EUI Working Papers ECO 2007/07

Transcription

1 EUI Working Paper ECO 2007/07 Fixing the Quorum: Repreentation veru Abtention Sanne Zwart

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3 EUROPEAN UNIVERSITY INSTITUTE DEPARTMENT OF ECONOMICS Fixing the Quorum: Repreentation veru Abtention SANNE ZWART EUI Working Paper ECO 2007/07

4 Thi text may be downloaded for peronal reearch purpoe only. Any additional reproduction for other purpoe, whether in hard copy or electronically, require the conent of the author(), editor(). If cited or quoted, reference hould be made to the full name of the author(), editor(), the title, the working paper or other erie, the year, and the publiher. The author()/editor() hould inform the Economic Department of the EUI if the paper i to be publihed elewhere, and hould alo aume reponibility for any conequent obligation(). ISSN Sanne Zwart Printed in Italy European Univerity Intitute Badia Fieolana I San Domenico di Fieole (FI) Italy

5 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION SANNE ZWART MAY 2007 Abtract. The majority of the participating voter in referenda doe not necearily reflect the majority of the whole population ince voter can abtain. Thi paper how that a quorum exit for which the outcome of the referendum coincide with the population preference. However, a econd equilibrium can exit in which the propoal i alway rejected. When inufficient information make the optimal quorum unknown, it i in general more harmful to et the quorum too high than too low. Robutne of the reult i analyzed by allowing preure group to encourage or dicourage participation after the quorum i et. Keyword. deciion, voting rule. Electoral engineering, quorum, referendum, voting/not-voting JEL Claification. D72. Sanne.Zwart@eui.eu. Department of Economic, European Univerity Intitute, Via della Piazzuola 43, Florence, Italy. I would like to thank Giancarlo Coretti, Aitor Erce, Ayşen İaoğlu, Karel Merten, Marku Pochke, Pontu Rendahl, Karl Schlag and Nicola Sauger for contructive uggetion. All remaining error are mine.

6 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 2 1. Introduction In June 2005 a referendum wa held in Italy to block a fertility law. The law banned reearch uing tem cell from embryo and impoed tringent requirement on tet-tube pregnancie. Adverarie of the law initiated the referendum to aim for it abrogation. To ucceed, a quorum of 50% wa to be met and a majority of the participating voter had to upport the abrogation. Thi gave advocate of the law two different poibilitie to avoid abrogation: i) encouraging no-voter to take the effort to vote o that they would form the majority; ii) dicouraging no-voter from voting o that the quorum would not be met and the referendum would be invalid. In Italy, the advocate of the law choe for the econd option, for example the peaker of the Senate and the Chamber of Deputie a well a the Roman Church dicouraged people from voting. The New York Time (2005) write that Italian prelate have told parihioner to head to the beach intead of the polling place on Sunday and Monday, o that the quorum will not be met. The trategy ucceeded, the turnout wa too low and the referendum wa invalid. Since 26% of the population voted, while almot 90% of the participating voter were in favor, 23.4% of the population wa in favor and voted. When a handful of people in favor wa dicouraged by the forecat of an invalid referendum, encouraging no-voter to cat their ballot could indeed have led to a valid referendum in which the abrogation would have been approved. Referenda are becoming increaingly widepread in democratic countrie (Water (2003) and Matuaka (2005) dicu recent trend, ee alo the web ite of The Initiative & Referendum Intitute). One of the main reaon i the wih to give voter a direct ay in the iue at take. An additional reaon might be that direct democracy would contribute to voter involvement with and trut in the political ytem. However, referenda are known to be imperfect deciion making tool in the ene that a counter-intuitive relationhip between the voter preference and the outcome can occur. Nurmi (1998) lit variou voting paradoxe, including problem temming from multiple propoal or multiple alternative and the poibility of conflicting opinion between the majoritie of the voter and their repreentative. A the referendum in Italy how, a quorum give rie to an additional potential problem by giving opponent of change an additional tool to reach their aim. Fihburn and Brahm (1983) call thi the no-how paradox. The objective of thi paper i twofold. In the firt part we addre the quetion whether there i any theoretical upport for impoing a quorum in a referendum. The focu of the econd part i on the robutne of the reult. More pecifically, we firt look at the magnitude of the ditortion when the quorum i et either too low or too high and then at the impact of preure group which can affect the voter turnout after the quorum i et. The role of the quorum i analyzed in a tylized referendum model with heterogenou voter. The exitence of a quorum make the turnout a deciive variable for determining the outcome. But even for referenda without a quorum, the voting/not-voting deciion i

7 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 3 an important apect of explaining the outcome. Analyzing thi deciion uually lead to the concluion that people who vote do not form a repreentative ubet of the population. For example, Fort and Bunn (1998) find for referenda concerning nuclear power that actual participation ha more explanatory power for the ye/no deciion than both economic and preference variable. Succefully navigating the hurdle of regitering, going to the booth etc. made a no-vote more likely. In the model, thi aymmetry between opponent and proponent i reflected by their poibly different probabilitie of voting. In the firt part of the paper we how that with the appropriate choice of the quorum and the default outcome that occur if the quorum i not met, the population majority outcome can be attained. To ee how the referendum hould be deigned, uppoe that proponent are more likely to cat their ballot then opponent. In order to offet the bia toward accepting, the default outcome need to be rejection. A higher quorum need more participating voter. To be precie, it need a higher fraction of ye-voter in the population ince they are more likely to vote. A higher quorum thu reduce the cae where the majority of participating voter i in favor while the majority of the population i not. The population majority outcome i attained for the quorum for which they equal. Interetingly, when voter care more about the outcome when they are participating, the optimal quorum doe not necearily lead to the population majority outcome. A econd equilibrium can exit in which the default outcome alway occur. In thi cae, the referendum clearly i an imperfect tool for deciion making. The econd part of the paper analyze the robutne of the reult in two way. When the ocial planner ha inufficient knowledge about the population parameter or inufficient political power to et the quorum at it optimal level, a non-optimal quorum can arie. When the default outcome i et correctly, we how that etting the quorum too low i le harmful than etting it too high. The reaon i that the default outcome will alway occur when the quorum i too high, while when the quorum i too low both outcome might till occur. Since in mot real-life application there i not much flexibility in etting the quorum, thi finding implicate that only topic for which both ide have a high expected turnout hould be ubjected to referenda. A non-optimal quorum can alo arie when preure group have the poibility to affect the turnout after the quorum i et, like in the Italian referendum dicued above. When the default outcome i rejecting the propoal, ye-preure group hould alway encourage people to vote. For no-preure group it i optimal to encourage voter to participate only if it i likely that there are relatively many no-voter, otherwie they hould be dicouraged from voting. Since the bai of democracy i that all people are equally important, we conider the preference of the population majority a the benchmark outcome. We thu abtain from ocial welfare conideration that balance an optimal outcome with the cot of repreentation. The model can eaily be adapted to addre different intenitie of voter

8 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 4 preference. In cae a quorum i exogenouly impoed to guarantee a certain level of repreentativene, the econd part of the paper can be read a analyzing the difference between the referendum outcome and the population majority outcome. We aume that participation i voluntary, a compulory voting would trivially reult in the population majority outcome (however, Franklin (1999) and Jakee and Sun (2006) raie argument againt compulory voting). Theoretical upport for the importance of the population majority outcome follow from the axiomatization of May (1952) a the only voting rule that i deciive, anonymou, notfavoring any of the outcome and poitively reponding (i.e. when one voter change opinion then the group deciion become more favorable toward that opinion). However, when voter can abtain from participating, Côrte-Real and Pereira (2004) find that in general no voting rule that i independent of the abtainer preference can achieve the population majority outcome. They how that thi outcome can be achieved if in the cae of a turnout below the quorum, the underlying reaon determine the outcome. In the equilibrium etting of thi paper model, thi interpretation of an inufficient turnout i done ex ante when the referendum i deigned. The model i baed on the deciion-theoretic approach initiated by Down (1957). Voter participate in the referendum when they receive a poitive net utility from voting. Following Riker and Ordehook (1968) and in line with empirical evidence dicued extenively by Blai (2000), the net utility of voting depend on the outcome of the referendum, the cot of cating the ballot and a conumption benefit that repreent the fulfillment of a voter civic duty. The main difference between their and our model i how a voter derive utility from the outcome of the referendum and from participating. In their model, they conider the benchmark of a utility function that i linear in the outcome of the referendum. However, there might be nonlinearitie involved with repect to the outcome and participation. More pecifically, the utility of the referendum outcome might depend on whether a voter ha participated or not. On top of thi, when there are many potential voter, the probability that a particular voter action i deciive i almot zero. Myeron (2000) derive etimate of the order Hence, unle the utility difference between the outcome i extremely large relative to the cot of voting, the nonlinear effect might be far more important. It i not clear what the direction of thi nonlinear effect hould be: there are convincing argument for all poibilitie. When it i zero the outcome of the referendum doe not affect a voter participation deciion. When it i negative, a voter exhibit an underdog-mentality: the le likely her preferred outcome, the more likely he will vote. When the nonlinear effect i poitive, a voter like to be part of the winning ide. In thi paper we conider all type. Moreover, we how that if all type can occur imultaneouly, the average type drive the reult.

9 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 5 Although the literature on voting i vat, there are few paper on referenda. Herrera and Mattozzi (2007) dicu a group-baed referendum model. A in thi paper, the turnout of each group i endogenou. However, intead of having the referendum outcome directly affecting the voter utility, their group weigh the cot of increaing the turnout with it effect on the referendum outcome. They find a quorum paradox : the equilibrium turnout might only exceed the quorum if the quorum i not impoed. Myatt (2007) dicue a model in which a finite number of privately informed voter have to choe between two alternative that are preferred to the tatu quo. In contrat with the model of thi paper, trategic voting can occur when a voter fear that her mot preferred alternative will not receive ufficient upport. Marquette and Hinckley (1988) and Kanazawa (1998) ugget that a voter recall of previou election i alo relevant for current turnout. Cloely related to the model of thi paper, Kanazawa (1998) propoe to ubtitute the Riker- Ordehook probability regarding the current election with the probability that the voter preferred outcome occurred when he participated in pat election. Hence, intead of computing the probability that her preferred candidate win a in thi paper model, a voter ue an etimation baed on pat experience. The outline of the paper i a follow: Section 2 preent the model, Section 3 how that there i a quorum for which the population majority outcome can occur and analyze it propertie, Section 4 addree the robutne of the reult by conidering a notoptimally et quorum and allowing for preure group. Appendice A and B contain precie formulation of claim made in the main text. Proof are deferred to Appendix C. 2. The Referendum Model 2.1. The Referendum. A referendum i held in order to decide whether a propoal hould be accepted or rejected. Each voter ha three option: i) to vote in favor of the propoal; ii) to vote againt it; iii) not to vote. Voter who do indeed vote are called participating voter. The referendum i only valid if a quorum i met, that i if more than a certain fraction of the voter i indeed voting. The propoal i accepted if the referendum i valid and if the majority of the participating voter i in favor. 1 When the quorum i met but a majority of the participating voter i againt, the propoal i rejected. In cae the referendum i invalid, a preet default outcome determine whether the propoal i accepted or not. Although in ome real-life referenda the default outcome i not explicitly et, in mot cae it i rather clear what will happen when the referendum i not valid. For example, in the referendum about the European Contitution in the Netherland there wa no formal default outcome. Though, all major political partie were in favor and it wa clear that the European Contitution would be accepted in cae the quorum would 1 When the intenitie of the voter preference differ, a qualified majority can be ued to protect a minority from the majority, ee Appendix A for detail.

10 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 6 not be met. In thi paper, deigning a referendum i thu chooing the quorum and default option. There i a continuum of voter with meaure one. Each voter know whether he i in favor of the propoal or againt it, but there i uncertainty about the overall fraction of voter in favor of the topic. 2 The aumption that the preference of voter are endogenouly determined i rather tandard. However, Roema (2004) dicue the pychology of voting and find that poible election outcome are ued in the deciion what to vote. Making voter preference endogenou though, jutifie reearch on it own and i outide the cope of thi paper. Hence, denote by y the proportion of voter in favor of the propoal. The very reaon that a referendum i needed, i that the value of y i unknown. Hence, y i a random variable which take it value in an interval [y, y] [0, 1]. The ditribution of y i common knowledge. Thi can be the cae if for example forecating agencie provide correct projection when not everyone ha made up her mind yet. The proportion of voter in favor ha full upport on [y, y]. The model i not relevant when the majority i either alway in favor or alway againt, o it i aumed that y < 1 2 < y. When the proportion of ye-voter y were obervable, no referendum i needed to have the propoal accepted or rejected according to the majority of the voter. Thi benchmark cae i referred to a the population majority outcome. To be precie, denote by A the event that the propoal i accepted and by R = A c the event that it i rejected. The population majority outcome i then defined a A when y > 1 2 and R when y < 1 2. When y = 1 2, the population majority outcome precribe both A and R with probability 1 2. However, for notational convenience A i precribed but we aume that thi cae doe not occur, i.e. P[y = 1 2 ] = 0. Since voter have the poibility to abtain from voting, the proportion of ye-voter y i not directly obervable. Thi paper analyze whether a referendum can be deigned in uch a way that the population majority outcome alway occur The Voter. A voter who i in favor of the propoal i referred to a a ye-voter, a voter who i againt the propoal a a no-voter. The typical ye-voter will be indicated by index i and the typical no-voter by index j. Whether a voter will indeed participate depend on her net benefit of doing o. A voter participate in the referendum if her net utility of doing o i poitive. In our model, thi net utility of voting ha the form propoed by Riker and Ordehook (1968). A in their model, the net utility conit of three term: i) a cot of voting; ii) a conumption benefit from the act of voting and iii) a utility from the outcome of the referendum depending on it probability of occurrence. The main 2 It i poible to allow for voter who are indifferent with repect to the propoal by auming that thi group ha a fixed ize and that due to a lack of motivation thee voter do never participate in the referendum.

11 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 7 difference between their model and mine i how the utility depend on the outcome of the referendum. A voter who decide to indeed cat her vote, incur a cot c > 0 repreenting the effort to go to the ballot box. Since there i a continuum of voter, the impact of a ingle voter on the outcome i nil. If voter were only concerned about the trategic benefit of voting and it cot, thi would lead to the well-known paradox that none of the voter would take the effort to cat a ballot. Cultural theorie of voting argue that the incorporation of civic engagement eliminate the paradox. In an empirical tudy, Blai, Young and Lapp (2000) find upport for thi hypothei. In explaining voter turnout, the cot of voting and a return depending on the outcome of the referendum matter, but only among the voter with a relatively weak civic engagement. In the model thi civic engagement i a moral preure to vote that differ acro voter. Let m i be the moral preure of ye-voter i. The moral preure of a ye-voter ha a uniform ditribution on the interval [ m y α 2, my + α 2 ] o that the average moral preure of ye-voter i given by m y. Similarly, aume that the moral preure of no-voter ha a uniform ditribution on the interval [ m n α 2, mn + α 2 ]. The moral preure i felt a a diutility when a voter i not voting. Since there are no trong argument why ye- and no-voter hould have differently haped moral preure ditribution, they are taken a identical. Hence, the caling parameter α that determine the within-group heterogeneity i the ame for both ide. The average moral preure though can be different. Thi allow for the propoal to unequally affect the ye- and no-voter, o that one ide might be more inclined to vote. Different average moral preure can thu caue a bia toward accepting or rejecting the propoal. The dependence on the outcome i modeled in the following way. A ye-voter want the propoal to be accepted and derive utility in thi cae. The utility a ye-voter derive from acceptance of the propoal can depend on whether the voter indeed participate in the referendum or not. Let the utility of an accepted propoal for a participating ye-voter be γ v, while it i γ nv for a non-participating ye-voter. 3 Similarly, when the propoal i rejected, a participating no-voter derive utility γ v while a non-participating no-voter derive utility γ nv. For γ v > γ nv, voter derive more utility from their preferred outcome when they have participated. When the revered inequality hold, a voter like her preferred outcome bet when it occur without coting her any effort. If γ v γ nv, the additional bia toward accepting or rejecting the propoal might either offet or trengthen the bia temming from different average moral preure. 3 Thi i equivalent to the more elaborate modelling where diutility i derived from rejection of the propoal. For example, when participating ye-voter derive utility β va P[A] in cae of acceptance and β vr P[R] in cae of rejection, the total utility i (β va β vr )P[A] β vr. Defining γ v a β va β vr and noting that the contant can be aborbed by recaling of m y, a will be made clear below, give the reult.

12 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 8 voting not voting net utility of voting Riker-Ordehook thi model utility of ye-voter i γ v P v [A] c γ nv P nv [A] m i γ v (P v [A] P nv [A]) + (γ v γ nv )P nv [A] c + m i γ v (P v [A] P nv [A]) + (γ v γ nv )P nv [A] c + m i γ v (P v [A] P nv [A]) + (γ v γ nv )P[A] nv c + m i Table 1. Riker and Ordehook (1968) aume that the utility of the outcome doe not depend on participation, o γ v = γ nv. However, when the impact of a ingle voter i nihil, the probability of acceptance P[A] i independent of voter i participation and the outcome only affect the participation deciion through difference between γ v and γ nv. The outcome of the referendum i unknown when the voter have to make their deciion. The ex ante expected utility thu depend on the probability of acceptance or rejection. Theoretically thee probabilitie can depend on whether a voter participate or not, o denote the probability of acceptance by P v [A] when a voter participate and by P nv [A] when he doe not. For a ye-voter, the expected utility derived from the outcome of the referendum i thu γ v P v [A] or γ nv P nv [A] depending on whether he i participating or not. The utilitie of a ye-voter are ummarized in Table 1, for a no-voter identical expreion hold when the probability of acceptance i replaced by the probability of rejection. 4 The net utility of voting i hown in the third line. The firt term i a utility difference caued by voter i impact on the outcome, the econd term i a utility difference due to different valuation of the outcome when a voter participate or not. Econometrician would call the latter an interaction effect. It capture nonlinearitie that arie from the participation and the outcome. Riker and Ordehook (1968) aume that the utility of the outcome doe not depend on the voter deciion, o γ v = γ nv. The outcome thu only affect voter deciion through different probabilitie of acceptance. However, the probability that a particular voter i pivotal i extremely mall when the population i large. For example, conider a population of 5 million voter of which 50.1% i expected to be in favor. Fedderen (2004) ue a formula derived by Myeron (2000) to find etimate for the probability of a pivotal vote of the order Thi how that even when γ v and γ nv are cloe, different valuation of the outcome may be far more important than the utility difference caued by the voter impact. Although voter tend to overetimate 4 We implicitly aume that whenever a voter cat her ballot, he vote according to whether he i in favor or againt. In other word, all voter are incere. It i neceary to aume thi ince each voter i atomitic and her deciion i not affecting the outcome. However, incere voting i guaranteed when the voter morality lead to a large negative utility when he vote for the non-preferred outcome.

13 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 9 their impact, a for example found be Blai et al. (2000), their biae hould be of a very high order to outweigh the effect of different valuation. To focu on how different valuation affect the referendum outcome, we abtain from the mall impact of a ingle voter by auming a continuum of voter. Hence, no trategic concern are incorporated in the deciion making proce at the individual level. 5 The probability of acceptance doe not depend on the voter action and i denoted by P[A]; the probability of rejection i then P[R] = 1 P[A]. The expreion of the net utility how that the level of the utilitie derived from acceptance or rejection are not relevant for the behavior of the voter, only their difference matter. Define γ = γ v γ nv a the exce utility of the preferred outcome of voting relative to not-voting. It i not clear what the ign of γ hould be, or even whether it hould be non-zero. We hence do not make any aumption and dicu the model for all poible value of γ. When γ = 0, the outcome of the referendum i not relevant for the deciion of a voter whether to vote or not. For thi reaon we refer to thee voter a imple-hearted voter. When γ < 0, the outcome of the referendum will give a higher utility when the voter doe not cat her vote. Thi capture the feeling of a voter who like her preferred outcome bet if he doe not have to do anything for it to occur. A higher probability of her preferred outcome make a voter le willing to vote. Thi reemble the underdog effect reported by Levine and Palfrey (2007) in a laboratorial experiment: voter upporting the le popular alternative have higher participation rate. Another way of interpreting thi behavior i uggeted by Haan and Kooreman (2003). For a finite number of voter they how that the ide with the highet number of upporter can till be the mot likely to loe due to free-riding behavior. When γ < 0 voter balance their moral preure with the outcome of the referendum, and we therefore refer to them a calculating voter. When γ > 0, the more likely it i that the preferred outcome will occur, the more likely a voter will participate. Thi repreent a voter who want to be part of the winning team: the higher the probability of winning, the more likely he want to take action to upport it. Thi i in line with the expreive voting model of Schueler (2000) in which benefit from attachment to a collective lead to a preference for the winning party. For example, Ahworth, Gey and Heyndel (2006) find evidence that although in Belgian municipal election turnout i highet when the larget party obtain a mall majority, turnout i again timulated when there i a clear winner with at leat two third of the vote. Further upport that ome voter want to be a winner i given by Bartel (1988) who how that the public opinion before US preidential election tend toward the winner of the mot recent primary election. Remarkably, Clauen (1968) find that in pot-election recall urvey the winning candidate upport i overetimated and conclude that apparently 5 In Section 4 we will give interet group the poibility to coordinate the individual. Thi allow individual to indirectly trategically affect the outcome.

14 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 10 too many people remember to have contributed to the victory. together when γ > 0, we refer to them a affectionate voter. Since voter cluter The above expreion how that the cot c of cating the ballot can be aborbed in the mean moral preure m y and m n. Without lo of generality, the expoition of the model can thu focu on the cae c = 0. It alo how that there i an alternative interpretation of the model in which all voter have the ame moral preure, but differ in their cot of voting Equilibrium. Since all voter have the ame information, they make the ame inference about P[A] and P[R]. For notational convenience we aume that when a voter i indifferent between voting or abtaining will vote. An equilibrium can then be characterized by two witching point γp and γr uch that ye-voter i only vote if m i γp, no-voter j only vote if m j γr, P[A] = p and P[R] = r. Since p + r = P[A] + P[R] = 1, an equilibrium i fully characterized by p. To find the equilibria, it thu uffice to analyze for all p [0, 1], whether p P[A] = 0 when the ye- and no-voter witching point are γp and γ(1 p) repectively. Let Y = P[m i γp] denote the probability that ye-voter i will vote. Invoking the law of large number, ee Judd (1985), Y alo denote the proportion of ye-voter who are voting. Hence, Y will be referred to a the propenity to vote of ye-voter. Similarly, define the propenity to vote of no-voter N = P[m j γ(1 p)]. Then { { m y + α 2 Y = min max + γp } }, 0, 1 = 12 { { m y α + min + γp max, 1 }, 1 }. (1) α 2 2 A imilar expreion hold for N. Note that Y and N are both function of p. When the proportion of ye-voter equal y, the meaure of participating ye-voter i given by yy and the meaure of participating no-voter by (1 y)n. The participation rate i thu given by yy + (1 y)n. When q [0, 1] denote the quorum, the referendum i valid if yy + (1 y)n q. Thi i the quorum condition. When the referendum i valid, the propoal i accepted if the majority of the participating voter i in favor, o if yy (1 y)n (for notational convenience the propoal i accepted when exactly half of the voter i in favor). Thi i the majority condition. In cae the referendum i not valid, the preet default outcome D {A, R} determine the outcome. Table 2 relate the probabilitie of accepting the propoal with the propenitie to vote and the quorum. Suppoe that the default outcome i rejecting the propoal, D = R (the cae D = A follow from ymmetric argument). Firt uppoe that ye-voter are more likely to participate than no-voter, o Y > N. A higher proportion y of ye-voter make a valid referendum more likely ince more voter will actually vote (a ye-voter i more likely to vote than a no-voter), and it make it more likely that the propoal i accepted (there are more participating ye-voter). When the quorum i below 2N Y/(Y + N), the quorum i relatively eaily met and the majority condition determine the probability

15 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 11 condition contraint P[A] Y > N and q 2NY Y +N majority P[y N Y +N ] Y > N and q 2NY Y +N quorum P[y q N Y N ] Y = N both P[y 1 2 ]1 {Y q} Y < N both P[ q N Y N y N Y +N ] Table 2. Binding contraint and the probability of accepting the propoal when the default outcome i rejection. of acceptance (note that for q = 2NY/(Y + N) the majority and quorum contraint coincide). For a higher quorum intead it i determined by the quorum contraint. Now uppoe that Y < N. A higher fraction of ye-voter y make a valid referendum le likely ince le voter will actually vote (a ye-voter i le likely to vote than a no-voter), but if the referendum i valid it i more likely that the propoal i accepted (there are more participating ye-voter). Both contraint are binding, the quorum contraint from above, the majority contraint from below. Note that when Y = N, the quorum can only be met if q Y = N. In thi cae the probability of accepting i determined by the majority condition. An equilibrium in cae D = R i thu a olution of p P[A] = 0, where P[A], Y and N are a dicued above. Thi equilibrium characterization i at the core of the analyi. 3. The Quorum and the Population Majority Outcome 3.1. Simple-Hearted Voter. Suppoe that the voter are imple-hearted, o γ = 0. The expectation about the outcome of the referendum do not affect the voter deciion whether to vote or not. Thi implie that the choice of the quorum doe not affect the propenitie to vote. Any bia that tem from different average moral preure can thu be directly addreed by a quorum. The following propoition tate that with the right choice of the quorum and the default option, the population majority outcome occur. Propoition 1. (Simple-Hearted Voter and the Population Majority Outcome) Aume that γ = 0 and m y, m n ( α 2, α 2 ). i) When m y = m n, the population majority outcome i only achieved in the unique equilibrium of the referendum with a quorum of at mot q = my + m n 2α and default outcome D {A, R}. ii) When m y m n, the population majority outcome i only achieved in the unique equilibrium of the referendum with quorum q = my + m n 2α and default outcome D = R if m y > m n and D = A otherwie.

16 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 12 In order to dicu the implication of the propoition, it i inightful to look firt at the propenitie to vote. The condition that m y and m n are contained in ( α 2, α 2 ) implie that they are given by Y = my /α and N = mn /α and that they are contained in (0, 1), ee Equation (1). Thi aure that on each ide ome voter do abtain from voting while other cat their vote. It hence exclude the le relevant cae where all voter of a ide vote or all of them do not vote. The firt tatement of the propoition aume that the propenitie to vote are equal for ye- and no-voter. Obviouly, a majority of ye-voter in the whole population, y 1 2, will then lead to a majority of ye-voter among the participating voter. The participation rate i contant and equal to yy + (1 y)n = Y = N. In thi cae, any quorum below or equal to the propenity Y or N i automatically met and the default outcome i free to chooe (in the propoition the average propenity 1 2 (Y +N ) i ued to tre the imilarity with the optimal quorum in the econd tatement). Since the majority of the participating voter perfectly reflect the majority among the population, the population majority outcome i achieved. Note epecially that the quorum q = 0 i allowed, which i identical to the cae of not having a quorum. Intuitively, when the propenitie to vote are equal, there i no bia toward accepting or rejecting the propoal and no quorum i needed. However, ince the participation rate i contant, any ufficiently low quorum doe no harm. The econd tatement aume that the propenitie to vote are different. With the found expreion for Y and N, the optimal quorum can be expreed a the average propenity to vote 1 2 (Y + N ). To ee why thi i the cae, aume that m y > m n (ymmetric argument hold for the oppoite cae). Thi aumption implie that Y > N. Ye-voter are more likely to vote and without a quorum there i a bia toward accepting the propoal. When a quorum i introduced, it can only offet thi bia if the default outcome i rejecting the propoal, D = R. The participation rate yy + (1 y)n i trictly increaing in y. Thi how that a majority of the population i in favor of the propoal, y 1 2, if and only if the participation rate i higher than 1 2 (Y + N ). The population majority outcome can thu be achieved by the quorum q = 1 2 (Y + N ). Note that the majority contraint i redundant: whenever the referendum i valid, a majority of the participating voter i in favor of the propoal. Intead of the fraction of participating voter in favor, the participation rate i the deciive variable. The model thu ha a trong prediction: for a correctly et quorum the default outcome will never occur a the outcome of a valid referendum. At firt ight it might eem counterintuitive that the optimal quorum i increaing in the propenity to vote of both ye- and no-voter: the bia toward accepting i increaed when ye-voter become more likely to vote, but it i decreaed when no-voter become more likely to vote. An increaed bia might need a higher quorum and a decreaed bia a lower quorum. Thi reaoning correctly aee the effect on the bia in the abence of a

17 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 13 quorum. However, when the optimal quorum i impoed, the previou paragraph howed that the majority contraint i redundant. An increae in the propenity to vote of yevoter ha an identical effect on the quorum contraint a an increae in the propenity to vote of no-voter. More voter will indeed vote, o the quorum i more likely to be met and the probability of accepting the propoal i increaed. To achieve the population majority outcome, an increae in the quorum i needed Calculating Voter. Now uppoe that the voter are calculating, o γ < 0. The potential diutility of an unneceary vote make that le voter indeed take the effort to cat their ballot compared to the imple-hearted voter. Ceteri paribu, thi lead to a lower optimal quorum. To contruct a referendum that achieve the population majority outcome, the probability of a majority of ye-voter among the whole population i needed. Let ξ denote thi probability, o ξ = P[y 1 2 ]. From the aumption on the ditribution of y it follow that ξ (0, 1). The following propoition tate that with the right deign of the referendum, the population majority outcome occur. Propoition 2. (Calculating Voter and the Population Majority Outcome) Aume that γ < 0 and m y, m n ( α 2 γ, α 2 ). i) When m y = m n + γ(1 2ξ), the population majority outcome i only achieved in the unique equilibrium of the referendum with a quorum of at mot q = my + m n +γ 2α and default outcome D {A, R}. ii) When m y m n + γ(1 2ξ), the population majority outcome i only achieved in the unique equilibrium of the referendum with quorum q = my + m n +γ 2α and the default outcome D = R if m y > m n + γ(1 2ξ) and D = A otherwie. The intuition for the propoition follow again from firt looking to the propenitie to vote. In the population majority outcome the probability that the propoal i accepted i given by ξ. The probability that the propoal i rejected i then given by 1 ξ. Thi mean that the propenitie to vote of ye-voter and no-voter are given by Y = 1 2 +( my +γξ)/α and N = ( mn + γ γξ)/α repectively. The condition that m y and m n are contained in ( α 2 γ, α 2 ) implie that for all ξ (0, 1) the propenitie to vote Y and N are between 0 and 1. In other word, the condition enure that for a fraction γ/α of the voter indeed their voting deciion depend on their expectation (that γ < α follow from the ame condition). The firt tatement of the propoition now claim that when the propenitie to vote are equal for ye- and no-voter, the referendum with a quota below or equal to 1 2 (Y + N ) achieve the population majority outcome. The reaon i the ame a for the imple-hearted voter: with equal propenitie to vote the fraction of ye- and no-voter among the participating voter are identical to the population fraction. No quorum i needed, but a ufficiently mall quorum doe not affect the outcome of the referendum ince the participation rate i contant.

18 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 14 When the propenitie are not equal, according to the econd tatement a quorum i needed to achieve the population majority outcome. In fact, the optimal quorum i again the average of the propenitie to vote, but now evaluated at the equilibrium, q = 1 2 (Y + N ). To get more intuition, aume that m y > m n + γ(1 2ξ) (ymmetric argument hold for the oppoite cae). Thi implie that Y > N. Similar to the model with imple-hearted voter, a quorum with rejecting a default outcome, D = R, i needed to offet the bia toward accepting. The participation rate yy + (1 y)n i trictly increaing in y. The majority of the population i in favor if and only if the participation rate i higher than 1 2 (Y + N ). Since in thi cae the ye-voter contitute a majority, the quorum q = 1 2 (Y + N ) achieve the population majority outcome. Compared to the model with imple-hearted voter, there are two important difference. Firtly, ceteri paribu the optimal quorum i lower in cae of calculating voter. Comparing the expreion for q in the econd tatement of Propoition 1 and 2 how that in the model with calculating voter the quorum i γ/α lower. Some of the voter who would have cat their ballot when they would have been imple-hearted, prefer not to do o when they are calculating. A lower quorum i needed to offet a lower participation rate. Thi how that when the referendum i deigned for a population of imple-hearted voter while intead the voter are calculating, the quorum i et too high. In cae m y > m n + γ(1 2ξ), the quorum will only be met when the true proportion of ye-voter i at leat y for y > 1 2. The propoal i thu rejected for y [ 1 2, y ). When P[y [ 1 2, y )] > 0, the referendum with the incorrectly et quorum will not achieve the population majority outcome and there i a tendency toward the default outcome R. A econd difference compared to the model with imple-hearted voter i that the deign of the optimal referendum require knowledge of ξ = P[y 1 2 ]. Somewhat urpriingly, thi knowledge i not needed for etting the optimal quorum. Intead, the knowledge of ξ i needed for etting the default outcome optimally. Intuitively, for the optimal quorum only the um of the reduction in voter matter, while for the optimal default outcome the difference matter. When γ = 0 the propenity to vote i independent of the expectation. However, when γ < 0 the propenitie to vote will in general depend on γ. Only when a population majority of ye- and no-voter i equally likely, o ξ = 1 2, the default outcome coincide with thoe in cae of imple-hearted voter. When ξ 1 2, there will be fewer participating ye- and no-voter in equilibrium than in cae of imple-hearted voter. When ξ > 1 2, the decreae in ye-voter i larger than the decreae in no-voter. The choice of the default outcome need to take account of thi effect. The term γ(1 2ξ) in the condition accomplihe thi. Thi effect i increaing in the extent to which voter calculate, γ. Note that the model with imple-hearted voter can be een a the limiting cae of the model with calculating voter and γ 0.

19 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION Affectionate Voter. Now conider the model with affectionate voter, o γ > 0. The expectation about the outcome of the referendum again matter. But now the higher the probability that the preferred outcome occur, the more likely that a voter indeed cat her ballot. Ceteri paribu, thi lead to more participating voter and hence to a higher optimal quorum than in cae of imple-hearted voter. Compared to thoe voter, the affectionate voter have a tendency to behave in a coordinated way. Thi give raie to the poibility of multiple equilibria. The following propoition tate that although the referendum can be deigned uch that the population majority outcome occur, under a certain condition there i indeed another equilibrium. Propoition 3. (Affectionate Voter and the Population Majority Outcome) Aume that γ > 0 and m y, m n ( α 2, α 2 γ). i) The population majority outcome i achieved in an equilibrium of the referendum deigned a pecified in Propoition 2. ii) For the quorum q, the equilibrium mentioned in i) i the unique equilibrium when m y m n γ, otherwie there i a ingle alternative equilibrium which i characterized by P[D] = 1. The firt tatement how that the expreion for the optimal quorum in cae of calculating voter alo hold for affectionate voter. Compared to the model with imple-hearted voter, the optimal quorum i higher with affectionate voter ince voter are more likely to participate. Comparing the expreion for the optimal quorum of the three model how that the quorum i increaing in the extent of affection γ (or decreaing in the extent voter calculate γ). The propoition tate that multiple equilibria can indeed arie. The econd tatement claim that when m y and m n are ufficiently cloe to each other, the optimal quorum doe not necearily lead to the population majority outcome. 6 In fact, thi quorum can dicourage the opponent of the default outcome from voting, an effect that i aggravated by the tendency to coordinate. Thi might give raie to an equilibrium where none of the voter expect the quorum to be met and becaue the voter adapt their behavior to thi expectation, the quorum will indeed never be met. When m y m n < γ the fact that voter bae their deciion to vote on expectation together with their tendency to coordinate give rie to elf-fulfilling equilibria. When intead the difference between m y and m n i ufficiently big, the equilibrium with P[D] = 1 i not feaible anymore. To ee why, uppoe m y m n + γ. Even when P[R] = 1 the propenity to vote of ye-voter i (weakly) higher a that of no-voter. There will be a poitive probability of accepting the propoal, which i a contradiction. 6 In cae m y = m n + γ(1 2ξ) and q < q the equilibrium can be unique, but there can alo be two other equilibria, ee Appendix B for detail.

20 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 16 1 p P[A] 1 p P[A] 0 ξ Y (p) = N(p) 0 ξ ξ 0 1 p Figure 1. In cae of calculating voter the optimal quorum lead to a unique equilibrium with the population majority outcome (P[A] = ξ). ξ Y (p) = N(p) 0 1 p Figure 2. In cae of affectionate voter the optimal quorum can alo lead to a econd equilibrium in which the propoal i never accepted (P[A] = 0). A graphical repreentation provide additional inight in why the equilibrium i necearily unique for the calculating voter but not for the affectionate voter. In Figure 1 and 2, p P[A] i hown a function of p for calculating and affectionate voter repectively. Recall that in equilibrium p P[A] = 0. In cae of calculating voter, γ < 0, the propenity to vote Y = ( my + γp)/α i decreaing in p. The propenity to vote N = ( mn + γ γp)/α i increaing in p at the ame rate. The participation rate for y = 1 2 i thu independent of p. But a dicued above, for the optimal quorum only the quorum contraint i binding. Thi implie that for all p the quorum contraint i alo atified if and only if y 1 2. For mall p the probability of accepting the propoal i then ξ until the no-voter are more likely to participate than ye-voter. In thi cae the quorum contraint and the majority contraint cannot be imultaneouly met and P[A] = 0. The function p P[A] i thu trictly increaing and ha a un upward jump. Since it i increaing, i croe the x-axi at mot once. The choice of the default outcome implie that the jump i after ξ, o that indeed an equilibrium exit. In cae of affectionate voter, γ > 0, Y i increaing in p and N decreaing. Argument oppoite to the one above how that P[A] i zero for mall p, while it jump to ξ for larger p. Thi implie that p P[A] i not trictly increaing in p. There can be two equilibria: one with P[A] = 0 and one with P[A] = ξ. The choice of the default outcome guarantee that the latter equilibrium exit. When m y m n < γ, ye-voter have a lower propenity to vote than no-voter for p = 0. Thi implie that the quorum contraint and the majority contraint cannot be imultaneouly. Since then P[A] = 0, there i a econd equilibrium in which the default outcome alway occur.

21 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION Heterogenou Voter Type. We now allow for heterogenou voter. To be more pecific, the population can conit of imple-hearted, calculating and affectionate voter. Moreover, the parameter α and γ can differ acro voter. Thi mean that a voter k i defined by her preference, i.e. in favor or againt the propoal, and the parameter ( m k, α k, γ k ). Define the parameter et P = R (0, ) R. Now define the ubet ˆP of P a follow ˆP = { ( m, α, γ) P ( m α 2 + max{0, γ}, α ) } 2 + min{0, γ}. Note that thi retriction reemble the aumption on m y and m n in Propoition 1-3. In fact, for any parameter ( m k, α k, γ k ) ˆP the aumption in the propoition indicated by γ k i atified for m k, α k and γ k. Denote the ditribution function of the parameter of ye-voter i by Φ y and of no-voter j by Φ n. By the law of large number, Φ y and Φ n are alo the population ditribution. Denote the denity function by φ y and φ n repectively. The firt condition on the denity function i that φ y ( m k, α k, γ k ) = φ n ( m k, α k, γ k ) = 0 if ( m k, α k, γ k ) / ˆP. Thi aure that of all the ye- or no-voter with a type ( m k, α k, γ k ) that can occur, ome will indeed vote while other will not. average parameter of the ye-voter [ ] m y mi = E y = α i [ ] γ y γi = E y α i = ˆP ˆP m i α i dφ y ( m i, α i, γ i ), γ i α i dφ y ( m i, α i, γ i ). Now define the following Denote the counterpart for the no-voter by m n and γ n. The econd condition on the denity function i that γ y = γ n. Since thi i equivalent to E y [γ i /α i ] = E n [γ j /α j ], thi condition i atified if for example m k and (α k, γ k ) are independently ditributed and the denity function for (α k, γ k ) i independent of being in favor or againt the propoal. The common denity function i the analogue of the aumption made in the previou ection that γ i a population parameter and that the caling parameter α of the moral preure ditribution i equal for both voter group. Although thi aumption i mainly made to keep the model tractable, there are no reaon to aume that γ y and γ n are very different. When they are cloe to each other, the outcome will be imilar to when they are identical. Define γ = γ y = γ n. The econd condition implie that both the average type, i.e. implehearted, calculating or affectionate, and the extent of the affection (or the extent to which voter are calculating) caled by α are equal among ye- and no-voter. The following propoition claim that knowledge of thee average parameter together with ξ = P[y 1 2 ] i ufficient to deign a referendum that achieve the population majority outcome.

22 FIXING THE QUORUM: REPRESENTATION VERSUS ABSTENTION 18 Propoition 4. (Heterogenou Voter and the Population Majority Outcome) Aume that the upport of Φ y and Φ n are contained in ˆP and that E y [ γ i α i ] = E n [ γ j α j ]. Then, the quorum, default outcome and uniquene of the population majority outcome are a in the model with only the repreentative voter type defined by ( m y, 1, γ) and ( m n, 1, γ). The propoition tate that when the population conit of imple-hearted, calculating and affectionate voter and when the other parameter are allowed to vary acro the voter, the quorum and default option hould be et a for the population that only conit of the repreentative voter type ( m y, 1, γ) and ( m n, 1, γ). Hence, the analyi in the firt three ubection i not a implification but intead decribe model with heterogenou voter type a well. When the ign and ize of individual γ k can be different, an increae in p ha different effect on voter with different γ k. In cae of different ign, it make ome voter more willing to vote and other le. Only the average effect count for etting the optimal quorum. Note that the repreentative voter type alo determine whether the optimal quorum necearily reult in the population majority outcome or that the equilibrium with P[D] = 1 can occur a well. 4. A Non-Optimal Quorum In thi ection we analyze the conequence of a non-optimal quorum. There are two reaon why a non-optimal quorum can arie. Firtly, the quorum could have been et non-optimally due to inufficient knowledge about the relevant parameter or for political reaon. Secondly, after the quorum i et, whether optimally or not, preure group have incentive to affect the behavior of voter in order to make their preferred outcome more likely. Throughout it i aumed that the proportion of ye-voter y ha a uniform ditribution on [y, y] with y < 1 2 < y. Let φ denote the denity, o φ = (y y) 1. The probability of accepting the propoal according to the population majority i then given by ξ = φ(y 1 2 ). The analye for the default outcome A and R are ymmetric. We aume D = R o the propoal can only be accepted when the referendum i valid and when a majority of the participating voter i in favor A Not-Optimally Set Quorum. Firt conider the imple-hearted voter with γ = 0. The outcome of the referendum doe not affect the behavior of the voter o the propenitie to vote Y and N are fixed. When it i known which contraint are binding, the probability of accepting the propoal can be computed in a traightforward manner uing the three cae conidered in Subection 2.3. Denote thi probability by p m when only the majority contraint i binding, by p q when only the quorum contraint i binding and by p b when both contraint are binding. Let denote the um of the propenitie to vote, o = Y + N. Thee probabilitie of accepting the propoal given the binding