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

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 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

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

Condition 2: Run-off pivotAssume 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 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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Positions of the candidates

(EL) L C R (ER)

Profile I / 4 11 13 /

Profile I bis 1 4 11 13 20

Profile II / 3 8 15 /

Profile II bis 0 3 8 15 20

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

Type Type 1 Type 2 Type 3 Type 4

Condition Direct pivot

Run-off pivotEqual strengthL or C leaders

Run-off pivot≠ strengthR leaderC preferred to R

Run-off pivot≠ strength M leaderR preferred to C

Nb of cases 314 178 159 42

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 reasoningType 1 Type 2 Type 3 Type 4

Direct pivot Run-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 IType 1 Type 2 Type 3 Type 4

Direct pivot Run-off pivotEqual strength

Run-off pivot≠ strengthStronger challenger preferred to leader

Run-off pivot≠ strengthLeader preferred to stronger challenger

Any configuration

mM~C

Cm~M

Mm~C

Mm~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 typeType Type 1 Type 2 Type 3 Type 4Condition Direct pivot Run-off pivot

Equal strengthm or C leader

Run-off pivot≠ strengthM leader,C preferred to M

Run-off pivot≠ strength M leaderM preferred to C

Nb of cases 314 178 159 42

Among which % of correct predictions

80% 65% 62% 12%

Nb of cases where strategic rec. sincere

231 159 81 0

Among which % correct 93% 70% 93% /

Nb of cases where strategic rec. non sincere

83 19 78 42

Among which % correct 46% 21% 31% 12%