A Leader’s Judgment

The equilibrium strategies (m _{− j}, y_j) given any chosen leader j are easily derived from the analysis leading to Corollary 1 in Galeotti, Ghiglino, and Squintani (Reference Galeotti, Ghiglino and Squintani2013). Given the received messages $\mathbf {\hat{m}}_{-j}$ and her signal s _j, by sequential rationality, the leader j chooses $\hat{y}$ to maximize her expected utility. Because of the quadratic loss specification of players’ payoffs, she chooses

(1) $$\begin{equation} y_{j}(s_{j},\mathbf {\hat{m}}_{-j})=b_{j}+E[\theta |s_{j},\mathbf {\hat{m}} _{-j}]. \end{equation}$$

Let d_j(m _{− j}) be the number of politicians willing to truthfully advise j, were she to lead the group. These politicians form the group of trustworthy associates of j. We prove (in the Online Appendix) that the profile m _{− j} is an equilibrium if and only if, whenever i is truthful to j,

(2) $$\begin{equation} \left|b_{i}-b_{j}\right|\le \frac{1}{2\left[ d_{j}(\mathbf {m}_{-j})+3 \right] }. \end{equation}$$

An important consequence of the equilibrium condition Equation (2) is that truthful communication from politician i to leader j is possible only if the ideological positions of i and j are sufficiently close. We use this result to derive how informed politician j would be in the event where she becomes leader.

First we note that the term d_j(m _{− j}) is a function of the equilibrium communication strategies m _{− j} deployed by group members. In particular, whenever i can be truthful to j in equilibrium, then there is another equilibrium in which i “babbles” when communicating with j: since she babbles j will ignore her, and given this response there exists no profitable deviation for i. Fixing j’s leadership, the equilibria (m _{− j}, y_j) that provides the highest expected payoffs to all politicians are such that j’s information d_j(m _{− j}) is maximal. That is, we consider the most information that j could obtain when any politician who could communicate truthfully in equilibrium will in fact do so.Footnote ⁹ This allows us to define d*_j as the maximal size of the group of politician who form j’s trustworthy associates. Straightforwardly, we can relate the maximal size of this group to a leader’s equilibrium judgment.

Next we derive this leadership characteristic from first principles. In particular, we can define it as a consequence of j’s ideological position relative to that of other politicians in her party. To do so, we define the function $N_{j}:\mathbb {R}\rightarrow \mathbb {N}$ as the ideological “neighborhood” function of politician j. For any real number b, the quantity N _j(b) is the number of politicians whose ideology is within distance b of her own; that is, the number of politicians whose ideology is in j’s ideological neighborhood of size b. To formally define the function N _j, we exploit the fact that politicians are ordered according to their bias, so that

$$\begin{eqnarray*} N_{j}\left( b\right) & = & \max \lbrace i\in N:|b_{i}-b_{j}|\le b\rbrace \\ && - \, \min \lbrace i\in N:|b_{i}-b_{j}|\le b\rbrace , \end{eqnarray*}$$

for any real number b. For example, if the group of players who are truthful to leader j = 5 is {3, 4, 6, 7}, then N _j(b) = 7 − 3 = 4. We use the function N _j(·) combined with the equilibrium condition Equation (2) to calculate the maximal size of any politician j’s network of trustworthy associates:

(3) $$\begin{equation} d_{j}^{\ast } = \max \left\lbrace d \in \mathbb {N} : N_{j}\left( \frac{1}{2\left( d+3\right) }\right) \le d \right\rbrace . \end{equation}$$

This provides a simple rule to calculate d*_j by counting the number of politicians other than j that are ideologically close to her. For example, suppose that b _j = 0 and the three politicians closest to j have bias distance less than 1/12 from b _j; that is, they have a bias in the interval (− 1/12, 1/12). These politicians would provide truthful advice to j were she to be selected as leader. For j to have one more trustworthy associate, it must be that no member of that circle has a bias further from b _j = 0 than 1/14. Interestingly, the size of the ideological neighborhood of leader j (to which a politician needs to belong to be trustworthy to j) decreases in the number of associates truthful to j. For example, a politician i with bias b _i of distance within 1/10 and 1/8 to b _j will be truthful to j if and only if j has no other trustworthy associate.Footnote ¹⁰

The Optimal Leader

We first show that optimal leader selection involves trading off a politician’s ideological moderation and her judgment. To formalize this insight, we denote politician j’s moderation as −|b _j − ∑_{i ∈ N}b _i/n|, the contrary of the distance between b _j and the average ideology $\bar{b} \equiv \sum _{i\in N}b_{i}/n.$ We have defined d*_j as the maximal size of a leader’s network of trustworthy associates. It is but a small step to relate this number to her judgment, the second critical and endogenous leadership characteristic. When combining the information obtained from others with her own view, a leader forms an independent judgment of the best course of action. Thus a leader’s judgment is (strictly) increasing in the number of informative signals she obtains from her trustworthy associates.

In fact, and armed with these definitions, we can prove that the equilibrium ex ante sum of players’ payoffs W*(j) can be rewritten as

(4) $$\begin{equation} W^{\ast }\left( j\right) =-\sum _{i\in N}(b_{i}-b_{j})^{2}-n\frac{1}{ 6(d_{j}^{\ast }+3)}. \end{equation}$$

Equation (4) decomposes the welfare function into two elements: the aggregate ideological loss ∑_{i ∈ N}(b _i − b _j)² associated with the decision taken by j, and the aggregate residual variance of her decision $n\frac{1}{6(d_{j}^{\ast }+3)}$. Evidently, a more moderate leader, whose bias is closer to average ideology $\bar{b}$, makes the aggregate ideological loss ∑_{i ∈ N}(b _i − b _j)² smaller. Further, the residual variance $\frac{1}{6(d_{j}^{\ast }+3)}$ is inversely related to the size of the leader’s maximal informant set d*_j and hence to her judgment.Footnote ¹¹ The optimal leader j maximizes W*(j). Thus, optimal leader selection takes into account each politician’s moderation and her endogenous judgment that are related to the core primitives of our model, namely the ideologies of members of the group.

Leader j’s moderation can be understood spatially as the relative position of j’s bias b _j with respect to the whole ideology distribution b = {b ₁, . . ., b_n} in the group. In fact, every element of b, even extreme ones, matters for the determination of the average ideology $\bar{b}$. In this sense, moderation is a “global” property of j’s ideology b _j with respect to the distribution b = {b ₁, . . ., b_n}.

On the other hand, judgment is a “local” property of j’s ideology b _j within b = {b ₁, . . ., b_n}: it depends only on how many other politicians are ideologically close to j, in the sense defined by Equation (3). The leader’s understanding is thus defined by those close to the leader, or adopting Machiavelli’s text, by “the men he has around him.” This analysis of the role played by the local ideological distribution is, to our knowledge, novel in the large contemporary and formal literature on collective choice; though it echoes the insights of Machiavelli made in Il Principe 500 years ago.

We summarize our findings as follows.

Electing the Leader

We now determine which politician is elected as leader by a simple majority decision taken within the group. Each player i’s utility as a function of the leader’s identity j is

(5) $$\begin{equation} U^*_{i}(j)=-(b_{i}-b_{j})^{2}-\frac{1}{6(d_{j}^{\ast }+3)}. \end{equation}$$

As in Equation (4), the first term on the right-hand side illustrates the ideological loss −(b _i − b _j)² suffered by each member of the group i when j is chosen as leader. The second term illustrates player i’s preference for an informed leader j, as it increases in the judgment d*_j. We note that player utilities are not single-peaked with respect to a leader’s identity: a politician who is ideologically distant may in fact be better informed, and so have better judgment, than one who is ideologically similar. While Black’s theorem does not apply in this setting, so leadership choice under majority rule is far from straightforward, we can, nevertheless, make progress by showing that utility functions are single-crossing.

As a consequence of this result, we appeal to Theorem 2 by Gans and Smart (Reference Gans and Smart1996) to show that the player with median ideology will determine the outcome of the election. The unique Condorcet winner of the election game is the politician j who maximizes the expected payoff of the median player.

Having established the outcome of a majority election, we can compare it with the optimal leader selection by inspecting Equations (4) and (5), the latter for i = m. As in the earlier case, there is a tradeoff between moderation and judgment: the Condorcet winner j keeps both the ideological loss (b _m − b _j)² and the residual variance $\frac{1}{6(d_{j}^{\ast }+3)}$ as low as possible. Just as with optimal leadership, the majority choice involves a tradeoff between the desire for a moderate leader and that for a leader with good judgment, which, in turn, stems from having a large group of close-minded associates. Beyond this similarity, there is a critical difference and it is this: whereas a majority preferred leader makes this tradeoff decision by considering only her own payoff, by contrast, an optimally selected leader would consider the preferences of the entire group. Straightforwardly, and as the weights placed on these two features of good leadership are different in our key expressions, the majority choice of leader may not be optimal. As we shall see, the implications are surprising in that we identify instances in which the median politician’s utility U*_m places less weight on moderation (and more on judgment) than the group’s welfare W*. Thus, majority choice may be inefficient because it places too much weight on the leader’s judgment.

Moderate Leadership

A natural question to ask our model is thus: Under which conditions on the primitive parameters (the ideology distribution b) is the most moderate politician the optimal leader and the majority-preferred one? Evidently, this is the case when there are only three politicians in the group. For then the median is the most moderate politician and she cannot be less informed than either of the others: If she is willing to communicate truthfully with her neighbors, then at least one of them is willing to be truthful to her.Footnote ¹²

Moving beyond the three-player case, we illustrate sufficient conditions such that the optimal leader is also the most moderate politician. Doing so, we consider the situation in which politicians are distributed at even distances with respect to their ideology on the line. Because n is odd, and by symmetry of the ideological distribution, the median m is the most moderate and has at least as many trustworthy advisers as any other politician. Then there is no tradeoff between a leader’s moderation and her judgment. As the median politician is as informed as anyone else she should take the decision on behalf of the group and, indeed, she would be the unique choice of the majority.

We formalize this insight in the following proposition that proves a stronger result. We show that the median politician m is elected by the majority as leader and is also the optimal leader when the ideology distribution is symmetric around m and “single peaked” at m, in the sense that politicians are weakly more ideologically clustered as they get closer to m. Formally, we define the ideology distribution b as single peaked and symmetric at m when for every i = 1, . . ., m − 1, β_i = b _{i + 1} − b _i weakly increases in i and β_i = β_{n − i}. Evidently, the case in which politicians’ ideologies are evenly distributed on the line (i.e., there is a constant β > 0 such that b _{i + 1} − b _i = β for all i = 1, . . ., n − 1) is a limit case covered by the definition of b as single peaked and symmetric at m.

The result is depicted in Figure 1 for n = 5 and ordered left-right biases b ₁ to b ₅. In the figure, for each of three politicians j = 2, 3, 4, their maximal amount of equilibrium information is d*_j = 2. Then the optimal leader, and the one who is indeed chosen by the group, is player 3.

Proposition 3 relates the core characteristics of leadership to reveal that, with biases single-peaked and symmetric at m (as in the case of equidistant biases depicted in Figure 1) the most moderate politician is also (weakly) better informed and so has better judgment. The significance of our result lies in the fact that, when ideologies in the group are single-peaked and symmetric at m, there is no tradeoff between moderation and the capacity to gather reliable advice. This underlines that a necessary condition for a politician who is not the most moderate to be the optimal leader or the elected one is that ideologies are clustered away from the median.Footnote ¹³ We explore this tradeoff next, focusing on the case of five politicians for ease of exposition.

FIGURE 1.

Leadership Choice with Equidistant Bias. This figure depicts the case where b _{i + 1} − b _i = β for all politicians i = 1, . . ., 4 and 1/12 < β ≤ 1/10. An arrow linking two politicians illustrates that truthful communication is sustained between them.

The Case with Five Politicians

To explore the tradeoff between moderation and judgment, we study in-depth the case of five politicians, parametrized by the bias differences β_i = b _{i + 1} − b _i for players i = 1, . . ., 4. Our case is suitably rich (allowing us to identify properties of the ideology distribution that provide novel and interesting findings) yet simple (so that we can do so in a clean and clear manner).

First we note that it can never be optimal that 1 and 5 lead the group and neither will they be elected by the group as leaders. Players 2 and 4 can be chosen as leaders if and only if they have better judgment than player 3. Also, interchanging β₁ with β₄ and β₂ with β₃ then players 2 and 4 are symmetric to each other. Hence, it is with no loss of generality that we restrict attention to parameter values for which W*(2) ≥ W*(4), so that welfare is strictly greater when 2 is the leader rather than 4, and for which U*₃(2) ⩾ U ₃*(4), so that the only possible Condorcet winners are 2 and 3.

Player 2 has better judgment than 3 when she can count on more trustworthy associates, that is when d*₂ > d ₃*. Using condition Equation (3), we calculate (in the Online Appendix) all cases for which the condition d*₂ > d ₃* holds, and determine the restrictions each one of them imposes on the parameters β_i, i = 1, . . ., 4. Here, we illustrate our findings in the case in which d*₂ = 2 and d*₃ = 1. This holds when β₁ ≤ 1/10, β₂ ≤ 1/10 but β₃ > 1/10. In this case, players 1 and 3 are sufficiently close to 2 to be trustworthy, whereas the median politician 3 can trust only 2, but not 4. So politician 2 has better judgment, while 3 is more moderate. This scenario is illustrated in Figure 2, where β₃ = b ₄ − b ₃ is such that (in contrast to Figure 1) player 4 can no longer communicate truthfully with 3. As the player ideologies are not distributed at even distances there may be a tradeoff between judgment and moderation and, moreover, surprising consequences of ideological shifts (such as that of player 4 in Figure 2 relative to Figure 1), which we discuss below.

FIGURE 2.

Leadership Choice with Nonequidistant Bias. In this figure, β₁ ≤ 1/10, β₂ ≤ 1/10 and β₄ ≤ 1/10 < β₃. All immediate neighbor pairs truthfully communicate, with the exception of 3 and 4.

The Tradeoff between Judgment and Moderation

Whether the choice between the more moderate politician 3 and politician 2, who has better judgment, is resolved in favor of either depends upon β₂ = b ₃ − b ₂. This determines how extreme 2 is relative to 3. This pins down the majority choice: if 2 will make a more informed decision than 3, then 3 would be happy for her to do so, so long as the ideological distance between them is not too large. This allows us to define a majority threshold with respect to β₂. In fact, we can relate the optimal choice and the majority choice between politician 2 and 3 according to the size of β₂ relative to the other primitives of the model, as demonstrated by the lemma below that allows us to define a welfare threshold. Let δ = b ₅ − b ₃ + b ₄ − b ₃ − (b ₂ − b ₁) be the sum of the ideological distance from 3 of the players to the right of 3 (that is, 4 and 5) net of the distance from 2 of player 1, the only player to the left of 2.

The result defines a welfare threshold, τ(δ). The group is better off when player 2 takes the decision if and only if β₂, the ideological distance between players 2 and 3, is below τ(δ). Intuitively, it is optimal that 2 leads the group when her better judgment, combined with the benefits to those the left of the spectrum (b ₂ − b ₁) are not outweighed by the ideological loss incurred by those to the right (b ₄ − b ₃ + b ₅ − b ₃). We prove in the Online Appendix that τ is strictly decreasing in δ, with τ(1/10) > 1/10 and τ(δ) → 0 for δ → ∞.

Lemma 2 also defines the majority threshold $\frac{1}{2\sqrt{30}}$. This takes a simpler form as it depends only on the median player. She may obtain a more informed outcome when 2 takes the decision and this yields a constant addition to her utility. This comes at an ideological cost β₂. Thus the group chooses 2 as leader if and only if β₂ is below a threshold given by the constant $\frac{1}{2\sqrt{30}}$.

The final part of Lemma 2 reveals that, in equilibrium, the welfare threshold τ(δ) can either be larger or smaller than the majority threshold $\frac{1}{2\sqrt{30}}$, depending on how large δ is. In the former case, we can distinguish three possibilities on the basis of β₂: for small β₂, i.e., $\beta _2 < \frac{1}{2\sqrt{30}},$ the non-moderate politician 2 is both the Condorcet winner and the optimal leader; for large β₂, specifically, β₂ > τ(δ), the most moderate politician 3 is both the Condorcet winner and the optimal leader; in the intermediate case $\frac{1}{2\sqrt{30}} < \beta _2 < \tau (\delta )$, the optimal leader is the most informed politician 2, whereas the majority elects politician 3. This identifies an instance in which majority voting leads to a moderate but inefficient choice of leader.

A perhaps more interesting and unexpected case arises when $\tau (\delta ) < \frac{1}{2\sqrt{30}}$. Again, for small β₂, the non-moderate politician 2 is both the Condorcet winner and the optimal leader, and for large β₂ it is politician 3. Now inefficiency arises in the intermediate case in which $\tau (\delta ) < \beta _2 < \frac{1}{2\sqrt{30}}$. Although the optimal leader is the most moderate politician 3, the majority elects instead a relatively extreme leader in politician 2.

The logic behind this result is simple. The median politician may trade off moderation and judgment in a way that differs from the optimal choices made by a social planner. Starting from her ideal point, she may sacrifice a policy more in line with her bias for a more informed outcome. In our five-player example, she will indeed do so when β₂ is sufficiently small and d*₂ > d ₃*. Choosing a leftist leader then benefits the median and of course leftist members of the group. But it harms the right-wing members 4 and 5, who bear costs (b ₄ − b ₂)² and (b ₅ − b ₂)² respectively. Because the ideological loss function (b _i − b _j)² is convex in the ideological distance |b _i − b _j|, the leadership move from 3 to 2 is more harmful to rightwing politicians than it is beneficial to the leftist ones. Then it may be the case that a social planner would force the median politician to take the decision, if only she could.Footnote ¹⁴

Comparative Statics

Further analysis of the five-player case uncovers some interesting comparative statics results. Our comparative static exercise is as follows. We first consider leadership choice with a given set of primitives (players ideological biases) and then consider what happens to leadership choice when a player changes her ideology from right (left) to left (right). We show that this can in fact induce a shift in leadership in the opposite direction.Footnote ¹⁵

To illustrate, consider a benchmark case with players ideologies evenly distributed apart and where politicians 2, 3, and 4 can all count on the truthful advice of their ideological neighbors, so that β_i is constant in i = 1, . . ., 4, and 1/12 < β_i ≤ 1/10. Then following Proposition 3, politician 3 is (strictly) most moderate and has (weakly) better judgment among the five; hence she is elected as leader and this choice is also optimal for the group. Suppose now that the ideology of the center-right player 4 moves rightward and so away from that of the median player 3 and that, as a consequence, they are no longer truthful to one another (that is, suppose that β₃ increases so as to become larger than 1/10). Now politician 3 has lost a (previously) trustworthy associate. It is now possible that the center-left politician 2 is the Condorcet winner of the election game—3 will delegate authority to her, despite not being the most moderate politician. Indeed, by Lemma 2, we know that this is the case when $\beta _2 < \frac{1}{2\sqrt{30}}.$ Hence, the ideological movement of a player towards a more extreme position may induce a leadership change in the opposite direction.

Conversely, suppose that the five politicians are such that, in the benchmark case with evenly distributed ideologies, there is no truthful communication across players; that is, β_i is constant across i < n and β_i > 1/8. Suppose now that the leftist politicians 1 and 2 become more moderate, so that now politician 2 can count on the truthful advice of players 1 and 3 (formally, suppose that β₁ and β₂ decrease, so that they both become smaller than 1/10). Because player 4 is still not truthful to 3, leadership switches from the median player 3 to the center-right player 2, again, when $\beta _2 < \frac{1}{2\sqrt{30}}.$ Here, the rightward ideological movement of leftist players moves leadership choice (and hence policy outcomes) in the opposite direction. Thus moderation allows them to capture control of leadership of the group.

In both cases, and unexpectedly, a change in the ideological distribution by which politicians (weakly) move right leads to a shift in the group decision to the left.

We summarize our findings for this section in the following result.

Model

Suppose that there are two parties, A and B. The top politicians in party A consist of the set of players N _A and those in B consist of politicians N _B. At the beginning of the game, parties choose leaders {a, b}. To make our results general, we do not commit to a specific leader choice model. We assume only that each party selects as leader the strongest possible candidate, defined as the politician within the party who defeats the largest possible number of candidates from the other party in the general election.Footnote ¹⁸ To simplify the exposition, we consider only ideology and party profiles such that there is only one such politician in each party. The eventual winner j ∈ {a, b} of the general election then implements the final policy $\hat{y}\in \mathbb {R}$. Candidates cannot commit to electoral promises, and so the winner implements her preferred policy after consultation with her party’s top politicians.

There is a continuum of citizens, which includes the finite set of politicians N _A∪N _B. The preferences of each citizen k, including politicians, are expressed by

$$\begin{equation*} u_{k}\left( \hat{y},\theta \right) =-\left( \hat{y}-\theta -b_{k}\right) ^{2}, \end{equation*}$$

where b _k is the ideological bias of citizen k relative to the median voter in the general election, who we assume to have bias equal to zero, without loss of generality. As before, the utility of k depends on how well $\hat{y}$ matches an unknown state of the world θ together with her ideological bias b _k. We single out politicians who belong to the set N _A∪N _B, denote them with indexes i, and maintain the assumption that b _i is increasing in i and, therefore, that all politicians in A are to the left of all politicians in B.

The remainder of our model is as before. Each top politician i has some private information on θ. After the general election takes place, each i observes a signal s _i ∈ {0, 1} of θ such that $\Pr \left( s_{i}=1|\theta \right) =\theta$. And before the elected policy maker j chooses $\hat{y},$ each politician i can communicate by sending a message $\hat{m}_{ij}\in \lbrace 0,1\rbrace$ to j. We assume that there are social conventions stifling communication across parties, and that the elected politician has truthful associates only within her own party. We thus consider equilibria in which the politicians from the opposite party do not reveal any information to her.

As in the previous section, each voter k evaluates a candidate j on the basis of both j’s ideological proximity (b _k − b _j)² and her judgment, identified by the number of j’s trustworthy party fellows d*_j, according to the, by now usual, decomposition:

$$\begin{equation*} U_k^{\ast }(j)=-(b_k-b_{j})^{2}-\frac{1}{6(d_{j}^{\ast }+3)}. \end{equation*}$$

Because each voter k evaluates a candidate j’s judgment favorably, regardless of her ideology, we can think of judgment as valence. Here, it is endogenously determined by j’s network of trustworthy party fellows.

As a consequence of Lemma 1, preferences satisfy the single crossing condition with respect to the choice of leader in the general election. Moreover, the play of weakly undominated strategies in (the subgame that represents) the election implies that each voter chooses her preferred candidate j ∈ {a, b}. As a consequence, candidate a will be elected with certainty if and only if U ₀(a) > U ₀(b), where we take 0 to be the index of the median voter.

Policy Divergence

A natural benchmark for comparison is an otherwise identical model in which players do not communicate to the elected leader before she chooses the policy $\hat{y}$. As no communication can take place, so no information about θ can be aggregated, only the vector of ideologies b are relevant to votes cast in either primary or general election. As these are common knowledge, the game then boils down to a simple one of perfect information. It is then simple to prove that U ₀(a) > U ₀(b) only when the policy bias of leader a is closer to zero than that of b, so that the most moderate candidate in each party is chosen as leader.

Whenever politicians are sufficiently ideologically distant from each other that they can never communicate truthfully (even to their closest ideological neighbor), then the winner of the general election is the candidate with bias closest to zero. Beyond this simple case, it is immediate that, in our model of electoral competition, party leaders need not be moderate. The analysis follows our earlier logic: politicians with a large network of truthful informants may be preferred by the median voter even if they have relatively extreme ideologies. In fact, we can reveal new insights. First, we show that the winner of the general election need not be the most moderate politician (that is, the politician whose ideology is closest to the median voter), even in circumstances in which the politicians’ ideologies are evenly distributed in the ideological spectrum. Thus, our finding stands in sharp contrast with Proposition 3. Why so? When politicians are partitioned in competing parties, the most moderate politician (with respect to the electorate as a whole) is at the extreme end of the ideological spectrum within her own party. This constrains the pool of trustworthy associates she can rely upon and so hampers her ability to take informed decisions if elected to office.

This insight is demonstrated by the six-player example depicted in Figure 3. There are six politicians, with ideologies such that b _{i + 1} − b _i = β for all i = 1, . . ., 5, arranged symmetrically around the median ideology zero, so that b ₃ = −β/2 and b ₄ = β/2. The leftist politicians 1, 2, and 3 (lighter shading) belong to party A and the others (darker shading) to party B.

FIGURE 3.

Party Competition with Equidistant Bias. The figure illustrates a case where politicians 1,2,3 belong to party A and 4,5,6 to party B. Within each party, each politician is informed by her immediate neighbor.

Following our earlier analysis, unless politicians 2 and 5 can count on more trustworthy advisers than 3 and 4, in equilibrium, the latter will be selected by their parties and tie the general election. Because of the symmetry of b we can focus attention on the challenge between 2 and 3 for leadership of party A. Politician 2 has better judgment when d*₂ = 2 and d*₃ = 1, which requires that β ≤ 1/10 and that 2β > 1/10.

It is then relatively straightforward to identify a condition on β such that the median voter in the general election would prefer that candidate 2 is chosen by party A, that is U ₀(2) > U ₀(3). Specifically, we show in the Online Appendix that this is the case when $1/20 < \beta < \frac{1}{4\sqrt{15}} \approx 0.0645$. By symmetry, and since the median has zero bias, it must also be that U ₀(5) > U ₀(4). Hence, in the unique equilibrium, party A chooses politician 2 as leader, and party B chooses politician 5. In sum, the chosen leaders are not the most moderate candidates 3 and 4.

This result provides a new take on the documented divergence among candidates in two party elections. Even if two parties compete for power, they will not necessarily be the most moderate candidates, so that convergence to the median will not take place.Footnote ¹⁹

Moderation and Party Cohesiveness

The second novel insight highlights the value of a party’s ideological cohesiveness. Interpreting a party’s cohesiveness as the ideological distance among its top politicians, we find that a more cohesive party can defeat a larger, less cohesive one in a general election. This can occur even though the larger party can draw information from a larger set of informed politicians. And it can occur even though the leader of the larger party has ideological views that are closer to those of the median voter. Why? The leader of the more cohesive party can count on more trustworthy associates than her opponent. The median voter anticipates that, as a consequence, she will have better judgment.

Finally, we find that the outcome of the election may depend on the whole ideological distribution and often in a very subtle way. For example, suppose that (repeating the comparative static exercise from earlier) the ideology of the leader of a party becomes more moderate. Should she be elected then she will implement her preferred policy which, in turn depends upon the advice she obtains. Surprisingly she may in fact lose the general election as a result of her new found moderation. (Of course, the opposite can happen: a politician may lose the election if her views become more extreme). Intuitively, this occurs because, as the leader’s views become more moderate, the ideological distance with others in the party increases. As a consequence, the leader may lose the benefit of truthful advice from erstwhile allies. This result is, to the best of our knowledge, both novel and unsupported by any variant of the standard spatial model found in the literature.

The result can be demonstrated by means of the following example, illustrated in Figure 4. Suppose that there are five politicians, with b ₂ < 0 < b ₃ and |b ₃| < |b ₂|. Politicians 1 and 2 belong to party A and 3,4,5 belong to party B. In this example, there are two senses in which party B is advantaged: there is a larger set of top politicians from whom the leader could draw upon for information; and in player 3, it has a potential leader whose views are closest to those of the median voter, whose ideal point is equal to zero. This implies that, were there no possibility of communicating private information, party B would always win the election by selecting politician 3 as its leader. Party B is not only larger, it is also “ideologically majoritarian” in that, in contrast to party A, it is able to put forward candidates whose ideological perspective appeals to a majority of the electorate.

FIGURE 4.

Party Competition with a Majoritarian Party. Illustrates a case where politicians 1 and 2 are in party A and 3,4,5 are in party B; b ₂ < 0 < b ₃, b ₃ < |b ₂|, β₁ ≤ 1/8, β₃ > 1/8 and β₄ > 1/8. Politicians truthfully communicate within party A but not within party B.

Given the advantage of B in fielding more moderate candidates, following Lemma 1, a politician from party A can only be elected due to her better judgment. For this to be the case it must be that d*₂ = 1 > d ₃* = 0, as there is only one other informed politician in party A. This situation requires that β₁ ≤ 1/8, β₃ > 1/8 and β₄ > 1/8, and is depicted in Figure 4. It is then not difficult to find conditions under which the median voter prefers politician 2 from party A to any politician from party B, so that, in equilibrium, B will lose the election despite its advantage. Specifically, we find in the Online Appendix that this is the case if and only if b ²₂ − b ₃² < 1/72: this condition is satisfied when |b ₂| is not too much farther than |b ₃| from zero, the median voter’s ideal point.

To see that a leader can lose the election if her ideology moves closer to the median voter, consider politician 2 as leader of party A. Suppose that we start from a situation in which β₁ is smaller than but close to 1/8. Politician 1 is a trustworthy associate of 2, but if the ideal platform of politician 2 moves closer to the median voter’s, and 1’s views remains fixed, then the condition β₁ ≤ 1/8 will no longer be satisfied. Thus, candidate 2’s judgment is no longer better than that of politician 3 and so, were they to contest the election, 2 would lose.