Information in elections: Do third inflexible candidates always promote truthful behavior?

Abstract

We model an election between two Downsian mainstream candidates and a third inflexible politician. There is uncertainty about the state of the world. Candidates receive signals on the state and propose a policy to implement. There are two classes of voters: ideological, who are biased towards the policy proposed by the third candidate; and non-ideological, who want the policy implemented to correspond to the state of the world. We study two cases: (1) one in which the third candidate supports the most popular policy (in terms of the electorate’s prior); (2) another one in which he supports the less popular policy. We obtain that the presence of a third candidate facilitates equilibria in which the two mainstream politicians make informative announcements, specially when the third candidate is biased towards the most popular policy. We also obtain that many of the informative equilibria are sustained by a coalition government, however the coalition is never between the two mainstream candidates. Last, we observe that in equilibrium, the third inflexible candidate has significant chances of winning office.

1 Introduction

In systems with a representative democracy, the question of whether the electoral process can aggregate the information the politicians have and transmit it to the voters is of primary importance. Relevant contributions that date back to Harrington (1993), Roemer (1994), Schultz (1995, (1996, (2002), and more recently Heidhues and Lagerlöf (2003), Morelli and Weelden (2013), Loertscher (2012), and Kartik et al. (2015) agree that in the presence of asymmetric information between voters and (better informed) political parties, electoral processes may fail to be informative.¹

This abundant literature on the policy distortions that arise when politicians have superior information, has so far focused on the classic two-party competition model.² However, in light of the recent results in the 2015 Spanish Municipal and Regional Elections and the 2014 European Elections, this focus seems a bit narrow and unrealistic. In fact, the 2014 European Elections showed the erosion of majoritarian politics and the irruption of third extremist parties all over Europe, from France (where Front National won 25 $\%$ of votes), UK (where UKIP got the biggest share of votes) and Spain (where Podemos, a party only 4 month old got an 8 $\%$ of votes), to Austria, Denmark, Greece and Italy. More recently, the 2015 Spanish Municipal and Regional Elections confirmed the end of our bipartisan system, giving Spain’s biggest cities, Madrid and Barcelona, to extreme-left parties. A swift that also experienced other important cities, say A Coruña, Cádiz, Valencia and Zaragoza.

Have nontraditional parties come to stay? According to a study conducted by Pew Research in May 2015, the answer is yes. In fact, respondents of four out of six European countries said that the rise of nontraditional political parties is a good thing. This is the case in Germany, Italy, UK and Spain, where an overwhelmingly 70 $\%$ of respondents answered that the rise of Podemos is good for the Spanish democracy.³

Based on the recent empirical evidence, we propose a model that studies the incentives of two mainstream office-motivated politicians to reveal their private information in the presence of a third ideological candidate and an electorate that is heterogeneous.

To this aim, we consider an adaptation of the model in Heidhues and Lagerlöf (2003) (hereafter HL), in which two office-seeker candidates receive imperfect and correlated signals on the state of the world, which is unknown to voters.⁴ Following their specification we consider a binary world, with two states and two possible policies. For expositional purposes, and inspired by the current EU economic crisis, we identify the two policies as (1) an expansionary fiscal policy, that aims to stimulate aggregated demand through public spending; and (2) a policy of austerity, that aims to reduce budget deficits in order to foster the return of sustainable growth. Accordingly, in our words, the state of the world will be either pro-stimulus or pro-austerity. We assume that a policy consisting of expansionary measures or fiscal stimulus is the best fit for the economy if the state is pro-stimulus, and that an austerity policy is the appropriate one if the state is pro-austerity.⁵ Following HL, we assume that the two mainstream candidates observe the signals and propose the policies so as to win office.

We extend the HL set-up in two directions. (1) First, we consider the existence of a third candidate that runs for office. Based on casual observation, we assume that the third candidate is ideological and always proposes the same policy (in our case, the expansive fiscal policy). This is an assumption that aids considerably in the tractability of the problem and is grounded in the real world observation that third parties are usually more dogmatic than mainstream ones.⁶ Laver and Hunt (1992) find strong evidence to support this idea. In a study for 25 countries, they asked people to locate in a scale 1–20 the position of the most important parties in their country regarding the following question: “forced to make a choice, would party leaders give up policy objectives in order to get into government or would they sacrifice a place in government in order to maintain policy objectives?”. They observed that systematically, citizens of different countries perceive mainstream candidates as more willing to give up policy objectives (then, as more strategic) and third parties as more likely to give up a place in government (then, as more deterministic).⁷ (2) Second, we consider the existence of heterogeneous voters, divided into two groups. A majority consisting of non-ideological voters, who want the implemented policy to correctly match the state of the world; and a minority of ideological voters, who have a fixed preference for the policy that the third candidate stands for (the expansive fiscal policy).

Thus, our election game incorporates three candidates and two groups of voters. Candidates observe a signal about the state of the world and propose a policy to implemented if elected. Voters observe the policy proposals, update beliefs about which policy is best (if they need to) and cast their vote. Because with three candidates it could be that no politician wins an absolute majority of votes, we need to define the government coalition formation game and the policy implemented by the resulting government. In the paper we consider that there is a function that maps electoral outcomes (vote shares) into a probability distribution over the set of government outcomes (single-party governments and coalition governments). That is, for every vector of vote shares, this function determines the probability that each possible government outcome occurs. We also assume that when a coalition government forms, a lottery (between the policies proposed by the candidates in the coalition) yields the final implemented policy. This is known to everyone and taken into account by the voters when deciding their vote. After casting the votes, votes are counted and the electoral outcome determined. A winner of the election results and the corresponding policy is implemented. Candidates and voters receive their payoff.

The aim of this paper is twofold. First, to analyze whether in this new scenario with a third inflexible candidate and a group of ideological voters, the two mainstream candidates can credibly transmit their information to the voters through their choice of policy platforms. Thus, we focus on fully revealing equilibria. This is our information transmission game. Second, to analyze how Bayesian (non-ideological) voters decide their vote in the presence of the new actors; namely a third inflexible candidate that always proposes the expansive fiscal policy, and a group of ideological voters in favor of this policy. This is our voting game.

We solve the game by backward induction. We start with the voting game. We consider a platform profile that corresponds to a particular class of fully revealing equilibria. Given the platform profile, we analyze the voters’ optimal voting behavior. This allows us to obtain the vector of electoral outcomes or vote shares. Then, according to a function, we can obtain the probability that each single-party government and coalition government occurs. We obtain that, in equilibrium, all coalitions that receive positive probability are politically-akin coalitions. That is, coalitions of candidates that propose the same policy. Then, we go backwards and analyze whether the set of government outcomes (that receive positive probability) allows for information transmission from the two mainstream candidates. We obtain that the presence of a third inflexible candidate might actually help ease the information transmission game. This result is in line with Felgenhauer (2012), who in a simpler set-up with one voter and a less sophisticated signal technology, obtains that a third uninformed candidate can help restore efficiency.⁸ We also show that for the mainstream candidates to credibly communicate their information to the voters, it is best that the third inflexible candidate be biased towards the most popular policy (in terms of the electorate’s prior). That is, that both ideological and non-ideological voters have the same (a-priori) like bias. This adds a new insight to Felgenhauer (2012)’s conclusion about the informational benefit of an inflexible third candidate, namely that as important as the existence of a third candidate it is his bias. Last, we obtain that for the mainstream candidates to honestly reveal their information, we need voters to have a strict preference for one of these two candidates. That is, that whenever mainstream candidates propose the same policy, voters always vote for the same candidate. Otherwise, that is if under indifference they flip a coin, no information transmission can be achieved. In this sense, voters’ captivity seems to be good.

The remainder of the paper is organized as follows. Section 2 describes the model. Section 3 analyzes the voting game and Sect. 4 studies the information transmission game. Then, Sect. 5 presents a discussion, where we examine the implications that the two main features of the model: third candidate and heterogenous voters, have on the results. Finally, Sect. 6 concludes.

2 The model

An election is to be held. There are three political candidates and two groups of voters. There are two stages of the world, $w_E$ and $w_A$. Throughout the paper, $w_E$ refers to an economic situation where expansionary or fiscal stimulus policies are required to return economic growth. We will say that in this case, the state is pro-stimulus. On the other hand, $w_A$ refers to a situation where austerity measures are to be implemented first if we want to foster long-run sustainable growth. Here, we say that the state is pro-austerity. Let q be the prior probability that the state is $w_E$, i.e., $P(w=w_E)=q\in (0,1)$.

The candidates: Candidates are labeled 1, 2 and 3. Candidates 1 and 2 are Downsian and want to get into office. We refer to them as the mainstream candidates. Candidate 3 is considered to have a preferred policy that he always proposes. This assumption is intended to capture the empirical observation that third parties are usually more dogmatic than the mainstream ones. We refer to the third candidate as the ideological or inflexible candidate.⁹

There are two policy alternatives E and A, where E stands for expansionary fiscal policies and A for austerity measures. We assume E is the correct policy in state $w_E$ and A is the correct one in state $w_A$. Candidates 1 and 2 choose which policy to propose so as to win office. Let $x_i\in \{E,A\}$ be the policy proposed by candidate $i\in \{1,2,3\}$. Without loss of generality, we assume $x_3=E$. Rents from office are K.

The information structure of the candidates: Each candidate 1 and 2 receives a signal $s_1,s_2\in \{e,a\}$, on the state of the world.¹⁰ Signals are correlated, with $\rho \in [0,1)$ being a measure of the degree of correlation between the signals. Hence, the higher $\rho$, the greater the correlation. Additionally, signals are not perfectly informative about the state of the world. With probability $(1-\varepsilon )$, a candidate’s signal is equal to the state; with probability $\varepsilon$ he receives an incorrect signal. We assume $\varepsilon \in (0,1/2)$. Table 1 summarizes the signal technology.¹¹

Table 1 Signal technology

Upon receiving a signal, candidates choose the policy to campaign on and to implement if elected.

The voters: There are two groups of voters: a group of ideological voters and a group of non-ideological voters. All voters in a group are identical. Non-ideological voters want the policy to be appropriate to the state. The utility of a non-ideological voter, denoted by 4, is $U_4(E,w_E)=U_4(A,w_A)=1$ and $U_4(E,w_A)=U_4(A,w_E)=0$. Ideological voters always prefer fiscal stimulus policies to cutbacks and other stiff austerity measures. The utility of an ideological voter, denoted by 5, is $U_5(E,w_i)=1$ and $U_5(A,w_i)=0$, for $i\in \{E,A\}$. Total mass of voters is equal to one. Let $\beta$ and $1-\beta$ be the fraction of non-ideological and ideological voters, respectively. We assume $\beta \in (\frac{1}{2},1)$, i.e., we consider a majority of non-ideological voters.¹²

The election: Voters observe the vector of proposed policies $(x_1,x_2,x_3)\in \{E,A\}^3$. Then, voters update beliefs about which policy is best (if they need to) and cast their vote.¹³ This determines the vector of vote shares $v=(v_1,v_2,v_3)$, where $v_i$ denotes the fraction of votes of candidate $i\in \{1,2,3\}$, with $v_1+v_2+v_3=1$.

The winner and the implemented policy: Let $V=\{(v_1,v_2,v_3)\in [0,1]^3 \mid v_1+v_2+v_3=1\}$ be the set of electoral outcomes (or vote shares), and let v be an element of this set. Additionally, let $G=\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\}\}$ be the set of government outcomes (collection of all single-party governments and coalition governments), and let g be an element of this set. Note that G does not include the grand coalition.¹⁴

Now we introduce a function f: $V\rightarrow \bigtriangleup G$, that maps electoral outcomes into government outcomes. More precisely, for each $v\in V$,

$$\begin{aligned}&f(v)=\big (p_{\{1\}},p_{\{2\}},p_{\{3\}},p_{\{1,2\}},p_{\{1,3\}},p_{\{2,3\}}\big )\\&\qquad \text {with } p_g\in [0,1] \text { and } \sum _{g\in G}p_g=1, \end{aligned}$$

where component $p_g$ gives the probability that government outcome $g\in G$ occurs.

Our formulation introduces one assumption: a strict majority of votes is required to govern. Thus, given $v\in V$, our definition of f satisfies:

$$\begin{aligned} \begin{array}{ll} p_{\{i\}}=1 &{}\quad \text {if } v_i>1/2\\ p_{\{1,2\}}+p_{\{1,3\}}+p_{\{2,3\}}=1, \text { with } p_{\{1,2\}},p_{\{1,3\}},p_{\{2,3\}}>0 &{}\quad \text {if } \max \{v_1,v_2,v_3\}<1/2\\ p_{\{i,j\}}+p_{\{i,k\}}=1, \text { with } p_{\{i,j\}},p_{\{i,k\}}>0 &{}\quad \text {if } v_i=1/2,v_j,v_k>0\\ p_{\{i,j\}}=1 &{}\quad \text {if } v_i=v_j=1/2,v_k=0 \end{array} \end{aligned}$$

with $(i,j,k)\in \{1,2,3\}^3$, $i\ne j\ne k$.

Two important comments:

• When there is a candidate, say $i\in \{1,2,3\}$, that wins an absolute majority of votes, our formulation implies that this candidate governs alone. That is when $v_i>1/2$, $p_{\{i\}}=1$; thus $g=\{i\}$. In this case, we say there is a single-party government. Here, the policy implemented is the one proposed by the winning candidate i, that is $x_i$. Candidate i gets the entire rents K from holding office.

• When no candidate receives a strict majority of votes, our formulation considers that any pair of candidates $(i,j)\in \{1,2,3\}^2$ such that $v_i+v_j>1/2$ can form a coalition government.¹⁵ That is, voters assign positive probability to any pair (i, j) that satisfies $v_i+v_j>1/2$ to enter government. In this case, the final government will necessarily be a coalition government.¹⁶ Because all coalitions are minimal, we assume that a candidate in a coalition receives half of the rents K from holding office. Next, we define what the implemented policy will be in this case. To this, we first have to define what the (expected) implemented policy of a coalition, say $g=\{i,j\}$, will be. We denote by $x_{\{i,j\}}$ the expected implemented policy of this coalition. Given $g=\{i,j\}$, with $(i,j)\in \{1,2,3\}^2$ and $i\ne j$, the expected implemented policy $x_{\{i,j\}}\in \{E,A\}$ is assumed to be the result of a lottery between the policies proposed by the candidates in the coalition, and probabilities that are strictly positive for each event. More precisely, given the vector of outcomes $(x_i,x_j)$, with $(x_i,x_j)\in \{E,A\}^2$, policy $x_{\{i,j\}}$ is given by the lottery

  $$\begin{aligned} (x_i,x_j;p_i,p_j) \end{aligned}$$

  (1)

  that yields policy $x_k$ with probability $p_k$, with $k\in \{i,j\}$, $(p_i,p_j)\in (0,1)^2$ and $p_i+p_j=1$.¹⁷ Now, because when $\max \{v_1,v_2,v_3\}<1/2$ any pair of candidates (i, j), with $(i,j)\in \{1,2,3\}^2$, can form a coalition government, the expected implemented policy in this case will be a compound lottery between the policies implemented by the winning coalitions (with $x_{\{i,j\}}$ defined by the lottery above), and probabilities that are strictly positive for each event. That is, given the vector of simple lotteries $(x_{\{1,2\}},x_{\{1,3\}},x_{\{2,3\}})$, the compound lottery

  $$\begin{aligned} (x_{\{1,2\}},x_{\{1,3\}},x_{\{2,3\}};p_{\{1,2\}},p_{\{1,3\}},p_{\{2,3\}}) \end{aligned}$$

  (2)

  is the risky alternative that yields the simple lottery $x_{\{i,j\}}$ with probability $p_{\{i,j\}}$, with $p_{\{i,j\}}\in (0,1)$, $\sum _{\{i,j\}}p_{\{i,j\}}=1$ and $\{i,j\}\in \{\{1,2\},\{1,3\},\{2,3\}\}$.¹⁸

Timing: The sequence of events is as follows. First, candidates receive a private signal on the stage of the world and choose the policy to propose. Voters observe the policy proposals, update beliefs about which policy is best (if they need to), and cast their vote. Votes are counted and according to function f, the government outcome is determined. It also determines the final implemented policy.

Candidates’ strategies: Note that for the ideological candidate, $x_3=E$. Hence, we restrict notation to exclusively account for the behavior of the two mainstream candidates. We define $\sigma _i^j$ as the probability that candidate $i\in \{1,2\}$ proposes policy E after observing signal $j\in \{e,a\}$, and $\sigma _c=(\sigma _1^{e},\sigma _1^{a};\sigma _2^{e},\sigma _2^{a})$ as the vector of strategies for the (mainstream) candidates. Henceforth, we refer to $(x_1,x_2)\in \{E,A\}^2$ as a platform profile.

Voters’ strategies: We assume that all voters in a group use the same voting strategy, which implicitly means that voters in each group can perfectly coordinate in their voting strategies and choose a common strategy (that they will all follow) that is a best response to the other group strategy. This is a simplifying assumption that is of much help in our analysis. In fact, note that because with a continuum of voters no voter is pivotal, any voter’s strategy would be a best response, so our voting game would be multiplicity of Nash equilibria.¹⁹

Note that under this assumption, it is sufficient to define the following two strategies. Let $\sigma _{4,i}^{x_1 x_2}$ be the probability that each non-ideological voter votes for $i\in \{1,2,3\}$, after having observed the policy proposal $(x_1,x_2)\in \{E,A\}^2$. Similarly, let $\sigma _{5,i}^{x_1 x_2}$ be the probability that each ideological voter votes for $i\in \{1,2,3\}$, after having observed the policy proposal $(x_1,x_2)\in \{E,A\}^2$. Additionally, let $\sigma _v^{x_1 x_2}=(\sigma _4^{x_1 x_2};\sigma _5^{x_1 x_2})$, with $\sigma _4^{x_1 x_2}=(\sigma _{4,1}^{x_1 x_2},\sigma _{4,2}^{x_1 x_2},\sigma _{4,3}^{x_1 x_2})$ and $\sigma _5^{x_1 x_2}=(\sigma _{5,1}^{x_1 x_2},\sigma _{5,2}^{x_1 x_2},\sigma _{5,3}^{x_1 x_2})$, denote the vector of voting strategies for voters of groups 4 and 5, respectively.

Thus, for example, if $\sigma _4^{x_1 x_2}=(1/2,1/2,0)$ and $\sigma _5^{x_1 x_2}=(0,0,1)$, we have a situation in which each non-ideological voter votes for each of the two mainstream candidates with probability 1 / 2, and each ideological voter votes for the third candidate with probability one. Now, because the mass of non-ideological and ideological voters is $\beta$ and $1-\beta$, respectively, we have that in this case, the (expected) vector of vote shares is $v=(\frac{1}{2}\beta ,\frac{1}{2}\beta ,1-\beta )$.²⁰

The equilibria: We are interested in studying whether in the presence of a third inflexible candidate, the two mainstream candidates can credibly communicate their information to the voters through their choice of policy platforms. Thus, our analysis focuses on equilibria with full revelation. In the following, we say that an equilibrium is fully revealing if the two mainstream candidates perfectly signal their information at all stages of the nature. Note that if we restrict our attention to this class of equilibria, candidate $i\in \{1,2\}$ has two strategies: (1) a faithful strategy, defined by $(\sigma _i^e,\sigma _i^a)=(1,0)$, and (2) a reversed strategy, defined by $(\sigma _i^e,\sigma _i^a)=(0,1)$. Hence, in the fully revealing class of equilibrium we focus on, the vector of the candidates’ strategies is $\sigma _c\in \{(1,0;1,0),(1,0;0,1),(0,1;1,0),(0,1;0,1)\}$.

The equilibrium concept that we use is perfect Bayesian equilibrium.

Note that although our model considers a one-stage election game, the fact that there are three candidates and two groups of voters makes the analysis of the information transmission game and the voting game a bit complex. Thus, in order to be clear in the exposition of the analysis and the results, we take the following approach. Using a backward induction argument, we analyze the election game in two steps: (1) first, we study the voting game. That is, we analyze how a voter that observes the candidates’ platform profile $(x_1,x_2)$ and understands the mechanism that transforms votes into government outcomes (function f), casts her vote. We do this for any platform profile coming from a fully revealing class of equilibrium. (2) Then, we go backwards and analyze the information transmission game. That is, we investigate whether the resulting government outcomes allow the two mainstream candidates to fully reveal their information to the voters during the election campaign.

3 Voting

Consider the following example. Suppose an election campaign in which the two mainstream candidates propose cutbacks and other stiff austerity measures and the third candidate proposes an expansionary fiscal policy. Which candidate/s would voters vote for?

If the voter is ideological the answer is quite straightforward. Because she has a strict preference for expansionary fiscal policies, it is a dominated strategy for her to vote for any of the two mainstream candidates. Thus, let us suppose she (and all the other ideological voters) votes for candidate 3. Now, what about non-ideological voters?

To answer this question we first have to analyze how non-ideological voters would change their belief about which policy is best, if they were able to learn the content of the two candidates’ signals. Note that this is always the case in the fully revealing class of equilibria we focus on. There are two cases: (1) the prior distribution of the state is in favor of the expansionary fiscal policy ($q>1/2$); and (2) the prior distribution of the state is in favor of the austerity policy ($q<1/2$).²¹

First, suppose $q>1/2$. In this case, when a voter has no other information than her prior, she thinks that policy E is the best. Now, let us consider that the voter knew the content of the two signals. If both were in favor of policy E, the voter would be reinforced in her opinion that policy E is the best.²² If one signal indicated e and the other a, the content of the two signals would cancel out and the voter would still prefer E, her prior.²³ Lastly, suppose the two signals indicated a. Applying Bayes rule, we obtain that $P(w=w_A \mid s_1=a,s_2=a)>P(w=w_E \mid s_1=a,s_2=a)$ if and only if $(1-q)[(1-\varepsilon )^2+\rho \varepsilon (1-\varepsilon )]>q[\varepsilon ^2+\rho \varepsilon (1-\varepsilon )]$, which can be rewritten as $q<\widetilde{q}=\frac{(1-\varepsilon )[1-\varepsilon (1-\rho )]}{1-2\varepsilon (1-\varepsilon )(1-\rho )}.$ In order for the problem to be interesting, when analyzing the case $q>1/2$, we will assume that after two a signals, the voter changes her mind and prefers A. This requires the prior probability that E is the best policy being not too large, i.e., $q\in (\frac{1}{2},\widetilde{q})$.²⁴

Analogously, if $q<1/2$, a voter with no other information than her prior would vote for policy A. Now, if the voter knew that at least one signal indicates a, she would prefer A. Otherwise, we assume she changes her mind and prefers E. Similar to the previous case, this requires the prior probability that A is the best policy being not too large, i.e., $q\in (1-\widetilde{q},\frac{1}{2})$.

Assumption 1

(Not too large prior) $q\in (1-\widetilde{q},\widetilde{q})$, with $\widetilde{q}=\frac{(1-\varepsilon )[1-\varepsilon (1-\rho )]}{1-2\varepsilon (1-\varepsilon )(1-\rho )}$.

Let $\hat{q}=P(w_E \mid s_1,s_2)$ be the (non-ideological) voters’ posterior belief that the state is $w_E$, conditional on candidates 1 and 2 observing signals $s_1$ and $s_2$, respectively. Assumption 1 guarantees that, independently of which policy, E or A, the prior distribution of the state is in favor of, a non-ideological voter may change her mind and end up with a belief that differs from her initial one. That is, if $q\in (1-\widetilde{q},\widetilde{q})$, $\hat{q}$ has support in all the interval (0, 1). Additionally, note that the voter’s updated belief may favor a policy that is not proposed by any of the candidates.²⁵ In this case, we assume that non-ideological voters are bound to vote for a candidate that supports their non-preferred policy.

Let us now come back to the example. Remember we were considering a scenario in which the two mainstream candidates are proposing austerity measures, this is $(x_1,x_2)=(A,A)$. Also, suppose we are in a situation in which the posterior belief on the state of the world is $\hat{q}<1/2$.²⁶ Finally, note that we consider that (non-ideological) voters can coordinate their voting strategies and choose a common strategy (that they will all follow) that is a best response to the other group strategy. In this case, which candidate/s would non-ideological voters vote for?

Because $\hat{q}<1/2$, voters believe that policy A is the best. In this case, it is natural to think that non-ideological voters should vote for either one of the two mainstream candidates, 1 or 2. In fact, if they all were to vote for 1, that is $\sigma _4^{AA}=(1,0,0)$, the electoral outcome would be $v=(\beta ,0,1-\beta )$. Since $\beta >1/2$, the government outcome would be $g=\{1\}$ and the policy implemented A, in which case the utility to each non-ideological voter would be $1-\hat{q}$. On the other hand, if they all were to vote for candidate 3 (let us say with probability 1), the electoral outcome would be $v=(0,0,1)$, the government outcome $g=\{3\}$ and the policy implemented E. Because the utility to each non-ideological voter in this case would be $\hat{q}$ and $\hat{q}<1/2$, the result follows.

Now, what about a mixed strategy in which each non-ideological voter were to vote for each of the two mainstream candidates with probability 1 / 2? Is it an optimal strategy? To answer this question, suppose $\sigma _4^{AA}=(\frac{1}{2},\frac{1}{2},0)$. Note that in this case, $v=(\frac{1}{2}\beta ,\frac{1}{2}\beta ,1-\beta )$. Since $\beta \in (1/2,1)$, $max\{v_1,v_2,v_3\}<1/2$. That is, we are in a situation in which any pair of two candidates can form a coalition. In this case, because all coalitions are possible, the implemented policy is given by the compound lottery:

$$\begin{aligned} \big (x_{\{1,2\}},x_{\{1,3\}},x_{\{2,3\}};p_{\{1,2\}},p_{\{1,3\}},p_{\{2,3\}}\big ) \end{aligned}$$

with $p_{\{i,j\}}\in (0,1)$, $\sum _{\{i,j\}}p_{\{i,j\}}=1$ and $\{i,j\}\in \{\{1,2\},\{1,3\},\{2,3\}\}$; with outcome $x_{\{1,2\}}$ being equal to policy A,²⁷ and outcome $x_{\{i,3\}}$, with $i\in \{1,2\}$, being the simple lottery:

$$\begin{aligned} \big (A,E;p_i,p_3\big ) \end{aligned}$$

with $(p_i,p_3)\in (0,1)^2$, $p_i+p_3=1$, and $i\in \{1,2\}$.

Now, let us denote by $EU_4(x_{\{i,j\}})$ the expected utility to any non-ideological voter when the policy implemented is the simple lottery $x_{\{i,j\}}$. Then, the expected utility to each of these voters in the present compound lottery is:

$$\begin{aligned}&p_{\{1,2\}}EU_4(x_{\{1,2\}})+\sum _{i\in \{1,2\}}p_{\{i,3\}}EU_4(x_{\{i,3\}})\\&\quad =p_{\{1,2\}}EU_4(A)+\sum _{i\in \{1,2\}}p_{\{i,3\}}\big (p_i EU_4(A)+p_3 EU_4(E)\big )\\&\quad = p_{\{1,2\}}(1-\hat{q})+\sum _{i\in \{1,2\}}p_{\{i,3\}}\left( p_i(1-\hat{q})+p_3\hat{q}\right) . \end{aligned}$$

Note that $(p_i(1-\hat{q})+p_3\hat{q})$ is a convex combination of $(1-\hat{q})$ and $\hat{q}$. Also, remember that if $\sigma _{4,1}^{AA}=1$, the payoff to each of these voters is $1-\hat{q}$. Then, since $\forall \hat{q}<1/2$, $1-\hat{q}>\hat{q}$, the payoff to each non-ideological voter if they all follow $\sigma _4^{AA}=(1,0,0)$ is higher than her payoff if they all use $\sigma _4^{AA}=(\frac{1}{2},\frac{1}{2},0)$. Thus, we can conclude that using a mixed strategy that gives no mainstream candidate an absolute majority of votes is not a best common strategy for this group of voters.

From the analysis above we learn that in equilibrium, no pair of candidates proposing different policies will get enough votes to form a coalition government. That is, all possible winning coalitions are going to be politically-akin coalitions.²⁸

Lemma 1 characterizes the voters’ optimal voting strategies, given a platform profile and a posterior belief about the state of the world. To simplify the characterization, we assume that ideological voters never vote for a candidate proposing policy A.²⁹

Lemma 1

Consider a platform profile $(x_1,x_2)\in \{E,A\}^2$. In the voting game, the optimal voting behavior of the voters, $\sigma _v^{*x_1 x_2}$, satisfies the following conditions:

• 1. Consider $(x_1,x_2)=(E,E)$. For any $\hat{q}\in (0,1)$:

  • (a) $(\sigma _{4,1}^{*EE},\sigma _{4,2}^{*EE},\sigma _{4,3}^{*EE})\in [0,1]^3$, is such that $\sum _{i=\{1,2,3\}}\sigma _{4,i}^{*EE}=1$,

  • (b) $(\sigma _{5,1}^{*EE},\sigma _{5,2}^{*EE},\sigma _{5,3}^{*EE})\in [0,1]^3$, is such that $\sum _{i=\{1,2,3\}}\sigma _{5,i}^{*EE}=1$.

• 2. Consider $(x_1,x_2)=(E,A)$.

  • (a) When $\hat{q}<1/2$:

    • (i) $(\sigma _{4,1}^{*EA},\sigma _{4,2}^{*EA},\sigma _{4,3}^{*EA})$, is such that $\beta \sigma _{4,2}^{*EA}>\beta (\sigma _{4,1}^{*EA}+\sigma _{4,3}^{*EA})+(1-\beta )$,

    • (ii) $(\sigma _{5,1}^{*EA},\sigma _{5,2}^{*EA},\sigma _{5,3}^{*EA})$, is such that $\sigma _{5,2}^{*EA}=0, \sum _{i=\{1,3\}}\sigma _{5,i}^{*EA}=1$.

  • (b) When $\hat{q}>1/2$:

    • (i) $(\sigma _{4,1}^{*EA},\sigma _{4,2}^{*EA},\sigma _{4,3}^{*EA})$, is such that either $\sigma _{4,2}^{*EA}>0$ and $\beta \sigma _{4,i}^{*EA}+(1-\beta )\sigma _{5,i}^{*EA}>\beta (\sigma _{4,2}^{*EA}+\sigma _{4,j}^{*EA})+(1-\beta )\sigma _{5,j}^{*EA}$, for $(i,j)\in \{1,3\}^2, i\ne j;$ or $\sigma _{4,2}^{*EA}=0$ and $\sum _{i\in \{1,3\}}\sigma _{4,i}^{*EA}=1$,

    • (ii) $(\sigma _{5,1}^{*EA},\sigma _{5,2}^{*EA},\sigma _{5,3}^{*EA})$, is such that $\sigma _{5,2}^{*EA}=0, \sum _{i=\{1,3\}}\sigma _{5,i}^{*EA}=1$.

• 3. Consider $(x_1,x_2)=(A,E)$.

  • (a) When $\hat{q}<1/2$:

    • (i) $(\sigma _{4,1}^{*AE},\sigma _{4,2}^{*AE},\sigma _{4,3}^{*AE})$, is such that $\beta \sigma _{4,1}^{*AE}>\beta (\sigma _{4,2}^{*AE}+\sigma _{4,3}^{*AE})+(1-\beta )$,

    • (ii) $(\sigma _{5,1}^{*AE},\sigma _{5,2}^{*AE},\sigma _{5,3}^{*AE})$, is such that $\sigma _{5,1}^{*AE}=0, \sum _{i=\{2,3\}}\sigma _{5,i}^{*AE}=1$.

  • (b) When $\hat{q}>1/2$:

    • (i) $(\sigma _{4,1}^{*AE},\sigma _{4,2}^{*AE},\sigma _{4,3}^{*AE})$, is such that either $\sigma _{4,1}^{*AE}>0$ and $\beta \sigma _{4,i}^{*AE}+(1-\beta )\sigma _{5,i}^{*AE}>\beta (\sigma _{4,1}^{*AE}+\sigma _{4,j}^{*AE})+(1-\beta )\sigma _{5,j}^{*AE}$, for $(i,j)\in \{2,3\}^2, i\ne j;$ or $\sigma _{4,1}^{*AE}=0$ and $\sum _{i=\{2,3\}}\sigma _{4,i}^{*AE}=1$,

    • (ii) $(\sigma _{5,1}^{*AE},\sigma _{5,2}^{*AE},\sigma _{5,3}^{*AE})$, is such that $\sigma _{5,1}^{*AE}=0, \sum _{i=\{2,3\}}\sigma _{5,i}^{*AE}=1$.

• 4. Consider $(x_1,x_2)=(A,A)$.

  • (a) When $\hat{q}<1/2$:

    • (i) $(\sigma _{4,1}^{*AA},\sigma _{4,2}^{*AA},\sigma _{4,3}^{*AA})$, is such that $\beta \sigma _{4,i}^{*AA}>\beta \sigma _{4,j}^{*AA}+(1-\beta )$, for $(i,j)\in \{1,2\}^2, i\ne j$,

    • (ii) $(\sigma _{5,1}^{*AA},\sigma _{5,2}^{*AA},\sigma _{5,3}^{*AA})=(0,0,1)$.

  • (b) When $\hat{q}>1/2$:

    • (i) $(\sigma _{4,1}^{*AA},\sigma _{4,2}^{*AA},\sigma _{4,3}^{*AA})$ is such that $\beta \sigma _{4,3}^{*AA}+(1-\beta )>\beta (\sigma _{4,1}^{*AA}+\sigma _{4,2}^{*AA})$,

    • (ii) $(\sigma _{5,1}^{*AA},\sigma _{5,2}^{*AA},\sigma _{5,3}^{*AA})=(0,0,1)$.

Proof

In the Appendix.$\square$

For each platform profile $(x_1,x_2)\in \{E,A\}^2$, and posterior $\hat{q}\in (0,1)$, let $\Gamma _v^{* x_1 x_2}$ be the set of the voters’ optimal voting strategies that satisfy conditions in Lemma 1. Let $\sigma _v^{* x_1 x_2}\in \Gamma _v^{* x_1 x_2}$ be an element of this set.

Having characterized the voters’ optimal voting strategies for a platform profile $(x_1,x_2)\in \{E,A\}^2$ and the posterior $\hat{q}\in (0,1)$, the next steep is to determine, for each platform profile and posterior, what electoral outcomes $v\in V$ will result and then, given function f, what probability distribution over government outcomes we will have. This will allow us to determine, for each platform profile $(x_1,x_2)\in \{E,A\}^2$, what government outcomes (single-party governments and/or coalition governments) will have strictly positive probability to form. Prior to this, we introduce a new definition.

Abusing a bit of notation, for each platform profile $(x_1,x_2)\in \{E,A\}^2$, we denote by $G^{x_1 x_2}\subseteq G$ the set of government outcomes such that, for each $g^{x_1 x_2}\in G^{x_1 x_2}$, there exists optimal strategies for the voters $\sigma _4^{*x_1 x_2}$ and $\sigma _5^{*x_1 x_2}$ that, given function f, result in $g^{x_1 x_2}$ forming a government with (strictly) positive probability when the platform profile $(x_1,x_2)$ is proposed. This definition will be very useful in the analysis of the next section.³⁰

To clarify this concept, consider the example above, where $(x_1,x_2)=(A,A)$ and $\hat{q}<1/2$. According to Lemma 1, in this case $(v_1,v_2,v_3)=(\beta \sigma _{4,1}^{*AA},\beta \sigma _{4,2}^{*AA},1-\beta )$, with $\beta \sigma _{4,i}^{*AA}>\beta \sigma _{4,j}^{*AA}+(1-\beta )$, for $(i,j)\in \{1,2\}^2, i\ne j$. Thus, in the equilibrium of the voting game, either $v_1>1/2$ or $v_2>1/2$ occurs. Now, from the definition of function f, we know that if $v_1>1/2$, then $p_{\{1\}}=1$, and if $v_2>1/2$, then $p_{\{2\}}=1$. As a consequence, $G^{AA}=\{\{1\},\{2\}\}$. That is, given the platform profile $(x_1,x_2)=(A,A)$, when $\hat{q}<1/2$, there are two government outcomes that, a priori, have (strictly) positive probability: one in which candidate 1 governs alone, $g^{AA}=\{1\}$, and another one in which candidate 2 does it, $g^{AA}=\{2\}$.

Next Corollary characterizes, for every platform profile $(x_1,x_2)\in \{E,A\}^2$, the set $G^{x_1 x_2}\subseteq G$.³¹

Corollary 1

Given a platform profile $(x_1,x_2)\in \{E,A\}^2$, the set of voters’ optimal voting strategies, $\Gamma _v^{*x_1 x_2}$, and the function f, the set $G^{x_1 x_2}$ is:

• 1. Consider $(x_1,x_2)=(E,E)$. Then, $\forall \hat{q}\in (0,1), G^{EE}=\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\}\}$.

• 2. Consider $(x_1,x_2)=(E,A)$.

  • (a) Then $\forall \hat{q}<1/2, G^{EA}=\{\{2\}\}$,

  • (b) Then $\forall \hat{q}>1/2, G^{EA}=\{\{1\},\{3\},\{1,3\}\}$.

• 3. Consider $(x_1,x_2)=(A,E)$.

  • (a) Then $\forall \hat{q}<1/2$, then $G^{AE}=\{\{1\}\}$,

  • (b) Then $\forall \hat{q}>1/2, G^{AE}=\{\{2\},\{3\},\{2,3\}\}$.

• 3. Consider $(x_1,x_2)=(A,A)$.

  • (a) Then $\forall \hat{q}<1/2$, then $G^{AA}=\{\{1\},\{2\}\}$,

  • (b) Then $\forall \hat{q}>1/2$, then $G^{AA}=\{\{3\}\}$.

Corollary 1 shows that in the equilibrium of the voting game, only the candidates that propose the policies that, according to the posterior belief $\hat{q}$, are more likely to match the state of the world, have chances of entering a government. Thus, as already anticipated, no coalition of candidates proposing different policies will ever form. Additionally, we observe that except for the case $(x_1,x_2)=(E,E)$, the two mainstream candidates will never get enough votes to form a winning coalition together.

To have an intuition for the proof, consider $(x_1,x_2)=(E,E)$. In this case, any voting strategy for the voters constitute a best response. Then, any $v\in V$ is possible and thus, any government outcome has strictly positive probability. Accordingly, $G^{EE}=\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\}\}$. Now suppose $(x_1,x_2)=(E,A)$ and $\hat{q}<1/2$. In this case, non-ideological voters prefer policy A. According to Lemma 1, any vector of voting strategies $\sigma _v^{EA}$ such that the resulting electoral outcome is $v_2>v_1+v_3$ is optimal. Because in this case $v_2>1/2$, applying function f, we have $p_{\{2\}}=1$; thus $G^{EA}=\{\{2\}\}$. Suppose now $(x_1,x_2)=(E,A)$ and $\hat{q}>1/2$. Here, voters agree that the best policy is E. Then, any voting strategy $\sigma _4^{EA}$ that guarantees that policy E is implemented with probability 1 is a best response. This occurs when the vector of voting strategies $\sigma _v^{EA}$ results in either $v_i>1/2$, for $i\in \{1,3\};$ or $v_2=0$ and $v_1=v_3=1/2$. Thus, $G^{EA}=\{\{1\},\{3\},\{1,3\}\}$. Last, suppose $(x_1,x_2)=(A,A)$. When $\hat{q}>1/2$, voters agree that the best policy is E. In this case, any vector of voting strategies $\sigma _v^{AA}$ that results in $v_3>1/2$ is optimal. Then, $G^{AA}=\{\{3\}\}$.³²

4 Information transmission

We now return to the question of whether the two mainstream candidates can credibly communicate their information to the voters through their choice of policy platforms. To this aim, we go backwards and analyze whether for a given platform profile $(x_1,x_2)\in \{E,A\}^2$, the optimal voting behavior and the resulting government outcomes $g^{x_1 x_2}\subseteq G^{x_1 x_2}$, allow to sustain platform profile $(x_1,x_2)\in \{E,A\}^2$ as part of a fully revealing equilibrium.

We restrict attention to fully revealing equilibria and differentiate two cases, according to which state of the world the prior is in favor of.³³

Like biases

Let us start considering the case in which politicians know that expansionary fiscal policies are more popular with the electorate than cutbacks and other stiff austerity measures. In terms of our analysis, this means $P(w_E)=q>1/2$. Here, note that non-ideological voters with no other information than their prior prefer policy E. Since ideological voters are in our model biased towards policy E, we say we have in this case a scenario with like biases.

Perhaps somewhat surprisingly, our results show that candidates can, in this scenario, credibly communicate their information to the voters. More interestingly, we obtain that the number of government outcomes that allows for full revelation is greater here than in the case of opposing biases, which is analyzed next.³⁴

How can it be that the number of equilibria in which the two mainstream candidates behave honestly is higher when the electorate is homogeneous than when voters have different points of view? To see the intuition for this result, first note that in both scenarios, the mainstream candidates have an incentive to bias their message towards the electorate’s preferred policy, as in HL.³⁵ However, the presence of a third candidate who dogmatically supports fiscal stimulus policies introduces a second effect, that reduces the attractiveness of proposing E.³⁶ In the case of like biases, this second effect counterbalances the first one, hence facilitating equilibria in which the two mainstream candidates behave honestly. In contrast, in the case of opposing biases, the first and the second effects go in the same direction, therefore making information transmission more difficult to hold.

Prior to the results, given a platform profile $(x_1,x_2)\in \{E,A\}^2$, we define $\hat{\Gamma }_v^{* x_1 x_2}\subset \Gamma _v^{* x_1 x_2}$ as the set of voters’ optimal voting strategies that satisfy conditions (1)–(3) of anonymity:

• 1. If when $(x_1,x_2)=(E,A), p_{\{1\}}=1$; then when $(x_1,x_2)=(A,E), p_{\{2\}}=1$.

• 2. If when $(x_1,x_2)=(E,A), p_{\{3\}}=1$, then when $(x_1,x_2)=(A,E), p_{\{3\}}=1$.

• 3. If when $(x_1,x_2)=(E,A), p_{\{1,3\}}=1$, then when $(x_1,x_2)=(A,E), p_{\{2,3\}}=1$.

Note that the conditions on $\hat{\Gamma }_v^{* x_1 x_2}$ only restrict the voting behavior when the two mainstream candidates support different policies (in which case, the voters’ preferred policy is E). That is, when the non-ideological voters’ choice is between the inflexible candidate and the mainstream candidate that proposes E. With respect to these situations, we say that any $\sigma _v^{* x_1 x_2}\in \hat{\Gamma }_v^{* x_1 x_2}$ holds the anonymity condition, which means that the identity of the mainstream politician (whether he is 1 or 2) does not affect the voter’s optimal voting behavior. The result in the Proposition 1 considers $\sigma _v^{* x_1 x_2}\in \hat{\Gamma }_v^{* x_1 x_2}$, which allows us to simplify the analysis.³⁷

Last, for $i,j\in \{e,a\}$, let us define $P_{i \mid j}=P(s_2=i \mid s_1=j)$. Also, we will write $g^{x_1 x_2}(\hat{\Gamma }_v^{* x_1 x_2})=\{g\}$ when we want to refer to a situation in which given a platform profile $(x_1,x_2)\in \{E,A\}^2$, and the set of the voters’ optimal voting strategies $\hat{\Gamma }_v^{* x_1 x_2}$, applying f we obtain $p_{\{g\}}=1$, with $g\in G$. That is, given $(x_1,x_2)\in \{E,A\}^2, g^{x_1 x_2}(\hat{\Gamma }_v^{* x_1 x_2})=\{g\}$ is equivalent to $p_{\{g\}}=1$.

Proposition 1

$(\textit{Like biases})$ Consider $q\in (\frac{1}{2},\widetilde{q})$. A configuration $(\sigma _c^*,\sigma _v^{* x_1 x_2};$ $g^{x_1x_2}(\hat{\Gamma }_v^{* x_1 x_2}))_{(x_1,x_2)\in \{E,A\}^2,\sigma _v^{* x_1 x_2}\in \hat{\Gamma }_v^{* x_1 x_2}}$ constitutes a fully revealing equilibrium if and only if:

• (i) $\sigma _c^*=(1,0;1,0)$ and for any $(x_1,x_2)\in \{E,A\}^2, \sigma _v^{*x_1 x_2}\in \hat{\Gamma }_v^{* x_1 x_2}$ is such that either:

  • (a) $g^{EA}(\hat{\Gamma }_v^{*EA})=g^{AE}(\hat{\Gamma }_v^{*AE})=\{3\},$ $g^{AA}(\hat{\Gamma }_v^{*AA})=g^{EE}(\hat{\Gamma }_v^{*EE})=\{1\};$ or

  • (b) $g^{EA}(\hat{\Gamma }_v^{*EA})=g^{AE}(\hat{\Gamma }_v^{*AE})=\{3\},$ $g^{AA}(\hat{\Gamma }_v^{*AA})=\{1\},$ $g^{EE}(\hat{\Gamma }_v^{*EE})=\{1,3\},$ with $P_{e \mid e}\ge 2P_{a \mid e};$ or

  • (c) $g^{EA}(\hat{\Gamma }_v^{*EA})=g^{AE}(\hat{\Gamma }_v^{*AE})=\{3\},$ $g^{AA}(\hat{\Gamma }_v^{*AA})=g^{EE}(\hat{\Gamma }_v^{*EE})=\{2\};$ or

  • (d) $g^{EA}(\hat{\Gamma }_v^{*EA})=g^{AE}(\hat{\Gamma }_v^{*AE})=\{3\},$ $g^{AA}(\hat{\Gamma }_v^{*AA})=\{2\},$ $g^{EE}(\hat{\Gamma }_v^{*EE})=\{2,3\}$, with $P_{e \mid e}\ge 2P_{a \mid e};$ or

• (ii) $\sigma _c^*=(1,0;0,1)$ and for any $(x_1,x_2)\in \{E,A\}^2, \sigma _v^{*x_1 x_2}\in \hat{\Gamma }_v^{* x_1 x_2}$ is such that $g^{AE}(\hat{\Gamma }_v^{*AE})=\{1\}, g^{AA}(\hat{\Gamma }_v^{*AA})=\{3\}$ and either:

  • (a) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{1\}, g^{EA}(\hat{\Gamma }_v^{*EA})=\{3\};$ or

  • (b) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{3\}, g^{EA}(\hat{\Gamma }_v^{*EA})=\{1\};$ or

  • (c) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{3\}, g^{EA}(\hat{\Gamma }_v^{*EA})=\{1,3\}, P_{e \mid e} >2P_{a \mid e} ;$ or

  • (d) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{1,3\}, g^{EA}(\hat{\Gamma }_v^{*EA})=\{1,3\};$ or

  • (e) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{1,3\}, g^{EA}(\hat{\Gamma }_v^{*EA})=\{1\}, P_{a \mid a} >2P_{e \mid a} ;$ or

• (iii) $\sigma _c^*=(0,1;1,0)$ and for any $(x_1,x_2)\in \{E,A\}^2, \sigma _v^{*x_1 x_2}\in \hat{\Gamma }_v^{* x_1 x_2}$ is such that $g^{EA}(\hat{\Gamma }_v^{*EA})=\{2\}, g^{AA}(\hat{\Gamma }_v^{*AA})=\{3\}$ and either:

  • (a) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{2\}, g^{AE}(\hat{\Gamma }_v^{*AE})=\{3\};$ or

  • (b) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{3\}, g^{AE}(\hat{\Gamma }_v^{*AE})=\{2\};$ or

  • (c) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{3\}, g^{AE}(\hat{\Gamma }_v^{*AE})=\{2,3\}, P_{e \mid e} >2P_{a \mid e} ;$ or

  • (d) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{2,3\}, g^{AE}(\hat{\Gamma }_v^{*AE})=\{2,3\};$ or

  • (e) $g^{EE}(\hat{\Gamma }_v^{*EE})=\{2,3\}, g^{AE}(\hat{\Gamma }_v^{*AE})=\{2\}, P_{a \mid a} >2P_{e \mid a} ;$ or

• (iv) $\sigma _c^*=(0,1;0,1)$ and for any $(x_1,x_2)\in \{E,A\}^2, \sigma _v^{*x_1 x_2}\in \hat{\Gamma }_v^{* x_1 x_2}$ is such that $g^{AA}(\hat{\Gamma }_v^{*AA})=g^{EA}(\hat{\Gamma }_v^{*EA})=g^{AE}(\hat{\Gamma }_v^{*AE})=g^{EE}(\hat{\Gamma }_v^{*EE})=\{3\}$.

Proof

In the Appendix.$\square$

Proposition 1 yields a number of comments. First and foremost, that the most robust equilibria are the ones in which candidates use a faithful strategy (case (i)). Nonetheless, these are the only equilibria in which the two mainstream candidates have a chance of winning office (although not together in a coalition, as discussed next). Apart from this scenario, one of the mainstream candidates (the one using the reversed strategy) never gets elected, which is the reason why we can sustain these equilibria. In this sense, the equilibria in (ii)–(iv) are more fragile than those in (i).

Let us therefore comment in more detail the results in (i). An important conclusion is that for the mainstream candidates to use a faithful strategy, voters must penalize any disagreement between the policies proposed by these two candidates. That is, information transmission requires that the ideological candidate forms a single-party government whenever the mainstream candidates announce different policies. But not only this, we also obtain that one of the two mainstream candidates is always excluded from the government. A direct implication of this result is that no coalition between the two mainstream candidates can ever form.³⁸ To have an intuition for this result, note that when $(x_1,x_2)=(A,A)$ and $\widehat{q}<\frac{1}{2}$, we showed that non-ideological voters had to vote for one of the mainstream candidates with probability one.³⁹ We now observe that for information transmission to occur, after observing $(x_1,x_2)=(E,E)$, non-ideological voters have to vote again, with probability one, for the same mainstream candidate. Otherwise, the candidate who previously received no votes would find it profitable to deviate to E.

Thus, from the analysis we learn that information transmission requires of voters having a strict preference for one of the two mainstream candidates, that they use whenever are indifferent between them. Otherwise, that is if under indifference voters flip a coin (thus they may swing from candidate 1 to 2 and viceversa), no information transmission can be achieved.

There is a last remark to make. It refers to the effect of signal quality and the correlation between signals on the probability that transmission of information occurs. To this respect, we observe that $P_{i \mid i} \ge 2P_{i \mid j}$ is a necessary condition for some of the government outcomes to sustain information transmission. Since $P_{i \mid i}$ is increasing in $\rho$ and decreasing in $\varepsilon$, and $P_{i \mid j}$ is decreasing in $\rho$ and increasing in $\varepsilon$, we obtain that the higher the correlation between the candidates’ signals ($\rho$) and the lower the signal’s error ($\varepsilon$), the higher the number of government outcomes that sustain information transmission. Corollary 2 formalizes this idea.

Corollary 2

The number of government outcomes that allow for information transmission increases in:

• (i) the correlation between the candidates’ signals,

• (ii) the quality of the signals.

Proof

From the signal technology described in Table 1, we obtain $P_{i \mid i} =(1-\varepsilon )^2+\varepsilon ^2+2\rho \varepsilon (1-\varepsilon )$ and $P_{i \mid j} =2(1-\rho )\varepsilon (1-\varepsilon )$. Since $\partial P_{i \mid i} /\partial \rho >0, \partial P_{i \mid i} /\partial \varepsilon =2(2\varepsilon -1)(1-\rho )<0, \partial P_{i \mid j} /\partial \rho <0$ and $\partial P_{i \mid j} /\partial \varepsilon =2(1-\rho )(1-2\varepsilon )>0$, the proof follows. $\square$

Opposing biases

Let us now consider the case of non-ideological voters ex ante thinking that austerity measures are more appropriate than expansionary fiscal policies to return sustainable economic growth. In terms of our analysis, this means $P(w_E)=q<1/2$. Because ideological and non-ideological voters differ in this case in their (a priori) preferred policy, we say we have here an scenario of opposing biases.

As already pointed out, our result here shows that having an electorate with opposing views does not facilitate information transmission, but instead makes it harder. We base this conclusion on two facts. First, the number of equilibria in which candidates can credibly communicate their information to voters is smaller than in the case of like biases. The reason being that with opposing biases, the incentive to go for the electorate’s prior (now A) is no longer counterbalanced, but even reinforced, by the existence of a third inflexible candidate that dogmatically supports policy E. Second, the equilibria in the present case are quite fragile, in that they always embed one candidate losing the election (the one using a reversed strategy).⁴⁰

Proposition 2

$(\textit{Opposing biases})$ Consider $q\in (1-\widetilde{q},\frac{1}{2})$. A configuration $(\sigma _c^*,\sigma _v^{*x_1 x_2}; g^{x_1x_2}(\Gamma _v^{*x_1 x_2}))_{(x_1,x_2)\in \{E,A\}^2,\sigma _v^{*x_1 x_2}\in \Gamma _v^{* x_1 x_2}}$ constitutes a fully revealing equilibrium if and only if:

• (i) $\sigma _c^*=(1,0;0,1)$ and for any $(x_1,x_2)\in \{E,A\}^2, \sigma _v^{*x_1 x_2}$ is such that $g^{AA}(\Gamma _v^{*AA})=g^{AE}(\Gamma _v^{*AE})=g^{EA}(\Gamma _v^{*EA})=g^{EE}(\Gamma _v^{*EE})=\{1\};$ or

• (ii) $\sigma _c^*=(0,1;1,0)$ and for any $(x_1,x_2)\in \{E,A\}^2, \sigma _v^{*x_1 x_2}$ is such that $g^{AA}(\Gamma _v^{*AA})=g^{AE}(\Gamma _v^{*AE})=g^{EA}(\Gamma _v^{*EA})=g^{EE}(\Gamma _v^{*EE})=\{2\}$.

Proof

In the Appendix.$\square$

The comparison between the results in Propositions 1 and 2 draws an important conclusion. It is neither the fact that voters are homogeneous, nor even the existence of an inflexible third politician, that facilitates credible transmission of information by mainstream candidates. The crucial feature is the combination of both. That is, having an inflexible third candidate that caters to the (ex-ante) most popular policy. In this sense, our results suggest that for third candidates to help indeed restore the informativeness of electoral processes, the electorate should be like biased.

5 Discussion

We now discuss how the two main assumptions in the model affect our results.

5.1 Two-candidate competition

We first relax the assumption of three candidates competing for office and analyze a scenario with, exclusively, two office-motivated candidates. Note that in this case, there is no room for coalitions, as winning office is simply a matter of getting more votes than your opponent. Given ${(x_1,x_2)\in \{E,A\}^2}$, let $\sigma _v^{x_1 x_2}=(\sigma _4^{x_1 x_2};\sigma _5^{x_1 x_2})$, with $\sigma _i^{x_1 x_2}=(\sigma _{i,1}^{x_1 x_2},\sigma _{i,2}^{x_1 x_2})$ and $i\in \{4,5\}$ be the vector of voting strategies for voters 4 and 5 in this case.

As in the main body of the paper, we posit two scenarios: like biases and opposing biases.⁴¹ Interestingly, our results now show that there is no information transmission in the case of like biases, but that there is in the case of opposing biases. Note this is in contrast to the results in the previous section, where the presence of an electorate with a like bias facilitated equilibria in which the two mainstream candidates behaved honestly. This means that the result in the present set-up follows much in line with the stark intuition that heterogeneity of views within the voters reduces the incentives to go for the electorate’s prior. In fact, this reason explains the next result.

Proposition 3

$(\textit{Two-candidate election})$ Suppose $q\ne 1/2$. A configuration $(\sigma _c^*,\sigma _v^{*x_1x_2})_{(x_1,x_2)\in \{E,A\}^2}$ constitutes a fully revealing equilibrium if and only if $q<1/2, \beta =1/2$, $\sigma _v^{*AE}=(1,0;0,1)$, $\sigma _v^{*EA}=(0,1;1,0)$, $\sigma _v^{*AA}=\sigma _v^{*EE}=(\frac{1}{2},\frac{1}{2};\frac{1}{2},\frac{1}{2})$ and either $\sigma _c^*=(1,0;1,0)$ or $\sigma _c^*=(0,1;0,1)$.

Proof

In the Appendix.$\square$

Proposition 3 shows that with two candidates competing for office, the electoral process can be informative, but only if voters have opposite views. Otherwise, the incentive to go for the popular belief prevents any information flow (as in HL). Additionally, it also requires that ideological and non-ideological voters are equal in size, and that when the two candidates choose the same platform, they share votes equally. Under these conditions, our result shows that introducing heterogeneity between voters restores the possibility of an informative equilibrium in a two-candidate context, thus breaking with HL’s result.

5.2 One type of voter

Let us now consider that voters are only of one type: non-ideological. That is, there is an unanimous consensus in the electorate that the best policy is that fitting the state of the world.

To be consistent with the main body of the paper, we consider that the third candidate is biased in favor of a policy of fiscal stimulus. Additionally, we assume $q>1/2$, i.e., with no other information, voters think that expansionary fiscal policies are the most appropriate ones to return economic growth.⁴²

Proceeding as in the main body of the paper, we can analyze the voters’ optimal voting strategy. Note that in this case the analysis is simpler, as non-ideological voters no longer have to worry about what a second group of voters vote. We obtain the next result:

Lemma 2

Consider a platform profile $(x_1,x_2)\in \{E,A\}^2$. In the voting game with one group of voters, the optimal voting behavior of the voters, $\sigma _4^{*x_1 x_2}$, satisfies the following conditions:

• 1. Consider $(x_1,x_2)=(E,E)$. For any $\hat{q}\in (0,1), (\sigma _{4,1}^{*EE},\sigma _{4,2}^{*EE},\sigma _{4,3}^{*EE})\in [0,1]^3,$ is such that $\sum _{i=\{1,2,3\}}\sigma _{4,i}^{*EE}=1$.

• 2. Consider $(x_1,x_2)=(E,A)$.

  • (a) When $\hat{q}<1/2, (\sigma _{4,1}^{*EA},\sigma _{4,2}^{*EA},\sigma _{4,3}^{*EA}),$ is such that $\sigma _{4,2}^{*EA}>\sigma _{4,1}^{*EA}+\sigma _{4,3}^{*EA}$.

  • (b) When $\hat{q}>1/2, (\sigma _{4,1}^{*EA},\sigma _{4,2}^{*EA},\sigma _{4,3}^{*EA}),$ is such that either $\sigma _{4,2}^{*EA}>0$ and $\sigma _{4,i}^{*EA}>\sigma _{4,2}^{*EA}+\sigma _{4,j}^{*EA},$ for $(i,j)\in \{1,3\}^2, i\ne j;$ or $\sigma _{4,2}^{*EA}=0$ and $\sum _{i\in \{1,3\}}\sigma _{4,i}^{*EA}=1$.

• 3. Consider $(x_1,x_2)=(A,E)$.

  • (a) When $\hat{q}<1/2, (\sigma _{4,1}^{*AE},\sigma _{4,2}^{*AE},\sigma _{4,3}^{*AE}),$ is such that $\sigma _{4,1}^{*AE}>\sigma _{4,2}^{*AE}+\sigma _{4,3}^{*AE}$.

  • (b) When $\hat{q}>1/2, (\sigma _{4,1}^{*AE},\sigma _{4,2}^{*AE},\sigma _{4,3}^{*AE}),$ is such that either $\sigma _{4,1}^{*AE}>0$ and $\sigma _{4,i}^{*AE}>\sigma _{4,1}^{*AE}+\sigma _{4,j}^{*AE},$ for $(i,j)\in \{2,3\}^2, i\ne j;$ or $\sigma _{4,1}^{*AE}=0$ and $\sum _{i=\{2,3\}}\sigma _{4,i}^{*AE}=1$.

• 4. Consider $(x_1,x_2)=(A,A)$.

  • (a) When $\hat{q}<1/2, (\sigma _{4,1}^{*AA},\sigma _{4,2}^{*AA},\sigma _{4,3}^{*AA}),$ is such that either $\sigma _{4,3}^{*AA}>0$ and $\sigma _{4,i}^{*AA}>\sigma _{4,j}^{*AA}+\sigma _{4,3}^{*AA},$ for $(i,j)\in \{1,2\}^2, i\ne j;$ or $\sigma _{4,3}^{*AA}=0$ and $\sum _{i=\{1,2\}}\sigma _{4,i}^{*AE}=1$.

  • (b) When $\hat{q}>1/2, (\sigma _{4,1}^{*AA},\sigma _{4,2}^{*AA},\sigma _{4,3}^{*AA})$ is such that $\sigma _{4,3}^{*AA}>\sigma _{4,1}^{*AA}+\sigma _{4,2}^{*AA}$.

Analogously to the case with two groups of voters, for each platform profile $(x_1,x_2)\in \{E,A\}^2$, let $\Gamma _4^{* x_1 x_2}$ be the set of the voters’ optimal voting strategies that satisfy conditions in Lemma 1. Let $\sigma _4^{* x_1 x_2}\in \Gamma _4^{* x_1 x_2}$ be an element of this set. We next write the electoral outcomes in terms of government outcomes.

Corollary 3

Given a platform profile $(x_1,x_2)\in \{E,A\}^2$, the set of voters’ optimal voting strategies, $\Gamma _4^{*x_1 x_2}$, and the function f, in the game with one group of voters the set $G^{x_1 x_2}$ is:

• 1. Consider $(x_1,x_2)=(E,E)$. Then, $\forall \hat{q}\in (0,1), G^{EE}=\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\}\}$.

• 2. Consider $(x_1,x_2)=(E,A)$.

  • (a) Then $\forall \hat{q}<1/2, G^{EA}=\{\{2\}\}$,

  • (b) Then $\forall \hat{q}>1/2, G^{EA}=\{\{1\},\{3\},\{1,3\}\}$.

• 3. Consider $(x_1,x_2)=(A,E)$.

  • (a) Then $\forall \hat{q}<1/2, G^{AE}=\{\{1\}\}$,

  • (b) Then $\forall \hat{q}>1/2, G^{AE}=\{\{2\},\{3\},\{2,3\}\}$.

• 4. Consider $(x_1,x_2)=(A,A)$.

  • (a) Then $\forall \hat{q}<1/2, G^{AA}=\{\{1\},\{2\},\{1,2\}\}$,

  • (b) Then $\forall \hat{q}>1/2, G^{AA}=\{\{3\}\}$.

A comparison between the scenario with two groups of voters (Lemma 1 and Corollary 1) and the present case (Lemma 2 and Corollary 2), shows that because in the current situation the non-ideological voters no longer have to worry about the ideological ones, those cases in which the existence of the latter class of agents obliged the former to carefully cast their vote, are now altered. We observe that this is the case when $\hat{q}<1/2$. Otherwise, all voters had a preference for the expansionary fiscal policy, so whether there is one or two groups of voters is irrelevant for the results. Additionally, we observe that for the number of groups of voters to make a difference, we need that more than one candidate proposes the austerity policy. Otherwise, it is clear which candidate should non-ideological voters vote for. This means that the difference is to be found in the case $(x_1,x_2)=(A,A)$ and $\hat{q}<1/2$. Here, we obtain that a coalition between the two mainstream candidates can now form. The reason is that with no votes for the third candidate, any voters’ voting strategy such that $\sigma _{4,2}^{AA}=0$ is a best response. This is in contrast to the scenario with two groups of voters, where the coalition between the two mainstream candidates was not possible.

Taking into account this new electoral outcome, we go backwards and analyze the information transmission game.

Proposition 4

$(\textit{One type of voter})$ A configuration $(\sigma _c^*,\sigma _4^{* x_1 x_2};$ $g^{x_1x_2} (\hat{\Gamma }_4^{* x_1 x_2}))_{(x_1,x_2)\in \{E,A\}^2,\sigma _4^{* x_1 x_2}\in \hat{\Gamma }_4^{* x_1 x_2}}$ constitutes a fully revealing equilibrium if and only if it satisfies one of the conditions (i)–(iv) of Proposition 1, or condition

• (v) $\sigma _c^*=(1,0;1,0)$ and for any $(x_1,x_2)\in \{E,A\}^2, \sigma _v^{* x_1 x_2}\in \hat{\Gamma }_4^{* x_1 x_2}$ is such that either:

  • (a) $g^{EA}(\hat{\Gamma }_4^{*EA})=\{1,3\},g^{AE}(\hat{\Gamma }_4^{*AE})=\{2,3\}, g^{AA}(\hat{\Gamma }_4^{*AA})=\{1,2\},g^{EE}(\hat{\Gamma }_4^{*EE})=\{3\};$ or

  • (b) $g^{EA}(\hat{\Gamma }_4^{*EA})=g^{AE}(\hat{\Gamma }_4^{*AE})=\{3\}, g^{AA}(\hat{\Gamma }_4^{*AA})=\{1,2\}, g^{EE}(\hat{\Gamma }_4^{*EE})=\{1,2\}$.

Proof

In the Appendix.$\square$

We obtain that a scenario with just one group of voters facilitates information transmission by the two mainstream candidates. The reason is that in addition to the equilibria characterized in Proposition 1, we now have two new government configurations that sustain honest behavior. Interestingly, we obtain that in these new government configurations, mainstream candidates are better off (compared to the benchmark scenario), while the third inflexible candidate is worst off. In fact, any disagreement between the two mainstream candidates does no longer result in these candidates being always excluded from office. We also observe that under this new scenario, if the mainstream candidates happen to announce the same policy, they can now form a coalition government.

6 Conclusion

It is little wonder that the past 2014 European Elections will be remembered as the rejection election.⁴³ Indeed, poll results all over Europe showed the end of the seemingly settled two-party political system of the mid-20th century. From Britain to Denmark, France, Italy or Spain, “the center-left and center-right groups that form the core of national and European Union politics have seen their power eroded by the rise of extremist parties very different from one another, united only by their rejection of the way things are, both at home and at the European Union” (“As goes Greece, so goes Europe?”, The New York Times, 28th May 2014).

The rise of third parties presents an unprecedented challenge to mainstream political parties and to society as a whole. What consequences will this threat have for the economic model that will govern us in the near future is something that we will learn in due course. Working within a small area of this topic, this paper presents results that move towards showing the consequences that this threat can have on the resulting electoral outcomes and the informativeness of the electoral processes.

In this work we argue that the presence of a third inflexible candidate might help ease the information transmission game, specially, when all voters are biased in the same direction and so there are no conflicting views in the electorate. This result, which a priori may look a bit contradicting, is robust. We observe it when in a posterior stage we consider the existence of only one group of voters, and obtain that the number of government outcomes that sustain information transmission is greater here than in the previous case. Another interesting result is that for the mainstream candidates to honestly reveal their information, we need voters to have a strict preference for one of these two candidates. That is, that whenever mainstream candidates propose the same policy, voters always vote for the same candidate. Otherwise, that is if under indifference they flip a coin, no information transmission can be achieved. In this sense, voters’ captivity seems to be good.

Appendix

Proof of Lemma 1:

Proof

In the analysis that follows, we consider that ideological voters never vote for a candidate proposing policy A. Next, we differentiate four cases:

• (i) Consider $(x_1,x_2)=(EE)$. In this case only expansionary fiscal policies are being proposed, therefore the final policy will always be $x=E$. This means that any voting strategy is optimal for voters. That is, the optimal strategies of the non-ideological and ideological voters satisfy, respectively, $(\sigma _{4,1}^{*EE},\sigma _{4,2}^{*EE},\sigma _{4,3}^{*EE})\in [0,1]^3$, such that $\sum _{i=\{1,2,3\}}\sigma _{4,i}^{*EE}=1$; and $(\sigma _{5,1}^{*EE},\sigma _{5,2}^{*EE},\sigma _{5,3}^{*EE})\in [0,1]^3$, such that $\sum _{i=\{1,2,3\}}\sigma _{5,i}^{*EE}=1$.

• (ii) Consider $(x_1,x_2)=(EA)$. In this case $(\sigma _{5,1}^{*EA},\sigma _{5,2}^{*EA},\sigma _{5,3}^{*EA})$ is such that $\sigma _{5,2}^{*EA}=0$, with $\sum _{i\in \{1,3\}}\sigma _{5,i}^{*EA}=1$. First, consider $\hat{q}<1/2$. In this case, if non-ideological voters were to vote for candidate 2 with probability 1, $\sigma _{4,2}^{EA}=1$, the electoral outcome would be $(v_1,v_2,v_3)=((1-\beta )\sigma _{5,1}^{*EA},\beta ,(1-\beta )\sigma _{5,3}^{*EA})$. Since $\beta >1/2, g=\{2\}$, in which case the payoff to each non-ideological voter would be $1-\hat{q}$. Now, if all non-ideological voters where to vote in such a way that candidate 2 did not get a strict majority of votes, their individual payoff would be smaller, as the policy implemented by the resulting government would be E with positive probability. Then, the optimal strategy of non-ideological voters, $(\sigma _{4,1}^{*EA},\sigma _{4,2}^{*EA},\sigma _{4,3}^{*EA})$, satisfies $\beta \sigma _{4,2}^{*EA}>\beta (\sigma _{4,1}^{*EA}+\sigma _{4,3}^{*EA})+(1-\beta )$. Now, consider $\hat{q}>1/2$. Here, if $G\subseteq \{\{1\},\{3\},\{1,3\}\}$, the payoff to each non-ideological voter would be $\hat{q}$, whereas it would be smaller otherwise.⁴⁴ Hence, non-ideological voters must prevent candidate 2 from entering the government. This means that their voting strategy must satisfy that if neither candidate 1 nor 3 gets an absolute majority of votes, candidate 2 cannot form a government. Mathematically, $v_2+v_i<1/2$, for $i\in \{1,3\}$. But this is a contradiction, as whenever $\max \{v_1,v_2,v_3\}<1/2$, any pair of two candidates gets an absolute majority of votes. Thus, the optimal strategy of non-ideological voters, $(\sigma _{4,1}^{*EA},\sigma _{4,2}^{*EA},\sigma _{4,3}^{*EA})$, is such that either $\sigma _{4,2}^{*EA}>0$ and $\beta \sigma _{4,i}^{*EA}+(1-\beta )\sigma _{5,i}^{*EA}>\beta (\sigma _{4,2}^{*EA}+\sigma _{4,j}^{*EA})+(1-\beta )\sigma _{5,j}^{*EA}$, for $(i,j)\in \{1,3\}^2, i\ne j;$ or $\sigma _{4,2}^{*EA}=0$ and $\sum _{i\in \{1,3\}}\sigma _{4,i}^{*EA}=1.$

• (iii) Consider $(x_1,x_2)=(AE)$. In this case $(\sigma _{5,1}^{*AE},\sigma _{5,2}^{*AE},\sigma _{5,3}^{*AE})$, such that $\sigma _{5,1}^{*AE}=0$, with $\sum _{i\in \{2,3\}}\sigma _{5,i}^{*AE}=1$. Now, analogously to the previous case, we obtain: When $\hat{q}<1/2, (\sigma _{4,1}^{*AE},\sigma _{4,2}^{*AE},\sigma _{4,3}^{*AE})$, such that $\beta \sigma _{4,1}^{*AE}>\beta (\sigma _{4,2}^{*AE}+\sigma _{4,3}^{*AE})+(1-\beta )$. When $\hat{q}>1/2, (\sigma _{4,1}^{*AE},\sigma _{4,2}^{*AE},\sigma _{4,3}^{*AE})$, such that either $\sigma _{4,1}^{*AE}>0$ and $\beta \sigma _{4,i}^{*AE}+(1-\beta )\sigma _{5,i}^{*AE}>\beta (\sigma _{4,1}^{*AE}+\sigma _{4,j}^{*AE})+(1-\beta )\sigma _{5,j}^{*AE}$, for $(i,j)\in \{2,3\}^2, i\ne j;$ or $\sigma _{4,1}^{*AE}=0$ and $\sum _{i\in \{2,3\}}\sigma _{4,i}^{*AE}=1.$

• (iv) Last, consider $(x_1,x_2)=(AA)$. In this case $(\sigma _{5,1}^{*AA},\sigma _{5,2}^{*AA},\sigma _{5,3}^{*AA})=(0,0,1)$. First, consider $\hat{q}<1/2$. In this case, if non-ideological voters were to vote for one of the two mainstream candidates (let us say candidate 1) with probability $1, \sigma _{4,1}^{AA}=1$, the electoral outcome would be $(v_1,v_2,v_3)=(\beta ,0,1-\beta )$, the government outcome $g=\{1\}$, and the payoff to each of these voters $1-\hat{q}$. In contrast, if they all voted in such a way that candidate 1 (nor 2) no longer obtained a majority of votes, their individual expected utility would be smaller, as policy E would be implemented with positive probability. Then, the optimal strategy of these voters, $(\sigma _{4,1}^{*AA},\sigma _{4,2}^{*AA},\sigma _{4,3}^{*AA})$, satisfies that either one of the two mainstream candidates receive a strict majority of votes. This is, $\beta \sigma _{4,i}^{*AA}>\beta \sigma _{4,j}^{*AA}+(1-\beta )$, for $(i,j)\in \{1,2\}^2, i\ne j$. Last, consider $\hat{q}>1/2$. In this case, if non-ideological voters were to vote for candidate 3 with probability $1, \sigma _{4,3}^{AA}=1$, the electoral outcome would be $(v_1,v_2,v_3)=(0,0,1)$, the government outcome $g=\{3\}$, and the payoff to each of these voters $\hat{q}$. In contrast, if they all voted in such a way that the third inflexible candidate no longer obtained a majority of votes and one of the two mainstream candidates were to enter the government, their individual expected utility would be smaller, as policy A would be implemented with positive probability. Then, the optimal strategy of these voters, $(\sigma _{4,1}^{*AA},\sigma _{4,2}^{*AA},\sigma _{4,3}^{*AA})$, satisfies $\beta \sigma _{4,3}^{*AA}+(1-\beta )>\beta (\sigma _{4,1}^{*AA}+\sigma _{4,2}^{*AA})$. This completes the proof.$\square$

Proof of Proposition 1:

Proof

We have to analyze four cases:

• (i) Suppose $\sigma _c=(1,0;1,0)$ is part of an equilibrium. In equilibrium, choosing E when having observed e must be a best response for candidates 1 (Eq. 3) and 2 (Eq. 4), respectively:

  $$\begin{aligned}&P_{e \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. + P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad \ge P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{1\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{2\} \end{array} \right. \end{aligned}$$

  (3)
  $$\begin{aligned}&P_{e \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. + P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AE}=\{2,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AE}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad \ge P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{1\} \end{array} \right. \end{aligned}$$

  (4)

  Analogously, choosing A when having observed a must be a best response for candidates 1 (Eq. 5) and 2 (Eq. 6), respectively:

  $$\begin{aligned}&P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{1\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{2\} \end{array} \right. \ge P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad + P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \end{aligned}$$

  (5)
  $$\begin{aligned}&P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{1\} \end{array} \right. \ge P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AE}=\{2,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AE}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad + P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. \end{aligned}$$

  (6)

  Now, let us first suppose $g^{EA}=\{1\}$. Then, we have $g^{AE}=\{2\}$. In this case, a necessary condition for (5) to hold is $g^{AA}=\{1\}$, but then (6) cannot hold. Similarly, let us suppose $g^{EA}=\{1,3\}$. Then, we have $g^{AE}=\{2,3\}$. Again, $g^{AA}=\{1\}$ is necessary for (5) to hold, but then (6) cannot hold. Hence, this is not an equilibrium. Last, let us suppose $g^{EA}=\{3\}$. Then, we have $g^{AE}=\{3\}$. Here we have two cases: (a) Consider $g^{AA}=\{1\}$. A necessary condition for (6) to hold is $g^{EE}\in \{\{1\},\{3\},\{1,3\}\}$, and a necessary condition for (3) to hold is either $g^{EE}\in \{\{1,2\},\{1,3\}\}$ and $P_{e \mid e} \ge 2P_{a \mid e}$, or $g^{EE}=\{1\}$. Note that if $g^{EE}\in \{\{1\},\{1,3\}\}$, conditions (4) and (5) hold. Hence, if either $g^{EE}=\{13\}$ and $P_{e \mid e} \ge 2P_{a \mid e}$, or $g^{EE}=\{1\}$, there is an equilibrium in this case. (b) Analogously to the previous case, there is an equilibrium in which $g^{AA}=\{2\}$ and either $g^{EE}=\{2,3\}$ and $P_{e \mid e} \ge 2P_{a \mid e}$, or $g^{EE}=\{2\}$.

• (ii) Suppose $\sigma _c=(1,0;0,1)$ is part of an equilibrium. In equilibrium, choosing E when having observed e must be a best response for candidate 1 (Eq. 7), and choosing A when having observed e must be a best response for candidate 2 (Eq. 8):

  $$\begin{aligned}&P_{e \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \quad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \quad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. + P_{a \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \nonumber \\&\quad \ge P_{a \mid e} K. \end{aligned}$$

  (7)
  $$\begin{aligned} 0 \ge P_{e \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. \end{aligned}$$

  (8)

  Analogously, choosing A when having observed a must be a best response for candidate 1 (Eq. 9), and choosing E when having observed a must be a best response for candidate 2 (Eq. 10):

  $$\begin{aligned} P_{a \mid a} K\ge & {} P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \nonumber \\&+ P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. \end{aligned}$$

  (9)
  $$\begin{aligned} P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{2,3\}\} \end{array} \right. \ge 0 \end{aligned}$$

  (10)

  First, note that condition (10) always hold, and that a necessary condition for (8) to hold is $g^{EE}\in \{\{1\},\{3\},\{1,3\}\}$. Then, let us first consider $g^{EE}=\{1\}$. In this case, a necessary condition for (9) to hold is $g^{EA}=\{3\}$, in which case (7) holds. Hence, there is an equilibrium in this case. Second, let us consider $g^{EE}=\{3\}$. A necessary condition for (7) to hold is either $g^{EA}=\{1,3\}$ and $P_{e \mid e} >2P_{a \mid e}$, or $g^{EA}=\{1\}$. Note that in both cases condition (9) holds. Hence, there is an equilibrium in this case. Last, let us consider $g^{EE}=\{1,3\}$. Again, a necessary condition for (7) to hold is either $g^{EA}=\{1,3\}$ or $g^{EA}=\{1\}$. (a) If $g^{EA}=\{1\}$, a necessary condition for (9) to hold is $P_{a \mid a} >2P_{e \mid a}$. Hence, there is an equilibrium in this case. (b) If $g^{EA}=\{1,3\}$, (9) always hold. Hence, there is an equilibrium in this case.

• (iii) The case $\sigma _c=(0,1;1,0)$ is analogous to the previous one.

• (iv) Last, suppose $\sigma _c=(0,1;0,1)$ is part of an equilibrium. In equilibrium, choosing A when having observed e must be a best response for candidates 1 (Eq. 11) and 2 (Eq. 12), respectively:

  $$\begin{aligned} 0\ge & {} P_{e \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. \nonumber \\&+ P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \end{aligned}$$

  (11)
  $$\begin{aligned} 0 \ge P_{e \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{AE}=\{2\}\\ K/2 &{} \quad \mathrm{if}\quad g^{AE}=\{2,3\}\\ 0 &{} \quad \mathrm{if}\quad g^{AE}=\{3\} \end{array} \right. + P_{a \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. \nonumber \\ \end{aligned}$$

  (12)

  For (11) and (12) to hold, a necessary condition is $g^{EA}=g^{AE}=g^{EE}=\{3\}$. Additionally, note that if $(x_1,x_2)=(AA)$, the non-ideological voter thinks policy E is the best one, then $g^{AA}=\{3\}$. Hence, candidates 1 and 2 never get votes, and so this is an equilibrium. This completes the proof.$\square$

Proof of Proposition 2:

Proof

We have to analyze four cases:

• (i) Suppose $\sigma _c=(1,0;1,0)$ is part of an equilibrium. In equilibrium, choosing E when having observed e must be a best response for candidates 1 (Eq. 13) and 2 (Eq. 14), respectively:

  $$\begin{aligned} P_{e \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \ge P_{e \mid e} K+ P_{a \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{AA}=\{1\}\\ 0 &{} \quad \mathrm{if}\quad g^{AA}=\{2\} \end{array} \right. \nonumber \\ \end{aligned}$$

  (13)
  $$\begin{aligned} P_{e \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. \ge P_{e \mid e} K+ P_{a \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{AA}=\{2\}\\ 0 &{} \quad \mathrm{if}\quad g^{AA}=\{1\} \end{array} \right. \nonumber \\ \end{aligned}$$

  (14)

  A necessary condition for (13) to hold is $g^{EE}=\{1\}$. But then (14) cannot hold. Hence, this is not an equilibrium.

• (ii) Suppose $\sigma _c=(1,0;0,1)$ is part of an equilibrium. In equilibrium, choosing E when having observed e must be a best response for candidate 1 (Eq. 15), and choosing A when having observed e must be a best response for candidate 2 (Eq. 16):

  $$\begin{aligned}&P_{e \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \quad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \quad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. + P_{a \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \quad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \nonumber \\&\quad \qquad \ge P_{e \mid e} \left\{ \begin{array}{ll} K &{} \quad \mathrm{if}\quad g^{AA}=\{1\}\\ 0 &{} \quad \mathrm{if}\quad g^{AA}=\{2\} \end{array} \right. + P_{a \mid e} K \end{aligned}$$

  (15)
  $$\begin{aligned} P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{1\} \end{array} \right. \ge P_{e \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. \nonumber \\ \end{aligned}$$

  (16)

  Analogously, choosing A when having observed a must be a best response for candidate 1 (Eq. 17), and choosing E when having observed a must be a best response for candidate 2 (Eq. 18):

  $$\begin{aligned}&P_{a \mid a} K + P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{1\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{2\} \end{array} \right. \nonumber \\&\qquad \ge P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \nonumber \\&\qquad \quad +P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. \end{aligned}$$

  (17)
  $$\begin{aligned} P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{2,3\}\} \end{array} \right. \ge P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{1\} \end{array} \right. \nonumber \\ \end{aligned}$$

  (18)

  First, note that if $g^{AA}=\{2\}$, condition (18) cannot hold. Hence, this is not an equilibrium. Now, consider $g^{AA}=\{1\}$. A necessary condition for (16) to hold is $g^{EE}\in \{\{1\},\{3\},\{1,3\}\}$, and the necessary conditions for (15) to hold are $g^{EA}=g^{EE}=\{1\}$. Note that in this case, both (17) and (18) hold. Hence, there is an equilibrium in this case.

• (iii) The case $\sigma _c=(0,1;1,0)$ is analogous to the previous one.

• (iv) Last, suppose $\sigma _c=(0,1;0,1)$ is part of an equilibrium. In equilibrium, choosing E when having observed a must be a best response for candidates 1 (Eq. 19) and 2 (Eq. 20), respectively:

  $$\begin{aligned} P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \ge P_{a \mid a} K \end{aligned}$$

  (19)
  $$\begin{aligned} P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. \ge P_{a \mid a} K \end{aligned}$$

  (20)

  For (19) to hold, a necessary condition is $g^{EE}=\{1\}$. But then (20) cannot hold. Hence, this is not an equilibrium. This completes the proof.$\square$

Proof of Proposition 3:

Proof

We prove it in two steps: like biases and opposing biases.

• 1. Let us first consider the case of like biases, i.e. $q>\frac{1}{2}$. We have to analyze four cases:

  • 1.i) Suppose $\sigma _c=(1,0;1,0)$ is part of an equilibrium. Here, choosing A when having observed a must be a best response for candidates 1 and 2:

    $$\begin{aligned} P_{a \mid a} [\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}]\ge P_{a \mid a} +P_{e \mid a} [\beta \sigma _{4,1}^{EE}+(1-\beta )\sigma _{5,1}^{EE}]\\ P_{a \mid a} [\beta \sigma _{4,2}^{AA}+(1-\beta )\sigma _{5,2}^{AA}]\ge P_{a \mid a} +P_{e \mid a} [\beta \sigma _{4,2}^{EE}+(1-\beta )\sigma _{5,2}^{EE}] \end{aligned}$$

    Adding inequalities we obtain $0 \ge P_{a \mid a} +P_{e \mid a} =1$, which is impossible. Hence, there is no equilibrium in this case.

  • 1.ii) Suppose $\sigma _c=(1,0;0,1)$ is part of an equilibrium. Here, choosing A when having observed a must be a best response for candidate 1:

    $$\begin{aligned} P_{a \mid a} \beta +P_{e \mid a} [\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}]\ge P_{a \mid a} [\beta \sigma _{4,1}^{EE}+(1-\beta )\sigma _{5,1}^{EE}]+P_{e \mid a} \end{aligned}$$

    Analogously, choosing A when having observed e must be a best response for candidate 2:

    $$\begin{aligned} P_{a \mid e} [\beta \sigma _{4,2}^{AA}+(1-\beta )\sigma _{5,2}^{AA}]\ge P_{e \mid e} [\beta \sigma _{4,2}^{EE}+(1-\beta )\sigma _{5,2}^{EE}]+P_{a \mid e} (1-\beta ) \end{aligned}$$

    Since, for $i,j\in \{E,A\}$ and $k\in \{4,5\}, P_{i \mid j} =P_{j \mid i}$ and $\sigma _{k,1}^{ii}+\sigma _{k,2}^{ii}=1$, adding inequalities and rearranging, we obtain $0\ge P_{a \mid a} (1-\beta )+P_{e \mid a} (1-\beta )$. Since for $i,j\in \{E,A\}, P_{j \mid i} +P_{i \mid i} =1$, there is no equilibrium in this case.

  • 1.iii) The case $\sigma _c=(0,1;1,0)$ is analogous to the previous one.

  • 1.iv) Last, suppose $\sigma _c=(0,1;0,1)$ is part of an equilibrium. Here, choosing A when having observed e must be a best response for any of the two mainstream candidates, let us say, candidate 1. That is to say:

    $$\begin{aligned} P_{e \mid e} [\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}]\ge P_{e \mid e} +P_{a \mid e} [\beta \sigma _{4,1}^{EE}+(1-\beta )\sigma _{5,1}^{EE}] \end{aligned}$$

    Since $\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}\le 1$, there is no equilibrium in this case.

• 2. Let us now consider the case of opposing biases, i.e., $q<\frac{1}{2}$. Again, there are four cases to analyze:

  • 2.i) Suppose $\sigma _c=(1,0;1,0)$ is part of an equilibrium. Here, choosing E when having observed e must be a best response for candidates 1 and 2:

    $$\begin{aligned}&P_{e \mid e} [\beta \sigma _{4,1}^{EE}+(1-\beta )\sigma _{5,1}^{EE}]+P_{a \mid e} (1-\beta ) \ge P_{e \mid e} \beta \nonumber \\&\quad +P_{a \mid e} [\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}] \end{aligned}$$

    (21)
    $$\begin{aligned}&P_{e \mid e} [\beta \sigma _{4,2}^{EE}+(1-\beta )\sigma _{5,2}^{EE}]\nonumber \\&\quad +P_{a \mid e} (1-\beta ) \ge P_{e \mid e} \beta +P_{a \mid e} [\beta \sigma _{4,2}^{AA}+(1-\beta )\sigma _{5,2}^{AA}] \end{aligned}$$

    (22)

    Adding inequalities we obtain $P_{e \mid e} (1-2\beta )+P_{a \mid e} (1-2\beta )\ge 0 \Leftrightarrow \beta \le 1/2$. Similarly, in equilibrium, choosing A when having observed a must be a best response for candidates 1 and 2:

    $$\begin{aligned}&P_{a \mid a} [\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}]+P_{e \mid a} \beta \ge P_{a \mid a} (1-\beta )\nonumber \\&\quad +P_{e \mid a} [\beta \sigma _{4,1}^{EE}+(1-\beta )\sigma _{5,1}^{EE}] \end{aligned}$$

    (23)
    $$\begin{aligned}&P_{a \mid a} [\beta \sigma _{4,2}^{AA}+(1-\beta )\sigma _{5,2}^{AA}]+P_{e \mid a} \beta \ge P_{a \mid a} (1-\beta )\nonumber \\&\quad +P_{e \mid a} [\beta \sigma _{4,2}^{EE}+(1-\beta )\sigma _{5,2}^{EE}] \end{aligned}$$

    (24)

    Adding inequalities we obtain $0\ge P_{a \mid a} (1-2\beta )+P_{e \mid a} (1-2\beta ) \Leftrightarrow \beta \ge 1/2$. Hence, both requirements can only meet if $\beta =1/2$. Now, if $\beta =1/2$, and taking into account that for $k\in \{4,5\}$ and $i\in \{E,A\}, \sigma _{k,1}^{ii}+\sigma _{k,2}^{ii}=1$, inequalities (21)–(24) can be rewritten as:

    $$\begin{aligned} P_{a \mid e} (1-\sigma _{4,1}^{AA}-\sigma _{5,1}^{AA})=P_{e \mid e} (1-\sigma _{4,1}^{EE}-\sigma _{5,1}^{EE})\\ P_{a \mid a} (1-\sigma _{4,1}^{AA}-\sigma _{5,1}^{AA})=P_{e \mid a} (1-\sigma _{4,1}^{EE}-\sigma _{5,1}^{EE}) \end{aligned}$$

    Now, since $P_{i \mid i} >P_{j \mid i}$, there is a unique solution for the system above, which is $\sigma _{4,1}^{AA}=\sigma _{5,1}^{AA}=\sigma _{4,1}^{EE}=\sigma _{5,1}^{EE}=1/2$. Hence, in this case, we have an equilibrium.

  • 2.ii) Suppose $\sigma _c=(1,0;0,1)$ is part of an equilibrium. In this case, the payoffs coincide with those in case 1.ii). Hence, this is not an equilibrium.

  • 2.iii) Analogously, there is neither an equilibrium in the case $\sigma _c=(0,1;1,0)$.

  • 2.iv) Last, suppose $\sigma _c=(0,1;0,1)$ is part of an equilibrium. Here, choosing A when having observed e must be a best response for candidates 1 and 2. That is to say:

    $$\begin{aligned}&P_{e \mid e} [\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}]+P_{a \mid e} \beta \ge P_{e \mid e} (1-\beta )\nonumber \\&\quad +P_{a \mid e} [\beta \sigma _{4,1}^{EE}+(1-\beta )\sigma _{5,1}^{EE}] \end{aligned}$$

    (25)
    $$\begin{aligned}&P_{e \mid e} [\beta \sigma _{4,2}^{AA}+(1-\beta )\sigma _{5,2}^{AA}]+P_{a \mid e} \beta \ge P_{e \mid e} (1-\beta )\nonumber \\&\quad +P_{a \mid e} [\beta \sigma _{4,2}^{EE}+(1-\beta )\sigma _{5,2}^{EE}] \end{aligned}$$

    (26)

    Adding inequalities we obtain $0\ge P_{e \mid e} (1-2\beta )+P_{a \mid e} (1-2\beta ) \Leftrightarrow \beta \ge 1/2$. Similarly, in equilibrium, choosing E when having observed a must be a best response for candidates 1 and 2. That is to say:

    $$\begin{aligned}&P_{a \mid a} [\beta \sigma _{4,1}^{EE}+(1-\beta )\sigma _{5,1}^{EE}]+P_{e \mid a} (1-\beta ) \ge P_{a \mid a} \beta \nonumber \\&\quad +P_{e \mid a} [\beta \sigma _{4,1}^{AA}+(1-\beta )\sigma _{5,1}^{AA}] \end{aligned}$$

    (27)
    $$\begin{aligned}&P_{a \mid a} [\beta \sigma _{4,2}^{EE}+(1-\beta )\sigma _{5,2}^{EE}]+P_{e \mid a} (1-\beta ) \ge P_{a \mid a} \beta \nonumber \\&\quad +P_{e \mid a} [\beta \sigma _{4,2}^{AA}+(1-\beta )\sigma _{5,2}^{AA}] \end{aligned}$$

    (28)

    Adding inequalities we obtain $P_{a \mid a} (1-2\beta )+P_{e \mid a} (1-2\beta )\ge 0 \Leftrightarrow \beta \le 1/2$. Hence, both requirements can only meet if $\beta =1/2$. Now, if $\beta =1/2$, and taking into account that for $k\in \{4,5\}$ and $i\in \{E,A\}, \sigma _{k,1}^{ii}+\sigma _{k,2}^{ii}=1$, inequalities (25)–(28) can be rewritten as:

    $$\begin{aligned} P_{a \mid e} (1-\sigma _{4,1}^{EE}-\sigma _{5,1}^{EE})=P_{e \mid e} (1-\sigma _{4,1}^{AA}-\sigma _{5,1}^{AA})\\ P_{a \mid a} (1-\sigma _{4,1}^{EE}-\sigma _{5,1}^{EE})=P_{e \mid a} (1-\sigma _{4,1}^{AA}-\sigma _{5,1}^{AA}) \end{aligned}$$

    Now, since $P_{i \mid i} >P_{j \mid i}$, there is a unique solution for the system above, which is $\sigma _{4,1}^{AA}=\sigma _{4,1}^{EE}=\sigma _{5,1}^{AA}=\sigma _{5,1}^{EE}=1/2$. Hence, in this case, there is an equilibrium of the type conjectured.$\square$

Proof of Proposition 4:

Proof

First, suppose an equilibrium in which either $\sigma _c=(1,0;0,1), \sigma _c=(1,0;0,1)$ or $\sigma _c=(0,1;1,0)$. In these cases, note that when the platforms profile observed is $(x_1,x_2)=(AA)$, applying Bayes’ rule we obtain $\hat{q}>1/2$. Hence, $g^{AA}=\{3\}$, as it was the case in the analogous situations of Proposition 1. The analysis of these cases are therefore identical to those in Proposition 1, thus omitted.

Let us now suppose $\sigma _c=(1,0;1,0)$ is part of an equilibrium.

In equilibrium, choosing E when having observed e must be a best response for candidates 1 (Eq. 29) and 2 (Eq. 30), respectively:

$$\begin{aligned}&P_{e \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. + P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad \ge P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AA}=\{1,2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{2\} \end{array} \right. \end{aligned}$$

(29)

$$\begin{aligned}&P_{e \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. + P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AE}=\{2,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AE}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad \ge P_{a \mid e} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AA}=\{1,2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{1\} \end{array} \right. \end{aligned}$$

(30)

Analogously, choosing A when having observed a must be a best response for candidates 1 (Eq. 31) and 2 (Eq. 32), respectively:

$$\begin{aligned}&P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AA}=\{1,2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{2\} \end{array} \right. \ge P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EA}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EA}=\{1,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EA}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad +P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{1\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{1,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{2\},\{3\},\{2,3\}\} \end{array} \right. \end{aligned}$$

(31)

$$\begin{aligned}&P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AA}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AA}=\{1,2\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AA}=\{1\} \end{array} \right. \ge P_{a \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{AE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{AE}=\{2,3\}\\ 0 &{} \qquad \mathrm{if}\quad g^{AE}=\{3\} \end{array} \right. \nonumber \\&\qquad \qquad +P_{e \mid a} \left\{ \begin{array}{ll} K &{} \qquad \mathrm{if}\quad g^{EE}=\{2\}\\ K/2 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1,2\},\{2,3\}\}\\ 0 &{} \qquad \mathrm{if}\quad g^{EE}\in \{\{1\},\{3\},\{1,3\}\} \end{array} \right. \end{aligned}$$

(32)

Now, let us first suppose $g^{EA}=\{1\}$. Then, we have $g^{AE}=\{2\}$. In this case, a necessary condition for (31) to hold is $G^{AAF}=\{1\}$, but then (32) cannot hold.

Similarly, let us suppose $g^{EA}=\{1,3\}$. Then, we have $g^{AE}=\{2,3\}$. Now, $g^{AA}\in \{\{1\},\{1,2\}\}$ is necessary for (31) to hold. If $g^{AA}=\{1\}$ we are in the previous situation, therefore no equilibrium exists. Let us consider $g^{AA}=\{1,2\}$. Then, a necessary condition for (31) to hold is $g^{EE}\in \{\{2\},\{3\},\{2,3\}\}$, and a necessary condition for (32) to hold is $g^{EE}\in \{\{1\},\{3\},\{1,3\}\}$. Note that if $g^{EE}=\{3\}$, conditions (29) and (30) hold. Hence, there is an equilibrium in this case.

Last, let us suppose $g^{EA}=\{3\}$. Then, we have $g^{AE}=\{3\}$. Here we have three cases: (a) consider $g^{AA}=\{1\}$. A necessary condition for (32) to hold is $g^{EE}\in \{\{1\},\{3\},\{1,3\}\}$, and a necessary condition for (29) to hold is either $g^{EE}\in \{\{1,2\},\{1,3\}\}$ and $P_{e \mid e} \ge 2P_{a \mid e}$, or $g^{EE}=\{1\}$. Note that if $g^{EE}\in \{\{1\},\{1,3\}\}$, conditions (30) and (31) hold. Hence, if either $g^{EE}=\{1,3\}$ and $P_{e \mid e} \ge 2P_{a \mid e}$, or $g^{EE}=\{1\}$, there is an equilibrium of the type conjectured. (b) Analogously to the previous case, there is an equilibrium in which $g^{AA}=\{2\}$ and either $g^{EE}=\{2,3\}$ and $P_{e \mid e} \ge 2P_{a \mid e}$, or $g^{EE}=\{2\}$. (c) Last, consider $g^{AA}=\{1,2\}$. A necessary condition for (29) to hold is $g^{EE}\in \{\{1\},\{1,2\},\{1,3\}\}$, and a necessary condition for (30) to hold $g^{EE}\in \{\{2\},\{1,2\},\{2,3\}\}$. Note that if $g^{EE}=\{1,2\}$, conditions (31) and (32) hold. Hence, there is an equilibrium in this case. This completes the proof. $\square$