Policy Bundling:
A Model of Party Strategy in Multi-Issue Elections
Chitralekha Basu
Ph.D. Candidate, University of Rochester
Visiting Fellow at NICEP
[email protected]
∗
Last Updated: February 26, 2016
Abstract
This paper develops a theory of issue selection by parties when voters’ issue
priorities may be influenced by their campaigns. I write a model in which parties
are constrained in their policy choices by both activists and voters, and so, each
party wishes to de-emphasize the issue on which the preferences of its activists are
relatively non-centrist. However, parties also face a separate incentive to emphasize
issues that are more salient to voters, to clarify their positions on these issues
for sympathetic voters. The relative strength of these two incentives determines
whether, in equilibrium, we observe parties ‘talk past each other’, or we see ‘policy
bundling’, when at least one party addresses both issues. In the latter, a party
‘bundles’ popular and unpopular policies while disproportionately emphasizing the
former. This may explain why we particularly observe ‘issue engagement’ on salient
issues, but also see parties focusing disproportionately on particular issues.
∗
I am grateful to Bing Powell, Bonnie Meguid, Jim Johnson, Avi Acharya, Jim Adams, Sergio Ascencio, Rob Carroll, Mike Gibilisco, Tasos Kalandrakis, Matthew Knowles and Michael Thies for their
suggestions and thoughtful comments on earlier versions of this paper. This paper has also benefited
from suggestions made by seminar participants at the 2015 Southern Political Science Association Annual Meeting, the 2015 Midwest Political Science Association Annual Meeting, the 4th Annual Graduate
Student Conference in Comparative Politics at UCLA, the 2015 American Political Science Association
Annual Meeting, and the URDPS Women’s Working Group. Any errors that remain are my own.
1
1
Introduction
A vast body of work on what might variously be described as ‘heresthetics’, ‘issue competition’, ‘saliency theory’ or ‘issue ownership theory’ (Robertson 1976; Budge and Farlie
1983; Riker 1993; Petrocik 1996; Green-Pedersen 2007) has argued that parties primarily
compete by drawing voters’ attention to particular issues, in an effort to alter the dimensions on which they are evaluated.1 To date, researchers have amassed considerable
evidence from a wide range of countries that parties do focus disproportionately on issues that favor them.2 However, the incentives described in these studies cannot entirely
explain issue selection by parties in campaigns for two reasons.
First, although positional, or non-consensual, issues are a substantial focus of parties’
campaigns,3 ownership-based explanations of issue selection by parties largely deal with
valence or consensual issues, and do not focus on how a party may come to ‘own’ a
positional issue. In a positional context, an important determinant of a party’s advantage
on an issue is presumably the relative popularity of its position on the issue. Then, to fully
understand issue selection in campaigns, we must consider how parties come by particular
issue positions – and in particular, what binds certain parties to winning positions, and
others to losing ones. Although a large formal literature has identified various ‘centrifugal’
forces that might lead parties to take divergent policy positions4 , research on the sources
of parties’ issue ownership has not focused on explaining party ownership of positional
issues per se, and so has not incorporated insights from this literature.5
1
A large empirical and experimental literature on the importance of “priming effects” argues that
political advertising has a significant effect on voters’ issue priorities (Iyengar and Kinder 1987; Krosnick
and Kinder 1990).
2
For instance, Green and Hobolt (2008) observe that during the 2005 British elections, both Labour
and the Conservatives campaigned predominantly on their respective ‘owned’ issues, while GreenPedersen and Mortensen (2010) note that during the period of left-wing governments in Denmark between
1993 and 2001, the right-wing opposition continually drew attention to immigration, an issue on which
it was favored by voters. Other studies with similar findings include Druckman, Jacobs and Ostermeier
(2004), Vavreck (2009), Dolezal et al. (2014) and de Sio and Weber (2014).
3
In their analysis of whether issue engagement varied between consensual (or valence) and nonconsensual (or positional) issues in U.S. Senate campaigns between 1998 and 2002, Kaplan, Park and
Ridout (2006) found that 31 of the 43 issues addressed by candidates were non-consensual.
4
For a review of the formal literature that finds non-convergence in equilibrium, see Grofman (2004).
Grofman notes that if any of a large number of assumptions in the standard Downsian model are relaxed,
we find parties taking divergent positions in equilibrium. Whether and how much parties’ platforms
diverge is found to depend on features such as the proportionality of the electoral system, the scale of
abstention by extreme voters, and the importance of non-policy considerations such as partisanship to
voters.
5
Research on the sources of parties’ issue ownership has identified factors such as communication
by parties, party performance, and association with particular social groups—such as the poor—as
important (Walgrave, Lefevere and Tresch 2012; Walgrave, Lefevere and Nuytemans 2009; Brasher 2009;
2
Second, contrary to what might be expected under saliency or ownership theory, it
is well-established that parties actually spend much of their campaigns focusing on the
same issues – and in particular, on issues which are already salient to voters.6 Most
commonly, this has been explained as resulting from the importance of particular issues
to voters. It is reasoned, for instance, that parties may not want to ignore issues of public
concern that are the subject of extensive media coverage (Ansolabehere and Iyengar 1994;
Aldrich and Griffin 2003), as this may relinquish control over the framing of the issue to
their opponents. It is also possible that parties may be forced to confront unfavorable
but salient issues by their political opponents and by the media.7 Yet, few studies have
satisfactorily explained why, if a party is able to influence the salience of a preferred
issue, it will devote any time to an issue on which it is disadvantaged, even if this is an
important issue for voters.8
The objective of this paper is to formally develop a theory of issue selection that can
account for these dynamics. Building on existing research (Aldrich 1983; Whiteley and
Seyd 1994; Miller and Schofield 2003; Schofield and Miller 2007), I suggest that whether a
party ‘owns’ a positional issue may relate to the policy preferences of its activists, donors
and core supporters (hereafter, ‘activists’). It is reasoned that activists provide crucial
financial and logistical support in parties’ electoral campaigns, increasing their vote share
among all voters; further, activists’ campaign contributions are likely to be larger when
parties accommodate their policy preferences. Consequently, even if able to choose any
position on each issue, each party prefers to locate as close to their activists as possible,
leading it to take relatively unpopular positions on some issues. Then, if able to influence
the importance of different issues for voters through its emphasis, each party will choose
to emphasize the issue on which the policies preferred by its activists, on average, are
more centrist, in order to raise the electoral salience of the issue. I formalize this logic
Holian 2004; Stubager and Slothuus 2013). Meanwhile, on an individual level, partisanship and policy
attitudes have been identified as important for explaining a party’s perceived ownership of an issue by
voters.
6
This has been particularly noted in U.S. presidential and congressional campaigns (Kahn and Kenney
1999; Aldrich and Griffin 2003; Damore 2004, 2005; Sigelman and Buell 2004; Kaplan, Park and Ridout
2006; Sides 2006; Milita, Ryan and Simas 2014), but has also been observed in multiparty contexts like
Austria and Denmark (Green-Pedersen and Mortensen 2010; Dolezal et al. 2014; Meyer and Wagner
2015). For instance, when analyzing presidential campaigns in the U.S., Sigelman and Buell (2004)
found that both candidates spoke on the same issue, on average, a staggering 73% of the time.
7
In keeping with this logic, Sides (2006) notes that both the Democrats and the Republicans focused
on Social Security, education and health care in campaigns for U.S. House and Senate races in 1998—
the issues most prominent on the public’s agenda at the time—while Kaplan, Park and Ridout (2006)
identify a sizeable effect of issue salience on candidates’ emphases in Senate campaigns.
8
For an important exception, see Minozzi (2014), who argues that disadvantaged parties will choose
to campaign on salient issues in order to improve their reputation on such issues.
3
in a two-party model of party competition. In the model, each party can choose both a
position and a level of emphasis on each of two issues, but are constrained in their policy
choices by the preferences of their activists as well as of voters.
However, I observe that the extent to which a party emphasizes an issue can have
two effects on voters: it may influence the importance of the issue for voters, but it may
also influence voters’ certainty regarding the party’s position on the issue. This implies
an incentive for parties to emphasize an issue in order to clarify their position on it even
when they are not preferred by the majority of voters on the issue. In particular, for more
salient issues, parties are concerned to ensure that the voters who would prefer them
if they knew their stance on the issue are made aware of their position. As a result, in
their joint decisions of which issues to emphasize, and which positions to take on different
issues, parties compete by trying to focus voters’ attention on issues where their activists’
preferences are more popular, while simultaneously being compelled to emphasize issues
on which voters’ attention is already focused.
I find that if voters’ priorities are sufficiently responsive to parties’ emphases, parties
will simply campaign on the issue on which their activists’ preferred policy stance is more
popular with the majority of voters than their opponents’ activists’ stance, consistent
with saliency theory. Otherwise, we observe ‘policy bundling’, where at least one party
addresses both issues in its public statements, ‘bundling’ popular and unpopular policies
in its campaign while disproportionately emphasizing its popular policies. All else equal,
a party is more likely to address its less preferred issue if it is more salient to voters,
and if the issue is sufficiently salient to voters, it may even place more emphasis on that
issue than the issue on which its position is more popular with voters. To the best of my
knowledge, this is one of only a few spatial models of issue selection which find opposing
parties campaigning on the same issue in equilibrium, and the only model which finds
that parties will campaign on unfavorable, or non-owned, issues if these are especially
salient to voters – as is consistent with the empirical evidence.
The remainder of this paper proceeds as follows. Section 2 situates the paper within
the relevant literature. Section 3 introduces the modeling framework, and Section 4
presents analytical results. Section 5 concludes.
2
Related Literature
This paper joins a small but growing literature which analyzes the relationship between
the positions parties occupy on issues, and their decisions over which issues to emphasize
4
in their public statements (Meguid 2008; Tavits 2008; Wagner 2012; de Sio and Weber
2014). This paper, along with much of this literature, differs from traditional saliency
theory in assuming that parties can choose both a position and a level of emphasis on
each issue. This contrasts with early work in saliency theory, which viewed parties’
positions on each issue as largely inflexible (Budge and Farlie 1983, 279). In making this
distinction, it also differs from work which equates a party taking an extreme position on
an issue with emphasis, clarity or ‘intensity’ on the issue. This equivalence is implicit in
directional theory (Rabinowitz and Macdonald 1989), and also apparent in more recent
work relating to parties’ salience strategies (van der Brug 2004; Rovny 2012).9 This
overlooks the possibility that parties may be able to take a relatively extreme position
on an issue while placing little emphasis on that issue in their public statements and,
further, that this behavior may reduce the electoral salience of the issue in some cases.10
Among recent studies seeking to extend saliency theory, this paper relates most closely
to de Sio and Weber (2014), who argue that parties will want to emphasize issues with
high ‘issue yield’, where their preferred position is popular with existing supporters as
well as in the general electorate. A key implication of my model is that parties will
want to emphasize the issues on which the preferences of their activists, donors, or core
supporters are more popular, while de-emphasizing those issues on which these groups’
preferences are unpopular with the general electorate. This implication is consistent with
their findings. However, the model developed in this paper extends their theory in several respects. First, I emphasize the implications of activists’ and donors’ preferences
for parties’ incentives, in addition to those of core party supporters. Second, I develop
a strategic theory of parties’ emphasis strategies, where their opponents’ emphasis decisions also feature in each party’s calculus – whereas de Sio and Weber. Third, I explicitly
distinguish between parties’ positional and emphasis strategies, and discuss the responsiveness of each to activist and voter preferences, whereas de Sio and Weber see parties’
positions as essentially fixed in the short term. Fourth, I also introduce an incentive for
9
For instance, Rovny argues that “[o]utlying positions are more distinguishable and capture attention,
making the issue more prominent and the party more visible”(Rovny 2012, 5).
10
This is not inconsistent with the claim that issues on which parties are more polarized are frequently
more electorally salient for voters – as has repeatedly been observed in the literature (Rabinowitz and
Macdonald 1989; Carmines and Stimson 1989; Riker 1993; Green and Hobolt 2008). For instance, if voters
attach varying levels of importance to each issue, then issues on which parties take distinct positions
may naturally emerge as more significant in voters’ decision-making. If, further, parties face an incentive
to address salient issues (e.g., in order to clarify their position on such issues, as in this paper), then
I conjecture that parties may face an additional and competing incentive to emphasize issues on which
parties’ positions are relatively dispersed, rather than those on which parties have converged, even if to
the median. A formalization of this argument is left to future work.
5
parties to address issues on which their activists’, donors’ and core supporters’ preferred
policies are less popular with the average voter than those of their opponents’, but which
are salient issues for voters.
Since at least Petrocik (1996), researchers have obliquely observed that the issues
a party owns relate to the policy preferences of its ‘constituency’ or ‘coalition’ – typically referring to a party’s core voters. For instance, it has been argued that the U.S.
Democratic Party’s ownership of the civil rights issue relates to its overwhelming support
among African-American voters.11 However, this paper pins down a precise mechanism
linking the policy preferences of some of a party’s constituents to the issues it emphasizes,
which can explain how a party comes to ‘own’ some positional issues and not others.12
In arguing for the constraining role of a party’s activists on its policy choices, I draw on
a large and rich literature (McKenzie 1963; Schlesinger 1994; Aldrich 1983; Miller and
Schofield 2003). However, no other study, to my knowledge, sees the policy preferences of
a party’s activists as directly influencing its level of emphasis on different issues, or as indirectly influencing the importance of issues for the electorate. Most similarly, Miller and
Schofield (2003) explain partisan realignment over the 20th century in the U.S. as driven
by candidates seeking to woo disaffected activists to the existing activist coalitions backing their parties by taking a non-centrist position on a previously dormant issue. In their
study, the implications of such actions by parties, if any, for parties’ salience strategies
and for the electoral salience of issues are not made explicit.13
The mechanism I propose to explain why a party might address an issue it does not
own is that placing some emphasis on such an issue, when it is salient, is important for
clarifying the party’s position on the issue for sympathetic voters. This is in keeping
with a sizable literature which argues that ‘the more uncertain a voter is about candidate
positions, the less likely she is to support the candidate (Alvarez 1998).’14 At the same
time, particularly of late, several studies have suggested that positional ambiguity may
11
See Petrocik, Benoit and Hansen (2003, 625) for a list of issues they classify as owned by the
Democrats and Republicans, respectively.
12
The model leaves unanswered how a party comes by its particular combination of activists—in that
each party’s activists preferences are exogenously given in the model. Indeed, any explanation of issue
ownership that hinges on the character of a party’s ‘constituency’ is subject to this complaint.
13
In cases where a party taking a non-centrist position on an issue increases the salience of an issue,
and further, when taking a non-centrist position equates to emphasizing an issue, the mechanisms driving partisan realignment in the US presented by Miller and Schofield (2003) are consistent with those
presented in this paper. However, the analysis presented here suggests that these conditions may not
hold generically, i.e. that parties may de-emphasize issues that they hold a non-centrist position on, and
that this may reduce the electoral salience of the issue in some cases.
14
For other studies that argue similarly, see Enelow and Hinich (1981), Shepsle (1972), Bartels (1986),
Gill (2005) and Ezrow, Homola and Tavits (2014).
6
be electorally beneficial for parties or candidates (Campbell 1983; Alesina and Holden
2008; Tomz and van Houweling 2009; Rovny 2012; Kartik, van Weelden and Wolton
2015; Somer-Topcu 2015). However, most of these studies have focused on estimating
the effect of candidate or party ambiguity on vote choice, rather than of voter uncertainty
regarding parties’ true positions – or discussing the two interchangeably. While closely
related, these attributes are conceptually distinct: uncertainty is ‘a psychological state in
which voters are unsure about the policy positions of candidates’, while ambiguity is ‘an
attribute of candidate [or party] position taking’ (Tomz and van Houweling 2009, 83).15
These distinctions are important. As a positional strategy, ‘ambiguity’ is qualitatively
different from placing little emphasis, or remaining silent, on an issue – which is the action
I focus on instead. Indeed, when defining ‘ambiguity’, these studies refer to parties
taking ‘vaguely broad positions’ on issues, or presenting a ‘mixture of positions’ (Rovny
2012, 3).16 Instead, de-emphasizing an issue, or staying silent on an issue, more closely
resembles the ‘dismissive’ strategy analyzed by Meguid (2008). Quite reasonably, such
studies argue that ambiguity may not necessarily increase uncertainty among voters or
prove costly for parties, due to ‘projection’ by partisan voters, or a perception that
ambiguity by candidates on an issue indicates ‘flexibility’. It may also help convince
diverse groups of voters that the party is closer to their preferred position (Somer-Topcu
2015, 842). However, it remains probable that if a party were to instead avoid discussion
of an issue, this might elicit suspicion from voters who care about the issue, and will
ultimately choose between parties on the basis of their positions on this issue, but remain
uncertain regarding said party’s policies on the issue. That Tomz and van Houweling
(2009) find risk-averse voters who are certain about their own position least likely to
embrace ambiguous candidates seems consistent with this claim.17
Last but not least, this paper adds to a small literature seeking to formally model parties’ issue selection strategies (Austen-Smith 1993; Simon 2002; Amorós and Puy 2013;
Ascencio and Gibilisco 2014; Aragonês, Castanheira and Giani 2015; Egorov 2015; Dragu
15
Other studies that reference such distinctions include Aldrich, Ley and Schober (2013) and Milita,
Ryan and Simas (2014).
16
For instance, Alesina and Holden (2008) and Kartik, van Weelden and Wolton (2015) both explicitly
model ambiguity as candidates choosing an interval on an issue dimension rather than a single point,
while in their survey experiment, Tomz and van Houweling (2009) ask respondents to consider candidates
who take a position within some specified range. In Somer-Topcu’s analysis, a ‘broad-appeal’ strategy
can encompass parties taking clear but multiple positions on various issues, or a party selecting centrist
candidates while releasing an extreme election manifesto (Somer-Topcu 2015, 843).
17
Tomz and van Houweling (2009, 96) further qualify their findings with the statement that “[v]oters
may, for example, accept ambiguity within the range we studied but shun candidates who are totally
vague.”
7
and Fan 2015). I extend this literature in two respects. First, none of these papers simultaneously model parties’ positional choices and emphasis decisions. To my knowledge,
only Amorós and Puy (2013) and Dragu and Fan (2015) model parties choosing between
positional issues, and both take parties’ issue positions as exogenously given. By contrast, parties choose both a position and a level of emphasis on each issue in this paper.
Second, and most significantly, most of these studies do not find parties addressing the
same issues in equilibrium, and none of these studies have been able to explain why parties might address issues they do not already ‘own’ – contrary to the empirical evidence.
Studies only find parties campaigning on the same issue when parties have roughly equal
abilities on both issue dimensions (Egorov 2015), when parties share ownership of an
issue (Ascencio and Gibilisco 2014), or when a party is favored by voters on both issues
(Amorós and Puy 2013). Meanwhile, in the model presented by Aragonês, Castanheira
and Giani (2015), while parties may ‘invest’ in the quality of their proposals on the same
issue in equilibrium, parties never devote time to more than one issue in their campaigns.
Similarly, Dragu and Fan (2015) find that parties will never advertise the same policy
issue in equilibrium.18 By contrast, I find that so long as voters’ priorities are not too
responsive to parties’ emphases, at least one party will address both issues, including the
issue on which its chosen position is less popular with voters than that of its opponent.
3
The Model
This section presents a model of electoral competition with two vote-maximizing parties
and two issues. I characterize party and voter strategies in turn, before discussing their
joint implications for the electoral outcome.
3.1
Parties
There are two parties – denoted L and R – which compete for votes over issues X and
Y . Each issue is represented by a unit interval [0, 1] of possible policies on that issue,
and each party simultaneously chooses a position and a level of emphasis on each issue.
In choosing a position, each party takes into account both voter preferences and the
preferences of its activists on the issue. Let xj and yj denote the positions chosen by each
party j on each issue, and xaj and yaj denote the position preferred by party j’s activists,
18
Though, they qualify this finding with the statement that parties may still advertise the same policy
area, but when doing so, they will emphasize different aspects of that policy domain (Dragu and Fan
2015, 16).
8
on average, on each issue.19 We assume that 0 < xaL , yaL < 21 and 12 < xaR , yaR < 1,
implying that party L’s activists are left of center on both issues, and party R’s activists
are right of center on both issues. Additionally, we assume that |xaL − xm | < |xaR − xm |
and |yaR − ym | < |yaL − ym |, where xm = ym = 21 is the position of the median voter on
each issue, indicating that party L’s activists are closer to the median voter on issue X
than on Y , and party R’s activists are closer to the median voter on issue Y than on X.
Y
Meanwhile, eX
L and eR denote the level of emphasis placed by each party on each
Y
issue, respectively, where 0 ≤ eX
L ≤ 1 and 0 ≤ eR ≤ 1. These constraints capture
the notion that both parties have limited resources, and therefore cannot increase their
emphasis on an issue indefinitely. Additionally, by increasing their emphasis on one issue,
parties must devote less time to the other issue. This is formalized in the requirement
Y
Y
X 20
that eX
The extent to which a party emphasizes each
L = 1 − eL and eR = 1 − eR .
issue has two consequences: it influences the salience of issues X and Y for voters, and
also influences the certainty with which voters observe the party’s position on each issue.
That is, if a party increases its emphasis on one issue, say issue X, this increases the
probability that each voter will observe the party’s campaign on this issue and increases
the probability that each voter will observe both parties’ positions on the issue.21
3.2
Voters
There exists a continuum of voters, with voters’ ideal points uniformly distributed on
each issue. Some fraction of voters only care about issue X and some fraction only care
about issue Y . These fractions are determined endogenously, as will be discussed below.
Voters who care only about issue X have ideal points uniformly distributed over the [0, 1]
interval for issue X, and similarly for those voters who care only about issue Y . Voters’
utility functions are as follows:
Uij (xj , xaj ) = Aij (xj ) + hj (xj , xaj )
Uij (yj , yaj ) = Aij (yj ) + hj (yj , yaj )
19
As mentioned previously, the category ‘activists’ is meant to include a party’s donors and core party
supporters as well as its activists in the literal sense.
20
Alternatively, we may model parties as choosing their level of emphasis on each issue independently,
but requiring that their total emphasis on both issues not exceed one. It is easy to show that parties
will never choose a level of emphasis on each issue such that this constraint does not bind.
21
In the model, a voter may observe a party’s position on an issue without observing its campaign on
the issue. When a voter observes one party’s campaign on an issue, she observes that party’s position
with probability 1 and its opponent’s position with probability 12 . I elaborate on this assumption in more
detail in the following section.
9
where Uij (·) measures how much voter i likes party j. Here, xj and yj denote the positions
chosen by each party j on issues X and Y respectively, and xaj and yaj denote the position
preferred by party j’s activists, on average, on each issue. Voter utility is modeled as
having two components, a policy component, denoted Aij (·), and an issue-specific valence
component, denoted hj (·). More precisely,
Aij (xj ) = −(x̂i − xj )2
Aij (yj ) = −(ŷi − yj )2
where x̂i and ŷi denote the ideal points of voter i on issues X and Y respectively. Thus,
each individual voter’s support for a party is decreasing in her squared distance from the
party’s position on the issue she cares about.
Meanwhile, following Miller and Schofield (2003), the utility each voter i receives from
party j is argued to have a valence component, which is influenced by the time and money
party j receives from its activists. The justification for including such a term in voter
utility is as follows: the greater the financial and logistical contributions a party receives
from its activists, donors and core supporters, the greater the resources the party is able to
invest in buying media space, door-to-door campaigning, and in holding rallies, which in
sum, serve to improve its reputation for competence among voters.22 However, as policymotivated actors, how much activists contribute to a party’s campaign depends on the
policies it promises to implement when in government. Consequently, it is assumed that
activists’ campaign contributions will be larger when a party takes a position closer to its
activists on the issue those activists care about. I therefore model the valence component
of voter utility as decreasing in the distance between the party and its activists, on
average, on each issue:
hj (xj , xaj ) = λX 1 − (xj − xaj )2
hj (yj , yaj ) = λY 1 − (yj − yaj )2
Here, the parameters λX and λY , where λX , λY ∈ ( 12 , 1)23 , can be thought of as indicating
22
This follows a large literature which demonstrates the contribution of activist and donors to parties’
vote share. For instance, Whiteley and Seyd (1994) show that local party campaigning by Labour party
members in Britain significantly increased its vote share in the 1987 election, and not (just) via its effect
on turnout.
23
These constraints on λX , λY eliminate the possibility of ‘leapfrogging’ by parties. When λX or λY
are sufficiently low, the loss of voters from moving away from its activists is small enough that each
party has a profitable deviation from the equilibrium strategy profile to a position more extreme than
10
the value of activist support in parties’ campaigns: the higher λX and λY , the more
effective are activists’ logistical and financial contributions in improving a party’s valence.
This valence component in voter utility is modeled as issue-specific, in that hj (xj , xaj )
is the same for all issue X voters, and hj (yj , yaj ) is the same for all issue Y voters. This
assumes that activists’ contributions to a party are only effective in improving reputation
for competence on the issue that those activists care about. This additional assumption
increases the tractability of the model, as it mandates that parties’ optimal positional
choices on each issue are independent of their positional choice on the other issue.
Thus, the proportion of voters preferring each party on each issue is a function of
parties’ platforms and (indirectly) of the preferences of their activists. Without loss of
generality, let ψX denote the proportion of voters who care about issue X and prefer L
on issue X, with 1 − ψX denoting the proportion of issue X voters that prefer R on issue
X. Similarly, let ψY denote the proportion of issue Y voters that prefer R on issue Y ,
with 1 − ψY of issue Y voters preferring L’s position. ψX and ψY emerge as follows:
Z
1
ψX =
1{UL (xL , xaL ) > UR (xR , xaR )} dxi
0
Z
ψY =
1
1{UL (yL , yaL ) < UR (yR , yaR )} dyi
0
where 1{·} denotes the indicator function, 0 ≤ ψX ≤ 1 and 0 ≤ ψY ≤ 1.
There are two types of voters: ‘impressionable’, and ‘non-impressionable’. Impressionable voters are those whose issue priorities are influenced by the extent to which
parties emphasize each issue, whereas non-impressionable voters are those whose issue
priorities are inflexible. The proportion of impressionable voters in the populace is given
by α, where 0 < α < 1.24 For impressionable voters, the salience of issues X and Y is
given by πX and πY , where
X
πX = γ1 (1 + eX
L )(1 + eR )
πY = γ2 (1 + eYL )(1 + eYR )
and πX , πY ∈ [0, 1]. The parameters γ1 and γ2 are chosen such that πX + πY ≤ 1
Y
for any eX
L and eR in the interval [0, 1]. These allow for the possibility that voters’
priorities may, in general, be more rigid on one issue than the other. Meanwhile, for
its opponent. As we do not observe this in reality, I omit this case.
24
In this formulation, we can also interpret α as capturing the sensitivity of the electoral salience of
each issue to changes in party emphasis.
11
non-impressionable voters, the salience of issues X and Y is given by π̄X and π̄Y , where
π̄X , π̄Y ∈ [0, 1]. Consequently, π̄X and π̄Y capture the “exogenous” or “natural” salience
of issues X and Y , respectively. The salience parameters πX , πY , π̄X and π̄Y measure
the proportion of impressionable and non-impressionable voters who will vote according
to their preferences on issue X or Y (and issue X or Y alone), respectively.25 This is
equivalent to a formulation that specifies that, conditional on being an impressionable
voter, each voter will vote on the basis of their preferences on X with probability πX , and
on the basis of their preferences on Y with probability πY . I also assume that π̄X +π̄Y = 1,
and that proportion 1 − πX − πY of impressionable voters consider neither issue salient
and vote for each party with probability 12 .
The extent to which both impressionable and non-impressionable voters observe a
party’s campaign on an issue depends on the degree to which a party emphasizes that
issue. First, consider impressionable voters. The proportion of impressionable voters who
observe the campaign of party j on issues X and Y is given by ηjX and ηjY , respectively,
where
ηjX =
2eX
j
1 + eX
j
ηjY =
2eYj
1 + eYj
with ηjX , ηjY ∈ [0, 1]. For non-impressionable voters, the proportion who observe the
campaign of party j on issues X and Y is instead given by η̄jX and η̄jY , respectively,
where
2
η̄jX = 1 − (1 − eX
j )
η̄jY = 1 − (1 − eYj )2
with η̄jX , η̄JY ∈ [0, 1]. Then, the probability that a voter observes a party’s campaign on
an issue is always concave and increasing in parties’ emphasis on that issue, but the same
Y
level of eX
j or ej leads to a smaller proportion of impressionable voters observing parties’
campaigns on an issue than non-impressionable voters.
Finally, we assume that voters are highly risk averse. Recall that voters are issue
voters, each voter basing her voting decision on only one of the two issues. It is assumed
that if a voter observes one party’s position on the issue that voter cares about, but
25
This approach is similar to the way issue salience is modelled by Ascencio and Gibilisco (2014).
12
does not observe the other party’s position, then the voter will always vote for the party
whose position she observes. That is, voters always chooses to vote for ‘the devil they
know’ rather than for a party whose position is unknown. Without loss of generality,
consider issue X. If a voter who will vote on issue X observes both parties’ campaigns on
the issue, she observes both parties’ positions on that issue, and will vote for the party
she prefers on that issue. However, if a voter observes only one party’s campaign on the
issue, she will observe that party’s position with probability 1 and its opponent’s position
with probability 21 . The implicit assumption here is that observing one party’s campaign
improves a voter’s understanding of both the issue positions taken by that party and the
positions taken by their opponents. Lastly, if such a voter sees neither party’s campaign
on issue X, she will vote for each party with probability 21 .
3.3
Election Outcome
Y
Each party j chooses xj , yj , eX
j and ej to maximize its vote share. Consider party L.
Among impressionable voters who care about issue X, party L’s vote share, denoted VLIX ,
is given by
VLIX
ηRX (1 − ηLX ) (1 − ηLX )(1 − ηRX )
X
= (ψX ) ηL +
+
2
2
X
X
X
ηL (1 − ηR ) (1 − ηL )(1 − ηRX )
+
+ (1 − ψX )
2
2
X
1 − ηRX
1 − ηL
X
= (ψX ) ηL +
+ (1 − ψX )
2
2
Here, ψX and ψY depend on the parties’ positional choices, as explained above. This
expression signifies that, of the impressionable voters who are sympathetic to L’s position
on X and care about issue X, all of those who observe L’s campaign on issue X and half
of those that do not will vote for L. The voters that vote for L without having observed
its campaign are composed from two groups: voters who observed R’s campaign on X
and so observed L’s position, and voters who observed neither party’s campaign on X.
Similarly, of the impressionable voters who are not sympathetic to L’s position X but
care about issue X, half of those who only observed L’s campaign on X will vote for L.
Likewise, among impressionable voters who care about issue Y , party L’s vote share
13
is denoted VLIY , and given by
VLIY
1 − ηLY
1 − ηRY
Y
= (1 − ψY ) ηL +
+ (ψY )
2
2
It follows that, of the impressionable voters who are sympathetic to L’s position on Y
and care about issue Y , all of those who observe L’s campaign on issue Y and half of
those that do not will vote for L, as will half of those impressionable voters who care
about issue Y , prefer R on issue Y , and only observe L’s campaign on issue Y . The
remaining 1−πX2−πY of impressionable voters that consider neither issue salient will split
evenly for both parties.
Non-impressionable voters respond to parties’ campaigns analogously. Party L’s vote
share among non-impressionable voters who care about issue X is denoted VLN X , whereas
her vote share among those non-impressionable voters who care about issue Y is denoted
VLN Y , with
VLN X
VLN Y
1 − η̄RX
1 − η̄LX
= (ψX )
+ (1 − ψX )
+
2
2
Y
1 − η̄RY
1 − η̄L
Y
+ (ψY )
= (1 − ψY ) η̄L +
2
2
η̄LX
Then, party L’s total vote share function can be written as follows:
VL = α[πX VLIX + πY VLIY +
1 − π X − πY
] + (1 − α)[π̄X VLN X + π̄Y VLN Y ]
2
Party R’s total vote share function is symmetric.
4
4.1
Analytical Results
Separability of Positional and Emphasis Strategies
In this model, a party’s optimal strategy is three-dimensional, in that each party chooses
a position and a level of emphasis on each issue. Moreover, in making these choices,
parties trade off as many as five competing incentives: choosing a position on each issue
that is popular with voters, choosing a position on each issue that is popular with its
activists, clarifying its position on salient issues, increasing the salience of issues on which
its activists’ preferences are more popular, and clarifying its position on issues on which
its activists’ preferences are more popular. Nevertheless, the model is rendered highly
14
tractable, as each party’s choice of position on each issue and its optimal level of emphasis
on each issue can be thought of as presenting three distinct optimization problems for
each party, and therefore can be considered separately.
This is guaranteed by four features of the modeling framework. First, parties’ positional choices only influence their vote share through their implications for ψX and ψY , or
the proportion of voters that prefer each party on X and Y . Second, the extent to which
each party chooses to emphasize an issue does not influence activists’ propensity to support a party. This is justified on the grounds that activists – and core party supporters
– are less susceptible to priming, and less reliant than ordinary voters on parties’ public
messages for information on a party’s position on an issue. Third, activists and voters
are both assumed to only care about parties’ positions on one issue in choosing whether
to support a party, rather than taking parties’ positions on both issues into account.
Finally, activists’ contributions to a party are assumed to only be effective in increasing
its support among voters on the issue that those activists care about. As such, we find
that each party’s vote share is weakly increasing in the proportion of voters that prefer
it on each issue. Therefore, each party chooses its position on each issue to maximize the
proportion of voters that prefer it on that issue, and conditional on the positions chosen
by both parties, chooses a level of emphasis on each issue to maximize its vote share.
This result is stated formally in Lemma 1.
Lemma 1. Party L chooses xL and yL to maximize ψX and 1 − ψY , and conditional
on ψX and ψY , chooses eX
L to maximize VL . Likewise, party R chooses xR and yR to
maximize 1 − ψX and ψY , and conditional on ψX and ψY , chooses eYR to maximize VR .
Proof. As the model is symmetric, it is sufficient to consider party L’s optimal strategy.
First, we show that for all eX
L and all actions by party R, party L’s vote share is increasing
in ψX and decreasing in ψY . Taking the derivative of VL with respect to ψX , we obtain:
1
1
dVL
= πX (α)(ηLX + ηRX ) + π̄X (1 − α)(η̄LX + η̄RX )
dψX
2
2
Y
which is weakly positive for all eX
L and eR , and trivially, for all xR and yR . Likewise,
taking the derivative of VL with respect to ψY , we obtain:
dVL
1
1
= − πY (α)(ηLY + ηRY ) − π̄Y (1 − α)(η̄LY + η̄RY )
dψY
2
2
Y
which is weakly negative for all eX
L and eR (and trivially, for all xR and yR ). As xL
and yL can only influence party L’s vote share through their implications for ψX and
15
ψY , respectively, it follows that party L chooses her positions on issues X and Y to
maximize ψX and to minimize ψY . Therefore, we can treat party L as facing three
distinct maximization problems: choosing xL and yL to maximize ψX and 1 − ψY , and
conditional on ψX and ψY , choosing eX
L to maximize VL .
Consequently, we solve for parties’ optimal positional and emphasis decisions separately, treating ψX and ψY as exogenous parameters when solving for parties’ emphasis
decisions.
4.2
Positional Strategies in Equilibrium
In any equilibrium, parties will choose a position on each issue that is closer to its opponent’s position than are its activists, with party L choosing a position to the left of
party R on each issue. Moreover, parties will locate the same distance away from their
activists’ preferences on each issue, with this distance shrinking as the importance of
activist support in parties’ campaigns and the importance of the issue to activists grows,
and increasing as the distance between parties’ activists on each issue increases. Proposition 1 formally characterizes parties’ optimal positional strategies.
Proposition 1. The only possible positional strategies in equilibrium are the following.
On issue X, party L chooses x∗L = xaL + r, and party R chooses x∗R = xaR − r, where
1
r = 21 [ 1+λ
(xaR − xaL )]. On issue Y , party L chooses yL∗ = yaL + r̄, and party R chooses
X
1
(yaR − yaL )]. Parties will always choose positions such
yR∗ = yaR − r̄, where r̄ = 21 [ 1+λ
Y
that x∗L < x∗R and yL∗ < yR∗ , and converge to the median voter if and only if their activists
do.
A proof of this result is presented in the Appendix. In equilibrium, parties never
choose to locate at the median voter’s ideal point, as on each issue, the party whose
activists are closer to the median voter, on average, stands to gain all of the voters. To
see intuitively why this result holds, note that the argument with issue Y is symmetric,
and consider issue X. Recall that on this issue, on average, party L’s activists are located
closer to the median voter than party R’s activists. As proximity to one’s activists boosts
a party’s support on an issue by increasing activists’ support for its campaign, when both
parties locate at the median voter on issue X, all voters will support party L over R on
X. Then, party R has a profitable deviation closer to its activists’ preferred position on
X. Rather, due to the importance of activists’ contributions in each party’s campaign,
parties positional choices on each issue are anchored by their activists’ preferences on
16
that issue. By the symmetry of the problem, each party locates the same distance away
from its activists on an issue as its opponent in equilibrium. Indeed, it is never optimal
for the parties to locate at the same position on an issue, as a profitable deviation is then
available to the party further away from its activists. Unlike in the Downsian model, the
resulting pivotal voter on each issue – or the voter who is then indifferent between the
two parties – is not the median voter, but the voter whose preferences is at the midpoint
of both parties’ activists.
xL
xaL
xR
xm
xL
xaL
xm
xaR
xR
xaR
Figure 1: Parties’ positional choices in equilibrium
This figure illustrates parties’ optimal positional choices on issue X for two possible configurations of
activist preferences, with λX fixed at 0.75 in both cases. xL and xR denote the positions chosen by
parties L and R on issue X, and xaL and xaR denote the position preferred by each party’s activists, on
average, on the issue. The median voter’s ideal point is denoted by xm .
Figure 1 illustrates parties’ optimal positional choices on issue X for two possible
configurations of activist preferences. As voters’ loss functions are quadratic, each party
optimally locates between the pivotal voter and its own activists’ preferred position on
an issue. Parties never locate either at the pivotal voter’s preferred position, or at the
position preferred by their activists, as in this case, each party gains more voters than
it loses by moving slightly closer to its activists, or towards the pivotal voter. Moreover,
each party never chooses a position on an issue that is more extreme than its activists’
preferred position, as it can be closer to the pivotal voter and to its activists, on average,
by locating at its mean activist’s ideal point. Then, in an interesting contrast with the
Downsian model, parties’ choices of position on each issue are only influenced by their
activists’ preferences on the issue, and not by the preferences of the median voter. This
is because each party’s positional choice on an issue in equilibrium is a weighted average
of its activists’ preferred position and preferences of the pivotal voter, where the weight
is a function of λX . This means that as the distance between parties’ activists on an issue
increases, so does the distance between each party’s activists and the pivotal voter. This
17
allows both parties to move proportionately further away from their activists, as well as
from the pivotal voter, in equilibrium.
Each party’s optimal position on an issue only converges to the median voter’s position
as both parties’ activists converge to the median voter, rendering the median voter the
pivotal voter. Indeed, it is possible that both parties may locate either to the right or the
left of the median voter on an issue in equilibrium – for instance, if one party’s activists
are located very close to the median voter on an issue, and its opponents’ activists are
very extreme. Finally, we find that parties locate closer to their activists on each issue
as the importance of activists’ contributions to parties’ campaigns increases. This is
because, for each party, this increases the marginal return from proximity to its activists
– in terms of support among voters – relative to the return from proximity to the pivotal
voter. These results reaffirm the importance of activists’ contributions within parties in
explaining why parties do not appear to fully converge to the median voter in reality, and
may explain why, at points, both parties may even locate on the same side of the median
voter on an issue.
Recall that on each issue, the pivotal voter is at the midpoint of both parties’ activists.
It is then trivial to show that in any equilibrium, on the issue on which its activists’
preferred position is closer to the median voter, a party will always obtain the support of
the majority of voters on that issue. Consequently, conditional on its activists’ preferences
on both issues, each party benefits from one of the two issues being more electorally salient
for voters. Henceforth, I refer to the issue on which a party’s activists are closer to the
median voter as their ‘preferred issue’.
4.3
Emphasis Strategies in Equilibrium
In equilibrium, we observe one of two outcomes: either both parties focus solely on their
preferred issue in their campaigns, or at least one party addresses both issues, ‘bundling’
popular and less popular positions in the same platform. I term the former a ‘single issue
equilibrium’, and the latter a ‘policy bundling equilibrium’. These are defined formally
below.
∗
∗
Y
Definition 1. An equilibrium is a single issue equilibrium if eX
L = 1 and eR = 1, and
∗
Y∗
a policy bundling equilibrium if either eX
L ∈ (0, 1) or eR ∈ (0, 1).
Either a policy bundling or single issue equilibrium exists, and is unique, for all
parameter values.26 Proposition 2 characterizes the conditions under which we observe
26
For a proof of equilibrium existence and uniqueness, see the Appendix.
18
each of these outcomes.
Proposition 2. There exist α∗ ∈ (0, 1) and α0 ∈ (0, 1) such that a single issue equilibrium occurs if and only if α ≥ max {α∗ , α0 }. Otherwise, we observe a policy bundling
∗
Y∗
equilibrium. In any equilibrium, eX
L > 0 and eR > 0.
The proof of this result, and a detailed construction of parties’ emphasis strategies in
equilibrium when α < max {α∗ , α0 }, are presented in the Appendix. We find that a single
issue equilibrium occurs if voters’ priorities are sufficiently responsive to parties’ emphases
– that is, if enough voters are impressionable. When this is the case, both parties’ gains
from increasing the salience of their preferred issue for impressionable voters will outweigh
the benefits from clarifying their position on the more salient issue for non-impressionable
voters. Such an equilibrium – in which parties ‘talk past each other’ – allows parties to
maximize the electoral salience of their preferred issue, minimizing the electoral fallout
from relatively unpopular position they are anchored to on their less preferred issue. On
the other hand, when the proportion of impressionable voters falls below the threshold
required to ensure a single issue equilibrium, we observe a policy bundling equilibrium, as
at least one party feels compelled to clarify its position on its less preferred issue. Further,
we find that parties never focus solely on the issue on which their activists’ preferred
position is less popular: even as the proportion of impressionable voters converges to
zero, each party always has some voters to gain by clarifying its position on the issue on
which its preferred position is more popular.
Figure 2 plots parties’ optimal emphasis choices against α – the proportion of impressionable voters in the electorate – for various configurations of parameter values. These
illustrate three features of parties’ emphasis strategies in equilibrium. First, comparing
Figures 2a and 2b, it is evident that a policy bundling equilibrium persists for higher α
as each party’s lead with voters on its issue shrinks. Then, each party has fewer voters
to gain by increasing the salience of its preferred issue, and a stronger incentive to clarify
its position on its less preferred issue for the benefit of an increasing minority of sympathetic voters. Second, it is notable that although ψX = 0.695 and ψY = 0.55 in both
Figure 2a and Figure 2c – implying a considerably larger lead for party L on its issue
relative to Party R – the threshold for a single issue equilibrium is higher in Figure 2c
than in Figure 2a. Further, for a given α, both parties devote (weakly) more attention to
issue Y in Figure 2c than in Figure 2a. This results from the higher salience of issue Y
for non-impressionable voters in the former scenario than the latter, creating a stronger
incentive for both parties to clarify their position on issue Y for these voters. Consequently, the proportion of impressionable voters in the electorate must be higher before
19
Figure 2: Parties’ emphasis choices in equilibrium
1
1
∗
0.5
0
0.5
0
0.5
0
1
(a) ψX = 0.695, ψY = 0.55, π̄X = 0.5
1
0.5
0.5
0
0.5
0
0.5
1
(b) ψX = 0.6, ψY = 0.545, π̄X = 0.5
1
0
eX
L
∗
eYR
0
1
(c) ψX = 0.695, ψY = 0.55, π̄X = 0.3
0
0.5
1
(d) ψX = 0.6, ψY = 0.545, π̄X = 0.3
This figure plots parties’ optimal emphasis on their preferred issue as a function of α, the proportion
of impressionable voters in the electorate, for several configurations of parameter values. γ1 and γ2 are
fixed at 0.2 in all calculations. In all cases, a single issue equilibrium, where each party focuses solely on
its preferred issue, occurs for sufficiently high α.
20
party L’s incentive to emphasize issue X in order to raise its electoral salience dominates
its incentive to clarify its position on the issue more salient for non-impressionable voters.
Last but not least, in all four scenarios presented in Figure 2, each party emphasizes
its preferred issue more than is warranted by its importance to non-impressionable voters.
Indeed, Corollary 1 states that this holds true in any policy bundling equilibrium – even
as α converges to zero and ψX and ψY converge to a half, implying that parties are
virtually unable to alter the salience of issues for voters through their emphases, and
neither party has a large lead with voters on its preferred issue. In this case, each
party will still place slightly more emphasis on its preferred issue than the priorities of
non-impressionable voters would warrant, as it has more voters to gain by clarifying its
position on its preferred issue than by clarifying its position on the issue on which its
activists’ preferences are less popular. Consequently, it is never the case that parties
respond proportionately to voter priorities in their issue emphases.
∗
∗
∗
X
Y
Corollary 1. For all α, ψX and ψY , either eX
L > π̄X or eL = 1, and either eR > π̄Y
∗
or eYR = 1.
∗
∗
Y
Proof. The result is shown for eX
L ; the result for eR follows from a symmetric argument.
∗
X∗
It is sufficient to consider the case eX
L < 1. Then, eL satisfies the best response function
π̄X (ψX )
given in the proof of Proposition 2. Then, as α > 0, eX
L > π̄X (ψX )+π̄Y (1−ψY ) . For a proof
)
by contradiction, suppose that for some parameter values, π̄X ≥ π̄X (ψXπ̄X)+π̄(ψYX(1−ψ
. This
Y)
rearranges to the requirement that ψX ≤ 1 − ψY , a contradiction. This establishes that
∗
X∗
either eX
L = 1 or eL > π̄X for all α, ψX and ψY .
4.4
When Do Parties ‘Talk Past Each Other’ ?
We have already demonstrated that a single issue equilibrium occurs when voters’ priorities are sufficiently responsive to parties’ issue emphases, or when there are sufficiently
many impressionable voters. Proposition 3 identifies necessary and sufficient conditions
for such an equilibrium.
Proposition 3. There exist cutpoints α̂ and ᾱ > α̂ such that a single issue equilibrium
never occurs when α < α̂, and never occurs for α ∈ [α̂, ᾱ) if either π̄X = 1 or π̄Y = 1. For
any α ∈ (0, 1), if both parties’ activists are sufficiently close to the median voter on both
issues, a single issue equilibrium will not occur. However, for any values of the remaining
parameters, there exists a sufficiently high α such that a single issue equilibrium occurs.
21
A proof of this result is included in the Appendix. We find that, for any values
of the remaining parameters, a single issue equilibrium always occurs when voters are
sufficiently impressionable. Conversely, when there are sufficiently few impressionable
voters in the electorate, or if both parties’ activists are sufficiently close to the median
voter on both issues, it emerges that at least one party will always ‘bundle’ policies. When
most voters are non-impressionable, or when voters’ priorities are not very sensitive to
parties’ issue emphases, parties have more voters to gain by clarifying their position on
the issue on which their preferred position is less popular than by seeking to increase the
electoral salience of their preferred issue. This holds true even if each party commands a
very large majority of voters on its preferred issue, as then while parties may still want
to disproportionately emphasize their preferred issue, they will simultaneously wish to
clarify their position on their less preferred issue for the benefit of the vast majority of
non-impressionable voters. In a similar vein, if the leads commanded by both parties on
their preferred issues are sufficiently small, then regardless of the sensitivity of voters’
priorities to parties’ issue emphases, parties will want to clarify their position on their
less preferred issue for the large minority of voters who would be sympathetic to their
policies on the issue were they aware of them.
Moreover, Proposition 3 reveals that a single issue equilibrium is less likely to occur
when one issue is disproportionately salient than when both issues are similarly salient
for non-impressionable voters: when one of the two issues is already disproportionately
salient, α – or the proportion of impressionable voters in the electorate – must exceed
a higher threshold than otherwise before we observe a single issue equilibrium. This is
because, until the fraction of impressionable voters is high enough that the incentive to
raise its electoral salience dominates, the party whose preferred issue is hardly salient
for non-impressionable voters will be more concerned to clarify its position on the more
salient issue – even though this is the issue on which its preferred position is less popular.
5
Conclusion
How do parties select which issues to emphasize in their campaigns? This paper addresses
this question with a spatial model of party competition with two parties and two issues
in a context where voters’ issue priorities may be influenced by campaigns. Parties are
able to choose both a position and a level of emphasis on each issue. I deviate from
existing models of parties’ emphasis decisions in two key respects. First, I assume that
each party’s level of emphasis on each issue can not only influence the electoral salience of
22
issues, but also serves to clarify parties’ positions on each issue to voters. This provides an
incentive for parties to emphasize an issue even if they are not preferred by the majority
of voters on the issue, especially when the issue concerned is more salient. Second, I
assume that, while parties are able to choose positions on each issue, each party prefers
to choose a position as close to its activists, donors, and core party supporters as possible.
Therefore, in their joint decisions of which issues to emphasize and which positions to
take on each issue, parties compete by trying to focus voters’ attention on issues on which
their activists’ preferences are more popular, while simultaneously being compelled to
emphasize issues on which voters’ attention is already focused.
It emerges that parties locate closer to each other than their activists on each issue,
and only converge to the median voter when their activists do so as well. Moreover, parties
locate closer to their activists on each issue as the importance of activists’ contributions to
parties’ campaigns increases, or as the importance of the issue to activists increases. These
findings reaffirm the importance of activists’ contributions within parties in explaining
why parties do not appear to converge to the median voter in reality, a relationship
which had already been argued for by Aldrich (1983) and Miller and Schofield (2003). In
a further interesting contrast with the Downsian model, we find that each party’s choice
of position on each issue only depends on its activists’ preferences on the issue, and has no
relationship with the preferences of the median voter. Indeed, if one party’s activists are
very close to the median voter on an issue, and its opponent’s activists are very extreme,
we find that both parties may even locate on the same side of the median voter.
Meanwhile, when choosing a level of emphasis on each issue, parties balance three
competing incentives. First, each party has an incentive to emphasize issues in proportion
to their salience for voters in order to clarify its position on salient issues. Second, each
party prefers to increase its emphasis on the issue on which its activists’ preferences
are more popular, in order to increase its electoral salience. Third, each party has an
incentive to emphasize less the issue on which its activists’ preferences are less popular, as
it has fewer voters to gain by clarifying its position on that issue. The strength of these
incentives, and therefore, the extent to which parties are forced to respond to voters’
priorities rather than able to shape them, is determined by three factors: the sensitivity
of voters’ priorities to parties’ campaigns, the prior salience of issues, and the distance
between each party’s activists and the median voter on each issue.
If voters’ priorities are sufficiently responsive to parties’ emphases, we find that a
single issue equilibrium—where parties simply ‘talk past each other’—occurs. However,
in many contexts, parties instead choose to ‘bundle’ popular and unpopular policies
23
in their campaigns, while disproportionately emphasizing the issue on which a party’s
preferred position is more popular with voters. This strategy allows a party to limit its
punishment at the polls, even as it feels compelled to clarify its position on an issue when
this position is not popular with the majority of voters. Although, in a policy bundling
equilibrium, parties are more responsive to the priorities of voters than in a single issue
equilibrium, it is never the case that parties will mirror voters’ initial priorities in their
emphases exactly: even when parties are virtually unable to alter the salience of issues for
voters through their emphases, and when both parties’ activists are almost at the median
on both issues, each party still places slightly more emphasis on its preferred issue than
voter priorities would warrant.
To the best of my knowledge, this paper constitutes the only spatial model of issue
competition which finds that opposing parties will campaign on the same issue in equilibrium, and that parties will campaign on unfavorable issues if these are especially salient
to voters. This is consistent with an growing consensus among researchers analyzing
campaign agendas from a range of countries, who find, almost without exception, that
parties spend much of their campaigns focusing on the same issues, and in particular, on
issues that are already salient to voters. However, thus far, spatial models have failed to
predict that parties might address issues they do not already ‘own’.
Additionally, by highlighting the importance of activists in parties’ campaign efforts,
this paper identifies and models an incentive for parties to emphasize, or de-emphasize,
positional issues. Although several studies have examined how activists may constrain the
positional choices of parties (Aldrich 1983; Miller and Schofield 2003; Schofield and Miller
2007), none of these studies, to my knowledge, have also explored activists’ influence on
parties’ salience strategies, or on the importance of issues for the electorate. Meanwhile,
spatial models analyzing parties’ salience strategies with respect to positional issues have
refrained from endogenizing parties’ issue positions alongside their salience for voters
(Amorós and Puy 2013; Dragu and Fan 2015). Given the considerable time devoted by
parties to non-consensual, or positional, issues in their campaigns, the value of salience
strategies for positional issues merits explanation. This analysis suggests one possible
explanation for why and when we may observe parties’ using salience strategies with
respect to such issues.
This analysis presents several promising avenues for future research, both formal and
empirical. For instance, an extension of the model to three or more issues may be able
to help us account for the well-established finding that issues on which parties have converged are frequently less salient for voters – and conversely, issues on which parties
24
are polarized are frequently more salient for voters (Rabinowitz and Macdonald 1989;
Carmines and Stimson 1989; Riker 1993; Green and Hobolt 2008). Additionally, an extension of the model which introduces disunity among activists may create an additional
incentive for parties in their emphasis decisions, as parties may want to de-emphasize
issues on which their activists are divided, even if these are issues on which their activists
are, on average, closer to the median voter than their opponents’ activists. The importance of activists to parties’ salience strategies also invites discussion on the implications
of primaries for parties’ campaign agendas. Furthermore, the model has clear testable
empirical implications that deserve investigation in the data. For example, the model
leads us to expect that parties will want to de-emphasize issues on which their activists’
preferences are relatively non-centrist; also, we would expect that parties will tend to increase their emphasis on more salient issues, even if these are issues that are unfavorable
to them. Last but not least, the implications of such strategies for parties’ responsiveness to voters’ policy preferences, and when and how the prior salience of issues may
in turn feed back into parties’ positional choices, also present fertile territory for future
researchers.
25
6
6.1
Appendix
Proof of Proposition 1
Proposition 1 (Restated): The only possible positional strategies in equilibrium are the
following. On issue X, party L chooses x∗L = xaL + r, and party R chooses x∗R = xaR − r,
1
(xaR − xaL )]. On issue Y , party L chooses yL∗ = yaL + r̄, and party R
where r = 12 [ 1+λ
X
1
(yaR − yaL )]. Parties will always choose positions
chooses yR∗ = yaR − r̄, where r̄ = 12 [ 1+λ
Y
∗
∗
∗
∗
such that xL < xR and yL < yR , and converge to the median voter if and only if their
activists do.
Proof. Let c represent the voter who is indifferent between party L and party R on issue
X, and xc denote her position on issue X. We can then re-write ψX as follows:


xc





 1 − xc
ψX =
0


1



2


1
if
if
if
if
if
x∗L
x∗L
x∗L
x∗L
x∗L
< xR
> xR
= xR > 12 (xaL + xaR )
= xR = 12 (xaL + xaR )
= xR < 12 (xaL + xaR )
For the moment, we assume that x∗L < xR ; we will later show that this condition holds
in equilibrium. Then,
−(xc − xL )2 − λX (xL − xaL )2 = −(xc − xR )2 − λX (xR − xaR )2
Solving for xc , we find:
λX [(xL − xaL )2 − (xR − xaR )2 ] + x2L − x2R
2xL − 2xR
2
λX [(xL − xaL ) − (xR − xaR )2 ] 1
+ (xL + xR )
=
2xL − 2xR
2
xc =
Taking the derivative of xc with respect to xL , we find that xc achieves its maximum
when
r
λX
xL = xR ±
[(xR − xaL )2 − (xR − xaR )2 ]
1 + λX
q
λX
As x∗L < xR , it follows that x∗L = xR − 1+λ
[(xR − xaL )2 − (xR − xaR )2 ].
X
Note that as the model is symmetric, R’s optimal strategy is analogous. Then, we
can derive each party’s best response function for x∗L < xR and xL < x∗R . By symmetry,
26
the following conditions must hold in equilibrium:
(xL − xR )2 (1 + λX ) = λX [(xR − xaL )2 − (xR − xaR )2 ]
(xR − xL )2 (1 + λX ) = λX [(xL − xaR )2 − (xL − xaL )2 ]
By rearranging, these yield the following equilibrium condition:
x∗L − xaL = xaR − x∗R
Let this quantity be denoted r. Then, x∗L = xaL + r, and x∗R = xaR − r. By substituting
R’s optimal strategy into L’s best response function, we find that
2
2r
λX
2r
+1 =
+1
xaL − xaR
1 + λX xaL − xaR
1
1
1
=⇒ r = (xaR − xaL ) or r =
(xaR − xaL )
2
2 1 + λX
However, it is never the case that r = 12 (xaR − xaL ) in equilibrium. To see this, note that
when r = 21 (xaR − xaL ), x∗L = 12 xaL + 21 xaR and x∗R = 12 xaR + 12 xaL . Then, x∗L = x∗R , a
contradiction.
Next, we demonstrate that it is never the case that x∗L > x∗R in equilibrium. First, we
show that in equilibrium, x∗L ≤ 12 (xaL + xaR ) and x∗R ≥ 21 (xaL + xaR ). To see this, suppose
otherwise: x∗L > 21 (xaL + xaR ) or x∗R < 12 (xaL + xaR ). Consider x∗L > 12 (xaL + xaR ); the
argument for x∗R < 12 (xaL + xaR ) is symmetric. Then, party R always has a profitable
deviation to its opponent’s position on X, so that both parties are at the same position
on issue X but R is closer to its activists. Then, R would be preferred by all voters to L
on issue X, implying that ψX = 0. It follows from this that in equilibrium, x∗L ≤ x∗R .
Now, we show that it is never the case that x∗L = x∗R . There are two cases to be
considered: (1) x∗L = x∗R > 21 (xaL + xaR ) and (2) x∗L = x∗R = 12 (xaL + xaR ); the argument
for x∗L = x∗R < 21 (xaL + xaR ) is symmetric to (1). First, consider x∗L = x∗R > 12 (xaL + xaR );
then, party L has a profitable deviation to xL = xaL , which increases ψX from 0 to
−λX (xR −xaR )2
+ 12 (xaL + xR ) > 0. Next, consider x∗L = x∗R = 12 (xaL + xaR ). Then, ψX = 12 .
2xaL −2xR
Now, consider xL = x∗L − for some > 0. Then, ψX = xc = 12 λX (xaR − xaL − ) +
1
(xaR + xaL − ). As → 0, ψX → 21 λX (xaR − xaL ) + 21 (xaR + xaL ). Note that as
2
|xaL − xm | < |xaR − xm |, 12 (xaR + xaL ) > 12 and ψX > 12 for all xaL , xaR and λX , and a
profitable deviation exists for party L. Therefore, for x∗L = x∗R , at least one party has
a profitable deviation for all parameter values. It follows from the above that it is also
27
never the case that x∗L = x∗R = xm , as then R would have a profitable deviation to xaR ,
λ ( 12 −xaL )2
+ 12 xaR > 0.
increasing its support on X from 0 to 14 + X1−2x
aR
Finally, we show that it is never optimal for party L to choose x∗L = 0 and for party
R to choose x∗R = 1. We consider deviations for party L from x∗L = 0; the case with party
R is symmetric. Recall, again, that in any equilibrium, x∗R ≥ 12 (xaL + xaR ). We now
show that for any xR ≥ 21 (xaL + xaR ), xL = 0 is strictly dominated by xL = xaL . Note
λ [(x −x )2 −x2aL ]
that as |xaL − xm | < |xaR − xm |, xR > 12 . When xL = 0, ψX = X R 2xaR
+ 12 xR .
R
−xaR )2
However, for any xR > 12 , a deviation to xaL increases ψX to λX2x(xRR−2x
+ 21 (xaL + xR ).
aL
This establishes
h that
h party
i the only possible equilibrium is the following:
i L chooses
1
1
1
1
∗
∗
xL = xaL + 2 1+λX (xaR − xaL ), and party R chooses xR = xaR − 2 1+λX (xaR − xaL ).
1
1
1
∗
Note that xL rearranges to [1 − 1+λX ]xaL + 2 1+λX (xaL + xaR ), and likewise, x∗R
1
1
1
]x
+
(xaL + xaR ). Note, also, that as xaR , xaL → 12 ,
rearranges to [1 − 1+λ
aR
2 1+λX
X
(xaR − xaL ) → 0 and therefore, x∗L , x∗R → 12 . However, for all λX , xaL and xaR , x∗L > xaL
1
and x∗R < xaR . Moreover, as xaL → 12 and xaR → 1, x∗L → 12 + 14 [ 1+λ
] > 12 , with x∗L < x∗R .
X
As the model is symmetric, an analogous argument holds for parties’h optimal
i choices of
1
1
∗
positions on issue Y : in equilibrium, party L chooses yL = yaL + 2 1+λY (yaR − yaL ),
h
i
1
(yaR − yaL ).
and yR∗ = yaR − 12 1+λ
Y
6.2
Proof of Proposition 2
∗
Proposition 2 (Expanded): In any equilibrium, party L chooses eX
L = 1 if and only
∗
if α ≥ α , and chooses
∗
eX
L
=
α
[γ (1
1−α 1
1
1
Y
+ 2eX
R )(ψX − 2 ) + γ2 (1 + 2eR )(ψY − 2 )] + π̄X (ψX )
π̄X (ψX ) + π̄Y (1 − ψY )
∗
otherwise. Likewise, party R chooses eYR = 1 if and only if α ≥ α0 , and chooses
eYR
=
α
[γ (1
1−α 1
1
1
Y
+ 2eX
L )(ψX − 2 ) + γ2 (1 + 2eL )(ψY − 2 )] + π̄Y (ψY )
π̄Y (ψY ) + π̄X (1 − ψX )
otherwise. The thresholds α∗ and α0 are defined by the following equations:
α∗
π̄Y (1 − ψY )
=
1
1
Y
1 − α∗ γ1 (1 + 2eX
R )(ψX − 2 ) + γ2 (1 + 2eR )(ψY − 2 )
α0
π̄X (1 − ψX )
=
X
0
1 − α γ1 (1 + 2eL )(ψX − 12 ) + γ2 (1 + 2eYL )(ψY − 21 )
28
∗
∗
Y
It is never optimal for party L to choose eX
L = 0 and for party R to choose eR = 0.
Proof. The result is shown for party L; the result for party R follows from a symmetric
argument. In choosing eX
L to maximize VL , given ψX and ψY , the maximization problem
faced by party L is as follows:
1 − ηRX
+ πX (1 − ψX )
2
1 − ηLY
1 − ηRY
+ πY (1 − ψY ) ηLY +
+ πY (ψY )
2
2
1 − π X − πY
]
+
2
1 − η̄LX
1 − η̄RX
X
+ (1 − α)[π̄X (ψX ) η̄L +
+ π̄X (1 − ψX )
2
2
Y
Y
1 − η̄L
1 − η̄R
+ π̄Y (1 − ψY ) η̄LY +
]
+ π̄Y (ψY )
2
2
max VL = α[πX (ψX )
eX
L
ηLX
1 − ηLX
+
2
s.t. eX
L ≥ 0
eX
L ≤ 1
As our constraints are linear, the constraint qualification always holds. Further, the
objective function is continuous and the choice set is compact, so a solution to this
maximization problem exists. Additionally, the objective function is strictly concave
everywhere, so if a solution exists, for each set of parameter values, it is unique and given
by the Lagrangian method. The corresponding Lagrangian and first order conditions are
as follows:
X
L(eX
L , λ1 ) = VL − λ1 (eL − 1)
X
Y
Y
1. LX = α γ1 (ψX )(1 + eX
R ) − γ1 (1 − ψX )(eR ) − γ2 (1 − ψY )(1 + eR ) + γ2 (ψY )(eR )
X
+ (1 − α)[π̄X (ψX )(1 − eX
L ) − π̄Y (1 − ψY )eL ] − λ1 ≤ 0
2. eX
L ≥ 0
3. eX
L (LX ) = 0
4. eX
L ≤ 1
5. λ1 ≥ 0
6. λ1 (eX
L − 1) = 0
29
∗
∗
There are three candidate solutions to consider: (1) eX
= 1, (2) eX
= 0, and (3)
L
L
∗
∗
X
X
eL ∈ (0, 1). First, consider eL = 1. Then, from condition (3), it follows that LX = 0.
As λ1 ≥ 0, this implies, by substitution, that:
X
Y
Y
α γ1 (ψX )(1 + eX
)
−
γ
(1
−
ψ
)(e
)
−
γ
(1
−
ψ
)(1
+
e
)
+
γ
(ψ
)(e
)
1
X
2
Y
2
Y
R
R
R
R
+(1 − α)[−π̄Y (1 − ψY )] = λ1 ≥ 0
Y
By rearranging the above inequality, and using eX
R = 1 − eR , we obtain the following
condition:
π̄Y (1 − ψY )
α
≥
X
1−α
γ1 (1 + 2eR )(ψX − 21 ) + γ2 (1 + 2eYR )(ψY − 12 )
Since ψX , ψY ∈ ( 21 , 34 ) and π̄Y , eYR ∈ [0, 1], it follows that the right hand side is weakly
α
positive. Note that 1−α
is an increasing function of α. It follows that party L will choose
X
Y
eL = 1 and eL = 0 in equilibrium if and only if this condition obtains – that is, α is
sufficiently large. Note that the necessary level of α is decreasing in ψX – the proportion
of voters who prefer party L on issue X – and increasing in π̄Y – the salience of issue Y
for non-impressionable voters. Henceforth, I will refer to this level of α as α∗ .
∗
Next, consider eX
= 0. Then, from condition (6), it follows that λ1 = 0. By
L
substitution, this implies:
α γ1 (1 +
2eX
R )(ψX
1
1
Y
− ) + γ2 (1 + 2eR )(ψY − ) + (1 − α)π̄X (ψX ) ≤ 0
2
2
which rearranges to:
α
−π̄X (ψX )
≤
X
1−α
γ1 (1 + 2eR )(ψX − 21 ) + γ2 (1 + 2eYR )(ψY − 12 )
Since the numerator of the right hand side is weakly negative and the denominator
α
strictly positive for all parameter values, this inequality requires 1−α
≤ 0, a contradiction
for strictly positive α. Therefore, it is never optimal for party L to choose eX
L = 0 in
equilibrium.
Finally, consider an interior solution. Then, from conditions (3) and (6), it follows
30
that LX = 0 and λ1 = 0. By substitution, this implies:
α γ1 (1 +
2eX
R )(ψX
1
1
Y
− ) + γ2 (1 + 2eR )(ψY − )
2
2
X
+(1 − α)[π̄X (ψX )(1 − eX
L ) − π̄Y (1 − ψY )eL ] = 0
∗
X
Rearranging for eX
L , we find that eL is defined by the following expression:
∗
eX
L =
α
[γ (1
1−α 1
1
1
Y
+ 2eX
R )(ψX − 2 ) + γ2 (1 + 2eR )(ψY − 2 )] + π̄X (ψX )
π̄X (ψX ) + π̄Y (1 − ψY )
∗
∗
X
Since a solution exists and is unique, one of the two cases eX
L ∈ (0, 1) or eL = 1 must
be a solution. Since the first order conditions provide necessary and sufficient conditions
for a solution, it follows that all possible solutions have been characterized.
6.3
Proof of Proposition 3
Proposition 3 (Restated): There exist cutpoints α̂ and ᾱ > α̂ such that a single issue
equilibrium never occurs when α < α̂, and never occurs for α ∈ [α̂, ᾱ) if either π̄X = 1 or
π̄Y = 1. For any α ∈ (0, 1), if both parties’ activists are sufficiently close to the median
voter on both issues, a single issue equilibrium will not occur. However, for any values
of the remaining parameters, there exists a sufficiently high α such that a single issue
equilibrium occurs.
∗
Proof. From the proof of Proposition 2, we know that party L will choose eX
L = 1 if and
∗
only if α ≥ α , where
π̄Y (1 − ψY )
α∗
=
X
∗
1−α
γ1 (1 + 2eR )(ψX − 12 ) + γ2 (1 + 2eYR )(ψY − 21 )
∗
while party R will choose eYR = 1 if and only if α ≥ α0 , where
α0
π̄X (1 − ψX )
=
X
0
1−α
γ1 (1 + 2eL )(ψX − 12 ) + γ2 (1 + 2eYL )(ψY − 21 )
Y
Note that in a single issue equilibrium, eX
L = 1 and eR = 1. By substitution, we find that
31
such an equilibrium occurs if and only if α ≥ max{α∗ , α0 }, where
α∗
π̄Y (1 − ψY )
=
∗
1−α
γ1 (ψX − 12 ) + 3γ2 (ψY − 21 )
α0
π̄X (1 − ψX )
=
0
1−α
3γ1 (ψX − 21 ) + γ2 (ψY − 21 )
Since the left hand side of these equations approaches infinity as α∗ , α0 → 1, it follows
that α∗ and α0 are strictly less than one for any value of the other parameters. Therefore
a single issue equilibrium always occurs for sufficiently high α.
Additionally, note that the right hand side of both of these equations is decreasing
in ψY and ψX . The right hand side approaches infinity as ψX , ψY → 21 , so, for any α, a
single issue equilibrium does not occur if both parties’ activists are sufficiently close to
the median voter on both issues. Moreover, the right hand side of each equation attains
a minimum when ψX = ψY = 43 . Substituting in these values, it follows that a necessary
condition for a single issue equilibrium is:
α
≥ max
1−α
Since max{π̄Y ; π̄X } ≥
1
2
π̄X
π̄Y
;
γ1 + 3γ2 3γ1 + γ2
it follows that a single issue equilibrium can only occur if
α
≥ max
1−α
1
1
;
6γ1 + 2γ2 2γ1 + 6γ2
which is only true if
α ≥ max
1
1
;
1 + 6γ1 + 2γ2 1 + 2γ1 + 6γ2
Finally, note that when either π̄X = 1 or π̄Y = 1, this threshold converges to
α
≥ max
1−α
6.4
1
1
;
1 + γ1 + 3γ2 1 + 3γ1 + γ2
Existence and Uniqueness of Equilibria
Proposition 4. A pure strategy Nash equilibrium exists and is unique for all parameter
values.
32
Proof. The proof makes use of the following lemma.
Lemma 2. Recall that λX , λY ∈ ( 12 , 1). Then, there exists a unique (x∗j , yj∗ )j=L,R such
∗
that for each j, (x∗j , yj∗ ) is a best response to (x∗−j , y−j
).
Proof. Consider a strategy profile s∗ = (x∗L , yL∗ , x∗R , yR∗ ) which satisfies the equilibrium
conditions stated in Proposition 1. For the moment, we relax the assumptions that
|xaL − xm | < |xaR − xm | and |yaR − ym | < |yaL − ym |; this ensures perfect symmetry in
the profitable deviations available to each party. Then, we prove that neither party has
a profitable deviation from s∗ , only considering profitable deviations available to party
L. First, we demonstrate that x̃L = 0 is never a profitable deviation from x∗L < x∗R for
λ [(x∗ −x )2 −x2aL ]
+ 12 x∗R . This provides
party L. To see this, note that for x̃L = 0, ψX = X R 2xaR
∗
R
a profitable deviation if and only if the following inequality holds:
1
λX [(x∗R − xaR )2 − x2aL ] 1 ∗
+ xR > (xaL + xaR )
∗
2xR
2
2
This rearranges to λX (xaR − xaL h− x∗R ) i> x∗R . From the proof of Proposition 1, we know
1
(xaR − xaL ). By substituting xaR − r for x∗R in
that x∗R = xaR − r, where r = 12 1+λ
X
the above inequality condition, we obtain:
2λX + 1
(xaR − xaL ) − λX (xaL ) > xaR
2λX + 2
X +1
Note that 2λ
< 1 for all λX , and by assumption, xaR > 12 . Then, this inequality is
2λX +2
violated for all parameter values that satisfy the stipulated conditions.
λ [(1−xaL )2 −(x∗R −xaR )2 ]
−
Second, consider a deviation by L to x̃L = 1. Then, ψX = 1 − X
2−2x∗R
1
(1 + x∗R ). This provides a profitable deviation if and only if the following inequality
2
holds:
−λX (1 − xaL − xaR + x∗R )(1 − xaL + xaR − x∗R ) > (x∗R + xaL + xaR )(1 − x∗R )
As 12 (xaL + xaR ) < x∗R < 1, −λX (1 − xaL − xaR + x∗R )(1 − xaL + xaR − x∗R ) < 0 for all λX ,
a contradiction.
It is immediate that party L never has an incentive to deviate to x̃L = x∗R : as
x∗R > 21 (xaL + xaR ), the corresponding ψX = 0 < 12 (xaL + xaR ). Finally, consider a
deviation by L to x̃L > x∗R . This provides a profitable deviation if and only if the
33
following inequality holds:
λX [(x̃L − xaL )2 − (x∗R − xaR )2 ]
< 2 − (x̃L + xaL + x∗R + xaR )
∗
x̃L − xR
By rearranging, we find that a profitable deviation exists for sufficiently small λX :
λX <
(2 − x̃L − xaL − x∗R − xaR )(x̃L − x∗R )
(x̃L − xaL )2 − (xaR − x∗R )2
2
∗
Note that such a deviation can only exist
for
h
i 3 < xaL + xaR < 1. Let a = x̃L − xR , and
1
(xaR − xaL ). Note, also, that we know from
recall that x∗R = xaR − r, where r = 12 1+λ
X
the proof of Proposition 1 that ψX achieves a local maximum for xL > xR when
r
xL = xR +
λX
[(xR − xaL )2 − (xR − xaR )2 ]
1 + λX
implying that a profitable deviation for L to xL > xR would most likely be to this quantity.
λX
(xaR − xaL ) = x̃L − x∗R . By substitution and
From the above, it follows that a = 1+λ
X
rearrangement, we can then rewrite the above inequality condition as follows:
xaR − 12
1
λX < −
2 xaR − xaL
As xaL < 12 < xaR , this inequality is always satisfied for λX > 12 . As deviations for
party R and on issue Y follow symmetrically, this demonstrates that neither party has a
profitable deviation from x∗L , x∗R , yL∗ and yR∗ .
Let the correspondence χj (x−j , y−j ) denote party j’s optimal choice of position (xj , yj ),
given the position chosen by the other party. For each (xj , yj )j=L,R , let the correY
X Y
spondence Ej (eX
−j e−j |xL , yL , xR , yR ) denote party j’s optimal choice of emphasis (ej , ej )
given the choice of emphasis by the other party, and both parties’ positional choices
(xL , yL , xR , yR ). Let χ(·) ≡ (χL (·), χR (·)) and E(·|·) = (EL (·|·), ER (·|·)). Then, the four∗
Y∗
∗ ∗
tuple (x∗j , yj∗ , eX
j , ej )j=L,R is a Nash equilibrium if and only if (xj , yj )j=L,R is a fixed
∗
∗
Y
∗ ∗
point of χ(·), and (eX
j , ej )j=L,R is a fixed point of E(·|(xj , yj )j=L,R ).
Lemma 2 establishes that χ(·) has a fixed point (x∗j , yj∗ )j=L,R when the conditions
stipulated by Proposition 4 are satisfied. Due to the concavity of the objective functions, E(·|·) is upper-hemicontinuous, by Berge’s Maximum Theorem. Moreover, it maps
the closed interval [0, 1]2 onto itself. Therefore, for any (xj , yj )j=L,R , the correspondence
E(·|(xj , yj )j=L,R ) satisfies the conditions of Kakutani’s Fixed Point Theorem, and so a
34
∗
∗
Y
∗ ∗
fixed point (eX
j , ej )j=L,R exists for the function E(·|(xj , yj )j=L,R ). Therefore, an equilibrium exists.
To prove that for each set of parameter values, the equilibrium is unique, define the
function g(·) : [0, 1] → [0, 1], where, for each eX
L ∈ [0, 1],
∗
∗
X
Y
X
X
g(eX
L ) = eL (eR (eL )) − eL
∗
∗
Y
and the functions eX
L (·), eR (·) are the best response functions found in Proposition 2.
∗
Y
That is, for each eYR ∈ [0, 1], eX
L (eR ) is given by:
∗
Y
eX
L (eR )
if
=
α
[γ (1
1−α 1
1
1
Y
+ 2eX
R )(ψX − 2 ) + γ2 (1 + 2eR )(ψY − 2 )] + π̄X (ψX )
π̄X (ψX ) + π̄Y (1 − ψY )
α
π̄Y (1 − ψY )
≤
X
1−α
γ1 (1 + 2eR )(ψX − 21 ) + γ2 (1 + 2eYR )(ψY − 12 )
∗
∗
Y
Y
and is eX
L (eR ) = 1 otherwise. The function eR (·) is defined analogously.
Note that in equilibrium, g(·) = 0. As we know an equilibrium exists for all parameter
values, it is sufficient to show that g 0 (·) < 0 for all parameter values. Then for each set of
parameter values, the function g(·) can only intersect zero once. Note that this requires
that
∗
∗
Y∗ X
∂eYR (eX
∂eX
L)
L (eR (eL ))
×
−1<0
g 0 (eX
)
≡
∗
L
∂eYR
∂eX
L
∗
∂eX (eY )
∗
∂eY (eX )
Y
L
R
Then, it is sufficient to show that ∀eX
× R∂eX L ≤ 0. Differentiating
L , eR ,
∂eY
R
L
each party’s best response function with respect to its opponent’s response, we get:
∗
α
[γ (1 − 2ψX ) − γ2 (1 − 2ψY )]
∂eX
1−α 1
L
=
Y
π̄X (ψX ) + π̄Y (1 − ψY )
∂eR
α
Y∗
−
[γ1 (1 − 2ψX ) − γ2 (1 − 2ψY )]
∂eR
= 1−α
X
π̄Y (ψY ) + π̄X (1 − ψX )
∂eL
Since the denominators π̄X (ψX ) + π̄Y (1 − ψY ) and π̄Y (ψY ) + π̄X (1 − ψX ) are positive
∗
∂eX (eY )
given the constraints on the parameter values, it is immediate that the product L∂eY R ×
R
∗
X
∂eY
R (eL )
X
∂eL
is always weakly negative, which completes the proof.
35
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