The E ects of District Magnitude on Voting Behaviour

Transcription

1 The E ects of Dstrct Magntude on Votng Behavour Smon Hx London School of Economcs and Poltcal Scence Rafael Hortala-Vallve London School of Economcs and Poltcal Scence Gullem Rambau Yale - NUS College July 23, 2013 Abstract. Is there less strategc votng n mult-member dstrcts than n sngle-member dstrcts? Exstng research on ths queston s nconclusve, at least n part because t s d cult n observatonal data to solate the e ect of dstrct magntude on votng behavor ndependently from voters preferences or partes postons. Hence, we nvestgate ths ssue n a laboratory experment, where we vary dstrct magntude whle keepng voters preferences and partes postons constant. We fnd that votng for the preferred party (sncere votng) ncreases wth dstrct magntude and we are able to explan ths n terms of a mechancal e ect and a psychologcal one. We also fnd a hgh ncdence of votng for the frontrunner n all our electons, even when there are no ncentves for dong so. For helpful comments we thank Manuel Arellano, André Blas, Raymond Duch, Iván Fernández-Val, Alex Fournaes, Smon Hug, Becky Morton, Matt Shugart and audence members at the LSE/NYU conference n Poltcal Economy, APSA 2011, EPSA 2012, and MPSA

2 2 1. Introducton The desgn and reform of electoral systems s a salent polcy concern for new democraces as well as many advanced democraces. One key ssue n the desgn of electoral systems s the deal dstrct magntude: the number of canddates to be elected n each dstrct. In 2012, for example, Romana was consderng swtchng from large mult-member dstrcts to sngle-member dstrcts, Israel was consderng swtchng from a sngle natonal mult-member dstrct to smaller multmember dstrcts, whle Tunsa ntroduced small mult-member dstrcts for ts frst democratc electons. What are the consequences of dstrct magntude n terms of the behavor of voters, the e ectve number of poltcal partes, and the overall qualty of representaton and democracy? We stll have only partal answers to these questons, and the answers that we do have are manly based on observatonal data, where the underlyng theoretcal mechansms are d cult to dentfy clearly. In ths paper we analyze the behavor of voters n a controlled laboratory settng, n whch we change dstrct magntude but keep all other relevant poltcal varables constant. Cox (2007) suggests that as dstrct magntude ncreases, the proporton of voters that behave strategcally decreases, whle the proporton who votes sncerely for ther most preferred party ncreases. Indeed, Cox (2007, p. 100) clams that: strategc votng ought to fade out n multmember dstrcts when the dstrct magntude gets much above fve. Ths argument s smlar to a clam made by Sartor (1968, p. 279) much earler: The general rule s that the progresson from maxmal manpulatve mpact [va strategc votng] to sheer ne ectveness follows, more than anythng else, the sze of the consttuency. These ntutons mght explan the patterns observed n aggregate electon outcomes data, where voters n small mult-member dstrcts seem to strategcally coordnate around larger partes, whch leads to a low number of wasted votes (and a closer relatonshp between vote-shares and seat-shares of partes) as well as a low number of partes elected to parlaments and fewer partes n government. In contrast, n large mult-member dstrcts, where most voters smply vote sncerely for ther most preferred party, votng behavor fragments and government formaton then becomes more d cult. In other words, voters behave smlarly, n terms of ther ablty to strategcally coordnate around vable canddates, n small mult-member dstrcts and sngle-member dstrcts, but behave qute d erently n large mult-member dstrcts, overwhelmngly votng for ther most preferred partes (Carey and Hx, 2011).

3 We stll do not know what exactly mght be drvng these expected emprcal regulartes. Followng Duverger (1954), but wth a contemporary twst, we should ask ourselves whether the e ect of dstrct magntude on votng behavor s purely mechancal or psychologcal. The former s due to the fact that as dstrct magntude ncreases the proporton of voters whose strategc and sncere 3 motvatons concde ncreases as the number of vable partes/canddates ncreases. Te latter nstead would capture changes n voters strateges possbly due to the ncreased complexty of the electoral system. It s mpossble to solate such mcro-level e ects usng actual votng data, whch may explan why the presence of strategc (non-sncere) votng does not seem to vary wth dstrct magntude see Abramson, Aldrch, Blas, Damond, Dskn, Indrdason, Lee, and Levne (2010). In cross-country research, varatons n dstrct magntude covary wth a number of other factors whch nfluence how voters behave, such as the number of partes, socetal cleavage structures, nsttutonal e ects such as regme type, the level of poltcal and economc development of a country, and so on. In wthn-country research, dstrct magntude varatons also correlate wth other poltcal varatons. In Span, Brazl or Swtzerland, for example, where dstrct magntude vares between larger urban dstrcts and smaller rural dstrcts, electons are held at the same tme under the same poltcal nsttutons and poltcal contexts, but the number and type of canddates and partes competng n each dstrct vary consderably and may be endogenous to expected votng behavor. Smlarly, n presdental prmary electons n the Unted States, where there s varaton n whether delegates are rewarded n proporton to vote-shares or on a wnner-takes-all bass, the e ects of the electoral rule covary wth the tmng of the electons. It s also d cult to nvestgate these partcular mcro-level votng processes wth formal models. Ths s manly because of the problem of multple equlbra n mult-canddate and mult-seat electons. 1 Intutvely, t seems that should voters focus ther attenton n the close races for the last seat, strategc votng should be nvarant to dstrct magntude (see Gerber, Morton, and Retz 1998). We consequently nvestgate the e ect of dstrct magntude on votng behavor va a laboratory experment desgned to solate the motvatons behnd voter choces. Throughout we keep constant the dstrbuton of voters preferences and the number and polcy locaton of partes. Ths allows 1 Cox (1994) analyzes votng equlbra under sngle non transferable vote and shows support for hs now classc M +1 rule. We nstead analyze voters behavor n non majortaran mult member electons.

4 4 us to clearly observe the behavor of voters under d erent dstrct magntude treatments. We buld our analyss on two stylzed types of behavor: (1) sncere behavor, where a person votes for the party that yelds the hghest utlty regardless of nformaton about the electoral chances of the party; and (2) sophstcated behavor, where a person takes nto account the votes partes have receved n prevous electons when decdng whom to vote for. Here we defne voters beng strategc or sophstcated when they act n accordance wth both ther preferences for the canddates and ther perceptons of the relatve chances of varous pars of canddates beng n contenton for vctory. 2 Just as pre-electon polls serve to nform the electorate about the relatve chances of the canddates (Fey, 1997), n our mult-electon settng past votng behavor helps voters form expectatons on ther chances of nfluencng the outcome. We also observe a large proporton of subjects who do not vote sncerely even when ther sncere and sophstcated acton concdes.e. when there s no tenson between the honest expresson of ther preferences and the consderaton of castng a useful vote. To our surprse we fnd that among the subjects who do not vote for ther preferred party many vote for the party that obtaned most votes n the prevous electon. Ths further motvates the characterzaton of a thrd type of behavor: frontrunner behavor, where a person votes for the party that obtaned most votes n the prevous electon. Recently, Morton, Muller, Page, and Torgler (2013) observe frontrunner votng n French presdental electons. In the formal lterature, we are only aware of Callander (2008), who suggests that when voters care about the wnnng canddate a unque responsve equlbrum exsts, although the addton of a desre to wn creates multple equlbra. The paper s organzed as follows. We frst descrbe the expermental setup n Secton 2. We then summarze the man fndngs, n Secton 3, by showng the proporton of sncere, sophstcated and frontrunner votes n our d erent expermental treatments. We fnd that sncere votng ncreases wth dstrct magntude, sophstcated votng decreases, and frontrunner votng can never be dscarded n any of our treatments. Later, n Secton 4, we analyze ndvdual votng behavor and classfy each subject as one of our three types. We show that the ncrease n sncere votng wth dstrct magntude s due to both mechancal and psychologcal factors. 2 Myerson and Weber (1993, p. 135)

5 5 2. The experment Our experment conssts of four treatments, each correspondng to a d erent dstrct magntude: a sngle-member dstrct (M=1); a two-member dstrct (M=2); a three-member dstrct (M=3); and pure proportonal representaton (M=PR). Subjects partcpate n 60 electons by castng a sngle vote for one of fve partes. 3 In the M=1 treatment, a canddate from the party that receves the most votes s elected, and each subject receves a payo from the electon equvalent to hs or her utlty for that party. In the M=2 and M=3 treatments, we apply a form of closed-lst proportonal representaton, where seats are allocated to the partes n proporton to ther vote-shares (usng the Sante-Laguï 1 2 dvsor method), and each subject receves a pay-o from the electon equvalent to hs or her utlty for the party of each canddate that s elected. Fnally, n the PR treatment, each subject receves a pay-o n drect proporton to the share of votes each party receves. Table 1 shows how we allocated 212 subjects to our four treatments. 4 Gven that each subject partcpated n 60 electons we have 12,720 observatons. treatment M=1 M=2 M=3 PR total number of groups partcpants per group 24,24 25,25 22,24,24 20, Table 1. Partcpants and Treatments In all treatments, the utlty that subjects derved for each of the partes was prvately announced. Every fve perods, subjects were randomly assgned a preferred polcy so that the overall dstrbuton n a bounded two dmensonal polcy space was unform. Poltcal partes had fxed postons and a subject s utlty for each of the fve partes was assgned by assumng quadratc (sngle-peaked and symmetrc) preferences. Voters were only told the utlty they derved for each party and dd 3 Castng a vote for a sngle party s the most common ballot-structure n sngle-member as well as mult-member dstrcts n natonal parlamentary electons n democraces (Reynolds and Steenbergen, 2006). 4 No subject partcpated n more than one sesson. Students were recruted through the onlne recrutment system ORSEE (Grener, 2004) and the experment took place on networked personal computers n Centre for Expermental Socal Scences at Nu eld College, Oxford n November The experment was programmed and conducted wth the software z-tree (Fschbacher, 2007). The data and program code for the experment are avalable upon request.

6 6 not know ther polcy locaton relatve to the locatons of the other voters, nor the relatve postons of the partes n the polcy space. 5 In our experment, voters never observe other voters preferences; they can only nfer them by observng past votng behavor. The same procedure was used n all sessons. Instructons 6 were read aloud and questons answered n prvate. Students were asked to answer a questonnare to check that they fully understood the expermental desgn, the seat-allocaton method, and t he pay-o structure for ther partcular treatment group. If any of ther answers were wrong, we referred the partcpant to the secton of the nstructons where the correct answer was provded. Students were solated and could not communcate wth each other. In the frst electon each partcpant was shown a screen wth ther utlty from each of the fve partes and was asked to cast a sngle vote for one of the partes. Abstenton was not allowed. The partcpants were then nformed of the outcome of the frst electon: the number of votes each party receved; whch canddate(s) was (were) elected; and the payo they receved from the electon. The partcpants were then asked to vote agan for one of the partes. Ths procedure - n whch we counted the votes for each party, we assgned seats, and we nformed partcpants about the outcome of the electon and ther payo - was repeated for fve electons. Then, after fve electons, the partcpants preferences were redrawn and the partcpants nteracted for a further fve electons, after whch the preferences were redrawn agan. In other words the experment was organzed as 12 sets of fve rounds (60 electons n total) and for each set of electons, partcpants preferences and party labels were redrawn. At the end of the last electon, the computer randomly selected four electons and subjects were pad the profts they obtaned n those four electons (n Pence Sterlng). In addton, subjects receved a show-up fee of 3 GBP for takng part n the experment. At the end of each sesson, partcpants were asked to fll n a questonnare on the computer and were gven ther fnal payment n prvate. Sesson length, ncludng watng tme and payment, was around 90 mnutes. The average payment was GBP (approxmately 24 USD). 5 We mposed lttle knowledge on the underlyng structure of the experment, to avod favorng subjects that are comfortable wth spatal models of electoral competton and can deduce the vablty of each party from ts spatal locaton. 6 See the Appendx for the nstructons for the M=2 groups and varous screenshots of our program,.

7 7 3. Aggregate Results As an llustraton of our results, Table 2 shows the outcomes of electons 11 to 15 for one of the groups n each treatment (we report the votes receved by each of the fve partes and the canddates assgned to each party). The results wth M=1 shows voters coordnatng around the frst two partes A and B, wth support for the other three partes declnng over tme. Ths suggests a hgh proporton of sophstcated behavor, wth voters whose preferred party was C, D or E realzng that ther most preferred party had no chance of wnnng. When M=2, voters appear to coordnate around three partes (A, B and C) as Cox (2007) would have predcted, wth a non-neglgble proporton of voters stll supportng the two uncompettve partes (D and E); n contrast, when M=3 party C ran away wth the electon after a few rounds. Fnally, n the fully proportonal (control) treatment group, there were consderable shfts n votng patterns, despte the fact that the optmal behavor for each partcpant n ths treatment was to vote sncerely. Most strkngly we observe a tendency to vote for the frontrunner canddate. We leave ths aspect asde for now and wll revst t at the end of ths Secton. electon M=1 M=2 M=3 PR (6,7*,4,4,3) (7*,6,6*,3,3) (5*,5*,6*,4,4) (5,6,3,6,4) 12 (7,12*,4,1,0) (8*,7,8*,1,1) (4*,7*,10*,1,2) (6,7,3,5,3) 13 (8,12,*3,1,0) (7,8*,7*,1,2) (4*,3,16**,0,1) (9,7,2,4,2) 14 (8,14,*2,0,0) (7,8*,8*,0,2) (6*,3,13**,1,1) (11,6,1,5,1) 15 (9,14*,1,0,0) (7,8*,8*,1,1) (3,4*,14**,2,1) (13,5,2,3,1) Table 2. Sample of electon results for each treatment. In each cell we ndcate the votes receved by partes A, B, C, D, and E (resp.) and we dentfy wth one or two stars (* or **) the partes that obtaned 1 or 2 canddates, respectvely. In what follows we classfy a vote as sncere when the subject votes for hs/her most preferred party, the one that yelds maxmum payment. 7 A sophstcated vote s nstead a vote n whch the subject not only consders hs/her preferences for all partes but also the lkelhood that hs or her vote wll be pvotal. In the Appendx we o er a detaled explanaton of the computaton of expected utltes when votng for each party. For ths purpose we buld on Myerson and Weber (1993), and assume that our subjects best respond to the probablty that each voter wll vote 7 Ths knd of behavor s often also referred as expressve, honest or straghtforward vote. See Feddersen and Sandron (2006) or Fscher (1996).

8 8 for each of the partes when these probabltes concde wth the prevous perod frequency of votes. 8 Note that our defnton of a sophstcated vote d ers from the one n the electoral studes lterature. Followng game theoretc conventons, we defne a sophstcated vote smply as the vote whch maxmzes expected utlty (takes nto account the utlty for each canddate as well as the probablty tat the vote s pvotal). Instead, the emprcal electoral studes lterature often treats a sophstcated (or strategc) vote as the vote whch maxmzes expected utlty and does not concde wth an expressve vote. In our defnton, sncere and sophstcated votng can ether occur unquely or can be present at the same tme. The frequency of sncere and/or sophstcated votng behavor n our four treatments s shown n Table 3. The long-standng hypothess n the electoral studes lterature that sncere votng should ncrease wth dstrct magntude seems to fnd lttle support n our aggregated data. M=1 M=2 M=3 PR %sncere % sophstcated % observed both % predcted both Table 3. Frequency of Types of Behavor by Treatment. Observed Both : a subject s both sncere and sophstcated. Predcted Both : the sncere vote and the sophstcated vote concde Most surprsngly, we see a decrease n sophstcated votng as we ncrease dstrct magntude from 1 to 3. We expected the opposte because as we ncrease dstrct magntude, the lkelhood that a sophstcated vote wll concde wth a sncere vote ncreases. Ths ncreased concdence n both types of votng s captured n the last row of the table, where the predcted values are the percentage of observatons where votng sncerely for the most preferred party n an electon can also be classfed as a sophstcated vote for the party that maxmzes the expected payo. We dentfy the e ects of dstrct magntude on votng behavor va the classc dstncton between mechancal and psychologcal e ects of electoral rules. We defne the mechancal e ect of dstrct magntude on votng behavor as the change n sncere behavor not caused by a voters changng 8 Forsythe, Myerson, Retz, and Weber (1993) s the only expermental work we are aware of that uses past electon nformaton as the cue from whch voters form expectatons about ther probablty of beng pvotal. From a game theoretcal perspectve, we are assumng that subjects engage n a type of fcttous play, where subjects myopcally best respond to prevous perod play.

9 9 ther strategy but due to the ncreased lkelhood a sophstcated vote s sncere. Instead, the changes n strategy are coned the psychologcal e ect of dstrct magntude on votng behavor. Consder, for nstance, a sophstcated voter whose preferred party s the thrd ranked n number of votes: when dstrct magntude s 1 hs/her vote s less lkely to concde wth hs/her sncere vote than when dstrct magntude s 2. Note that the strategy of the voter s not changng wth dstrct magntude yet the way we classfy hs/her actons s. Followng ths dstncton, the results n Table 3 suggest that the ncrease n sncere votng (as we ncrease dstrct magntude) s not drven by a mechancal e ect but by a psychologcal one. Ths s possbly due to the hgher complexty of computng the correct sophstcated acton when dstrct magntude s large, whch leads voters to be less lkely to vote n a sophstcated way and to vote sncerely nstead as dstrct magntude ncreases. Somethng that seems puzzlng n Table 3, though, s the large d erence between the percentage of observatons that are both sncere and sophstcated and the stuatons that are predcted to be so. It seems that n stuatons n whch both sncere and sophstcated actons concde, the voter should have no conflct about supportng hs preferred party. However, we observe that around a 20 percent of subjects fal to choose ths acton when t s optmal to do so! For whom are they votng? To our surprse we see that 50 percent of the subjects who dd not vote for ther most preferred party (when sncere and sophstcated actons concde) voted nstead for the party that obtaned the most votes n the prevous electon round. We consequently defne ths type of behavor as frontrunner votng: when a subject s votng for the party that obtaned the most votes n the prevous perod of play. 9 Together wth ths thrd classfcaton, our three types of behavor descrbe more than 90 percent of all vote choces. Further evdence towards frontrunner behavor s found n our control treatment, wth a fully proportonal electoral system, where votng sncerely s the domnant strategy. In Fgure 1 we depct the data of our control treatment aggregated by each set of fve electons (recall that preferences are redrawn every fve perods) whenever the frontrunner acton does not concde wth the sncere one. That s, we drop observatons n whch the subject s preferred party was the most voted party 9 Frontrunner votng s only defned for electon rounds 2,3,4 and 5 gven that n the frst electon (round 1) preferences have been redrawn and there s no prevous perod of play wth the same preferences. In common value stuatons, votng for the wnner can be understood n terms of herdng Nageeb and Kartk (2012), nformaton aggregaton ((Feddersen and Pesendorfer, 1997), or favorng a stable governng party Rambau (2013a). There s no room for such ratonalzatons n our setup.

10 10 n the prevous electon. We see that most vote choces are sncere, yet a large proporton (around 11 percent) of votes are for the frontrunner of the prevous electon when ths s clearly defned n rounds 2, 3, 4 and 5. Fgure 1. Frequences of Votes by Electon Round when PR and Votng Sncerely s not Votng for the Frontrunner A key problem when analyzng our data (as wth actual votng data) s that many observatons can be smultaneously classfed as more than one type. Consder for nstance a subject whose preferred party s also the one that obtaned most votes n the prevous perod when dstrct magntude s 1. In ths stuaton, when the subject votes expressvely, s/he s also votng for the frontrunner and most lkely her/hs vote also concdes wth hs sophstcated acton. Table 4 hence shows the subsample of observatons n whch the three types of actons do not concde. 10 M=1 M=2 M=3 %sncere % sophstcated % frontrunner % other observatons Table 4. Frequency of Types of Behavor by Treatment when Sncere, Sophstcated and Frontrunner do not concde. The last row n the table shows yet another manfestaton of the mechancal e ect of dstrct magntude: as dstrct magntude ncreases t s more lkely that the sncere and sophstcated actons concde, thus our sample becomes thnner. Even when we now only consder less than 7 percent of our observatons, the patterns we observed earler are preserved n ths subsample: sncere 10 Ths s ndeed the approach followed by many studes n the lterature, see for example (Duch, May, and Armstrong II, 2010)

11 votng s greater when dstrct magntude s 3 rather than 1, and sophstcated votng decreases. Possbly due to the ncreased complexty of the votng rule we see frontrunner votng and other behavor ncreasng wth dstrct magntude. 11 By constructon, the dsjont set never ncludes the frst round of electons when subjects vote just after preferences have been redrawn. In these cases there s no prevous nformaton, so sncere and sophstcated votng concdes n all treatments. As we dscussed above when analyzng our control treatment, n such electons t s domnant to vote sncerely for the preferred party. Indeed, we observe a hgh ncdence of such behavor n our data from the very frst electon 82 percent of our frst electon observatons are sncere and such behavor ncreases as the experment unfolds reachng 90 percent of sncere observatons n the frst electon of the last round of fve electons (electon 56). The patterns we observe are clearly suggestve of both mechancal and psychologcal e ects on votng behavor n our experment. The ncrease n sncere votng s (at least partly) drven by the mechancal e ect of dstrct magntude. However, f only mechancal e ects are present, we should also be observng an ncrease n the number of votes that are qualfed as sophstcated. Moreover, frontrunner votng should not ncrease wth dstrct magntude. Overall, there seems to be a tendency to move away from sophstcated actons, towards non-ratonal behavor (as expressed by sncere and frontrunner votng) as we ncrease dstrct magntude. A possble explanaton s that as dstrct magntude ncreases t becomes exponentally d cult to compute the correct sophstcated acton. Tables 3 and 4 both ndcate the heterogeneous e ects of dstrct magntude n our populaton. If all subjects were sncere we should observe 100 percent of observatons as sncere, whle sophstcated votng should ncrease wth dstrct magntude due to the mechancal e ect, and frontrunner votng should reman unchanged. Instead, f all subjects were sophstcated, sncere votng should ncrease wth dstrct magntude, sophstcated votng should always be at 100 percent, and frontrunner votng should decrease wth dstrct magntude, because more partes become vable so less voters need to favor the frontrunner canddate. In the next secton we consequently analyze n detal ndvdual votng decsons, to understand whether dstrct magntude has a systematc e ect on the heterogeneous behavor of subjects. Our goal, here, s to measure the relatve power of the

12 12 three types of behavor for a representatve voter and to see how dstrct magntude nfluences the relatve weght of the d erent motvatons. 4. Indvdual behavor Our ntal specfcaton assumes that subjects are ether sncere or sophstcated: the (unobserved) ndcator functon z soph t (zt sn ) takes value 1 f n round t subject behaves n a sophstcated (sncere) way, and 0 otherwse. Our goal s to estmate z t =(z soph t,zt sn ) for all subjects and rounds,.e. for each of our treatments we want to dentfy what s the uncondtonal probablty that a subject s of each type. 11 Ths uncondtonal probablty can also be nterpreted as the proporton of subjects who are basng ther choce solely on ther preferences (sncere types), and the proporton who are also takng nto nto account ther probablty of beng pvotal (sophstcated types). We wll use both nterpretatons ndstnctly throughout the text. Let u jt be the utlty subject derves from party j n electon t; and v jt be the expected utlty derves from votng for party j n electon t. y jt s a dummy varable that takes value 1 when subject votes for party j n round t: y jt =1fu jt y jt =1fv jt max(u 1t,u 2t,...,u 5t) and s sncere) max(v 1t,v 2t,...,v 5t) and s sophstcated) y jt = 0 otherwse where u jt =ū j + u jt +" u jt when zssn t = 1 and v jt = v j + v jt +" v jt when zsoph t = 1. We assume " u jt, "v jt type I extreme value. We model the probablty that subject votes for j n perod t as a multnomal logt (MNL): 12 p jt (u jt,v jt ; z t )= p jt (u jt,v jt ; z t )= eūj+ u jt P J k=1 eūk+ u kt e v j+ v jt P J k=1 e v k+ v kt f z soph t =0 f z soph t =1 11 Note that n ths frst specfcaton we have that z soph t =1 zt sn. 12 The reported results mpose that the constants ūj and v j are equal 8j. That s, the mean propensty of votng for each party s the same regardless of the type (we fnd analogous results when we relax ths condton).

13 Note that modelng utltes wth a MNL allows us to take nto account cardnalty of preferences: the lkelhood of votng for one party depends not only on the rankng of the party n an ordnal settng, but on the cardnal utlty ths party yelds relatve to all other partes. Gven that the type z =(z sn,z soph ) s unobserved, we can at most nfer the probablty that each subject s of each type. We estmate z s usng the Expectaton-Maxmzaton algorthm Arcdacono, Seg, and Sloan (2007) elegantly summarzes what we do: gven ntal values of the parameters, we frst calculate the condtonal probablty that an ndvdual s a partcular type; usng these condtonal probabltes as weghts, we treat types as observed and we maxmze the (now) addtvely separable log-lkelhood functon; gven the new parameter estmates, we then update the condtonal probabltes of beng each of the types and terate untl convergence (Arcdacono, Seg, and Sloan, 2007). 14 Snce the lkelhood of beng a partcular type s always strctly postve, we fnd that even when a vote seems to unambguously be of type z, the algorthm can only assgn t a probablty of beng of type z whch s arbtrarly close to 1, but never 1 (and, n some cases, gven the avalable nformaton, even not necessarly close to one). In the Appendx we specfy all computatonal detals of the EM algorthm. Throughout, we do a double exercse. Frst, we pool all observatons for each dstrct magntude, and run the EM algorthm just once per dstrct magntude.e. we run t four tmes (M=0,1,2,3). For each acton we fnd ẑ t and report the average across all ndvduals and perods (panels A n tables below). In other words, we frst assume that all observatons are ndependent. Second, gven that for each subject we have many observatons, we run the algorthm once per subject. 15 In ths case we also report the averages across all subjects (panels B n tables below). Ths second set of regressons can be nterpreted as controllng for ndvdual fxed e ects gven that we are estmatng z s for each subject separately. We can advance that results d er slghtly among our two types of regressons but the patterns are the same regardless of whch specfcaton we use. Table 5 reports the results for the two types model. For each dstrct magntude we report the percentage of ndvduals classfed as sncere and those who are classfed as sophstcated. We 13 See Dempster, Lard, and Rubn (1977) and Frühwrth-Schnatter (2006) for detaled descrptons of the algorthm. 14 Our strategy s smlar to that of Duch, May, and Armstrong II (2010): n both cases types are unobserved and from the data we can at most nfer the probablty that each subject s of each type at each round. Nevertheless, whereas Duch, May, and Armstrong II (2010) use Bayesan methods, we use the computatonally smpler EM algorthm, as n Rambau (2013b). 15 Gven that we do not nclude frst round observatons when sophstcated and sncere concdes we have 48 observatons per subject.

14 14 also report the percent of correct vote predctons usng ẑ t and the parameters of the model. In both panels A and B the estmated proporton of sncere types sharply ncreases wth dstrct magntude. Ths s a strong result, yet we have to remember that for a large proporton of our observatons, sncere and sophstcated actons are observatonally equvalent. Ths means that we are forcng our algorthm to decde between two types when both are potentally correct. Note that there are no results for our fully proportonalty treatment because n these cases both sncere and sophstcated behavor are equvalent. Note also that the percentage of observatons we correctly classfy decreases wth dstrct magntude but always reaches a hgh score. Panel A: Whole sample M=1 M=2 M=3 %sncere % sophstcated % correct observatons 2,304 2,400 3,360 Panel B: Average across subjects %sncere % sophstcated % correct observatons 2,256 2,352 3,312 Table 5. Proporton of subjects of each type (2 types) We check statstcal sgnfcance on the ncrease n the proporton of sncere voters by runnng a Kolmogorov-Smrnov test: for each dstrct magntude, the null hypothess s that the cumulatve dstrbuton functon of the dstrbuton of zjt sn s for M + 1 s not larger than the same one for M. We do not report the results but we fnd ths hypothess rejected n both cases. That s, the dstrbuton of z sn t s ncrease wth dstrct magntude. 16 Followng our prevous dscusson, we next ntroduce a thrd type that captures whether subjects vote for the frontrunner party n the prevous electon. We denote t z FR =1 z sn z soph. Formally, for ths type y jt s equal to 1 f n the prevous round party j obtaned the most votes, 16 Note that panel B has slghtly less observatons than panel A. Ths s because n each treatment we had to drop one subject to reach convergence n our algorthm.

15 15 and s equal to 0 otherwse. The utlty of ths type of voter s gven by: p jt (u jt,v jt,fr jt ; z t )= e j+ FR jt P J k=1 e k+ FR kt f z FR t =1 where FR j s a dummy that takes value 1 f j was the party wth most votes n the prevous round and 0 otherwse. Table 6 shows the results for the three types model. We consstently fnd the same patterns: n both panels A and B, the proporton of sncere actons ncreases wth dstrct magntude. Ths result s statstcally sgnfcant when usng the Kolmogorov-Smrnov test. Interestngly, t s worth notng that around one n every ten votes s classfed as frontrunner. Ths s ndeed the proporton we observed n our control treatment wth a fully proportonal electoral system. Panel A: Whole sample M=1 M=2 M=3 PR %sncere %sncere % sophstcated % frontrunner % correct observatons 2,304 2,400 3,360 2,112 Panel B: Average across subjects %sncere %sncere % sophstcated % frontrunner % correct observatons 2,256 2,352 3,312 2,112 Table 6. Proporton of subjects of each type (3 types) As was noted n Secton 3, one of the man ssues wth our expermental desgn s that many actons are observatonally equvalent. As a robustness check we now look at the subsample of observatons n whch the three types of acton do not concde. Note that there s no self-selecton nto ths subsample: utltes are randomly assgned every fve electons. Besdes, expected utltes are computed gven the behavor of all subjects n the prevous round so there s no way a subject can choose a partcular orderng of partes accordng to utltes or expected utltes. The medan number of observatons per ndvdual s 16, 9 and 5 n the treatments wth dstrct magntude 1, 2, and 3, respectvely. Ths once agan captures the mechancal e ect of dstrct magntude. Table 7

16 16 reports results for the subsample of dsjont observatons (the observatons for full proportoanlty are consdered dsjont when sncere does not concde wth frontrunner): once agan we observe that the uncondtonal probablty that an acton s sncere ncreases. Even when the average values do not seem to confrm ths last statement (for M=2), the Kolmogov-Smrnov test rejects that the cumulatve dstrbuton functon for (ẑ sn M = 3) s not larger than the c.d.f for (ẑ M = 2) ths contradcts the reported average levels because the dstrbuton of ẑ sn s s more sparse when M = 2. A key lesson we take from ths last table s the fact that the cases M = 2 and M = 3 are almost equvalent and both are dstnctly d erent than M = 0 and full PR. Fnally, t s surprsng to see that the propensty to cast a frontrunner vote s now much larger than we observed n the full sample. Panel A: Whole sample M=1 M=2 M=3 PR %sncere %sncere % sophstcated % frontrunner % correct observatons Panel B: Weghted average %sncere %sncere % sophstcated % frontrunner % correct observatons Table 7. Proporton of subjects of each type (3 types) when M=1,2, or 3 and Sncere, Sophstcated and Frontrunner do not concde or when PR and Sncere and Frontrunner do not concde. In Panel B, observatons are weghted by the number of tmes the subject appears n the dsjont sample Robustness Check. We fnally show that our results are not drven by the partcular choce of the number of types or the EM algorthm. We do not run a condtonal probt as such a model would estmate the values of the parameters for the average subject and we want to allow for subject heterogenety the mxed logt s our model of choce gven that we can estmate parameters for each ndvdual and report the dstrbuton of such parameters among our populaton.

17 In our approach we have mplctly assumed that the value of each of our parameters s the same wthn each type and we have then assgned each of our subjects a probablty of beng each 17 type (z s). Instead, the mxed logt assumes that all subjects smultaneously take nto account utltaran, sophstcated and frontrunner consderatons (and allows subjects to place d erent weght on each component). Formally we have that: p jt (u jt,v jt,fr jt )= e j+ sn u jt+ soph v jt+ FR FR jt P J k=1 e k+ sn u kt+ soph v kt+ FR FR kt where 0 sn soph FR 0 1 C A N sn soph, snsoph 2 sn snsoph snf R 2 soph FR snf R sophf R sophf R 2 FR 1 C A We assume that the dstrbuton of the vector of parameters follows a multvarate normal dstrbuton. Below we report the estmated means of our mxed logt computatons (covarance matrces are avalable upon request). M=1 M=2 M=3 PR Mean sn (0.50) (0.26) (0.32) (1.20) Mean soph (9.57) (6.14) (6.68) Mean FR (0.15) (0.12) (0.11) (0.88) % correct observatons 2,304 2,400 3,360 2,112 Table 8. Mxed Logt Results (standard errors n brackets) Note that the scale of the utlty and expected utlty values are not the same so parameters n our four d erent specfcatons are not drectly comparable. However, we can observe smlar trends as those we observed earler: sncere consderatons (as captured by sn ) ncrease wth dstrct magntude n all cases apart from M = 1 to M = 2; sophstcated consderatons (as captured

18 18 by soph ) decrease wth dstrct magntude n all cases; and fnally, frontrunner consderatons are always present. In terms of the percentage of correct observatons we see that our EM algorthm has a much greater predctve power than the mxed logt model. Duch, May, and Armstrong II (2010) run a smlar comparson wth the mxed logt model and defne the Proportonal Reducton n Error (PRE) as: PRE = #{correct types model} #{correct mxed logt } N #{correct mxed logt} where N s the total number of observatons. We can compute ths score for each of our treatments and we fnd that the relatve mprovement of the EM algorthm wth respect to the more standard mxed logt s always above 30 percent and reaches ts maxmum for the M=1 treatment n whch PRE s 60.1 percent Conclusons A wdely-held assumpton n poltcal scence s that non-sncere votng should be lower n hghermagntude dstrcts. Yet, formal work on the probablty of beng pvotal as well as actual votng data from electons suggests that voters are just as sophstcated n mult-member dstrcts are they are n sngle-member dstrcts. One problem for emprcal research, however, s that t s almost mpossble to solate the e ect of dstrct magntude on votng behavor ndependently of voters preferences or partes postons. We hence desgned a lab experment to solate ths e ect, by varyng dstrct magntude whle keepng voters preferences and partes postons constant. We do fnd a decrease n sophstcated votng as dstrct magntude ncreases. But, we also fnd evdence of a mechancal e ect of dstrct magntude on the propensty to vote sncerely for a voter s most preferred party. As dstrct magntude ncreased, the proporton of voters who found that ther most preferred party would also now yeld them the hghest expected utlty (and hence that ther sncere and sophstcated motvatons concded) also ncreased. Nevertheless, not as many partcpants voted for ther most preferred party when ther sncere and sophstcated motvatons concded as we expected. We also found that sophstcated votng does not ncrease wth dstrct magntude and that there was a hgh ncdence of votng for the wnner of the prevous electon (the frontrunner) n all our 17 PRE values are 60.1%, 30.2%, 34.3% and 39.7% for DM=1,2,3 and PR, respectvely.

19 treatments (we even fnd the latter to be the case n our control wth fully proportonal electons - above 10 percent of such observatons). Followng our classfcaton of mechancal and psychologcal e ects, we have ndeed seen that as we ncrease dstrct magntude we fnd that there s an ncrease n sncere votng possbly due to a mechancal e ect. However, our results suggest that the decrease n sophstcated votng and the ncrease n frontrunner votng can only be classfed as a psychologcal e ect of the ncrease n dstrct magntude, possbly due to the rsng complexty of calculatng the correct sophstcated acton. Ths last aspect, the presence of frontrunner votng, has not receved much attenton n the emprcal 19 and theoretcal studes of votng. However, as Hnch (1981) ponts out: votng for the wnner s no less plausble than the assumpton that voters beleve they can be pvotal. References Abramson, P. R., J. H. Aldrch, A. Blas, M. Damond, A. Dskn, I. H. Indrdason, D. J. Lee, and R. Levne (2010): Comparng Strategc Votng Under FPTP and PR, Comparatve Poltcal Studes, 1(43), Arcdacono, P., H. Seg, and F. Sloan (2007): Lvng Ratonally Under the Volcano? An Emprcal Analyss of Heavy Drnkng and Smokng, Internatonal Economc Revew, 48(1), Austen-Smth, D., and J. S. Banks (1988): Electons, Coaltons and Legslatve OutcomeS, Amercan Poltcal Scence Revew, 2(82), (1996): Informaton Aggregaton, Ratonalty and the Condorcet Jury Theorem, Amercan Poltcal Scence Revew, 1(90), Bargstad, M. A., and O. Kedar (2009): Coalton-Targeted Duvergeran Votng: How Expectatons A ect Voter Choce under Proportonal Representaton, Amercan Journal of Poltcal Scence, 2(53), Blas, A., J. Aldrch, I. Indrdason, and R. Levne (2006): Do Voters Vote for Government Coaltons? Testng Downs Pessmstc Concluson, Party Poltcs, 6(12), Borman, S. (2009): The Expectaton Maxmzaton Algorthm A short tutoral, mmeo. Callander, S. (2008): Majorty Rule when Voters lke to Wn, Games and Economc Behavour, 2(64),

20 20 Carey, J., and S. Hx (2011): The Electoral Sweet Spot: Low-Magntude Proportonal Electoral Systems, Amercan Journal of Poltcal Scence, 2(55), Cox, G. W. (1994): Strategc Votng Equlbra Under the Sngle Nontransferable Vote, Amercan Poltcal Scence Revew, 3(88). (2007): Makng Votes Count: Strategc Coordnaton n the World s Electoral Systems. Cambrdge Unversty Press, Cambrdge. Dempster, A., N. M. Lard, and D. B. Rubn (1977): Maxmum Lkelhood from Incomplete Data va the EM Algorthm, Journal of Royal Statstcal Socety, Seres B (Methodologcal), 1(39), Duch, R., J. May, and D. A. Armstrong II (2010): Coalton-drected Votng n Multparty Democraces, Amercan Poltcal Scence Revew, 104, Duverger, M. (1954): Poltcal Partes: Ther Organzaton and Actvty n the Modern State. John Wley, New York, NY. Feddersen, T. J., and W. Pesendorfer (1996): The Swng Voter s Curse, Amercan Economc Revew, 3(86), (1997): Votng Behavor and Informaton Aggregaton n Electons Wth Prvate Informaton, Econometrca, 5(65), Feddersen, T. J., and A. Sandron (2006): A Theory of Partcpaton n Electons, Amercan Economc Revew, 4(96), Fey, M. (1997): Stablty and Coordnaton n Duverger s Law: A Formal Model of Preelecton Polls and Strategc Votng, Amercan Poltcal Scence Revew, 1(91), Fschbacher, U. (2007): Z-Tree: Zurch Toolbox for Ready-Made Economc Experments, Expermental Economcs, 2(10), Fscher, A. J. (1996): A Further Expermental Study of Expressve Votng, Publc Choce, 1(88), Forsythe, R., R. B. Myerson, T. A. Retz, and R. J. Weber (1993): An Experment on Coordnaton n Mult-canddate Electons: The Importance of Polls and Electon Hstores, Socal Choce and Welfare, 3(10), Frühwrth-Schnatter, S. (2006): Fnte Mxture and Markov Swtchng Models. Sprnger Seres n Statstcs, New York.

21 21 Gerber, E. R., R. B. Morton, and T. A. Retz (1998): Mnorty Representaton n Mult- Member Dstrcts, Amercan Poltcal Scence Revew, 1(92), Grener, B. (2004): The Onlne Recrutment System ORSEE 2.0: A Gude for the Organzaton of Experments n Economcs, Workng Paper Seres n Economcs - Unversty of Cologne, 10. Hnch, M. J. (1981): Votng as an Act of Contrbuton, Internatonal Economc Revew, 36(1), McCuen, B., and R. B. Morton (2006): Tactcal Coalton Votng and Informaton n the Laboratory, Electoral Studes, 3(29), Morton, R., D. Muller, L. Page, and B. Torgler (2013): Ext Polls, Turnout, and Bandwagon Votng: Evdence from a Natural Experment, mmeo. Myerson, R. B., and R. J. Weber (1993): A Theory of Votng Equlbra, Amercan Poltcal Scence Revew, 1(87), Nageeb, A., and N. Kartk (2012): Herdng wth Collectve Preferences, Economc Theory, 51, Pketty, T. (2000): Votng as Communcatng, Revew of Economc Studes, 1(67), Reynolds, A., and M. Steenbergen (2006): The Poltcal Consequences of Ballot Desgn, Innovaton and Manpulaton, Electoral Studes, 3(25), Rambau, G. (2013a): Bandwagon n Israel? Note on Bargsted and Kedar (2009), mmeo. (2013b): Votng for Partes or Polces? Evdence from Israel, mmeo. Sartor, G. (1968): Poltcal Development and Poltcal Engneerng. Cambrdge Unversty Press, New York, NY.

22 22 6. Appendx 6.1. Instructons. (treatment M=2) Thank you for agreeng to partcpate n our votng experment. The sum of money you wll earn durng the sesson wll be gven prvately to you at the end of the experment. From now on (and untl the end of the experment) you cannot talk to any other partcpant. If you have a queston, please rase your hand and one of the nstructors wll answer your questons prvately. Please do not ask anythng aloud! You belong to a group of 25 partcpants wth whom you wll nteract for 60 electons. The rules are the same for all partcpants and for all electons. In each electon the group wll vote to elect two canddates. The wnnng canddates wll be selected by a form of proportonal representaton, where each party wll wn seats n proporton to ther share of the vote. After each electon you wll be announced the outcome and your proft n such electon. At the end of the experment you ll be asked to answer a questonnare Votng procedure. The party wth the most votes wns the frst seat, and ts vote-total s then dvded by 3. The party wth the hghest remanng votes wns the second seat. In the case of a te, the wnner s determned randomly. As an llustraton consder the followng example: Party A B C D E Votes Votes 3 As a result, partes B and D each obtan a canddate because 9 and 7 are the hghest numbers. Now consder a d erent example where partes obtan the followng number of votes: Party A B C D E Votes Votes 5 In ths second example, Party A obtans 2 canddates because 15 and 5 are the hghest numbers Profts n each electon. The profts you receve n each electon depend on the canddates elected by the group regardless of whether you voted for any of them. Your proft wll be equal to the sum of your valuaton of the party of each elected canddate. The table below shows fve hypothetcal valuatons for each of the fve partes: Party A B C D E Your valuatons So, f the 2 canddates from party A are elected, you obtan a proft of 1000 ( = ). Alternatvely, f one canddate from party C s elected, and one canddate from party D s elected you obtan a proft of 1900 (= ). It s mportant to note that (a) your valuatons are d erent from the valuatons of all other voters; and (b) that no other voter knows the valuatons of any other voter Fnal Payment. At the end of the last electon, the computer wll randomly select 4 electons and you wll earn the sum of the profts on those electons n pennes. Addtonally you wll be pad three pounds for takng part n the experment Questonnare. (pror to the begnnng of the sesson) 1. When the wnners of an electon are known, do you know your proft n such electon? YES/NO 2. When the wnners of an electon are known, do you know the proft of any other partcpant? YES/NO 3. Imagne that party A obtans more votes than party B. Could t ever be the case that party B obtans more canddates than party A? YES/NO 4. Imagne a stuaton where the votes obtaned by each party are gven by the table below. What would the outcome of the electon be? Party A B C D E Votes Two canddates from party A One canddate from party D and one from party E One canddate from party A and one from party D Two canddates from party B

23 5. Consder a stuaton where your valuatons and the votes obtaned by each party are gven by the table below. Imagne you voted for party B, what would your proft be? 1000 or 1200 or 2400 or 0 or 2300 Party A B C D E Votes Your Valuatons Who s gong to be pad at the end of the experment? No one 1 person accordng to hs/her proft n varous electons 2 people accordng to ther proft n varous electon Everyone accordng to hs or her proft n four electons Everyone accordng to hs or her average proft throughout the experment 23! Screenshot!at!the!begnnng!of!electon!1!!! Screenshot!after!electon!1!! Screenshot!at!the!begnnng!of!electon!2!! Fgure 2. Screenshots of the Ztree program for the treatment M=2.

24 Locaton of Poltcal Partes. In all our sessons, the locaton of the partes and the voters were re-drawn after each set of fve electons. We alternated between two types of party locatons, shown n the Fgure below. So, electons 1 to 5, 11 to 15, 21 to 25, 31 to 35, 41 to 45, and 51 to 55 where held wth the Type A locatons of the partes. And, electons 6 to 10, 16 to 20, 26 to 30, 36 to 40, 46 to 50, and 56 to 60 where held wth the Type B locatons of the partes. Note that n the Type B electons we assumed radal symmetry so that there were many more players that were nd erent between varous partes. The labellng of partes (from A to E) was randomly allocated n each set of fve electons. Gven that the partcpants were not aware of the two-dmensonal locaton of ther preferences and the relatve locaton of the partes, when we swtched between the two types of party locatons we smply announced that the preferences had been redrawn. We clearly stated that the next fve electons were ndependent from the prevous fve Fgure 3. Locaton of poltcal partes. In all treatments, subjects preferred polces are unformly dstrbuted n the two dmensonal polcy space [0, 100] [0, 100] Sophstcated votng. A subject s sophstcated vote takes nto account not only hs or her utlty but also the probablty hs or her vote wll be pvotal. The latter component depends on the subject s belefs about the dstrbuton of votes. Usng the theory of votng equlbra Myerson and Weber (1993) we wll assume that any subject assumes that the probablty that any other subject votes for party k s equal to the v vote share obtaned by each party n the prevous round of electon. That s, p k = k v A +v B +v C +v D +v E,where (v A,v B,v C,v D,v E ) are the votes receved by each of the fve partes n the prevous round of electons. 18 Knowng these probabltes, we can compute the probablty that party A gets a votes, canddate B gets b votes, and so on, out of n-1 voters usng the followng multnomal probablty: (n 1)! f(a, b, c, d, e) = a!b!c!d!e! pa A p b B p c C p d D p e E where a + b + c + d + e = n 1. Gven a dstrbuton of votes n the prevous electon, there s then a functon that assgns seats to each of the fve partes (takng nto account the dstrct magntude and the seat allocaton formula, n our case Sante-Laguë for the M=2 and M=3 treatments). We denote the seats assgned to party k gven a partcular vote dstrbuton as k (a, b, c, d, e). The expected utlty of a subject wth preferences Î when votng for party A can then be expressed as follows: 0 1 X X E (u (vote for party A) k (a +1,b,c,d,e) k f(a, b, c, d, e) A (a,b,c,d,e)2v k=a,b,c,d,e where s the type of the voter (the utlty he assgn to each party) and V s any possble combnaton of votes for the fve partes so that they add up to (n-1). In our expermental settng, we restrct our attenton to stuatons n whch beng sophstcated s solely drven by the probablty of beng pvotal. There are, however, many motvatons that may lead voters to vote non sncerely n real electons. Voters may use ther votes to communcate ther preferences (Pketty, 2000). They may antcpate the best canddate to be elected gven other ctzens votes (Austen-Smth and Banks, 1996; Feddersen and Pesendorfer, 1996, 1997). Or, they may try to nfluence the coalton formaton n the legslature (Austen-Smth and Banks, 1988; Blas, Aldrch, Indrdason, and Levne, 2006; Bargstad and Kedar, 2009; Duch, May, and Armstrong II, 2010; McCuen and Morton, 2006). 18 In the frst round of electons, just after preferences have been redrawn, there s no prevous round on whch to condton the votng decson so the sophstcated vote concdes wth the sncere one: votng for most preferred party.