Who emerges from smoke-filled rooms? Political parties and candidate selection

Abstract

In many countries political parties control who can become a candidate for an election. In this gatekeeping role parties may be tempted to put their own interests first, particularly when voters have little information about candidates. This paper uses a theoretical model to demonstrate that electoral incentives can discipline parties to nominate high-quality candidates even when voters are initially unable to observe quality themselves. In equilibrium voters elect candidates that are ex-ante preferred by the party leader with lower probability. This effectively neutralises the bias of the party leader and induces her to use her superior information to select candidates according to the preferences of the median voter. This result requires that electoral competition is sufficiently strong. If competition is weak, nothing can prevent the party leader from following her own preferences. As ideological alignment between the median voter and a party reduces the degree of competition that this party faces, the median voter can be better off when parties are polarized. Excessively strong competition can be harmful, however, as some politicians cease to be viable candidates and the party leader is less able to select on quality. Allowing the party leadership to nominate candidates strategically makes the benefits of introducing primaries less clear than previously argued in the literature.

Appendix

1.1 A Proofs

Lemma 2

Fix some $p \in \{M,E\}$. An equilibrium in which $\eta _p(q_C)=0$ for all $q_C \in Q$ satisfies Universal Divinity if and only if $\bar{\pi }_p=1$.

Proof

If politician p is nominated unexpectedly, Universal Divinity requires that voters believe that politician p has quality $q_p \in q_C$ such that $\Lambda _p(q'_C)\subset \Lambda _p(q_C)$ for any possible $q'_C$, where $\Lambda _p(q_C)$ denotes the set of election strategies of the median voter such that the party leader benefits from the deviation conditional on having type $q_C$. As the utility of the party leader is increasing in the election probability of her nominated candidate it holds that $\Lambda _p(q_C) =(\lambda _p(q_C),1]$, where $\lambda _p(q_C)$ is the election probability that makes the party leader indifferent between deviating and sticking to her equilibrium strategy. The proof therefore needs to demonstrate which $q_C$ minimises $\lambda _p(q_C)$. Let $p'$ denote the competitor for the party nomination of politician $p \in \{M,E\}$. The interim utility of the party leader under a strategy profile $\sigma =(\eta _p,r)$ where $\eta _{p}(q)=0$ for all $q \in Q$ (politician p is nominated only off the equilibrium path) is given by

$$\begin{aligned} r(p')[-\,(p'-i_L)^2+ w\cdot q_{p'}+Y] + (1-r(p'))[-\,(I-i_L)^2 + w\cdot q_I]. \end{aligned}$$

Suppose politician p would be elected with probability $\lambda$ if nominated. The utility of the party leader from nominating p would then be

$$\begin{aligned} \lambda [-\,(p-i_L)^2+ w\cdot q_p+Y] +(1-\lambda ) [-\,(I-i_L)^2+ w\cdot q_I]. \end{aligned}$$

Equating the two utilities and solving for $\lambda$ yields the probability of electing politician p that makes the party leader indifferent between nominating either politician:

$$\begin{aligned} \lambda _p(q_C)= \frac{r(p')[-\,(p'-i_L)^2+(I-i_L)^2+ w (q_{p'}-q_I)+Y]}{[-\,(p-i_L)^2+(I-i_L)^2+w(q_p-q_I)+Y]}. \end{aligned}$$

As $q_p$ only shows up in the denominator of this expression, the minimum of $\lambda _p(q_C)$ can only be attained for $q_p$ equal to one. Universal Divinity therefore implies $\bar{\pi }_p=1$. $\square$

Proof of Proposition 1

Suppose that $r(M)=r(E)=1$ in equilibrium. Then the party leader prefers to nominate the extremists whenever the moderate and the extremist have the same quality as well as when $q_M=0$ and $q_E=1$. When $q_M=1$ and $q_E=0$ the party leader nominates the extremist if

$$\begin{aligned} -\,(M-i_L)^2+w \le -\,(E-i_L)^2 \end{aligned}$$

and nominates the moderate otherwise. In the former case the posterior belief of the median voter over the quality of the extremist $\bar{\pi }_E$ is equal to $\pi$ while in the latter case Bayes’ rule implies $\bar{\pi }_M=1$ and

$$\begin{aligned} \bar{\pi }_E=\frac{\pi }{\pi +(1-\pi )^2}. \end{aligned}$$

According to Lemma 2$\bar{\pi }_M=1$ under any of the two possible strategies of the party leader. Equilibrium is satisfied if the median voter prefers any nominated candidate of party C over the incumbent under these beliefs. The extremist is preferred over the incumbent if

$$\begin{aligned} -\,E^2+\bar{\pi }_E \ge \mathcal {I}, \end{aligned}$$

which is equivalent to the condition given in the statement of the proposition. This condition is also sufficient, as the median voter prefers a moderate of high quality over the incumbent whenever she prefers the extremist over the incumbent. $\square$

Proof of Proposition 2

If only one politician of party C is elected with positive probability, the party leader always nominates this politician. Nominating the politician who loses for sure could only be optimal if the party leader prefers the incumbent over a member of her party. This would require

$$\begin{aligned} -\,(I-i_L)^2+w \ge -\,(M-i_L)^2 + Y, \end{aligned}$$

(5)

as the lowest possible utility for the party leader from one of her own candidates winning is if this candidate is a moderate of low quality. As the moderate is located closer to the party leader than the incumbent and $Y\ge 1$ Inequality (5) is never satisfied.

Suppose $r(M)=0$ and $r(E)>0$ and the party leader accordingly always nominates the extremist. By Lemma 2, Universal Divinity requires $\bar{\pi }_M=1$. But if this is the case the utility of the median voter from electing the moderate must be strictly larger than the utility from electing the extremist, contradicting either that $r(M)=0$ or that $r(E)>0$. The only possible case is therefore $r(M)>0$ and $r(E)=0$, in which case the party leader must always nominate the moderate. This implies $\bar{\pi }_M=\pi$. The condition that the median voter at least weakly prefers the moderate over the incumbent is then

$$\begin{aligned} \mathcal {I} \le -\, M^2+\pi . \end{aligned}$$

Generically this inequality will be strict, implying $r(M)=1$. On the other hand, the median voter must at least weakly prefer the incumbent over the extremist, which is satisfied if

$$\begin{aligned} -\,E^2+1 \le \mathcal {I}. \end{aligned}$$

Solving the last two conditions for M and E, respectively, yields the conditions given in the statement of the proposition. $\square$

Proof of Lemma 1

Consider an election strategy r such that the party leader is indifferent between nominating the moderate and the extremist for some realization of qualities $(q_M,q_E)$. That is

$$\begin{aligned}&r(M)[-\,(M-p_L)^2 + w\cdot q_M] + (1-r(M)) [-\,(I-i_L) +w\cdot q_I]\nonumber \\&\quad = r(E)[-\,(E-p_L)^2 + w\cdot q_E] + (1-r(E)) [-\,(I-i_L) +w\cdot q_I]. \end{aligned}$$

(6)

Generically,

$$\begin{aligned} -\,(M-i_L)^2 + w\cdot q_M \ne -\,(E-i_L)^2 + w\cdot q_E \end{aligned}$$

and indifference thus requires $r(M)\ne r(E)$. Now suppose there was a second realization of qualities $(q'_M,q'_E)\ne (q_M,q_E)$ such that

$$\begin{aligned}&r(M)[-\,(M-p_L)^2 + w\cdot q'_M] + (1-r(M)) [-\,(I-i_L) +w\cdot q_I] \\&\quad = r(E)[-\,(E-p_L)^2 + w\cdot q'_E] + (1-r(E)) [-\,(I-i_L) +w\cdot q_I]. \end{aligned}$$

Adding and subtracting $r(M)\cdot w \cdot q_M$ and $r(E)\cdot w \cdot q_E$ to the left-hand and right-hand side of this equality, respectively, and using Equality (6) yields

$$\begin{aligned} r(M)\cdot w (q'_M-q_M) = r(E)\cdot w (q'_E-q_E). \end{aligned}$$

(7)

Since it must be generically true that $r(M)\ne r(E)$, this equality is violated for any combination $(q'_M,q'_E)\ne (q_M,q_E)$. $\square$

Proof of Proposition 3

First of all, it is generically impossible that $0<r(M)<1$ and $0<r(E)<1$ simultaneously. This would require that the median voter is indifferent between all candidates, that is

$$\begin{aligned} -\,M^2+\bar{\pi }_M\ =\ \mathcal {I}\ =\ -\,E^2 + \bar{\pi }_E. \end{aligned}$$

However, by the law of iterated expectations it also needs to be true that

$$\begin{aligned} \eta _M \bar{\pi }_M + (1-\eta _M) \bar{\pi }_E\ =\ \pi , \end{aligned}$$

where $\eta _p$ denotes the unconditional nomination probability of politician p. Both conditions can be satisfied simultaneously only in knife-edge cases and a Full Competition equilibrium must therefore generically take the shape $r(M)=1$ and $0<r(E)<1$ or $0<r(M)<1$ and $r(E) = 1$.

Next, assume that the politician getting elected with certainty was the extremist. This would imply that the moderate either never gets nominated or is chosen only in the case $q_C=(1,0)$, depending on the value of w. Both cases lead to the posterior belief $\bar{\pi }_M=1$. But if the median voter is willing to elect the extremist then she must certainly prefer a moderate of high quality over the incumbent as well, contradicting that $r(M)+r(E)<2$.

It must therefore be true that $r(M)=1$ and $0<r(E)<1$. This can only hold if the median voter is indifferent between the incumbent and the extremist, which requires

$$\begin{aligned} \bar{\pi }_E = \mathcal {I}+E^2. \end{aligned}$$

(8)

To generate this posterior expected quality of the extremist the party leader must be playing a mixed strategy. By Lemma 1, mixing is only possible for one particular realization of qualities. As the moderate gets elected with certainty the expected utility of the party leader from nominating the moderate is

$$\begin{aligned} -\,(M-i_L)^2+w \cdot q_M+Y \end{aligned}$$

while nominating the extremist gives

$$\begin{aligned} r(E)[-\,(E-i_L)^2+w \cdot q_E+Y] +(1-r(E)) [-\,(I-i_L)^2+w\cdot q_I]. \end{aligned}$$

Equating the two utilities it is possible to derive the following identity:

$$\begin{aligned} r(E)= \frac{[-\,(M-i_L)^2+w \cdot q_M+Y]-[-\,(I-i_L)^2+w\cdot q_I]}{[-\,(E-i_L)^2+w \cdot q_E+Y]-[-\,(I-i_L)^2+w\cdot q_I]}. \end{aligned}$$

(9)

Given the restrictions on parameters the expression on the right-hand side is always positive. In the case of $q_M=q_E=0$ the numerator is smaller than the denominator and accordingly there exists an election probability r(E) that leaves the party leader indifferent between nominating either a moderate or an extremist of low quality.

Indifference between politicians of low quality implies that under the quality combinations (1, 0) and (0, 1) the party leader nominates the politician of high quality, while in the case of both having high quality the party leader strictly prefers to nominate the moderate. The last point can be seen by recognizing that in this case the utility from nominating the moderate is equal to the utility of nominating a moderate of low quality plus w and the utility from nominating the extremist equal to the utility of nominating an extremist of low quality plus r(E)w. Hence, indifference in the (0, 0)-case implies that the difference in utilities from nominating the moderate and the extremist is equal to $w(1-r(E))$ in the (1, 1)-case, which is positive. Given this strategy of the party leader, posterior expectations are given by

$$\begin{aligned} \bar{\pi }_M = \frac{\pi }{\pi \ +\ (1-\pi )^2 (1-\eta _E(0,0))} \end{aligned}$$

(10)

and

$$\begin{aligned} \bar{\pi }_E=\frac{\pi }{\pi \ +\ (1-\pi ) \eta _E(0,0)}. \end{aligned}$$

Solving this last equality for $\eta _E(0,0)$ and using Eq. (8) to substitute for $\bar{\pi }_E$ gives

$$\begin{aligned} \eta _E(0,0) = \frac{\pi (1-\mathcal {I}-E^2)}{(1-\pi )(\mathcal {I}+E^2)}. \end{aligned}$$

(11)

For this expression to be no greater than 1, it must be true that $\mathcal {I} \ge -\,E^2+\pi$. This first necessary condition for the existence of this equilibrium implies that the denominator is positive. The second condition, which ensures that the numerator is non-negative, is $\mathcal {I}\le -\,E^2+1$. Finally, it has to be true that the median voter weakly prefers the moderate over the incumbent: $\mathcal {I}\le -\,M^2+\bar{\pi }_M$. After substituting Eq. (11) into Eq. (10) this condition can be written as

$$\begin{aligned} \mathcal {I} \le -\,M^2 +\frac{\pi (\mathcal {I}+E^2)}{\mathcal {I}+E^2-\pi (1-\pi )}. \end{aligned}$$

(12)

If the election strategy of the median voter was such that the party leader was indifferent if $q_M=0$ and $q_E=1$, then the party leader would strictly prefer to nominate the moderate whenever the quality of the extremist is zero. This implies $\bar{\pi }_E=1$ and contradicts that the median voter could be indifferent between the incumbent and the extremist.

Indifference under $q_M=1$ and $q_E=0$, on the other hand, is possible only if w is sufficiently small. As a consequence the extremist would be nominated whenever she has high quality and when both politicians have low quality. The posterior beliefs are then

$$\begin{aligned} \bar{\pi }_M = 1 \end{aligned}$$

and

$$\begin{aligned} \bar{\pi }_E= \frac{\pi }{\pi +(1-\pi )^2+(1-\pi )\pi \eta _E(1,0)}. \end{aligned}$$

Solving this last equality for $\eta _E(1,0)$ and using Eq. (8) to substitute for $\bar{\pi }_E$ gives

$$\begin{aligned} \eta _E(1,0) = \frac{\pi -[\pi +(1-\pi )^2] (\mathcal {I}+E^2)}{(1-\pi )\pi (\mathcal {I}+E^2)}. \end{aligned}$$

(13)

The necessary and sufficient conditions for this expression to be positive and no greater than one are

$$\begin{aligned} -\,E^2 + \pi \le \mathcal {I} \le -\,E^2 + \frac{\pi }{\pi +(1-\pi )^2}.\ \end{aligned}$$

The requirement that the median voter at least weakly prefers the moderate over the incumbent in this case is equivalent to the condition $\mathcal {I} \le -\,M^2+1$, which is satisfied whenever $\mathcal {I} \le -\,E^2 + \frac{\pi }{\pi +(1-\pi )^2}$ as $\frac{\pi }{\pi +(1-\pi )^2} < 1$.

Finally, suppose the party leader is indifferent between nominating either politician if both are of high quality. Proceeding as before, an equilibrium with this feature can be shown to exists under the same conditions as in the previous paragraph.

Thus, $-\,E^2 + \pi \le \mathcal {I}$ is a necessary condition for any Full Competition equilibrium. The condition

$$\begin{aligned} \mathcal {I} \le -\,E^2 + \frac{\pi }{\pi +(1-\pi )^2} \end{aligned}$$

required for the existence of the Full Competition equilibria under high-quality and mixed-quality indifference is satisfied whenever

$$\begin{aligned} \mathcal {I} \le -\,E^2 + 1, \end{aligned}$$

which is required for the existence of the equilibrium under low quality indifference. The necessary conditions for the existence of the former equilibria are therefore always satisfied when the conditions for the existence of the latter equilibrium hold. Inequality (12) combined with

$$\begin{aligned} -\,E^2 + \pi \le \mathcal {I} \le -\,E^2 + 1 \end{aligned}$$

are accordingly jointly sufficient for the existence of a Full Competition equilibrium. Solving these expressions for M and E, respectively, yields the conditions provided in the statement of the proposition. $\square$

Proof of Proposition 4

Suppose Condition (2) is satisfied and let

$$\begin{aligned} \mathcal {I}=-\,E^2+1. \end{aligned}$$

It then follows from Propositions 1, 2, and 3 and Appendix B that the only equilibria that exist in this case are Full Competition under low-quality indifference and Limited Competition. For the Full Competition equilibrium the existence condition

$$\begin{aligned} -\,E^2+1\ge \mathcal {I} \end{aligned}$$

is binding, while the same is true for the existence condition

$$\begin{aligned} -\,E^2+1\le \mathcal {I} \end{aligned}$$

of the Limited Competition equilibrium. The second existence condition for the Limited Competition equilibrium

$$\begin{aligned} -\,M^2+\pi \ge \mathcal {I} \end{aligned}$$

is satisfied as a strict inequality due to the above assumptions. The second existence condition for the Full Competition equilibrium, on the other hand, is

$$\begin{aligned} M \le \sqrt{\frac{\pi (\mathcal {I}+E^2)}{\mathcal {I}+E^2-\pi (1-\pi )}-\mathcal {I}}. \end{aligned}$$

Using $\mathcal {I}=-\,E^2+1$ this can be rewritten as

$$\begin{aligned} \mathcal {I} \le -\,M^2+\frac{\pi }{1-\pi (1-\pi )}. \end{aligned}$$

This condition is satisfied as a strict inequality since

$$\begin{aligned} \mathcal {I}< -\,M^2+\pi <-\,M^2+\frac{\pi }{1-\pi (1-\pi )}. \end{aligned}$$

There thus exists $\mathcal {I}_1=\mathcal {I}-\delta$ with $\delta >0$ and sufficiently small such that the Full Competition equilibrium under low quality indifference is the unique equilibrium under $\mathcal {I}_1$. On the other hand, there exists $\mathcal {I}_2=\mathcal {I}+\epsilon$ with $\epsilon >0$ and sufficiently small such that the Limited Competition equilibrium is the unique equilibrium under $\mathcal {I}_2$. As the moderate is always nominated and elected in this equilibrium, the expected utility of the median voter is equal to $-\,M^2+\pi$. For $\delta \rightarrow 0$ the utility of the median voter under $\mathcal {I}_1$ converges to the utility under $\mathcal {I}=-\,E^2+1$ due to the continuity of the utility function of the median voter and the continuity of the strategy of the party leader in $\mathcal {I}$ in the Full Competition equilibrium under low-quality indifference. This latter utility is equal to

$$\begin{aligned} (1-\pi (1-\pi ))\left( -\,M^2+\frac{\pi }{1-\pi (1-\pi )}\right) +\pi (1-\pi )(-\,E^2+1) \end{aligned}$$

as the median voter is indifferent between the extremist and the incumbent in this equilibrium and the extremist is only nominated in the case where $q_M=0$ and $q_E=1$ when $\mathcal {I}=-\,E^2+1$. To complete the proof it needs to be shown that the utility of the median voter under $\mathcal {I}_1$ in the limit as $\delta \rightarrow 0$ is strictly larger than the utility in the Limited Competition Equilibrium. The condition for this is

$$\begin{aligned} (1-\pi (1-\pi ))\left( -\,M^2+\frac{\pi }{1-\pi (1-\pi )}\right) +\pi (1-\pi )(-\,E^2+1) >-\,M^2+\pi \end{aligned}$$

which simplifies to

$$\begin{aligned} -\,E^2+1>-\,M^2. \end{aligned}$$

As $0<M<E\le 1$ this condition is always satisfied. $\square$

Proof of Proposition 5

If $w \le -\,E^2+M^2$ a party leader with ideal policy equal to zero always nominates the moderate under No Competition. The utility of the median voter in this case is equal to

$$\begin{aligned} -\,M^2+\pi \end{aligned}$$

while the utility under Full Competition with a party leader with ideal policy equal to one is

$$\begin{aligned} \tilde{\eta }_M (-\,M^2+\bar{\pi }_M) + \tilde{\eta }_E {[}r(E)(-\,E^2+\bar{\pi }_E)+(1-r(E))\mathcal {I}], \end{aligned}$$

where $\tilde{\eta }_p$ denotes the ex-ante probability that politician p gets nominated. Replacing all strategies and beliefs with their equilibrium expressions, some tedious but straightforward algebra shows that the difference in the utilities can be written as $-\,\mathcal {I}-\,M^2$. $\square$

Proof of Proposition 6

Suppose that in the absence of primaries the equilibrium is such that $r(M)=r(E)=0$. Then the utility of the party is at its lowest possible level and the introduction of primaries cannot hurt the party leader.

Next, suppose that the equilibrium in the absence of primaries is the No Competition case, which implies that the utility of the party leader is at its highest possible level. Then the introduction of primaries lowers the utility of the party leader if the preferred candidate of the party leader conditional on a particular realisation of qualities would not be elected under primaries. The weakest condition for this to be the case is that the median voter prefers the incumbent over an extremist of low quality, i.e. $-\,E^2 < \mathcal {I}$. If this condition is satisfied the median member will be forced to nominate the moderate conditional on $q_C=(0,0)$ and the utility of the party leader is strictly lower under primaries.

If the equilibrium in the absence of primaries is the Limited Competition equilibrium, then the introduction of primaries would not increase the electability of the extremist even if the extremist is revealed to be of high quality. This is the case as the Limited Competition equilibrium exists only if $-\,E^2+1<\mathcal {I}$, as can be seen after slightly rearranging Inequality (1). In contrast, the electability of a moderate of low quality will drop to zero after the introduction of primaries if $-\,M^2<\mathcal {I}$, lowering the utility of the party leader.

Finally, suppose that the equilibrium that applies in the absence of primaries is one of the equilibria labelled as Full Competition. In this case an extremist of low quality cannot get elected under primaries since the existence of any Full Competition equilibrium requires $\sqrt{\pi -\mathcal {I}}\le E$ by Proposition 3, which implies $-\,E^2<-\,E^2+\pi \le \mathcal {I}$. Any candidate of high quality, on the other hand, will be elected under primaries as the existence of any Full Competition equilibrium requires $E \le \sqrt{1-\mathcal {I}}$ by Proposition 3, which implies $\mathcal {I}\le -\,E^2+1 < -\,M^2+1$. Finally, the condition $-\,M^2<\mathcal {I}$ may or may not be satisfied in any Full Competition equilibrium, indicating that a moderate of low quality may or may not be elected under primaries. Under primaries the median member will therefore nominate the extremist if and only if the extremist has high quality. Keeping in mind that the utility of the party leader conditional on observed qualities is equal to

$$\begin{aligned} -\,(M-i_L)^2+w\cdot q_M+Y \end{aligned}$$

whenever the party leader mixes in any Full Competition equilibrium, the change in the utility of the party leader after the introduction of primaries can then be calculated.

Consulting Table 1 shows that the only effective change in the utility of the party leader after the introduction of primaries conditional on mixed-quality indifference comes from the fact that the moderate will be nominated and potentially not even elected for $q_C=(0,0)$ under primaries, while the extremist would be nominated and elected with positive probability without primaries. The introduction of primaries therefore always harms the party leader under mixed-quality indifference.

Under high-quality indifference combined with $-\,M^2\ge \mathcal {I}$ the condition that the utility of the party leader is higher without primaries can be written as

$$\begin{aligned}&\pi ^2 [-\,(M-i_L)^2+w+Y] + (1-\pi )^2 [-\,(E-i_L)^2+Y]\\&\quad >\ \pi ^2 [-\,(E-i_L)^2+w+Y] + (1-\pi )^2 [-\,(M-i_L)^2+Y], \end{aligned}$$

as the identity of the elected politician changes only if $q_C = (0,0)$ and if $q_C = (1,1)$. This simplifies to $\pi <0.5$. If $-\,M^2<\mathcal {I}$, on the other and, the relevant condition is

$$\begin{aligned}&\pi ^2 [-\,(M-i_L)^2+w+Y] + (1-\pi )^2 [-\,(E-i_L)^2+Y]\\&\quad >\ \pi ^2 [-\,(E-i_L)^2+w+Y] + (1-\pi )^2 [-\,(I-i_L)^2+w\cdot q_I], \end{aligned}$$

which can be rewritten as

$$\begin{aligned} \pi ^2 [-(M-i_L)^2&+(Ei_ L)^2]\\&>\ (1-\pi )^2 [-(Ii_ L)^2+w\cdot q_I+(E-i_L)^2-Y].\end{aligned}$$

Under low quality indifference, the utility of the party leader does not decrease under primaries if $-\,M^2\ge \mathcal {I}$, as a moderate of low quality can still get elected. If $-\,M^2< \mathcal {I}$, the condition that the party leader is strictly worse off under primaries is

$$\begin{aligned}&\pi ^2 [-\,(M-i_L)^2+w+Y] + (1-\pi )^2 [-\,(M-i_L)^2+Y]\\&\quad >\ \pi ^2 [-\,(E-i_L)^2+w+Y] + (1-\pi )^2 [-\,(I-i_L)^2+w\cdot q_I], \end{aligned}$$

which can be rewritten as

$$\begin{aligned} \pi ^2 [-(M-i_L)^2&+(Ei_ L)^2]\\&>\ (1-\pi )^2 [-(I-i_L)^2+w \cdot q_I+(M-i_L)^2-Y]\ .\end{aligned}$$

$\square$

Proof of Proposition 7

Under primaries the electability of candidates of party C does not depend on the nomination strategy of the median member as there is no asymmetric information. Fix a combination $(q_M,q_E)$ of candidate qualities. Generically there then exists a unique election strategy for the median voter in the general election as the median voter is indifferent only in knife-edge cases. If neither potential candidate can get elected, the nomination choice is inconsequential and the utility of the party leader conditional on realized qualities is flat in $i_P$. Suppose both potential candidates of party C get elected if nominated. In this case the median member is indifferent between nominating either of them if

$$\begin{aligned} -\,(M-i_P)^2+w\cdot q_M = -\,(E-i_P)^2+w\cdot q_E. \end{aligned}$$

Solving this expression for $i_P$ shows that there exists a unique real number such that the median member strictly prefers the nomination of the moderate (the extremist) if her ideal point lies below (above) this number. The ex-post utility of the party leader is therefore flat in $i_P$ with a single discontinuity at the threshold where the nomination choice of the median member changes. Next, consider the case where only one potential candidate of party C can get elected. Without loss of generality, let this be the moderate. The median member is indifferent between nominating either politician if

$$\begin{aligned} -\,(M-i_P)^2+w\cdot q_M + Y = -\,(I-i_P)^2 +w\cdot q_I. \end{aligned}$$

As in the previous case there exists a unique value of $i_P$ for which equality holds and the ex-post utility of the party leader is again flat in $i_P$ with a single discontinuity. Now consider the ex-ante utility of the party leader. For $i_P=i_L$ the utility functions of the median member and the party leader are identical and the choices of the median member maximise the utility of the party leader. As $i_P$ shifts away from $i_L$, however, the utility of the party leader drops whenever a threshold is reached where the median member changes her nomination choice for a particular quality combination. $\square$

1.2 B Weakly dominated strategies

This appendix derives bounds on the existence of equilibria where the incumbent is re-elected with certainty. When no politician of party C is elected with positive probability the party leader is indifferent between any of her pure strategies. Given the restrictions on equilibrium strategies, whether this case can be an equilibrium crucially depends on which posterior beliefs can be generated by weakly undominated strategies.

Fix an arbitrary nomination strategy $\eta$ and let $m(\eta )$ be the ex-ante probability that the moderate gets nominated under $\eta$. A second strategy $\eta '$ weakly dominates $\eta$ only if $m(\eta )=m(\eta ')$: In the case $m(\eta )>m(\eta ')$ the expected utility of the party leader under $\eta$ would be strictly higher under $\eta$ than under $\eta '$ given that $r(M)=1$ and $r(E)=0$, i.e. the median voter elects the moderate for sure and never elects the extremist. Similarly, if $m(\eta )>m(\eta ')$$\eta$ gives a strictly higher utility for $r(M)=0$ and $r(E)=1$.

Given this first result, the intuition for which strategies are weakly dominated can be given as follows: A strategy $\eta$ is weakly dominated if and only if it is possible to find a second strategy $\eta '$ such that $m(\eta )=m(\eta ')$ and $\eta '$ nominates politician p more frequently when this politician is of high quality and less frequently when this politician is of low quality, relative to $\eta$. The remainder of the proof formalizes this idea.

It is claimed that any nomination strategy that features $\eta _M(0,1)>0$ and $\eta _M(1,1)<1$ is weakly dominated. Construct a second strategy $\eta _M'$ by setting $\eta _M'(1,1)=\eta _M(1,1) + \varepsilon$ and $\eta _M'(0,1) =\eta _M(0,1) -\frac{\pi }{1-\pi } \varepsilon$ with $\varepsilon >0$ and leaving all other nomination probabilities unchanged relative to $\eta _M$. Choosing $\varepsilon$ sufficiently small ensures that all probabilities in the new strategy $\eta _M'$ are well defined. By construction, both politicians ex-ante get nominated with the same probability under $\eta _M$ and $\eta '_M$. The only difference between the two strategies is that for the quality combination (1, 1) the moderate is nominated more frequently under $\eta _M'$ than under $\eta _M$, while for the quality combination (0, 1) the moderate is nominated less frequently. The expected utility of the party leader under the strategy $\eta _M$ can be written as

$$\begin{aligned}&\sum _{q\in Q} Pr[q_C=q]\ \Big \{\eta _M(q) \big [r(M) (-\,(M-i_L)^2+Y+w\cdot q_M) \\&\quad + (1-r(M)) (-\,(I-i_L)^2+w\cdot q_I)\big ]\\&\quad + (1-\eta _M(q)) \big [ r(E) (-\,(E-i_L)^2+Y+w\cdot q_E)\\&\quad + (1-r(E)) (-\,(I-i_L)^2+w\cdot q_I)\big ] \ \Big \}. \end{aligned}$$

Define $U_M \equiv -\,(M-i_L)^2+Y$, $U_E \equiv -\,(E-i_L)^2+Y$, and $U_I \equiv -\,(I-i_L)^2+w\cdot q_I$. The difference in the expected utilities under $\eta _M'$ and $\eta _M$ is

$$\begin{aligned}&\pi ^2\ \varepsilon \ \Big \{ r(M) (U_M+w) + (1-r(M)) U_I \\&\quad - r(E) (U_E+w) - (1-r(E)) U_I \Big \} \\&\quad -\pi (1-\pi )\frac{\pi }{1-\pi }\ \varepsilon \ \Big \{r(M)U_M+(1-r(M)) U_I \\&\quad - r(E) (U_E+w) - (1-r(E)) U_I \Big \}, \end{aligned}$$

which is equal to $\pi ^2\ \varepsilon \ r(M)\ w$ and non-negative for any election strategy r. This shows that $\eta _M'$ weakly dominates $\eta _M$.

By analogous arguments any strategy such that either $\eta _M(0,0)>0$ and $\eta _M(1,0)<1$, $\eta _M(0,0)<1$ and $\eta _M(0,1)>0$, or $\eta _M(1,0)<1$ and $\eta _M(1,1)>0$, is weakly dominated as well. Now consider a strategy such that $\eta _M(1,0)<1$. For this strategy not to be weakly dominated it must be true that $\eta _M(0,0)=0$ and $\eta _M(1,1)=0$ by the second and fourth rule above, which in turn leads to the requirement $\eta _M(0,1)=0$ by the third rule. Any resulting strategy is not weakly dominated, as the construction of a weakly dominating strategy would require reducing the probability of nominating a high quality moderate.

Next, consider a strategy such that $\eta _M(1,0)=1$ and $\eta _M(0,1)>0$. By the first and third rule given above it must hold that $\eta _M(1,1)=1$ and $\eta _M(0,0)=1$ for this strategy to not be weakly dominated. Similar to before, to find a strategy that could weakly dominate this strategy it would be necessary to reduce the probability of nominating a high quality extremist, which would reduce utility against most strategies of the party leader.

Finally, let $\eta _M(1,0)=1$ and $\eta _M(0,1)=0$. None of the conditions above imposes any restrictions on $\eta _M(0,0)$ and $\eta _M(1,1)$. Furthermore, any strategy of this kind is not weakly dominated. Raising the probability of nominating a high quality politician while keeping the ex-ante nomination probabilities constant necessarily implies reducing the probability of nominating the second politician when she is of high quality by an equivalent amount.

To summarize, there are only three different types of nomination strategies that are not weakly dominated:

• $\eta _M(1,0)=1$, $\eta _M(0,1)=0$, $0 \le \eta _M(0,0) \le 1$, $0 \le \eta _M(1,1) \le 1$

• $\eta _M(1,0)=1$, $\eta _M(0,1)>0$, $\eta _M(0,0)=1$, $\eta _M(1,1)=1$

• $\eta _M(1,0)<1$, $\eta _M(0,1)=0$, $\eta _M(0,0)=0$, $\eta _M(1,1)=0$.

The second of these strategies nominates the extremist only if she has high quality and consequently $\bar{\pi }_E=1$ in this case. For the moderate this strategy implies

$$\begin{aligned} \bar{\pi }_M = \frac{\pi }{\pi +\pi (1-\pi )\eta _M(0,1)+(1-\pi )^2}. \end{aligned}$$

This expression achieves its minimum of $\pi$ for $\eta _M(0,1)=1$. The conditions $\mathcal {I}>-\,M^2+\pi$ and $\mathcal {I}>-\,E^2+1$ are therefore jointly sufficient for the existence of an equilibrium where $r(M)=r(E)=0$. Similarly, the third strategy nominates the moderate only if she has high quality and $\bar{\pi }_M=1$ must hold, while the lowest posterior expectation over the quality of the extremist that this strategy can generate is $\pi$ for $\eta _M(1,0)=0$. This implies the joint sufficient conditions $\mathcal {I} > -\,M^2 +1$ and $\mathcal {I}>-\,E^2+\pi$, where the second condition is satisfied whenever the first condition holds.

For the first of the weakly undominated strategies given above the posterior expectations are

$$\begin{aligned} \bar{\pi }_M= \frac{\pi (1-\pi )+\pi ^2\eta _M(1,1)}{\pi (1-\pi )+\pi ^2\eta _M(1,1)+(1-\pi )^2\eta _M(0,0)} \end{aligned}$$

(14)

and

$$\begin{aligned} \bar{\pi }_E =\frac{\pi (1-\pi )+\pi ^2(1-\eta _M(1,1))}{\pi (1-\pi )+\pi ^2(1-\eta _M(1,1)) +(1-\pi )^2 (1-\eta _M(0,0))}. \end{aligned}$$

(15)

This strategy generates $\bar{\pi }_E=1$ if and only if $\eta _M(0,0)=1$ and the lowest value of the posterior expectation $\bar{\pi }_M$ that can be achieved in this case is $\pi$, which implies the same sufficient conditions as the first set of conditions given in the previous paragraph. On the other hand, the lowest value that the right-hand side of Eq. (15) can take is $\pi$. Together with the previous results this shows that no undominated strategy can lead to a posterior expected quality below $\pi$ for any politician. It remains to show which sufficient conditions the current strategy yields if E is such that $-\,E^2+\pi \le \mathcal {I} \le -\,E^2+1$. This requires for any such E to find the lowest M such that the median voter is indifferent between the incumbent and both politicians of party C. This M satisfies $\mathcal {I}=-\,M^2+\bar{\pi }_M^*$, where $\bar{\pi }_M^*$ is the solution to the minimization problem

$$\begin{aligned} \begin{array}{rl} \mathop {\min }\limits _{0 \le x,y \le 1} &{} \frac{\pi (1-\pi )+\pi ^2x}{\pi (1-\pi )+\pi ^2x+(1-\pi )^2y}\\ s.t. &{} -\,E^2+\frac{\pi (1-\pi )+\pi ^2(1-x)}{\pi (1-\pi )+\pi ^2(1-x) +(1-\pi )^2 (1-y)}=\mathcal {I} \end{array} \end{aligned}$$

1.3 C Non-additive quality

This appendix provides a proof for the claim in Sect. 4.1 that the median voter theorem applies if the utility function of voters is given by

$$\begin{aligned} -\,(i-x)^2 + h(|x-i|)\cdot q \end{aligned}$$

as long as the following assumptions are satisfied: the function $h:\mathbb {R}_+ \rightarrow \mathbb {R}$ is decreasing and concave, there exists a positive constant d such that $h(d)\ge 0$, and all voters are located in the interval $[-d,d]$. It needs to be shown that either all voters to the left or all voters to the right of the median voter agree with the preference ordering of the median voter for all possible combinations of candidates. Without loss of generality, assume that the median voter prefers the extremist over the incumbent: For i equal to zero it then holds that

$$\begin{aligned} -\,(E-i)^2+h(|E-i|)\cdot \bar{\pi }_E\ >\ -\,(I-i)^2+h(|I-i|)\cdot q_I. \end{aligned}$$

(16)

Consider voters such that $i \in [0,E]$. As $I<0$ and $E>0$, Inequality (16) must hold for these voters: The right-hand side of the expression is decreasing in i while the left-hand side is increasing on this interval.

Now consider voters located in the interval (E, d] in the case where $d>E$. These voters clearly prefer the extremist over the incumbent on ideological grounds. As $h(|E-i|)\ge 0$ for any of these voters, the only way that they could prefer the incumbent over the extremist was if the quality $q_I$ of the incumbent was larger than the expected quality $\bar{\pi }_E$ of the extremist. But this, together with the result shown above that a voter located at $i=E$ must prefer the extremist over the incumbent, implies that all voters in the interval (E, d] must prefer the extremist as well. To see this note that it follows from h being concave and decreasing, $q_I>\bar{\pi }_E$, and $I<E$ that the function $h(|I-i|)\cdot q_I$ decreases at least as fast as the function $h(|E-i|)\cdot \bar{\pi }_E$ in i on the interval (E, d]. It is then clear that Inequality (16) holds for all $i \in (E,I]$.

1.4 D Uncertainty over candidate locations

This appendix considers a generalised version of the basic model where the ideal policies of the moderate and the extremist are drawn from distributions $F_M$ and $F_E$, respectively, and only observed by the party leader, but not voters. To keep things reasonably simple, the disutility from policy will now be given by the absolute value, rather than the square, of the difference between policy and ideal position of an agent. Furthermore, assume that $i_L=1$ and that the party leader expects that the moderate would get elected with certainty while the extremist would get elected with probability r(E), as in the Full Competition case above. The decision rule of the party leader is then to nominate the moderate if and only if

$$\begin{aligned}&-|M-1|+ w\cdot q_M+Y\ \ge \ r(E)[-|E-1|+ w\cdot q_E+Y]\ \\&\quad + (1-r(E)) [-|I-1|+w\cdot q_I] \end{aligned}$$

or equivalently

$$\begin{aligned} M-r(E)E\ \ge&\ r(E)[w\cdot q_E+Y]\ +\ (1-r(E)) (I+w\cdot q_I) - w\cdot q_M -Y\\ \equiv&\ K(q_C). \end{aligned}$$

This choice rule implies that under different quality combinations politicians will be nominated with different probabilities and the nomination choice can therefore still be a signal of quality. The expected quality of a moderate nominated according to this rule is

$$\begin{aligned} \bar{\pi }_M=\frac{\sum _{q\in \{0,1\}} \pi \ Pr[q_E=q]\ Pr[M-r(E)E\ge K(q_C)|q_C=(q,1)]}{\sum _{q\in Q} Pr[q_C=q]\ Pr[M-r(E)E\ge K(q_C)|q_C=q]}, \end{aligned}$$

which is simply the probability that the moderate gets nominated conditional on being of high quality divided by the unconditional nomination probability. One way to find an expression for $Pr[M-r(E)E\ge K(q_C)]$ is to first derive the density of the random variable $M-r(E)E$ at some point t. This is given by

$$\begin{aligned} \int _{\text {supp}(F_E)} f_E(e) f_M(t+r(E)e)\ de. \end{aligned}$$

Appropriately integrating over this density one obtains the desired probability. The expression for the posterior quality of the extremist can be derived analogously.

Beyond quality the nomination choice can now also be a signal of the policy position of a candidate. Considering the decision rule of the party leader, one observation is immediate: If all possible candidates are closer to the median than the party leader, then it is impossible that the expectation of the posterior distribution of the policy position of a nominated politician is below the expectation of the prior distribution. If the party leader prefers to nominate the moderate for a given M then she must ceteris paribus prefer to nominate the moderate for any higher M as well, implying that the posterior distribution first order stochastically dominates the prior distribution. The same holds for the extremist. Therefore, if a nomination tells voters anything about the policies a candidate stands for then that these are more extreme than previously thought. In other words, politically extreme parties are bad for the median voter in terms of the political views of the candidates they select.

To find an expression for the expected policy position of a moderate nominated according to the decision rule above, first note that according to Bayes’ rule the posterior probability density over M conditional on a certain quality combination q is given by

$$\begin{aligned} f_{M|q}(m) \equiv f_M(m)\frac{Pr[M-r(E)E\ge K(q_C)|M=m,q_C=q]}{Pr[M-r(E)E\ge K(q_C)|q_C=q]} \end{aligned}$$

with

$$\begin{aligned} Pr[M-r(E)E\ge K(q_C)|M=m,q_C=q]=F_E\left( [m-K(q_C)]/ r(E)\right) . \end{aligned}$$

The unconditional expected policy position of a nominated moderate is then given by the weighted sum of the conditional expectations:

$$\begin{aligned} \frac{\sum _{q\in Q} Pr[q_C=q]\ Pr[M-r(E)E\ge K(q_C)|q_C=q] \int _{\text {supp}(F_M)}\ m\ f_{M|q}(m)\ dm}{\sum _{q\in Q} Pr[q_C=q]\ Pr[M-r(E)E\ge K(q_C)|q_C=q]}. \end{aligned}$$

Again, the expected policy position of the extremist follows analogously.

Giving a general description of equilibrium is beyond the scope of this paper. Instead, a specific example will be given to illustrate that the characteristics of the Full Competition equilibrium emphasized above remain unchanged in the extended model. It is assumed that both M and E are uniformly distributed with support [0.2, 0.5] and [0.4, 0.7], respectively, while incumbent is located at -0.8 and has high quality. Note that the moderate is expected to be closer to the median than the extremist, but the opposite might be the case in actuality. In addition, $\pi =w=0.5$, $Y=1$ and $i_L=1$ will be used.

Figure 3 plots the expected utility of the median voter from electing either politician of party C, which can be calculated using the expressions above, as a function of the probability r(E) that the extremist will get elected. The dashed line represents the utility that the median voter receives in case the incumbent is re-elected.

Fig. 3

Expected utilities with uncertain policy positions

For low values of r(E) the party leader always selects the moderate and both expected utilities are flat in this region.¹⁶ As r(E) increases the party leader finds it worthwhile to nominate the extremist for high values of E in the case where the extremist has high quality and the moderate has low quality, and eventually also for lower values of E. This makes the extremist less extreme in expectation and explains the initial increase in the expected utility from electing her. For even higher values of r(E) the extremist gets nominated under other quality combination as well, which lowers her expected quality and results in a decrease in utility for the median voter. The increase in the expected utility from electing the moderate, on the other hand, stems from the fact that her expected quality increases as it becomes more attractive to nominate the extremist.

The figure shows that there is a unique election probability of the extremist such that the median voter is indifferent between the extremist and the incumbent while strictly preferring the moderate. This point in the graph thus represents an equilibrium—an equilibrium that is equivalent to the case of Full Competition described above.