

###### Abstract

This study examines the relationship between ambiguity and the ideological
positioning of political parties across the political spectrum. We identify a
strong non-monotonic (inverted U-shaped) relationship between party ideology
and ambiguity within a sample of 202 European political parties. This pattern
is observed across all ideological dimensions covered in the data. To explain
this pattern, we argue that centrist parties are perceived as less risky by
voters compared to extremist parties, giving them an advantage in employing
ambiguity to attract more voters at a lower cost. We support our explanation
with additional evidence from electoral outcomes and economic indicators in
the respective party countries.

Keywords: Political parties, Ambiguity, Extremism, Centrism
JEL classification: D72.

##  1 Introduction

Political parties often adopt ambiguous and inconsistent positions, a
phenomenon that spans various countries, party systems, and time periods. This
practice can limit voters’ knowledge of the policies potential leaders intend
to implement if elected, thus posing a considerable challenge to democracy.

The primary objective of this study is to delve into the factors driving such
electoral behavior, with a specific focus on a relatively intuitive yet
insufficiently explored aspect: the degree of extremism or centrism inherent
in political party ideologies. We specifically investigate the relationship
between ambiguity and the ideological positioning of political parties across
the political spectrum.

Our main finding is the identification of a non-monotonic (inverted U-shaped)
relationship between party ideology and ideological blurriness (or
ambiguity)111In this study, ambiguity is defined as a deliberate strategy
employed by political parties to avoid taking a clear position on a particular
issue, effectively making it synonymous with strategic ambiguity, position-
blurring, issue clarity, or ideological clarity. While there are some relevant
differences between these terms (for instance, see the crucial distinction
between ambiguity and vagueness highlighted by Praprotnik and Ennser-
Jedenastik, 2023, or the important variations among avoidance, ambiguity, and
alternation as studied by Koedam, 2021), for the purposes of this study, we
believe that the general definition provided earlier is sufficient, given the
data used. In addition, note that we assume ambiguity is strategic, in line
with existing literature (e.g. Somer-Topcu, 2015; Bräuninger and Giger, 2018).
within a sample of 202 European political parties. Specifically, our results
indicate that political parties with a centrist ideology tend to possess a
more blurred ideology. This relationship holds true for each ideological
dimension covered in the available data. Furthermore, the relationship remains
robust across different econometric specifications, providing strong evidence
of its existence not only within countries but also within parties. While this
study is descriptive in nature and a perfect identification strategy is
challenging given the research question’s nature, the broad range of
correlational results, along with some instrumental variable (IV) estimates,
provides support for the interpretation that the ideological position
influences the degree of blurriness.

We also offer an explanation for this pattern and present additional empirical
evidence to support it. Our explanation builds upon Glazer (1990) but expands
on it. It is based on the notion that centrist political parties are perceived
as less risky by voters compared to extremist parties. One possible reason is
that policies proposed by extremist parties are often untested and
unconventional, whereas centrist parties often propose more established
policies. If the median voter is risk-averse, this suggests that centrist
parties enjoy an ex ante advantage over extremist parties. Consequently, if
ambiguity facilitates parties in aligning with the median voter, centrist
parties have stronger incentives than extremist parties to strategically
utilize ambiguity for electoral success.

To strengthen the plausibility of our explanation, we analyze additional data
on electoral outcomes and economic indicators from the respective countries of
the studied parties. Our analysis reveals that in countries where extremist
parties have recently held power and experienced significant economic
fluctuations, centrist parties not in government tend to adopt more ambiguous
positions. This finding aligns with our theoretical framework, which suggests
that incumbent extremist parties in such contexts should be perceived as
riskier, thereby reducing the potential benefits of ambiguity for them.

Furthermore, we explore the empirical plausibility of two alternative
explanations for the observed non-monotonic inverted U-shaped relationship
between ideology and political position. Firstly, we examine whether this
relationship could be attributed to the presence of single-issue parties that
adopt centrist and ambiguous policies to attract a broader voter base, given
that those policies are not their primary focus (e.g., Rovny, 2012; Somer-
Topcu, 2015). Secondly, we investigate whether this relationship can be
explained by the greater interest that centrist parties may have in having
more flexibility while in office to choose policies that better deal with new
information not available during elections (e.g., Aragones and Neeman, 2000;
Kartik et al., 2017). Our empirical analyses, however, fail to provide support
for either of these alternative explanations.

This paper contributes to an extensive body of literature investigating the
relationship between ambiguity and the electoral behaviour of political
parties. This literature can be categorized into two groups. First, there is a
wide range of theoretical studies exploring why political parties may choose
to be ambiguous in presenting their electoral programs to voters. These
studies consider factors such as voters’ risk aversion or the intensity of
their preferences (Shepsle, 1972; Alesina and Cukierman, 1990; Aragones and
Neeman, 2000; Aragones and Postlewaite, 2002; Laslier, 2006), the importance
of maintaining flexibility in office (Aragones and Neeman, 2000; Kartik et
al., 2017), the significance of electoral competition in multi-party systems
(Bräuninger and Giger, 2018), context-dependent voting patterns (Callander and
Wilson, 2008), the influence of policy-motivated donors (Alesina and Holden,
2008), and whether political parties possess information about the median
voter’s position (Glazer, 1990). While the primary contribution of this paper
is empirical, it connects with this theoretical literature in two ways.
Firstly, it provides a theoretical explanation for the main empirical result
by extending Glazer (1990)’s model. Secondly, the paper explores whether
certain theoretical models from this literature can account for the primary
empirical result.

In addition to the theoretical literature, there is a vast body of empirical
research examining position blurring by political parties. For instance, Han
(2020) used data on party positions and public opinion on major political
issues in Western Europe and found that, in polarized environments, political
parties present clearer positions on the issues they primarily focus on, but
less clear positions on secondary issues. Bräuninger and Giger (2018)
estimated ambiguity from electoral manifestos and found evidence that
platforms become more ambiguous as the preferences of two key stakeholders,
the voting public and the party’s core constituency, diverge. Some studies
have also utilized experimental data and found evidence that ambiguity can be
a winning strategy (Tomz and Van Houweling, 2009) and that it is more popular
among non-centrist voters when one of the candidates is a known centrist
(Tolvanen et al., 2022). While these papers offer valuable evidence regarding
the factors influencing position blurring by political parties, they do not
directly address how this blurring relates to the ideological position of
parties on the political spectrum. Only a few papers, to our knowledge, Rovny
(2012, 2013), Lo et al. (2016), and Praprotnik (2017), have explored this
relationship directly, making them closely related to our study.

Rovny (2012, 2013) explores how political parties strategically employ
ambiguity in elections where multiple issues are at play. Rovny argues that
parties tend to emphasize dimensions where they hold extreme positions while
blurring their stance on others. Rovny supports this theory with cross-
sectional empirical evidence from over 100 political parties across 14
countries in 2006. In comparison to Rovny, although our paper also examines
the relationship between ambiguity and centrism and finds an inverse
relationship between the two, it differs on three important points: it expands
the analysis to cover various time periods, more parties, and more countries;
it proposes much more robust econometric evidence regarding the existence of
this relationship, and it presents an alternative theory that does not focus
on the notion of single-issue parties, which we argue is consistent with more
robust and broader evidence.

Praprotnik (2017) examines ambiguity in electoral competition in Austria and
suggests that extremist parties, being niche parties, are penalized for
presenting vague and ambiguous programs. In contrast, centrist parties, not
being niche parties, have more incentives to be ambiguous. Praprotnik links
this hypothesis to the government status of parties, arguing that government
parties have incentives to decrease clarity in their campaign strategies
compared to parties in opposition. Consequently, extremist parties in
government might be equally or even more ambiguous than centrist parties in
opposition. In comparison to Praprotnik, our paper also examines the
relationship between ambiguity and centrism, but expands the analysis beyond
Austrian political parties and niche extremist parties.

Lo et al. (2016) propose a method to estimate the ideological clarity of
political parties based on party manifestos, and apply it to 74 parties in
four countries. They find a positive correlation between their estimates of
ideological clarity and the level of ideological extremism of political
parties in those countries. They suggest that centrist parties may find
ambiguous positions advantageous because they can appeal to a larger segment
of the electorate located in the center of the political spectrum. This study
expands upon Lo et al.’s research by including more countries and a larger
sample of over 200 parties, resulting in more robust findings. Additionally,
we explore alternative explanations and provide empirical evidence to support
a new explanation that complements and extends the previous work.222It is
worth noting that Lo et al. do not extensively delve into their primary
explanation. Specifically, they observe that extreme voters tend to view
ideological ambiguity as a sign of weakness or inadequate commitment to their
cause, resulting in reduced support for extreme parties when they adopt
ambiguous positions. The rationale behind why this behaviour appears to be
specific to extreme voters warrants further investigation and remains an
intriguing aspect to explore.

The paper follows the following structure: Section 2 provides a comprehensive
overview of the data utilized in the study and outlines our empirical
approach. Section 3 presents the key findings of our analysis. In Section 4,
we put forth our primary explanation for the observed results. Section 5 then
explores various alternative explanations. Lastly, Section 6 concludes.

##  2 Data and Empirical Strategy

###  2.1 Data

This paper’s analysis primarily relies on the Chapel Hill Expert Survey (CHES)
dataset (Jolly et al., 2022), which provides party position data on ideology
for numerous national parties across various European countries.333This data
is publicly available and can be found at https://www.chesdata.eu The CHES
dataset incorporates information from multiple surveys conducted in 1999,
2002, 2006, 2010, 2014, 2017 and 2019. These surveys involve a consistent
assessment of party positions by a substantial number of experts, encompassing
general left-right ideology, economic left-right orientation, and social
values (GAL-TAN).444The CHES survey invites each expert to assess the position
of each political party across three dimensions: (i) its overall ideological
stance, (ii) its ideological stance on economic issues, involving
classifications based on the party’s position on matters such as
privatization, taxes, regulation, government spending, and the welfare state,
and (iii) its views on social and cultural values. The first set of variables
employed in this study is derived from these assessments of party positions.
In the more recent surveys (2017 and 2019), the CHES introduced a series of
questions to gauge the extent of ambiguity in the establishment of these
positions by political parties. The second set of variables used in this study
is based on these measures of blurriness.555As a complement to our blurriness
measure, we also consider the standard deviation of expert assessments. I
thank a referee for suggesting this extension. While this alternative measure
allows for a larger sample size, it is employed solely as a robustness check
for our baseline results. We see it as reflecting variance among expert
assessments rather than directly measuring blurriness. Though significant
differences in expert assessments plausibly indicates blurred ideology, there
are instances where experts differ greatly without perceiving platforms as
very blurred. This may explain the relatively low correlation between the two
blurriness measures: 0.5 for the economic dimension and 0.14 for the social
and cultural values dimension. Nonetheless, our baseline results remain robust
with the use of this alternative measure.

Given our focus on examining the relationship between ideology and blurriness,
our analysis is restricted to the years 2017 and 2019. We merge the available
data on party positions in the economic and social values dimensions with
information on the associated degree of blurriness. The resulting sample
consists of 202 political parties, spanning across two time periods. This
comprehensive dataset incorporates assessments from a minimum of 25 experts,
covering at least two ideological dimensions.

All the data in this study is available at the expert-party-year level. In our
primary analysis, we aggregate this information by averaging across experts
for the same party-year. However, all our results remain robust when examined
at the individual expert-party-year level.666Due to changes in the identity of
experts across surveys, it is not feasible to construct a panel structure at
the expert-party-year level. Consequently, in terms of identification, the
econometric specification at the party-year level is equivalent to that at the
expert-party-year level. The information on party positions is measured on a
scale ranging from 0 to 10. In the economic dimension, a value of 0 represents
an extreme left position, while a value of 10 indicates an extreme right
position.777In the economic dimension, the CHES defines left and right as
follows: parties on the economic left advocate for an active role of the
government in the economy, while parties on the economic right support a
reduced role for the government. In the social values dimension, a value of 0
corresponds to a pro-libertarian/postmaterialist stance, whereas a value of 10
represents a pro-traditional/authoritarian perspective.888The CHES defines
these terms as follows: while ‘Libertarian’ or ‘postmaterialist’ parties favor
expanded personal freedoms, for example, abortion rights, divorce, and same-
sex marriage, ‘traditional’ or ‘authoritarian’ parties reject these ideas in
favor of order, tradition, and stability, believing that the government should
be a firm moral authority on social and cultural issues. Similarly, the data
on blurriness is also measured on a scale ranging from 0 to 10. A value of 0
indicates no blurriness at all, whereas a value of 10 signifies extreme
blurriness.999Specifically, the CHES asks each expert ‘how blurry was each
party’s position on …’.

In addition to the main dataset, we incorporate supplementary databases to
gain deeper insights into the underlying mechanisms that drive our primary
findings. These additional databases comprise country-level data on economic
outcomes, such as GDP per capita sourced from the International Monetary Fund
(IMF), as well as party-level data on electoral outcomes, including the seat-
share of parties in each election obtained from the ParlGov project (Doring et
al., 2022). A more detailed description of these databases will be presented
in subsequent sections of the study.

###  2.2 Empirical Strategy

In our initial econometric specification, we model the outcome
$blurriness_{ipct}$, which represents the average expert opinion on the level
of blurriness in the political position of party $p$ regarding issue $i$ in
country $c$ during year (or wave) $t$, as

$blurriness_{ipct}=\beta_{0}+\beta_{1}\times position_{ipct}+\beta_{2}\times
position^{2}_{ipct}+\gamma_{ct}+\epsilon_{ipct}$ (1)

where $position_{ipct}$ is the average opinion of experts on the position of
party $p$ regarding issue $i$ in country $c$ during period $t$.
$position^{2}_{ipct}$ refers to the square of this position. The term
$\gamma_{ct}$ are country $\times$ year fixed effects, and $\epsilon_{ipct}$
is the error term.101010We also estimate and report results from a model using
data at the expert level. However, as mentioned earlier, since the experts
change with each survey, including expert fixed effects is not possible.
Consequently, for identification purposes, this model is equivalent to the one
specified in Equation (1). As we will explain later, the inclusion of
$\gamma_{ct}$ is crucial in our identification strategy, as it allows us to
control for various country-related factors that may change over time, such as
population, quality of national institutions, characteristics of party
systems, and political polarization. We also introduce political party fixed
effects in certain specifications to account for party-related factors that
remain constant over time. Including these fixed effects greatly strengthens
the robustness of our results, indicating that the relationship exists not
only within countries but also within parties.111111It is worth noting that
the inclusion of these fixed effects significantly reduces the sample size,
which is why we do not use this specification as our main model. However, all
our results remain robust even when these fixed effects are included.

The coefficients of interest are $\beta_{1}$ and $\beta_{2}$, which capture
the relationship between the parties’ positions on each issue and the
perceived blurriness of these positions by experts.

To facilitate the analysis of mechanisms, we also estimate different versions
of the following equation:

$blurriness_{ipct}=\alpha_{0}+\alpha_{1}\times
centrism_{ipct}+\gamma_{ct}+\eta_{ipct}$ (2)

where $centrism_{ipct}$ represents the difference between 5 and the level of
extremism of party $p$ on issue $i$ in country $c$ during period $t$.
Extremism is measured as the absolute value of the difference between 5 and
the average expert opinion on each party’s position. The coefficient of
interest, $\alpha_{1}$, captures the relationship between the level of
ideological centrism of party positions on each issue and how experts perceive
the blurriness of these positions.121212An alternative way to define centrism
is by considering the median of the position distribution within each
dimension for every party system. I thank a referee for suggesting this
alternative. In the following section we show that estimates using either
measure are identical.

This paper is descriptive in nature, and achieving a perfect identification
strategy is challenging given the research question’s nature. Specifically,
the models presented in Equations (1) and (2) may not establish causal effects
due to at least two endogeneity concerns.

Firstly, there could be omitted factors driving the association between
blurriness and party position. While we control for any time-varying variables
at the country level and include party fixed effects, certain time-varying
variables at the party level might still influence the results. For instance,
the level of internal dissent within each party could be one such variable:
centrist parties may experience more internal dissent, and internal dissent
may lead to more blurred positions. Additionally, the age of a party might
matter, with older parties possibly adopting more centrist positions and
having less blurred stances. Government status is another potential factor:
centrist parties might be more frequently in government, and governing parties
may be rewarded for presenting clear ideological stances. In the next section,
we show that the main results remain robust even after controlling these
potential time-varying party-level factors, which helps alleviate the first
concern.

Secondly, a more critical issue is the potential presence of simultaneity bias
in the models of Equations (1) and (2). This bias could occur if the
perception of a party’s position as blurred (by experts) increases the
likelihood of identifying that position as centrist. To address this concern,
we adopt an instrumental variables (IV) approach by using lagged values of
position (or centrism) as instruments. The effectiveness of this estimation
strategy hinges on two primary assumptions. Firstly, the relevance assumption,
which we verify with our data. Secondly, and more importantly, the assumption
that conditional on fixed effects and controls, lagged values of position (or
centrism) affects blurriness only through their contemporaneous values (i.e.
the exclusion restriction assumption). Although our IV strategy does not
resolve all endogeneity concerns, we think it provides a reliable approach as
long as the lagged values of position do not influence blurriness through
unaccounted time-varying party-level factors. In addition to providing
supplementary evidence that aligns with this assumption, it is important to
highlight that we could not identify any such factors that could undermine the
validity of this IV strategy (given the substantial number of controls and
fixed effects included).

##  3 Main Results

Figure I presents the relationship between experts’ opinions on each party’s
position on economic issues (Fig. 1(a)) and social values (Fig. 1(b)) and the
perceived blurriness of these positions. Both figures exhibit a clear non-
monotonic inverted U-shaped pattern. Notably, the peak of these curves occurs
around the midpoint of the ideological position distribution (approximately
5), which corresponds to centrist political parties. Complementing these
results, Figures 2(a) and 2(b) show a strong positive association between the
level of centrism of each party’s position (as defined in the previous
section) and the level of blurriness of these positions, consistent with the
non-monotonic inverted U-shaped relations found in Figures 1(a) and 1(b).

Columns (1) to (6) in Table Figures and Tables provide the estimates of
Equations 1 and 2 for the outcomes analyzed in Figures I (Panel A) and 2
(Panel B). Columns (1) to (4) report the estimates of Equation 1, showing
positive and negative coefficients for $\beta_{1}$ and $\beta_{2}$,
respectively. All coefficients are individually and jointly statistically
significant at conventional levels. Additionally, the peak points of the
curves fall between 4.1 and 5.3, well within the range of ideological
positions (from 1 to 10). These results confirm the inverted U-shaped
relationships observed in Figures 1(a) and 1(b). Columns (5) and (6) present
the estimates of Equation (2), showing positive and statistically significant
effects, which is consistent with the strong positive correlations seen in
Figures 2(a) and 2(b). Finally, columns (7) and (8) in Table Figures and
Tables report fixed effects OLS estimates of a hypothetical monotonic
relationship between a party’s position and blurriness. However, all these
estimates are statistically insignificant at conventional levels, providing
evidence for the existence of a non-monotonic relation.131313Tables
LABEL:tableA1 and LABEL:tab_centrismalt in Appendix A.1 present estimates of
Equations (1) and (2) using expert-level data and an alternative centrism
measure based on the median position distribution (instead of the midpoint of
the scale). As expected, these estimates are virtually the same as those in
Table Figures and Tables.

As a complement to the previous findings, Figure A1 and Table
LABEL:tab_baselineSD in Appendix A.1 show estimates of Equation (1) for an
alternative outcome: the standard deviation in expert assessments relative to
party positions on the economic dimension (Figure 1(a) and columns (1) to (4)
of Table LABEL:tab_baselineSD) and the social values dimension (Figure 1(b)
and columns (5) to (8) of Table LABEL:tab_baselineSD). As discussed earlier
(see footnote 5), while we do not view this alternative measure of blurriness
as a substitute for the baseline results, it serves as an important robustness
check. Notably, the results in Figure A1 and Table LABEL:tab_baselineSD align
with our baseline findings, confirming the previously observed inverted
U-shaped relationship.

As discussed in the previous section, the estimates in Table Figures and
Tables may suffer from serious endogeneity bias. To address this concern, I
re-estimate the models in Equations (1) and (2), while (i) controlling for
potential confounding time-varying variables at the party level, and (ii)
instrumenting position and centrism with lagged values.

Table LABEL:tableA2 in Appendix A.1 presents estimates of the baseline models
while incorporating additional controls, namely, the level of internal dissent
within each political party, the age of each party, and whether the party is
in government. The results shown in Table LABEL:tableA2 are statistically
indistinguishable from those displayed in Table Figures and Tables. As
previously explained, this supplementary evidence reduces the likelihood of
omitted variables bias in our baseline results. Table LABEL:tableA3 in
Appendix A.1 provides instrumental variable (IV) estimates for the models
described in Equations (1) and (2), employing lagged position (and centrism)
as instruments. The table presents the IV estimates for the second stage,
along with the F-statistic pertaining to the first stage. In addition to
providing evidence consistent with the relevance assumption (e.g. showing a
F-statistic consistently exceeding 10), Table LABEL:tableA3 also shows
estimates that are statistically equivalent to those presented in Table
Figures and Tables. As previously argued, this supplementary evidence reduces
the likelihood of simultaneous bias in our baseline results and offers
additional evidence against the existence of omitted variables bias.141414As
discussed in Section 2.2, a pivotal assumption in our instrumental variable
(IV) specification is the exclusion restriction assumption. This assumption
entails that, given fixed effects and controls, the influence of lagged
centrism on blurriness operates solely through their contemporaneous values.
As argued in Section 2.2, given the extensive number of fixed effects and
controls included, we find it unlikely that a mechanism exists through which a
party’s centrism influences its blurriness level via a channel not captured by
such fixed effects and controls. However, to increase the plausibility of this
assumption, Table LABEL:tableA3aa in Appendix A.1 offers fixed effects OLS
estimations based on Equation (2), but incorporating lagged centrism. Notably,
in alignment with the exclusion restriction assumption, our findings show that
lagged centrism exhibits no substantial correlation with blurriness when
controlling for contemporaneous centrism (see columns (3), (4), (7), and (8)
of Table LABEL:tableA3aa).

While we acknowledge that the findings in Tables LABEL:tableA2, LABEL:tableA3
and LABEL:tableA3aa do not entirely eliminate bias in the main results of this
section, nor do they establish causality, at the very least, they indicate
that any potential bias is likely to be small.

##  4 Explanation: Uncertainty of Extremes

So far we have documented a non-monotonic relationship in the form of an
inverted U-shaped curve between the ideology of the parties and the degree of
ideological blurriness or ambiguity. Moreover, through the use of various
econometric models, we have presented arguments suggesting that the
ideological position not only correlates with blurriness but likely exerts an
influence on it. This result is important in and of itself. However, this
result can be explained in several ways. In this section, we propose an
explanation that we consider the most plausible and empirically examine its
validity. While the analysis is exploratory in nature, it holds the potential
to offer valuable insights.

Our proposed explanation is based on the idea that centrist political parties
are perceived as a safer choice for voters compared to extremist parties. This
perception stems from the tendency of extremist parties to propose untested
and unconventional policies, while centrist parties tend to advocate for more
established and moderate policies. Given the general risk aversion of the
median voter, centrist parties inherently possess an advantage over
extremists. As a result, centrist parties strategically exploit this advantage
by adopting more ambiguous positions than extremist parties. This strategic
approach enables them to align more closely with the ideal point of the median
voter (a point often unknown by political parties), thereby increasing the
probability of winning elections while incurring relatively lower
costs.151515Note that if ambiguity facilitates political parties in aligning
their positions with the median voter, this explanation shares similarities
with a finding in Aragones and Neeman (2000) (see Theorem 1.(ii) and 1.(iii)).
We thank a referee for suggesting this connection. Aragones and Neeman (2000)
establishes that, in equilibrium, only extremist parties, competing against
each other, can afford to adopt ambiguous platforms. As we will elaborate
below, our model diverges from that proposed by Aragones and Neeman (2000) as
it incorporates an initial asymmetry between extremist and centrist parties.

Before introducing the formal model that serves as the basis for this
explanation, it is worthwhile to delve into its implications and assumptions.
Firstly, the explanation is highly general and stylized, aiming to capture a
widespread phenomenon observed in a relatively large sample. In this sense, we
acknowledge the possibility of alternative explanations for more specific
contexts.161616In connection with this, it is worth noting that this
explanation aligns with the hypothesis that centrist political parties,
exhibiting ambiguity, can be seen as catch-all parties. Thus, an
interpretation of our main results based on centrists predominantly being
catch-all parties is consistent with our explanation.

Secondly, regarding the assumptions, it’s noteworthy that some are consistent
with existing empirical evidence. Research on the party-voter linkage
indicates that voters are often uninformed about political parties’ issue
positions (Adams et al., 2011, 2014). Additionally, voters tend to dislike
ambiguity from parties (Martin, 2019). There is also evidence that the median
voter’s position is usually not well-known to parties (Abou-Chadi and
Stoetzer, 2020; Lindvall et al., forthcoming).

Thirdly, regarding the assumption that all voters are risk-averse, while
evidence shows variations in voters’ attitudes towards certain relevant
dimensions Ehrlich and Maestas (2010); Kam and Simas (2010); Kam (2012);
Eckles et al. (2014), to our knowledge, there is no evidence suggesting
differences fundamentally linked to voters’ ideological orientation.
Importantly, relevant evidence suggests that voting for change or populist
options may depend on disparities in voters’ traits related to their risk
attitudes Steenbergen and Siczek (2017); Morisi (2018).171717This aspect,
solely rooted in the demand-side and not in equilibrium, remains unexplored in
our model and presents an interesting potential extension.

Finally, concerning the assumption that political parties intentionally employ
ambiguity as an electoral strategy, there is indirect yet substantial evidence
backing this claim. For example, Somer-Topcu (2015) show empirically that
broad appeal strategies help parties win votes by convincing voters they align
closely with their preferences. Moreover, additional empirical evidence
employing diverse methodologies support this premise Bräuninger and Giger
(2018); Rovny (2012); Lo et al. (2016).181818While most of the existing
literature supports the idea that ambiguity is deliberate and strategic, some
non-strategic sources for ambiguity have also been proposed. An example is the
existence of intra-party divides (Lehrer and Lin, 2020), which we believe does
not affect our results as they are robust to controlling for this factor.

###  4.1 Model

We now present a straightforward formalization of our previous intuition. The
model we propose extends Glazer (1990)’s model to scenarios where uncertainty
associated with ambiguous political party behaviour is asymmetric.

#### 4.1.1 Description of the game

In this model, we consider two political parties, competing to maximize their
probability of winning. One party is centrist ($C$), and the other is
extremist ($E$). Each party can propose a policy $s_{J}$ to the voters, where
$J\in\\{C,E\\}$ and $s_{J}\in\mathbb{R}$. Both parties are uncertain about the
median voter’s ideal point, but they know its probability distribution.
Similarly, voters are uncertain about the actual policy that will be
implemented by the winning party, knowing only the probability distribution
for this policy. For simplicity, we assume these distributions are uniform.
Parties also have the option to be ambiguous, where not being ambiguous means
proposing a policy $s_{J}\in S_{J}$, and being ambiguous means refraining from
selecting any value from a potentially different set $S^{a}_{J}$, where
$S_{J}\subseteq S^{a}_{J}$ and $\\#S_{J}\leq\\#S^{a}_{J}<\infty$ (with $\\#S$
denoting the cardinality of $S$).

Crucially, we assume that $\\#S^{a}_{C}<\\#S^{a}_{E}$, indicating that if both
types of parties are ambiguous, voters are ex ante more uncertain about the
policy implemented by the winning party if it is the extremist party.
Additionally, we assume that $S_{C}=S^{a}_{C}$, which is consistent with our
main assumption, and that $S_{C}=S_{E}$, meaning that if both parties decide
not to be ambiguous, the uncertainty experienced by voters is ex ante the
same. To simplify the calculations, we define
$S_{C}=S_{E}=S^{a}_{C}=\\{-k,-1,0,1,k\\}$ and
$S^{a}_{E}=\\{-l,-k,-1,0,1,k,l\\}$, where $1<k<l$. Finally, we assume that the
outcome of the election is determined by the preferences of a risk-averse
median voter, whose preferences can be represented by the utility function
$u(x)=-x^{2}$.191919This model can be extended in several dimensions, such as
the spaces of possible alternatives or the preferences of the median voter.
However, for simplicity, and given the primary empirical focus of this paper,
we present the simplest version of the model.

#### 4.1.2 Equilibrium

In Appendix A.2, we show that in equilibrium, when the centrist party is
unable to adopt highly extreme policies, the centrist party chooses ambiguity,
while the extremist party chooses not to be ambiguous. Specifically, we
establish that (i) if $k^{2}<\frac{3}{2}$, the centrist party opts to be
ambiguous while the extremist party decides not to be ambiguous, and (ii) if
$k^{2}>\frac{3}{2}$, both parties choose not to be ambiguous.

The main idea of the proof and the underlying rationale for this result arises
from the strategic decisions of parties to either adopt an ambiguous stance or
not. To delve into these scenarios, it is crucial to assess the median voter’s
expected utility in each situation, considering that both parties share the
primary objective of maximizing their probability of victory, and the median
voter theorem applies. In the first scenario, both parties choose ambiguity.
Here, the median voter’s expected utility is
$u(p_{C}|a)=-\frac{2(k^{2}+1)}{5}$ when a centrist party wins, and
$u(p_{E}|a)=-\frac{2(l^{2}+k^{2}+1)}{7}$ when an extremist party wins.
Importantly, $u(p_{C}|a)>u(p_{E}|a)$, indicating the median voter’s preference
for the centrist party when both parties are ambiguous. A second scenario
arises when exactly one party specifies a position. If a party $J$ declares a
position, given that $S_{J}=\\{-k,-1,0,1,k\\}$, with a probability of
$\frac{3}{5}$, this party selects a position within one unit of the median
voter’s ideal point, ensuring the median voter a payoff of at least $-1$. A
third scenario involves both parties specifying a position. In this case,
since $S_{C}=S_{E}$ (implying the median voter’s indifference), each party has
an equal probability of winning.

Building upon this analysis, Appendix A.2 explores how equilibrium actions
depend on the parameter $k$ in each of the aforementioned scenarios. We find
that for sufficiently small values of $k$—when the policy alternatives
available to the centrist party are not excessively radical—we expect centrist
parties to exhibit more ambiguity than extremist parties. This expectation
arises because, for sufficiently small values of $k$, the extremist party can
only defeat the centrist party by specifying a position, given the higher
uncertainty faced by the median voter if the extremist party wins. In
contrast, the centrist party, benefiting from lower uncertainty associated
with its potential victory, faces less pressure to specify a position and, by
being ambiguous, aligns itself closer to the median voter’s ideal point.

Our model, along with the explanation of the results in Section 3 that we have
proposed, centers on how voters perceive extremist political parties.
Specifically, it hinges on the uncertainty that voters face when considering
the consequences of choosing extremist parties compared to centrist parties.
Since extremist parties often propose untested and unconventional policies,
while centrist parties present more established and moderate policies, voters
perceive centrist political parties as less risky than extremist parties. We
argue that due to the inherent asymmetry concerning this uncertainty and the
fact that the median voter is generally risk-averse, centrist parties have
more incentives than extremist parties to strategically use ambiguity to win
elections.

###  4.2 Evidence supporting the uncertainty of extremes explanation

We now assess the empirical plausibility of the explanation presented in the
preceding subsection. Our analysis focuses on instances where extremist
political parties are plausibly perceived to entail greater risks when
compared to centrist parties. Our objective is to investigate whether, in
accordance with the proposed mechanism, centrist parties also exhibit a
heightened level of ambiguity. We center our analysis on countries marked by
substantial fluctuations in recent economic growth, particularly those where
an extremist party has held power in the recent past. While in such scenarios
it may be expected that all parties are, on average, less ambiguous, we
hypothesize that in such circumstances, voters anticipate lower economic risks
under the governance of a centrist party. Consequently, this anticipation
induces extremist parties to adopt less ambiguous stances than the centrist
parties.

To empirically test this hypothesis, we present estimates derived from various
specifications detailed in Table LABEL:tab4_uncertainty. Columns (1) to (3)
examine economic issues, while columns (4) to (6) focus on social values. In
all the specifications, we introduce an interaction term between each party’s
level of economic or social centrism and the lagged GDP growth variance in
each country.202020We employ data on real GDP growth from the IMF, publicly
available at
https://www.imf.org/external/datamapper/NGDP_RPCH@WEO/OEMDC/ADVEC/WEOWORLD.
Additionally, in columns (2), (3), (5), and (6), we further interact this
specification with a dummy variable indicating parties not in the
government.212121To construct this variable, we use party-level data on
electoral outcomes from the ParlGov project, publicly available at
https://www.parlgov.org/data-info/. We use the fact of not obtaining the
largest number of seats in national parliaments as a proxy for parties not
being part of the government cabinet. We assume that if a party does not
secure the most seats, it is less likely to join a government coalition and be
viewed as responsible for government actions. Although this measure is clearly
noisy (as there are several cases where parties with the most votes do not
govern), we believe the bias created by this noise is downward.

The result in column (1) shows an estimated coefficient of the first
interaction that is positive but statistically insignificant. Column (2)
indicates that for the economic dimension, in countries with larger GDP growth
variation in the last year, this effect becomes more pronounced and
statistically different from zero when centrist parties were not part of the
government in the last year. Column (3) demonstrates that these results remain
robust and have larger magnitudes when including party fixed effects. Finally,
columns (4) to (6), which focus on social values, do not show estimated
coefficients of the interactions that are statistically significant. This is
not surprising, given the emphasis on scenarios primarily related to economic
aspects, such as countries experiencing high variation in economic
growth.222222In Table LABEL:tab_uncertaintyGDPalt in Appendix A.1, we explore
the robustness of the results in Table LABEL:tab4_uncertainty using an
alternative measure of GDP growth variance, a dummy variable equal to one if
the GDP growth variance of each country in each year is greater than the
median of the distribution of GDP growth variances of all countries in that
same year. We thank a referee for suggesting this alternative measure. Table
LABEL:tab_uncertaintyGDPalt shows statistically identical results to those in
Table LABEL:tab4_uncertainty.

The findings in Table LABEL:tab4_uncertainty support the ‘uncertainty of
extremes’ hypothesis we have put forth. Specifically, they suggest that the
increased ambiguity of centrist parties may be linked to the perception that
extreme parties pose higher risks for voters. While these results, along with
those of the previous subsections, strengthen our preference for the proposed
hypothesis, they do not entirely rule out alternative explanations that could
also contribute to the results of Section 3. Nevertheless, the results in
Table LABEL:tab4_uncertainty indicate that the hypothesis presented in this
section could be a valuable component of a comprehensive explanation for the
findings obtained in Section 3.

##  5 Other Possible Explanations

###  5.1 Single-issue politics

An alternative explanation for our findings is rooted in the idea that in
multidimensional political competitions, certain parties strategically adopt
blurred and centrist positions on specific issues in order to appeal to a
broader voter base. Simultaneously, these parties may prioritize other,
potentially single, issues. If these ‘single-issue’ parties prioritize issues
outside of the economic or social dimensions, it is likely that they would
display decreased ambiguity and less centrism on those particular
issues.232323This electoral strategy has been referred to as a ’broad-appeal
strategy’ by Somer-Topcu (2015), and there is ample evidence of its
implementation by several political parties in Europe. Notable studies
examining this phenomenon include Rovny (2012, 2013), Han (2020), and Rovny
and Polk (2020).

To assess the plausibility of this explanation, we examine one of its
immediate implications: centrist parties that are identified as blurred in the
economic and social dimensions should not exhibit centrist positions in the
dimension(s) they prioritize.242424 As discussed in Section 2.1, we have data
on blurriness only for the economic and social dimensions. Therefore, if
single issue parties prioritize a dimension beyond these two, we cannot
observe the level of ambiguity of these parties regarding that specific
dimension.This is clearly a limitation of our analysis. However, as we will
elaborate later, we do possess information about the degree of centrism
exhibited by these parties across a wide range of dimensions. Thus, our
analysis stands valid on the assumption that these parties will exhibit less
centrism in their prioritized dimensions. Although empirically assessing this
hypothesis is challenging due to the difficulty in identifying single-issue
political parties, we can utilize information from several additional and
significant policy dimensions to explore whether the findings from the
previous section can be explained by the presence of single-issue political
parties that focus on these dimensions. By including a diverse set of
dimensions in the analysis,252525These dimensions encompass a wide range of
issues, such as immigration policy, multiculturalism policy, economic
redistribution, environmental sustainability, spending vs. taxes, deregulation
of markets, intervention in the economy, civil liberties vs. law and order,
social lifestyle, religious principles in politics, ethnic minority rights,
nationalism, urban/rural interests, protectionism, decentralization, anti-
Islam rhetoric, anti-elite rhetoric, and European integration. it is unlikely
that, on average, single-issue political parties would concentrate on a
dimension that is not correlated with any of them.

Figures III to V illustrate the correlation between the level of centrism of
political parties in the economic and social dimensions, as well as their
correlation with four other ideological dimensions not directly related to
economics or social values. Figure III specifically focuses on the
relationship between the centrism of parties in the economic and social
dimensions, demonstrating a robust positive correlation. Similarly, Figures IV
and V depict the correlation between the centrism of parties in the economic
and social dimensions with four additional dimensions: immigration policy,
environmental sustainability, decentralization policy, and anti-elite
rhetoric. Importantly, all the figures demonstrate significant positive
correlations.

To further support these findings, Table A9 in Appendix A.1 expands upon the
analysis presented in Figures III to V, encompassing 18 additional policy
dimensions and reporting the statistical significance of the correlations. The
table reveals two key findings. Firstly, the correlations observed in Figures
III to V are statistically significant. Secondly, and notably, the pattern
described in these figures extends to 15 other policy dimensions.262626These
other policy dimensions are: multiculturalism policy, economic redistribution,
spending vs taxes, deregulation of markets, intervention in the economy, civil
liberties vs. law and order, social lifestyle, religious principles in
politics, ethnic minority rights, nationalism, and protectionism.

The results presented in Figures III to V and Table A9 challenge the single-
issue hypothesis, providing evidence that centrist parties in both economic
and social dimensions exhibit centrist ideological positions across 15 policy
dimensions. This suggests that maintaining a centrist and blurred stance is
not indicative of prioritizing other issues. However, it is important to
acknowledge that these results are not consistent across all dimensions for
which data is available. Specifically, Table A9 reveals two noteworthy
observations. Firstly, the degree of centrism displayed by political parties
in the economic dimension does not correlate with their level of centrism in
the dimensions of European integration, urban/rural interests, and anti-Islam
rhetoric. Secondly, the level of centrism demonstrated by parties in the
social dimension does not align with their level of centrism in the dimension
of European integration.272727Figure VI plots these relations, and confirms
the lack of correlation. These findings suggest the possibility that centrist
parties in the economic dimension may be single-issue’ in any of the
aforementioned three policy dimensions, or that centrist parties in the social
dimension may be single-issue’ in the dimension of European
integration.282828These results are consistent with the extensive literature
that has examined the issue of the dimensionality of party spaces,
particularly questioning the extent to which more than one dimension is
relevant (Benoit and Laver, 2006; Marks et al., 2006; Albright, 2010; Stoll,
2011; Benoit and Laver, 2012; Rovny and Edwards, 2012; Bakker et al., 2012;
Rovny and Polk, 2019). While there is no consensus in this literature on how
many dimensions are necessary to analyze political issues without losing
significant information, there is agreement on the correlation between many
dimensions, especially within the European context. Some dimensions, like
European Integration and immigration, are considered exceptional (in this
regard, see Marks et al., 2006; Rovny and Edwards, 2012; Otjes and Katsanidou,
2017; Toshkov and Krouwel, 2022).

To investigate whether the variation in the dimensions of European
integration, urban/rural interests, and anti-Islam rhetoric plays a crucial
role in explaining the correlation between centrism and blurriness in the
economic and/or social dimensions (as reported in Section 3), we introduce an
interaction term in Equation (2). This interaction term captures the
relationship between the level of centrism in the economic or social dimension
and the level of centrism in each of these three additional dimensions. Our
hypothesis is that if centrist parties adopt a single-issue approach in any of
these additional dimensions, and if it is due to this single-issue focus that
they exhibit blurriness in the economic or social dimensions, then the
positive correlation between centrism and blurriness found in Section 3 should
be stronger when these parties demonstrate less centrism on the issues they
prioritize. In other words, the more extreme the single-issue stance of
parties in these dimensions, the more blurred their positions in the economic
or social dimensions would need to be to effectively attract voters.

Table LABEL:tab2_singleissue presents the results of the specified model,
where columns (1) and (2) focus on European integration, columns (3) and (4)
on urban/rural interests, and columns (5) and (6) on anti-Islam rhetoric.
Importantly, among all the specifications, only one interaction term is
statistically significant, and surprisingly, it exhibits a positive sign,
contrary to what the single-issue hypothesis would suggest.292929Specifically,
the significant coefficient corresponds to the interaction between the level
of centrism in the economic dimension and the level of centrism in the
European integration dimension (column (1)), indicating that the more centrist
(or neutral) the orientation of party leadership towards European integration,
the more blurred centrist parties’ positions are in the economic dimension.

Additionally, we consider another relevant characteristic of political
parties, for which we have data, which is their emphasis on reducing political
corruption. It is plausible that certain single-issue political parties
prioritize the fight against corruption and, in an effort to attract votes,
strategically blur their economic and social positions. In columns (7) and (8)
of Table LABEL:tab2_singleissue, we empirically analyze the plausibility of
this hypothesis using data on how salient corruption is on the platform of
each political party. The hypothesis is that the more single-issue the
political parties are in the ’dimension’ of corruption (i.e., the more
significant reducing political corruption is for the parties), the more
correlated their centrism in the economic and social dimensions should be with
their blurriness. However, the interaction term in this last specification is
not statistically significant for either the economic dimension or the social
dimension.

The results in this subsection provide evidence against the single-issue
hypothesis, as the ideological position of political parties across the
dimensions examined does not seem to explain the correlation between centrism
and blurriness in the economic and/or social dimensions reported in the
previous section. Although this evidence is insufficient to completely rule
out the single-issue hypothesis (since political parties might still be
‘single-issue’ in dimensions for which data is unavailable, or this hypothesis
could apply to a subset of our sample), considering the diverse range of
dimensions included in this analysis, we find this explanation less
plausible.303030An important study related to ours, conducted by Rovny (2012),
empirically investigates this issue. Using CHES data from 2006, the author
finds correlations suggesting that parties further from the center on a
particular dimension emphasize that dimension while also adopting centrist and
blurred positions on other dimensions. Table LABEL:tab_singleissuealt in
Appendix A.1 extends this analysis to more periods and employs econometric
specifications that include country-year and party fixed effects. Notably, the
findings of Rovny (2012) do not appear robust when these new specifications
are employed, particularly when using our preferred outcome measure of party
blurriness (columns (1) and (3)). While similar results to Rovny (2012) are
obtained for the dimension of social and cultural values when the outcome is
the standard deviation of expert assessments regarding the ideology of each
party (column (4)), the estimates are not jointly statistically different from
zero. This suggests that while the hypothesis of Rovny plausibly explains the
behavior of some ’single-issue’ political parties, it may not adequately
capture the possibly more general effect found in Section 3.

###  5.2 Post-electoral policy bias mitigation

Another potential alternative explanation for the results in Section 3 is
based on the idea that certain policy-relevant information is only revealed to
politicians after elections. As a result, political parties may strategically
blur their positions to adapt policies based on new information discovered
while in office (Aragones and Neeman, 2000; Kartik et al., 2017). If centrist
political parties are more concerned than extremist parties about policy
adaptability, their platforms might exhibit higher levels of blurriness,
potentially explaining the correlations observed in Section 3.

Empirically assessing this hypothesis is challenging, primarily because it is
difficult to identify ex ante which political parties prioritize policy
adaptability. As an alternative approach, we explore a crucial implication of
this hypothesis: if the correlation between centrism and blurriness reported
in Section 3 is indeed driven by parties’ concerns about policy adaptability,
then this correlation should be stronger when such adaptability becomes more
relevant. Economic crises represent such critical moments, where uncertainty
regarding the effectiveness of policies is heightened, and it is plausible
that parties particularly concerned about policy adaptability would be
especially blurred during such crises.313131In this particular hypothesis, our
focus is on economic crises. However, an alternate perspective could posit
that situations characterized by substantial GDP growth variance might also
qualify as critical moments wherein parties with a distinct emphasis on policy
adaptability could exhibit heightened ambiguity. Should this proposition hold
true, one could infer that the empirical evidence presented in Section 4 would
be congruent with this hypothesis, thereby diminishing the rationale for
favouring the explanation presented in that section. Nevertheless, despite
acknowledging the plausibility of this alternate hypothesis, there is a
crucial aspect regarding the evidence presented in Section 4 which appears
less congruent with the hypothesis currently under scrutiny: that centrist
parties are more ambiguous not only within countries undergoing pronounced GDP
growth variation in the preceding year but notably when they did not hold
power during that same year. If high GDP growth variance itself engenders
increased ambiguity in centrist political parties, why would it be so
important that these parties were not in power during the occurrence of such
events? This aspect explains our inclination to prefer the hypothesis advanced
in Section 4 over the alternative discussed in this subsection.

Table LABEL:tab3_postpolicybias presents the results of a specification in
which we include an interaction term between the level of centrism of each
party in the economic or social dimension and a dummy variable that equals 1
if the real GDP growth of the country (contemporaneous and one-year lagged)
falls below the median, and 0 if it exceeds the median. Notably, none of the
interaction coefficients are statistically different from zero.323232As for
the estimates provided in Table LABEL:tab4_uncertainty, we derived these
estimates using data on real GDP growth from the IMF. In Appendix A.1, Panel B
of Table LABEL:tabA6_crisisrob reports estimates similar to those in Table
LABEL:tab3_postpolicybias, but using a continuous measure of real GDP growth
and the proportion of years in the past in which each country experienced
negative GDP growth. The results are similar to those in Table
LABEL:tab3_postpolicybias: in neither specification is the interaction
coefficient statistically different from zero.

To address concerns about the GDP growth measure’s ability to capture economic
crises accurately, we conduct a robustness analysis in Panel B of Table
LABEL:tabA6_crisisrob in Appendix A.1. We use data on financial crises from
Laeven and Valencia (2020) to examine whether the relationship between
centrism and blurriness is more pronounced in countries that have experienced
a greater number of financial crises in the past (e.g., banking crises or
currency crises). However, even in this analysis, the interaction coefficients
remain statistically insignificant, consistent with the results in Table
LABEL:tab3_postpolicybias.

In summary, the results of Tables LABEL:tab3_postpolicybias and
LABEL:tabA6_crisisrob do not provide support for the post-electoral policy
bias mitigation hypothesis. While this does not entirely rule out the
possibility of the hypothesis holding true in specific contexts, there is
currently insufficient evidence in its favor.

##  6 Conclusion

In this study, we have explored the relationship between ideological ambiguity
and the political positioning of European political parties. Our key finding
is a robust non-monotonic (inverted U-shaped) relationship between party
ideology and ideological blurriness, observed consistently across all
dimensions we analyzed. The results indicate that party ideology influences
the degree of blurriness.

To explain this pattern, we have proposed a simple and novel theory based on
voters perceiving centrist political parties as less risky than extremist
parties. This perception arises because extremist parties often propose
untested and unconventional policies, while centrist parties present more
established and moderate policies. Given that the median voter is generally
risk-averse, centrist parties have a natural advantage over extremists.
Consequently, centrist parties are incentivized to strategically employ
ambiguity to approach the median voter’s ideal point more closely and secure
electoral victories, given the lower relative cost of ambiguity for them. Our
explanation finds support in additional evidence, including data on electoral
outcomes and economic indicators from the countries where the studied parties
operate. Furthermore, we have explored the empirical plausibility of two
alternative explanations for the observed non-monotonic relationship between
ideology and political position. However, our analysis does not support these
alternative explanations; instead, the evidence seems to contradict them.

Looking ahead, there are several promising avenues for future research. One
area is to further improve the identification of causal effects for the
reported patterns. Additionally, investigating whether these patterns hold
true for political parties in other regions, such as Latin America and Africa,
could provide valuable insights. Moreover, delving deeper into the empirical
examination of theories explaining these patterns, particularly identifying
scenarios where certain theories apply better than others, presents an
intriguing direction for future study. Lastly, exploring strategies or
policies that limit or counter political parties’ use of ambiguity, under the
assumption that such ambiguity might be detrimental to democracy, could offer
valuable insights for improving democratic governance.

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  * Steenbergen and Siczek (2017) Steenbergen, Marco R and Tomasz Siczek, “Better the devil you know? Risk-taking, globalization and populism in Great Britain,” European Union Politics, 2017, 18 (1), 119–136.
  * Stoll (2011) Stoll, Heather, “Dimensionality and the number of parties in legislative elections,” Party Politics, 2011, 17 (3), 405–429.
  * Tolvanen et al. (2022) Tolvanen, Juha, James Tremewan, and Alexander K. Wagner, “Ambiguous Platforms and Correlated Preferences: Experimental Evidence,” American Political Science Review, 2022, 116 (2), 734–750.
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## Figures and Tables

Figure I: Party position on economic and social issues and blurriness

((a)) Economic issues ((b)) Social values

.

Figure II: Centrism on economic and social issues and blurriness

((a)) Economic issues ((b)) Social issues

.

Table I: Party position and blurriness: baseline results

| Dep. variable: Blurriness of each party’s position
---|---
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
_Panel A :_ | Economic issues |
Position on economic issues |  1.848∗∗∗ |  1.826∗∗∗ |  1.844∗∗∗ |  1.347∗∗ |  |  |  0.072† | -0.234
| (0.127) | (0.147) | (0.151) | (0.611) |  |  | (0.049) | (0.200)
Position on economic issues sq. |  -0.186∗∗∗ |  -0.184∗∗∗ |  -0.186∗∗∗ |  -0.142∗∗∗ |  |  |  |
| (0.014) | (0.016) | (0.017) | (0.050) |  |  |  |
Centrism on economic issues |  |  |  |  |  0.810∗∗∗ |  0.877∗∗∗ |  |
|  |  |  |  | (0.084) | (0.293) |  |
Joint significance | 0.000 | 0.000 | 0.000 | 0.005 |  |  |  |
Vertex | 4.981 | 4.965 | 4.957 | 4.735 |  |  |  |
R-squared | 0.303 | 0.494 | 0.517 | 0.895 | 0.443 | 0.897 | 0.264 | 0.888
Observations | 406 | 406 | 406 | 212 | 406 | 212 | 406 | 212

_Panel B :_ Social and cultural values Position on social values 1.722∗∗∗
1.632∗∗∗ 1.664∗∗∗ 0.520 0.022 -0.200 (0.095) (0.092) (0.091) (0.415) (0.032)
(0.207) Position on social values sq. -0.163∗∗∗ -0.156∗∗∗ -0.159∗∗∗ -0.063∗∗
(0.009) (0.009) (0.009) (0.030) Centrism on social values 0.980∗∗∗ 0.702∗∗∗
(0.053) (0.154) Joint significance 0.000 0.000 0.000 0.038 Vertex 5.277 5.227
5.234 4.111 R-squared 0.462 0.601 0.627 0.894 0.655 0.902 0.229 0.890
Observations 406 406 406 214 406 214 406 214 Country fixed effects N Y - - - -
- - Year fixed effects N Y - - - - - - Country $\times$ Year fixed effects N N
Y Y Y Y Y Y Political party fixed effects N N N Y N Y N Y  Notes: Columns (1)
to (4) report fixed effects OLS estimates from Eq (1). Columns (5) and (6)
report fixed effects OLS estimates from Eq (2). Columns (7) and (8) report
fixed effects OLS estimates from a monotonic relationship between a party’s
position and blurriness. The dependent variable in all columns is each party’s
blurriness on economic issues (Panel A) and social and cultural values (Panel
B). The sample is limited to the years 2017 and 2019. In this sample, the
average party position on economic issues is 4.916 (with s.d. 2.167), the
average position on social and cultural values is 5.004 (with s.d. 2.724), the
average level of blurriness in economic issues is 3.675 (with s.d. 1.731) and
the average level of blurriness on on social and cultural values is 2.824
(with s.d. 1.548). Robust standard errors clustered by political party are in
parentheses. * denotes results are statistically significant at the 10% level,
** at the 5% level, and *** at the 1% level.

Y Y Y Y Y Y Political party fixed effects N N Y N N Y  Notes: All columns
report fixed effects OLS estimates from an specification in which we introduce
into Eq. (2) interaction terms between the level of economic or social
centrism of each party and (i) the one-year lagged GDP growth variance of the
country in which each respective party operates, and (ii) a dummy variable
denoting whether the party held no governmental position during the same year.
The dependent variable in columns (1) to (3) is each party’s blurriness on
economic issues. The dependent variable in columns (4) to (6) is each party’s
blurriness on social and cultural values. The sample is limited to the years
2017 and 2019. Robust standard errors clustered by political party are in
parentheses. * denotes results are statistically significant at the 10% level,
** at the 5% level, and *** at the 1% level.

Figure III: Centrism across policy dimensions: economic issues vs social
values

((a)) Simple correlation ((b)) Partial correlation

.

Figure IV: Centrism across policy dimensions: economic issues vs other issues

((a)) Immigration policy

((b)) Enviromental issues

((c)) Decentralization policy

((d)) Anti-elite rhetoric

Figure V: Centrism across policy dimensions: social values vs other issues

((a)) Immigration policy

((b)) Enviromental issues

((c)) Decentralization policy

((d)) Anti-elite rhetoric

Figure VI: Centrism across policy dimensions: role of European integration,
urban/rural interests, anti-Islam rhetoric

((a)) Economic issues vs European integration

((b)) Social values vs European integration

((c)) Economic issues vs urban/rural interests

((d)) Economic issues vs anti-Islam rhetoric

Y Y Y Y Y Y Y Y  Notes: All columns report fixed effects OLS estimates from an
specification in which we introduce into Eq. (2) an interaction term between
the level of economic or social centrism of each party and the position of
each party regarding the European integration, urban/rural interests, anti-
Islam rhetoric and the salience of corruption. The dependent variable in all
columns is each party’s blurriness on each party’s issue. The sample is
limited to the years 2017 and 2019. Robust standard errors clustered by
political party are in parentheses. * denotes results are statistically
significant at the 10% level, ** at the 5% level, and *** at the 1% level.

Y Y Y Y Y Y Y Y Political party fixed effects N Y N Y N Y N Y  Notes: All
columns report fixed effects OLS estimates from an specification in which we
introduce into Eq. (2) an interaction term between the level of economic or
social centrism of each party and the GDP growth of each country. The
dependent variable in columns (1) to (4) is each party’s blurriness on
economic issues. The dependent variable in columns (5) to (8) is each party’s
blurriness on social and cultural values. The sample is limited to the years
2017 and 2019. Robust standard errors clustered by political party are in
parentheses. * denotes results are statistically significant at the 10% level,
** at the 5% level, and *** at the 1% level.

##  Appendix A Appendix

###  A.1 Additional Tables

Social and cultural values Position on social values 1.554∗∗∗ 1.530∗∗∗
1.536∗∗∗ 1.267∗∗∗ 0.033 0.149∗∗∗ (0.055) (0.057) (0.056) (0.065) (0.030)
(0.049) Position on social values sq. -0.147∗∗∗ -0.145∗∗∗ -0.146∗∗∗ -0.112∗∗∗
(0.005) (0.005) (0.005) (0.006) Centrism on social values 0.800∗∗∗ 0.625∗∗∗
(0.030) (0.036) Joint significance 0.000 0.000 0.000 0.000 Vertex 5.295 5.263
5.262 5.666 R-squared 0.283 0.313 0.319 0.432 0.323 0.429 0.054 0.345
Observations 3527 3527 3527 3510 3527 3510 3527 3510 Country fixed effects Y N
Y - - - - - Year fixed effects Y N Y - - - - - Country $\times$ Year fixed
effects Y N N Y Y Y Y Y Political party fixed effects N N N Y N Y N Y  Notes:
Columns (1) to (4) report fixed effects OLS estimates for estimates from Eq
(1). Columns (5) and (6) report fixed effects OLS estimates for estimates from
Eq (2). Columns (7) and (8) report fixed effects OLS estimates for estimates
from a monotonic relationship between a party’s position and blurriness. The
dependent variable in all columns is each party’s blurriness on economic
issues (Panel A) and social and cultural values (Panel B). The sample is
limited to the years 2017 and 2019. Robust standard errors clustered by
political party are in parentheses. * denotes results are statistically
significant at the 10% level, ** at the 5% level, and *** at the 1% level.

Y Y Y Y Political party fixed effects N Y N Y  Notes: All columns report fixed
effects OLS estimates for estimates from Eq (2), using the alternative
definition of centrism described in footnote 13. The dependent variable in all
columns is each party’s blurriness on economic issues (columns (1) and (2))
and social and cultural values (columns (3) and (4). The sample is limited to
the years 2017 and 2019. Robust standard errors clustered by political party
are in parentheses. * denotes results are statistically significant at the 10%
level, ** at the 5% level, and *** at the 1% level.

Figure A1: Party position on economic and social issues and standard deviation
(SD) in expert opinions

((a)) Economic issues ((b)) Social values

.

N Y - - N Y - - Year fixed effects N Y - - N Y - - Country $\times$ Year fixed
effects N N Y Y N N Y Y Political party fixed effects N N N Y N N N Y  Notes:
All columns report fixed effects OLS estimates for estimates from Eq (1). The
dependent variable is each party’s standard deviation (SD) in expert opinions
on economic issues (columns (1) to (4)) and social and cultural values
(columns (5) to (8). The sample includes the years 2006, 2010, 2014, 2017 and
2019. Robust standard errors clustered by political party are in parentheses.
* denotes results are statistically significant at the 10% level, ** at the 5%
level, and *** at the 1% level.

Social and cultural values Position on social values 1.276∗∗∗ 1.725∗∗∗
1.618∗∗∗ 1.362∗∗∗ (0.144) (0.096) (0.102) (0.154) Position on social values
sq. -0.121∗∗∗ -0.164∗∗∗ -0.155∗∗∗ -0.129∗∗∗ (0.014) (0.009) (0.010) (0.015)
Centrism on social values 0.809∗∗∗ 1.009∗∗∗ 0.968∗∗∗ 0.837∗∗∗ (0.087) (0.056)
(0.059) (0.092) Joint significance 0.000 0.000 0.000 0.000 Vertex 5.271 5.261
5.211 5.276 R-squared 0.655 0.640 0.626 0.657 0.686 0.663 0.661 0.690
Observations 267 392 369 239 267 392 369 240 Control for internal dissent Y N
N Y Y N N Y Control for party age N Y N Y N Y N Y Control for government
status N N Y Y N N Y Y Country $\times$ Year fixed effects Y Y Y Y Y Y Y Y
Notes: Columns (1) to (4) report fixed effects OLS estimates for estimates
from Eq (1). Columns (5) and (8) report fixed effects OLS estimates for
estimates from Eq (2). The dependent variable in all columns is each party’s
blurriness on economic issues (Panel A) and social and cultural values (Panel
B). The sample is limited to the years 2017 and 2019. Robust standard errors
clustered by political party are in parentheses. * denotes results are
statistically significant at the 10% level, ** at the 5% level, and *** at the
1% level.

Social and cultural values Position on social values 1.861∗∗∗ 1.897∗∗∗
1.862∗∗∗ 1.838∗∗∗ 0.017 0.022 (0.126) (0.121) (0.119) (0.098) (0.039) (0.031)
Position on social values sq. -0.180∗∗∗ -0.183∗∗∗ -0.179∗∗∗ -0.177∗∗∗ (0.012)
(0.011) (0.011) (0.009) Centrism on social values 1.078∗∗∗ 1.081∗∗∗ (0.062)
(0.052) Joint significance 0.000 0.000 0.000 0.000 Vertex 5.183 5.187 5.189
5.206 Observations 328 328 328 328 328 328 328 328 R-squared 0.527 0.526 0.527
0.527 0.576 0.576 0.001 0.001 KP F statistic (first stage) 320.928 26.359
20.684 10.856 20.247 11.661 74.456 43.343 Country fixed effects N Y - - - - -
- Year fixed effects N Y - - - - - - Country $\times$ Year fixed effects N N Y
Y Y Y Y Y Political party fixed effects N N N Y N Y N Y  Notes: All columns
report second-stage IV estimates using lags of position and centrism as
instruments. Columns (1) to (4) report the second-stage IV estimates from Eq
(1). Columns (5) and (6) report the second-stage IV estimates from Eq (2).
Columns (7) and (8) report the second-stage IV estimates from a monotonic
relationship between a party’s position and blurriness. The dependent variable
in all columns is each party’s blurriness on economic issues (Panel A) and
social and cultural values (Panel B). The sample is limited to the years 2017
and 2019. Robust standard errors clustered by political party are in
parentheses. * denotes results are statistically significant at the 10% level,
** at the 5% level, and *** at the 1% level.

Y Y Y Y Y Y Y Y Political party fixed effects N Y N Y N Y N Y  Notes: All
columns report fixed effects OLS estimates from Eq. (2). The dependent
variable in columns (1) to (4) is each party’s blurriness on economic issues.
The dependent variable in columns (5) to (8) is each party’s blurriness on
social and cultural values. The sample is limited to the years 2017 and 2019.
Robust standard errors clustered by political party are in parentheses. *
denotes results are statistically significant at the 10% level, ** at the 5%
level, and *** at the 1% level.

Y Y Y Y Y Y Political party fixed effects N N Y N N Y  Notes: All columns
report fixed effects OLS estimates from an specification in which we introduce
into Eq. (2) interaction terms between the level of economic or social
centrism of each party and (i) a dummy variable equal to one if the one-year
lagged GDP growth variance of each country is greater than the median of the
distribution of GDP growth variances of all countries in that same year, and
(ii) a dummy variable denoting whether the party held no governmental position
during the same year. The dependent variable in columns (1) to (3) is each
party’s blurriness on economic issues. The dependent variable in columns (4)
to (6) is each party’s blurriness on social and cultural values. The sample is
limited to the years 2017 and 2019. Robust standard errors clustered by
political party are in parentheses. * denotes results are statistically
significant at the 10% level, ** at the 5% level, and *** at the 1% level.

Y Y Y Y Political party fixed effects Y Y Y Y  Notes: All columns report fixed
effects OLS estimates from Eq (1). The dependent variable is each party’s
blurriness on economic issues (columns (1) and (2)) and social and cultural
values (columns (3) and (4)). Columns (1) and (3) use our baseline measure of
a party’s ambiguity (the assessment of each expert regarding such ambiguity).
Columns (2) and (4) measure a party’s ambiguity using the standard deviation
of each expert’s assessment relative to the ideology of each party. The sample
in columns (1) and (3) includes the years 2017 and 2019, while the sample in
columns (2) and (4) encompasses the years 2006, 2010, 2014, 2017, and 2019.
Robust standard errors clustered by political party are in parentheses. *
denotes results are statistically significant at the 10% level, ** at the 5%
level, and *** at the 1% level.

Table A9: Centrism across policy dimensions: simple correlations Variables | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16) | (17) | (18) | (19) | (20)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Economic issues | 1.000 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Social values | 0.256 | 1.000 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| (0.000) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Immigration | 0.330 | 0.686 | 1.000 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Multiculturalism | 0.220 | 0.689 | 0.852 | 1.000 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Redistribution | 0.817 | 0.223 | 0.275 | 0.183 | 1.000 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Environment | 0.358 | 0.503 | 0.519 | 0.482 | 0.368 | 1.000 |  |  |  |  |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Spending vs taxes | 0.826 | 0.197 | 0.264 | 0.162 | 0.843 | 0.397 | 1.000 |  |  |  |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |  |  |  |
Deregulation | 0.866 | 0.238 | 0.288 | 0.176 | 0.805 | 0.353 | 0.827 | 1.000 |  |  |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |  |  |
Econ intervention | 0.858 | 0.254 | 0.289 | 0.217 | 0.852 | 0.360 | 0.835 | 0.889 | 1.000 |  |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |  |
Civil liberties | 0.369 | 0.762 | 0.775 | 0.770 | 0.325 | 0.550 | 0.302 | 0.343 | 0.344 | 1.000 |  |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |  |
Lifestyle | 0.247 | 0.815 | 0.528 | 0.547 | 0.289 | 0.448 | 0.257 | 0.276 | 0.321 | 0.648 | 1.000 |  |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |  |
Religion in politics | 0.403 | 0.579 | 0.261 | 0.291 | 0.469 | 0.349 | 0.457 | 0.411 | 0.448 | 0.461 | 0.679 | 1.000 |  |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |  |
Minority rights | 0.221 | 0.625 | 0.668 | 0.714 | 0.222 | 0.433 | 0.178 | 0.179 | 0.214 | 0.685 | 0.539 | 0.398 | 1.000 |  |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |  |
Nationalism | 0.239 | 0.760 | 0.757 | 0.804 | 0.141 | 0.484 | 0.123 | 0.171 | 0.177 | 0.773 | 0.637 | 0.414 | 0.752 | 1.000 |  |  |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.001) | (0.000) | (0.004) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |  |  |
Urban/Rural | 0.068 | 0.254 | 0.082 | 0.106 | 0.063 | 0.141 | 0.037 | 0.133 | 0.117 | 0.146 | 0.206 | 0.253 | 0.090 | 0.268 | 1.000 |  |  |  |  |
| (0.058) | (0.000) | (0.023) | (0.003) | (0.081) | (0.000) | (0.296) | (0.000) | (0.006) | (0.000) | (0.000) | (0.000) | (0.012) | (0.000) |  |  |  |  |  |
Protectionism | 0.422 | 0.385 | 0.506 | 0.419 | 0.231 | 0.267 | 0.216 | 0.376 | 0.367 | 0.469 | 0.294 | 0.221 | 0.302 | 0.467 | 0.359 | 1.000 |  |  |  |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |  |  |  |  |
Decentralization | 0.122 | 0.226 | 0.195 | 0.267 | 0.088 | 0.192 | 0.070 | 0.120 | 0.123 | 0.260 | 0.174 | 0.171 | 0.418 | 0.413 | 0.089 | 0.224 | 1.000 |  |  |
| (0.001) | (0.000) | (0.000) | (0.000) | (0.014) | (0.000) | (0.051) | (0.001) | (0.004) | (0.000) | (0.000) | (0.005) | (0.000) | (0.000) | (0.013) | (0.000) |  |  |  |
Anti-Islam | 0.063 | 0.202 | 0.004 | 0.075 | 0.087 | 0.166 | 0.090 | 0.040 | 0.087 | 0.183 | 0.312 | 0.198 | 0.247 | 0.224 | 0.104 | 0.055 | 0.051 | 1.000 |  |
| (0.305) | (0.001) | (0.952) | (0.217) | (0.151) | (0.006) | (0.141) | (0.510) | (0.155) | (0.002) | (0.000) | (0.001) | (0.000) | (0.000) | (0.087) | (0.367) | (0.410) |  |  |
Anti-elite | 0.180 | 0.241 | 0.395 | 0.386 | 0.093 | 0.253 | 0.089 | 0.176 | 0.141 | 0.374 | 0.095 | 0.018 | 0.334 | 0.344 | 0.078 | 0.365 | 0.348 | -0.006 | 1.000 |
| (0.000) | (0.000) | (0.000) | (0.000) | (0.123) | (0.000) | (0.142) | (0.003) | (0.019) | (0.000) | (0.116) | (0.765) | (0.000) | (0.000) | (0.195) | (0.000) | (0.000) | (0.922) |  |
European integration | -0.058 | -0.036 | -0.032 | -0.036 | -0.101 | -0.025 | -0.058 | -0.020 | -0.085 | 0.031 | 0.013 | -0.053 | 0.045 | 0.064 | 0.108 | 0.150 | -0.005 | 0.126 | 0.001 | 1.000
| (0.056) | (0.234) | (0.281) | (0.234) | (0.005) | (0.483) | (0.107) | (0.581) | (0.047) | (0.387) | (0.725) | (0.379) | (0.178) | (0.139) | (0.002) | (0.013) | (0.890) | (0.038) | (0.981) |
Notes: The p-values are reported in parentheses.

Y Y Y Y Y Y Y Y Political party fixed effects N Y N Y N Y N Y  _Panel B :_
Centrism on economic issues 0.756∗∗∗ 0.835∗∗∗ 0.792∗∗∗ 0.792∗∗∗ (0.274)
(0.082) (0.079) (0.079) Centrism $\times$ years with banking crisis 1.956
(9.854) Centrism $\times$ years with currency crisis -2.245 (4.499) Centrism
$\times$ years with debt crisis 4.320 (9.188) Centrism $\times$ years with
debt restructuring 4.320 (9.188) Centrism on social values 1.056∗∗∗ 1.012∗∗∗
0.953∗∗∗ 0.953∗∗∗ (0.149) (0.060) (0.057) (0.057) Centrism $\times$ years with
banking crisis -2.779 (4.413) Centrism $\times$ years with currency crisis
-2.699 (2.029) Centrism $\times$ years with debt crisis 5.711 (5.025) Centrism
$\times$ years with debt restructuring 5.711 (5.025) R-squared 0.442 0.442
0.442 0.442 0.656 0.657 0.657 0.657 Observations 404 404 404 404 404 404 404
404 Country $\times$ Year fixed effects Y Y Y Y Y Y Y Y Political party fixed
effects N N N N N N N N  Notes: All columns report fixed effects OLS estimates
from an specification in which we introduce into Eq. (2) an interaction term
between the level of economic or social centrism of each party and the number
of years with positive (or negative) GDP growth in each country (Panel A) and
the number of years with banking or debts crisis in each country (Panel B).
The dependent variable in columns (1) to (4) is each party’s blurriness on
economic issues. The dependent variable in columns (5) to (8) is each party’s
blurriness on social and cultural values. The sample is limited to the years
2017 and 2019. Robust standard errors clustered by political party are in
parentheses. * denotes results are statistically significant at the 10% level,
** at the 5% level, and *** at the 1% level.

###  A.2 Model for the main mechanism: proofs

In this section, we prove the main results outlined in Section 4.1.
Specifically, we delve into the strategic decision-making process of political
parties concerning the adoption of an ambiguous stance. Our analysis shows
that the equilibrium actions critically hinge on whether $k^{2}<\frac{3}{2}$
or $k^{2}>\frac{3}{2}$.

Using the payoffs that the median voter obtains in each possible scenario (as
detailed in Section 4.1), we proceed to examine the actions and best responses
of each party in these scenarios. We begin by considering the scenario where
the centrist party is ambiguous, and investigate what happens if the extremist
party does not follow suit. If the extremist party also chooses to be
ambiguous, as mentioned earlier, it will consistently end up losing. However,
if the extremist party decides to specify a position, two distinct scenarios
arise: (i) When $|-1|<|-\frac{2(k^{2}+1)}{5}|$, that is, when
$k^{2}>\frac{3}{2}$, the extremist party will defeat the centrist party with a
probability of $\frac{3}{5}$. (ii) When $k^{2}<\frac{3}{2}$, the extremist
party will only defeat the centrist party if the extremist party announces a
position that aligns with the median voter’s ideal point, and this occurs with
a probability of $\frac{1}{5}$. _Thus, regardless of $k$, the best response
for the extremist party when the centrist party is ambiguous is to avoid
ambiguity and instead specify a position._

Let’s shift our focus to the scenario where the extremist party adopts an
ambiguous stance. If the centrist party also remains ambiguous, as mentioned
earlier, the centrist party will always win. However, if the centrist party
specifies a position, we encounter a situation similar to the one studied
before: if $|-1|<|-\frac{2(l^{2}+k^{2}+1)}{7}|$, that is, if
$k^{2}+l^{2}>\frac{5}{2}$, the centrist party will defeat the extremist party
with a probability of $\frac{3}{5}$; and if $k^{2}+l^{2}<\frac{5}{2}$, the
centrist party will defeat the extremist party only if the centrist party
announced a position coincident with the median voter’s ideal point, and this
occurs with a probability of $\frac{1}{5}$. _Thus, if the extremist party opts
for an ambiguous stance, the best response for the centrist party is to also
remain ambiguous, regardless of the specific values of $k$ or $l$._

Now, let’s consider the scenario in which the extremist party specifies a
position. If the centrist party also specifies a position, then, as previously
mentioned, each party will have an equal probability of winning, with a chance
of $\frac{1}{2}$ for each. However, if the centrist party chooses instead to
remain ambiguous, we encounter a situation similar to the one analyzed
earlier: (i) When $|-1|<|-\frac{2(k^{2}+1)}{5}|$, meaning that
$k^{2}>\frac{3}{2}$, the centrist party will defeat the extremist party with a
probability of $\frac{2}{5}$. Notably, this probability is smaller than the
$\frac{1}{2}$ probability when both parties specify positions. (ii) When
$k^{2}<\frac{3}{2}$, the extremist party will only defeat the centrist party
if the extremist party announces a position coinciding with the median voter’s
ideal point, and this occurs with a probability of $\frac{1}{5}$. _Thus, when
the extremist party specifies a position: (i) If $k^{2}>\frac{3}{2}$, the best
response by the centrist party is to specify a position, as this yields better
odds of winning than remaining ambiguous. (ii) If $k^{2}<\frac{3}{2}$, the
best response by the centrist party is to remain ambiguous, given that this
increases their chances of winning compared to specifying a position._

Finally, let’s now consider the scenario in which the centrist party specifies
a position. If the extremist party also specifies a position, then, as
previously mentioned, each party will have an equal probability of winning,
with a chance of $\frac{1}{2}$ for each. However, if the extremist party
chooses instead to remain ambiguous, we encounter a situation similar to the
one studied before: (i) When $|-1|<|-\frac{2(l^{2}+k^{2}+1)}{7}|$, that is, if
$k^{2}+l^{2}>\frac{5}{2}$, the centrist party will defeat the extremist party
with a probability of $\frac{3}{5}$. (ii) When $k^{2}+l^{2}<\frac{5}{2}$, the
centrist party will defeat the extremist party only if the centrist party
announced a position coincident with the median voter’s ideal point, and this
occurs with a probability of $\frac{1}{5}$. _Thus, if the centrist party
specifies a position: (i) If $k^{2}+l^{2}>\frac{5}{2}$, the best strategic
response by the extremist party is to specify a position, as this yields
better odds of winning than remaining ambiguous. (ii) If
$k^{2}+l^{2}<\frac{5}{2}$, the best response by the extremist party is to
remain ambiguous, given that this increases their chances of winning compared
to specifying a position._

The previous results can be summarized as follows:

  * (i)

When $k^{2}<\frac{3}{2}$, the equilibrium outcome is that the centrist party
remains ambiguous, while the extremist party adopts a non-ambiguous stance.

  * (ii)

When $k^{2}>\frac{3}{2}$, since this implies that $k^{2}+l^{2}>\frac{5}{2}$,
the equilibrium outcome is that both the centrist and extremist parties choose
not to be ambiguous.

  * (iii)

There is no equilibrium in which both parties are ambiguous or where only the
extremist party is ambiguous.