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Social Choice and Welfare

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07.05.2022 | Original Paper

Electoral Institutions with impressionable voters

verfasst von: Costel Andonie, Daniel Diermeier

Erschienen in: Social Choice and Welfare | Ausgabe 3/2022

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Abstract

We use a model of impressionable voters to study multi-candidate elections under different electoral rules. Instead of maximizing expected utility, voters cast their ballots based on impressions. We show that, under each rule, there is a monotone relationship between voter preferences and vote measures. The nature of this relationship, however, varies by electoral rule. Vote measures are biased upwards for socially preferred candidates under plurality rule, but biased downwards under negative plurality. There is no such bias under approval voting or Borda count. Voters always elect the socially preferred candidate in two-way races for any electoral rule. In multi-candidate elections, however, the ability to elect a Condorcet winner varies by rule. The results show that multi-candidate elections can perform well even if voters follow simple behavioral rules. The relative performance of specific electoral institutions, however, depends on the assumed behavioral model of voting.

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The nature of the ballot depends on the electoral rule. It corresponds to a “vote” under plurality rule, or in run-off elections, an “approval” under approval voting, a “negative vote” under negative plurality, and a ranking over candidates under the Borda rule.

See Bendor 2010 and Bendor et al. 2011 for detailed discussions.

Recent research (e.g., Spenkuch 2018) has argued that real electorates are best understood as heterogeneous, where some voters are rational, others impressionable or sincere. Modeling such electorates is valuable but very challenging. It also goes beyond the purpose of our paper.

The two remaining rules, Borda and runoff, are discussed later in Sect. 6.

For examples of such factors see (Gasper and Reeves 2011; Cole et al. 2012; Achen and Bartels 2002) and Wolfers (2002).

Casting a “favorable ballot for candidate i” corresponds to voting for i under plurality rule, approving of i under approval voting, or not voting in opposition to i under negative plurality.

The propensity to abstain \(p_{\theta }^{t}(A)\) can be computed from the vote propensities \(p_{\theta }^{t}\) and so does not need to be tracked explicitly. We will show how to compute \(p_{\theta }^{t}(A)\) from \(p_{\theta }^{t}\) when we analyse the stationary distributions.

This disadvantage is present only under approval voting. To see this, suppose that, under approval voting, there are three candidates and only one type of voters. Consider the following two distributions over the set B: (1) a distribution where ballots (1, 1, 0) and (0, 1, 1) each have probability \(\frac{1}{2}\) of being cast; (2) a distribution where ballots (1, 1, 1) and (0, 1, 0) each have probability \(\frac{1}{2}\) of being cast. Both of these distributions yield vote propensities \((p(1),p(2),p(3))=(\frac{ 1}{2},1,\frac{1}{2})\), i.e., an outcome where candidate 2 gets a measure of 1 of approval votes, while candidates 1 and 3 get a measure of \(\frac{1}{2}\). If we were to work directly with the probability distribution \(\{r_{\theta }^{t}(b)\}_{b\in B}\), we would be able to differentiate between the two cases, while the distribution of vote propensities \(\{p_{\theta }^{t}(i)\}_{i\in N}\) is not able to do so.

Alternatively, we could consider a more complex updating rule where updating varies with the magnitude of the voter’s impressions. That is, a voter with a “very negative” impression of a candidate may decrease that candidate’s propensity more heavily compared to the case of “moderately negative” impressions, etc. Adding such complexity, however, does not change the qualitative results of the analysis.

As under plurality rule, we could allow an updating process where the increase or decrease of propensities depends also on the magnitude of impressions. But again, this will not change the qualitative results.

It is easy to see that if \(v_{\theta }^{i}>v_{\theta }^{j}\), then \(q_{\theta }(i)>q_{\theta }(j)\), \(E[\frac{1}{1+I_{\theta }(-i)}]>E[\frac{1}{1+I_{\theta }(-j)}]\) and \(E[\frac{1}{1+J_{\theta }(-i)}]<E[\frac{1}{1+J_{\theta }(-j)}]\), and therefore \(S_{\theta }(i)>S_{\theta }(j)\) under each rule.

Similar implications hold when comparing differences in vote measures across groups. Under plurality rule, controlling for the differences \(q_{\theta }(i)-q_{\theta }(j)\) between two groups, the difference in vote measures \(S_{\theta }(i)-S_{\theta }(j)\) will be larger for the group for which, on average, the number of candidates, excluding candidates i and j, that voters of that group like is lower. Under negative plurality, in contrast, the difference in vote measures \(S_{\theta }(i)-S_{\theta }(j)\) will be larger for the group for which, on average, the number of candidates, other than i and j, that voters of that group like is larger.

A Condorcet winner beats all other candidates in pair-wise comparison. A Condorcet loser is beaten by all other candidates in pair-wise competition.

Different aspirations may shift all probabilities of positive impressions \(\{q_{\theta }(i)\}_{i\in N}\) either up or down, possibly in a non-linear fashion. However, this does not change the qualitative results of Propositions 1, and 2.

See Andonie and Diermeier (2019) for the details under plurality rule. The same holds for the other rules.

Two-candidate elections under plurality rule with impressionable voters have been analysed in detail in Andonie and Diermeier (2019). Here, we focus on the comparison among the three rules. We omit the proof of Proposition 3. The proof follows immediately by noting that the difference in the measures of votes of the two candidates, by Proposition 2, is \(S(1)-S(2)=\sum _{\theta }s_{\theta }(q_{\theta }(1)-q_{\theta }(2))\) in all three rules, and then using the same argument as in plurality rule.

Under approval voting, a voter that decides to vote will never vote for (or approve) both candidates, therefore (1, 1) can be eliminated from the set \(B^{a}\). In the negative rule, choosing action (1, 1) (i.e., abstention) is the same as choosing action (0, 0) in the other rules. Therefore, all sets B are the same in the three rules, and so their equilibria and associated behavior of voters will also be the same.

Under plurality rule, in the rational voter model there may be equilibria where voters vote for two of the candidates that excludes the uniformly preferred candidate. See, e.g., Proposition 1 in Myerson (2002).

Under negative plurality, in the rational voter model there may be equilibria where all candidates have a positive probability of winning. See, e.g., Proposition 2 in Myerson (2002).

Under approval voting, the concept of sincere voting is ambiguous. See, e.g., Myerson and Weber (1993) (footnote 11, p. 113), or Nunez (2014), for possible definitions of “sincere” voting under approval voting.

The assumptions that the measures of groups 1 and 2 of voters are the same, and that the component of impressions voters of groups 1 and 2 associate with candidate 3, i.e., parameter v, is the same for the two groups, are made for simplicity. The ordering of the three rules characterized in Lemma 1 does not change if, e.g., the parameters v of the two groups differ, but are close to one another.

Other cases that may be interesting are as follows. (1) If \(v>1\) and \(w>0\) then candidate 3 is “ranked” first by all voters. The analysis of this case is similar to that of \(v>1\) from the previous section, with similar conclusions. (2) If \(v<0\) and \(w<0\) then candidate 3 is “ranked” last by all voters. Again, the analysis of this case is similar to that of \(v<0\) from the previous section. (3) If \(v>1\) and \(w<0\) then voters of groups 1 and 2 “rank” candidate 3 first, while voters of group 3 “rank” candidate 3 last. The analysis and insights are similar to the case we discuss in this section.

The threshold \(w^{\phi }\) depends on various parameters such as v, F(.), \(\alpha\), etc. To simplify notation, we will write \(w^{\phi }\). It is possible that a value of w with this property does not exist. For example, if \(s\in (\frac{1}{4},\frac{1}{3})\), then in the plurality rule, a \(w^{p}\) with this property exists for any \(v<0\); in the approval rule, a \(w^{a}\) exists only for v with \(v^{a}<v<0\); while in the negative rule, a \(w^{n}\) exists only for v with \(v^{n}<v<0\). Moreover, the two cutoffs of v for the existence in the approval and negative rules are such that \(v^{a}<v^{n}<0\).

Specifically, the probabilities of positive impression of candidate 2 are such that: \(q_{1}(2),q_{2}(2)>\frac{1}{2}\) and \(q_{3}(2)<\frac{1}{2}\). This follows because v is large (in absolute value), and therefore the probabilities of positive impression of candidate 2 for groups 1, and 2 are: \(q_{1}(2),q_{2}(2)>\frac{1}{2}\). At the same time, the corresponding probability for group 3 is \(q_{3}(2)<\frac{1}{2}\) because group 3 of voters rank candidate 2 last.

Indeed, Lemma 1 implies that, if, for a given set of values of the model’s parameters (e.g., fixed components v, or w, distribution F(.), weight \(\alpha\), etc.), candidate 3 wins under negative plurality, or approval voting, then he must also win under plurality rule. However, this allows the possibility that candidate 3 wins under plurality rule, but loses under negative plurality, or approval voting, for some values of the parameters that fall under the case considered. In other words, the set of parameters’ values under which candidate 3 wins is largest under plurality rule, followed by approval voting, and is smallest under negative plurality.

In the election considered by Myerson and Weber (1993), the utilities (which correspond to the vectors of fixed components in the impressionable voter model) of the three groups of voters are: \(u_{1}=\{10,9,0\}\), \(u_{2}=\{9,10,0\}\) and \(u_{3}=\{0,0,10\}\). As the differences over candidates 1 and 2 of voters in groups 1 and 2 are small relative to their strong opposition of candidate 3, we consider the division between groups 1 and 2 to be “small”.

If \(\alpha \rightarrow 0\), then it can be shown that \(S(1)-S(3)\rightarrow -0.1\) under plurality rule, and \(S(1)-S(3)\rightarrow 0.2\) under approval voting. On the other hand, if \(\alpha \rightarrow +\infty\) then \(S(1)-S(3)\) converges to zero, from above, under both plurality and approval rules. Therefore, if the distribution function F() is continuous, then, e.g., if \(\alpha\) is sufficiently small, then \(S(1)-S(3)<0\) under plurality rule, and \(S(1)-S(3)>0\) under approval voting.

In a recent paper, Bouton and Ogden (2021) consider a model of ethical voting under plurality, and runoff rules, and find that the set of equilibria of their model depends on the magnitude of voters’ preferences as well. We discuss the runoff rule in the next section.

The analysis can be obtained from the authors upon request.

The most common threshold is \(h=0.5\), but other thresholds are also used, typically below 0.5 (Bouton 2013).

If \(n=2\) then there is no difference between the runoff and plurality rule.

We are assuming that the socially preferred candidate wins when confronting a less preferred candidate in the second round. However, under some particular cases this may not hold, e.g., a Condorcet winner can lose to a less preferred candidate in the second round, if the preferences of voters preferring the Condorcet winner are substantially “weaker” than those of voters preferring the opponent candidate (this can be seen by writing the difference of measures of votes in the second round, \(S^{II}(1)-S^{II}(2)\) from above, in terms of the distribution function F(.), and impression components \(v_{\theta }^{1}\) and \(v_{\theta }^{2}\).).

We note that differences in impression probabilities \(q_{\theta }(1)-q_{\theta }(i)\) also influence the sign of \(S^{I}(1)-S^{I}(i)\). However, they are relevant only when the differences in impression components \(v_{\theta }^{1}-v_{\theta }^{i}\) differ substantially across types of voters, and the socially preferred candidate corresponds to the Condorcet winner. In all other cases e.g., when the differences in impression components \(v_{\theta }^{1}-v_{\theta }^{i}\) are the same magnitude across types, or the socially preferred candidate corresponds to the utilitarian welfare maximizing candidate, the influence of the differences \(q_{\theta }(1)-q_{\theta }(i)\) is irrelevant.

To break the tie between candidates 2 and 3, we can assume, for example, that voters of group 2 marginally prefer candidate 2 to candidate 3, i.e., their vector of fixed components is \(u_{2}=\{0,1+\eta ,1\}\), where \(\eta >0\) and small. The qualitative results of Proposition 6 will be the same.

They also apply to the election with four candidates, and three groups of voters we discuss below.

For voters of group 1: \(v_{1}^{3}=0\) and \(a_{1}=\frac{1}{3}\), while for voters of group 2: \(v_{2}^{3}=1\) and \(a_{2}=\frac{2}{3}\).

As in the previous example, to break the tie between candidates 3 and 4, we can assume, for example, that a majority of voters marginally prefers candidate 3 to candidate 4. Again, the qualitative results will be the same as in Proposition 7.

For some common distributions, e.g., \(U[-1,+1]\), \(\alpha ^{*}=+\infty\). So they do not require restrictions on the values of \(\alpha\).

For voters of group 1: \(v_{1}^{2}=0\), \(v_{1}^{4}=0\) and \(a_{1}=\frac{1}{4}\), while for voters of group 3: \(v_{3}^{2}=0\), \(v_{3}^{4}=1\) and \(a_{3}=\frac{1 }{2}\).

For voters of group 2 we have that \(q_{3}(1)=q_{3}(3)\), and therefore their votes do not influence the difference \(S^{I}(1)-S^{I}(3)\).

We discuss laboratory experiments in the Conclusion. Here we focus on multi-candidate elections. The absence of studies of two candidate elections under different electoral rules may be due to the fact that in a rational voter model all voting rules lead to equivalent outcomes.

See Andonie and Diermeier (2017) for details.

In the run-off elections we studied previously only two candidates can run in the second round. However, it is straightforward to modify the model and its analysis so that three candidates can run in the second round, like in the French elections.

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Titel: Electoral Institutions with impressionable voters
verfasst von: Costel Andonie
Daniel Diermeier
Publikationsdatum: 07.05.2022
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 3/2022
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-022-01406-6

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