The Effects of Dstrct Magntude on Votng Behavor Smon Hx London School of Economcs and Poltcal Scence Rafael Hortala-Vallve London School of Economcs and Poltcal Scence Gullem Rambau Yale - NUS College September 8, 2014 Abstract. Is there more sncere votng n mult-member dstrcts than n sngle-member dstrcts? Exstng research on ths queston s nconclusve, at least n part because t s dffcult wth observatonal data to solate the effect 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 effect and a psychologcal effect. 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, Yale Poltcal Economy Semnar, Toronto Poltcal Behavor Workshop, McMaster Unversty Publc Lecture, and MPSA 2013. 1
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. Between 2012 and 2014, for example, Romana consdered swtchng from large mult-member dstrcts to snglemember dstrcts, Israel consdered swtchng from a sngle natonal mult-member dstrct to smaller mult-member dstrcts, the Chlean government proposed to replace two-member dstrcts wth large mult-member 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 effectve 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 dffcult 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 who 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 neffectveness follows, more than anythng else, the sze of the consttuency. These ntutons mght explan the patterns observed n electons, where voters n small multmember dstrcts seem to coordnate around larger partes, whch leads to a low number of wasted votes (and a closer relatonshp between vote-shares and seat-shares) as well as a low number of partes elected to parlaments and fewer partes n government. In contrast, n large mult-member dstrcts, we observe ctzens overwhelmngly votng for ther most preferred partes (Carey and Hx, 2011). 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 effect 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. The latter meanwhle captures changes n voters strateges possbly due to the ncreased complexty of the electoral system. It s mpossble to solate such mcro-level effects 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, lke the number of partes, socetal cleavage structures, 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 effects of the electoral rule covary wth the tmng of the electons. It s also dffcult 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 effect 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 us to clearly observe the behavor of voters under dfferent 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 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 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 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. In our data we also observe a large proporton of subjects who do not vote sncerely even when ther sncere and sophstcated actons concde.e. when there s no tenson between the honest expresson of ther preferences and the consderaton of castng a useful vote. We fnd that subjects not votng for ther preferred party (when ths vote s also the sophstcated acton) 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. There s a large lterature on the bandwagon effect. 3 However n our expermental desgn ths behavour cannot be ratonalsed by subjects extractng nformaton from observng a party ganng the largest support. Ths type of behavoural pattern s analysed n Callander (2008), who suggests that when voters care about supportng 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 dfferent expermental treatments. We fnd that sncere votng ncreases wth dstrct magntude, sophstcated votng (weakly) 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) 3 See Smon (1954) and the large lterature that followed.
5 2. The experment Our experment conssts of four treatments, each correspondng to a dfferent 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 belong to a group of 25 subjects 4 and partcpate n 60 electons by castng a sngle vote for one of fve partes. 5 In the M=1 treatment, a canddate from the party that receves the most votes s elected, and each subject receves a payoff 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-Laguë dvsor method), and each subject receves a pay-off 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-off n drect proporton to the share of votes each party receves. Ths specfcaton of payoffs allows us to dentfy the strategc ncentves of voters as we modfy dstrct magntude. Needless to say t abstracts from the coalton formaton game that would most lkely arse n non-majortaran electoral systems. 6 Table 1 shows how we allocated 212 subjects to our four treatments. 7 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 2 2 3 2 9 partcpants per group 24,24 25,25 22,24,24 20,24 212 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 preferences were redrawn. We leave the specfc detals of the overall 4 Due to some partcpants not showng up on tme some groups had slghtly less partcpants (see Table 1) obvously, nstructons were modfed accordngly. 5 Castng a vote for a sngle party s the most common ballot-structure n sngle-member as well as multmember dstrcts n natonal parlamentary electons n democraces (Reynolds and Steenbergen, 2006). 6 Strategc votng when takng nto account post electons coaltons has receved growng nterest 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 but s out of the scope of ths paper. 7 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 Nuffeld College, Oxford n November 2011. 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 dstrbuton of preferences to the Appendx. For our purposes what really matters s that subjects never observe other voters preferences; they can only nfer them by observng past votng behavor. Implct n ths partcular feature of our experment s that we want subjects to condton ther votng decsons solely on ther own preferences and aggregate past votng behavour. The same procedure was used n all sessons. Instructons 8 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 the payoff 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 payoff 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 payoff - 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 15.71 GBP (approxmately 26 USD). 8 See the Appendx for the nstructons for the M=2 groups and varous screenshots of our program.
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. In contrast, when M=3 party C ran away wth the electon after a few rounds. In these 3 cases, we always observe some voters supportng uncompettve partes (D and E). 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 s to vote sncerely. We also observe a tendency to vote for the frontrunner canddate we wll leave ths aspect asde for now and wll revst t at the end of ths Secton. electon M=1 M=2 M=3 PR 11 11 (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. 9 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 we compute expected utltes and assume that a sophstcated subject votes for the party that maxmses expected utlty. In the Appendx we offer a detaled explanaton of the computaton of expected utltes when votng for each party. For ths purpose we buld on Myerson 9 Ths knd of behavor s often also referred as expressve, honest or straghtforward vote. See Feddersen and Sandron (2006) or Fscher (1996).
8 and Weber (1993), and assume that our subjects best respond to prevous perod play by assumng that the probablty that any player votes for each of the partes concdes wth the prevous perod frequency of votes. 10 Our defntons allow a voter to be smultaneously sophstcated and sncere: the optmal acton may consst of votng towards the subjects preferred party. 11 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 when comparng our three non-fully proportonal treatments. M=1 M=2 M=3 PR % sncere 70.5 72.4 72.7 89.8 % sophstcated 84.2 74.1 70.5 89.8 % observed both 64.5 67.0 66.6 89.8 % predcted both 72.5 85.5 91.2 100 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 dstrct magntude ncreases, the lkelhood that a sophstcated vote concdes 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 payoff. We dentfy the effects of dstrct magntude on votng behavor va the classc dstncton between mechancal and psychologcal effects of electoral rules. We defne the mechancal effect of dstrct 10 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, by myopcally best respondng to prevous perod play. 11 There s not an agreement n the lterature on what s meant by a voter actng strategcally, hence our use of the word sophstcated. For nstance, smlar to us, Kawa and Watanabe (2013) and Abramson, Aldrch, Blas, Damond, Dskn, Indrdason, Lee, and Levne (2010) consder a voter to be strategc when hs/her acton maxmzes expected utlty (regardless of ths acton beng sncere or not); nstead, the poltcal behavour lterature only classfes a voter to be strategc when hs/her vote maxmses expected utlty and s not sncere.
magntude on votng behavor as the change n sncere behavor not caused by a voter changng 9 ther strategy but due to the ncreased lkelhood that a sophstcated vote s sncere. Instead, the changes n strategy are coned the psychologcal effect of dstrct magntude on votng behavor. Consder, for nstance, a sophstcated voter whose preferred party s ranked thrd: 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 but the way we classfy hs/her actons s. Followng ths dstncton, the results n Table 3 suggest that the small ncrease n sncere votng (as we ncrease dstrct magntude) s not drven by a mechancal effect but by a psychologcal one: the ncrease n dstrct magntude makes more partes vable so subjects are more lkely to vote for ther preferred party, even when takng nto account consderatons of pvotablty. Somethng that seems puzzlng n Table 3, though, s the huge dfference 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: the frontrunner. 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 s usually referred to as bandwagon votng and can be understood n terms of herdng (Nageeb and Kartk, 2012), nformaton aggregaton (Feddersen and Pesendorfer, 1997), or favorng a stable governng party (Rambau, 2013). However, there s no room for such ratonalzatons n our prvate values setup. 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 (.e. sncere and sophstcated votes always concde). In Fgure 1 we depct the data of our control treatment aggregated by each set of fve electons whenever the frontrunner acton does not concde wth the
10 sncere one. That s, we drop observatons n whch the subjects preferred party was the most voted party n the prevous electon and 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. Fgure 1. Frequences of Votes by Electon Round n the PR treatment when 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 her/hs sophstcated acton. Table 4 shows the subsample of observatons n whch the three types of actons do not concde. 12 M=1 M=2 M=3 % sncere 14.4 12.4 22.5 % sophstcated 70.5 53.7 42.8 % frontrunner 7.4 24.8 22.5 % other 7.7 9.2 12.3 observatons 312 218 138 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 effect 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 are now only consderng less than 7 percent of our observatons, the patterns we observed earler are somehow preserved n ths 12 Most studes n the lterature restrct ther analyss to the set of observatons n whch the type of behavour s dsjont (e.g. Duch, May, and Armstrong II, 2010)
subsample: sncere votng s greater when dstrct magntude s 3 rather than 1, and sophstcated 11 votng decreases. Possbly due to the ncreased complexty of the votng rule we see frontrunner votng and other behavor ncreasng wth dstrct magntude. 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 concde t s a 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 effects on votng behavor n our experment. The ncrease n sncere votng s (at least partly) drven by the mechancal effect of dstrct magntude. However, f only mechancal effects 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. Tables 3 and 4 both ndcate the heterogeneous effects 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 effect, 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 end up favorng the frontrunner canddate. In the next secton we analyze n detal ndvdual votng decsons, to understand whether dstrct magntude has a systematc effect on the ndvdual behavor of subjects. Our goal s to measure the relatve power of the three types of behavor for a representatve voter and to see how dstrct magntude nfluences the relatve weght of the dfferent motvatons.
12 4. Indvdual behavor In ths Secton we follow the same strategy that Duch, May, and Armstrong II (2010) use to account for coaltonal voters n 86 parlamentary democraces. Our ntal specfcaton assumes that subjects are ether sncere or sophstcated: the (unobserved) ndcator functon z soph t (sncere) way, and 0 otherwse. (zt sn ) takes value 1 f n round t subject behaves n a sophstcated Our goal s to estmate z t = (z soph t, z sn ) for all subjects and rounds,.e. for each of our treatments we want to dentfy what s the estmated probablty that t an acton s of each type. 13 The average of all z t across types and rounds s the uncondtonal probablty that each acton s of each type, whch could 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. Then: y jt = 1 f u jt max(u 1t, u 2t,..., u 5t) and s sncere y jt = 1 f v jt max(v 1t, v 2t,..., v 5t) and s sophstcated y jt = 0 otherwse where u jt = ū j + αu jt + ε u jt when zsn t = 1 and v jt = v j + βv jt + ε v jt when zsoph t = 1. We assume ε u jt, εv jt type I extreme value. We model the probablty that subject votes for j n perod t as a multnomal logt (MNL): p jt (u jt, v jt ; z t ) = p jt (u jt, v jt ; z t ) = eūj+αu jt J k=1 eūk+αu kt e v j+βv jt J k=1 e v k+βv kt f z soph t = 0 f z soph t = 1 Note that modelng utltes wth a MNL allows us to take nto account the 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. 13 Note that n ths frst specfcaton we have that z soph t = 1 z sn t.
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 (EM) algorthm. 14 Arcdacono, Seg, and Sloan (2007) elegantly summarzes what we do: gven ntal values of the parameters, we frst calculate the condtonal probablty that an acton 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, 13 Seg, and Sloan, 2007). 15 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. 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 assume that all observatons are ndependent. We also run the algorthm separately for each subject (panels B n tables below). 16 In ths case we also report the averages across all subjects. Ths second set of regressons are run as a robustness check and can be nterpreted as controllng for ndvdual fxed effects (we are estmatng z s for each subject separately). We can advance that results dffer 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 also report the percentage 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 14 See Dempster, Lard, and Rubn (1977) and Frühwrth-Schnatter (2006) for detaled descrptons of the algorthm. 15 As mentoned earler, our approach s analogous to Duch, May, and Armstrong II (2010) wth the only dfference that these authors use Bayesan methods whle we use the computatonally smpler EM algorthm. 16 Gven that we do not nclude frst round observatons when sophstcated and sncere concdes we have 48 observatons per subject.
14 algorthm to decde between two types when both are potentally correct. Note that there are no results for our fully proportonal treatment because n these cases both sncere and sophstcated behavor are equvalent (we also dropped the frst rounds when sncere and sophstcated votes concde). Note also that the percentage of observatons we correctly classfy s always qute hgh yet decreases wth dstrct magntude. Panel A: Whole sample M=1 M=2 M=3 % sncere 21.98 33.49 55.99 % sophstcated 78.02 66.51 44.01 % correctly predcted votes 87.24 76.79 68.60 observatons 2,304 2,400 3,360 Panel B: Average across subjects % sncere 30.56 36.24 51.50 % sophstcated 69.44 63.76 48.50 % correctly predcted votes 92.07 81.30 76.29 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. 17 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 F R = 1 z sn z soph. Formally, y jt s equal to 1 f n the prevous round party j obtaned the most votes, and s equal to 0 otherwse. The utlty of ths type of voter s gven by: p jt (u jt, v jt, F R jt ; z t ) = e γ j+αf R jt k=j f z F R k=1 e γ k+αf R t = 1 kt 17 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.
where F R j s a dummy that takes value 1 f j was the party wth most votes n the prevous round and 0 otherwse. 15 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 19.99 20.41 35.86 85.91 % sophstcated 71.04 63.73 53.89 % frontrunner 8.97 15.86 10.25 14.09 % correctly predcted votes 91.32 79.58 78.24 91.10 observatons 2,304 2,400 3,360 2,112 Panel B: Average across subjects % sncere % sncere 27.30 29.41 41.62 86.95 % sophstcated 63.67 55.82 38.50 % frontrunner 9.03 14.77 19.88 13.05 % correctly predcted votes 95.30 87.07 85.45 94.41 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 that 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 pattern n the number of observatons captures the mechancal effect of dstrct magntude: as dstrct magntude ncreases, there s a decrease n the proporton of observatons where sophstcated and and sncere actons do not concde. Table 7 reports results
16 for the subsample of dsjont observatons (the PR treatment observatons 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 Kolmogorov-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 dfferent than M = 1 and full PR. Panel A: Whole sample M=1 M=2 M=3 PR % sncere % sncere 14.47 21.16 20.24 86.34 % sophstcated 79.39 57.17 59.63 % frontrunner 6.14 21.67 20.13 13.66 % correctly predcted votes 79.10 75.69 63.77 92.17 observatons 312 218 138 1405 Panel B: Weghted average % sncere % sncere 13.95 21.96 20.67 86.67 % sophstcated 80.14 57.24 57.75 % frontrunner 5.91 20.80 21.57 13.33 % correctly predcted votes 78.71 76.46 63.55 91.87 observatons 312 218 138 1405 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 n the case of PR, when Sncere and Frontrunner do not concde. In Panel B, observatons are weghed by the number of tmes the subject appears n the dsjont sample. 4.1. 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. 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 dfferent weght on each component). Formally we have that: where p jt (u jt, v jt, F R jt ) = β sn β soph β F R e α j+β sn u jt+β soph v jt+β F R F R jt k=j k=1 eα k+β sn u kt+β soph v kt+β F R F R kt β sn σsn 2 σ snsoph σ snf R N β soph, σ snsoph σsoph 2 σ sophf R β F R σ snf R σ sophf R σf 2 R We assume that the dstrbuton of the vector of parameters β follows a multvarate normal dstrbuton. Below we report the estmated means of our mxed logt computatons: M=1 M=2 M=3 PR Mean β sn 2.97 1.92 3.78 14.99 (0.50) (0.26) (0.32) (1.20) Mean β soph 90.84 80.24 63.61 (9.57) (6.14) (6.68) Mean β F R 1.66 0.77 1.02 1.98 (0.15) (0.12) (0.11) (0.88) % correctly predcted votes 78.26 70.75 66.90 85.23 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 dfferent specfcatons are not drectly comparable. However, we can observe smlar trends to those we observed earler: sncere consderatons (as captured by β sn ) ncrease wth dstrct magntude n all cases apart from M = 1 to M = 2; sophstcated consderatons (as captured by β soph ) decrease wth dstrct magntude n all cases; and fnally, frontrunner consderatons are always present. Fnally, we would lke to hghlght the advantage of our types model and EM algorthm by comparng the predctve power (percentage of correct observatons) of dfferent approaches. Analogously to Duch, May, and Armstrong II (2010) we defne the Proportonal Reducton n Error from model A to model B (PRE) as:
18 PRE = #{correct predctons model A} #{correct predctons model B} N #{correct predctons model B} where N s the total number of observatons. We compare our approach to the more standard and prevalently used condtonal logt model, and also to the mxed logt model. We fnd that our approach does a much better job: PRE s consstently above 30% for all dstrct magntudes and both models, and up to 60% when DM=1 (see Table 9, below). M=1 M=2 M=3 PR Condtonal Logt 58.50% 30.50% 34.22% 47.21% Mxed Logt 60.07% 30.19% 34.26% 39.74% 2 types model 31.97% 12.02% 30.70% - Table 9. Proportonal Reducton n Error of our 3 types model wth respect to three other specfcatons. 5. Who s sophstcated? What drves sophstcated behavour? Followng our prevous dscussons, t seems reasonable that subjects n the frst past the post treatments would be the most lkely to be sophstcated gven that t s easer to dentfy where a vote s most pvotal. As we ncrease dstrct magntude, the number of stuatons n whch a vote can be pvotal ncreases and wth t the complexty of dentfyng the sophstcated acton. Below (see Table 10) we run a smple lnear regresson (wth clustered standard errors by ndvdual) where the dependent varable s the z soph we estmated for each subject n each electon. 18 We drop the PR and frst round observatons where, by constructon, sncere and sophstcated actons concde. We see that subjects are more lkely to be sophstcated as we decrease dstrct magntude. The postve (and sgnfcantly dfferent than 0) coeffcent n the varable perod captures the exstence of some learnng along each of the sessons, subjects learn to take nto account the pvotablty of ther vote and act accordngly. Towards the end of the experment subjects are almost 20% more lkely to act sophstcated than n the ntal perods. The varable tme captures the number of seconds subjects take n decdng ther vote: subjects that take longer tend to be more sophstcated. 18 We use the same specfcaton as n Table 6 panel A.
19 varables (1) (2) (3) (4) DM2-0.073*** -0.088*** -0.090*** -0.087*** (0.02) (0.02) (0.03) (0.02) DM3-0.171*** -0.166*** -0.167*** -0.155*** (0.02) (0.02) (0.020) (0.02) round 0.003 0.003 0.003 0.002 (0.003) (0.003) (0.003) (0.003) perod 0.001*** 0.001*** 0.001*** 0.001*** (0.0002) (0.0002) (0.0002) (0.0002) tme 0.002** 0.002** 0.002** (0.001) (0.001) (0.001) herfndhal 0.117 0.133 (0.086) (0.085) margn vote -0.008*** -0.007*** (0.002) (0.002) age -0.006** (0.002) male 0.003 (0.018) experence -0.007 (0.005) economcs -0.012 (0.033) non FPP 0.002 (0.023) Brtsh -0.029 (0.025) R 2 0.08 0.08 0.09 0.10 Observatons 8064 8064 8064 8064 Table 10. Lnear regresson on the estmated probablty of beng sophstcated the dependent varable s z soph as estmated n Table 6 panel A, standard errors (n brackets) are clustered at the ndvdual level, ***p<0.01, **p<0.05, *p<0.1 In column (3) we add two varables that capture the overall vote dstrbuton. On the one hand the coeffcent on the Herfndhal Index 19 shows that more concentrated votes n the prevous electon allow subjects to be more sophstcated. Ths mght be due to the ncreased salence of the sophstcated acton. Smlarly, the varable vote margn captures how contentous the fght for the last seat s and s negatvely correlated wth beng sophstcated. 20 Once agan, the smaller the vote margn, the more salent the sophstcated acton and the more lkely subjects are to target ther vote to the race between those two partes. Fnally, n the last column we add the socodemographc characterstcs we elcted at the end of the experment. 21 The only effect we observe s that younger subjects tend to be more sophstcated than the rest of subjects. 19 Ths ndex s commonly used to capture the concentraton n votes: t s the sum of each party s squared vote share. 20 Vote margn s the dfference of votes between the party that obtans a seat wth least votes and the party that does not obtan a seat wth most votes. 21 Most varables we nclude are self-explanatory: age, male, experence (number of experments the subject has partcpated n), economcs (whether the students has/s studyng economcs), non FPP (whether the
20 We are also nterested n fndng out whether actng n a sophstcated manner mpacts postvely n the payoff that subjects acheve. The fact that subjects are seldom pvotal makes us a lttle skeptcal about the fndngs of such an analyss, however, n table 11 we fnd that actng n a sophstcated way has a sgnfcantly postve effect on the payoff subjects obtan. varables (1) (2) z soph 10.578*** 10.076*** (2.516) (2.322) lagged payoff 0.679*** 0.698*** (0.016) (0.016) round -0.448-1.174** (0.481) (0.555) perod -0.115*** -0.109*** (0.038) (0.041) tme 0.285** 0.275*** (0.114) (0.102) DM2 4.378** (2.149) DM3 8.188*** (2.048) herfndhal 18.502 (13.256) margn vote 0.172 (0.513) age 0.136 (0.156) male 0.288 (1.434) experence 0.362 (0.416) economcs -1.114 (2.614) non FPP 1.623 (2.371) Brtsh 2.001 (1.955) Indvdual fxed effects YES NO R 2 0.52 0.52 Observatons 6864 6864 Table 11. Lnear regresson on payoff subjects obtan standard errors (n brackets) are clustered at the ndvdual level, ***p<0.01, **p<0.05, *p<0.1 In the frst column we run an ndvdual fxed effects regresson (wth clustered standard errors at the ndvdual level) and fnd that actng n a sophstcated manner (as captured by our estmated varable z soph ) has a postve and sgnfcant mpact on the payoff obtaned n that perod. Note that the fxed effects regresson mples that the postve coeffcent reflects that beng more sophstcated electoral system of student s country of orgn s not frst past the post), or Brtsh (54% of our subjects are Brtsh.
n a partcular electon correlates postvely wth a hgher payoff. We are not sayng that more sophstcated ndvduals make hgher profts but rather that per perod small varatons on the lkelhood of beng sophstcated leads to varatons n that perod s payoff. The postve and sgnfcant coeffcent on the lkelhood of beng sophstcated remans unchanged when we look at an alternatve specfcaton wthout fxed effects but wth soco-demographc and electoral system controls (column 2). We see that subjects that take more tme to vote usually obtan a hgher payoff. 21 6. Concluson 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 as they are n sngle-member dstrcts (see for nstance Abramson, Aldrch, Blas, Damond, Dskn, Indrdason, Lee, and Levne, 2010). One problem for emprcal research, however, s that t s almost mpossble to solate the effect of dstrct magntude on votng behavor ndependently of voters preferences, partes postons or expectatons about post-electon coalton formaton. We hence desgned a lab experment to solate ths effect, by varyng dstrct magntude whle keepng voters preferences and partes postons constant and excludng post-electon coalton calculatons. We found an ncrease n sncere votng as dstrct magntude ncreased. Ths was due to a mechancal effect: as dstrct magntude ncreased, the proporton of voters who found that ther most preferred party yelded the hghest expected utlty ncreased. However, not all of ths ncrease n sncere votng was due to a mechancal effect because sophstcated votng dd not ncrease wth dstrct magntude and many voters faled to vote sncerely. In other words, part of the ncrease n sncere votng was due to psychologcal effects. The source of these effects remans a frutful area for future research. We also found that there was a hgh ncdence (above 10% of our observatons) of votng for the wnner of the prevous electon (the frontrunner) n all our treatments. We even found ths effect n our control treatment wth fully proportonal electons. The presence of frontrunner votng has not receved much attenton n the emprcal and theoretcal studes of votng. However, as Hnch
22 (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), 61 90. 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), 37 65. Austen-Smth, D., and J. S. Banks (1988): Electons, Coaltons and Legslatve OutcomeS, Amercan Poltcal Scence Revew, 2(82), 405 422. (1996): Informaton Aggregaton, Ratonalty and the Condorcet Jury Theorem, Amercan Poltcal Scence Revew, 1(90), 35 45. Bargstad, M. A., and O. Kedar (2009): Coalton-Targeted Duvergeran Votng: How Expectatons Affect Voter Choce under Proportonal Representaton, Amercan Journal of Poltcal Scence, 2(53), 307 323. Blas, A., J. Aldrch, I. Indrdason, and R. Levne (2006): Do Voters Vote for Government Coaltons? Testng Downs Pessmstc Concluson, Party Poltcs, 6(12), 691 705. 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), 393 420. Carey, J., and S. Hx (2011): The Electoral Sweet Spot: Low-Magntude Proportonal Electoral Systems, Amercan Journal of Poltcal Scence, 2(55), 383 339. 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),
23 1(39), 1 38. Duch, R., J. May, and D. A. Armstrong II (2010): Coalton-drected Votng n Multparty Democraces, Amercan Poltcal Scence Revew, 104, 698 719. 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), 408 424. (1997): Votng Behavor and Informaton Aggregaton n Electons Wth Prvate Informaton, Econometrca, 5(65), 1029 1058. Feddersen, T. J., and A. Sandron (2006): A Theory of Partcpaton n Electons, Amercan Economc Revew, 4(96), 1271 1282. 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), 135 147. Fschbacher, U. (2007): Z-Tree: Zurch Toolbox for Ready-Made Economc Experments, Expermental Economcs, 2(10), 171 178. Fscher, A. J. (1996): A Further Expermental Study of Expressve Votng, Publc Choce, 1(88), 171 180. 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), 223 247. Frühwrth-Schnatter, S. (2006): Fnte Mxture and Markov Swtchng Models. Sprnger Seres n Statstcs, New York. Gerber, E. R., R. B. Morton, and T. A. Retz (1998): Mnorty Representaton n Mult- Member Dstrcts, Amercan Poltcal Scence Revew, 1(92), 127 144. 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), 135 140. Kawa, K., and Y. Watanabe (2013): Inferrng Strategc Votng, Amercan Economc Revew, 103, 624 62.
24 McCuen, B., and R. B. Morton (2006): Tactcal Coalton Votng and Informaton n the Laboratory, Electoral Studes, 3(29), 316 328. Myerson, R. B., and R. J. Weber (1993): A Theory of Votng Equlbra, Amercan Poltcal Scence Revew, 1(87), 102 114. Nageeb, A., and N. Kartk (2012): Herdng wth Collectve Preferences, Economc Theory, 51, 601 626. Pketty, T. (2000): Votng as Communcatng, Revew of Economc Studes, 1(67), 169 191. Reynolds, A., and M. Steenbergen (2006): The Poltcal Consequences of Ballot Desgn, Innovaton and Manpulaton, Electoral Studes, 3(25), 570 598. Rambau, G. (2013): Bandwagon n Israel? Note on Bargsted and Kedar (2009), mmeo. Sartor, G. (1968): Poltcal Development and Poltcal Engneerng. Cambrdge Unversty Press, New York, NY. Smon, H. A. (1954): Bandwagon and underdog effects and the possblty of electon predctons, Publc Opnon Quarterly, 18, 245 253.
25 7. Appendx 7.1. Instructons. (treatment M=2 wth 25 subjects) 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. 7.1.1. 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 2 9 2 7 5 Votes 3 As a result, partes B and D each obtan a canddate because 9 and 7 are the hghest numbers. Now consder a dfferent example where partes obtan the followng number of votes: Party A B C D E Votes 15 1 4 2 3 Votes 5 In ths second example, Party A obtans 2 canddates because 15 and 5 are the hghest numbers. 7.1.2. 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 500 1200 100 1800 500 So, f the 2 canddates from party A are elected, you obtan a proft of 1000 ( = 500 + 500). Alternatvely, f one canddate from party C s elected, and one canddate from party D s elected you obtan a proft of 1900 (= 100 + 1800). It s mportant to note that (a) your valuatons are dfferent from the valuatons of all other voters; and (b) that no other voter knows the valuatons of any other voter. 7.1.3. 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. 7.1.4. 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 15 2 1 7 0 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
26 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 15 2 1 7 0 Your Valuatons 500 1200 100 1800 500 6. 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 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.