Strategic voting in run-off elections Jean-François L ASLIER (Ecole Polytechnique, France) Karine V AN DER S TRAETEN (Toulouse School of Economics, France) PRELIMINARY VERSION Social Choice and Welfare, Moscow, July

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.

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

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.

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

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

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

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 2 nd and the 3 rd 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

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 2 nd and the 3 rd 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

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

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

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

Positions of the 21 voters Left-right axis labelled from 0 to 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

Positions of the candidates (EL)LCR(ER) Profile I/41113/ Profile I bis Profile II/3815/ Profile II bis

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.)

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

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

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

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

Performance of strategic model by type TypeType 1Type 2Type 3Type 4 Condition Direct pivot Run-off pivot Equal strength L or C leaders Run-off pivot ≠ strength R leader C preferred to R Run-off pivot ≠ strength M leader R preferred to C Nb of cases Among which % of correct predictions 80%65%62%12%

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

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”

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

Typology of strategic reasoning Type 1Type 2Type 3Type 4 Direct pivotRun-off pivot Equal strength Run-off pivot ≠ strength Stronger challenger preferred to leader Run-off pivot ≠ strength Leader preferred to stronger challenger Vote for leader if preferred to first follower Vote for the preferred follower Vote for the stronger run-off challenger Vote for the weaker run-off challenger

Typology in profile I Type 1Type 2Type 3Type 4 Direct pivotRun-off pivot Equal strength Run-off pivot ≠ strength Stronger challenger preferred to leader Run-off pivot ≠ strength Leader preferred to stronger challenger Any configuration mM~CmM~C Cm~MCm~M Mm~CMm~C Mm~CMm~C Vote for the leader if he is preferred to immediate follower m supporters vote C M supporters vote M C supporters vote m (or M if C leader & m least preferred candidate) m and C supporters vote C M supporters vote m

Performance of strategic model by type TypeType 1Type 2Type 3Type 4 Condition Direct pivotRun-off pivot Equal strength m or C leader Run-off pivot ≠ strength M leader, C preferred to M Run-off pivot ≠ strength M leader M preferred to C Nb of cases Among which % of correct predictions 80%65%62%12% Nb of cases where strategic rec. sincere Among which % correct 93%70%93%/ Nb of cases where strategic rec. non sincere Among which % correct 46%21%31%12%