Strategic candidacy for political compromise in party politics

Abstract

We study political competition among parties with policy-motivated candidates and demonstrate two-party competition where one of them has no chance of winning in the general election. If there are multiple candidates in a party, then the party can nominate the representative contingent on who are the rivals in the general election. Given that the winning party takes such a strategy, the opponent party is willing to run even with no chance of winning the general election since policy compromises can be induced from the winning party. We further demonstrate that introducing primary elections into the nomination process may induce more ideological polarization or more moderation.

Keywords

citizen candidates party competition political compromise primary elections strategic candidacy

1. Introduction

Empirical and anecdotal evidence reports that a unique opposed candidate often runs for election without a chance of winning. For instance, an established incumbent with a dominant advantage often faces a fresh and weak challenger nominated from the opposing party in a wide range of elections in the United States, for a various offices such as state legislature (Jewell and Breaux, 1988, 1991) and House (Brady et al., 2007). Another example is observed in authoritarian regimes, where an election is often held and an elite sometimes stands uniquely as opposed to the incumbent ruling party that usually wins.¹ In addition to decisions made by electoral candidates, there have been various countries in which there was a unique effective party that has not won general elections against the ruling party.² All of these minor candidates should have strategic motives to uniquely run against the dominant rival.

The citizen-candidate approach, initiated by Osborne and Slivinski (1996) and Besley and Coate (1997), embeds endogenous entry decisions in electoral competition and allows us to investigate strategic candidacy, by which hereafter we mean running for election without any chance of winning. In both models, the winner implements her unique most preferred policy in equilibrium so that the implemented policy may be changed only if the winner in the election is changed. Provided that entering the election is costly, a potential candidate attempts strategic candidacy only if entry into the election can alter the winner.³ It in turn implies that if there exists a candidate attempting strategic candidacy, then there must be at least two other rivals in the election. Thus, despite the evidence mentioned above, strategic candidacy can never happen in two-candidate competition.

The purpose of this article is to propose a political mechanism that supports strategic candidacy in two-candidate competition. A key mechanism in our model is that candidacy behaviour induces political compromise from the winning rival caused through nomination of the party representative. Even in two-party competition, the party who is expected to lose for certain may be willing to run for election since it induces the winning party to implement a policy more preferred to the losing party. We consider that the electoral competition consists of two selection processes after deciding to stand for the general election: nomination of the representative from each party and competition among the representatives from the parties in the general election. As the nomination of the representative depends on whether the rival parties stand for the general election, entry to the general election may alter the representative nominated from the rival party. Consequently, the implemented policy may also be altered by the losing party’s entry.

To illustrate the idea, consider the following example.⁴ There are two parties, a left party expected to win and a right party expected to lose. Furthermore, suppose that the left party has two party members, a partisan leader and a moderate candidate, while the right party consists of one member. If each leader chooses to stand for election or not, then the leader of the left party may nominate the representative contingent on whether the rival party also stands for election or not. Specifically, suppose that the left leader nominates the moderate candidate when and only when the right party also stands for election. Given that the left party takes such a strategy, the right party’s entry alters the left party’s representative from the partisan leader to the moderate candidate. In other words, the right party’s entry can induce a compromised policy from the left party. It motivates the right party to stand for election even when it has no chance of winning and no possibility of changing the identity of the winning party.

This intuition is captured by a necessary condition for the existence of equilibria on which strategic candidacy emerges in two-party competition. The existence condition (Proposition 2) states that in order to support strategic candidacy in two-party competition, (i) the loser’s entry must induce a political compromise the benefit of which outweighs the cost of entry, and (ii) the policy implemented by the winner must be supported by a strict majority of voters. In the aforementioned example, condition (i) motivates the right party to enter since the entry can induce a better outcome than exit, and condition (ii) guarantees that the moderate candidate in the left party is supported by a strict majority of the voters in the general election and then the left party can certainly win. In order to guarantee condition (i), the leader in the left party should take a nomination strategy contingent on the right party’s entry. Then, we additionally obtain the necessary condition (Corollary 1) that the winning party must have multiple members with different ideal policies.⁵

As an application of our framework, we discuss the impact of primary elections on the political consequence. Scholars in political science have developed the discussion that primary elections lead to partisan polarization especially in U.S. politics. Nevertheless, recent empirical articles (Hirano et al., 2010; McGhee et al., 2013; Rogowski and Langella, 2014: e.g.) report that there is little evidence that the introduction of primary elections is systematically associated with partisan polarization. Based on these discussions, we consider a variant of our model where, compared with the baseline model in which the representative is nominated dictatorially by the party’s leader, the representative is determined by the primary election within the party. As consistent with the above empirical studies suggesting little systematic evidence between primary elections and partisan polarization, we demonstrate that when the primary election is introduced, the representative in the winning party may be more partisan or less partisan than when the representative is nominated dictatorially. We also identify two key factors that make the political outcome either moderate or partisan: the distribution of the party members’ ideological positions and whether the voters in the primary elections can coordinate their vote for the nomination.

The main objective of the present paper is to investigate when strategic candidacy emerges. In the canonical citizen-candidate approach such as in Osborne and Slivinski (1996) and Besley and Coate (1997), strategic candidacy never emerges if there are only two candidates. Our model demonstrates that strategic candidacy may emerge even in two-candidate competition by incorporating party politics and internal selection within the parties. The following articles demonstrate a similar result to ours in a different setup. Hummel (2013) considered a similar two-step election, primaries and general elections, and showed that, different from our result, two candidates within a party run and one of them loses certainly in the primary election. The reason for the losing candidate entry is somewhat similar in that the entry leads to a preferable policy for the loser. Nevertheless, the result relies on two important assumptions that each entrant can commit to his/her policy before voting in the primary election and the potential candidates are vertically differentiated in terms of their valence, neither of which is assumed in our model. Asako (2015) studied a citizen-candidate model with ‘betrayal cost’ that is increasing in the distance between the promise made before vote and the actual implementing policy. While he pointed out the possibility of strategic candidacy in two-candidate competition with betrayal cost, our argument is based on different factors, that is, multiple party members and representative nomination contingent on the rival parties. It should also be noted that our result and intuition on the condition for emergence of strategic candidacy in two-party competition do not depend on the assumption of one-dimensional spatial competition, which is assumed both in Hummel (2013) and Asako (2015). Ishihara and Miura (2017) considered a variant of political competition, where a finite number of potential candidates make a decision to enter the election or not sequentially. They pointed out that the first mover’s commitment to enter may crowd out entry by a less-preferred citizen and then induce strategic candidacy in two-candidate competition. Nevertheless, their logic behind the result is definitely different from ours.

Our model is also related to the literature that investigates the role of parties in political competition. An important aspect of party politics is that the winning representative is selected through two-step processes: nomination within each party, followed by the general election. Cadigan and Janeba (2002), Jackson et al. (2007), Crutzen et al. (2010), Snyder and Ting (2011), and Hortala-Vallve and Mueller (2015) considered such two-step political selection processes, where, like in our model, candidates cannot commit to a policy platform before voting.⁶ Our contribution to this literature is that we demonstrate a novel phenomenon in party competition that a party may strategically choose to be active in the general election even in two-party competition since strategic candidacy may induce political compromise through the rival party’s choice of representative.⁷

The rest of this paper is organized as follows. Section 2 describes our baseline model. Section 3 investigates strategic candidacy emerging on a two-party equilibrium in a simple example and characterizes a general condition for the existence of such equilibria. Section 4 introduces primary elections into the nomination process of representatives and discusses the impact of the primary elections. Section 5 provides several remarks and conclusions. The Appendix contains the proofs of the formal results.

2. The model

We formulate party politics as a two-step electoral process, the internal selection within parties, followed by the general election. The internal selection decides the representative of the party for the general election. The representative of the winning party in the general election implements a policy.

Let $I \equiv {1, 2, \dots, I}$ be the set of (finite) potential politicians (called agents), and $P \equiv {p^{1}, p^{2}, \dots, p^{N}}$ be a partition of $I$ , where $p^{n} \subset I$ for $n = 1, \dots, I$ , $p^{n} \cap p^{m} = \emptyset$ for $n \neq m$ , and $⋃_{n = 1}^{N} p^{n} = I$ . An element $p^{n} \in P$ is a party, implying that there are N parties and each agent belongs to a party. For notational convenience, given a subset of parties $C \subset P$ , denote the set of the party indexes by $N (C)$ .⁸ Each party $p^{n}$ has a unique agent $λ^{n} \in p^{n}$ , called the leader of party $p^{k}$ . Let $Λ \equiv {λ^{1}, \dots, λ^{N}}$ be the set of the leaders. The timing of the model is as follows.

(i) (Entry stage): Each leader $λ^{n} \in Λ$ chooses whether to stand for the general election.

(ii) (Nomination stage): Each party that announced to stand for the general election nominates the representative from the agents of the party.

(iii) (Voting stage): The voters in the general election vote for a party and the party with the greatest vote share becomes the winning party.

(iv) (Policy stage): The representative of the winning party in the general election chooses an implementing policy from the compact set $A$ .

Given that policy $a \in A$ is implemented, agent i obtains political benefit $v^{i} (a) \in R$ . If a party stands for the general election in Stage 1, then the leader in the party must incur a utility cost $d > 0$ . Accordingly, each agent’s net payoff is the sum of the expected political benefit and the disutility from standing for election. More specifically, when policy $a \in A$ is implemented, the ex post payoff for agent i is given by $v^{i} (a) - d$ if she is the leader in a party and decides to stand for the general election and $v^{i} (a)$ otherwise. Throughout the analysis, we assume $# \underset{a}{\arg \max} v^{i} (a) = 1$ for all $i \in I$ , meaning that each agent has a unique ideal policy, hereafter denoted by ${\bar{a}}^{i} \in \underset{a}{\arg \max} v^{i} (a)$ . The backward induction immediately implies that if agent i is the representative of the winning party, then she chooses ${\bar{a}}^{i}$ in Stage 4.

There are several assumptions on decision making as follows. First, if no party stands for the general election after Stage 1, then the status quo policy $a^{ϕ} \in A$ is immediately implemented and the game ends. Second, if there is a unique party standing for the general election, then the party wins for sure. Third, in Stage 2, the representative of the party is nominated by the leader’s choice. The third assumption is relaxed in Section 4.

If agent $i \in p^{n}$ is not the leader of party $p^{n}$ , then she makes a decision only when she becomes the representative of party $p^{n}$ and her party wins in the general election, in which she chooses policy ${\bar{a}}^{i}$ . Given the representative’s policy choice, the winner in the general election is determined as follows. Let $C \subset P$ be the set of the parties standing for the general election, $r^{n} \in p^{n}$ be the representative of party $p^{n} \in C$ , and $r = (r^{n})_{n \in N (C)}$ be the profile. We define the probability function $Δ (p^{n} | r, C) \in [0, 1]$ , meaning that party $p^{n}$ (with its representative $r^{n}$ ) wins the general election with probability $Δ (p^{n} | r, C)$ .⁹ Consistent with the previous assumption, we assume $Δ (p^{n} | r, {p^{n}}) = 1$ .

The model is interpreted as a game played by the leaders in $Λ$ . To define strategies, let $P^{n} \equiv {C^{'} \subset P | p^{n} \in C^{'}}$ be the collection of non-empty subsets of P including party $p^{n}$ . Leader $λ^{n}$ ’s strategy specifies (i) $γ^{n} \in {0, 1}$ , meaning whether party $p^{n}$ enters the general election or not; and (ii) $ω^{n} \equiv (ω_{C}^{n})_{C \in P^{n}}$ such that $ω_{C}^{n} \in p^{n}$ , meaning that $ω_{C}^{n}$ is the representative chosen in the nomination process given that party $p^{n}$ entered and the set of the parties that entered is C.¹⁰ Given the entry decisions $(γ^{n})_{n \in N (P)}$ , denote the set of the parties standing for the general election by $Γ \equiv {p^{n} | γ^{n} = 1}$ . We look at subgame perfect equilibria of the game.

3. Strategic candidacy in two-party competition

3.1. Definition

Our interest is whether there exists a subgame perfect equilibrium such that two parties run for the general election and one of them wins with probability one. Specifically, this phenomenon is defined as follows.

Definition 1. A strategy profile $(γ^{n}, ω^{n})_{n \in N (P)}$ causes $(p^{k}, p^{l})$ strategic candidacy (hereafter ‘ $(p^{k}, p^{l})$ -SC’) if $γ^{n} = 1$ for $n = k, l$ , $γ^{n} = 0$ for $n \neq k, l$ and $Δ (p^{k} | (ω_{Γ}^{k}, ω_{Γ}^{l}), Γ) = 1$ .

3.2. No internal selection

The canonical citizen-candidate models of Osborne and Slivinski (1996) and Besley and Coate (1997) are also described in our model by assuming that every party has a unique agent.

Assumption 1. We assume $p^{n} = {n}$ for all $n \in N (P)$ .

In the canonical citizen-candidate models, an entry decision is made by each individual candidate, who implements the policy after winning. In other words, the players who makes an entry decision must coincide with the representative who implements the policy after winning the election. Under Assumption 1, as the leader is the unique member in each party, there is no variation in the internal selection and accordingly the leader of the party must be the representative.

Under Assumption 1, there is never an equilibrium in which there are two parties to stand for the general election and one of them has no chance of winning.

Proposition 1. Suppose that Assumption 1 holds. Then, for any $p^{k}, p^{l} \in P$ , no strategy profile that causes $(p^{k}, p^{l})$ -SC is a subgame perfect equilibrium.

Provided that party $p^{k}$ has a unique possible representative, say agent i, if party $p^{k}$ wins, then agent i’s ideal policy ${\bar{a}}^{i}$ is implemented whether other parties stand for the general election or not. Hence, if parties $p^{k}$ and $p^{l}$ are the candidates and party $p^{l}$ has no chance of winning in the general election, then the implemented policy is ${\bar{a}}^{i}$ as well. Since party $p^{l}$ ’s entry decision does not affect the implemented policy, party $p^{l}$ is willing to quit to save the entry cost d. In other words, whenever there are exactly two parties in the general election and each of the parties has only a single member, both of the parties must have positive probability of winning and then there is no possibility of strategic candidacy.¹¹

The key idea behind this result is that the winning party implements the unique policy whatever happened in the past. It is due to the assumption that the party has no option to choose another representative and then there is no other policy that is credibly implemented after the voting stage. We next see that strategic candidacy in our setup may emerge if the winning party has two or more members.

3.3. An example

We first look at the following illustrative example with three agents: $I = 3$ , $A = [- 1 / 2, 1 / 2]$ and $v^{i} (a) = - | a - {\bar{a}}^{i} |$ , where ${\bar{a}}^{1} = - 1 / 3$ , ${\bar{a}}^{2} = - 1 / 6$ , ${\bar{a}}^{3} = 1 / 4$ . Consider a two-party system $P = {p^{1}, p^{2}}$ , where $p^{1} = {1, 2}$ , $p^{2} = {3}$ and $λ^{1} = 1$ , which means that agents 1 and 2 form a party and agent 1 makes a decision whether party $p^{1}$ stands for the general election or not and who the representative of the party is.

Assume that there is a median voter whose ideal position is 0 so that the winning probability in the general election with two parties is specified as

Δ (p^{1} | (r^{1}, r^{2}), {p^{1}, p^{2}}) = {\begin{matrix} 1 & if | a^{r^{1}} | < | a^{r^{2}} | \\ 0 & if | a^{r^{1}} | > | a^{r^{2}} | \end{matrix}

Figure 1 illustrates the policy space and the ideal positions.

Figure 1.

The spatial example with three agents.

Consider the following decision in the nomination stage:

ω_{C}^{1} = {\begin{matrix} 2 & if C = P \\ 1 & otherwise \end{matrix}

and $ω_{C}^{2} = 2$ for any $C \in P^{2}$ . Given that both parties decided to stand for the general election, when agent 1 is the representative of $p^{1}$ , she is beaten by agent 3 from party $p^{2}$ in the general election, and policy $a = {\bar{a}}^{3} = 1 / 4$ is implemented consequently. On the other hand, if agent 2 is the representative of $p^{1}$ , then she wins against agent 3 in the general election and implements policy ${\bar{a}}^{2} = - 1 / 6$ . Although ${\bar{a}}^{2}$ is not the ideal policy for the leader in party $p^{1}$ , it is possible to confirm that the leader strictly prefers to nominate agent 2 as the representative (i.e. $ω_{P}^{1} = 2$ ) since ${\bar{a}}^{2}$ is closer to the leader’s ideological position ${\bar{a}}^{1}$ than ${\bar{a}}^{3}$ .

Given $(ω^{1}, ω^{2})$ specified previously, for sufficiently small d, there exists an equilibrium such that both parties $p^{1}$ and $p^{2}$ stand for the general election in Stage 1 and party $p^{2}$ certainly loses. If agent 3 stands for the general election, then agent 2 nominated from party $p^{1}$ is elected and implements $- 1 / 6$ with probability one, which yields the payoff $- | - 1 / 6 - 1 / 4 | - d = - 5 / 12 - d$ for agent 3. If agent 3 withdraws instead, then, since there is no rival party in the general election, the leader in $p^{1}$ prefers to choose herself as the representative and to implement policy ${\bar{a}}^{1} = - 1 / 3$ consequently, which yields the payoff $- | - 1 / 3 - 1 / 4 | = - 7 / 12$ for agent 3. The comparison between these implies that agent 3 enters if $- 5 / 12 - d \geq - 7 / 12$ or, equivalently, $d \leq 1 / 6$ .

Intuitively, $p^{2}$ chooses to stand for the general election even with no chance of winning in order to induce political compromise from the rival party $p^{1}$ through changing the representative of $p^{1}$ from agent 1 to 2. The condition $d \leq 1 / 6$ expresses that the benefit of the policy compromise dominates the cost of entry for party 2. It is important to note that the nominated representative in party $p^{1}$ is opponent-dependent. In order for the nomination process to be opponent-dependent, it is necessary for the winning party to hold multiple members. As seen in Section 3.2, under the assumption that each party has only a single member, the political compromise is not induced and then strategic candidacy never emerges in two-candidate competition.

3.4. A necessary and sufficient condition

The insight in the above example with three agents can be characterized by the conditions for the existence of equilibria causing strategic candidacy in generalized environments.

Proposition 2. If there exists a pure strategy subgame perfect equilibrium causing $(p^{k}, p^{l})$ -SC, where $λ^{n}$ nominates $r^{n} \in p^{n}$ for each $n = k, l$ after both parties stand for the general election, then the following conditions are satisfied:

v^{λ^{k}} ({\bar{a}}^{r^{k}}) - d \geq v^{λ^{k}} ({\bar{a}}^{λ^{l}})

(1)

v^{λ^{l}} ({\bar{a}}^{r^{k}}) - d \geq v^{λ^{l}} ({\bar{a}}^{λ^{k}})

(2)

Δ (p^{k} | (r^{k}, r^{l}), {p^{k}, p^{l}}) = 1

(3)

r^{k} \in \underset{j \in p^{k}}{\arg \max} {Δ (p^{k} | (j, r^{l}), {p^{k}, p^{l}}) v^{λ^{k}} ({\bar{a}}^{j}) + [1 - Δ (p^{k} | (j, r^{l}), {p^{k}, p^{l}})] v^{λ^{k}} ({\bar{a}}^{r^{l}})}

(4)

r^{l} \in \underset{j \in p^{l}}{\arg \max} {Δ (p^{k} | (r^{k}, j), {p^{k}, p^{l}}) v^{λ^{l}} ({\bar{a}}^{r^{k}}) + [1 - Δ (p^{k} | (r^{k}, j), {p^{k}, p^{l}})] v^{λ^{l}} ({\bar{a}}^{j})}

(5)

Conversely, given $N = 2$ , if (1), (2), (3), (4) and (5) are satisfied, then there exists a pure strategy subgame perfect equilibrium causing $(p^{k}, p^{l})$ -SC, where $λ^{n}$ nominates $r^{n} \in p^{n}$ for each $n = k, l$ after both parties stand for the general election.

Suppose that there exists an equilibrium that causes $(p^{k}, p^{l})$ -SC and nominates $r^{k}$ and $r^{l}$ in case of the two-party competition between $p^{k}$ and $p^{l}$ . If condition (3) is not satisfied, then it is obvious to see that, given the pair of the representatives $(r^{k}, r^{l})$ , party $p^{k}$ does not certainly win, implying that party $p^{l}$ ’s entry is not purely strategic. Since the leader in each party is willing to become the representative given that there is no rival party in the general election, conditions (1) and (2) state that given agents $r^{k}$ and $r^{l}$ become the representatives, the leaders in both parties are willing to decide to stand for the general election. For causing strategic candidacy in two-party competition, there are additional conditions due to strategic interaction in the nomination stage: given that both parties stand for the general election, the best representative for each party in general depends on the representative from the other party. Conditions (4) and (5) are required in order to guarantee the incentive to nominate agent $r^{n}$ in party $p^{n}$ for $n = k, l$ in the nomination stage. Besides the above conditions, we need to additionally consider whether the parties other than $p^{k}$ and $p^{l}$ have incentive to enter the general election. Nevertheless, when $N = 2$ , since there is no other party, under the conditions from (1) to (5), we can construct an equilibrium $(γ^{n}, ω^{n})_{n \in Λ}$ causing $(p^{k}, p^{l})$ -SC such that $ω_{P}^{n} = r^{n}$ for $n = k, l$ .

When $(p^{k}, p^{l})$ -SC is caused, although the probability of party $p^{l}$ ’s winning is zero, the party has an incentive to stand for the general election since the entry alters the implemented policy from ${\bar{a}}^{λ^{k}}$ to ${\bar{a}}^{r^{k}}$ . In particular, condition (2) in Proposition 2 implies that ${\bar{a}}^{r^{k}}$ must be strictly preferred to ${\bar{a}}^{λ^{k}}$ by the leader in party $p^{l}$ . This obviously implies that, consistent with Proposition 1, party $p^{k}$ must have multiple members: otherwise, party $p^{k}$ cannot credibly implement two different policies ${\bar{a}}^{r^{k}}$ and ${\bar{a}}^{λ^{k}}$ .

Corollary 1. If there exists a subgame perfect equilibrium causing $(p^{k}, p^{l})$ -SC, then $# p^{k} \geq 2$ .

3.5. Spatial environments

In order to obtain intuitive insights on conditions (4) and (5) in Proposition 2, we now consider spatial environments that satisfy the following properties.

Assumption 2. (Two-party spatial environments) We make the following assumptions:

(i) $P = (p^{1}, p^{2})$ , where $p^{1} = {1, \dots, \hat{I}}$ for some $\hat{I} \in {1, \dots, I - 1}$ ;

(ii) $A = [- 1 / 2, 1 / 2]$ ;

(iii) For all $i \in I$ , $v^{i} (a)$ is decreasing in $| a - {\bar{a}}^{i} |$ for some ${\bar{a}}^{i} \in [- 1 / 2, 1 / 2]$ ;

(viii) ${\bar{a}}^{i} < {\bar{a}}^{i + 1}$ for $i = 1, \dots, I - 1$ ;

(v) $| {\bar{a}}^{i} | \neq | {\bar{a}}^{j} |$ for $i, j = 1, \dots, I$ with $i \neq j$ ;

(vi) for each $r \equiv (r^{k}, r^{l}) \in p^{k} \times p^{l}$ , $Δ (p^{k} | r, C) = 1$ if $| {\bar{a}}^{r^{k}} | < | {\bar{a}}^{r^{l}} |$ .

Under Assumption 2, as in the standard spatial model, each agent i is willing to implement the policy close to his ideal policy ${\bar{a}}^{i}$ . Assumption 2 implicitly presumes that the voters of the general election are uniformly distributed over $A$ , and each voter sincerely votes for the most preferred candidate. Accordingly, the winning probability $Δ$ is equivalent to that there is a median voter, who has the ideal policy $a = 0$ and chooses the winning party. Assumption 2 also imposes a reasonable two-party structure under which each party consists of members with similar ideologies, typically interpreted as the left-wing and right-wing parties.¹²

If Assumption 2 is satisfied, then a necessary and sufficient condition for the existence of equilibria causing strategic candidacy is obtained in a more intuitive way.

Proposition 3. Suppose Assumption 2 holds.

(i) There exists a pure strategy subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC, where $λ^{1}$ nominates $r^{1} \in p^{1} \ {1}$ after both parties stand for the general election, if and only if for $k = 1$ and $l = 2$ , (1), (2), and

| {\bar{a}}^{r^{1}} | < | {\bar{a}}^{\hat{I} + 1} | < | {\bar{a}}^{r^{1} - 1} |

(6)

are satisfied.

(ii) Suppose furthermore $d < v^{λ^{n}} ({\bar{a}}^{i}) - v^{λ^{n}} ({\bar{a}}^{λ^{m}})$ for each $i \in p^{n}$ , $n = 1, 2$ , and $m \neq n$ . Then, there exists a pure strategy subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC if and only if

| {\bar{a}}^{λ^{1}} | > min_{i \in p^{2}} | {\bar{a}}^{i} | > min_{i \in p^{1}} | {\bar{a}}^{i} |

(7)

(iii) There does not exist a pure strategy subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC, where $λ^{1}$ nominates agent 1 after both parties stand for the general election.

(iv) Suppose that there exists a pure strategy subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC in which the representative in $p^{1}$ is $r^{1} \in p^{1}$ .

(a) There is no pure strategy subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC in which the representative in $p^{1}$ is other than $r^{1}$ .

(b) There is no pure strategy subgame perfect equilibrium causing $(p^{2}, p^{1})$ -SC.

The condition in Proposition 3(i) does not specify $r^{2}$ , who is expected to be the representative nominated from the losing party $p^{2}$ . Under Assumption 2, if a party stands for the general election without any chance of winning, then the leader is willing to nominate the agent whose ideal policy is the closest to the median. The reason is somewhat similar to the well-known median voter theorem established in the standard Downsian environment in that nomination of the agent closest to the median by the losing party induces the winning party to nominate the agent who is also close to the median.

To illustrate the idea in Proposition 3(i) intuitively, consider another example, in which, as in Figure 2, there are four agents, $({\bar{a}}^{1}, {\bar{a}}^{2}, {\bar{a}}^{3}, {\bar{a}}^{4}) = (- 1 / 4, - 1 / 6, 1 / 5, 1 / 3)$ , $p^{1} = {1, 2}$ , and $Λ = (1, 4)$ . This environment satisfies the conditions in Proposition 3 for $r^{1} = 2$ , and has an equilibrium that causes $(p^{1}, p^{2})$ -SC, where each leader nominates another member in the party, agents 2 and 3, respectively, as each representative. Different from the previous example with three agents, each leader has two alternatives to nominate the representative. Agent 4 in the losing party is willing to nominate agent 3 since agent 3’s ideal policy is closer to the median than ${\bar{a}}^{4}$ is and then $p^{2}$ ’s winning probability in the general election increases. Given party $p^{2}$ ’s entry, agent 1 in the winning party rationally expects agent 4’s attempt and then nominates agent 2, whose ideal policy is also close to the median, to win the general election against party $p^{2}$ with agent 3 as the representative. In other words, in the spatial environments under Assumption 2, both parties tend to nominate agents whose ideal policy is close to the median, and more specifically, the losing party must nominate an agent whose ideal policy is the closest to the median in the party. Note that strategic candidacy emerges only if the benefit of the policy compromise dominates the entry cost for the losing party. Proposition 3(ii) implies that given that the entry cost is sufficiently small, the condition for strategic candidacy is simply expressed only by the position of the ideal policies.

Figure 2.

The spatial example with four agents.

4. Nomination via primary elections

The baseline model makes a simplifying assumption that the leader dictatorially chooses the representative in the nomination process. The internal selection of the representative is often more democratic through primary elections. The literature on primary elections in party politics is often interested in policy impacts of primary elections, in particular, whether the policy is more moderated by introducing primary elections. To keep within the scope of this paper, we investigate the impacts of primary elections on policy moderation by focusing on the situations where strategic candidacy in two-party competition emerges.

We now modify the model to incorporate primary elections into the nomination process. Specifically, instead of assuming the leaders’ dictatorial nomination of the representative, we consider an arbitrary choice rule in the nomination stage by defining functions that indicate the outcome of the internal selection. For $C \in P^{n}$ , let $Δ^{n} (r^{n} | C)$ be the probability that agent $r^{n} \in p^{n}$ is chosen as the representative in the primary election given the set of the parties for general election C. We assume that functions $Δ^{n} (r^{n} | C)$ are exogenously given.

The modified model, hereafter called the party competition with primary elections, is interpreted as a game played by the leaders in $Λ$ , where each leader chooses only the entry decision $γ^{n} \in {1, 0}$ . Given that parties $p^{k}$ and $p^{l}$ enter, the probability that party $p^{k}$ wins in the general election is expressed as $\sum_{r^{k} \in p^{k}} \sum_{r^{l} \in p^{l}} Δ^{k} (r^{k} | {p^{k}, p^{l}}) Δ^{l} (r^{l} | {p^{k}, p^{l}}) Δ (p^{k} | (r^{k}, r^{l}), {p^{k}, p^{l}})$ . Accordingly, the definition of strategic candidacy is modified as follows.

Definition 2. In the party competition with primary elections, a strategy profile $(γ^{n})_{n \in N (P)}$ causes $(p^{k}, p^{l})$ strategic candidacy if $γ^{n} = 1$ for $n = k, l$ , $γ^{n} = 0$ for $n \neq k, l$ , and $\sum_{r^{k} \in p^{k}} \sum_{r^{l} \in p^{l}} Δ^{k} (r^{k} | Γ) Δ^{l} (r^{l} | Γ) Δ (p^{k} | (r^{k}, r^{l}), Γ) = 1$ .

As in Proposition 2, under a two-party system (i.e. $N = 2$ ), a necessary and sufficient condition for the existence of equilibria causing strategic candidacy is simply stated.

Proposition 4. Suppose $N = 2$ . In the party competition with primary elections, there exists a pure strategy subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC if and only if

\sum_{r^{1} \in p^{1}} Δ^{1} (r^{1} | P) v^{λ^{1}} ({\bar{a}}^{r^{1}}) - d \geq \sum_{r^{2} \in p^{2}} Δ^{2} (r^{2} | {p^{2}}) v^{λ^{1}} ({\bar{a}}^{r^{2}})

(8)

\sum_{r^{1} \in p^{1}} Δ^{2} (r^{1} | P) v^{λ^{2}} ({\bar{a}}^{r^{1}}) - d \geq \sum_{r^{1} \in p^{1}} Δ^{1} (r^{1} | {p^{1}}) v^{λ^{2}} ({\bar{a}}^{r^{1}})

(9)

\sum_{r^{k} \in p^{k}} \sum_{r^{l} \in p^{l}} Δ^{k} (r^{k} | P) Δ^{l} (r^{l} | P) Δ (p^{k} | (r^{k}, r^{l}), P) = 1

(10)

In contrast to Proposition 2, as the nominated representative is exogenously determined by $Δ^{n} (r^{n} | Γ)$ , it is enough to express the condition for guaranteeing the leaders’ incentive entry by (8) and (9) and certainty of party 1’s winning by (10).

In the two-party spatial environments specified by Assumption 2, when the representative is determined dictatorially by the leader, Proposition 3 argues that if there is a subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC, then the representative of the winning party $p^{1}$ on such equilibria must be unique. When the representative is determined by probability function $Δ^{n} (r^{n} | Γ)$ , there could be multiple agents who are ex post nominated as representative of $p^{1}$ . To avoid such multiplicity, we make the following assumption for simplification.

Assumption 3 We make the following assumptions:

(i) for each n and $C \subset P$ , there exists ${\hat{r}}^{n} (C) \in p^{n}$ such that $Δ^{n} ({\hat{r}}^{n} (C) | C) = 1$ ;

(ii) $d < | v^{λ^{n}} ({\bar{a}}^{i}) - v^{λ^{n}} ({\bar{a}}^{j}) |$ for each $i \neq j$ and $n = 1, 2$ ;

(iii) $v^{λ^{1}} ({\bar{a}}^{i}) - d > v^{λ^{1}} ({\bar{a}}^{j})$ for each $i \in p^{1}$ and $j \in p^{2}$ .

Assumption 3(i) requires that outcomes of primary elections are deterministic. Assumption 3(ii) is a stronger condition on d than the condition imposed in Proposition 3(ii). Roughly speaking, this assumption holds as long as d is sufficiently small. Assumption 3(iii) implies that if the parties compete in the general election, then the leader in party $p^{1}$ always prefers her own party’s victory to the rival’s winning. In the spatial environment, this assumption holds whenever there is no member in $p^{1}$ (the left party) who prefers an extremely partisan (left) policy.¹³ With Assumption 3, the equilibrium condition is characterized by the following proposition.

Proposition 5. Consider the party competition with primary elections, and suppose that Assumptions 2 and 3 hold. There exists a pure strategy subgame perfect equilibrium causing $(p^{1}, p^{2})$ -SC if and only if ${\hat{r}}^{1} (P) > {\hat{r}}^{1} ({p^{1}})$ and $| {\bar{a}}^{{\hat{r}}^{1} (P)} | < | {\bar{a}}^{{\hat{r}}^{2} (P)} |$ .

Under Assumption 3, if there exists an equilibrium causing $(p^{1}, p^{2})$ -SC, then the nominated representative from party $p^{1}$ is ${\hat{r}}^{1} (P)$ . By contrast, when the leader dictatorially chooses the representative in the nomination process, given that (7) is satisfied, on the equilibrium causing $(p^{1}, p^{2})$ -SC, the nominated representative is expressed by $r^{1 *} \equiv \underset{i \in W^{1}}{\arg \min} {\bar{a}}^{i}$ , where $W^{1} \equiv {i \in p^{1} | | {\bar{a}}^{i} | < min_{j \in p^{2}} | {\bar{a}}^{j} |}$ . In general, their nominated representatives ${\hat{r}}^{1} (P)$ and $r^{1 *}$ are different. More specifically, compared with the case of the leaders’ dictatorial nomination of the representative, the party competition with primary elections may result in moderation ( ${\hat{r}}^{1} (P) > r^{1 *}$ ) or polarization ( ${\hat{r}}^{1} (P) < r^{1 *}$ ).

To obtain the insight behind the result, consider another spatial example, in which, as in Figure 3, there are six agents, $({\bar{a}}^{1}, {\bar{a}}^{2}, {\bar{a}}^{3}, {\bar{a}}^{4}, {\bar{a}}^{5}, {\bar{a}}^{6}) = (- 2 / 5, - 3 / 10, - 1 / 5, - 1 / 10, 1 / 4, 3 / 8)$ , $p^{1} = {1, 2, 3, 4}$ , $Λ = (1, 6)$ , and d is assumed to be sufficiently small.¹⁴ Since this example satisfies inequality (7), there exists an equilibrium that causes $(p^{1}, p^{2})$ -SC, where each leader chooses $r^{1} = r^{1 *} = 3$ and $r^{2} = 5$ .

Figure 3.

The spatial example with six agents.

In the party competition with primary elections, depending on ${\hat{r}}^{1} (P)$ and ${\hat{r}}^{2} (P)$ , there also exists an equilibrium that causes $(p^{1}, p^{2})$ -SC, where ${\hat{r}}^{1} (P)$ may differ from $r^{1 *} = 3$ . For instance, by Proposition 5, there is an equilibrium that causes $(p^{1}, p^{2})$ -SC in the party competition with primary elections if ${\hat{r}}^{1} (P) = 4 > {\hat{r}}^{1} ({p^{1}})$ . As $r^{1 *} = 3 < 4 = {\hat{r}}^{1} (P)$ , this case implies that the internal selection through the primary election induces a more moderate policy than through the dictatorial nomination. To understand the insight behind this observation, suppose furthermore ${\hat{r}}^{1} ({p^{1}}) = 3 > λ^{1}$ , implying that the representative chosen through the primary election is also more moderate than through the dictatorial nomination even without party 2’s entry. This situation is consistent with the environment where the distribution of voters’ ideal policies in the primary elections within party $p^{1}$ is skewed toward the moderate policy. In addition, as the outcome of the primary election depends on the other party’s entry decision, the voters within the party must be able to coordinate their votes in the primary election contingent on other parties’ entry.

There could also be another kind of equilibrium that causes $(p^{1}, p^{2})$ -SC in the party competition with primary elections if ${\hat{r}}^{2} (P) = {\hat{r}}^{2} ({p^{2}}) = 6$ , ${\hat{r}}^{1} (P) = 2$ , and ${\hat{r}}^{1} ({p^{1}}) = 1$ . In this case, as $r^{1 *} = 3 > 2 = {\hat{r}}^{1} (P)$ , the internal selection through the primary election induces a less-moderate policy than through the dictatorial nomination. Recall that in the case of dictatorial nomination, if party $p^{1}$ stands for the general election, then the leader in party $p^{2}$ nominates agent 5, who is more moderate than the leader herself, as the representative to induce policy compromises from party $p^{1}$ . By contrast, the primary election in party $p^{2}$ fails to nominate agent 5, which guarantees party $p^{1}$ to win in the general election even if the representative of $p^{1}$ is less moderate. In other words, the losing party’s coordination failure in the primary election induces an extreme policy from the winning party.

Although we admit that it is difficult to show more general results, the above examples suggest that democratic nomination procedures of party representatives may result in a more moderate or ideological political outcome.¹⁵ There are two key factors to determine the political outcome. First, the skewness of the distribution of the voters of the primary election within parties may alter the nominated representative so that the political consequence may also change. Second, whether the voters can coordinate their votes or not matters. More specifically, the above example illustrates that coordination failure in the losing party has a possibility to nominate a less-moderate agent so that the winning party does not need to nominate a moderate representative.

5. Concluding remarks

As in empirical observations pointed out in the introduction, strategic interaction of entry decisions may happen even when there are only two potential candidates. In this article, we have investigated a model of political competition with entry decisions in party politics. We have shown that strategic candidacy in two-party competition may emerge when each party nominates a representative for the general election contingent on the rival parties.

We can modify various assumptions that have been imposed so far. It is quite easy to introduce heterogeneity of the entry cost among the parties, as denoted by $d^{n}$ . Furthermore, as in Besley and Coate (1997), we may incorporate each agent’s capability for a politician such that when agent j wins the election, she chooses a policy from her own policy space $A^{j} \subset A$ and the political benefit of agent i is generalized to $v^{i} (a, j)$ . This generalization may allow us to consider a realistic setting in which a weak party in a moderate position attempts strategic candidacy against a strong and ideological party.

Throughout the paper, we have focused on equilibria in which there are exactly two parties and one of them is sure to lose. For capturing the idea of strategic candidacy to induce political compromises, it is enough to focus on two-party equilibria. Nevertheless, equilibria with three or more standing parties might include two kinds of strategic candidacy simultaneously, one for changing the winning party, which the literature has pointed out, and one for inducing political compromise as studied in this paper. The interaction of these motives could have interesting implications.

Footnotes

Appendix

Acknowledgements

This paper is a substantially revised chapter of my Ph.D. dissertation, submitted to London School of Economics and Political Science. I am greatly indebted to Leonardo Felli and Andrea Prat for their advice and encouragement. I am grateful to Yasushi Asako, Ehud Lehrer, Hülya Eraslan, Erik Eyster, Kentaro Fukumoto, Martin Osborne, Ronny Razin, Francesco Squintani, Kimiko Terai, John Patty (Editor), two anonymous referees and participants at seminars and conferences for helpful comments and discussions. All errors are my own.

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Akifumi Ishihara

Notes

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