Abstract
The chapter studies individual-level strategic voting in run-off elections and makes two contributions. On the theoretical side, we propose a typology of the strategic situations a voter can face and of the kind of reasoning that the rational voter should perform in such situations. On the empirical side, we conduct laboratory experiments and show that voters follow such reasoning when it is simple enough, particularly when there is a serious possibility that a candidate reaches the absolute majority threshold in the first round. More complex conditional reasonings based on ‘backward induction’ are seldom followed.
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Notes
- 1.
According to the Institute for Democratic and Electoral Assistance (http://www.idea.int/index.cfm). See Blais et al. (1997) or Golder (2005).
- 2.
For a similar route, although in a quite different context, see Esponda and Vespa (2014). In a voting experiment with private information, they study heterogeneity in cognitive abilities across subjects. They aim at understanding when the strategic model fails to perform well in explaining actual behavior, and the sources of failure: failure of hypothetical thinking vs. failure of information extraction from other voters’ votes. Koriyama and Ozkes (2014) study bounded rationality in voting under imperfect information with a model of hierarchical rationality levels.
- 3.
In most cases, the two conditions will not hold together, and by her action the voter will only be able to induce at most two different first-round outcomes. Notice that there can be more complex situations, where the voter can induce through her action three or more possible first-round outcomes. This will be the case for example if conditions 1 and 2 simultaneously hold, or if three (or more) candidates are tied for the first rank (in which case at least three different run-offs are possible). Since our objective is to derive a simple typology, we restrict our attention to events in which the voter’s action can only produce two different first-round outcomes.
- 4.
In the terminology by Bouton (2013), the voter is said to be “threshold-pivotal”.
- 5.
Our restrictions about the set of events (as noted in footnote 4) imply that there is only one such candidate, who gets strictly less than the majority minus one vote.
- 6.
Note that if the voter prefers the main challenger to the leader, she prefers a run-off between the leader and his main challenger to a direct win of the leader. In that case all actions, except of course voting for the leader, yield the same outcome (the run-off between the leader and the main challenger) and thus the same payoff. In that case, the voter can, as well, decide to vote for any other candidate. This will not be considered as “Type 1” reasoning since, as explained above, we are interested in situations which yield unique best responses. Similarly, if it is certain that the leader will defeat the main challenger in case of a runoff, all actions yield the same payoff, and this situation is not considered as Type 1 reasoning either.
- 7.
Again, our restrictions about the set of possible first-round outcomes imply that there is only one leader, and only two challengers.
- 8.
The full instructions are available upon request.
- 9.
During one session, each subject participates in total to 32 elections. The participants are informed from the beginning that one of the 32 elections will be randomly drawn as the “decisive” election, the one which will actually determine payoffs. This is customary in experimental economics. This has the advantage of keeping subjects equally interested in all elections and to avoid insurance effects; see Davis and Holt (1993).
- 10.
The candidate positions in structure sII were fine-tuned in the first two sessions: in the first session, the positions were (0,5,14,18,20) and in the second (0,5,12,18,20). We then adopted positions (0,5,14,17,20), which were kept for the remaining sessions. Results are presented pooling all sessions. Treating them separately does not alter the results. Additional sessions were conducted in Paris in 2014 to study learning and are presented in Sect. 4.2.
- 11.
These assumptions about run-off outcomes will be similar in the perfect and myopic models of first-round expectations.
- 12.
To make a link with the “bounded rationality” literature, note that the models of “trembled anticipations” used in theory following Myerson (2000), or for individual data analysis in the present chapter, can be seen as specific instances of Tversky’s theory of choice by sequential elimination: the “aspects” to be sequentially considered by the individual in order to reach a (unique) decision are events of successive magnitudes. See Tversky (1972); Tversky and Sattah (1979).
- 13.
Denoting by A, B, C the candidates, let n A , n B , n C be the numbers of votes obtained from voters other than voter i (with n A + n B + n C = 20). What are the possibilities for one mistake? The n A voters voting for A could vote for B or C (2n A possibilities), the n B voters voting for B could vote for A or C (2n B possibilities), and the n C voters voting for C could vote for A or B (2n C possibilities), which yields a total or 2(n A + n B + n C ) = 40 possibilities.
- 14.
Indeed, in some (rare) cases, a recommendation may be of no well-defined type: as noted when defining the types, our typology does not cover events where a voter’s action can produce three or more different outcomes (for example, when three candidates are tied for first place).
- 15.
Therefore, among the 2484 observations for which a unique strategic recommendation was found, 2188, 88 %, have a well-defined type.
- 16.
Remember that in a run-off with two candidates, voting for the candidate closest to her position is a dominant strategy for the voter. We will assume that voters anticipate other voters to follow such a strategy, possibly with some small unbiased mistakes. The run-off outcomes described here are derived under this assumption. The details of our assumptions about how voters form beliefs about run-off outcomes, and more generally about other voters’ behavior, are provided in Sect. 3.2.
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Acknowledgments
The authors thank the Social Sciences and Humanities Research Council of Canada (Project ‘Making Electoral Democracy Work’) and the Center for the Study of Democratic Citizenship for their financial support, and the laboratories of experimental economics in Paris (LEEP) and Montréal (CIRANO). Thanks to Damien Bol for insightful comments.
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Appendix: The Choice of Candidate Positions (“Political Supply Structures”) in the Experiment
Appendix: The Choice of Candidate Positions (“Political Supply Structures”) in the Experiment
We explain here the precise choice of the positions of the candidates in the experiment. We have identified, in theory, four possible types of strategic reasoning. Our objective in the experiment is to check whether the subjects follow each of them. We therefore aim at designing political structures that are likely to generate all these possible types of reasoning. Because it was difficult to do so with one single profile of candidate positions, we chose to have four different candidate profiles: two under structure sI—one with three and one with five candidates, and two under structure sII—again, one with three and one with five candidates. We explain below which types of reasoning are expected to arise under the various political structures.
Structure sI with positions (1,4,11,13,20)
Our typology in part rests on the candidates’ relative strength in case of a run-off. Let us therefore start by describing these relative strengths.
Relative run-off strength of the candidates
Notice that the candidate in position 11 is a Condorcet winner: he gets at least 11 votes (the absolute majority) whoever his opponent, and is therefore bound to defeat any opponent in a run-off.Footnote 16 Consider now the case of the candidate in position 4: he is defeated by the candidates in positions 11 and 13 in case of a run-off, by about the same vote margin. Thus candidates in position 11 and 13 are equally strong when opposed to the candidate in position 4. Consider now the candidate in position 13: he loses the run-off against the Condorcet winner, but wins against the candidate in position 4. The extreme candidates are defeated by all candidates, except possibly by the other extreme candidate. Based on this analysis of the candidates’ relative strength in case of run-offs (Sect. 2), we will use the following terminology to describe the candidates: the candidate in position 11 will be called the centrist candidate, candidate in position 13 the strong moderate, and candidate in position 4 the weak moderate.
Types of reasoning
What type of reasoning are voters required to perform when facing such a political structure? To get a flavor of what can be induced by this type of political supply, imagine that all voters start by voting sincerely for the candidate closest to them.
Consider first the profile with three candidates. The centrist candidate gets votes from voters in position 8–11 or 12, that is about 4.5 votes. His opponents in position 4 and 13 should respectively get about 8 and 8.5 votes and thus both would be in the run-off. In the run-off, the strong moderate candidate (position 13) should win in front of his more extreme opponent (position 4). Then some voters who prefer the weak moderate to the strong moderate may desert the centrist candidate in order to try and provoke a first-round victory of the weak moderate (Type 1 reasoning). If the weak moderate appears to be leading (again, the precise assumptions we make about how voters form their anticipations are described in detail in Sect. 3.2), and if voters believe that the most likely event is that they are going to be decisive in determining who, between the centrist and the strong moderate, is going to be part of the run-off, then, Type 2 reasoning should be performed, since as noted above, both candidates are equally strong against the weak moderate. If now the strong moderate appears to be leading, and if voters believe that the most likely event is that they are going to be decisive in determining who, between the centrist and the weak moderate, is going to be part of the run-off, then, the choice is really between the centrist candidate and the strong moderate. Voters who prefer the centrist candidate to the strong moderate should vote for the centrist candidate (Type 3 reasoning), whereas some right-wing voters should vote for the weak candidate whom they dislike but who would secure a victory of their favorite candidate (Type 4 reasoning). All types of reasoning are thus possible under this structure.
What happens with five candidates? Again, imagine that all voters start by voting sincerely. The centrist candidate still gets 4.5 votes. His moderate opponents now share their votes with the extreme candidates. The weak moderate gets 5 votes, and his extreme neighbor 3, whereas the strong moderate gets 4.5 votes, and his extreme neighbor 4. First round winners become much less likely, thus Type 1 reasoning is less likely. The vote difference between the two moderate candidates and the centrist being small, we expect to see Type 2, 3 and 4 reasonings.
Structure sII with positions (0,3,8,15,20)
In terms of relative run-off strength of the candidates, one may check that a pattern similar to that described in structure sI is observed here. We will therefore use the same terminology to describe the candidates: the candidate in position 8 will be called the centrist candidate, the candidate in position 15 the strong moderate (who loses against the centrist but wins in case of a run-off against the candidate in position 3), and the candidate in position 3 the weak moderate.
Let us now describe the types of strategic reasoning which might emerge with such a structure. Let us start with the three-candidate profile. Again, imagine that all voters start by voting sincerely for the candidate closest to them. The centrist candidate, in position 8, gets votes from voters 6 to 11, that is 6 votes. The strong and weak moderate opponents respectively get 9 and 6 votes. In that case it is clear that the strong moderate will be in a run-off or win in the first round. If the strong moderate is in position to win in the first round, this calls for Type 1 reasoning. If not, in case a run-off is the most likely outcome, voters are pivotal in deciding whom, between the centrist and the weak moderate, is going to be part of the run-off. This is a typical situation that calls for reasonings of Type 3 and 4 (since the weak moderate and the centrist candidate are of unequal strength, when facing the strong moderate candidate). With such a structure, we expect to put voters in situations calling for reasonings of Type 1, 3 and 4.
What happens with five candidates? Again, imagine that all voters start by voting sincerely. The centrist candidate still gets 6 votes. The weak moderate gets 4 votes, and his extreme neighbor 2, whereas the strong moderate gets 6 votes, and his extreme neighbor 3. First round wins become much less likely, thus Type 1 reasonings are less likely. The vote difference between the two moderate and the centrist are small, so we expect to see situations with Type 2, 3 and 4 reasonings.
Conclusion about the candidate profiles
All types of reasonings can be generated by both political structures, but in different proportions: Type 1 is more likely in structure sII, Type 2 is more likely in structure sI, and Type 3 and 4 should be present under both structures. Furthermore, Type 1 situations are more likely to occur with three candidates than with five candidates.
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Van der Straeten, K., Laslier, JF., Blais, A. (2016). Patterns of Strategic Voting in Run-Off Elections. In: Blais, A., Laslier, JF., Van der Straeten, K. (eds) Voting Experiments. Springer, Cham. https://doi.org/10.1007/978-3-319-40573-5_11
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