Multiple votes, multiple candidacies and polarization

Under Approval Voting, every citizen can vote for as many candidates as she wishes, and the candidate with the most votes is elected. Approval Voting was popularized by Brams and Fishburn (1978). It is currently used by several professional and academic associations to elect their officers, and by the UN to elect its secretary general. For recent scholarship on this topic see Laslier and Sanver (2010) edited handbook.

The phenomenon we call multiple candidacies has been previously referred to as “duplicate candidacies” in Myerson (2002) and is related to the concept of “clone candidates” in Tideman (1987).

The assumption of a unidimensional policy space is made to facilitate comparison with related contributions (e.g., Cox 1987, 1990; Myerson and Weber 1993), in which the policy space is assumed to be a closed interval on the real line.

A continuum of citizens is assumed in order to be consistent with the sincere voting assumption. Indeed, a sincere voting profile (as any other voting profile) is then a Nash equilibrium given that no vote can ever be pivotal.

Formally, we endow \(\mathcal {N}\) with the structure of a probability space \((\mathcal {N}, \Sigma , \mu )\) with a sigma algebra \(\Sigma \) and a probability measure \(\mu \). Let a \(\Sigma \)-measurable function \(\chi {:}\, \mathcal {N} \rightarrow X\) assign to each citizen \(n \in \mathcal {N}\) an ideal policy \(x_n \in X\). The probability measure \(\mu \) then induces a cumulative distribution function \(F(\cdot )\) on X such that \(F(y)=\mu \{n \in \mathcal {N}:x_n \le y\}.\)

Lee et al. (2004) provides empirical support for this assumption. Moreover, our main conclusion—i.e., \(\left( s,t\right) \)-rules can support more, not less, policy polarization than the Plurality Rule—would become trivial if we were to relax this assumption. Indeed, if candidates can commit on implementing policies other than their ideal ones, then only policies close to the median are supported by equilibria under the Plurality Rule; polarization is then minimal in Plurality Rule elections. Relatedly, Dellis and Oak (2007) adopts the citizen-candidate approach, as in the current paper, but with strategic voting behavior. They show that when candidates cannot commit on implementing policies other than their ideal ones, Approval Voting supports less policy polarization than the Plurality Rule, while the reverse is true when candidates can commit on implementing any policy. The key feature underlying this result is again the possibility for multiple candidate clusters.

All the results and proofs for an arbitrary finite number of positions are available from the authors upon request.

Assuming a finite number of potential candidates provides a justification for why potential candidates are strategic when making their candidacy decision, but sincere when making their voting decision.

Observe that \(s\ge 1\) rules out complete abstention. Such abstention can be ignored here since voting is costless and information is complete; with a finite (possibly large) electorate, complete abstention would be weakly dominated by a sincere vote.

To give an example, if three candidates are standing for election (i.e., \( c=3 \)), then each voter must vote for exactly one candidate if \(s=t=1\), for exactly two candidates if \(t\ge s\ge 2\), and for at least one candidate and at most two candidates if \(s=1\) and \(t\ge 2\).

i.e., measurable with respect to \(\Sigma \), the sigma algebra defined on \(\mathcal {N}\).

In Sect. 5 of the paper we discuss alternatives to this definition of sincere voting.

Formally, if every citizen in a set A is indifferent between casting h votes among k (\(>h\)) candidates, then each of these k candidates receives a mass \(\frac{h}{k}\mu \left( A\right) \) of votes from the citizens in A.

If \(y<1/3\), he loses outright. If \(y=1/3\), all three candidates tie for first place. In this case, the candidate at \(x_{M}\) does not want to enter the race since his expected utility gain is equal to \(\left( 1/3\right) \left( 1/3\right) =1/9\), which is smaller than the candidacy cost \(\delta =1/5\).

This extreme case follows from the uniform distribution of ideal tax rates, which is used here for expositional purposes.

Observe that this cannot be possible under the Plurality Rule since \(t=1\) and, as noted above, there cannot be more than t candidates at a position. Hence, we already have the contradiction in the case of the Plurality Rule.

Example 1 in the supplementary material contains a 3-position equilibrium under Approval Voting.

This observation, together with the fact that the set of policies supported by 1- and 2-position equilibria is an interval centered around the median m, is proven formally in Lemma 4 in the Appendix.

Observe that, as stated in Proposition 1, an \(\left( s,t\right) \)-rule where \(1<s=t\le p\) can, for some distributions of citizens’ ideal policies, support the same level of polarization as the Plurality Rule. This happens for example when the distribution of ideal policies is so concentrated around the median m that only 1-position equilibria exist.

Observe that all \(\left( s,t\right) \)-rules with \(s=t>p\) support the same level of polarization. This is because their equilibrium sets are equivalent. This is easily seen given that: (1) their 1-position equilibrium sets are equivalent (by Lemma 1); (2) a pair \(\left\{ x_{L},x_{R}\right\} \) is supported by a 2-position equilibrium under any \(\left( s,t\right) \)-rule where \(s=t>p\) if and only if \(-\frac{u\left( \left| x_{L}-x_{R}\right| \right) }{2} \ge \delta >-u\left( \left| x_{L}-x_{M}\right| \right) \), with p candidates at each of the two positions, a condition which is identical for all \(\left( s,t\right) \)-rules with \(s=t>p\) (by Lemma 2); and (3) there are no 3-position equilibria (by Lemma 3). This observation, together with Proposition 1 (and Proposition 2 below), implies that the \(\left( s,t\right) \)-rules with \(s=t>p\) are the ones in the family of \(\left( s,t\right) \)-rules that support the least polarization.

Situations where \(p<t\) are easier to understand. Since \(p<t\), a candidate at \(x_{M}\) entering the race would receive a vote from every citizen and win the election outright. Only the less polarized 2-position equilibria can therefore be supported under the \(\left( t,t\right) \)-rule. It follows trivially that the \(\left( s,t\right) \)-rule can support more polarized 2-position equilibria.

Observe that the voting profile under the \(\left( t,t\right) \)-rule (as described in the discussion of Proposition 1) differs only in that the citizens in the middle vote for \(\left( t-1\right) \) of the t candidates at \(x_{L}\) (if they are centre-leftists) or for \(\left( t-1\right) \) of the t candidates at \(x_{R}\) (if they are centre-rightists); they do not vote for all the t candidates standing at that position.

Observe that the key difference between \(\left( s,t\right) \)-rules that allow partial abstention and \(\left( s,t\right) \)-rules that do not, and a driving force behind Proposition 2, is the greater multiplicity of sincere vote profiles under the former rules. Interestingly, the desirability of the greater multiplicity of voting profiles under Approval Voting as compared to the Plurality Rule has been the object of a heated debate. On the one hand, Donald Saari argues that the greater multiplicity of voting profiles is a detriment since it makes the election outcome under Approval Voting highly indeterminate (e.g., Saari and Newenhizen 1988; Saari 2001). On the other hand, Steven Brams argues that the greater multiplicity of voting profiles is beneficial since it makes Approval Voting responsive to voters’ preferences (e.g., Brams et al. 1988; Brams and Sanver 2006).

One may object that the result in Proposition 2 follows because the notion of sincere voting does not put enough restrictions on exactly how many candidates a citizen votes for. One may then want to consider an alternative definition of sincere voting, namely, pure sincerity (to use a terminology proposed in Merrill and Nagel 1987). A voting decision for a citizen is purely sincere if (1) she votes for as many as possible of the candidates from whom she gets at least as much utility as the average utility over the whole set of candidates, and 2) she votes for as few as possible of the candidates for whom she gets less utility than the average utility over the whole set of candidates. (Fishburn and Brams 1981 argues that this is the way citizens should be voting under Approval Voting). Under pure sincerity, we find that (1) the level of polarization supported by \(\left( s,t\right) \)-rules where \(p<t\), whether \(s<t\) or \( s=t \), is smaller than, or the same as, the level of polarization supported by the Plurality Rule, whereas (2) the level of polarization supported by \(\left( s,t\right) \)-rules where \(p\ge t>s=1\), is greater than, or the same as, the level of polarization supported by the Plurality Rule. (A complete characterization of equilibria under pure sincerity is available from the authors.) This is because in the second group the same logic as under the assumption of sincere voting applies, whereas in the first group a candidate at \(x_{M}\) running against candidates at \(x_{L}\) and at \(x_{R}\) would necessarily receive a vote from every citizen and win outright. Observe that Approval Voting belongs to the first group (\(t=3p>p\ge s=1\)). It follows that whether Approval Voting would support more or less polarization than the Plurality Rule depends on the way citizens would vote. Whether citizens would vote sincerely, purely sincerely or otherwise is an empirical question.

When \(\beta \ge 2\delta \), the differences in candidacy incentives across voting rules get (partly) dominated by the effect of office-motivation, muddying the analysis. With \(\beta \ge 2\delta \), the set of 1-position equilibria need no longer be equivalent under every voting rule. This is because two or more candidates want to stand for election. As a result, the characterization of 1-position equilibria involves sets of candidates of cardinality three or more (at least two candidates, plus one entrant). With three or more candidates, different \(\left( s,t\right) \)-rules may elect different candidates. Hence, 1-position equilibria need no longer be equivalent under all voting rules. Moreover, with \(\beta \ge 3\delta \), 2-position equilibria may involve candidates at \(x_{M}\), and 3-position equilibria may exist even under \(\left( s,t\right) \)-rules where \(s=t\).

An example illustrating the above discussion (Example 3) is provided in the online appendix.

The Alternative Vote Rule is a voting rule which is currently used for lower House elections in Australia and for presidential elections in Ireland. Moreover, it has recently gained popularity in the electoral reform debate, having been proposed as a replacement to the Plurality Rule (e.g., in British Columbia, Canada in 2009 and in the UK in 2011). Under the Alternative Vote Rule, every voter ranks the candidates from first to last. A candidate is elected if he is ranked first on a majority of ballots. Otherwise, the candidate who receives the smallest number of first-place votes is eliminated and his first-place votes are transferred to the candidates who are ranked next on those ballots. If one of the remaining candidates now receives a majority of first-place votes, he is elected. Otherwise, a second candidate is eliminated and the process is repeated until one candidate receives a majority of first-place votes.

Throughout the proofs we shall use the tilde to refer to a situation in which a potential candidate has deviated from his candidacy strategy.

This follows from \(c_{R}>c_{L}\ge s\) and the definition of sincere voting (according to which a citizen who is not indifferent between all candidates votes for as many of her most-favorite candidates and for as few of her least-favorite candidates as possible). This property is used repeatedly throughout the proof.

This follows straightforwardly if \(\widehat{c}_{L}=\widehat{c}_{R}=s\). If \( \widehat{c}_{L}=\widehat{c}_{R}>s\), it follows from \(\widehat{c}_{L}= \widehat{c}_{R}>1\) and \(-\frac{u\left( \left| \widehat{x}_{L}-\widehat{x} _{R}\right| \right) }{2\left( \widehat{c}_{L}+\widehat{c}_{R}-1\right) } \ge \delta \).

To see this, assume by way of contradiction that \(F\left( \overline{x} ^{\prime }\right) \ge \frac{1}{2}+\frac{t}{1+t}F\left( \underline{x} ^{\prime }\right) \). Observe that \(F\left( \overline{x}\right) \ge 1-F\left( \underline{x}\right) \) and \(F\left( \overline{x}\right) \le \frac{ 1}{2}+\frac{1}{t}\left[ \left( t+1\right) F\left( \underline{x}\right) - \frac{1}{2}\right] \) imply \(F\left( \underline{x}\right) \ge \frac{1}{2} \frac{t+1}{2t+1}\). Also, \(F\left( \overline{x}^{\prime }\right) <1-F\left( \underline{x}^{\prime }\right) \) and \(F\left( \overline{x}^{\prime }\right) \ge \frac{1}{2}+\frac{t}{1+t}F\left( \underline{x}^{\prime }\right) \) imply \(F\left( \underline{x}^{\prime }\right) <\frac{1}{2}\frac{t+1}{2t+1}\). Taken together, these two implications give \(F\left( \underline{x}\right) >F\left( \underline{x}^{\prime }\right) \); a contradiction.

To see this, assume by way of contradiction that \(F\left( \overline{x} ^{\prime }\right) \ge \frac{1}{2}+F\left( \underline{x}^{\prime }\right) \). Since \(F\left( \underline{x}\right) <F\left( \underline{x}^{\prime }\right) \) and \(F\left( \overline{x}\right) >F\left( \overline{x}^{\prime }\right) \), we then get \(F\left( \overline{x}\right) >\frac{1}{2}+F\left( \underline{x} \right) \). The latter, together with \(F\left( \overline{x}\right) \le \frac{ 1}{2}+\frac{1}{t}\left[ \left( t+1\right) F\left( \underline{x}\right) - \frac{1}{2}\right] \), implies \(F\left( \underline{x}\right) >1/2\); a contradiction.

This is illustrated for some \(\left( s,t\right) \)-rules in the examples presented in the paper and in the online appendix. More generally, this happens for example when: 1) \(s<t\le p\); 2) there is a 2-position equilibrium under the \(\left( s,t\right) \)-rule where \(s\le c_{L}=c_{R}<t\) under the \(\left( s,t\right) \)-rule; and 3) \(F\left( \overline{x}\right) < \frac{1}{2}+F\left( \underline{x}\right) \), \(F\left( \overline{x}\right) \ge 1-F\left( \underline{x}\right) \), \(F\left( \overline{x}\right) >\frac{1 }{2}+\frac{1}{t}\left[ \left( t+1\right) F\left( \underline{x}\right) -\frac{ 1}{2}\right] \) and \(\delta \le -u\left( \left| x_{L}-x_{M}\right| \right) \). In this case a potential candidate at \(x_{M}\) would want to enter against candidates at \(x_{L}\) and \(x_{R}\) when the election is held under the \(\left( t,t\right) \)-rule. As a result, a 2-position equilibrium would not exist under the \(\left( t,t\right) \)-rule.