Laboratory experiments on voting rules
André BLAIS
(Université de Montréal, Canada)
Jean-François LASLIER
(Ecole Polytechnique, France)
Nicolas SAUGER
(Sciences-Po Paris, France)
Karine VAN DER STRAETEN
(Toulouse School of Economics, France)
ESOP Workshop on Culture, Behavior and Distribution, Oslo, November 19 2010
Why democratic institutions matter?
• In shaping the rules of the games for politicians,
lobbying groups, voters, … democratic institutions
influence the choice of policies, the quality of
politicians, the level of corruption, …
• Some references include Downs, Persson and
Tabellini, Myerson, …
• This work: part of this positive approach, with a
narrower scope: study voting rules with a fixed set
of candidates
Comparing voting rules
• Long tradition in Political Science.
• Example: « The Duverger Law », in the 1950’s:
The plurality rule leads to a two-party system,
whereas proportional representation fosters the
development of a multi-party system.
Why should different voting rules
yield different outcomes?
• Mechanical effects: votes are aggregated in
different ways, even if voters vote in the same way
• Psychological effects: voters may behave
differently
Since then, the model of the instrumental rational
voter has been the main explanation for the
psychological effects: see e.g. Myerson and Weber
(1993), Myerson (1994), Cox (1997), Palfrey (1989)
Might offer some explanation for the Duverger law.
Our research questions
• 1. How can we quantify these mechanical and
psychological effects?
• 2. Regarding the psychological effect, is the
rational voter model a satisfactory behavioral
model of how voters vote? Is the sincere model a
better behavioral model?
Does it depend on the voting rule?
Methodology?
• Question 1: How can we quantify these
mechanical and psychological effects?
Difficult because one needs counterfactuals
• Question 2: Are voters as strategic as assumed in
formal game-theoretical models?
Not so easy to answer with electoral surveys
In order to predict strategic behavior, one needs
precise information on preferences + beliefs
regarding other voters behavior
• Answer: Rely on laboratory experiments
Lab experiments on voting rules
• Comparisons of voting rules: Forsythe,
Myerson, Rietz and Weber (1996). Overall
supports the conclusion that voters cast votes
strategically
• Other experiments on strategic voting:
Guarnaschelli, McKelvey and Palfrey (2000),
Fiorina and Plott (1978)
• For a survey on laboratory experiments in
political economy, see Palfrey (2005)
Outline of the talk
• Description of the experimental protocol
• Aggregate results
- Who wins?
- Decomposing the differences between 2R and
1R election outcomes into mechanical and
psychological effects
• Individual level analysis
- Do voters vote strategically or not?
Our experiments
• 23 sessions in Paris, Lille (Dec. 2006, Jan. 2007)
and Montreal (Feb. 2007)
• Groups of 21 voters (students)
• Incentive structure mimics one-dimensional politics
with 5 candidates
• Complete information setting
• In each session, we run 2 or 3 series of four
elections, each series under a different voting rule
Positions of the 5 candidates
A
B
C
D
E
01
6
10
14
19 20
These positions remain the same
through the whole session.
Positions of the 21 voters
0
1
1
2 3 4
5
6 7
6
8 9 10 11 12 13 14 15 16 17 18 19 20
10
14
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
19
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.)
Example : If you are in position 11, here
is your payoff if candidat B (position 6)
is elected.
Distance = 5
B
0
1
6
1011
15
19 20
• Distance = 11 - 6 = 5
• Payoff = 20 - Distance = 20 - 5 = 15 euros
Information given to the subjects about their
randomly drawn position:
You are in position 11.
Your payoff depending on who gets elected :
if A is elected: 10 euros
if B is elected : 15 euros
if C is elected : 19 euros
if D is elected : 17 euros
if E is elected : 12 euros
The voting rules
In each session:
- 1 series of 4 one-round elections
- 1 series of 4 two-round elections
• In 10 sessions, we add a third series of 4
elections, using either approval voting, or the
single transferable vote
• Within each series, voters’ positions remain the
same; after each election, the results of the vote
are publicly announced
●
One-round Voting Rule (Plurality)
• You vote by circling the name of one (and
only one) candidate.
• The candidate who gets the highest number
of votes is elected (Ties, if any, are broken
randomly).
Voting ballot (1-Round)
Voting ballot
A
B
C
D
E
Circle the candidate you vote for
December 19 2006, Group 1, Voter 11, Election 1
Two-round voting rule
• On the first round, you vote by circling the
name of one (and only one) candidate.
- If one candidate gets more than 50% of the
votes, he is elected.
- If no candidate gets more than 50% of the votes,
the two candidates with the highest two scores
proceed to a second round (Ties, if any, are
randomly broken).
• On the second round (if any), you vote by
circling the name of one (and only one)
candidate, among those two runoff candidates.
The candidate with the highest number of votes
is elected (Ties, if any, are randomly broken).
Approval voting
• You vote by circling the name of one or
several candidate(s): you can circle the
name of one, two, three, four or five
candidates.
• For each ballot: each candidate that has
been circled gets one vote.
• The candidate with the highest number of
votes is elected (Ties, if any, are randomly
broken).
The Single Transferable Vote
• Each voter has one vote.
• You vote by ranking all five candidates,
from rank 1 to rank 5.
• The elected candidate is the candidate that
gets more than 50% of the votes.
• The process that determines to which
candidate the vote of each voter is assigned
may be composed of several steps.
Voting ballot (STV)
Voting ballot (STV)
Rank 1:
Rank 2:
Rank 3:
Rank 4:
Rank 5:
Rank the candidates from rank 1 to rank 5
December 19 2006, Group 1, Voter 11, Election 1
The Single Transferable Vote (2)
• At the first step: Each voter’s vote is assigned to the
candidate that she ranked first on her ballot.
- If a candidate gets more than 50% of the votes, he is
elected.
- If not, one proceeds to a second step.
• At the second step: The candidate that got the smallest
number of votes at the first step is eliminated. The votes
that were assigned to him are now transferred to candidates
that are ranked second on the ballots. One counts the new
number of votes that the remaining candidates get.
- If a candidate gets more than 50% of the votes, he is
elected.
- If not, one proceeds to a third step.
• … One goes on similarly until a candidate gets more than
50% of the votes.
Timing of a session
• Presentation of the incentive structure
• Subjects randomly draw a position – Presentation of the
voting rule (e.g. one-round)
- election 1: one-round, vote, results are publicly announced
- election 2: one-round, vote, results
- election 3: one-round, vote, results
- election 4: one-round, vote, results
• Subjects randomly draw a new position– Presentation of a
new voting rule (e.g. two-round)
- election 5: two-round, vote, results
- election 6: two-round, results
- election 7: two-round, results
- election 8: two-round, results
● In some sessions, a third series of elections
• One election is randomly drawn to determine payoffs
Results
• First aggregate outcome results
• Then individual level analysis: Comparison
of different models of individual behavior
Winners and losers
C
One
Two
AV
STV
round
round
49 %
54 %
79%
0
B or D
51 %
45 %
21 %
100 %
A or E
0
0
0
0
total
92
92
24
16
Winners and losers
(last two elections)
C
One
Two
AV
STV
round
round
52 %
50 %
100 %
0
B or D
48 %
50 %
0
100 %
A or E
0
0
0
0
total
46
46
12
8
Decomposing the effects on 2R vs.
1R on the election of centrist C
• 1R and 2R deliver almost identical aggregate
results: centrist C elected about half of the times
• Mechanical effects of 2R versus 1R very
favorable to C: if one applies 2R system on
actual 1R votes, C wins in almost 75% of the
cases
• If psychological effects defined as the residual: it
means that psychological effects of 2R very
detrimental to C.
Evolution of the scores
of ranked candidates (1R)
Evolution of the scores
of ranked candidates (2R)
Evolution of the scores
of ranked candidates (AV)
Evolution of the (Borda) scores
of ranked candidates (STV)
Sincere and strategic voting
in 1R and 2R elections
• Sincere voting model: Individuals vote for
any candidate that yields the highest payoff if
elected.
• Strategic voting model : Individuals
maximize their expected utility, given their
beliefs about other voters’ vote
- Assumption 1: utility = payoff
- Assumption 2a: beliefs = current behavior
- (Assumption 2b: beliefs = past behavior)
For each model
• For each model, compute the predictions for
each individual at each election.
• Compare predictions to observations.
• Restrict to cases of unique predictions.
• For the Strategic model, when beliefs are
precise, we have too few unique predictions.
Use a « Trembling hand » model. With small
probability, one vote is changed to a randomly
drawn candidate
Tests for 1-round elections
Tests for 2-round elections
Why does the strategic model perform well in
1R elections, and not in 2R elections ?
• In both cases, prescribes the desertion from extreme
candidates (« I do not want to waste my vote on a
candidate who has no chance to be in a close race
anyway»)
• But in 2R elections, more involved reasoning (« I
do not need to vote for someone who will be at the
second round in any case » and « I should promote
a very bad candidate who will be more easily
beaten by my favorite »…)
A model of “heuristic voting”
for 2R elections
"Top three" model: Individuals vote for their
preferred candidate among the three candidates that
are expected to get the highest three numbers of
votes
Beliefs: Based on past or current behavior
Remark: In 1R elections, being strategic coincides
with a "Top two" model: Individuals vote for their
preferred candidate among the two candidates that
are expected to get the highest two numbers of votes
Tests for 2-round elections
Tests for 1-round elections
Coming back to the aggregate results
in 1R and 2R elections
• 1R and 2R deliver almost identical aggregate
results: centrist C elected about half of the times
• Mechanical effects of 2R versus 1R very
favorable to C (if one applies 2R System on actual
1R votes, C wins in almost 75% of the cases)
• If psychological effects defined as the residual:
means that psychological effects of 2R very
detrimental to C.
Why are the pshychological effects of
2R vs 1R so detrimental to C?
• With the Top 3 heuristics used in 2R, C is certain to
be viable. But supporters of the non viable extreme
candidates are more likely to move to one of the two
moderate candidates, weakening C’s chances of
making it to the second round.
• With the strategic voting (=Top 2 heuristics) used in
1R, there are only two viable candidates, and
supporters of the extreme candidates are willing to
vote for the centrist candidate, increasing C’s
chances of winning.
Results for Approval voting
• A strategic model: Vote par comparison with
the main candidate. This model works for
about 88% of the comparisons.
• Outperforms sincerity (if sincerity defined as
voting only for one’s preferred option)
• With this profile, strategic voting + AV
implies that the Condorcet Winner (Candidate
C) wins the election.
Results for Single Transferable Vote
• Sincerity test: yes at more than 90%
• With this profile, sincere voting + STV
implies that the Condorcet Winner
(Candidate C) is eliminated at the third
elimination step at the latest.
Conclusions: Aggregate outcome level
One-round and Two-Round elections show important
path-dependency effects.
Both elect the Condorcet winner about half of the
times. Large mechanical and psychological effects
cancel off.
Approval voting always elects the Condorcet winner
while STV never does.
Conclusions : individual level
Do voters vote sincerely or strategically… or
somewhere in between?
Strategic voting correctly predicts individual
behaviour in One-Round elections and AV.
In Two-Round elections, the theory correctly
predicts desertion of the extremes but is not a good
predictor of actual choices, “Top three” heuristics
is a better model.
In STV elections, sincere voting is good predictor of
voters’ choices.
Answer: Calls for a case by case analysis.