O R I G I N A L PA P E R
Bernardo Moreno . M. Socorro Puy
The scoring rules in an endogenous election
Received: 10 September 2002 / Accepted: 26 April 2004 / Published online: 23 November 2005 © Springer-Verlag 2005
Abstract Plurality rule is mostly criticized from being capable of choosing an alternative considered as worst by a strict majority. This paper considers elections in which the agenda consists of potential candidates strategically choosing whether or not to enter the election. In this context, we examine the ability of scoring rules to fulfil the Condorcet criterion. We show for the case of three potential candidates that Plurality rule is the only scoring rule that satisfies a version of the Condorcet criterion in two cases: 1) when preferences are single-peaked and, 2) when pref-erences are single-dipped.
1 Introduction
In some voting processes, a previous announcement of candidates or policy choices usually takes place. When it occurs, the agenda consists of potential candidates strategically choosing whether or not to enter the election. This is the case for instance, of the primaries in some political parties, the elections of representative members (as the head of some state universities, some labor unions), the election of a policy by a committee, etc. These voting procedures have in common that they can be viewed as a two-stage process. In the first stage or entry-decision stage the voting alternatives or candidates are self- declared and in a second stage, a voting rule elects one of these alternatives.
Osborne and Slivinski [13] and Besley and Coate [1] provide the first analysis of this type of scenario. Subsequently, Dutta, Jackson and Le Breton [4] show that the
B. Moreno . M. Socorro Puy (*)
Departamento de Teoría e Historia Económica, Universidad de Málaga, Campus el Ejido, 29013 Málaga, Spain
outcome of every non-dictatorial and unanimous1voting procedure is modified by strategic candidacy, i.e., the strategic behavior at the entry-decision stage. Therefore, exploring the equilibrium outcome of some commonly used voting rules when we account for strategic candidacy, has become an important issue. In this direction, Dutta, Jackson and Le Breton [5] explore the properties of the successive elimination voting procedure. They find that strategic candidacy provides, in direct contrast to the fixed agenda case, equilibrium outcomes which include Pareto dominated alternatives. The Condorcet Consistency criterion is the most commonly used principle for evaluating alternative voting rules. In an election, an alternative is a Condorcet winner (Condorcet [2]) when it is majority preferred to any other alternative. Thus, theCondorcet Consistency criterionrequires the election of the Condorcet winner when such an alternative exists. While a Condorcet winner may not always exists, there are some particular domains of preferences (as single-peaked preferences) for which we can guarantee its existence.
In a fixed agenda election, a well known result is that every scoring rule violates the Condorcet Consistency criterion even when the election involves three candi-dates and preferences are single-peaked. The scoring rules (among which Plurality rule and Borda rule are the best known) are a class of voting rules where the winner is determined by computing a score that depends on the rank of the alternatives in the voters’preference orders.
In this paper, we examine the ability of scoring rules to fulfill the Condorcet criterion when we account for strategic candidacy. We consider a finite set of voters. We study three different domains: 1) peaked preferences, 2) single-dipped preferences, and 3) separable preferences. We then examine the equilibrium outcomes of the two-stages voting game when there are three potential candidates. For the case of single-peaked preferences, we firstly find that no scoring rule satisfies Condorcet Consistency. We then propose a weaker property that we call Candidacy Condorcet Consistencywhich requires electing the Condorcet winner whenever the Condorcet winner alternative enters the contest. We then find that Plurality rule is the only scoring rule which satisfies this property. For the case of single-dipped preferences, we find that Plurality rule is the only scoring rule that satisfies Condorcet Consistency. And finally, for the case of separable preferences, no scoring rule satisfies the proposed Condorcet criteria.
The rest of the paper is organized as follows: Section2describes the model, Section3states the results and Section4just makes some concluding remarks.
2 Model
LetN= {1, . . . ,n} be a finite set of voters, wherenis an odd number.
Let A={a, b, c} where A⊂N be the set of potential candidates. The three candidates are different andσ= (a,b,c) is an ordering of the elements ofA.
Voters have strict preferences over the set of candidates. The preference relation ofiis denoted byPi. We say thataRibif eitheraPibora=b. LetP denote the set
of preference profiles andPdenotes a generic profile inP.
1Unanimity requires that if all voters find the same candidate most preferred out of the entering
Given anyC⊂A, letPi∣Cdenote the binary relation onCinduced byPiandP∣C
the profile of induced relations. We assume that every candidate strictly prefers an element ofAto the situation where no candidate is elected.
When the set of candidates isC=A, lets= (1,w, 0) be a vector of scores such that 0≤w≤1. And when the set of candidatesC⊂Ais such that #C=2, the vector of scores is given bys=(1, 0).2
Ascoring ruleis a function Sw:2An∅P !A such that for all C⊂Aand P2Peach voter gives a score to each candidate, and the total score of a candidate is obtained summing up the scores given by all voters to this candidate. The candidate elected by a scoring ruleSw(C,P)∈C, is the one obtaining the greatest
score. Some of the best known scoring rules are Plurality ruleS0wherew= 0, and
Borda ruleS1
2 wherew¼
1 2.
A scoring rule chooses a candidate from the set of available candidates. The candidate elected by a scoring rule just depends on voters’preferences over the set of feasible candidates. We assume that voters give scores to the candidate in a sincere way and that in case of ties a deterministic tie-breaking rule chooses one of the candidates.
The electoral mechanism follows two stages. In the first stage or entry decision stage, each potential candidate announces whether or not he wants to become candidate. This entry stage determines the set of self-declared candidatesC⊆A. In a second stage, given a vector of scores which is defined for some 0≤w≤1, each voter gives a score to each of the candidates inC, and the candidate obtaining the greatest score becomes the winning candidate.
The entry decision is the only strategic decision made by the potential can-didates.3Thus, the equilibrium concept that we consider is Nash equilibrium. The setC⊆Ais anentry equilibriumrelative toSwandP2P ifSw(C,P)RjSw(C\{j},
P) for all j∈C and Sw(C, P)RjSw(C∪{j}, P) for all j∈A\C. Thus, an entry
equilibriumC requires that on the one hand no candidate inCstrictly improves withdrawing from the contest and on the other hand that no candidate in A\C strictly improves entering the race.
3 Condorcet consistency
We say that candidatej defeatscandidatek in pairwise majority comparisonif a strict majority of voters prefersjoverk.
TheCondorcet winnerthat we denote bym, wherem∈Ais the candidate that defeats every other candidate in pairwise majority comparisons. A strong Condorcet winner is a candidate ranked in first position by more than half of the voters.
In contrast, a candidate is Condorcet loser if it is defeated by each of the remaining candidates in pairwise majority comparisons.
Following Moulin [12], we define the property of Condorcet Consistency. A scoring ruleSwsatisfiesCondorcet Consistencyif in every entry equilibriumC, we
2Note that in two-candidate elections, all scoring rules coincide with Majority rule.
3This assumption is also made by Osborne and Slivinsky [13]. Besley and Coate [1] assume
have that Sw(C; P) =m for every P2P. Thus, a scoring rule is Condorcet
Consistent if every entry equilibrium is such that the Condorcet winner is elected. Taking into account that in an endogenous context the Condorcet winner may not become candidate, we next define a weaker concept of Condorcet Consistency. With this new property we try to keep the same spirit of the Condorcet criterion in the context of endogenous elections. A scoring rule Sw satisfies Candidacy
Condorcet Consistencyif in every entry equilibriumCwherem∈C, we have that Sw(C, P) =m for every P2P. Thus, a scoring rule is Candidacy Condorcet
Consistent if in every entry equilibrium where the Condorcet winner enters the race, the Condorcet winner is elected.
From the set of all scoring rules, we firstly analyze what rules satisfy Condorcet Consistency and/or Candidacy Condorcet Consistency. We study three different domains: single-peaked preferences, single-dipped preferences, and separable preferences.
3.1 Single-peaked preferences
LetPsp denote the set of single-peaked preference profiles with respect to ordering σandPdenotes a generic profile inPsp. Four groups of preference rankings are then admissible:
n1 aPibPic n2 bPiaPic n3 bPicPia n4 cPibPia
wheren1,n2,n3, andn4indicate the number of voters with each of the preference
rankings. A generic profilePcan be then described as a vector whereP= (n1,n2,
n3,n4).
We assume that candidateais of typen1, candidatebis either of typen2orn3,
and candidatecis of typen4.
Since preferences are single-peaked and the number of voters is odd, the Condorcet winner exists and is unique.
Theorem 1 In an endogenous election where preferences are single-peaked, there is no scoring rule that satisfies Condorcet Consistency.
Proof For all Swwhere 0≤w≤1 we next show that there exists P2Psp and an
entry equilibriumCsuch thatSw(C,P)≠m.
Claim 1 Plurality rule is such thatS0(C,P)≠mfor some entry equilibriumCand
someP2Psp:
LetP=(2, 1, 0, 2) and consider thatC={a,c}. Then,S0(C, P)=awhereas the
Condorcet winner isb. Since no candidate inChas incentives to withdraw, andb cannot win the election by means of entering the race,Cis an entry equilibrium.
Claim 2 Every scoring ruleSwwhere 0 <w≤1 is such thatSw(C,P)≠mfor some
LetP= (n1, 0,n3,n4) wheren1,n3,n4> 0 and consider thatC=A. Then, for every
0 <w≤1,Sw(C,P) =bif the following conditions (1) and (2) hold:
n3þw nð 1þn4Þ>n1 (1)
n3þw nð 1þn4Þ>wn3þn4: (2)
However, if the following condition (3) holds, candidateais Condorcet winner:
n1>n3þn4: (3)
Note that inequalities (1) and (3) imply (2). We next show that some voting numbers can be assigned such that inequalities (1) and (3) are satisfied. For instance letn1¼n 12þ"2
,n3¼n 12"
>0 andn4¼n"2 where"> 0 and close enough to zero and where nis the total number of votes.4Condition (3) clearly holds, substituting the voting numbers and simplifying Inequality (1), we have
" < w
32w: (4)
Therefore, for"satisfying (4), inequalities (1) and (3) hold which implies thatb is elected by every scoring rule Swwhere 0 <w≤1, whereas ais the Condorcet
winner. It is clear that candidate bhas no incentives to withdraw. Furthermore, neitheranorccan improve withdrawing since thenbbecomes their most preferred candidate, so thatCis an entry equilibrium.
▪
The above Theorem provides two counterexamples of entry equilibria showing that in an endogenous election all the scoring rules can fail to elect the Condorcet winner. Since Theorem 1 provides a negative result in a particular domain of preferences, the result can be extended to an unrestricted domain of preferences.
As it is shown in the proof, there are some entry equilibria where two candidates enter the contest that illustrate that Plurality rule does not satisfy Condorcet Consistency. The reason for this is that in equilibrium the Condorcet winner does not become candidate.
For all the scoring rules but Plurality rule, we provide an entry equilibrium where the three potential candidates enter the race. This example illustrates that even when a candidate is a strong Condorcet winner,5these scoring rules fail to elect such candidate.
Since all the scoring rules fail to satisfy Condorcet Consistency, we secondly analyze what scoring rules satisfy Candidacy Condorcet Consistency.
Theorem 2 In an endogenous election where preferences are single-peaked, Plurality rule is the only scoring rule that satisfies Candidacy Condorcet Consistency.
4Fornsufficiently large the voting numbers can be obtained as integer numbers.
Proof We make the proof in two steps.
Step 1 Let us show that Plurality rule is Candidacy Condorcet Consistent. Depending onC, we distinguish three cases:
Case 1: If the entry equilibrium is such thatC={m}, thenS0(C,P) =mfor every
P2Psp
Case 2: If the entry equilibrium is such that C=A\{j} where j≠m, then the Condorcet winner defeats in pairwise majority comparisons every other candidate so thatS0(C,P) =mfor everyP2Psp:
Case 3: If the entry equilibrium is such thatC=A, three cases can be distinguished:
3.1)
IfS0(C,P) =a, it implies thatn1>n2+n3andn1>n4. Candidateccannot be a
Condorcet winner since it requires thatn4>n1+n2+n3which contradicts that
n1>n4. Ifbis the Condorcet winner, Plurality rule may not satisfy Candidacy
Condorcet Consistency sinceS0(C,P)≠b. In this case however, candidatec
has incentives to withdraw since thenS0(C\{c},P) =bso thatS0(C\{c},P)
PcS0(C,P) which contradicts thatCis an entry equilibrium. It then implies
that in every entry equilibrium whereS0(C,P)=a, the Condorcet winner isa.
3.2)
IfS0(C,P)=b, it implies thatn2+n3>n1andn2+n3>n4so that neitheranorccan
be Condorcet winner. Thus,bis the Condorcet winner, and thenS0(C,P)=m.
3.3)
IfS0(C,P)=c, this case is symmetric (but equivalent) to that in 3.1) and a
similar argument shows thatcshould necessarily be the Condorcet winner.
Step 2 Let us show that for allSwwhere 0 <w≤1 there exists P2Psp such that
Sw(C,P)≠mfor some entry equilibriumCwherem∈C.
It directly follows from the counterexample of Claim 2 in Theorem 1.
▪
As we show, the only entry equilibrium where Plurality rule fails to elect the Condorcet winner is such that two candidates enter the contest and none of them is Condorcet winner. When we restrict attention to those entry equilibria where the Condorcet winner becomes candidate, we find that Plurality rule elects the Condorcet winner. For all the other scoring rules however, some entry equilibria always exist such that either candidate a or c are Condorcet winner whereas candidatebis the elected candidate. Note that this result is directly connected to the fact that Plurality rule is the only scoring rule that always selects a strong Condorcet winner whenever such candidate exists.As we next show, a crucial assumption for the results in Theorem 2 is restricting the set of potential candidates to three.
Proposition 1 If there are strictly more than three potential candidates, Plurality rule in an endogenous election where preferences are single-peaked fails to satisfy Candidacy Condorcet Consistency. Furthermore, Plurality rule can select a Con-dorcet loser.
Proof LetA′={a,b,c,d} be the set of potential candidates. Consider the following groups of single-peaked preference rankings
n1 aPibPicPid n2 bPiaPicPid n3 cPibPiaPid n4 dPicPibPia
wheren1=n2=n3=2 andn4= 5, so thatcis a Condorcet winner. LetC=A′be the set
of candidates, thendis elected by Plurality rule. Since no candidate can strictly improve withdrawing,Cis an entry equilibrium where the elected candidatedis a Condorcet loser.
▪
From the obtained results in Theorem 2 we also deduce that the Condorcet loser is never selected under Plurality rule in an endogenous election involving three potential candidates.6However, as it follows from Proposition 1, in four-alternative endogenous elections the Condorcet loser can be elected.
3.2 Single-dipped preferences
LetPsd denote the set of single-dipped preference profiles with respect to ordering σandPsd denotes a generic profile inP: Four groups of preference rankings are then admissible:
n1 aPibPic n2 aPicPib n3 cPiaPib n4 cPibPia
wheren1,n2,n3, andn4indicate the number of voters with each of the preference
rankings. A generic profilePcan be then described as a vector whereP=(n1,n2,
n3,n4).
We assume that candidateais either of typen1orn2, candidatecis either of
typen3orn4, and candidatebcan be of every type.
Since preferences are single-dipped and the number of voters is odd, the Condorcet winner exists and is unique.
Theorem 3 In an endogenous election where preferences are single-dipped, Plurality rule is the only scoring rule that satisfies Condorcet Consistency.
Proof Under single-dipped preferences it follows that only a or c can be Condorcet winner. We make the proof in two steps.
6Furthermore, it can be shown that in an endogenous election involving three candidates no
scoring rule elects a Condorcet loser. By contrast Lepelley et al. [10] show, in the classical three candidate election, that a Scoring rule can elect a Condorcet loser if and only if0w<1
Step 1 Let us show that Plurality rule is Condorcet Consistent.
Suppose without lost of generality thatais Condorcet winner. Thenadefeatsc in pairwise majority comparison so thatn1+n2>n3+n4. It then implies thatais
strong Condorcet winner. Since Plurality rule always elects the strong Condorcet winner when it exists, in every entry equilibrium candidateawill enter the contest and win the election.
Step 2 Let us show that every scoring rule Sw where 0 <w≤1 fails to satisfy
Condorcet Consistency (and Candidacy Condorcet Consistency). LetP= (n1,n2,n3,n4) wheren1,n2,n3,n4> 0, and consider thatC=A.
For every 0 <w≤1,Sw(C,P) =cif the following conditions hold:
n3þn4þwn2>n1þn2þwn3 (5)
n3þn4þwn2>w nð 1þn4Þ: (6)
However, if the following condition holds, candidateais Condorcet winner:
n1þn2>n3þn4: (7)
We next show that some voting numbers can be assigned such that inequalities (5), (6) and (7) are satisfied. For instance, letn1¼n"2;n2¼n 12"2
;n3¼n "2
; n4¼n 1232"
where"> 0 and close enough to zero and wherenis the total number of votes. Sincen1+n2>w(n1+n4), condition (5) implies condition (6). Condition (7)
clearly holds. Substituting these voting numbers in inequality (5) and simplifying, we have
" <w2 2ð þwÞ: (8)
Thus, for "satisfying (8), candidate cis elected by every scoring rule where 0<w≤1, whereas candidateais Condorcet winner. Furthermore, if candidateais of typen2and candidatebis of typen4, none of this candidates can strictly improve
withdrawing from the contest. Thus, C=A is an entry equilibrium where the Condorcet winner is not elected.
▪
Single-dipped preferences imply that there always exists a strong Condorcet winner. Therefore, in every entry equilibrium under Plurality rule the strong Condorcet winner is elected, and thus, in this domain of preferences Plurality rule is Condorcet consistent for any number of potential candidates.7
3.3 Separable preferences
We next show that there are some domains of preferences out of the scope of single-peaked and single-dipped preferences, such that Plurality rule fails to satisfy Candidacy Condorcet Consistency (and therefore, Condorcet Consistency).
LetPsep denote the set of separable preference profiles with respect to candidate a,so that the following groups of preference rankings are then admissible:8
n1 aPibPic n2 aPicPib n3 bPicPia n4 cPibPia
wheren1,n2,n3, andn4indicate the number of voters with each of the preference
rankings. A generic profilePcan be described as a vector whereP=(n1,n2,n3,n4).
We assume that candidateais either of typen1orn2, candidatebof typen3,
and candidatecis of typen4.
Since preferences are separable and the number of voters is odd, the Condorcet winner exists and is unique.
Theorem 4 In an endogenous election where preferences are separable, there is no scoring rule that satisfies Candidacy Condorcet Consistency.
Proof For allSwwhere 0≤w≤1 we next show that there existsP2Psep and an
entry equilibriumCsuch thatSw(C,P)≠m.
Claim 1 Plurality rule is such thatS0(C,P)≠mfor some entry equilibriumCand
someP2Psep:
Let P=(1, 3, 5, 4) and consider that C=A. Then, S0(C, P) =b whereas the
Condorcet winner isc. Neither candidatebnorchave incentives to withdraw, and if we suppose that candidateais of typen1, he cannot strictly improve withdrawing.
Therefore,Cis an entry equilibrium.
Claim 2 Every scoring ruleSwwhere 0 <w≤1 is such thatSw(C,P)≠mfor some
entry equilibriumCand someP2Psep:
It directly follows from the counterexample of Claim 2 in Theorem 1 (where note thatn2=0 implies that the proposed profile of single-peaked preferences also
qualifies as a profile of separable preferences).9
▪
Thus, we find that in the domain of separable preferences, no scoring rule satisfies the proposed Condorcet criteria.
4 Conclusions
While the property of Condorcet Consistency has been criticized by Dummett [3] and Saari [14] as it is a too strong requirement, these authors justify introducing some less restrictive requirements. In this way, several alternative criteria have been proposed in a fixed agenda election (see for instance Fishburn and Gehrlein [6], Lepelley and Merlin [9] and more recently Sanver [16] and Woeginger [17]) to deepen into the analysis of voting rules. While in a fixed alternative election these
8These preferences are separable since alternativeais never middle-ranked. 9This Theorem was suggested by an anonymous referee.
authors come up with additional properties, such as Condorcet efficiency,10 no property has been proposed so far in the analysis of endogenous alternative elections.
With this paper, we provide a first step studying the scoring rules when the agenda is endogenously determined. As a weaker property we have introduced the concept of Candidacy Condorcet Consistency. We analyze three different domains of preferences.
For the case ofsingle-peaked preferences, we have shown that Plurality rule is the only scoring rule that satisfies Candidacy Condorcet Consistency. It is important to note that this result is in sharp contrast with the classical results obtained when considering fixed agenda elections. In the specific case where 1) preferences are single-peaked and 2) the fixed agenda election involves only three candidates, it is well-known that no scoring rule satisfies Condorcet Consistency. Furthermore, in this specific case, Plurality rule is mostly criticized from being capable of choosing a Condorcet loser.11 This is why some authors have recommended abandoning Plurality rule in favor of other voting rules (such as Borda rule).
For the case ofsingle-dipped preferences, we have shown that Plurality rule is the only scoring rule that satisfies Condorcet Consistency. For the case ofseparable preferences no scoring rule is Candidacy Condorcet Consistent (and hence, Condorcet Consistent). These results coincide with the classical results obtained when considering fixed agenda elections.
With the obtained results, we have shown that strategic candidacy provides additional arguments in favor of Plurality rule since it guarantees the election of the Condorcet winner in two cases 1) when preferences are single-peaked and there is no more than three candidates, and 2) when preferences are single-dipped. In this sense, our results are in the same vein as the one obtained by Forsythe et al. [7].12 While our results establishes a first step in the study of the scoring rules in an endogenous election, we think that analysis which incorporate additional properties as well as additional voting rules deserve further research. Another possible ex-tension is using the techniques of Saari and Valognes [15] to identify the set of profiles of preferences where the Condorcet winner is in the entry equilibrium and selected by Plurality rule. For a description of all the entry equilibria when just three candidates enter the contest and where there is no restriction on the set of potential candidates, see Moreno and Puy [11].
Acknowledgements We would like to thank Pablo Amorós, Carmen Beviá, Vincent Merlin and Matthias Messner for helpful comments and suggestions. We also thank two anonymous referees for their helpful comments. Financial support from Fundación Ramón Areces and the research project SEC2002-01926 are gratefully acknowledged.
10Which is defined as the conditional probability that the rule selects the Condorcet winner. 11See Lepelley [8] (Sections 4 and 5) for further comments on this point. This author also
provides the exampleP=(3, 0, 2, 2) to illustrate that candidateais a Condorcet loser selected by Plurality rule.
12These authors show by means of experiments that, under Plurality rule, pre-election polls
References
1. Besley T, Coate S (1997) An economic model of representative democracy. Q J Econ 112:84–94
2. Condorcet M (1785) Éssai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix, Paris
3. Dummett M (1984) Voting procedures. Oxford University Press, Oxford
4. Dutta B, Jackson MO, Le Breton M (2001) Strategic candidacy and voting procedures. Econometrica 69:1013–1037
5. Dutta B, Jackson MO, Le Breton M (2002) Voting by successive elimination and strategic candidacy. J Econ Theory 103:190–218
6. Fishburn PC, Gehrlein WV (1976) Borda’s rule, positional voting and Condorcet’s simple majority principle. Public Choice 28:79–88
7. Forsythe R, Myerson RB, Rietz TA, Weber RJ (1993) An experiment on coordination in multi-candidate elections. Soc Choice Welf 10:223–248
8. Lepelley D (1993) On the probability of electing the Condorcet loser. Math Soc Sci 25: 105–116
9. Lepelley D, Merlin V (1998) Choix social positionnel et principe majoritaire. Ann Econ Stat 51:29–48
10. Lepelley D, Pierron P, Valognes F (2000) Scoring rules, Condorcet efficiency and social homogeneity. Theory Decis 49:175–196
11. Moreno B, Puy MS (2003) Plurality Rule Works in Three-Candidate Elections. CentrA, Documento de trabajo E2003/09
12. Moulin H (1988) Axioms of cooperative decision making. Chapter 9. Econometric Society Monographs no 15, Cambridge University Press, Cambridge
13. Osborne MJ, Slivinski A (1996) A model of political competition with citizen-candidates. Q J Econ 111:64–96
14. Saari DG (1994) Geometry of voting. Springer, Berlin Heidelberg New York
15. Saari DG, Valognes F (1999) The geometry of black’s single-peakedness and related conditions. J Math E 32:429–456
16. Sanver MR (2002) Scoring rules cannot respect majority in choice and elimination simultaneously. Math Soc Sci 43:151–155
17. Woeginger G (2003) A note on scoring rules that respect majority in choice and elimination. Math Soc Sci 46:347–354
