Multiple Votes, Ballot Truncation and the Two-party
System: An Experiment
September 2008
Abstract
Countries that elect their policy-makers by means of Plurality Voting tend to have
a two-party system. We conduct laboratory experiments to study whether alternative
voting procedures yield a two-party system as well. Plurality Voting is compared with
Approval Voting and Dual Voting, both of which allow voters to vote for multiple
candidates, but differ in whether voters are required to cast all their votes. We find
that both Plurality and Approval Voting yield a two-party system, whereas Dual
Voting may yield a multi-party system due to strategic voting. Voters’ ability to
truncate ballots (not cast all their votes) is essential for maintaining the two-party
system under Approval Voting.
Keywords: Strategic voting; Approval Voting; Ballot truncation; Duverger’s law.
JEL Classification: C72, C9, D72.
Arnaud Dellis∗
Université Laval and CIRPÉE
E-mail: [email protected]
Sean D’Evelyn∗ and Katerina Sherstyuk∗
University of Hawai‘i - Manoa
E-mail: [email protected], [email protected]
∗ Financial
support by the University of Hawaii Social Science Research Institute is gratefully
acknowledged. We would like to thank Thomas Palfrey, Thomas Rietz, Theodore Turocy, and the
participants of the 2007 ESA North American Regional Meetings, the 2008 International Meeting
of the Society for Social Choice and Welfare, and the 2008 ESA International Meetings for valuable
comments and discussion.
1
1
Introduction
Constitutional rules have a significant impact on economic policy-making. Empirical studies have
shown that a relationship exists between the effective number of parties and macro-economic
variables such as the levels of public deficit and income inequality or the composition of public
spending (Lizzeri and Persico, 2001; Persson and Tabellini, 2003). This paper contributes to
understanding how voting procedures affect the political party system. Using a laboratory setting,
we compare the effect several widely-discussed voting procedures have on the number of parties.
Countries that elect their policy-makers by means of Plurality Voting—i.e., the voting procedure where every voter votes for one candidate—tend to have a two-party system. This empirical
regularity is known as Duverger’s law, which states that “the simple-majority single-ballot system
[i.e., Plurality Voting] favors the two-party system” (Duverger 1954; 217). This is certainly true
in the U.S., with the Democratic and Republican parties.1
Some political activists and scholars advocate replacing Plurality Voting with other voting
procedures that would allow or force people to vote for more than one candidate. Such electoral
reforms have been passed in several countries and municipalities (e.g. New Zealand, San Francisco)
and have recently been the object of referenda in two Canadian provinces (British Columbia and
Ontario). Electoral reforms are also advocated by several citizens associations (e.g. Citizens for
Approval Voting). The argument in support of such an electoral reform is based on the following
wasting-the-vote effect of Plurality Voting. In Plurality Voting elections, a voter whose mostpreferred candidate has no chance of winning the election would waste her vote by casting it for
that candidate. Such a voter has therefore an incentive to vote for another candidate that she
may like less but whose electoral prospects are better. This wasting-the-vote effect of Plurality
Voting is said to create a barrier to third-party candidates given that voters tend to anticipate
that third-party candidates have no chance of winning the election.2 The argument then goes that
if voters were given several votes to cast for the different candidates, they would no longer fear
wasting a vote on a third-party candidate, which would therefore help such a candidate.
Others advise against letting people vote for multiple candidates. They warn that lifting
the barriers to third-party candidates would trigger a proliferation of parties and the dangers
of multipartism. For example, multipartism could give rise to coalition governments and the
inefficiencies that are often associated with this form of government, e.g. Roubini and Sachs
(1989). Since candidates can be elected with less votes in a multi-party system than in a two-party
1
There are a few exceptions however. For example, Canada has a multiparty system despite holding its
political elections under Plurality Voting. Riker (1982) explains the Canadian exception by the geographic
concentration of third-party voters.
2
For theoretical contributions explaining Duverger’s law by the wasting-the-vote effect of Plurality
Voting, see e.g. Palfrey 1989; Feddersen 1992; Myerson and Weber 1993; Fey 1997.
2
system, inequality could then be on the rise. This is because candidates would have an incentive
to target electoral promises to smaller groups of voters, e.g. Lizzeri and Persico (2005).
This paper contributes to the debate whether letting people vote for multiple candidates would
obliterate the two-party system. We say that there is a two-party system if no more than two
parties get their candidate elected with positive probability. Another common formal interpretation
of Duverger’s law is that in Plurality Voting elections, only two parties receive votes; e.g. Palfrey
(1989), Feddersen (1992), Myerson and Weber (1993). As Myatt (2007) points out however, it is
rare to observe a full concentration of votes on only two candidates; trailing candidates usually
receive a non-negligible share of the votes. This is confirmed by Hirano and Snyder (2007) who
provide empirical evidence that in Plurality Voting elections third-party candidates receive votes.
Our definition of a two-party system relies on the latter observation, and focuses on the upperbound on the number of viable competitors.
Contrary to conventional wisdom, the theoretical results in Dellis (2007) suggest that letting
people vote for multiple candidates need not obliterate the two-party system that characterizes
Plurality Voting elections. Dellis studies scoring rules, a large class of commonly used and studied
voting procedures (e.g. Cox 1987; Myerson 2002) that include Plurality Voting and many of its
frequently-discussed alternatives. Under a scoring rule, every voter has a set of scores to cast for
the different candidates and the candidate whose total score is the highest wins the election. All
scoring rules differ only in the type of ballot a voter can cast.3 Dellis identifies two classes of
scoring rules which, under standard assumptions on voter preferences, yield a two-party system.
One class includes all the scoring rules under which every voter has a unique maximal top-score
vote to cast; Plurality Voting belongs to this class.4 The other class of scoring rules that yield
a two-party system consists of all the scoring rules that admit truncated ballots, that is, do not
force voters to cast all their scores.5 An example of such scoring rule is Approval Voting, under
which a voter can vote for as many candidates as she wishes. Finally, all scoring rules that belong
to neither of these two classes risk the breakdown of the two party system. Examples are: Dual
Voting, under which a voter must vote for exactly two candidates; and Negative Voting, under
which a voter must vote for all except one of the candidates.
This paper reports the results of a laboratory experiment that compares three distinct scoring
3
Part of the appeal of the scoring rules for study purposes is related to the way a voting procedure is
defined. More specifically, a voting procedure is defined by three elements: (1) the ballot structure, i.e.,
the type of ballot a voter can cast; (2) the allocation rule, i.e., the way it transforms votes into seats; and
(3) the district magnitude, i.e., the number of seats in a district. By comparing scoring rules we can isolate
the effect of the ballot structure since scoring rules differ only in this variable.
4
Other examples of such scoring rules are the Borda Count and Cumulative Voting. We do not consider
these rules in the present study.
5
Ballot truncation is often referred to in the literature as partial absention.
3
rules in their effect on the two-party system: Plurality Voting, Approval Voting and Dual Voting.
Plurality Voting is chosen as a benchmark; it is also an example of a scoring rule that requires
unique maximal top-score vote to cast and thus belongs to the first class identified by Dellis (2007).
Approval Voting is chosen as one of the most often discussed alternatives to Plurality Voting (e.g.,
Brams 2007; Forsythe et al. 1996; Laslier and Van der Straeten 2008);6 it is also a scoring rule
that admits truncated ballots and thus belongs to the second class of scoring rules yielding a twoparty system as identified by Dellis (2007). Finally, Dual Voting is chosen as a scoring rule that
belongs to neither class and thus may lead to a multi-party system. It is also identical to Approval
Voting in the three-candidate setting that we study, except it does not admit truncated ballots.
This allows us to test whether Dual Voting may lead to a different number of winning parties as
compared to Approval Voting, and thus consider the role of ballot truncation in Approval Voting
procedure.
Most experimental work on voting in elections focuses on Plurality Voting, and studies voter
behavior under incomplete information and the informational or coordinating role of past elections
(e.g., Collier et al. 1987; Forsythe et al. 1993), pre-election polls, ballot position, campaign
contributions (e.g., Forsythe et al. 1993; Rietz et al. 1998) or sequential voting (e.g., Morton and
Williams 1999; Battaglini et al. 2007).7 Several studies compare voting rules in multi-candidate
elections; see Rietz (2003) and Palfrey (2006) for comprehensive surveys. Rapoport et al. (1991)
study sincere and strategic voting under Plurality and Approval Voting; they report that both types
of voting are common in their experiments. Forsythe et al. (1996) compare vote coordination in
three-candidate elections under Plurality Voting, Approval Voting and the Borda Count. They
find that three-way ties among candidates occur more frequently under Approval Voting and the
Borda Count than under Plurality Voting. Unlike Forsythe et al. (1996), our analysis considers
situations where all voters’ utility functions are concave over the policy space.8
The present paper yields several interesting results. First, we clearly observe that the voting
procedures differ in their effect on the two-party system. In our experiment both Plurality and
Approval Voting gave rise to a two-party system: in almost all elections at most two candidates
tied for the first place, and three-way ties among candidates occurred very rarely. By contrast,
6
Approval Voting is currently used by several academic and professional associations to elect their
officers.
7
Our focus here is on voting behavior in elections; we therefore do not review many other important
contributions to experimental political science such as candidate spatial competition, committee decision
making, information aggregation in juries, or voter turnout and participation; see, e.g., Palfrey (2006).
8
Blais et al. (2008) study strategic voting behavior in five-candidate elections under Plurality Voting,
Plurality Runoff, Approval Voting and Single Transferable Vote. They conclude that voters tend to behave
strategically provided strategic computations are not too involved, as when the election is held under
Plurality Voting or Approval Voting. Unlike Blais et al. (2008), our analysis considers how the possibility
to truncate one’s ballot affects voters’ strategic behavior and, in turn, the party structure.
4
Dual Voting gave rise to multipartism: three-way ties occurred in a significant number of elections.
We also observe that, under Plurality and Approval Voting elections, the same two candidates
were repeatedly among those who had a positive chance of winning elections; the third candidate
was almost never in the winning set. At the same time, all three candidates had a positive and
significant chance of being in the winning set when the elections were held under Dual Voting.
This gives additional evidence for the two-party system under Plurality and Approval Voting, and
the multiparty system under Dual Voting.
We further show that the breakdown of the two-party system in Dual Voting elections is the
result of strategic voting behavior. Dellis attributes the possibility of multipartism in Dual Voting
elections to the fact that those voters who decide to vote cannot truncate their ballots and are
thus forced to vote for two candidates. The voters who prefer a three-way tie to the outright
election of their second most-preferred candidate are then led to cast one of their two votes for
their most-preferred candidate and dump their second vote on their least-preferred candidate; thus,
an equilibrium that involves a three-way tie among candidates may exist. By contrast, such vote
dumping should not occur in Plurality and Approval Voting elections because here voters can
truncate their ballots and therefore are not forced to vote for more than one candidate. Consistent
with this argument, we find strong evidence of strategic vote dumping in Dual Voting elections,
but not in Plurality and Approval Voting elections. This establishes that the voters’ ability to
truncate ballots is essential for maintaining the two-party system under Approval Voting.
Our experiments also yield some unanticipated results. Notable among these is a strong presence of the middle candidate–i.e., the candidate whose platform lies between the platforms of the
other two candidates–among winners under all three voting procedures. Due to multiplicity of
equilibria, this focal effect of the middle candidate is not anticipated (even though it is not ruled
out either) by the Nash Equilibrium theory, even if the voters are assumed not to use weakly
dominated voting strategies. Interestingly, theories of boundedly rational behavior, such as the
Quantal Response Equilibrium and level-k reasoning, do not explain the overwhelming precedence
of the middle candidate in the winning sets for our setting either.
Finally, we gain some insights into individual voting behavior and voter ability to understand
different voting rules. A common practice in the formal voting literature is to assume that people
do not cast weakly dominated ballots, that is, ballots that can never affect the electoral outcome
in the person’s favor compared to some other ballot. Consistent with previous experiments (e.g.
Rapoport et al., 1991; Forsythe et al., 1996), we find strong evidence in support of this assumption
in the case of Plurality Voting. However, we find that sizeable numbers of weakly dominated
ballots were cast in Dual and Approval Voting elections. Moreover, consistent with Rapoport
et al. (1991), we find that this phenomenon was rather widespread and persistent with subject
5
experience in the case of Approval Voting. This contrasts with Laslier and Van der Straeten (2008),
who report, based on a field experiment in France, that the principle of Approval Voting was easily
understood and accepted by French voters. Our findings suggest that Approval Voting may be
more complex than it seems. Further, such evidence puts in question a common practice in the
formal comparative voting literature, the practice of imposing the weak undominance refinement
indiscriminately across all voting procedures.
As any study, this work has some limitations. First, we establish that in some cases the twoparty system may break down under voting procedures that do not admit truncated ballots, such as
Dual Voting. We do not address the question of how likely multipartism is to take place under Dual
Voting under a wide range of voter distributions and possible candidate positions. Our objective,
however, is to demonstrate that a single feature of a voting rule, such as admissibility of truncated
ballots, may have a large effect on the two-party system in some cases; thus, voters’ ability to
truncate ballots is essential for maintaining the two-party system under Approval Voting. Second,
we focus on voter behavior and not candidate behavior. Adding the candidate competition aspect
in comparing different voting procedures is important but is left for further research.9 Third, we
are looking at experimental elections with a relatively small number of voters. The implications of
our results for large elections are discussed in Section 5 at the end. Fourth, we assume all voters
have complete information about the candidates’ and other voters’ positions, and do not consider
a more realistic setting with incomplete information. All these issues are important but we feel
that an experimental test in a simple setting, such as the one we consider in this study, is essential
before one could move on to more complex and realistic settings.
The remainder of the paper is organized as follows. Section 2 outlines the theoretical model
that forms the basis for our experiment. Section 3 describes the experimental design. Section 4
presents the results. Section 5 concludes.
2
Model
A community of N citizens must elect a representative to implement a policy such as a tax rate or
the level of public spending. The set of policy alternatives X is a non-empty interval on R.10 We
assume that citizen ’s policy preferences can be represented by a strictly concave utility function
u (·), and we define x ≡ arg max u (x) as citizen ’s (unique) ideal policy. All citizens are expected
x∈X
utility maximizers.
9
A number of theoretical contributions explain Duverger’s law by the strategic behavior of parties/candidates: e.g. Palfrey, 1984; Weber, 1992 and 1998; Morelli, 2004; Callander, 2005; Fey, 2007.
10
Empirical works have shown that citizens’ preferences over the different issues tend to be strongly
correlated. It has therefore been suggested that assuming a unidimensional policy space need not be
restrictive, e.g. Cox (1990), Poole and Rosenthal (1991) and Hinich and Munger (1994).
6
Let C be a non-empty set of candidates, and let xi denote candidate i’s platform, i.e., the
policy that candidate i is committed to implement if elected. We assume that each candidate is
put on the ballot by a different party.
The citizens vote for candidates simultaneously. The election is held under a scoring rule,
where each citizen is given a set of point-scores {s1 , ..., sC } to cast for the different candidates,
with s1 ≥ s2 ≥ ... ≥ sC . The candidate who receives the most points wins the election. If
several candidates tie for first place, each of these candidates is elected with an equal probability.
We denote citizen ’s (pure) voting strategy by α (C) = α1 (C) , ..., αC
(C) , where αi (C) is the
point-score that citizen casts for candidate i. In this paper we consider three scoring rules: (1)
Plurality Voting (hereafter PV), under which a feasible ballot is any permutation of the vector
(1, 0, ..., 0); (2) Dual Voting (hereafter DV), under which a feasible ballot is any permutation of
the vector (1, 1, 0, ..., 0); and (3) Approval Voting (hereafter AV), under which a feasible ballot is
any permutation of any one of the following vectors (1, 0, ..., 0), (1, 1, 0, ..., 0), ..., (1, ..., 1). Citizens
can also abstain from voting, in which case the ballot is the null vector.11
Since in our model each candidate is identified with a different party, we say that a voting
procedure yields a two-party system if in any Nash equilibrium, defined in the usual way, at most
two candidates (parties) tie for first place. The following proposition, which follows from Dellis
(2007), identifies sufficient conditions for a scoring rule to yield a two-party system.
Proposition 1 Consider a non-empty set of candidates. A scoring rule yields a two-party system
if the voter
1. has a unique top-score vote to cast, i.e., s1 = s2 ; or
2. can truncate his ballot, i.e., a ballot is feasible if it is a permutation of any of the following
vectors (0, ..., 0), (s1 , 0, ..., 0), (s1 , s2 , 0, ..., 0), ..., (s1 , ..., sC ).
PV (AV) satisfies the first (second) condition of Proposition 1 and, therefore, yields a twoparty system. By contrast, DV satisfies none of the conditions of Proposition 1 and may yield
multipartism, as illustrated next. Consider an example displayed in Figure 1. There are three
candidates located on the interval [1, 10]. One candidate is located at 1 (labeled “Green”), another
one at 3 (labeled “Orange”) and the third one at 9 (labeled “Blue”). There are six voters whose
ideal points are indicated by crosses on the figure. There are two voters with ideal point 2, two
voters with ideal point 6, and two voters with ideal point 10. The voters at 2 are indifferent
between Green and Orange, and prefer these candidates to Blue; the voters at 6 are indifferent
11
Observe that under AV, voting for all candidates – i.e., the ballot (1, ..., 1) – is equivalent to abstaining
from voting.
7
between Orange and Blue, and prefer them to Green; and the voters at 10 prefer Blue to Orange
to Green.
INSERT FIGURE 1 ABOUT HERE
It is straightforward to check that in any Nash equilibrium under Plurality or Approval Voting,
at most two candidates tie for first place, implying a two-party system. In contrast, under Dual
Voting, an equilibrium exists where all three candidates tie for first place, implying multipartism.
This equilibrium under Dual Voting may occur due to strategic behavior on the part of the voters
located at 10. To see this, note that the voters at 2 and 6 cannot benefit from misrepresenting their
preferences and, therefore, should vote for the two candidates who are closest to their position;
this generates two votes for Green, four votes for Orange, and two votes for Blue. Voter at 10 may
vote for Blue and Orange, in which case Orange wins outright. Or, each voter at 10 may vote for
Blue and Green, in which case all three candidates tie for first place. For a payoff structure as
in Table 1, voters at position 10 get a higher expected payoff from the three-way tie than from
Orange winning outright and, therefore, have an incentive to cast one of their votes for Blue, their
most-preferred candidate, and “dump” their other vote on Green, their least preferred candidate.
Note that this incentive does not exist under Plurality and Approval Voting since neither of these
two voting rules forces people to vote for multiple candidates.
While this example appears specific, it is illustrative of the general properties of the voting
rules we study. Below, we describe an experiment that was designed to test the above predictions.
3
Experimental design
Experimental subjects in the roles of voters participated in sequences of experimental elections
over three candidates. There were three experimental treatments corresponding to three voting
procedures: Plurality Voting (vote for exactly one candidate), Dual Voting (vote for exactly two
candidates) and Approval Voting (vote for as many of the candidates as wished). The motivation
for the choice of these voting procedures is explained in detail in Sections 1 and 2 above. Under
all three voting procedures, voters were also given the option to abstain from voting.12
12
We gave this option to ensure symmetry across all three voting procedures. Indeed, without this
option, participants could have abstained from voting under Approval Voting, but not under Plurality
and Dual Voting. This is because under Approval Voting, voting for all candidates is equivalent to vote
abstention. In fact, our participants seldomly abstained, which suggests that our results would have been
similar had we not allowed participants to abstain from voting. In PV sessions abstention was chosen
in only 9 out of 1,440 voting decisions. In AV sessions abstention was chosen in 3 out of 1,392 voting
decisions while voting for all three candidates–a voting behavior which, as noted above, is equivalent to
vote abstention–was chosen 68 times. Finally, in DV sessions abstention was chosen in 62 out of 1,392
voting decisions.
8
There were twelve experimental sessions, with four independent sessions held under each of
the three voting procedures. The voting procedure was held constant within a session. There were
twelve subjects in each session. A session consisted of seven or eight voting series, with four election
periods per series.13 The division of a session into series was meant to provide some opportunity
for convergence within a series, while limiting dynamic strategic behavior. At the beginning of each
series of election periods, the participants were randomly matched into two groups of six voters
each. Within each group, the participants’ ideal points were randomly assigned among the voters’
locations as described in Figure 1. The colors associated with the candidates (Green, Orange and
Blue) were also randomly assigned among the candidates’ locations described in the figure.14 The
locations of the candidates and of the different participants in the group were common knowledge.
However, the identity and the individual voting history of the other participants in the group was
not revealed. The instructions (included in Appendix A) used a neutral terminology with the
candidates referred to as “alternatives”.
In each election period, an independent election was held within each group. Participants
placed votes for the candidates according to the given voting procedure. The candidate who
received the most votes in the group won the election. Ties were broken randomly. After all participants had placed their votes, each participant’s screen reported the vote totals of the candidates,
the winning candidate in their group, and the participant’s payoff from the election.
Table 1 reports the payoffs that a participant received from the election depending on his
location and on the location of the elected candidate.
INSERT TABLE 1 ABOUT HERE
The payoff matrix was constructed from strictly concave distance utility functions. Further,
the payoffs of a participant at 10 were constructed so that her expected payoff was larger when
all three candidates tied for first place than when Orange won outright; this was necessary for a
three-way-tie equilibrium to exist under DV. In addition, to address possible concerns for ‘equity’
across participants, affine transformations of the payoffs were taken so that at each position the
13
The first session under DV and AV and the first two sessions under PV had eight series of four election
periods. As it became obvious that there were no significant differences between the seventh and the eighth
series, we shortened all subsequent sessions to seven series of election periods so that no session lasts more
than two hours. We include in our analysis the data from these eighth series of election periods. Our
results are not sensitive to this inclusion.
14
During each session we also alternated the line-up described in Figure 1 with a line-up symmetric to
this, where candidates were located at 2, 8 and 10, and participants were located at 1, 5 and 9. The
two line-ups were used to keep the participants from getting accustomed to the same setting, and also to
check whether the results are different depending on whether the line-up involved a left extremist (i.e.,
a candidate located at 1) or a right extremist (i.e., a candidate located at 10). The data indicate no
difference in participants’ behavior across the two settings. We will therefore pool the data and present
the results in the context of the line-up described in Figure 1.
9
payoffs over all three candidates summed up to a similar number.15 The payoff matrix was common
knowledge to all participants.
The Nash equilibrium predictions for each treatment are summarized in Table 2.
INSERT TABLE 2 ABOUT HERE
The outcome of a Nash equilibrium where candidate x wins outright is denoted by {x}; the
outcome of a Nash equilibrium where candidates x and y tie for first place, and thus each is elected
with probability one-half, is denoted by {x, y}. Notice that under Plurality and Approval Voting
no Nash equilibrium exists where all three candidates are tying for first place. In fact, if we further
restrict attention to equilibria in weakly undominated voting strategies, only Orange and Blue are
elected with a positive probability.16 This contrasts with DV where an equilibrium exists with all
three candidates, {Green, Orange, Blue}, tying for first place (and no equilibrium exists where two
candidates are tying for first place).17
The experiment was computerized using the software z-tree (Fischbacher 2007). Figure 4 in
Appendix B reports the participant screen. It contained: (1) a representation of the locations,
as in Figure 1; (2) a payoff matrix, as in Table 1; (3) a test outcome calculator; (4) a ballot; (5)
a history that contained, for the current series of election periods, the distribution of vote totals
across the three candidates; and (6) the cumulative payoff of the participant.
Procedures Experimental participants were recruited among students at the University of
Hawai‘i at Manoa. Upon their arrival, participants were seated at computer terminals. Once all
twelve participants were seated, a hard copy of the instructions and an optional record sheet were
handed out and instructions were read aloud. Participants were invited to ask questions, which
were answered in public. After the instruction phase, participants were walked through a tutorial,
followed by two trial, unpaid elections.18 Paid elections then started. To minimize last period
effects participants were not told the number of election periods in the session.
15
The actual payoff sums are 193 for a participant at 2, 194 for a participant at 6 and 192 for a participant
at 10. We noticed that a few participants did compute these sums.
16
Under PV, a voting strategy is weakly undominated if the voter votes for a candidate other than his
least-preferred candidate. Under AV, a voting strategy is weakly undominated if the voter votes for all his
most-preferred candidates and does not vote for any of his least-preferred candidates. Finally, under DV
a voting strategy is weakly undominated if the voter votes for all his most-preferred candidates.
17
Note that under PV {Orange}, {Blue} and {Orange, Blue} are all supported by equilibria that survive
iterated deletion of weakly dominated voting strategies. By contrast, under AV (DV) only {Orange, Blue}
({Green, Orange, Blue}) is supported by an equilibrium that survives iterated deletion of weakly dominated
strategies.
18
In each session participants were offered the option of two additional trial elections; participants never
availed themselves of this option.
10
Each session lasted for around two hours including instruction. At the end of the session, each
participant was paid, in cash, the sum of his realized payoffs over all 28 or 32 election periods,
at the exchange rate of 1 U.S. dollar per 100 experimental tokens, plus a 5 dollar show-up fee.
Payments ranged from 16 to 33 dollars, with an average of 24.58 dollars and a standard deviation
of 2.70 dollars.
4
Results
A total of 144 subjects participated in the experiment, with about equal numbers of men and
women. Twelve independent sessions were held in total, with four sessions per each voting procedure, and twelve participants per session. There were a total of 240 elections and 1,440 voting
decisions in the PV treatment, and a total of 232 elections and 1,392 voting decisions in each of
the DV and AV treatments.
We shall proceed in two parts, examining election outcomes first, followed by individual voting
behavior. The results aggregated by voting rule are presented in Tables 3-8 in the main text.
Tables 11-14 with more detailed statistics by session are included in Appendix C. Unless stated
otherwise, we use session averages as independent units of observation for statistical tests. To trace
possible learning effects, we sometimes compare the data for early election periods (periods 1-12)
and late election periods (periods 13-28 or 13-32, depending on the session duration).
4.1
Election outcomes
We start by checking whether the actual election outcomes could be the result of random voting
behavior on the part of the participants or whether they are the result of purposeful voting behavior.
For this purpose, we compare the actual frequencies of election outcomes with the frequencies of
election outcomes that would obtain under the assumption that every participant acted randomly,
casting each admissible ballot with an equal probability. The frequencies are reported in Table 3.
INSERT TABLE 3 ABOUT HERE
For each voting procedure, the actual distribution of election outcomes is clearly different from
the distribution predicted by random voting (p-values for the Kolmogorov-Smirnov test are 0.000
for each session). We can therefore reject the hypothesis that our results are merely the outcome
of random voting.
We continue by checking whether the election outcomes are consistent with the Nash equilibrium predictions. Success rates for each voting procedure are reported in Table 4; Table 11 in
Appendix C gives more detailed statistics by session.
11
INSERT TABLE 4 ABOUT HERE
We find that under PV, 99.6% of election outcomes are consistent with the outcome of a Nash
equilibrium; the corresponding percentages are 95.7% and 90.5% for AV and DV, respectively.
These numbers are even higher if one considers late election periods. Indeed, under PV 99.3% of
late election outcomes (or 143 out of 144) are consistent with Nash equilibrium predictions, and
under both AV and DV 96.3% of late election outcomes (or 131 out of 136) are consistent with
Nash equilibrium predictions.
The predictive power of Nash equilibrium is still very strong if one restricts attention to Nash
equilibria in weakly undominated voting strategies (WUNE). A ballot is said to be weakly dominated if there exists another ballot that never makes the participant worse off and makes him
sometimes strictly better off.19 As Table 4 further reports, under PV 99.6% of election outcomes
are consistent with the outcome of a Nash equilibrium in weakly undominated voting strategies;
the corresponding percentages are 68.1% and 77.6% for AV and DV, respectively. These numbers
are even higher for late election periods: 99.3% for PV, 75% for AV, and 84.5% for DV.
Conclusion 1 summarizes these results.
Conclusion 1 For each voting procedure the actual distribution of election outcomes is significantly different from the distribution of election outcomes that corresponds to random voting.
Almost all election outcomes are consistent with (pure-strategy) Nash equilibrium predictions. An
overwhelming majority of election outcomes are also consistent with Nash equilibrium in weakly
undominated (pure) voting strategies. We now investigate the existence of a two-party system under each of the three voting procedures. In Section 2 above we said that a voting procedure yields a two-party system if in every
equilibrium no more than two candidates are tying for first place. It is obvious that even in a
tightly controlled laboratory experiment it is virtually impossible that it happens in all of the 232
or 240 elections that were conducted under each voting procedure; one would expect to see some
learning and experimentation on the part of participants under any voting rule.
INSERT TABLE 5 ABOUT HERE
Table 5 reports the frequencies of three-way ties under each voting procedure (see also Table 12
in Appendix C). Three-way ties occurred in 0.4% of elections held under PV, in 3% of elections held
19
Given our design, the following ballots are weakly dominated. Under PV, a ballot is weakly dominated
for a participant if and only if he abstains from voting or votes for his least-preferred candidate. Under
AV, a ballot is weakly dominated for a participant if and only if he abstains from voting, does not vote for
all his most-preferred candidates, or casts a vote for his least-preferred candidate. Under DV, a ballot is
weakly dominated for a participant if and only if he does not vote for his most-preferred candidate(s).
12
under AV, and in 13.8% of elections held under DV. Thus, three-way ties virtually never happened
in PV elections. Further, the frequency of three-way ties in AV elections is not statistically different
from the frequency of three-way ties in PV elections (p=0.1814 for a Wilcoxon Mann-Whitney rank
order test using session averages as units of observation). By contrast, the frequency of three-way
ties in DV elections is both statistically different from zero and statistically different from the
frequency of three-way ties in PV and AV elections at conventional significance levels (p=0.0286).
Based on these observations we cannot reject the hypothesis that PV and AV yield a two-party
system. At the same time, we can reject the hypothesis that DV yields a two-party system.
These conclusions are even stronger if one takes participants’ learning into account. From
Table 5, we observe no difference between early and late election periods for both PV and AV. In
the case of PV, three-way ties occurred in zero elections out of 96 in the early periods as compared
to one election out of 144 in the late periods. In the case of AV, three-way ties occurred in 3 out
of 96 early elections as compared to 4 out of 136 late elections. By contrast, under DV, three-way
ties occurred much more often in late elections than in early elections: 5.2% or 5 out of 96 early
elections, as compared to 19.9% or 27 out of 136 late elections.
These results are summarized in Conclusion 2.20,21
Conclusion 2 The frequencies of three-way ties in Plurality Voting and Approval Voting elections
are both negligible, and are not statistically different from each other. By contrast, the frequency
of three-way ties in Dual Voting elections is significantly above the frequencies of three-way ties
in Plurality Voting and Approval Voting elections. Thus, we can reject the hypothesis that Dual
Voting yields a two-party system, but we cannot reject the hypothesis that either Plurality Voting
or Approval Voting yields a two-party system. 20
Because we only re-assigned participants’ positions every four election periods, there is a chance that
we are catching some of the dynamic aspects of the game rather than the equilibrium results from a oneshot game. To help mitigate these elements, we separately looked at the results from the first election of
each series of four election periods. The above results are even stronger in this context.
21
In their experimental test of Duverger’s law Forsythe et al. (1996) use the average difference between
the vote totals of the second- and the third-placed candidates. There are several reasons why their test is
not adequate for our purposes. First, Forsythe et al. seek to test the concentration of votes on exactly two
candidates, while our purpose is to test the existence of at most two viable competitors. The second reason
why the average second to third place spread is not adequate here is most easily understood in the context
of PV elections. There are (at least) two cases where the spread is equal to zero: (1) a three-way tie; and
(2) a unanimous vote for Orange. While the latter case is consistent with PV yielding a two-party system,
the former case is not. The third reason why the average second to third place spread is not adequate
here follows from our comparative purposes. By letting or forcing voters to vote for more than one of
the candidates, the spread is necessarily smaller under Approval and Dual Voting than under PV. With
the above caveats in mind, we report the vote spreads under each voting procedure. The average second
to third place spread is equal to 31.9% under PV, 16.4% under AV, and 9.2% under DV. Moreover, the
spread in each PV session is higher than the spreads in any AV session, and the spread in each AV session
is higher than the spreads in any DV sessions; the differences in spreads among any two voting procedures
are statistically significant. Based on this test, the support for a two-party system is the strongest under
PV, followed in order by AV and DV.
13
So far our results are consistent with the prediction that Plurality and Approval Voting yield a
two-party system in the sense that in each election at most two candidates are tying for first place.
One may wonder however if across elections more than two candidates are viable. In other words,
one may ask whether the same one or two candidates are repeatedly in the winning set (i.e., the
set of candidates who are tying for first place) or whether each of the three candidates is regularly
in the winning set.
INSERT TABLE 6 ABOUT HERE
Table 6 reports, for every voting procedure, the frequency with which each candidate is in the
winning set (see also Table 13 in Appendix C). Orange, the middle candidate, is almost always
in the winning set: in 85.4% of the elections held under PV, 72% of the elections held under AV,
and 86.2% of the elections held under DV. Hence, Orange, the candidate located at 3, appears
focal. Blue, the candidate at 9, is often in the winning set as well – in 39.2%, 50.9% and 30.2%
of the elections held under PV, AV and DV, respectively. By contrast, Green, the candidate at 1,
is (almost) never in the winning set under PV (0.4% of the times), is rarely there under AV (in
only 4.3% of elections), but is frequently in the winning set under DV (in 20.7% of elections).22
The percentage of times Green is in the winning set is significantly higher under DV than under
either PV or AV (p=0.0286 for a Wilcoxon Mann-Whitney test). At the same time, the frequencies
with which Green is in the winning set under PV and under AV are not different at conventional
significance levels (p=0.1814).23 In late periods, Green is even less often in the winning set under
AV (3.7% of late election periods), and more often under DV (25% of late election periods). Thus,
we cannot reject the hypothesis that under both PV and AV only two candidates are viable across
elections. However, we can reject this hypothesis for DV.
These results are summarized in Conclusion 3.
Conclusion 3 Under both Plurality Voting and Approval Voting, only two out of three candidates
are viable across elections: their probability of being in the winning set is strictly positive. By
contrast, under Dual Voting, all three candidates are viable across elections. The middle candidate
is the candidate who is the most frequently in the winning set under all three voting procedures. 22
Observe that Orange and Blue are both Condorcet winners (i.e., candidates who defeat every other
candidate in a pairwise contest) while Green is a Condorcet loser (i.e., a candidate who is defeated by every
other candidate in a pairwise contest). Thus, the Condorcet loser is almost never elected under Plurality
and Approval Voting, but is regularly in the winning set under DV. However, this feature is design-specific
and need not hold in general.
23
A matched pairs test using sessions as units of observation shows that Green appears in the winning
set less frequently than Blue (p=0.0625 for both PV and AV). This is the best significance level we can
get with only four independent observations (sessions) under each voting procedure. In contrast, under
DV, the frequencies with which Green and Blue are in the winning set are not statistically different at
conventional significance levels (p=0.125).
14
Thus, Conclusion 3 extends across elections the within-election result stated in Conclusion 2.
4.2
Individual voting behavior
In this section, we look at behavioral regularities behind the election outcomes described above.
We start by considering whether the observed multipartism under DV may be attributed to random
factors, or if it is due to strategic voting. We say that a vote is strategic if it is neither sincere nor
weakly dominated; a vote is said to be sincere if whenever the voter votes for a candidate, he also
votes for all the candidates he prefers to that candidate.24
Recall, from Section 2, that in our experimental setting, an equilibrium with a three-way tie
occurs under DV only if the voters located at 10 behave strategically, casting one vote for their
most-preferred candidate, Blue, and dumping their second vote on their least-preferred candidate,
Green. (If, instead, they vote sincerely for the two candidates they most prefer, Blue and Orange,
then Orange wins outright.) By contrast, the same voters have no incentive to cast a vote for
Green if the election is held under PV or AV. Furthermore, casting a vote for Green is weakly
dominated for the voters located at 10 if the election is held under Plurality or Approval Voting.
We now investigate the voting behavior of our participants when they are located at 10.
INSERT TABLE 7 ABOUT HERE
Table 7 reports the frequencies with which each ballot was cast by the participants at 10 (see
also Table 14 in Appendix C).25 There are several things worth pointing out. First, among all
voting decisions made by the participants at 10, only 1.67% (8 out of 480) of the voting decisions
under PV, and only 8.41% (39 out of 464) of the voting decisions under AV, included a vote for
Green. By contrast, under DV, 50.44% of the voting decisions made by the participants in this
position (234 voting decisions out of 464) included a vote for Green. This number is significantly
higher than the numbers obtained under PV and AV. Second, the strategic ballot consisting of
a vote for Green and a vote for Blue was cast 43.97% of the times under DV, whereas the same
ballot was cast only 4.31% of the times under AV. Moreover, under DV, this ballot was cast
much more frequently in late election periods than in early election periods (39.06% of the times
in early periods compared to 47.43% in late periods), whereas under AV the frequency was left
24
For PV, this means that a voter votes sincerely if he votes for (one of) his most-preferred candidate(s).
For DV, this means that a voter votes sincerely if he votes for the two candidates he prefers. Finally, for
AV, this means that a voter votes sincerely if he votes either for his most-preferred candidate, or for the
two candidates he prefers (or for all three candidates). This definition of sincere voting corresponds to the
one proposed by Brams and Fishburn (1978).
25
For each voting procedure, Table 7 also indicates which ballots are strategic, and which are weakly
dominated. One may notice that none of the ballots are marked as strategic under AV. This is because
with three voting alternatives, every weakly undominated ballot under AV is sincere (Brams and Fishburn
1978, Theorem 3).
15
virtually unchanged (4.17% in early periods compared to 4.41% in late periods). Figure 2 further
demonstrates that in late DV elections many of the participants chose to cast the strategic ballot
each time they were located at 10.
FIGURE 2 ABOUT HERE.
Finally, note that in DV elections the strategic ballot (Green, Blue) was cast more often than
the sincere ballot (Orange, Blue) – 43.97% as compared to 41.38%. This difference is even more
pronounced if one takes participants’ learning into account; the corresponding percentages for
late election periods are 47.43% for the strategic ballot as compared to 38.97% for the sincere
ballot. This observation is quite remarkable since casting the sincere ballot (Orange, Blue) and
casting the strategic ballot (Green, Blue) are both equilibrium strategies for the voters at 10 under
DV; the sincere ballot is part of the Nash equilibrium where Orange wins outright, whereas the
strategic ballot is part of the three-way-tie equilibrium (both equilibria are in weakly undominated
strategies). However, the voters at 10 may prefer the three-way-tie equilibrium because of a higher
expected payoff.
These results are summarized in Conclusion 4.
Conclusion 4 Multipartism in the Dual Voting experimental elections emerged because of strategic
voting behavior on the part of participants. Voters in position 10 voted strategically almost half of
the time, casting one vote for their most-preferred candidate (Blue), and dumping their other vote
on their least-preferred candidate (Green). In contrast, under both Plurality Voting and Approval
Voting, participants at 10 cast very few votes for their least-preferred candidate (Green), yielding
a two-party system. We next consider how often our participants cast weakly dominated ballots. The casting
of only few weakly dominated ballots would provide evidence of sophistication on the part of
participants and their ability to understand the voting procedure. By contrast, the casting of
many weakly dominated ballots would suggest that the participants did not fully understand the
voting procedure.
INSERT TABLE 8 ABOUT HERE
The frequencies with which weakly dominated ballots were cast are reported in the first three
columns of Table 8. Weakly dominated ballots were rarely cast under PV (only 2.8% of ballots
overall), whereas a significant fraction of the ballots cast under DV (12.5%) and an extremely
high fraction of the ballots cast under AV (36.2%) were weakly dominated. For any two voting
16
procedures, the frequencies with which weakly dominated ballots were cast are statistically different
(p=0.0286 for a Wilcoxon Mann-Whitney test in all cases). Table 8 also shows that the percentage
of elections in which at least one participant cast a weakly dominated ballot was the highest under
AV (93.5% of all elections), followed by DV (54.7% of elections), followed by PV (15.4% of all
elections). Again, the differences between any two voting rules are all significant (p =0.0286).
It is possible that weakly dominated ballots were mostly cast in early elections with the participants learning not to cast weakly dominated ballots as the sessions progressed. Table 8 indicates
that some learning seem to have taken place under DV; the share of weakly dominated ballots cast
under DV dropped from 15.1% in early election periods to 10.7% in late periods. By contrast,
the share of weakly dominated ballots under both PV and AV remained rather stable: under PV,
3.5% in early election periods as compared to 2.4% in late periods; and under AV, 36.5% in early
election periods as compared to 36% in late periods.
We further checked for the presence of learning by testing for each voting procedure and for
each voter position, whether the election period was a good predictor of the likelihood that a
weakly dominated ballot was cast. We found that DV was the only voting rule among the three
studied here where the likelihood of casting a weakly dominated ballot decreased significantly as
periods progressed. For the other two voting rules, the election period was not a good predictor
of the likelihood with which a weakly dominated ballot was cast.
Conclusion 5 summarizes the above observations.
Conclusion 5 Among the three voting procedures, Plurality Voting had the smallest share of
weakly dominated ballots, followed in order by Dual Voting and Approval Voting. Under Plurality Voting, weakly dominated ballots were rarely cast, whereas under Approval Voting every third
ballot cast was weakly dominated. Under Approval Voting there was little learning on the part of
the participants to avoid casting weakly dominated ballots. By contrast, some learning took place
under Dual Voting. The widespread and persistent use of dominated strategies by experimental subjects under
AV but not under PV is consistent with Rapoport et al.’s (1991) findings, who attribute this
phenomenon, in part, to the novelty and complexity of AV as compared to PV. If we use the
frequency of weakly dominated ballots cast as an empirical measure of a voting rule complexity,
then, based on our data, PV appears the simplest, and AV appears the most complex, among the
three voting procedures that we study.
Finally, we look at subject heterogeneity among participants.
INSERT FIGURE 3 ABOUT HERE.
17
Figure 3 illustrates the distribution of weakly dominated ballots among participants. While many
participants rarely cast weakly dominated ballots, a few did cast weakly dominated ballots almost
every period. We reject the hypothesis that there was no individual heterogeneity using the
Kolmogorov-Smirnov one-sample test under all three voting rules (p-value = .000). A similar
result applies with regards to the casting of the strategic ballot in DV elections; see Figure 2,
above. Some participants were systematically casting the strategic ballot each time they were
located at position 10, starting early on in the session. By contrast, others chose never to cast the
strategic ballot. Again, these differences are highly significant (p-value = .000). We conclude:
Conclusion 6 Participants were heterogeneous both in their propensity to cast weakly dominated
ballots and in their propensity to cast the strategic ballot under Dual Voting. Interestingly, we observe that individuals casting weakly dominated ballots did not always
affect aggregate voting outcomes. Weakly dominated votes were cast by individual voters in many
elections (see Table 8); yet, most of the election outcomes were consistent with the outcome of
a Nash equilibrium in weakly undominated strategies (Conclusion 1 in Section 4.1 above). The
starkest example is AV, where 93.5% of (or 217 out of 232) elections included at least one weakly
dominated ballot, but 68.1% of electoral outcomes (or 158 out of 232) were still consistent with
Nash equilibria in weakly undominated strategies (Tables 4 and 8). This suggests that in aggregate,
people’s ‘mistakes’ largely canceled each other out, thereby yielding the same outcomes as if all
the ballots cast were weakly undominated.26 In contrast to “mistakes”, participants’ purposeful
(strategic) behavior often did have an effect on electoral outcomes: many outcomes under DV
corresponded to the Nash equilibrium under strategic voting (Conclusions 2 and 4).
4.3
Bounded Rationality
We now consider whether models of bounded rationality may add to explaining the participant
behavior in our experiment. We consider two popular models: Quantal Response Equilibrium and
level-k reasoning.
Quantal Response Equilibrium (QRE) is a model of bounded rationality first developed by
McKelvey and Palfrey (1995). In QRE, subjects’ beliefs of their expected payoffs, while correct on
26
The experimental evidence that outcomes of many institutions, especially markets, are robust to individual errors, has been used by experimental economists to justify the rationality approach to social
sciences. Charles Plott writes: “Claims about the irrelevance of models of rational choice and the consequent irrelevance of economics are not uncommon... If one looks at experimental markets for evidence,
the pessimism is not justified. Market models based on rational choice principles... do a pretty good job
of capturing the essence of very complicated phenomena” (1986, p. S325). Colin Camerer also notes that
“... better models of individual decision making may not improve market level predictions; whether they
do so is fundamentally an empirical question that economic experiments help answer...” (Camerer, 1995,
p. 589).
18
average, are subject to random error. In the most common form of QRE, logit QRE, the stochastic
error term is assumed to be drawn from a logit distribution. Because the error term can be any
real number, all strategies are chosen with positive probability, but strategies with higher true
expected payoff are played with higher frequency than strategies with lower expected payoffs.27 Of
particular importance in the QRE model, is the parameter λ, which is inversely proportional to
the spread of the stochastic error term. When the subjects’ expected payoffs are subject to very
large error terms, (i.e., λ is very low), subjects tend to randomize more with λ → 0 converging
to perfectly random strategy mixing. Conversely, as λ → ∞, subjects’ plays converge to a Nash
equilibrium. The predicted probabilities for select strategies given different levels of λ are expressed
in Figures 5 - 7 in Appendix D.28
Two observations regarding the QRE predictions are of particular interest for our experimental
results. First, even as the error term decreases to zero (i.e., λ → ∞), the limit strategies for QRE
equilibrium still contain weakly dominated strategies. This further suggests that assuming subjects
do not cast weakly dominated ballots may be an overly restrictive assumption. Second, under Dual
Voting, for voters in position 10 the only strategy that is consistent with the QRE limiting strategy
is the strategic vote (Green, Blue).29 Thus, the limit of the QRE model as λ → ∞ predicts that,
under Dual Voting, participants in position 10 should always vote strategically.
The maximum likelihood estimations for λ for each of the three voting rules, along with the
corresponding QRE predictions on different types of strategies used, are summarized in Table 9,
below.30
INSERT TABLE 9 ABOUT HERE
The QRE estimations make some correct qualitative predictions about the data. In general, as
per QRE predictions, subjects did cast many weakly dominated ballots, but they tended to cast
higher expected payoff ballots more frequently than low expected payoff ballots. For example, a
27
Specifically, the probability player i plays strategy sij ∈ Si , denoted σi (sij ) is given by σi (sij ) =
λEPi (sij )
e
sik ∈Si
28
eλEP (sik )
, where EP (sij ) is the equilibrium expected payoff from choosing strategy sij .
To our knowledge, Figure 7 features the first publicized game where the primary QRE branch contains
a backward bending section. Many thanks to Ted Turocy for his insight into QRE and for helping us verify
the stability of the ‘forward-moving’ branches.
29
The following Nash equilibrium strategies are the limit strategies for QRE equilibrium, as λ approaches
infinity: For PV, voters at 2 vote for Orange; voters at 9 vote for Blue; and voters at 6 vote for Orange, or
Blue or abstain; thus Orange or Blue may win. For AV, voters at 2 vote for Orange, or Green and Orange;
voters at 6 vote for Orange, or Blue, or Orange and Blue; and voters at 10 vote for Blue; thus Orange or
Blue may win. For DV, voters at 2 vote for Green and Orange; voters at 6 vote for Orange and Blue; and
voters at 9 vote for Green and Blue; thus a three-way tie occurs, and Green or Orange or Blue may win.
30
The table reports the frequencies of sincere, weakly dominated, and, when applicable, strategic ballots.
The QRE predicted frequencies of weakly dominated ballots are especially high for the participants in
position six. This is because the payoffs for each of the three candidates winning are very similar, making
players in position six largely indifferent between all their available strategies.
19
substantial number of subjects in position 2 under AV (23.3%) cast ballots only for their preferred
alternative that had a reasonable chance of being in the winning set (Orange), despite the fact
that voting for both Orange and Green weakly dominated it. QRE also provides us with another
indication that there was little to no learning under either PV or AV, but that there was substantial learning under DV.31 However, QRE drastically overpredicts the number of weakly dominated
ballots cast. This suggests that while our subjects may not have been rational enough to exclusively cast weakly undominated ballots, they were sophisticated enough to do it fairly often. One
possible reason why QRE may have produced substantial estimation errors is because the baseline
QRE model assumes that all subjects are homogeneous.32 Since our experimental participants
were highly heterogeneous both in their propensity to cast weakly dominated ballots as well as
their propensity to vote strategically (Conclusion 6, above), we next turn to a model of bounded
rationality that can take subject differences into consideration.
Starting with Nagel (1995), many studies have attempted to explain subjects’ choices based
on the ‘depth of reasoning’ made for their experimental choices. Following Costa-Gomez and
Crawford (2006), level-k reasoning model categorizes subjects into different types or ‘levels’. A
level 0 subject is one that perfectly randomizes among all available alternatives. A level k subject,
meanwhile, assumes that all other subjects are level (k − 1) and best respond most of the time, but
randomize their play with some finite probability. This randomization is assumed known by all
higher-level players. While not all players follow this typing exactly, the above papers and other
suggest that a large fraction of subjects do. Thus, it is not necessary to try to assign types to all of
our subjects, but instead we simply note the implications of having a large fraction of our subjects
be level-k reasoners. Table 10, below, summarizes the preferred ballots of level-k subjects. Note
that under all voting rules and for all positions, the preferred ballots for all subjects level 2 and
above are identical. Thus the table only reports on levels 1 and 2.
TABLE 10 ABOUT HERE.
One particularly appealing aspect of using level-k reasoning for this experiment is that it
gives a clear explanation for why some subjects cast predominantly sincere ballots while others
31
This is done by estimating λ separately for early and late periods under each voting rule. To the
extent that QRE’s λ parameter measures the degree of randomness of subjects’ voting, a higher λ could
be correlated with a higher degree of sophistication. Although MLE λ values increased from early to late
periods under both PV and AV (from .120 to .133 and from .128 to .130 respectively), the increase was
much more pronounced under DV where it increased from .185 to .321.
32
Camerer et al. (2008) provide a framework for a heterogeneous QRE model that also encompasses the
level-k reasoning model presented below. However, just as regular QRE requires parametric assumptions
to provide explanatory power (see Haile et al. 2008), so too does a heterogeneous QRE model. Here we
use a more computationally developed homogeneous QRE model.
20
cast predominantly strategic ballots when in the strategic position under DV. Specifically, level1 subjects cast sincere ballots because they fear Green winning outright, while level-2 subjects
cast strategic ballots because they believe that Orange will win outright otherwise. Furthermore,
unlike QRE, level-k reasoning is able to correctly predict the large frequency of weakly undominated
ballots under AV.
However, there are several drawbacks to using level-k reasoning to analyze the current data.
Aside from the theoretical issues with using it for repeated games, the limited strategy space of
this game makes it very difficult to distinguish between types, especially under AV where all types
have the same preferred ballot for each position. Further, level-k reasoning predicts a large number
of votes cast for Blue under PV, something that is not observed empirically.33 Instead, all sessions
had a very strong centrist effect, where the central candidate (Orange) received a lion’s share of
the votes.
Conclusion 7, below, summarizes our findings on the above two models of bounded rationality.
Conclusion 7 Quantal Response Equilibrium correctly predicts that voters tend to cast higher
expected payoff ballots more frequently than low expected payoff ballots, and that voters under Dual
Voting often vote strategically. At the same time, QRE drastically overpredicts the shares of weakly
dominated ballots cast. Level-k reasoning allows to explain a degree of observed subject heterogeneity
as well as a large share of weakly undominated ballots. At the same time, it fails to predict the
prevalence of ballots cast for the centrist candidate under Plurality Voting. 5
Discussion and conclusion
Duverger’s law states that countries where policymakers are elected under Plurality Voting tend
to have a two-party system. This empirical regularity has been explained by the wasting-the-vote
effect of Plurality Voting: fearing to waste their vote on an underdog, voters are led to behave
strategically and vote for the serious contender they prefer (instead of voting sincerely for the
contender they most prefer). Such strategic behavior on the part of voters triggers a desertion
of the trailing candidates and a concentration of the votes on only two candidates. Dellis (2007)
shows that this Duvergerian logic generalizes to some voting procedures other than Plurality Voting.
Approval Voting, in particular, is predicted to yield a two-party system as well. The key feature of
Approval Voting that allows to sustain the two-party system is admissibility of truncated ballots:
voters are allowed, but not forced to vote for multiple candidates. Forcing the voters to vote for
33
Specifically, level-1 subjects in position 6 are indifferent between voting for Orange and Blue, and are
generally assumed to randomize between the two. Level-2 subjects and higher, meanwhile, exclusively vote
for Blue. Therefore, if any subjects are level-2 or higher, then we should expect a larger share of subjects
casting ballots for Blue. Most studies predict at least 50% of subjects are level-2 or above.
21
more than one candidate, as under Dual Voting, may generate incentives for strategic vote dumping
that may give rise to multipartism.
Our experimental results provide evidence that is in line with the above predictions. In the
overwhelming majority of our Plurality Voting and Approval Voting elections, either one candidate
won the election outright, or two candidates tied for first place. In contrast, the Dual Voting
elections often resulted in three-way ties among the candidates, due to strategic voting in the form
of vote dumping by certain voters. We thus provide evidence that voters’ ability to truncate ballots
is essential for maintaining the two-party system under Approval Voting.
Our results help ease fears raised by many scholars – including the supporters of Approval
Voting – that using Approval Voting for (U.S.) political elections would obliterate the two-party
system. This fear was best captured by Cox (1997; 95) when he wrote: “... many believe that
[Approval Voting] would lead to multipartism, although there is no empirical evidence on this
score.” Notice that the emergence of a two-party system under Approval Voting is here the more
remarkable that our participants were unfamiliar with Approval Voting and that no education
effort had been undertaken prior the experiment.
We also observe some features of the individual voting behavior that have implications for
the implementability of Approval Voting and Dual Voting. One reasonable (and rather minimal)
requirement for a voting procedure to be implementable is that people understand it well enough
that they do not cast weakly dominated ballots. Our results suggest that contrary to Plurality
Voting, neither Approval Voting nor Dual Voting meet this requirement, even in a setting as simple
as in our experiment. Even more troublesome for Approval Voting is that people may not learn
over time to avoid casting weakly dominated ballots. This raises doubts on the claim made by the
supporters of Approval Voting that Approval Voting is simple for voters to understand. Despite
this conclusion, there is some hope that the problem would not be that serious if Approval Voting
was used for political elections. Indeed, party elites and media – two actors that are absent from
our experiment – may undertake an education effort toward the citizens. Also, political parties
may make suggestions to their supporters on how to vote in the election.34 If political parties
and interest groups guide the voters, then strategic voting behavior that was observed in our small
experimental elections may, under certain circumstances, also be encouraged and observed in larger
scale elections.
34
This is done for example in Australia, where legislative elections are held under Single Transferable
Vote (a voting rule which is much more complex than any of those we have considered here). In Lower
House elections, it takes the form of voting cards that parties distribute to voters when they enter the
polling station and that suggest voters a specific ballot to cast. In Senate elections, it takes the form of
political parties submitting beforehand a ballot and voters having the option to tick a box on the ballot
paper to select the ballot submitted by that party. For more details on the Australian electoral system
and the coordination role by political parties, see chapters 3 and 6 in Farrell (2001).
22
Another implication of our results concerns the usefulness of the weak undominance refinement
in the theoretical voting literature. If people are frequently casting weakly dominated ballots (as
was the case here in Approval Voting elections and, to a lesser extent, in Dual Voting elections),
then restricting attention to equilibria in weakly undominated voting strategies may yield misleading predictions. Moreover, if the frequency of weakly dominated ballots varies from one voting
procedure to another (as we have observed here), one may question the desirability of using this
refinement for comparing alternative voting procedures – a standard practice in the formal voting
literature. It is important to note, however, that despite the prevalence of participants casting
weakly dominated ballots under both Approval Voting and Dual Voting, most of our experimental elections delivered an outcome that was consistent with the outcome of a Nash equilibrium
in weakly undominated voting strategies. This suggests that in aggregate people ‘mistakes’ may
cancel each other out, thereby yielding the same electoral outcomes as if all the ballots cast were
weakly undominated.
As a word of caution, we emphasize that we see our results as preliminary evidence on the
effect of the three voting rules on the two-party system. Many robustness checks with varying
numbers and locations of voters and candidates are needed. Further, to make the setting more
realistic, it is important to incorporate strategic candidacy decisions into the analysis. Another
issue to consider is that of path dependence in electoral reforms. The current experiment shows
that multipartism may arise under Dual Voting, but it is silent about whether multipartism is
likely to occur following an electoral reform where Plurality Voting is replaced with Dual Voting.
More generally, in the presence of multiple equilibria – as is the case in our setting – we would
like to know which of them will become focal following the replacement of Plurality Voting with
another voting procedure. Our present evidence indicates that an equilibrium with the middle
candidate in the winning set might be focal. We are currently working on these questions.
23
Appendix A: Experimental Instructions
Experimental Instructions (PV)
Introduction
You are about to participate in an experiment in the economics of decision making in which you
will earn money based on the decisions you make. All earnings you make are yours to keep and will
be paid to you IN CASH at the end of the experiment. During the experiment all units of account
will be in experimental dollars. Upon concluding the experiment the amount of experimental
dollars you earn will be converted into dollars at the conversion rate of 1 U.S. dollar per 100
experimental dollars. Your earnings plus a lump sum amount of 5 U.S. dollars will be paid to
you in private.
Do not communicate with the other participants except according to the specific rules of the
experiment. If you have a question, feel free to raise your hand. An experimenter will come over
to you and answer your question in private.
In this experiment you are going to participate in a decision process along with 11 other
participants. From now on, you will be referred to by your ID number. Your ID number will be
assigned to you by the computer.
Periods and Groups
Decisions in this experiments will occur in a sequence of independent decision periods. At the
beginning of the the first decision period, you will be assigned to a decision group with 5 other
participant(s). You will not be told which of the other participants are in your decision group.
What happens in your group has no effect on the participants in other groups and vice versa. The
group composition will stay the same for 4 periods, but will change afterwards.
Decisions and Outcomes
During each period, you will be asked to choose, or vote for, one of the three alternatives, Green,
Orange, and Blue. All participants in your group will make their decisions at the same time,
without knowing the choices of others. The computer will then select as an outcome for your group
the alternative that most of the participants in your group voted for. If two or more alternatives
get the same highest number of votes, then an alternative among those with the highest number
of votes will be selected as an outcome randomly.
Example 1 Suppose the number of votes for each alternative in your group are: 1 vote for
Green, 2 votes for Orange, and 3 votes for Blue, then Blue will be the outcome for your group.
24
Example 2 Now suppose there are 3 votes for Green, 3 votes for Orange, and 0 votes for Blue.
Then the outcome will be chosen randomly between Green and Orange.
Voting Rules
In each period, you will be asked to choose (vote for) EXACTLY ONE of the alternatives. You
may also choose to vote for NONE, but if you decide to vote for any, then you must vote for exactly
one. The voting rule will stay the same for the whole experiment.
Payoffs
The value of decisions to you (your payoffs from each alternative) will be given to you by the
computer before you make your choices. (You will also know the payoffs of other participants in
your group before you make your decisions.) In order to understand how your payoffs depend on
the alternative chosen, suppose that the alternatives have positions on a number line, and you (as
well as other participants in your group) have your most preferred, or ideal position, on this line.
For example, the picture below indicates that Orange, Green and Blue correspond to positions 3,
6, and 9, respectively. The participant’s ideal positions, or locations, are marked with ”X” on the
graph, with your ideal position highlighted in red.
For example, on this picture your ideal position is at 5. The picture indicates that there is
another participant located at the same position, two other participants are located at 8, and the
other two participants are located at 10.
The amount of payoff you receive in a given period depends on how close the selected alternative, or the outcome for the period, is to your ideal position. The closer the outcome is to your
ideal position, the higher is your payoff. For example, the picture above indicates that Green and
Orange are closer to your position than Blue, and therefore they will correspond to a higher payoff
for you than Blue.
25
The exact payoffs will be given to you in a payoff table, that shows each participant’s payoff,
depending on their ideal position and the outcome in the period. Your payoffs are highlighted in
red. For example, the table below indicates your payoffs correspond to the ideal position at 5, and
your payoffs are:
Example 1 (continued) Suppose, again for example, your ideal position is at 5, and your
payoffs are as given in the payoff table above. Then, if the numbers of votes for each alternative
are as given in the example 1 above, that is: 1 vote for Green, 2 votes for Orange, and 3 votes for
Blue, then Blue will be the outcome for your group. As a result, your payoff in this period will be
1 experimental dollars.
Example 2 (continued) Now suppose that your payoff from alternatives are as before, and
there are 3 votes for Green, 3 votes for Orange, and 0 votes for Blue. Then alternatives Green and
Orange are equally likely to be selected. As a result, your payoff in this period will be either 10 or
5 experimental dollars, depending on the outcome (5 if Orange, 10 if Green).
To help you with your choice, you will be provided with an outcome calculator, that will help
you to assess the likely outcome, depending on the number of votes that each alternative receives.
After all participants make their choices, the computer will display the “Results” screen, which
will inform you about how many participants chose each alternative, the resulting outcome for the
group, and your payoff in this period. After all subjects have seen the results, the next period will
begin automatically.
This will continue for a number of periods. The group composition, and the participants’
payoffs will stay the same for 4 periods, but will change every four periods. You will be informed
about the new payoffs and the new number of alternatives you may vote for, when the changes
occur.
Your total payoff in the experiment will be equal to the sum of per period payoffs.
26
The first two periods in the experiment will be trial periods. They will not count toward your
earnings.
ARE THERE ANY QUESTIONS?
If you have a question during the experiment, please raise your hand and I will come by to
answer your question.
Exercises
Exercise 1 Suppose that your payoff from Green is 11, your payoff from Orange is 13, and your
payoff from Blue is 8 experimental dollars. Given the choices of the participants listed below, fill
in the alternative(s) that may selected and your corresponding payoff(s):
Number of votes for Green: 5
Number of votes for Orange: 4
Number of votes for Blue: 2
Can Green be selected? 2 yes 2 no
Can Orange be selected? 2 yes 2 no
Can Blue be selected? 2 yes 2 no
Your payoff will be:
Exercise 2 Now suppose the numbers of votes for each alternative are as listed below. Fill in
the alternative(s) that may selected and your corresponding payoff(s) for this period (assume that
your payoffs are the same as in Exercise 1 above):
Number of votes for Green: 4
Number of votes for Orange: 4
Number of votes for Blue: 4
Can Green be selected? 2 yes 2 no
Can Orange be selected? 2 yes 2 no
Can Blue be selected? 2 yes 2 no
Your payoff will be:
ARE THERE ANY QUESTIONS?
27
Appendix B: Subject Computer Screenshot
FIGURE 4 HERE
Appendix C: Additional Tables
TABLES 11-14 HERE
Appendix D: Additional Figures
FIGURES 5 - 7 HERE
28
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31
List of Tables
1. Experimental Design: Payoff Matrix
2. Equilibria
3. Frequency of Outcomes, by Voting Rule and Session
4. Nash and Weak Undominance Nash Equilibrium Outcomes by Voting Rule
5. Share of Three-Way Ties in Election Outcomes
6. Frequencies of Each Alternative Appearing in the Winning Set, by Voting Rule
7. Ballots Cast by Participants in Position 10, percent
8. Weakly Dominated Votes
9. Actual and Predicted Frequencies Under Quantal Response Equilibrium, by Position and
Voting Rule
10. Preferred Ballots Under Level-k Reasoning
11. Nash and Weak Undominance Nash Equilibria, by Voting Rule and Session
12. Share of Three-Way Ties in Election Outcomes, by Session
13. Frequencies of Each Alternative Appearing in the Winning Set, by Session
14. Ballots Cast by Participants in Position 10, by Session, percent
List of Figures
1. Experimental Design: Number Line Representation
2. Distribution of Strategic Votes among Voters in Position 10 Under Dual Voting
3. Distribution of Weakly Dominated Votes among Voters, by Voting Rule
4. Subject Computer Screenshot
5. Plurality Voting: QRE Predicted Strategies’ Frequencies vs. Observed Frequencies
6. Approval Voting: QRE Predicted Strategies’ Frequencies vs. Observed Frequencies
7. Dual Voting: QRE Predicted Strategies’ Frequencies vs. Observed Frequencies
32
Table 1: Experimental Design: Payoff Matrix
Position of
Participant
2
6
10
Payoff by Outcome
Green (located
Orange
Blue (located
at 1)
(located at 3)
at 9)
91
54
32
91
70
40
11
70
120
Table 2: Equilibria
Voting Rule
PV
Equilibrium Outcome
{G}
{O}*
{O,B}*
Supporting Strategy Profiles**
All weakly dominated***
{G,G,O,O,O,O}
{G,O,O,O,O,O}
{G,O,O,O,O,B}
{G,O,O,B,O,O}
{G,O,O,O,B,B}
{O,O,O,O,O,O}
{O,O,O,O,O,B}
{O,O,O,B,O,O}
{O,O,O,O,B,B}
{G,G,O,B,B,B}
{G,G,B,B,O,B}
{G,G,B,B,B,B}
{G,O,B,B,B,B}
{O,O,B,B,B,B}
{O,O,O,B,B,B}
{G}
{O}*
{B}
{O,B}*
All weakly dominated***
{(G,O),(G,O),(O,B),(O,B),(O,B),(O,B)}
All weakly dominated***
{(G,O),(G,O),(O,B),(O,B),B,B}
{B}*
AV
DV
{G}
All weakly dominated***
{O}*
{(G,O),(G,O),(O,B),(O,B),(O,B),(O,B)}
{B}
All weakly dominated***
{G,O,B}*
{(G,O),(G,O),(O,B),(O,B),(G,B),(G,B)}
The colored alternatives are located at: (G)reen -- 1, (O)range -- 3, (B)lue -- 9
* Indicates the outcome is a WUNE (Nash Equilibrium in Weakly
Undomainted Strategies).
** Only those not containing weakly dominated strategies are listed. The
first two entries list the ballot cast by the participants in position 2, the
second two by those in position 6, and the last two by those in position
10.
*** All supporting strategy profiles include weakly dominated strategies
Table 3: Frequency of Outcomes, by Voting Rule and Session
Session
PV-random
PV-actual -- all
PV1
PV2
PV3
PV4
Green
0.24
0.00*
0.00
0.00
0.00
0.00
Orange
0.24
0.61**
0.73
0.53
0.52
0.64
Blue
0.24
0.15**
0.03
0.28
0.16
0.11
Green and Orange
0.08
0.00
0.00
0.00
0.00
0.00
Green and Blue
0.08
0.00
0.00
0.00
0.00
0.00
Orange and Blue
0.08
0.24**
0.23
0.19
0.30
0.25
Three-way tie
0.05
0.00
0.00
0.00
0.02
0.00
AV-random
AV-actual -- all
AV1
AV2
AV3
AV4
0.23
0.00*
0.00
0.00
0.00
0.00
0.23
0.48**
0.64
0.43
0.50
0.34
0.23
0.28*
0.22
0.23
0.25
0.41
0.08
0.01
0.00
0.02
0.00
0.02
0.08
0.00
0.00
0.00
0.02
0.00
0.08
0.20**
0.14
0.29
0.18
0.20
0.06
0.03
0.00
0.04
0.05
0.04
DV-random
0.22
0.22
0.22
0.10
0.10
0.10
DV-actual -- all
0.04*
0.64**
0.09*
0.02
0.01
0.07
DV1
0.02
0.53
0.14
0.00
0.02
0.09
DV2
0.05
0.59
0.09
0.05
0.00
0.07
DV3
0.04
0.80
0.04
0.00
0.00
0.00
DV4
0.07
0.64
0.07
0.02
0.02
0.11
The rows that are marked "-random" show the frequency of each outcome if participants voted randomly.
The colored alternatives are located at: Green -- 1, Orange -- 3, Blue -- 9
* Consistent with Nash Equilibrium outcome
** Consistent with WUNE (Nash Equilibrium in Weakly Undomainted Strategies) outcome
0.05
0.14**
0.20
0.14
0.13
0.07
Table 4: Nash and Weak Undominance Nash Equilibrium Outcomes by Voting Rule
PV
AV
DV
NE Outcomes, share
all
early
late
0.996
1.000
0.993
WUNE Outcomes, share
all
early
late
0.996
1.000
0.993
(0.004)
(0.004)
(0.000)
(0.007)
(0.000)
(0.069)
0.957
0.948
0.963
0.681
0.583
0.750
(0.013)
(0.023)
(0.016)
(0.031)
(0.050)
(0.037)
0.905
0.823
0.963
0.776
0.677
0.846
(0.019)
(0.045)
(0.016)
(0.027)
(0.048)
(0.031)
Numbers in parenthesis are the standard errors
Table 5: Share of Three Way Ties in Election Outcomes
PV
Three-way ties, share
all
early
late
0.004
0.000
0.007
(0.004)
AV
DV
(0.000)
(0.007)
0.030
0.031
0.029
(0.011)
(0.018)
(0.014)
0.138
0.052
0.199
(0.023)
(0.023)
(0.034)
Numbers in parenthesis are the standard errors
Table 6: Frequencies of Each Alternative Appearing in the Winning Set, by Voting Rule
mean
(stdv)
PV
AV
DV
all
0.004
Green
early
0.000
late
0.007
all
0.854
Orange
early
0.854
late
0.854
all
0.392
Blue
early
0.438
late
0.361
(0.004)
(0.000)
(0.007)
(0.023)
(0.036)
(0.029)
(0.032)
(0.051)
(0.040)
0.043
0.052
0.037
0.720
0.625
0.787
0.509
0.594
0.449
(0.013)
(0.023)
(0.016)
(0.029)
(0.050)
(0.035)
(0.033)
(0.051)
(0.042)
0.207
0.146
0.250
0.862
0.833
0.882
0.302
0.302
0.301
(0.027)
(0.036)
(0.037)
(0.023)
(0.038)
(0.027)
(0.030)
(0.047)
(0.039)
Numbers in parenthesis are the standard errors
The colored alternatives are located at: Green -- 1, Orange -- 3, Blue -- 9
Table 7: Ballots Cast by Participants in Position 10, percent
PV
AV
Ballot
All
Early
Late
All
Early
Green only
1.67*
2.08*
1.39*
3.02*
1.04*
Orange only
14.38**
16.15**
13.19**
4.31*
2.60*
Blue only
82.50
79.69
84.38
73.92
77.08
Green and Orange
---1.08*
1.04*
Green and Blue
---4.31*
4.17*
Orange and Blue
---10.78
13.54
All Three
---2.37*
0.52*
Abstain
1.46*
2.08*
1.04*
0.22*
0.00*
Total
100.0
100.0
100.0
100.0
100.0
The colored alternatives are located at: Green -- 1, Orange -- 3, Blue -- 9
* The ballot is weakly dominated.
** The ballot is strategic.
Late
4.41*
5.51*
71.69
1.10*
4.41*
8.82
3.68*
0.37*
100.0
All
---6.47*
43.97**
41.38
-8.19*
100.0
DV
Early
---6.25*
39.06**
44.79
-9.90*
100.0
Late
---6.62*
47.43**
38.97
-6.99*
100.0
Table 8: Weakly Dominated Votes, percent
Rule & Session
PV
PV1
PV2
PV3
PV4
All
2.8
1.8
3.6
2.1
3.9
Votes
Early
3.5
2.1
4.9
4.2
2.8
Late
2.4
1.7
2.9
0.5
4.7
All
15.4
9.4
21.9
8.9
21.4
Elections*
Early
17.7
12.5
29.2
16.7
12.5
Late
13.9
7.5
17.5
3.1
28.1
AV
AV1
AV2
AV3
AV4
36.2
42.7
34.8
28.9
37.5
36.5
50.7
31.9
31.9
31.3
36.0
37.9
37.0
26.6
42.2
93.5
98.4
92.9
82.1
100.0
96.9
100.0
95.8
91.7
100.0
91.2
97.5
90.6
75.0
100.0
12.5
15.1
10.7
54.7
61.5
50.0
DV
DV1
8.9
18.8
2.9
37.5
70.8
17.5
DV2
16.4
16.7
16.1
67.9
70.8
65.6
DV3
8.9
6.3
10.9
48.2
37.5
56.3
DV4
16.4
18.8
14.6
67.9
66.7
68.8
* These columns measure the percent of elections where at least one participant
cast a weakly dominated vote.
Table 9: Actual and Predicted Frequencies Under Quantal Response Equilibrium, by Position and Voting Rule
Position
PV
AV
DV
QRE λ= 0.128145
LnLikelihood = -1326
QRE λ= 0.129069
LnLikelihood = -1986
QRE λ= 0.249838
LnLikelihood = -1204
2
Sincere
Weakly Dominated
Actual
0.956
0.044
QRE
0.925**
0.075**
Actual
0.591
0.409
QRE
0.574
0.426
Actual
0.886
0.114
QRE
0.951**
0.049**
6
Sincere
Weakly Dominated
0.990
0.010
0.561**
0.439**
0.476
0.524
0.177**
0.823**
0.886
0.114
0.515**
0.485**
10
Sincere
0.825
0.939**
0.847
0.786**
0.414
0.335**
Strategic
0.144
0.007**
--0.440
0.340**
Weakly Dominated
0.031
0.053*
0.153
0.214**
0.147
0.325**
* Indicates the difference between QRE estimation and observed frequencies are significant at the 5% level.
** Indicates the difference between QRE estimation and observed frequencies are significant at the 1% level.
Table 10: Preferred Strategies Under Level-k Reasoning
Position
PV
AV
DV
Level 1
Level 2+
Level 1
Level 2+
Level 1
Level 2+
2
G or O
O
G and O
G and O
G and O
G and O
6
O or B
B
O and B
O and B
O and B
O and B
10
B
B
B
B
O and B
G and B
The colored alternatives are located at: (G)reen -- 1, (O)range -- 3, (B)lue -- 9
Table 11: Nash and Weak Undominance Nash Equilibria, by Voting Rule and Session
PV
PV1
PV2
PV3
PV4
AV
AV1
AV2
AV3
AV4
DV
DV1
DV2
DV3
DV4
% NE Outcomes
late
all
early
0.996
1.000
0.993
% WUNE Outcomes
all
early
late
0.996
1.000
0.993
(0.004)
(0.000)
(0.007)
(0.004)
(0.000)
(0.069)
1.000
1.000
1.000
1.000
1.000
1.000
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
1.000
1.000
1.000
1.000
1.000
1.000
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
0.982
1.000
0.969
0.982
1.000
0.969
(0.018)
(0.000)
(0.033)
(0.018)
(0.000)
(0.033)
1.000
1.000
1.000
1.000
1.000
1.000
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
0.957
0.948
0.963
0.681
0.583
0.750
(0.013)
(0.023)
(0.016)
(0.031)
(0.050)
(0.037)
1.000
1.000
1.000
0.781
0.583
0.900
(0.000)
(0.000)
(0.000)
(0.052)
(0.101)
(0.048)
0.946
0.875
1.000
0.714
0.708
0.719
(0.030)
(0.068)
(0.000)
(0.060)
(0.093)
(0.085)
0.929
0.917
0.938
0.679
0.542
0.781
(0.034)
(0.056)
(0.046)
(0.063)
(0.102)
(0.078)
0.946
1.000
0.906
0.536
0.500
0.563
(0.030)
(0.000)
(0.055)
(0.067)
(0.102)
(0.094)
0.905
0.823
0.963
0.776
0.677
0.846
(0.019)
(0.045)
(0.016)
(0.027)
(0.048)
(0.031)
0.891
0.708
1.000
0.734
0.542
0.850
(0.039)
(0.093)
(0.000)
(0.055)
(0.102)
(0.057)
0.875
0.792
0.938
0.732
0.625
0.813
(0.044)
(0.083)
(0.043)
(0.059)
(0.099)
(0.074)
1.000
1.000
1.000
0.929
0.958
0.906
(0.000)
(0.000)
(0.000)
(0.034)
(0.041)
(0.055)
0.857
0.792
0.906
0.714
0.583
0.813
(0.047)
(0.083)
(0.052)
(0.060)
(0.101)
(0.074)
Numbers in parenthesis are the standard errors
Table 12: Share of Three Way Ties in Election Outcomes, by Session
PV
Three-way ties, share
all
early
late
0.004
0.000
0.007
(0.004)
PV1
PV2
PV3
PV4
AV
AV1
AV2
AV3
AV4
DV
DV1
DV2
DV3
DV4
(0.000)
(0.007)
0.000
0.000
0.000
(0.000)
(0.000)
(0.000)
0.000
0.000
0.000
(0.000)
(0.000)
(0.000)
0.018
0.000
0.031
(0.018)
(0.000)
(0.033)
0.000
0.000
0.000
(0.000)
(0.000)
(0.000)
0.030
0.031
0.029
(0.011)
(0.018)
(0.014)
0.000
0.000
0.000
(0.000)
(0.000)
(0.000)
0.036
0.083
0.000
(0.025)
(0.057)
(0.000)
0.054
0.042
0.063
(0.030)
(0.041)
(0.046)
0.036
0.000
0.063
(0.025)
(0.000)
(0.046)
0.138
0.052
0.199
(0.023)
(0.023)
(0.034)
0.203
0.083
0.275
(0.050)
(0.057)
(0.071)
0.143
0.083
0.188
(0.047)
(0.057)
(0.074)
0.125
0.000
0.219
(0.044)
(0.000)
(0.078)
0.071
0.042
0.094
(0.034)
(0.041)
(0.055)
Numbers in parenthesis are the standard errors
Table 13: Frequencies of Each Alternative Appearing in the Winning Set, by Session
mean
(stdv)
PV
all
0.004
Green
early
0.000
late
0.007
all
0.854
Orange
early
0.854
late
0.854
all
0.392
Blue
early
0.438
late
0.361
(0.004)
(0.000)
(0.007)
(0.023)
(0.036)
(0.029)
(0.032)
(0.051)
(0.040)
0.000
0.000
0.000
0.969
1.000
0.950
0.266
0.250
0.275
(0.000)
(0.000)
(0.000)
(0.022)
(0.000)
(0.035)
(0.055)
(0.089)
(0.071)
PV2
0.000
0.000
0.000
0.719
0.625
0.775
0.469
0.583
0.400
(0.000)
(0.000)
(0.000)
(0.056)
(0.099)
(0.066)
(0.062)
(0.101)
(0.078)
PV3
0.018
0.000
0.031
0.839
0.875
0.813
0.482
0.417
0.531
(0.018)
(0.000)
(0.031)
(0.049)
(0.068)
(0.069)
(0.067)
(0.101)
(0.088)
PV4
0.000
0.000
0.000
0.893
0.917
0.875
0.357
0.500
0.250
(0.000)
(0.000)
(0.000)
(0.041)
(0.057)
(0.059)
(0.064)
(0.102)
(0.077)
0.043
0.052
0.037
0.720
0.625
0.787
0.509
0.594
0.449
(0.013)
(0.023)
(0.016)
(0.029)
(0.050)
(0.035)
(0.033)
(0.051)
(0.042)
0.000
0.000
0.000
0.781
0.583
0.900
0.359
0.458
0.300
(0.000)
(0.000)
(0.000)
(0.052)
(0.101)
(0.048)
(0.060)
(0.102)
(0.073)
AV2
0.054
0.125
0.000
0.768
0.833
0.719
0.554
0.500
0.594
(0.030)
(0.068)
(0.000)
(0.057)
(0.076)
(0.080)
(0.067)
(0.102)
(0.087)
AV3
0.071
0.083
0.063
0.732
0.583
0.844
0.500
0.625
0.406
(0.034)
(0.057)
(0.043)
(0.059)
(0.101)
(0.064)
(0.067)
(0.099)
(0.087)
AV4
(0.054)
(0.000)
(0.094)
(0.589)
(0.500)
(0.656)
(0.643)
(0.792)
(0.531)
(0.030)
(0.000)
(0.052)
(0.066)
(0.102)
(0.084)
(0.064)
(0.083)
(0.088)
0.207
0.146
0.250
0.862
0.833
0.882
0.302
0.302
0.301
(0.027)
(0.036)
(0.037)
(0.023)
(0.038)
(0.027)
(0.030)
(0.047)
(0.039)
0.234
0.125
0.300
0.828
0.792
0.850
0.453
0.542
0.400
(0.053)
(0.068)
(0.073)
(0.047)
(0.083)
(0.057)
(0.062)
(0.102)
(0.078)
DV2
0.250
0.250
0.250
0.857
0.833
0.875
0.304
0.292
0.313
(0.058)
(0.089)
(0.077)
(0.047)
(0.076)
(0.059)
(0.062)
(0.093)
(0.082)
DV3
0.161
0.000
0.281
0.929
0.958
0.906
0.161
0.042
0.250
(0.049)
(0.000)
(0.080)
(0.034)
(0.041)
(0.052)
(0.049)
(0.041)
(0.077)
DV4
0.179
0.208
0.156
0.839
0.750
0.906
0.268
0.333
0.219
(0.051)
(0.083)
(0.064)
(0.049)
(0.089)
(0.052)
(0.059)
(0.097)
(0.073)
PV1
AV
AV1
DV
DV1
Numbers in parenthesis are the standard errors
The colored alternatives are located at: Green -- 1, Orange -- 3, Blue -- 9
Table 14: Ballots Cast by Participants in Position 10,by Session, percent
Ballot
Green only
Orange only
Blue only
Green and Orange
Green and Blue
Orange and Blue
All Three
Abstain
Total
All
2.34*
14.06**
81.25
----2.34*
100.0
PV1
Early
2.08*
18.75**
75.00
----4.17*
100.0
Late
2.50*
11.25**
85.00
----1.25*
100.0
All
5.47*
10.94*
64.84
2.34*
2.34*
14.06
0.00*
0.00*
100.0
AV1
Early
2.08*
6.25*
62.50
4.17*
0.00*
25.00
0.00*
0.00*
100.0
Late
7.50*
13.75*
66.25
1.25*
3.75*
7.50
0.00*
0.00*
100.0
All
---3.13*
60.16**
31.25
-5.47*
100.0
DV1
Early
---6.25*
50.00**
29.17
-14.58*
100.0
Late
---1.25*
66.25**
32.50
-0.00*
100.0
Ballot
Green only
Orange only
Blue only
Green and Orange
Green and Blue
Orange and Blue
All three
Abstain
Total
All
0.00*
22.66**
76.56
----0.78*
100.0
PV2
Early
0.00*
16.67**
81.25
----2.08*
100.0
Late
0.00*
26.25**
73.75
----0.00*
100.0
All
0.89*
0.89*
86.61
0.00*
8.93*
2.68
0.00*
0.00*
100.0
AV2
Early
0.00*
2.08*
83.33
0.00*
12.50*
2.08
0.00*
0.00*
100.0
Late
1.56*
0.00*
89.06
0.00*
6.25*
3.13
0.00*
0.00*
100.0
All
---14.29*
41.96**
33.93
-9.82*
100.0
DV2
Early
---8.33*
41.67**
39.58
-10.42*
100.0
Late
---18.75*
42.19**
29.69
-9.38*
100.0
Ballot
Green only
Orange only
Blue only
Green and Orange
Green and Blue
Orange and Blue
All Three
Abstain
Total
All
1.79*
8.93**
89.29
----0.00*
100.0
PV3
Early
4.17*
18.75**
77.08
----0.00*
100.0
Late
0.00*
1.56**
98.44
----0.00*
100.0
All
0.89*
4.46*
75.00
0.00*
0.00*
17.86
0.89*
0.89*
100.0
AV3
Early
0.00*
2.08*
79.17
0.00*
0.00*
18.75
0.00*
0.00*
100.0
Late
1.56*
6.25*
71.88
0.00*
0.00*
17.19
1.56*
1.56*
100.0
All
---5.36*
38.39**
45.54
-10.71*
100.0
DV3
Early
---8.33*
27.08**
58.33
-6.25*
100.0
Late
---3.13*
46.88**
35.94
-14.06*
100.0
Late
6.25*
0.00*
60.94
3.13*
7.81*
7.81
14.06*
0.00*
100.0
All
---3.57*
33.04**
56.25
-7.14*
100.0
DV4
Early
---2.08*
37.50**
52.08
-8.33*
100.0
Late
---4.69*
29.69**
59.38
-6.25*
100.0
PV4
AV4
Ballot
All
Early
Late
All
Early
Green only
2.68*
2.08*
3.13*
4.46*
2.08*
Orange only
10.71**
10.42**
10.94**
0.00*
0.00*
Blue only
83.93
85.42
82.81
70.54
83.33
Green and Orange
---1.79*
0.00*
Green and Blue
---6.25*
4.17*
Orange and Blue
---8.04
8.33
All Three
---8.93*
2.08*
Abstain
2.68*
2.08*
3.13*
0.00*
0.00*
Total
100.0
100.0
100.0
100.0
100.0
The colored alternatives are located at: Green -- 1, Orange -- 3, Blue -- 9
* Indicates the ballot is weakly dominated.
** Indicates the ballot is strategic.
Figure 1: Experimental Design: Number Line Representation
green
1
orange
2
X
X
3
blue
4
5
6
X
X
7
8
9
10
X
X
Figure 2: Distribution of Strategic Votes among Voters in Position 10 Under Dual Voting
Number of Participants
14
12
10
Early
8
Late
6
4
2
0
0.0
0.1-20.0
20.1-40.0
40.1-60.0
Percent Strategic Votes
60.1-80.0
80.1-100.0
Figure 3: Distribution of Weakly Dominated Votes among Voters, by Voting Rule
Plurality Voting
Number of Participants
40
35
30
25
20
15
10
5
0
Early
Late
0.0
0.1-20.0
20.1-40.0
40.1-60.0
60.1-80.0
80.1-100.0
Percent Weakly Dominated Votes
Approval Voting
Number of
Participants
40
35
30
25
20
15
10
5
0
Early
Late
0.0
0.1-20.0
20.1-40.0
40.1-60.0
60.1-80.0
80.1-100.0
Percent Weakly Dominated Votes
Dual Voting
Number of Participants
40
35
30
25
Early
Late
20
15
10
5
0
0.0
0.1-20.0
20.1-40.0
40.1-60.0
60.1-80.0
Percent Weakly Dominated Votes
80.1-100.0
Figure 5: Plurality Voting: QRE Predicted Strategies' Frequencies vs. Observed Frequencies
PV- Position 2
1.2
Green
Probability
1
G: Actual
Orange
0.8
O: Actual
0.6
Blue
0.4
B: Actual
Abstain
0.2
Abs: Actual
0
0.0001
0.001
0.01
0.1
1
10
Lambda
PV- Position 6
0.8
Green
0.7
G: Actual
Probability
0.6
Orange
0.5
O: Actual
0.4
Blue
0.3
B: Actual
0.2
Abstain
0.1
0
0.0001
Abs: Actual
0.001
0.01
0.1
1
10
Lambda
PV- Position 10
1.2
Green
Probability
1
G: Actual
Orange
0.8
O: Actual
0.6
Blue
0.4
B: Actual
Abstain
0.2
Abs: Actual
0
0.0001
0.001
0.01
0.1
Lambda
1
10
Figure 6: Approval Voting: QRE Predicted Strategies' Frequencies vs. Observed Frequencies
Green
AV - Position 2
G: Actual
Orange
0.9
O: Actual
0.8
Blue
Probability
0.7
B: Actual
0.6
0.5
Green & Orange
0.4
G&O: Actual
0.3
Green & Blue
0.2
G&B: Actual
0.1
Orange & Blue
0
0.0001
O&B: Actual
0.001
0.01
0.1
1
10
All Three
All3: Actual
Lambda
Abstain
Green
AV - Position 6
Probability
G: Actual
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.0001
Orange
O: Actual
Blue
B: Actual
Green & Orange
G&O: Actual
Green & Blue
G&B: Actual
Orange & Blue
O&B: Actual
0.001
0.01
0.1
1
10
All Three
All3: Actual
Lambda
Abstain
Abs: Actual
Green
AV - Position 10
G: Actual
Orange
1.2
O: Actual
1
Blue
Probability
B: Actual
0.8
Green & Orange
G&O: Actual
0.6
Green & Blue
0.4
G&B: Actual
Orange & Blue
0.2
O&B: Actual
0
0.0001
All Three
0.001
0.01
0.1
Lambda
1
10
All3: Actual
Abstain
Abs: Actual
Figure 7: Dual Voting: QRE Predicted Strategies' Frequencies vs. Observed Frequencies
DV - Position 2
1.2
Green & Orange
Probability
1
G&O: Actual
Green & Blue
0.8
G&B: Actual
0.6
Orange & Blue
0.4
O&B: Actual
Abstain
0.2
Abs: Actual
0
0.0001
0.001
0.01
0.1
1
10
Lambda
DV - Position 6
1.2
Green & Orange
Probability
1
G&O: Actual
Green & Blue
0.8
G&B: Actual
0.6
Orange & Blue
0.4
O&B: Actual
Abstain
0.2
Abs: Actual
0
0.0001
0.001
0.01
0.1
1
10
Lambda
DV - Position 10
1.2
Green & Orange
Probability
1
G&O: Actual
Green & Blue
0.8
G&B: Actual
0.6
Orange & Blue
0.4
O&B: Actual
Abstain
0.2
Abs: Actual
0
0.0001
0.001
0.01
0.1
Lambda
1
10