Experimental Evidence on Condorcet-Eciency and Strategic Voting: Plurality vs Approval Voting

Experimental Evidence on Condorcet-Eciency and Strategic Voting: Plurality vs Approval Voting Ðura-Georg Grani Abstract We report on the results of series of experimental 4-alternativeelections. Preference are induced by modeling the participants' expected payo conditional on the election outcomes. The induced preference prole is such that a unique Condorcet-Winner and a unique Condorcet-Loser exists. We asses the Condorcet-Eciency of Approval Voting in comparison with Plurality Voting. In addition, we vary the underlying informational structure independently from a completeinformation framework where participants are fully informed about the preference structure in the whole electorate to an incompleteinformation framework where participants only know their own payos and the election histories. For the induced preference prole, Approval Voting tends to select the Condorcet-Winner more often, respectively the Condorcet-Loser less often. Further, Approval Voting is less responsive to variations in the information structure. Under Plurality Voting with incomplete-information, the Condorcet-Winner almost never wins. Repeated interaction of the participants over several rounds allows us to study the participants' adaptive behavior, i.e. learning over time and strategic responses to previous elections. Department of Economics, University of Konstanz, Box 150, D-78457 Konstanz (Germany) 1

1 Introduction The majority/plurality voting system which underlies most voting methods in modern democratic societies frequently gives rise to questionable results. The reason for the problems with this voting system is that voters can only vote for their most preferred candidate (and cannot additionally express preferences for other candidates). Some countries have sought to avoid these problems by introducing other voting systems, including the Single Transferable Vote method (or 'Instant Runo`) employed in Australia, run-o ballots in France, and multiple votes in Germany. All these methods give voters additional opportunities to express their wishes. Other methods have been proposed requiring voters to provide a complete ranking of candidates, the main example being the Borda Count (see e.g. Saari, 1994). The object of study in this article is Approval Voting, a method rst analytically described by Brams and Fishburn (1978). Contrary to other methods which aim for voters to construct and reveal full rankings of candidates, this method merely requires that voters reveal which candidates are acceptable, i.e. each voter needs only mark the names of the candidates he approves of. The total number of approvals determines the ultimate winner. Arguments have been put forward in the literature that the Approval Voting method provides an accurate reection of voters' wishes and is not vulnerable to voter manipulation (see Brams and Fishburn, 1978; Fishburn, 1978a,b; Brams and Fishburn, 2005; Wolitzky, 2009). The main theoretical critiques against this voting method in the literature are (i) the multiplicity of potential outcomes for the same set of preferences (see Saari and Van Newenhizen, 1988a,b) and (ii) the possibility of selecting a Condorcet-Loser, or not selecting the Condorcet-Winner of an election respectively (see Brams and Sanver, 2005); notice that a Condorcet-Loser (Winner) is dened as an alternative that is beaten by (that can beat) any other alternative in pairwise comparison. In order to fully understand the implications of the Approval Voting method for society, it is essential to collect empirical data on its performance and on actual voter behavior when the method is adopted. Field experiments have been a rst step in this direction, providing us with invaluable data and with clear evidence on the actual feasibility of the method, see Alós-Ferrer and Grani (2011), Baujard and Igersheim (2009) and Laslier and Van der Straeten (2008). There are, however, a number of limitations associated with eld experiments. First, due to legal concerns (ensuring voter anonymity in actual elections) as well as methodological considerations, one is not able to fully identify the participants' actual preferences. Second, self-selection biases may occur, which cannot fully be accounted for. Hence, without know- 2

ing neither the preferences of the participants nor whether the samples are truly unbiased, certain properties of Approval Voting cannot be tested. As a complement to eld experimentation, the controlled laboratory environment typical of experimental economics allows us to induce preferences over candidates and hence to study theoretical properties of Approval Voting that cannot be tested in the eld. In the laboratory, preferences can easily be induced by modeling the participants' expected payo conditional on the election outcomes. This article reports the results of a series experimental elections conducted in the computer laboratory at the University of Konstanz, Germany. We intend to assess the Condorcet-Eciency of Approval Voting in comparison to Plurality Voting, empirically. We induce a set of preferences by modeling subjects' payments conditional on the election outcome such that there always exist a unique Condorcet winner and Condorcet loser. Additionally, we vary the information structure (complete-information framework where participants are fully informed about the preference structure in the whole electorate vs incomplete-information framework where participants only know their own payos and the election histories), and hence analyze its impact on the Condorcet-Eciency of the two voting methods. Closely related to our work are series of experiments on voting rules including Approval Voting. Forsythe et al. (1993) and Forsythe et al. (1996) analyze various coordination devices, such as polling or histories of repeated interaction, and their impact on election outcomes. Our experimental design is partially based on those experiments. Dellis et al. (2010) study various voting methods in a framework of single-peaked preferences. They show that voting procedures other than Plurality Voting, such as Approval Voting, can favor a two-party system as well. 1 Along a similar line of research, Van der Straeten et al. (2010) provide evidence that rational choice theory provides very good predictions of actual individual behavior in one-round and approval voting elections. The remainder of the article is structured as follows. The next section (2) introduces the experimental design in detail. Section 3 presents the results of the experiment. Admittedly, a large part of the analysis is still work in progress. As a consequence, Section 3 will only present mostly descriptive results of aggregated behavior. We plan to extend this section with a more detailed analysis on the election outcomes and a detailed analysis of the individual choice data. Since we actually induce the preferences of experimental subjects through explicit monetary incentives, it is possible to identify both 1 Duverger's law roughly states that Plurality Voting tends to favor a two-party system (Duverger, 1954). 3

insincere voting and other forms of strategic behavior, as e.g. selection of a best reply among the set of sincere voting ballots. One feature of the design is the interaction of the participants over several rounds, explicitly enabling them to acquire familiarity with the method and allowing for learning eects. This allows to study the participants' adaptive behavior, i.e. learning over time or strategic responses to previous elections. 2 Design of the experiment 2.1 The Voting Game The baseline game is a standard voting game with four available alternatives (neutrally labelled: A, B, C and D) and three dierent types of voters. Seven subjects form one group in which voting takes place. Each group itself consists of two voters of type I with preferences A D C B, three voters of type II with preferences B D C A and two voters of type III with preferences C D A B, where denotes the usual strict preference relation (i.e. strictly preferred to). For each election, subjects anonymously and independently submit their voting ballots and the winner of the election is determined according to a pre-specied voting method. Ties among two or more alternatives are broken randomly (e.g. by the roll of a die). Abstention, including the cast of empty ballots, is not allowed. Preferences were induced with means of monetary incentives by conditioning subjects' remuneration on the outcome of the election. The underlying payo schedule (in terms Experimental Currency Units, ECU) as well as the corresponding induced strict preferences over the set of alternatives are summarized in Table 1. Payos in ECU Number of Subjects A B C D Induced Preferences 2 100 40 60 80 A D C B 3 40 100 60 80 B D C A 2 60 40 100 80 C D A B Table 1: Payo schedule and induced preference prole. A weak majority of voters strictly prefers alternative B to all other alternatives. However, alternative B is a (strict) Condorcet-Loser, it is beaten by 4

every other alternative in a pairwise comparison. Further, it constitutes the worst possible outcome for an absolute majority of 4 voters. If everybody votes sincerely, alternative B will win plurality voting based elections. In order to avoid the worst outcome, the majority of voters has to coordinate their votes. Although never at the top of any individual's preferences, a natural candidate for coordination should be alternative D. It beats every other alternative in a pairwise comparison. That is, alternative D is a (strict) Condorcet-Winner. 2.2 Procedures The paradigm used as the basic experimental design is based on Forsythe et al. (1993) and Forsythe et al. (1996). The experiment follows a 2 between (informational structure: complete vs incomplete) 2 between (voting Method: Approval Voting vs Plurality Voting) design as described in Table 2. In the complete information treatment, all participants are informed about the payo schedule of the group, their own payos and the complete history of past elections. In the incomplete-information treatment the participants know their own payo and past election outcomes only. Under Approval Voting, subjects can approve of as many alternatives as wished. The total number of approvals determines the ultimate winner. Under Plurality Voting, subjects give their vote to one alternative only (One man, One vote), and the winner is determined by absolute number of votes received. We vary the treatment variables independently allowing us to asses the Condorcet-Eciency (i.e. the fraction of won elections by the Condorcet- Winner alternative) of Approval Voting in comparison with Plurality Voting and to investigate the impact of the dierent informational structures on the Condorcet-Eciency of the two voting methods. Information Structure Voting Method Full Information (FI) Incomplete Information (II) Approval Voting (AV) Treatment 1 (AV, FI) Treatment 2 (AV, II) Plurality Voting (PV) Treatment 3 (PV, FI) Treatment 4 (PV, II) Table 2: Experimental design. We ran a total of eight sessions (two per treatment) with 28 subjects participating in each session for a total of 224 subjects. In each session, subjects were randomly allocated to four dierent groups of equal size and 5

a random draw determined the voter type for each subject. During each round of the experiment, one election was held within each group. Thus, during each round, there were four distinct groups of 7 voters each and four distinct elections were held. At the end of each election the subjects were informed on the outcome of the election and the money they earned in this election round. The group composition and the induced preference proles were held xed for a series of eight election rounds. After eight elections rounds, we rematched the subjects into new groups and a dierent set of preferences was induced. Hereby we reshued the labels of the alternatives and extracted a new random draw on the types of voters within the dierent groups. Additionally, we introduced small perturbations on the payo scheme before the start of the experiment and after regrouping subjects, thus eliminating possible confound factors (equity and eciency concerns, focal points, etc). 2 This regrouping procedure preserved the ordinal structure of the preference prole, so that the overall design was unchanged, but every voter faced a qualitatively new situation. At the same any repeated game eect that could occur across groups was minimized. We repeated this procedure twice, so that in each session 4 8 3 = 96 elections were held. Upon arrival at the computer laboratory, subjects were randomly allocated to the 28 workstations. Printed version of the experimental instructions were then distributed and subjects had 10 minutes of time to read them. The instructions carefully explained the procedures of the experiment. More precisely, all subjects knew that they were going to participate in a total 24 elections and that the group composition and induced preference will be held xed for 8 election rounds. An extended example explained the conditional payment structure. It was explicitly mentioned that the payo of any election was ultimately determined by the winner alternative, independent of whether they actually voted for that particular alternative or not. In addition, the experimental instructions included a screenshot of the decision screen explaining each single element step by step. 3 A large box at the centre of the screen represented the voting ballot. Alternatives were vertically aligned and subject could mark the corresponding alternatives they wanted to vote for. However, the votes cast had to be conrmed by clicking a button. A second, smaller box contained information of past election outcomes. The left side of the screen was occupied by a box with the subject's payo 2 For each entry in the payo-matrix, the roll of a ten-sided die determined the direction and the amount of the payo perturbation. For outcomes from 6 to 10, we added 1 to 5 ECUs to and for outcomes 5 to 1, we subtracted 1 to 5 ECUs from the corresponding payos in Table 1. 3 The numbers presented for the payo information matched the ones from the extended example. 6

information and a second box containing the group's payo schedule for the full-information treatments. The experiment was conducted in the University of Konstanz' own computer laboratory (Lakelab). Subjects were recruited using an online recruitment system for economic experiments (Greiner, 2004) excluding students from related elds with basic knowledge in Game Theory and Social Choice. 100 ECUs were worth 60 Eurocents. Sessions lasted approximately 1h. No show-up fee was paid. The experiment was run using the computer software z-tree (Fischbacher, 2007). 2.3 Nash Equilibria Given the induced preference prole (see Table 1), all alternatives can be sustained as Nash-Equilibrium outcomes of the voting game. Consider the simple-most case where all voters vote for the same alternative. There exists no unilateral protable deviation. That is, neither player has a strict incentive to deviate from such a prole. 4 However, such proles include the use of (weakly) dominated strategies by some players. A common assumption in the literature is to restrict the set of strategies for each players to undominated strategies narrowing down the set of possible outcomes. Hereby, we follow Brams and Fishburn (1978) and qualify undominated strategies as admissible. Admissible strategies for Plurality Voting include all voting ballots except casting a vote for the least preferred alternative. The set of admissible strategies for Approval Voting is characterized by all possible ballots in which the most preferred alternative is approved of and the least preferred alternative is not approved of. 5 Under admissibility, alternatives A, C and D are still Nash Equilibrium outcomes for both voting methods. This is no longer true for alternative B, the Condorcet-Loser. Suppose B wins the election. B's votes received in such cases can only come from the three type III voter who vote for/approve of B. Admissibility guarantees the existence of another alternative with at least two votes/approvals. The Condorcet-Loser is never able to win the election with a margin of two or more. As a consequence, there always exists one voter among the four remaining ones who can alter the outcome of the election to his favor, either by giving rise to a new winner or by letting B tie with another alternative he strictly prefers (remember, for this type of voter B is the least preferred option). 4 Every voting situations in which one alternative is winning the election with a margin of more than two votes constitute a Nash Equilibrium. 5 E.g. with 4 alternatives approving of the most and second most preferred alternative or approving of the most and third most preferred alternative are both admissible strategies. 7

3 Results For the presentation of the experiment's results in this section, the labels of the alternatives were recoded to match the original preference prole as presented in Table 1. Alternative B, thus, represents the Condorcet-Loser and and alternative D the Condorcet-Winner. 3.1 Condorcet-Eciency Figure 1 shows the aggregated election outcomes across the four treatments. With full information, the Condorcet-Winner (alternative D) won 75.52% of the elections under Approval Voting and respectively 46.88% of the elections under Plurality Voting. 6 Similarly, the Condorcet-Loser won 15.63% percent of the elections under Approval Voting and 35.94% of the elections under Plurality Voting. These gures indicate that given the induced preference prole, coordination on the Condorcet-Winner is much harder to establish under a single-vote method than under a multiple-vote method. The limited amount of information that is transmitted through a Plurality Voting ballot hinders coordination. Failed coordination by voters with B as their least preferred alternative is represented by the higher fraction of won elections of the Condorcet-Loser. For both methods, the change from full-information to incomplete-information leads to a decrease in the Condorcet-Eciency and to an increase in the Condorcet-Ineciency. Never being at the top of an individual's preference, the Condorcet-Winner disappears in treatment (PV,II). The respective fractions of won elections drop from the above mentioned 46.75% to marginal 9.34%. At the same time, the introduced manipulation on the information structure has little to almost no eect for Approval Voting. Figure 2 breaks down the election outcomes into the respective series of elections for the Condorcet-Winner and the Condorcet-Loser. As a reminder, participants were randomly rematched into new groups after eight election rounds and received new payos (i.e. reshue and relabel of the original payo matrix from Table 1 with small payo perturbations). T01 represents the rst series of eight elections, T02 and T03 the second, respectively the third ones. In deed, as Figure 2 clearly shows, the procedures used for regrouping the participants (payo perturbations, reshuing and relabeling) generated qualitatively new situations for the subjects while preserving the ordinal structure of the payment schedule. A χ 2 goodness-of-t-test detects 6 In case of ties among k alternatives, each alternative in the winner-set was treated as having won the k th fraction of an election. For example, if alternatives A and B tied for the rst place, both A and B won half of an election. 8

0.8 Fraction of won elections 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 A B C D (AV,FI) (PV,FI) (AV,II) (PV,II) Alternatives Figure 1: Fraction of won elections for the four alternatives across treatments. Fraction of won elections 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T01 T02 T03 (AV,FI) (PV,FI) (AV,II) (PV,II) Fraction of won elections 0.5 0.4 0.3 0.2 0.1 0 T01 T02 T03 (AV,FI) (PV,FI) (AV,II) (PV,II) Series of elections Series of elections (a) Condorcet Eciency (b) Condorcet Ineciency Figure 2: Fraction of won elections Condorcet-Winner and Condorcet-Loser across series of elections. statistically signicant dierences between the series of elections only for the Plurality Voting, Full Information treatment (p-value=0.0286) as illustrated by the peaked orange curvature in Figure 2. The remaining treatments produce quantitatively similar outcomes across series (with signicant dierences between treatments). Table 3 below shows the fraction of won elections for each alternative for the dierent treatments. In addition, the fractions have been restricted to election outcomes of the last four election rounds and for the very last election round within a series. The rst row for each treatment block presents the fractions underlying Figure 1. Most importantly, the second and last column in Table 3 reveal an increasing pattern of Condorcet-Eciency (fraction of won elections by D) and a decreasing pattern of Condorcet-Ineciency (fraction of won elections by B). We observe pronounced learning eects within series of elections. Eventually, participants learn to play the more desirable 9

Alternative A B C D AVFI 2.08% 15.63% 6.77% 75.52% AVFI last 4 periods 3.13% 12.50% 3.13% 81.25% AVFI last 4.17% 0.00% 0.00% 95.83% AVII 7.81% 23.96% 6.77% 61.46% AVII last 4 periods 7.29% 23.96% 6.25% 62.50% AVII last 8.33% 20.83% 4.17% 66.67% PVFI 6.77% 35.94% 10.42% 46.88% PVFI last 4 periods 11.46% 23.96% 10.42% 54.17% PVFI last 8.33% 16.67% 16.67% 58.33% PVII 19.79% 44.27% 26.56% 9.38% PVII last 4 periods 23.96% 28.13% 30.21% 17.71% PVII last 20.83% 25.00% 25.00% 29.17% Table 3: Fraction of won election for each treatment for all elections, the last 4 elections within each series and the last election within each series. outcome and avoid the less desirable outcome (in the sense that a strong majority within each voting group strictly prefers D to B). In order to assess the observations from above statistically while controlling for group dierences, random-eects probit regressions were conducted on Condorcet-Eciency and Condorcet-Ineciency. The results of these analyses are reported in Table 4 and Table 5. The author is fully aware of the fact that the dierent observations across series are not statistically independent. First, the results below can be reproduced by means of non-parametric tests (e.g. Wilcoxon rank-sum test) or by using only the truly independent outcomes for group decisions from the rst eight series of elections. Second, given the observations from above we conclude that our rematching procedure enables us to treat observations across series as quasi-independent. Hence, each of the regression result represents data from a strongly balanced panel of 96 independent groups with eight observations per group. A switch from Plurality Voting to Approval signicantly increases the probability that the Condorcet-Winner is elected and reduces the winning 10

dy/dx Std. Err. z P > z [95% Conf. Interval] condt1.0691.0505 1.37 0.171 -.0299.1682 av.2385.0582 4.09 0.000.1243.3526 noinf -.3876.0934-4.15 0.000 -.5706 -.2046 t.0157.0273 0.58 0.565 -.0378.0693 period.0263.0097 2.70 0.007.0072.0454 av*noinf.3018.0953 3.17 0.002.1151.4885 ct1*noinf -.0382.0715-0.53 0.593 -.1784.1020 per*noinf -.0036.0143-0.25 0.802 -.0316.0244 Table 4: Regression results for panel probit estimates with random eects at group level. Dep. variable: Conde, 1 if Condorcet Winner won the election, 0 otherwise. Reported Eects: Average marginal eects (predicted probability Pr(Conde=1)). probability of the Condorcet-Loser (av is a dummy, = 1 if the voting method is Approval Voting). While incomplete information (noinf is dummy, = 1 if information structure is incomplete) decreases the likelihood of the ecient outcome, it has no inuence on the probability that B wins under Plurality Voting. In line with the observations from above, alternative D completely disappears in the treatment (PV,II). Participants essentially coordinate on the two remaining outcomes A and C more often. The pronounced learning eects within series are represented by the signicant coecients of period, a variable denoting the current election round. 11

dy/dx Std. Err. z P > z [95% Conf. Interval] condt1 -.0348.0542-0.64 0.521 -.1410.0714 av -.2197.0616-3.57 0.000 -.3405 -.0989 noinf.0504.0838 0.60 0.547 -.1137.2146 t -.0071.0257-0.28 0.782 -.0575.0433 period -.0401.0108-3.70 0.000 -.0614 -.0188 av*noinf -.0065.0907-0.07 0.942 -.1843.1712 ct1*noinf.1104.0801 1.38 0.168 -.0465.2673 per*noinf.0020.0147 0.13 0.894 -.0268.0307 Table 5: Regression results for panel probit estimates with random eects at group level. Dep. variable: Condine, 1 if Condorcet Loser won the election, 0 otherwise. Reported Eects: Average marginal eects (predicted probability Pr(Condine=1)). References C. Alós-Ferrer and Ð. G. Grani. Two Field Experiments on Approval Voting in Germany. Social Choice and Welfare, forthcoming, 2011. A. Baujard and H. Igersheim. Expérimentation du vote par note et du vote par approbation le 22 avril 2007. Premiers résultats. Revue Economique, 60:189201, 2009. S. J. Brams and P. C. Fishburn. Approval Voting. The American Political Science Review, 72(3):831847, 1978. S. J. Brams and P. C. Fishburn. Going from Theory to Practice: The Mixed Success of Approval Voting. Social Choice and Welfare, 25(2):457474, 2005. S. J. Brams and M. R. Sanver. Critical Strategies Under Approval Voting: Who Gets Ruled In and Ruled Out. Electoral Studies, 25(2):287305, 2005. A. Dellis, S. Da'Evelyn, K. Sherstyuk, et al. Multiple Votes, Ballot Truncation and the Two-Party System: An Experiment. Social Choice and Welfare, pages 130, 2010. M. Duverger. Political Parties. North B and North R Methuen and Company, London, 1954. 12

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