Strategic voting in run-off elections

1. Strategic voting in run-off elections Jean-François LASLIER (Ecole Polytechnique, France) Karine VAN DER STRAETEN (Toulouse School of Economics, France) PRELIMINARY VERSION Social Choice and Welfare, Moscow, July 21-24 2010

2. Run-off elections: definition • On the first round, voters vote for one candidate. • If one candidate gets more than 50% of the votes, he is elected. • If not, the two candidates with the highest two numbers of votes proceed to a second round. • On the second round (if any), voters vote for one candidate. The candidate with the highest number of votes is elected.

3. Run-off elections: Properties • Rarely used in legislative elections, but quite common in presidential elections • Aggregate properties? Duverger: Multiparty system (/ plurality where two parties dominate) • Voter behavior? - Duverger: sincere - Cox: strategic (instrumental voters reasoning on pivot-events), three candidates only get votes

4. Are voters strategic? • Our focus here. • Why is it important? Consequences on the party structure: affects the number of candidates receiving votes Qualitative consequences on who gets elected Ex.: Single-dimension politics with three candidates: a centrist Condorcet winner “squeezed” between a Condorcet loser on the left, and a rightist candidate. With sincere voting, the rightist candidate wins; with strategic voting, the centrist may win.

5. Empirical evidence on strategic voting in runoff elections • Election or survey data: pb = to compute strategic recommendation, one needs a lot of information about a voter’s preferences and beliefs • Lab experiments data: Blais et al. (SCW, forth.) • in a single-dimension five-candidate setting, voters neither (fully rational) strategic, nor sincere • behavior best explained by a top-three heuristics, whereby voters vote for their preferred candidate among the three candidates expected to get the most votes

6. This talk • Part 1: Typology of strategic reasoning Describe possible patterns of strategic reasoning in run-off elections • Part 2: Experiment A lab experiment to test whether subjects are able to perform any of the patterns of the strategic reasoning • Part 3: Analysis Analysis of the experimental data with the help of the typology

7. Part 1: Typology of strategic reasoning in run-off elections • Being strategic in run-off elections entails different kinds of reasoning, more or less complex. • We propose here a typology of such types of reasoning, based on the different pivot-events in which the voter may happen to be

8. When is a voter pivotal on 1rst round? A voter is pivotal if other voters’ votes are such that one of the following two conditions holds: • Condition 1: one candidate receives an absolute majority minus one vote: by voting for this candidate, the voter can make him a 1rst-round winner - Condition 2: no candidate gets an absolute majority and the vote margin between the 2nd and the 3rd ranked candidates is at most one vote: by voting for one of these candidates, the voter can make him be part of the run-off

9. When is a voter pivotal on 1rst round? A voter is pivotal if other voters’ votes are such that one of the following two conditions holds: • Condition 1: one candidate receives an absolute majority minus one vote: by voting for this candidate, the voter can make him a 1rst-round winner TYPE 1 - Condition 2: no candidate gets an absolute majority and the vote margin between the 2nd and the 3rd ranked candidates is at most one vote: by voting for one of these candidates, the voter can make him be part of the run-off

10. Condition 2: Run-off pivot Assume some candidate, say A is leading (with no majority), followed by B and C at equality If the voter votes for B: run-off(AB), with payoff u(A)+Pr[B wins/(AB)] × [u(B)-u(A)] If votes for C: u(A)+Pr[C wins/(AC)] × [u(C)-u(A)] If votes for any other candidate: run-off (AB) with probability ½ and a run-off (AC) with proba ½ → Optimal decision: voting B or C, depending on the utility derived from the election of each candidate, and the relative strength of the follower candidates B and C in case of a run-off against leader A If

11. Run-off pivot: comparing (AB) and (AC) If votes for B: u(A)+Pr[B wins /(AB)] × [u(B)-u(A)] If votes for C: u(A)+Pr[C wins/(AC)] × [u(C)-u(A)] Condition “equal strength”: Both followers are equally strong run-off candidates against A Recommend.: Vote for the preferred follower TYPE 2 Condition “different strength”: One follower is a stronger run-off candidate against A Recommend.: Vote for stronger run-off candidate if he is preferred to A TYPE 3 and for the weaker otherwise TYPE 4

12. Part 2: The experiment • Designed to test whether subjects follow the strategic recommendations described above • Groups of 21 voters (students) acting as voters • Incentive structure mimics one-dimensional politics with 3 or 5 candidates, with different candidate positions

13. Positions of the 21 voters Left-right axis labelled from 0 to 20. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 subjects in 21 positions: 1 voter in position 0, 1 voter in position 1, …, 1 voter in position 20. The distribution of positions is known to all voters. Positions are randomly assigned

14. Positions of the candidates

15. The payoffs • Depend on the distance between the subject’s position and the elected candidate’s position on the axis. • The smaller this distance, the higher the payoff. • Subjects receive 20 euros minus the distance between the subject’s position and the elected candidate’s position. • (At the end of the session, one election was randomly drawn and used to determine payoffs.)

16. Timing of a session • Explain the incentive structure and the voting rule • Series of four elections where positions of the candidates and voters’ preferences remain constant; after each election, the results of the election are publicly announced • After each series of 4 elections is completed, voters draw a new position, and the profile of candidates is changed • Complete information setting = distribtuion of voter positions is known, as well as candidate positions • So far, 5 sessions in Paris

17. Part 3: Analysis Computation of the strategic recommendation • For each voter in each election, compute her best response against other voters’ votes. • Assumptions: • Utility = payoff • Beliefs = The voter correctly anticipates other voters’ behavior, but assumes some possible (small) mistakes – “trembling hand assumption”, that yields unique strategic predictions even when the election is not so close that a single vote can indeed make a difference

18. Does the strategic recommendation coïncide with actual vote? • Preliminary results • Focus on three candidates elections • Does the strategic recommendation coïncide with actual vote? Yes in 68% of the cases

19. Performance of the strategic model by type • Does the performance of the strategic model vary across types? • For each voter in each election, trace which type of reasoning the voter needs to make to decide for which candidate to cast a vote

20. Performance of strategic model by type

21. Performance of the sincere model of individual behavior • The strategic recommendation coïncides with actual vote in 68% of the cases • To be compared with the sincere behavioral model, whereby voters simply vote for the candidate yielding the highest payoff if elected, which correctly predicts vote in 76% of the cases

22. Conclusion • In a lab experimental setting, we test strategic voting in run-off elections • In the three-candidate setting, little strategic voting is observed • Some recommendations of the strategic model are followed: e.g. “Vote for a candidate that might be a first-round winner” • But others are not: e.g. “Vote for a weak candidate which might be more easily defeated”

23. Next steps • Extend the analysis to the five-candidate elections • Run more sessions (5 more in Montreal are scheduled) • Correlate strategic voting with measures of cognitive skills

24. Typology of strategic reasoning

25. Typology in profile I

26. Performance of strategic model by type