Duverger s Hypothesis, the Run-Off Rule, and Electoral Competition

Advance Access publication May 5, 005 Political Analysis (005) 13:09 3 doi:10.1093/pan/mpi013 Duverger s Hypothesis, the Run-Off Rule, and Electoral Competition Steven Callander Kellogg School of Management, Northwestern University, Evanston, IL 6008 e-mail: scal@kellogg.northwestern.edu I analyze a model of electoral competition with entry under the run-off rule. I consider both two- and multiple-party systems. The principal result is that two-party systems may prove stable under the run-off rule: I show that a continuum of equilibria exists in which only two parties enter and subsequent entry is deterred. This finding conflicts with the accepted wisdom on the run-off rule encapsulated by Duverger s Hypothesis. The results of the model are then reconciled with Duverger s Hypothesis and a more precise formulation is proposed. 1 Introduction The run-off electoral rule is widely used throughout the world in environments as diverse as presidential elections (France), parliamentary elections (Australia), and gubernatorial primary elections (United States). Despite its apparent importance to electoral processes and outcomes, the theoretical properties of the run-off rule are little understood. This article is an attempt to partially fill this gap. Although the run-off rule has not come under the scrutiny of much formal analysis, the tendencies and properties of the rule have been informally documented by many commentators. Much of this attention has focused on the number of parties (or candidates) that enter and compete. Underlying most conclusions is the observation that the repeated nature of the process does not necessarily induce each party to maximize its vote share in the first round (in contrast to the plurality rule). As a result, it has long been conjectured that the run-off rule does not provide the incentives for a two-party system to develop. This prediction, which groups the run-off rule with proportional representation, goes at least as far back as Lowell (1896) and Holcombe (1911), although, as is also the case with the plurality rule, the idea is now most closely associated with Duverger (1954) and is referred to by Riker (198) as Duverger s Hypothesis. Substantial empirical observation supports the conclusions of Duverger s Hypothesis. The evidence, however, is not overwhelming, and the presence of counterexamples led Riker (198, p. 760) to conclude that we can therefore abandon Duverger s hypothesis in its deterministic form. 1 Riker does not present an explanation for the dissonance between Author s note: I thank three anonymous referees for very thoughtful and constructive comments; their input led to substantive improvement of the paper. This version is based on chapter of my 001 Caltech dissertation. For helpful comments I thank Jeff Banks, Richard McKelvey, Tom Palfrey, Larry Rothenberg, and Catherine Wilson. 1 Evidence for and against Duverger s Hypothesis is discussed in Section 6. Political Analysis Vol. 13 No. 3, Ó The Author 005. Published by Oxford University Press on behalf of the Society for Political Methodology. All rights reserved. For Permissions, please email: journals.permissions@oupjournals.org 09

10 Steven Callander observation and theory, leaving as an open question why two-party outcomes may arise under the run-off rule. In this article I propose a solution to this question. I develop a model of electoral competition under the run-off rule and characterize stable configurations of parties in which subsequent entry is deterred. I find that stable party systems may take on many different forms with differing numbers of parties participating in the electoral system. Principally, I find that two-party systems may prove stable: I show that two parties can choose policy platforms that deter additional entry and that such positions are supportable as equilibrium behavior. Thus, it is possible for electoral competition to induce two-party (or two-candidate) outcomes in line with the counterexamples to Duverger s Hypothesis. The equilibrium predictions of the model are then reconciled with Duverger s Hypothesis. Surprisingly, despite the conflict with the formal statement of Duverger s Hypothesis, I find that the equilibrium results are not, in fact, at odds with the original intuition of Duverger and earlier scholars. The difference between the earlier conclusions on the run-off rule and the predictions of my model turns on the issue of timing and the initial conditions that characterize particular political systems. The variety of initial political conditions implies that a complete description of political stability requires two components: predictions of both stable and unstable party structures. Duverger s original statement (perhaps in the interests of parsimony) addresses this task only partially and states that if more than two parties are competing then these parties will not necessarily have the incentive to exit the election; thus, multiple party stability. His statement is silent on the alternative prediction of whether two-party systems are necessarily unstable and induce additional entry. The results presented here deal with the previously ignored two-party question (as well as multiple-party stability). Thus, the model s findings and the arguments of Duverger are not mutually exclusive, and the multiplicity of real-world outcomes under the run-off rule support such a conclusion. I review evidence on the history of the Australian and French political systems and find it to be consistent with this conclusion. The results presented here imply that the statement of Duverger s Hypothesis covers a broader domain than is warranted by the original motivating observations. Consequently, two-party systems, despite the original observations and theorizing being silent on the question, have heretofore been interpreted as counterexamples to the hypothesis. To correct the shortcomings of the hypothesis and deal with the subtleties of timing uncovered here, I propose a more general statement of Duverger s Hypothesis as it relates to the runoff rule (thus separating theoretical predictions on the run-off rule from those for proportional representation). The only paper, to the best of my knowledge, that considers a spatial model of electoral competition under the run-off rule is the citizen-candidate model of Osborne and Slivinski (1996). Osborne and Slivinski contrast with the current work by assuming that parties are policy orientated and thus, most important, policy restricted. They find that the restriction on party movement leads, under certain conditions, to two-party equilibria in which the entry of additional parties is deterred. Their intuition applies equally to competition under the plurality rule. In this article I develop alternative intuition by allowing policy commitment (and, therefore, policy mobility) and showing that, in contrast to competition under the plurality rule, two-party entry-deterring equilibria exist even when parties are free to choose their policy platforms. 3 In that only two parties enter and have a chance of victory. The competitors are referred to exclusively here as parties, although the results apply equally well at the candidate level. 3 Osborne and Slivinski (1996) also characterize conditions under which three- and four-party equilibria may arise, as well as the equilibria themselves.

Two-party Systems and the Run-off Rule 11 Greenberg and Shepsle (1987) consider elections for K member districts in which parties seek to maximize their rank order (i.e., ensure they are in the top K vote getters); see also Shvetsova (1995). Wright and Riker (1989) claim that the case for K ¼ captures the incentives of a run-off election. However, such an interpretation ignores the fact that there can be only one winner in a run-off election (in contrast to the K winners in Greenberg and Shepsle s model), and therefore rank order maximization does not accurately capture the party incentives within the run-off system. Finally, Myerson (1993) also considers the run-off rule (in the form of the alternative vote), although the connection with the current work is minimal. He considers a distributive model of elections and does not concern himself with the equilibrium number of parties that is of interest here. The remainder of the article is organized as follows. After a brief literature review, Section describes the run-off rule and its different forms. Section 3 presents the model of electoral competition, and Sections 4 and 5 provide the results. Section 6 contains a discussion of the model and the restatement of Duverger s Hypothesis. In Section 7 I develop an extension of the model to simultaneous competition for multiple districts and find the equilibria robust to a limited amount of district heterogeneity. Section 8 concludes. The Run-Off Rule The run-off rule is a repeated process involving multiple rounds. As the rounds progress parties are eliminated until only one remains; the remaining party is then declared the winner. There are, however, many variations in how the rounds and the elimination process are conducted. The most basic line of division is whether the repeated rounds involve repeated voting or merely repeated counting (Riker 198). If voting is repeated then at each round voters are required to cast a new ballot to determine which parties shall progress. The most common form is the dual-ballot, although obviously there can be any number of ballots. In the dual-ballot system all parties stand in the first round and voters cast a single, nontransferable vote. 4 Unless one party gains a majority, in which case it is declared the winner immediately, the two leading parties progress to the second round and all other parties are eliminated. 5 The winner is decided from these remaining parties by a simple majority vote. This system is currently used for presidential elections in France and several Latin American countries and was used in Israel between 1996 and 001 to elect a prime minister. If counting, rather than voting, is repeated then voters cast only one ballot. Instead of indicating only a most preferred party, the voters will rank the parties in some way. A common form of this rule is the alternative vote, which is used for lower house state and federal elections in Australia. 6 Voters are required to list their preferences for all parties from first to last. The number of first preferences is tallied for each party and the one with the fewest is eliminated. The ballots for the eliminated party are then redistributed according to the next preference listed. The votes are again tallied and the party with the lowest total is eliminated. The process is repeated until there is only one remaining party, 4 Using the terminology of Cox (1997) throughout. 5 A further variant allows any parties to progress to the second round, making the first round purely demonstrative. 6 With the exception of the state of Tasmania (Wright 1986). In fact, the original system was characterized by Nanson (1900) at the time of implementation as the compression of a French double-ballot election into one round.

1 Steven Callander which is then declared the winner. With this counting rule the number of rounds must be one less than the number of parties. Of course, different counting rules can be developed that produce any number of rounds. 3 A Formal Model of Electoral Competition under the Run-Off Rule I develop a simple model of electoral competition for one single-member district under the run-off rule. Under the run-off rule votes are mapped into an outcome as follows. Once a point is reached such that there are only two competing parties then the one with the greatest vote share is the winner. If more than two parties contest the election then the two highest vote getters progress to the run-off stage. Ties in any round are decided randomly. The winner of the election is denoted by W. The model is consistent with a variety of the ways in which the rule can be implemented, encompassing the two-round alternative vote rule and the dual-ballot rules. 7 There is a continuum of voters with symmetric, single peaked preferences over the issue space. The issue space is the real line, R. The distribution of voter ideal points is given by the cdf F(x) for all x R, with corresponding pdf, f. I assume that f is continuous, symmetric around zero with support on some open interval (ÿa, A), and is weakly increasing on (ÿa, 0). The set of parties and the order of play are as follows. There are n dominant parties that move first and simultaneously choose to enter or stay out of an election; if they enter then they select policy positions. These parties can be thought of as describing the initial conditions in a political system. There also exists a potential entrant, who reacts to the platforms of the dominant parties and chooses whether to compete and, if so, a policy position. The entered parties then engage in the election. 8 As in Palfrey (1984), I model party entry decisions as sequential to capture the possibility for entry deterrence. This structure ensures in a tractable manner that if successful entry is possible in a given district, then the incumbent dominant parties lose that district to entry. 9 Parties are free to locate at any point in the policy space and must maintain this point for all rounds of the election. Denoting party labels by upper case and policy positions by lower case, for all dominant parties D j, j ¼ 1,,..., n, and the entrant E, the policy platform is d j fr,?g and e fr,?g, where? represents the decision to not enter the election. The objective of parties is to maximize their probability of victory they are Downsian and they enter the election only if this probability is strictly positive, capturing the belief that political parties are designed and compete to win elections. If the probability of victory is maximized for a set of points then the party chooses randomly from this set. 10 7 The flexibility of the model is the product of two assumptions: full information and sincere voting, as well as the implied assumption that parties cannot change their platforms between ballots. Note that with three parties the rule is equivalent to the Australian alternative vote system. 8 The ordering was chosen to enable comparison with the results of Palfrey (1984) and Callander (forthcoming) for the plurality rule, although it should be noted that the results presented here do not depend critically on this particular timing scheme. 9 Allowing for an arbitrary number of potential entrants does not affect most of the findings in the article; however, a complete specification of behavior after a deviation by a dominant party requires, in certain circumstances, a characterization of entry combinations by the potential entrants that is excessively and tediously detailed. 10 A more natural assumption may be that the party chooses the point that maximizes its vote share in order to maximize its exposure, mandate, or some other secondary objective. Such an assumption does not alter the results although it necessitates a more complicated equilibrium concept to deal with technical issues arising from the nonexistence of a maximum (see Callander 001).

Two-party Systems and the Run-off Rule 13 I assume voters are sincere and support the party closest to their ideal point in the policy space. If a voter is indifferent between the two dominant parties then they randomize, but if they are indifferent between a dominant party and the entrant then they vote for the dominant party. 11 This assumption prevents entrants from wanting to locate on top of a dominant party and is a formalization of the notion that voters have a preference for established parties if all else is the same. Voters are therefore not modeled as strategic actors. 1 Formally, a strategy for the entrant is a function, e : fr,?g n fi fr,?g, mapping the actions of the dominant parties into a policy decision of its own. The definition of equilibrium can now be stated. For simplicity and because voters do not act strategically, equilibrium is stated in terms of the strategies of the dominant parties and potential entrant only. Define P(D j j d, e) andv(d j j d, e) to be the probability of victory and vote share, respectively, for party D j, given the set of platforms d and e; P(E j d, e) and V(E j d, e) are defined analogously for E. Finally, write d ÿj ¼ (d 1,..., d jÿ1, d jþ1,..., d n ). Definition 1. A set of strategies, d * and e *, is an equilibrium if and only if: (i) P(D j j y, d * ÿj, e* ) P(D j j d * j, d * ÿj, e* ) for every j and all y; further, d * j ¼? if P(D j j d *, e * ) ¼ 0. (ii) P(E j d, e * ) P(E j d, y) for all d and y; further, e ¼? if P(E j d, e * ) ¼ 0. Condition (ii) requires that in equilibrium entrants maximize their probability of winning for all possible actions of the dominant parties; subgame perfection is thus imposed. As the parties have no ideological motivation, any permutation of equilibrium party strategies must also constitute an equilibrium. I adopt the convention that in equilibrium d * jþ1 R only if d * j R and that d * j d * jþ1 for all j. Finally, I say an equilibrium is a k party equilibrium if only k dominant parties enter and the potential entrant is deterred: that is, in a k party equilibrium P k j¼1 P(D j j d *, e * ) ¼ 1 and d * j ¼ e * ¼? for all j. k. Throughout the article the following notation is used: e ¼ x þ implies that the entrant locates arbitrary close to the right of policy position x R; similarly, x ÿ denotes a point arbitrarily to the left of x; x ÿþ denotes a set of points arbitrarily close to either side of x. 4 Results 4.1 Intuition and a Preliminary Result The run-off rule bears a resemblance to the plurality rule, and indeed the two rules are identical if only two parties compete. The differences between the two rules are subtle, yet they lead to significantly different incentives for parties and, consequently, substantively different equilibria. Under both rules a special role is played by symmetric locations of the dominant parties (around the median voter). The two rules differ in the incentives parties face to move to symmetric policy positions and their incentives to deviate from such positions. Before presenting the formal statement of results, I outline here the unique intuition underlying the run-off rule and develop a preliminary result for the case of two dominant parties (n ¼ ) that drives the equilibria of the model. 11 Alternatively, it could be assumed that ties in the overall election between a dominant party and the entrant are decided in favor of the dominant party, and that ties between dominant parties are decided randomly. 1 This assumption is later relaxed.

14 Steven Callander Consider first the incentives faced by the dominant parties if they are located asymmetrically. If the entrant stays out of the election then clearly the dominant party furthest from the median voter loses the election. Moreover, this dominant party must still certainly lose even if an optimizing entrant enters the election (maximizing either probability of victory or vote share). This fact, although apparently obvious, is striking as it does not hold under plurality rule (Palfrey 1984; Callander 001, forthcoming). In a run-off election, in contrast to plurality rule, the party furthest from the median, whether dominant or entrant, cannot win the election as it must lose at the run-off stage, regardless of its opponent. For the more extreme dominant party to win the election, therefore, the entrant must choose at least an equally extreme policy position. By the same logic as for the dominant party, the entrant does not locate at a more extreme position and, therefore, needs to be equally distant but on the opposite side of the median to the most divergent dominant party. Such a location, however, is dominated by locating on the immediate flank of the dominant party closest to the median and so the entrant locates more centrally than the extreme dominant party. 13 The intuition for the run-off rule is stated formally as Lemma 1. Lemma 1. Suppose n ¼ and jd 1 j, jd j. If the entrant responds optimally then P(D j d, e) ¼ 0 and the dominant party furthest from the center loses the election. Lemma 1 implies that the dominant parties have the incentive to move toward the median voter to ensure they are not the most extreme. This incentive, however, only extends to ensuring they are not the most extreme and does not imply that a party wishes to move closer to the median than the other dominant party. To see this, consider the problem faced by the entrant. Following the logic above, the challenge for the entrant under the run-off rule is to find a policy platform that is not more extreme than both dominant parties but that allows the entrant to squeeze one of the dominant parties out in the first round. If the dominant parties are located symmetrically then the achievement of both objectives will be difficult. To satisfy the first requirement (be less extreme than at least one dominant party) the entrant must locate between the dominant parties. If the dominant parties are not too far apart then the entrant will secure few votes and will itself be eliminated in the first round. The implication of this logic is that the dominant parties can deter successful entry if they are located symmetrically and not too far from the median, the same logic that applies to competition under the plurality rule. However, when considering possible deviations from symmetric entry-deterring locations, the equivalence between run-off and plurality rule again breaks down. Suppose a dominant party deviates from a symmetric entry-deterring location pair. If the deviation is away from the median then, by Lemma 1, the deviating party must lose the election and so to be profitable the deviation must be toward the median. However, for all inward deviations no matter how small the conditions required for the entrant to win the election are now satisfied: by locating just outside the deviating party the entrant is able to (1) simultaneously trap the deviator in the middle and eliminate it in the first round, and () locate at a policy point closer to the median than the other dominant party, thereby securing victory in the run-off stage. The run-off rule, by providing two stages of competition, allows an entrant locating on the flank to take votes from both dominant parties 13 Assuming the dominant parties are on opposite sides of the median; if not then locating at the median secures victory for the entrant and the same result holds.

Two-party Systems and the Run-off Rule 15 (from a different party in each round). Thus, if the dominant parties locate asymmetrically then not only must the party furthest from the median lose (as implied by Lemma 1), but so too must the party closer to the median voter. 14 The entrant s ability to enter successfully when the dominant parties are located asymmetrically implies that profitable deviations from symmetric locations are not forthcoming. In the following section I prove that this absence of profitable deviations translates into a multitude of dominant two-party equilibria. In contrast, the plurality rule limits considerably the capabilities of the entrant as the entrant is unable to simultaneously punish both dominant parties. Consequently, more profitable deviations for the dominant parties exist under plurality rule leading to the opposite conclusion (to that obtained under the run-off rule): no equilibrium existence. 4. Two-Party Equilibria From the above intuitions can be drawn two implications for dominant party behavior: (1) to deter successful entry the dominant parties will locate symmetrically about the median voter and not too far apart, and () deviations from symmetric location pairs will induce successful entry. In combination, the two implications provide an equilibrium result. Proposition 1 below states that a continuum of two-party equilibria exists under the run-off rule, that all equilibria require the dominant parties to locate symmetrically, and that the entrant never wins. The equilibria are demarcated into two domains: the equilibria of the first domain always exist and require the dominant parties to locate centrally, and the existence of equilibria in the second domain is dependent on the exact voter distribution as it allows for more extreme policy locations for the dominant parties. Therefore, under the run-off rule there exists a continuum of two-party entry-deterring equilibria and on all but a set of measure zero involve noncentrist platforms. All proofs have been relegated to the appendix. Proposition 1. If n ¼ then all two-party equilibria are of the form d 1 ¼ÿd, e ¼?, and P(D 1 ) ¼ P(D ) ¼ 1, where the domain of existence is: (a) For all d 1 [a 1, 0], where a 1 solves 1 ÿ F( a 1 ) ¼ F(a 1 ), an equilibrium exists; (b) Depending on F, additional equilibria may exist that satisfy the following necessary, but not sufficient, conditions: d 1 [F ÿ1 ( 1 4 ), a 1) and F( d 1 ). 1 3. The equilibrium regions are depicted in Figure 1. If the entrant locates on a flank then it squeezes one of the dominant parties out in the first round but is then defeated by the other dominant party in the run-off. Thus, in all equilibria the entrant has zero chance of victory and stays out of the election. The set of equilibria in category (b) is difficult to characterize as it depends critically on the particular distribution of voters in the electorate. For symmetric locations in (b) the entrant will not win locating arbitrarily close to either dominant party. However, there may exist a point e ¼ d 1 þ k, k. 0, distinctly separate from the dominant parties such that E has a strictly positive chance of winning the election. The existence of such a point depends on F and the locations of the dominant parties. The definition of a 1 precludes the existence of such points in (a); for locations not in (a) or (b) such a point must exist. 14 For some voter distributions there may exist asymmetric platforms for the dominant parties that preclude entry. However, these platforms cannot constitute equilibria as the dominant party furthest from the median always loses. The platforms require one dominant party to be relatively far from the median and cannot be reached by a single profitable deviation if both dominant parties are close enough to the median (this condition is satisfied by the equilibrium bounds of the following section).

16 Steven Callander Fig. 1 Two-party equilibria under the run-off rule. To highlight the incentives facing dominant parties and the nature of entry, I explore in Corollary 1 the case in which the entrant always enters even if it cannot win the election. I assume the entrant chooses a point that maximizes its probability of victory, and if victory is not possible then it maximizes its vote share. This environment differs from above in that entry is no longer a threat but a guarantee. Despite the difference, Lemma 1 extends to this environment, and the equilibrium behavior of the two dominant parties is unchanged. To deal with technicalities that arise in this environment (the existence of an optimum) a more complicated limit-equilibrium concept is required. This concept is similar in spirit to Definition 1 and is closely related to that used by Palfrey (1984); a formal statement is included in the appendix. Corollary 1. If n ¼ and the entrant always enters the election, then all limitequilibria are of the form d 1 ¼ÿd, and P(D 1 ) ¼ P(D ) ¼ 1, where the domain of existence and entrant strategy is: (a) For all d 1 [a 1, 0], where 1 ÿ Fð a 1 Þ¼F(a 1 ), an equilibrium exists, and e 1 fd ÿ 1, dþ g;15 (b) Depending on F, additional equilibria may exist that satisfy the following necessary, but not sufficient, conditions: d 1 [F ÿ1 ( 1 4 ), a 1) and Fð d 1 Þ. 1 3, and e 0ÿþ. By symmetry of the equilibria, entry ex ante affects each dominant party equally and each has an equal chance of winning the election. In the equilibria of category (a) the entrant knocks out one of the dominant parties in the first round but is itself eliminated in the run-off stage. A deviation outward by a dominant party would ensure that the entrant locates on the opposing flank, thereby eliminating the targeted party in the first round. By Lemma 1, however, this enables the entrant to win; such a deviation is not profitable and equilibrium is maintained. The equivalence of equilibria under the run-off rule for voluntary and involuntary third-party entry is striking as equivalence does not arise in plurality rule elections. The divergent equilibrium of Palfrey (1984) requires that third-party entry is guaranteed, whereas dominant party convergence is unabated when the entrant has the option to not compete (Callander forthcoming). 15 If x ¼ a 1 then e 1 fd 1 ÿ, d þ,0 ÿþ g for all F except when F is uniform where 0 ÿþ [ (d 1, d ).

Two-party Systems and the Run-off Rule 17 4.3 Multiple-Party Equilibria In this section I turn to the possibility of multiple-party political structures and whether they may also prove stable. If parties are required to locate at distinct locations then the answer to this question, stated as Proposition, is negative. The proposition shows that equilibrium cannot be sustained with parties at distinct points if three or more dominant parties compete. Thus, a two-party outcome is uniquely stable under these conditions. Proposition. If a k party equilibrium exists and d j 6¼ d h for all 1 j, h k, then k ¼. The key to this result is the inability of the entrant to make the run-off stage. In a two-party equilibrium the threat of entry is binding as the entrant can squeeze one dominant party out in the first round and make the run-off stage. With more than two parties, however, squeezing one party out is no longer sufficient to progress to the run-off. Thus, dominant parties on the flank are able to move their policy position toward their competitors without inducing entry, ultimately violating the equilibrium condition by squeezing out the center party. This intuition is similar to that inducing convergence in two-party elections under the plurality rule. Entry is unsuccessful in the one-stage plurality system as the entrant can punish only one of the dominant parties. The run-off rule affords the entrant the opportunity to punish two dominant parties with its entry. With three or more parties, however, this ability is insufficient for the threat of entry to constrain dominant party behavior and maintain an equilibrium with distinct policy positions. The nonexistence of multiple-party equilibria described in Proposition hinges on the requirement that parties choose distinct policy positions. The following results show that multiple-party equilibria are possible once this requirement is relaxed and, in fact, that configurations with any number of parties are possible. In these equilibria groups of parties offer policy platforms indistinguishable to voters; the parties at any point then divide equally the votes that are attracted to their policy position. Proposition 3 deals with the case of an even number of dominant parties entering in equilibrium, and Proposition 4 with an odd number of dominant parties. The propositions provide existence results and characterize symmetric equilibria only. Proposition 3. For k 4 and even, a continuum of symmetric k party equilibria exist. Define a as Fða ÞþkF a ¼ 1 þ k : The equilibria are d j ¼ÿd h for all j ¼ 1,,..., k, and h ¼ k þ 1,..., k, where d h [0, a ]. In equilibrium half of the entered parties locate at a single point on the right flank and the other half at the reflected position on the left flank. All parties receive equal vote shares in the first round of the election and possess equal chances of victory. Just as in the twoparty equilibria, entry is deterred as long as these two policy positions are symmetric and not too divergent; the bound on existence is given by a. As the number of parties grows the vote share of each shrinks and the dominant parties become increasingly susceptible to entry at the median voter. Thus, a fi 0askfi. Proposition 4. For k 5 and odd, a symmetric k party equilibrium exists. Define a 3 as F a 3 ¼ 1 k þ 1 : k

18 Steven Callander The equilibrium is dkþ1 ¼ 0, d j ¼ÿa 3 for j ¼ 1,,..., kÿ1, and d h ¼ a 3 for h ¼ kþ3 1,..., k. þ In the equilibrium one party locates at the median voter s ideal point and the remaining parties are split equally over two symmetric divergent locations. Equilibrium requires the entered parties to win equal vote shares in the first round and, given the number of parties and symmetric configuration, this is satisfied for a unique degree of divergence. It is easy to show that symmetric locations are not supportable as equilibria if three or more parties locate at the center of the distribution (as then entry arbitrarily close to zero would guarantee electoral success for the entrant). As in Proposition 3, the vote share of each party must decline as the number of parties grows; thus k fi implies that a 3 fi 0 and parties increasingly converge. In equilibrium all parties have equal vote share and hence equal chances of proceeding to the run-off stage. At the run-off stage, however, opportunities are no longer equal and the center party, Dkþ1, wins with certainty. Thus the ex ante probability of victory for the center party is k, and kÿ kðkÿ1þ for each of the other dominant parties. The election can, therefore, switch from a tied election in the first stage to a clear victory for the center party, or remain close at the run-off stage should the center party be eliminated in the first round. Two aspects of Propositions 3 and 4 are worthy of note. First, no restriction on n, the pool of available dominant parties, is made and the equilibrium configurations characterized can arise for any size pool of potential dominant parties (as long as n k). Second, the threat of entry is not required and the same equilibria exist even if there is no potential entrant. If a dominant party deviated (or an additional dominant party entered) then it would be squeezed in the center of the distribution or marginalized on the flanks (just as a potential entrant would). Thus, the weight of numbers is self-reinforcing in the multiple party structures described here. 16 The final result of this section addresses the potential for three-party equilibria. The negative result of Proposition implies that symmetric and divergent three-party equilibria do not exist and thus divergent equilibria must be asymmetric. Proposition 5 establishes existence and characterizes a family of asymmetric three-party equilibria. Proposition 5. For n ¼ 3 a continuum of asymmetric three-party equilibria exists: d 1 ¼ÿd ¼ÿd 3 where d [0, a 4 ] and F(a 4 ) þ 4F( a 4 ) ¼ 3. This equilibrium is similar to the two-party equilibria of Proposition 1, although the bounds on existence are more convergent here, as with parties doubled up on the right flank the fraction of votes required for successful entry between the dominant parties is lower. In equilibrium party D 1 wins all voters left of center and the other parties split voters right of center. D 1 is guaranteed of making the run-off stage but once there it faces a close election with the surviving right-wing party. Thus, domination of the vote share in the first round does not translate into electoral domination [more precisely, in equilibrium P(D 1 ) ¼ 1 and P(D ) ¼ P(D 3 ) ¼ 1 4 ]. In moving from the first round to the run-off stage, one of the right-wing parties is eliminated, forcing voters on the right to coordinate on the other party to oppose the left-wing party. This equilibrium, and the possibility of victory for a right-wing party, highlights the coordination function served by the run-off electoral rule. 16 The deterrence of additional dominant parties implies that the potential entrant could not enter profitably in these equilibria even if permitted to locate on top of a dominant party and attract an equal vote share.

Two-party Systems and the Run-off Rule 19 5 Strategic Voting The central behavioral assumption of the model is that voters vote sincerely. In this section I explore the application of strategic voting to multiple-party elections and the incentives that arise. The desire of voters to behave strategically is a powerful one. Yet to understand its manifestation in multiple-party elections requires the resolution of (at least) two questions about voter capabilities, one psychological and the other social. First, an understanding of the ability and willingness of voters to make rational strategic calculations remains elusive. Empirical estimates, albeit imprecise, are that a fraction of voters behave strategically in their vote choice but just as many do not. 17 Second, a voter s optimal action is dependent on the actions of others (often many others) as outcomes are determined by majority rule. Thus, effective strategic voting requires a large amount of voter coordination. Potential imperfections in the ability and willingness of voters to strategize and coordinate have important implications for electoral outcomes. If all voters strategize and are perfectly able to coordinate then the outcomes of electoral competition are clear: policy outcomes must be in the core. If voter capabilities are imperfect, however, then outcomes are less clear. In this section I consider two such possibilities. First, I allow for perfect coordination but assume that only a fraction of the electorate votes strategically. Second, I assume complete strategic voting but imperfect coordination and find that the results of Section 4 can be interpreted as a model of plurality rule elections. 5.1 Run-Off Rule Equilibria with Strategic Voting Expansion of the model to allow for strategic voting produces multitudes of contingencies for which behavior must be specified. Moreover, behavior often depends on utility parameters in ways that are avoided with sincere voting; for example, the degree of voter risk aversion. 18 In order to gain traction, I restrict attention in this section to examining the robustness of the two-party equilibria described in Proposition 1. I suppose that a fraction q of the population votes strategically, leaving (1 ÿ q) as sincere voters, and that voters willingness to vote strategically is uncorrelated with their ideal points. Strategic voters like sincere voters prefer the party closest to their ideal point to win, but unlike sincere voters they may cast a ballot for a different party in the hope of facilitating their optimal outcome or at least avoiding victory by the least attractive party. With an infinite population strategic considerations at an individual level are futile, but with the capability to coordinate groups of voters can alter electoral outcomes by voting strategically. I say that a group of voters possesses an incentive to vote strategically if by changing their voting behavior all members of the group can obtain a more preferred outcome. If a strategic voter benefits from some group voting strategically, then I say that voter wants to vote strategically (the difference between incentive and wants is whether strategic voting is sufficient to affect the outcome). The following lemma describes strategic behavior in several simple circumstances. The results follow from simple arguments and proofs are omitted. 17 See Alvarez and Nagler (000) for evidence and a discussion of the literature. 18 Risk aversion arises if voters have increasing marginal disutility as the distance from a party to the voter s ideal point increases, such as with a quadratic loss utility function.

0 Steven Callander Lemma. Suppose three parties, D j, D k, D l, compete in the election: (i) In the run-off stage no voter wants to vote strategically. (ii) If jd j j, jd k j, jd l j then P(D l ) ¼ 0. If P(D k ) ¼ 1 then voters whose preference order is D l _ D j _ D k want to vote strategically for D j in the first round. (iii) Suppose d j ¼ 0 and d k ¼ÿd l 6¼ 0. If P(D j ) ¼ 0 then P(D k ) ¼ P(D l ) ¼ 1. If voters are risk averse then this outcome is dominated for all voters by P(D j ) ¼ 1. All strategic voters want to vote strategically for D j. If 0, jd j j, jd k j¼jd l j then some strategic voters may not want to vote strategically for D j. In two-party elections sincere and strategic voting coincide, thus strategic considerations in run-off elections alter behavior only in the first round and affect outcomes by affecting the parties that proceed to the run-off stage. A key implication of this is that a Condorcet loser (a strictly majority least-preferred party) cannot win the election. In the context of the equilibria of Proposition 1, this implies that an entrant cannot succeed by locating on the flanks of two symmetric dominant parties, regardless of the level of strategic voting. In contrast, if the entrant locates between the dominant parties and it is the Condorcet winner, then it may benefit from strategic voting. The final two parts of Lemma address two such possibilities. In the previous sections it was established that centrist entry would not be successful if all voters were sincere. The question then is, can the addition of strategic voters allow the entrant to profit with centrist entry? Proposition 6 answers this in the affirmative and provides a bound on the fraction of strategic voters such that entry can be deterred and equilibrium sustained. Proposition 6. Suppose a fraction q of voters are strategic, perfect coordination is possible, and voters are risk averse. If n ¼ and q, 1 3 then a continuum of symmetric two-party equilibria exists: d 1 ¼ÿd and e ¼?, where d 1 [a 5, 0] and a 5 satisfies F(a 5 ) þ F ( a 5 ) ¼ 1ÿq 1. If q 1 3 the unique equilibrium is d 1 ¼ d ¼ 0 and e ¼?. Thus, a continuum of divergent two-party equilibria remains if the proportion of strategic voters is strictly less than 1 3 ; for a greater proportion of strategic voters convergence is induced. The bound of 1 3 is intuitive: in a three-party contest, a 1 3 vote share ensures a party progresses to the run-off stage, thus q 1 3 ensures that the Condorcet winner must win election when competing against two equally divergent competitors. The bounds on equilibrium existence are dependent on q, and as q fi 1 3 the boundary a 5 fi 0. As more voters are strategic, the vote share to a centrist entrant increases. By converging, the dominant parties reduce the fraction of sincere voters that are won by the entrant. Ultimately, this fraction cannot be reduced sufficiently to counteract the growth in the fraction of strategic voters. Proposition 6 confirms the (not surprising) result that equilibria under sincere voting are different from equilibria when strategic voting is allowed. However, the proposition also establishes that the sincere voting equilibria are not dependent on complete sincere voting and are robust to a significant but limited amount of strategizing. 5. Alternative Interpretation: Plurality Rule and Imperfect Coordination In the equilibria of Proposition 6 coordination by strategic voters on a centrist entrant is relatively easy as the entrant is strictly preferred by all voters to a lottery over the dominant parties. In general settings coordination is more difficult and the party on which to coordinate is often far from obvious. Coordination, therefore, may be imperfect. Under the

Two-party Systems and the Run-off Rule 1 plurality rule coordination is particularly troublesome as there is not a second stage to facilitate coordination and ensure that a Condorcet loser is not elected. In practice, external factors may be used as coordination devices and in elections preelection polls often fill this role (Fey 1997; McKelvey and Ordeshook 1985). Polls serve as signals of which parties are credible that is, have a legitimate chance of victory and are therefore worthy of a vote: the higher a party s standing in opinion polls the more credible its candidacy. A reasonable operationalization of imperfect voter coordination under plurality rule may be that voters sincerely respond to opinion polls and then strategically coalesce around the two leading parties. With the degree of coordination and rationality just described, electoral dynamics are composed of two stages: an initial multiple-party field is whittled down to two serious contenders, who then compete directly for election. Remarkably, this competitive situation mirrors exactly that facing sincere voters and strategic parties in a run-off rule election. Consequently, the results of Section 4 apply directly to the setting of plurality rule elections with strategic voters but imperfect coordination. Corollary formalizes this duality, the proof of which is immediate. Corollary. Suppose voters are strategic, coordination is imperfect, and the plurality rule determines elections. A continuum of symmetric two-party equilibria exists; equilibrium strategies are as characterized in Proposition 1. The equilibria in Corollary are of interest in their own right. The result shows that if voters are strategic and coordination is imperfect then divergent two-party entry-deterring equilibria exist (in fact, a continuum of them). This result is consistent with both policy divergence observed in U.S. and British plurality rule systems and Duverger s Law, the more famous and robust prediction of Duverger that plurality rule systems induce twoparty outcomes. Note that these equilibria do not arise if coordination is perfect (Feddersen et al. 1990) or if voting is sincere (Callander forthcoming). The result of Corollary highlights the coordination role played by the run-off rule. By extending the decision process to multiple rounds, the run-off rule ensures a degree of voter coordination and frees voters from having to achieve it independently. 6 Discussion Early commentary on the run-off rule focused on the observation that parties do not necessarily have the incentive to maximize their first-round vote share, as they would under the plurality rule. The implication of this incentive (or lack thereof) is that in a multiple-party system the parties do not necessarily merge into larger bodies in order to improve their electoral chances. Preceding Duverger, this conclusion was succinctly stated by Lowell (1896, p. 110): Where a number of groups exist, [the two stage majority system] tends to foster them, and prevent their fusing into larger bodies (quoted in Wright and Riker 1989). Notably absent from the statement by Lowell is whether under the run-off rule a twoparty system would expand to multiple parties. Despite the limited scope of observation, Duverger s Hypothesis has been stated and interpreted as claiming that multiple-party structures are the only stable states under the run-off rule. Consequently, when stable twoparty systems under the run-off rule are observed they are seen as counterexamples to the hypothesis; these counterexamples induced Riker (198) to abandon the hypothesis in its deterministic form.

Steven Callander The model and results developed herein show that initial conditions are critical to outcomes under the run-off rule. Moreover, it is shown that when the run-off rule is introduced to a two-party system (n ¼ ) the two dominant parties are able to deter subsequent entry and establish a stable duopoly on power. This possibility is overlooked in the statement of Duverger s Hypothesis, which address only whether multiple-party structures converge to two-party systems. Explorations of multiple-party systems in Section 4.3 confirm that this aspect of the hypothesis is consistent with theoretical prediction, but only if it is possible for party groupings to occupy a single policy position. To reconcile the model results with previous theory I propose the following restatement of Duverger s Hypothesis: Competition between two parties under the run-off rule is a stable configuration, although systems with more than two parties do not necessarily converge to this configuration. Broad evidence from the world s electoral systems supports this restatement. Several researchers have documented numerous instances (counterexamples according to Duverger s original formulation) of two-party, or two-candidate, outcomes under the run-off rule. Shugart and Taagepera (1994) calculate the effective number of candidates in presidential run-off elections worldwide and find many instances in which it is less than three. Wright and Riker (1989) detail examples of two-candidate outcomes in southern U.S. gubernatorial primaries (as does Canon 1978). For legislative elections Duverger (1954) presents the case of Australian parliamentary elections as a stable two-party system under the run-off rule. 19 Evidence in support of the dynamics underlying the model and the importance of initial conditions in political systems is more difficult to obtain, yet suggestive evidence can be found in the recent experience under the run-off rule of two stable democracies. In Australia the run-off rule was established soon after the founding of the federal government. Critically, however, the run-off rule was preceded by the plurality rule, which helped establish the two-party system that has persisted to the modern day. Thus, at the time the run-off rule was introduced there existed an established two-party system and there was no incentive, as explained in the current model, for a third party to enter. In contrast to the Australian experience stands the French Fifth Republic in which the run-off rule has supported a multiparty system. 0 For most national elections between 1945 and 1988 France employed a dual-ballot run-off rule for both parliamentary seats and the president. 1 The first system employed after the war, however, was proportional representation, which induced the entry of many parties. Thus, at the time the run-off rule was introduced multiple parties already existed, and as these parties did not necessarily have the incentive to exit, the electoral system of France was (and still is) characterized by a multiparty system. Within this multiple-party system, however, a notable feature of party evolution in France is that the effective number of parties competing and succeeding has been diminishing through time (Lijphart 1994, p. 161). 19 Although there are three major parties in Australia, one of these parties, the National Party, is effectively the rural arm of the conservative Liberal Party (see Riker 198). 0 See Lijphart (1994) or, for a more detailed analysis, Schlesinger and Schlesinger (1990). 1 For the presidential election only the top two candidates in the first round were permitted to compete in the runoff stage. For the parliamentary elections any candidate with more than 1.5% of the vote (the threshold was 10% for some elections) could proceed. This lower threshold would, most likely, give even greater incentive to rationalize into two parties as in this case the rule more closely resembles a plurality rule. See Cox (1997) for models and evidence of the tendency of multiple parties to enter proportional representation elections.