Ideology signaling in electoral politics

Abstract

This paper considers an electoral model in which an incumbent and a challenger have ideological policy preferences that are private information. The incumbent may bias pre-electoral policies to signal preferences to the electorate with the aim of affecting the outcome of the election. When the two candidates are of completely different types, such a policy bias can occur only in a moderate direction. However, when their possible types overlap, a policy bias can be created in either a moderate or an extreme direction.

Keywords

Election ideology private information signaling

1. Introduction

Although there are many cases in the political arena where politicians take moderate action to pander to median voters, there are also cases where they take action that may seem extreme, even to the eye of the vast majority of voters. On occasion, such extreme policies may ultimately enhance the popularity of the politician. Furthermore, there are cases where such phenomena emerge from a politically ‘murky’ situation in which the ideologies of ruling and opposition parties are not clearly divided.

For instance, after becoming prime minister in 2001, Junichiro Koizumi set out to enact market-oriented reforms in Japan. Although many Japanese, who had been fatigued by long-term economic stagnation, formed some degree of consensus around the need for decisive action to bolster the economy, a number of people were stunned by Prime Minister Koizumi’s reforms, which were accompanied by a slogan of ‘reform with no sacred cows’ and were radical by Japan’s standards (The Economist, 2006).

Astonishment reached its peak when he dissolved the House of Representatives after his bill to privatize Japan’s vast state-run postal service was rejected in the Upper House in 2005. Mr Koizumi called the general election, which essentially became a referendum on this privatization issue. People’s initial surprise turned to support for his administration, which manifested itself in the form of an electoral victory, and Mr Koizumi was re-elected prime minister.

An intriguing point is that the prime minister proceeded with the radical reforms even though he had not received monolithic support from the ruling parties, at least in his early days in office. Japan’s Liberal Democratic Party (LDP), to which Mr Koizumi belonged, had a considerable number of politicians who shared anti-market views with the bureaucrats at the main ministries (Mulgan, 2002: 55; Estévez-Abe, 2006). These conditions may have prompted him to make his declaration when taking office in 2001: ‘My reform plans would be tantamount to the destruction of the LDP (Mulgan, 2002: 49).’

Coincidentally, it seems that Mr Koizumi shared some of the political views of the opposition parties. There was a certain amount of policy commonality between him and some members of the opposition Democratic Party of Japan (DPJ), though they were not identical. Even though the LDP had traditionally been closer to business interests and the DPJ was closer to labor interests, the two major parties both contained a spectrum of views on the central issue of market-liberal reforms (Mulgan, 2002: 62). Out of such a muddled political environment emerged Mr Koizumi’s radical reform programs.

The purpose of this paper is to present a theoretical hypothesis that helps explain such extremism and sheds light on the relationship between interparty ideological differentiation and intraparty policy polarization. A fundamental question is this: Are pre-electoral policies more likely to become polarized when the ruling and opposition parties have distinct ideologies or when they share a degree of commonality? To address this question, this paper considers an electoral model in which an incumbent and a challenger compete against each other when their ideological preferences are private information. The focus is on pre-electoral policymaking by the incumbent that may be biased to signal preferences to voters. Behind this modeling selection is the author’s conjecture that Mr Koizumi might have been signaling his political philosophy to the electorate. Nonetheless, the model is completely abstract and does not deal with any concrete episodes.

The model makes a simple but novel assumption regarding the candidates’ ideological preferences. It allows the possible preferences of candidates from different parties to overlap. That is, it assumes that a candidate from a rightist party may have an ideal point that comes to the left of the most right-ideal point of a leftist party candidate. Under this assumption, it is shown that a policy bias can be created in either a moderate or an extreme direction, depending on conditions. By contrast, under a more standard assumption that the two candidates have completely different preferences, such a policy bias can occur only in a moderate direction.

To assume overlapping preferences to explain possible extreme actions may sound paradoxical. However, this assumption is actually not so unrealistic. Dewan and Spirling (2011) argue that the overlap condition is almost certainly present in all Westminster-type political systems. They estimate the ideal points for members of the British House of Commons by using their roll-call votes for the periods 1997–2001 and 2001–2005 that appear to have been free (i.e. not subject to party whips). With regard to the overlap, they report that approximately one third of the opposition party members were located to the left of the most right-wing member of the Labour Party, which was ruling during those periods.¹ Meanwhile, Kellermann (2012) uses early day motions (EDMs) instead of roll-call votes. EDMs can be interpreted as formal, non-binding expressions of opinion by members of the British House of Commons with parties exerting relatively little influence over signatures on them (Kellermann, 2012: 760–1). He develops a statistical model of EDM signing behavior and uses it to estimate the preferences of the legislators. The result exhibits a similar overlap pattern of ideal points.

Even in the United States, where the two major parties are usually supposed to have distinct ideologies, there are cases where congressional representatives from the two parties will take the same action on a legislative bill. In addition, from the theoretical standpoint of this model, even if there is the slightest possibility of overlapping ideologies, the result obtained below will not be affected as long as separating equilibrium is focused on. This point will be touched on at the end of Section 5.

There is a growing literature on candidate signaling behavior in elections. One strand of this literature focuses on pre-electoral platforms announced by candidates. Banks (1990) and Harrington (1992) derive conditions under which campaign promises are able to convey some sort of correct information about the candidate’s preferences. Kartik and McAfee (2007) develop an electoral model in which candidates may signal their character, an unobservable trait with which they implement what they believe without pandering to the electorate. Callander and Wilkie (2007) establish equilibria when candidates have different incentives to misinform voters about their preferences. In Callander’s (2008) model, politicians may signal that they are policy-motivated, rather than office-motivated.

Another strand of the literature, to which this paper contributes, analyzes electoral models in which incumbents seeking re-election signal some information via their pre-electoral behavior in office. There are several models in which politicians signal their competence (e.g. Rogoff and Sibert, 1988; Canes-Wrone et al., 2001; Ashworth and Shotts, 2010; Fox and Stephenson, 2011). Some politicians may choose ‘populist’ policies when they wish to be perceived as honest, rather than corruptible (Acemoglu et al., 2013). Morris (2001) and Maskin and Tirole (2004) analyze incentives to signal congruence with voter preferences. Morelli and Van Weelden (2013) extend the Maskin–Tirole model and find that policymakers are likely to pander to public opinion on ‘doorstep issues’, on which politicians do not have great informational advantages over voters.

Schultz’s (2002) model is most similar to the one in this paper in terms of its structure. He analyzes a situation in which policymakers have private information about two dimensions (i.e. their ideological preferences and the state of the economy) and may bias their policies to signal such information. Schultz shows that uncertainty about the state of the economy leads to extreme policies, while uncertainty about the incumbent’s preferences leads to moderate policies.

Thus, in Schultz’s model, the extreme-policy result is not generated by uncertainty about preferences. One of the factors behind this conclusion is in the following assumption which he makes with regard to ideological preferences: the ideology level of non-government candidates is less than that of moderate-type incumbents which is less than that of extreme-type incumbents. That is, candidate preference types are clearly divided.

The model developed in the current paper simplifies Schultz’s model in a certain sense. It narrows the focus to situations where a politician sends signals regarding unknown parameters of one dimension, namely, ideology, and examines in full detail the various possibilities related to such parameters. Specifically, it shows that extreme-policy results can be obtained without introducing two-dimensional uncertainties if we assume overlapping ideology types. To the best of the author’s knowledge, there is no electoral-competition model that makes this assumption. Cukierman and Tommasi (1998), like Schultz (2002), assume that ideology types do not overlap: candidate R is always perceived to be the right-wing candidate. By contrast, in many papers that analyze the informational role of platforms, such as Harrington (1992) and Banks (1990), the two candidates are treated symmetrically, so situations are often tackled where the sets of possible types of candidate are identical. It is worthwhile to analyze a case lying between these two polar scenarios.

The rest of the paper is organized as follows. Section 2 explains the set-up and Section 3 briefly examines the standard case with non-overlapping types. Section 4 analyzes voter decisions and Section 5 examines the incumbent’s signaling behavior and derives central results. Section 6 concludes the paper.

2. Set-up of the model

This paper considers a spatial electoral model that is admittedly very simple. There are N voters and two candidates (L from a leftist party and R from a rightist party). To avoid unnecessary complications related to a tie-breaking rule, N shall be an odd number. Without loss of generality, the incumbent is supposed to be from the rightist party (R) and the challenger from the leftist party (L).² We use female pronouns for the incumbent and challenger and male pronouns for voters. As will be specified below, the voters as well as candidates, L and R, have their respective ideological preferences for policies. As in Banks (1990), it is assumed that the ideologies that voters harbor are common knowledge but that the ideologies to which the two candidates adhere are private information.³

The timing of the game is as follows. Nature picks the two candidates’ ideology types independently. This is private information. Incumbent R implements a policy in the first period. Voters observe the policy action and update their beliefs concerning the incumbent’s ideology. They then vote for either candidate L or R. After the election, the candidate obtaining the majority of votes implements a second-period policy.

Because this is a two-period model, the winner is expected to adopt her ideal policy in the second period. The focal points of the model are the policy taken in the first period and the election results. The incumbent may bias the first-period policy, being conscious that her policy choice could transmit a signal regarding her ideology to the electorate and affect the upcoming election results.

The utility of voter $i \in {1, \dots, N}$ in period $t \in {1, 2}$ is given by

U^{i} (x_{t}) = - {(x_{t} - β^{i})}^{2}

(1)

where $x_{t}$ is a policy chosen by the policymaker in period t and $β^{i}$ is the ideal policy for this voter ( $β$ from ‘bliss point’). Each voter’s identity is labeled by his ideal policy from smallest to largest, with the median voter denoted as m: $β^{1} \leq, \dots, \leq β^{m} \leq, \dots, \leq β^{N}$ .

If the issue is about income redistribution, a micro-foundation behind equation (1) can be provided using a simplified version of a redistributive-politics model developed by Dixit and Londregan (1998). In this case, $x_{t}$ can be interpreted as a post-transfer economic disparity between the rich and poor groups. In Dixit and Londregan, voters are assumed to care about economic efficiency and inequality, in addition to their selfish concerns for consumption. Solving the simplified version of their model with only two income-bracket groups, it can be shown that if, for example, a voter cares greatly about efficiency and/or if he belongs to the rich group, then he tends to have a large $β^{i}$ (i.e. admits a large disparity).⁴

The pay-off for candidate P (P = L, R) in period t is assumed to have the form

W_{t}^{P} = V - {(x_{t} - β^{P})}^{2}

(2)

where V is a constant perquisite only a politician in office can receive, and $β^{P}$ is candidate P’s ideal policy and represents her type.

We now introduce uncertainty regarding the candidates’ ideologies. Candidates L and R have possible ideology types ${{\underline{β}}^{L}, {\bar{β}}^{L}}$ and ${{\underline{β}}^{R}, {\bar{β}}^{R}}$ , respectively, where ${\underline{β}}^{L} < {\bar{β}}^{L}$ and ${\underline{β}}^{R} < {\bar{β}}^{R}$ . Nature picks ${\bar{β}}^{L}$ with prior probability $l^{0}$ and ${\bar{β}}^{R}$ with prior probability $r^{0}$ , independently. Once determined by Nature, these types are kept the same throughout the game. They are private information and the voters (and the other candidate) know only the prior distributions.

Based on the motivation described in the Introduction, we mainly analyze the overlapping-type case: ${\underline{β}}^{L} < {\underline{β}}^{R} < {\bar{β}}^{L} < {\bar{β}}^{R}$ . That is, the moderate L is located to the right of the moderate R, as in Figure 1(a). An analysis under a more standard assumption of non-overlapping types (Figure 1(b)) is briefly mentioned in Section 3 and at the end of Section 5.

Figure 1.

Ideal policies.

The following assumption is not essential but is made to avoid unnecessarily increasing the number of situations to be considered.

Assumption 1: ${\bar{β}}^{L} - {\underline{β}}^{L} = {\bar{β}}^{R} - {\underline{β}}^{R}$ .

To investigate voter behavior in detail, the values A, B, C, D, and E are defined as (see Figure 1)

\begin{matrix} A \equiv ({\underline{β}}^{L} + {\underline{β}}^{R})] / 2; B \equiv ({\underline{β}}^{L} + {\bar{β}}^{L}) / 2; C \equiv ({\underline{β}}^{R} + {\bar{β}}^{L})] / 2 \\ D \equiv ({\underline{β}}^{R} + {\bar{β}}^{R})] / 2; E \equiv ({\bar{β}}^{L} + {\bar{β}}^{R})] / 2 \end{matrix}

From now on, a candidate with, say, bliss point ${\underline{β}}^{R}$ is referred to as type ${\underline{β}}^{R}$ or just as ${\underline{β}}^{R}$ , for convenience. We also call ${\bar{β}}^{R}$ the incumbent of the extreme type and ${\underline{β}}^{R}$ of the moderate type. In what follows, we show how the signaling motives of the incumbent about these types affect her first-period policy.

3. A non-overlapping-type case

Before starting the main analysis, we briefly look at the situations under the standard assumption: possible types of candidates L and R do not overlap (i.e. ${\underline{β}}^{L} < {\bar{β}}^{L} < {\underline{β}}^{R} < {\bar{β}}^{R}$ ). With this assumption, the policy bias, if any, emerges only in a moderate direction. The extreme-policy result is not derived.

The key idea is that the incumbent R would like to signal her own type to the median voter if doing so increases the chance of her re-election. In this respect, if the median voter favors a relatively small policy (i.e. $β^{m} < D$ in Figure 1(b)), then R may be able to win the election by signaling that she is moderate. Here, a potential imitator is the extreme incumbent, while a potential blocker of this imitation is the moderate one. The moderate type may then create a downward bias in the first-period policy (i.e. she may choose a policy lower than her ideal ${\underline{β}}^{R}$ ) to differentiate herself.

Suppose, by contrast, that the median voter is tilted toward the right (i.e. $β^{m} > D$ in Figure 1(b)). The extreme type would then look more attractive to the eye of the median voter if she signaled her true type. However, she does not have to do this by paying the cost of policy bias because, as long as $β^{m} > E$ , the median voter votes for the incumbent irrespective of her type. In short, the extreme incumbent is always a bystander in an election in the sense that whether she gets re-elected is predetermined, and thus the upward bias (i.e. a policy higher than ${\bar{β}}^{R}$ ) never emerges. Therefore, under this standard assumption about preferences, an extreme-policy result would require an additional assumption, such as uncertainty about the state of the economy, as in Schultz (2002).

4. Voter decisions at polls

Returning to our main assumption of overlapping types, voter decisions are examined. After observing the first-period policy, each voter updates his beliefs about incumbent R’s type from the prior $r^{0}$ to the posterior r. Because candidate L is the challenger and does not have any signaling opportunities,⁵ the public maintains its prior beliefs about the challenger without revision: $l = l^{0}$ .

The second-period policy is chosen by the winner of the election. This is done without further concern for elections. When candidate $β^{P}$ has won the election, she chooses $x_{2} = β^{P}$ to maximize $W_{2}^{P}$ in equation (2). In anticipation of such post-electoral policies and based on the pair of posterior beliefs r and l, each voter casts his vote for the candidate he prefers.⁶ Voter i votes for candidate L with certainty if and only if

(1 - l) U^{i} ({\underline{β}}^{L}) + l \cdot U^{i} ({\bar{β}}^{L}) > (1 - r) U^{i} ({\underline{β}}^{R}) + r \cdot U^{i} ({\bar{β}}^{R})

(3)

If the above inequality holds with <, then this voter votes for R with certainty. Replacing the inequality by an equality and using equation (1) yields an equation that represents a straight line in (r, l) space (Figure 2). If the posterior pair (r, l) falls on this line, voter i becomes indifferent about the two candidates and makes his decision by tossing a fair coin. Thus, this straight line is called voter i’s indifference line. If the posterior pair (r, l) falls in the shaded area in Figure 2, then voter i votes for candidate L; if it falls in the non-shaded area, they vote for candidate R.

Figure 2.

Voters’ decisions.

The following lemma tells us how the indifference line shifts when voter i’s ideal policy rises. The proof of the lemma, as well as the precise form of the indifference line, is in Appendix 1.

Lemma 1. Voter indifference line

Voter i’s indifference line rotates around a fixed point $(r^{*}, l^{*})$ counterclockwise as his ideal policy $β^{i}$ rises, as seen in Figure 2. Also $r^{*}$ and $l^{*}$ satisfy: $r^{*} \in (0, 1 / 2)$ ; $l^{*} \in (1 / 2, 1)$ ; $r^{*} + l^{*} = 1$ .

For example, Figure 2(d) illustrates voter i’s indifference line as a bold straight line when his bliss point comes to some $β^{i} \in (B, C)$ . If, in addition, the prior $l^{0}$ takes on the value $l'$ in that figure, then the belief about the challenger is represented by the horizontal line $l = l' (= l^{0})$ because the belief about her is not updated in this model. The incumbent R then wins this vote when she signals $r \in [0, r')$ but loses it when she signals $r \in (r', 1]$ .

The following lemma states that the outcome of the election depends upon the behavior of the median voter. A sketch of the proof is in Appendix 2.

Lemma 2. Median voter’s decisiveness

For the incumbent R who pursues majority votes, the median voter m is decisive in the sense that his voting behavior determines the outcome of the election.

This result is not trivial because the monotonicity in voters’ preferences for candidates is violated here. For instance, if the posterior pair (r, l) is close to the origin in Figure 2, the voter preference switches from candidate L to R as his bliss point $β^{i}$ rises from smallest to largest. However, if the posterior pair is close to (0,1), the converse is true. Still, Lemma 2 holds because, despite the violation of monotonicity, a voter’s preference for candidates never changes more than once, given a specific posterior, as shown in Appendix 2.

Taking Lemma 2 into account, we focus on the median voter’s behavior. The belief of the voters about the challenger’s type is summarized by $l = l^{0}$ , as explained above. Let $r^{m}$ denote the horizontal coordinate of the point at which the horizontal line $l = l^{0}$ crosses the median voter’s indifference line.

For instance, suppose that the median voter’s ideal policy satisfies $β^{m} \in (B, C)$ , as in Figure 2(d). If, in addition, $l^{0} = l'$ in that figure, then $r^{m} = r'$ . In this case, if the incumbent R transmits the signal $r \in [0, r')$ , she obtains the median voter’s vote, as discussed above. This result is also explained by returning to Figure 1(a). If $β^{m} \in (B, C)$ , then the median voter’s favorite is closer to ${\underline{β}}^{R}$ than to ${\bar{β}}^{R}$ and hence the incumbent obtains this vote by signaling that she has a high probability of being the moderate type. From Lemma 2, it follows that, by signaling this, she obtains a majority of votes and wins the election for certain.

However, if $β^{m} = E$ , as in Figure 2(g), and $l^{0} = l ″$ , then R gets re-elected when she signals $r \in (r ″, 1]$ . R thus has an incentive to signal that she is likely to be the extreme type in this situation. This is because $β^{m} = E$ is closer to ${\bar{β}}^{R}$ than to ${\underline{β}}^{R}$ . However, the median voter’s behavior is also affected by the prior $l^{0}$ . If $l^{0} = l'$ in the same figure (Figure 2(g)), then $r^{m} \notin [0, 1]$ and R wins the election no matter what she signals. This is because the median voter is closer to R, irrespective of R’s type, than to the average challenger L. A contrary example in which R fails to be re-elected, despite any signals she transmits, is found when $β^{m} = D$ , as in Figure 2(f), and $l^{0} = \hat{l}$ . R chooses her first-period policy taking these into consideration, as is discussed in the next section.

5. Incumbent R’s signaling behavior

In this electoral game, the incumbent faces the following trade-off. On the one hand, she wishes to be re-elected because, if she is, she can gain a perquisite and achieve her ideal policy in period two. Thus, she may wish to bias her first-period policy if such a bias increases the odds of re-election. On the other hand, she does not want to bias her first-period policy too much because she does not want to reduce her first-period pay-off excessively.

Incumbent $β^{R}$ ’s first-period pay-off is $W_{1}^{R} = V - {(x_{1} - β^{R})}^{2}$ . Her second-period expected pay-off is ${EW}_{2}^{R} = π V - (1 - π) [(1 - l) {({\underline{β}}^{L} - β^{R})}^{2} + l {({\bar{β}}^{L} - β^{R})}^{2}]$ , where $π = π (r)$ is the probability with which she will win the election. That is, if the incumbent gets re-elected, she chooses her ideal policy, $β^{R}$ , and receives the perquisite in period two. If she loses, however, the initiative of policy making, as well as the perquisite, is transferred to L, whose type is unknown. Then, letting $ρ$ denote the discount factor, $β^{R}$ ’s total pay-off is $W^{R} (x_{1}, r | β^{R}) = W_{1}^{R} + ρ {EW}_{2}^{R}$ .

The focus here is the policy in period one. If information were complete, $β^{R}$ would just choose her ideal policy, $β^{R}$ , because voters are certain about the incumbent’s type and the first-period policy would play no role as a signaling device. If information is incomplete, as assumed here, however, the policy can transmit a signal about r that may affect the probability of re-election.

There are three cases to be considered.

5.1. Case 1: when $r^{m} \notin [0, 1]$

If $r^{m} \notin [0, 1]$ , then the first-period policy cannot alter the median voter’s choice because his voting behavior is predetermined. An example of this case is found when the median voter’s bliss point corresponds to Figure 2(f) with $l^{0} = \hat{l}$ or to Figure 2(g) with $l^{0} = l'$ , as discussed in the last section. In this case, the incumbent will not bias her policy at all, as she would not under complete information. This consideration leads to the following proposition.

Proposition 1. Outcomes in Case 1

Suppose that $r^{m} \notin [0, 1]$ . Then, each type $β^{R}$ chooses policy $β^{R}$ in the first period. The outcome of the election is predetermined irrespective of the signal R transmits: if, in addition, $β^{m} < A$ , then R loses; if $β^{m} \geq C$ and $l^{0} < l^{*}$ , then R wins; if $β^{m} \geq C$ and $l^{0} > l^{*}$ , then R loses.

The last part of the proposition follows from the inspection of Figure 2.

5.2. Case 2: when $r^{m} \in (0, 1)$ and $β^{m} < D$

In this case, the extreme incumbent can have an incentive to mimic the moderate type, because she can be re-elected by signaling a small r. From Figures 2(a) to (f), it follows that in this case

π (r) = {\begin{matrix} 0 if r \in (r^{m}, 1] \\ 0.5 if r = r^{m} \\ 1 if r \in [0, r^{m}) \end{matrix}

(4)

An example of this case is in Figure 2(d), as seen above.

Based on the above consideration, Figure 3 illustrates the incumbent’s indifference loci in $(x_{1}, r)$ space. This type of figure is often used in the analysis of signaling games, with the horizontal axis measuring the sender’s signal and the vertical axis being the belief of the receiver. However, in contrast to continuous indifference curves typically found in standard signaling models, these loci consist of discontinuous straight lines. For instance, any points on the three line-segments ab, cd and ef give the same pay-off to the extreme incumbent. In the following paragraph, we derive the policy levels that correspond to the points d and f (and similar levels for the moderate type), which play a key role in deriving equilibrium.

Figure 3.

Candidate R’s indifference loci in Case 2.

In Case 2, we define ${\underline{x}}_{1}^{*} (β^{R})$ and ${\bar{x}}_{1}^{*} (β^{R})$ as the two values of $x_{1}$ that equate the following two pay-offs: (i) the pay-off that type $β^{R}$ receives when choosing her ideal policy, $β^{R}$ , in period one but losing the election with certainty by signaling $r \in (r^{m}, 1]$ and (ii) the pay-off that type $β^{R}$ receives when choosing a policy $x_{1}$ that is potentially different from $β^{R}$ but winning the election with certainty by signaling $r \in [0, r^{m})$ . Equating these pay-offs yields a quadratic equation in $x_{1}$ . The lower (respectively, upper) root for $x_{1}$ is denoted by ${\underline{x}}_{1}^{*} (β^{R})$ (respectively, ${\bar{x}}_{1}^{*} (β^{R})$ ). By construction, the three points $(β^{R}, 1)$ , $({\underline{x}}_{1}^{*} (β^{R}), 0)$ and $({\bar{x}}_{1}^{*} (β^{R}), 0)$ are on the same indifference locus in $(x_{1}, r)$ space (Figure 3).⁷ The specific forms of ${\underline{x}}_{1}^{*} (β^{R})$ and ${\bar{x}}_{1}^{*} (β^{R})$ are given in Appendix 3.

The solution concept used in the model is the perfect Bayesian equilibrium (PBE). Usually, when a signal-sender’s indifference loci have standard shapes of continuous curves and satisfy the single-crossing property, Cho and Kreps’s (1987) Intuitive Criterion selects the most Pareto-efficient separating PBE (i.e. the Riley outcome). In the current model, however, the sender’s indifference loci satisfy neither of the above conditions, and hence the Intuitive Criterion does not isolate a unique outcome.⁸ Nevertheless, this model focuses on the Riley outcome for the reason below. A sketch of the proof is in Appendix 4.

Lemma 3. Equilibrium

(i) When attention is restricted to separating equilibria, the Intuitive Criterion always selects a unique outcome (the Riley outcome). (ii) When the focus is on pooling equilibria, there exists no PBE per se under some parametric assumptions (e.g. $r^{0} > r^{m}$ ). (iii) When the focus is on hybrid equilibria, no equilibria per se exist in which ${\underline{β}}^{R}$ randomizes, if $r^{0} \geq r^{m}$ ; no equilibria in which ${\bar{β}}^{R}$ randomizes survive the Intuitive Criterion, if ${\underline{β}}^{R} < {\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ .

In short, because neither pooling nor hybrid equilibria can solve the problems of non-existence, this model focuses on separating equilibrium. A partial characterization of pooling and hybrid equilibria is provided in Appendix 4.

Another reason for this focus comes from the fact that if some uncertainty on the side of voters is introduced (under somewhat awkward assumptions), the sender’s indifference loci exhibit standard continuities and the Intuitive Criterion selects a unique separating outcome. This point is revisited at the end of this section.

The proof of part (i) of Lemma 3 directly derives the following proposition, in which the incumbent’s first-period policy and the election outcome in the intuitive separating outcomes are characterized. In what follows, the term bias in policy is used to indicate some difference in policy compared with that chosen under complete information.

Proposition 2. Outcomes in Case 2

Suppose that $r^{m} \in (0, 1)$ and $β^{m} < D$ . Then type ${\bar{β}}^{R}$ chooses her ideal ${\bar{β}}^{R}$ , losing the election; if ${\underline{β}}^{R} > {\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ (an envy case), type ${\underline{β}}^{R}$ chooses ${\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ (a downward bias), winning the election; if ${\underline{β}}^{R} \leq {\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ (a no-envy case), type ${\underline{β}}^{R}$ chooses her ideal ${\underline{β}}^{R}$ , winning the election.

Intuitively, in Case 2, the incumbent can be re-elected if she signals to the voters that she has a high probability of being moderate, as seen above. If, in addition, the incumbent’s types are relatively similar (i.e. ${\underline{x}}_{1}^{*} ({\bar{β}}^{R}) < {\underline{β}}^{R} < {\bar{β}}^{R}$ ), then an ‘envy case’ emerges, as in Figure 3. That is, whenever the moderate type ${\underline{β}}^{R}$ chooses ${\underline{β}}^{R}$ , which is her ideal policy under complete information, then the extreme type ${\bar{β}}^{R}$ would have an incentive to imitate the moderate type by taking the same policy, which would lead to victory. Put differently, the extreme type would desire to deviate from point a to point g in Figure 3. In any separating equilibrium, the moderate type has to differentiate herself from this potential imitator by biasing her policy downward: $x_{1} \leq {\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ . The Intuitive Criterion selects a unique outcome with the least bias, in which the moderate type chooses $x_{1} = {\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ .

Note that, from an inspection of equation (A.1) in Appendix 3, it follows that ${\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ is decreasing in $ρ$ , V, and $β^{L} - {\bar{β}}^{R}$ . That is, all other things being equal, if winning the election becomes more attractive than losing it, then the policy bias required to block the potential imitator becomes larger.

The above logic for a policy bias is the same as that under the standard assumption, as explained in Section 3. The importance of our novel assumption of overlapping types is highlighted in Case 3, as seen in the next subsection.

5.3. Case 3: when $r^{m} \in (0, 1)$ and $β^{m} > D$

In this case, the incumbent can be re-elected by signaling a large, not small, r. An example is found if the median voter’s bliss point is as in Figure 2(g) and if $l^{0} = l ″$ , as seen above. This case emerges when the median voter favors a relatively large policy ( $β^{m} > D$ ) and the average challenger L is likely to be a moderate type rather than an extreme left (i.e. has a somewhat large $l^{0}$ ).

In Case 3, the probability with which R wins the election is

π (r) = {\begin{matrix} 0 if r \in [0, r^{m}) \\ 0.5 if r = r^{m} \\ 1 if r \in (r^{m}, 1] \end{matrix}

in contrast to equation (4). This situation can be analyzed using Figure 4, which seems to be obtained by turning Figure 3 upside down. In Figure 4, ${\underline{x}}_{1}^{*} (β^{R})$ and ${\bar{x}}_{1}^{*} (β^{R})$ are defined analogously to those in the previous subsection. In addition, if the incumbent types are relatively similar (i.e. ${\underline{β}}^{R} < {\bar{β}}^{R} < {\bar{x}}_{1}^{*} ({\underline{β}}^{R})$ ), as in Figure 4, then an ‘envy case’ arises here: whenever the extreme type ${\bar{β}}^{R}$ chooses her bliss point ${\bar{β}}^{R}$ , the moderate type ${\underline{β}}^{R}$ would have an incentive to mimic the extreme type. In any separating equilibrium, the extreme type has to differentiate herself from this potential imitator ( ${\underline{β}}^{R}$ ) by creating an upward bias in policy. If we focus on separating equilibria for the same reasoning as in Lemma 3, the Intuitive Criterion selects a unique outcome with the smallest bias, leading to the following proposition, which is central to this paper.

Figure 4.

Candidate R’s indifference loci in Case 3.

Proposition 3. Outcomes in Case 3

Suppose that $r^{m} \in (0, 1)$ and $β^{m} > D$ . Then type ${\underline{β}}^{R}$ chooses her ideal ${\underline{β}}^{R}$ , losing the election; if ${\bar{β}}^{R} < {\bar{x}}_{1}^{*} ({\underline{β}}^{R})$ (an envy case), type ${\bar{β}}^{R}$ chooses ${\bar{x}}_{1}^{*} ({\underline{β}}^{R})$ (an upward bias), winning the election; if ${\bar{β}}^{R} \geq {\bar{x}}_{1}^{*} ({\underline{β}}^{R})$ (a no-envy case), type ${\bar{β}}^{R}$ chooses her ideal ${\bar{β}}^{R}$ , winning the election.

That is, a policy bias can also occur in the extreme (upward) direction when the possible types of the candidates overlap. The reason for this result is that we may have a situation in which even an extreme type, ${\bar{β}}^{R}$ , cannot be a bystander in an election. Indeed, when the median voter is tilted toward the right and the challenger is a centrist with a somewhat large probability, the extreme incumbent would lose the election unless she sends the correct message to the voters. That is, she would lose the median voter’s vote if she is mistaken for a moderate type. If, in addition, the possible types of incumbent are close to each other, the extreme type is required to bias her policy upward to send the correct message.

The above result contrasts with that under the standard assumption of non-overlapping types. An intuitive reason why a bias in extreme directions does not occur in the standard case was explained in Section 3. Technically, if the types do not overlap, as in Figure 1(b), then the central point, $(r^{*}, l^{*})$ , around which a voter’s indifference line rotates, goes outside the relevant square, as opposed to Figure 2: $r^{*} < 0$ ; $l^{*} > 1$ ; $r^{*} + l^{*} = 1$ . Then, the indifference line always has a downward slope when it crosses the square. Thus, situations like Figures 2(f) to (h) do not emerge.

Returning to the extreme-policy result, because the incumbent’s rightist policy is predicted when the median voter is tilted toward the right, it can be said that their interests are not completely in conflict. However, Proposition 3 shows that a small change in the median voter’s position can cause a magnified effect on a politician’s behavior through the politician’s signaling intention. Indeed, it is worth noting that her pre-electoral policy can be extreme to a degree that nobody desires. Suppose, for instance, that the largest ideal policy among the whole electorate, $β^{N}$ , is smaller than ${\bar{β}}^{R}$ . Even if this is the case, when the above conditions of an envy case are satisfied, a policy further right than ${\bar{β}}^{R}$ is created.

A partial justification for the assumption that the candidates’ types may overlap was given in the Introduction. We point out here that, from a theoretical viewpoint, even if the probability of such an overlap is very small, the result obtained above is not affected. To see this, note that the prior belief $r^{0}$ has no relevance in deriving the above separating equilibrium. Even if $r^{0}$ is close to unity (e.g. even if the incumbent is of type ${\bar{β}}^{R}$ with probability 0.99 and is of type ${\underline{β}}^{R}$ with probability 0.01), Propositions 1 to 3 still hold.⁹ Thus, the requirement to support the above results is the minimum.

This model focuses on separating equilibrium, instead of pooling or hybrid equilibria, using the reasoning in Lemma 3. Another justification for this focus comes from the result of a modified version of the model as follows. Suppose there is mutual uncertainty between candidates and voters, so that a voter preference for candidates is also private information. It might then be problematic to assume that every voter’s preference is uncertain because it may become difficult to determine the identity of the median voter. The simplest way to avoid this problem is to assume that there are only three voters, i =1, 2, 3, and that only voter number two has uncertainty. The probability of the incumbent being re-elected then depends upon the probability with which she can attract this median voter. Under this set-up, it is possible to construct a smooth indifference locus for the incumbent and show that the Intuitive Criterion selects a unique outcome, the Riley outcome.

Finally, if the incumbent is from a leftist party, the prediction is completely symmetric. An extremely leftist policy can be predicted when the opposite conditions to those in the envy case of Proposition 3 are satisfied.

6. Concluding remarks

This paper has shown that an incumbent’s signaling behavior is not limited to distorting her policy to moderate directions even if private information is introduced on only one-dimensional parameters: the candidate ideologies. This result contrasts with that of Schultz (2002), in which extreme policies were caused by uncertainty about an additional parameter: the state of the economy. The key assumption in our model is that possible ideology types of candidates from different parties overlap. Because this is a novel assumption, the model devoted a considerable amount of space to the proofs of basic results (e.g. the median voter’s decisiveness) with the help of figures.

One implication of the model is that the extremist tendency within any one party may become most active when party ideologies are overlapping. Conversely, pre-electoral signaling motives would inevitably generate moderation between parties when the ruling and opposition parties have completely distinct ideologies. Although this prediction can motivate empirical research, there will be at least one serious difficulty associated with such a study. Because extremism needs to be measured by policy divergence from an incumbent politician’s ideal, an estimate of her bliss point would be required. This is conceptually challenging because such bliss points cannot be gauged simply by reference to her actual actions. A possible way to estimate them may be to distinguish her pre-electoral and post-electoral policies, as implied by the model. Empirical analyses overcoming such difficulties may be a useful subject for future research.

Footnotes

Appendix 1. Proof of Lemma 1

The condition under which voter i votes for L with certainty (equation (3)) can be rewritten as follows, with the use of equation (1): $H \cdot l > K + J \cdot r$ , where $H \equiv {({\underline{β}}^{L} - β^{i})}^{2} - {({\bar{β}}^{L} - β^{i})}^{2}$ , $K \equiv {({\underline{β}}^{L} - β^{i})}^{2} - {({\underline{β}}^{R} - β^{i})}^{2}$ and $J \equiv {({\underline{β}}^{R} - β^{i})}^{2} - {({\bar{β}}^{R} - β^{i})}^{2}$ .

Voter i’s indifference line is then defined as $l = (K / H) + (J / H) r$ in (r, l) space. This equation can be rewritten as

l - l^{*} = (J / H) (r - r^{*})

where $l^{*} \equiv \frac{({\underline{β}}^{R} - {\underline{β}}^{L}) ({\bar{β}}^{R} - {\underline{β}}^{L})}{({\bar{β}}^{L} - {\underline{β}}^{L}) ({\bar{β}}^{R} + {\underline{β}}^{R} - {\bar{β}}^{L} - {\underline{β}}^{L})}$ $r^{*} \equiv \frac{({\underline{β}}^{R} - {\underline{β}}^{L}) ({\bar{β}}^{L} - {\underline{β}}^{R})}{({\bar{β}}^{R} - {\underline{β}}^{R}) ({\bar{β}}^{R} + {\underline{β}}^{R} - {\bar{β}}^{L} - {\underline{β}}^{L})}$

That is, any voter’s indifference line passes through the point $(r^{*}, l^{*})$ . From the above expression, it can be shown that $r^{*}, l^{*} \in (0, 1)$ when the types overlap ( ${\underline{β}}^{R} < {\bar{β}}^{L}$ ). In addition, under Assumption 1, $r^{*} + l^{*} = 1$ and $r^{*} < 1 / 2$ .

Next, if voter i’s ideal policy rises, the intercept of the indifference line for that voter decreases (Figure 2)

\frac{\partial (K / H)}{\partial β^{i}} = \frac{2 ({\underline{β}}^{R} - {\underline{β}}^{L}) ({\bar{β}}^{L} - {\underline{β}}^{L}) ({\underline{β}}^{R} - {\bar{β}}^{L})}{{[{({\underline{β}}^{L} - β^{i})}^{2} - {({\bar{β}}^{L} - β^{i})}^{2}]}^{2}} < 0

except for the point in which H becomes zero and K/H becomes discontinuous (i.e. when $β^{i} = B)$ . The strict inequality comes from ${\underline{β}}^{R} - {\bar{β}}^{L} < 0$ . On the basis of the above facts, voters’ indifference lines are illustrated in Figure 2. □

Appendix 2. Sketch of the proof of Lemma 2

First, we prove the following claim.

Claim 1. ${\frac{K}{H} |}_{β^{i} < A} < {\frac{K}{H} |}_{β^{i} > E}$

(Proof of Claim 1)

From the definitions of K and H in Appendix 1, $\frac{K}{H} = \frac{{({\underline{β}}^{L} - β^{i})}^{2} - {({\underline{β}}^{R} - β^{i})}^{2}}{{({\underline{β}}^{L} - β^{i})}^{2} - {({\bar{β}}^{L} - β^{i})}^{2}}$ .

By l’Hopital’s rule, $lim_{β^{i} \to - \infty} \frac{K}{H} = lim_{β^{i} \to - \infty} \frac{K' (β^{i})}{H' (β^{i})} = \frac{{\underline{β}}^{R} - {\underline{β}}^{L}}{{\bar{β}}^{L} - {\underline{β}}^{L}} = lim_{β^{i} \to + \infty} \frac{K}{H}$ . This fact, as well as Lemma 1, proves the claim. □

Claim 1 indicates that voter i’s indifference line rotates by less than $180 \circ$ , even if the bliss point of that voter goes up from smallest ( $β^{i} < A$ ) to largest ( $β^{i} > E$ ). Hence, the intercept of the line in Figure 2(a) is lower than that in Figure 2(h). Intuitively, this means that a voter’s preference for L or R does not switch more than once as long as his bliss point changes monotonically.

Next, we prove Lemma 2. There are four cases to be considered: (1) $β^{m} \in (- \infty, D)$ and $l^{0} < l^{*}$ ; (2) $β^{m} \in (- \infty, D)$ and $l^{0} \geq l^{*}$ ; (3) $β^{m} \in [D, + \infty)$ and $l^{0} < l^{*}$ ; (4) $β^{m} \in [D, + \infty)$ and $l^{0} \geq l^{*}$ . This proof considers only (1). All the other cases can be considered similarly. Let $r^{m}$ denote the horizontal coordinate of the point at which the horizontal line $l = l^{0}$ crosses the median voter’s indifference line. We assume for the moment that $r^{m} \in (0, 1)$ . Suppose, for example, that the median voter’s ideal, $β^{m}$ , is represented by Figure 2(d) (i.e. $β^{m} \in (B, C)$ ) and that $l^{0} = l'$ . Then the corresponding $r^{m} = r' \in (0, 1)$ is plotted in that figure.

Suppose, in addition, that R signals some $r \in [0, r^{m})$ in this situation.

As seen above, whether R can win the votes of voters who belong to (i) and (iv) by signaling some $r \in [0, r^{m})$ is not an ex ante certainty. However, Claim 1 implies that, for any $β^{i} \in [0, β^{m})$ , $β^{j} \in [D, + \infty)$ and $l^{0} < l^{*}$ , $ℜ^{W} (l^{0}, β^{i}) \cup ℜ^{W} (l^{0}, β^{j}) = [0, 1] ¹ [0, r^{m})$ . Hence, by signaling some $r \in [0, r^{m})$ , R may at worst lose all the votes in (i), which number (N– 1)/2, but in that case she wins all the votes from (iv). Adding the votes obtained from (i) and (iv) to votes from (ii) and (iii), the number of votes R wins is greater than or equal to (N+1)/2. Thus, R can obtain a majority of votes by signaling any $r \in [0, r^{m})$ .

Next, suppose that R signals $r = r^{m}$ . Then she wins (N–1)/2 votes with certainty from the voters in (iii) and (iv), loses (N– 1)/2 votes with certainty from the voters in (i) and wins the median voter’s vote with probability 0.5. Thus, R wins the election with probability 0.5.

Finally, suppose that R signals some $r \in (r^{m}, 1]$ . Then R loses the election with certainty because she loses all the votes from (i) and (ii).

If $r^{m} \notin [0, 1]$ , then the outcome of the election is predetermined, irrespective of the signal R sends. By use of Claim 1, it can be shown that, in this case, the candidate whom the median voter prefers wins the election with certainty. □

Appendix 3. Derivations of x ¯ 1 * ( β R ) and x ¯ 1 * ( β R )

The pay-off that candidate $β^{R}$ obtains when (i) in the text takes place is

$V - ρ [(1 - l) {({\underline{β}}^{L} - β^{R})}^{2} + l {({\bar{β}}^{L} - β^{R})}^{2}]$ . On the other hand, the pay-off she receives when (ii) in the text takes place is $V - {(x_{1} - β^{R})}^{2} + ρ V$ . The solutions for $x_{1}$ that equate these pay-offs are

(A.1)

x_{1} = β^{R} \pm \sqrt{ρ [V + (1 - l) {({\underline{β}}^{L} - β^{R})}^{2} + l {({\bar{β}}^{L} - β^{R})}^{2}]}

The smaller (respectively, larger) solution is denoted by ${\underline{x}}_{1}^{*} (β^{R})$ (respectively, ${\bar{x}}_{1}^{*} (β^{R})$ ).

Appendix 4. Sketch of the proof of Lemma 3

Here, $Ψ (β^{R})$ denotes a set of points $(x_{1}, r)$ that achieve, for each type $β^{R}$ , the pay-off equal to or greater than the pay-off she receives when choosing her ideal policy, $β^{R}$ , in period one and the voters interpret this policy as a signal of $r = 1$

Ψ (β^{R}) \equiv {\begin{matrix} (x_{1}, r) \begin{matrix} s . t . \end{matrix} W^{R} (x_{1}, r | β^{R}) \geq W^{R} (β^{R}, 1 | \end{matrix} β^{R})}

In addition, let $x_{1}^{e} (β^{R})$ denote a first-period policy that type $β^{R}$ chooses with some positive probability and $r^{e} (β^{R})$ the posterior induced by that policy. The proof of Lemma 3 uses the following claims.

Claim 2. A necessary condition for $(x_{1}^{e} (β^{R}), r^{e} (β^{R}))$ to constitute a part of the equilibrium is

(A.2)

(x_{1}^{e} (β^{R}), r^{e} (β^{R})) \in Ψ (β^{R})

(Proof of Claim 2)

In Figure 3, this claim implies that, for each type $β^{R}$ , any equilibrium path $(x_{1}^{e} (β^{R}), r^{e} (β^{R}))$ must fall on or within the two line-segments of the indifference locus (for $r^{e} \in [0, r^{m})$ ), or on the locus (for $r^{e} \in (r^{m}, 1]$ ) (that is, for type ${\bar{β}}^{R}$ , it must fall on or within the rectangle cdfe, or on the line segment ab in Figure 3). To see this, note that in a PBE, the posterior beliefs have to be formed for any signal $x_{1}$ , including both on- and off-the-equilibrium paths. Thus, there exists at least one $(x_{1}, r)$ that satisfies the above condition. Suppose that an equilibrium path $(x_{1}^{e} (β^{R}), r^{e} (β^{R}))$ does not satisfy (A.2). Then candidate R of that type would have an incentive to deviate from the proposed equilibrium to some $(x_{1}, r)$ that satisfies (A.2). □

Claim 3. ${\underline{x}}_{1}^{*} (β^{R})$ is increasing in $β^{R}$ .

(Proof of Claim 3)

Differentiating ${\underline{x}}_{1}^{*} (β^{R})$ in (A.1) with respect to $β^{R}$ gives

\frac{\partial {\underline{x}}_{1}^{*} (β^{R})}{\partial β^{R}} = 1 - \sqrt{ρ} (\frac{(1 - l) (β^{R} - {\underline{β}}^{L}) + l (β^{R} - {\bar{β}}^{L})}{\sqrt{V + (1 - l) {({\underline{β}}^{L} - β^{R})}^{2} + l {({\bar{β}}^{L} - β^{R})}^{2}}}) \equiv 1 - \sqrt{ρ} (λ / μ) .

As $μ^{2} - λ^{2} = V + l (1 - l) {({\bar{β}}^{L} - {\underline{β}}^{L})}^{2} > 0$ and $\sqrt{ρ} < 1$ , it follows that

$\partial {\underline{x}}_{1}^{*} (β^{R}) / \partial β^{R} > 0$ . This result implies ${\underline{x}}_{1}^{*} ({\underline{β}}^{R}) < {\underline{x}}_{1}^{*} ({\bar{β}}^{R})$ , as in Figure 3. □

Now we prove the lemma.

Acknowledgements

The author wishes to thank Claudio Mezzetti, Yoichi Hizen, Yuma Kawahara, Takashi Narihiro, Rodney Dunham, two anonymous referees and the editor for helpful comments. The usual disclaimers apply.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by JSPS KAKENHI (grant number 24530301).

Notes

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