Strategic Voting in Quorum Rules with Information Disclosure Theory and Descriptive Evidence

Transcription

1 Working Paper University of Chicago home.uchicago.edu/franava Strategic Voting in Quorum Rules with Information Disclosure Theory and Descriptive Evidence Francesco Nava 25/2/28 Abstract. The paper discusses optimal voting in quorum rules with endogenous timing of vote and information disclosure. When information is complete such rules favor voters opposing a reform, because they can manipulate outcomes by acting upon preferences in the committee. When information is incomplete and votes are unobservable to others, the ability of the opposing party to manipulate outcomes is limited. But when the timing of the vote is endogenous and participation is observable, opponents can again bias the consultation against the reform, even though no information on votes is ever disclosed. Such phenomenon can be mitigated by holding multiple consultations at once, since information on participation would no longer reveal information about preferences. Evidence from Italian referenda supports the predictions of the theory. A section explores why such rules are more susceptible to vote buying, when abstention is contractible. Contents Introduction Quorum Rules Equilibria with Information Disclosure Multiple Quorum Rules Abstention Buying Descriptive Evidence in Italian Referenda Conclusions Appendix

2 F. Nava Quorum Rules Introduction This paper investigates the extent to which strategic voting a ects outcomes of a quorum majority rule with endogenous timing of votes and information disclosure. A quorum majority rule is a binary rule in which a reform is enacted if a certain turnout requirement is met and if the majority of the votes submitted favors the reform. Such rules provide a simple analytical example of a strategic environment in which only a subset of voters in the committee can pro t from information disclosure. Turnout requirements are common practices both in plebiscites and in many parliamentary democracies. For instance, a quorum of 5 senators is required in the US senate for a vote to be valid. This analysis abstracts from the choice of the reform to focus upon optimal voting for a given proposed reform. Committee members endogenously choose when to vote at no cost. In this setup information on votes submitted early or turnout can potentially be disclosed before ballots close. I study how di erent information structures a ect strategic voting in such rules. Though several papers document the e ects of sequential voting with information disclosure, most these works take order of move in the committee as given. This study relaxes such assumption allowing players to choose when to vote. Committees considered are nite. Preferences of voters are uncertain and possibly correlated. To contain the multiplicity of equilibria, a symmetric perfect equilibrium re nement is adopted. The highest equilibrium ex-ante probability of rejection for the reform is used to measure manipulation in rules with di erent information structures. In quorum rules the maximal ex-ante equilibrium probability of rejection is weakly increasing in the level of disclosure, because the extent to which opponent can coordinate only can improve. If no information is disclosed before ballots close, such rules are susceptible to strategic abstention by voters opposing the reform. Indeed, conditions can be provided for which strategic abstention is the unique symmetric perfect equilibrium of this voting scheme. Sincere voting is not always an equilibrium in such rules. When either turnout or abstention are observable, the extent to which voters opposing the reform can manipulate outcomes increases. If those opposing the reform submit a ballot against the reform only when the quorum is met, the probability of rejection of the reform is maximized. Indeed, such strategy would cause any ballot submitted before the quorum is passed to reveal preferences in favor of the reform. Therefore, the extent to which ballots are secret would be compromised by equilibrium behavior if turnout were observable. Such phenomenon can be mitigated if committee members have to vote for several reforms at once and only turnout is observable. In such scenario the number of votes submitted conveys only partial information about whether the quorum passed. Therefore, those opposing a reform cannot perfectly condition their vote upon such information and will end up rejecting the reforms less often than if the two consultations were held separately. The extent to which manipulation is reduced depends on how correlated preferences for the two reforms are. If preferences among policies are perfectly correlated, manipulation is not reduced. However if preferences are not perfectly correlated, equilibrium manipulation falls. Quorum rules are susceptible to vote buying when abstention is observable. If voters have access to a transferable good and if abstention is contractible, there are incentives for those opposing the reform to pay some reformists to stay home. Such payments could be interpreted as reputational transfers amongst committee members playing an unspeci ed sequence of games. In such environment, equilibrium manipulation depends on the bargaining power of voters, which is determined by their preferences for the reform and their budget constraints. Strategic voting in quorum rules causes voting choices to di er from actual preferences. An updated versions will be posted regularly on home@uchicago.edu/~franava. 2

3 Quorum Rules F. Nava Even when sincere voting is an equilibrium of a quorum rule, the multiplicity of equilibria may make the identi cation of policy preferences form vote pro les impossible. Evidence from Italian quorum referenda held between 974 and 25 supports the conclusion of the theory. Speci cally, strategic abstention seems signi cant when a single consultation is held at a time. If several consultations are held at once strategic abstention is more pronounced when the topics of the di erent consultations are homogeneous. The empirical analysis is for the moment limited by of the relatively small sample size. Some stylized facts on quorum rules that the theory captures are: (i) disclosing information on turnout occasionally leads to reversals of relative majorities; (ii) incomplete information and lack of disclosure favor sincere voting; (iii) holding several consultations at once reduces the extent to which outcomes can be manipulated when turnout is observable; (iv) if utility is transferable and turnout observable, abstention buying can lead to reversals of absolute majorities; (v) the more information a vote reveals, the more turnout falls.. Related Literature [TBA].2 Road Map The paper is structured as follows: section 2 de nes notation and describes the Bayesian collective choice problems that lie in the scope of the analysis and characterizes quorum mechanisms with disclosure considered in the analysis. In section 3, behavior in perfect symmetric equilibria of the di erent rules is characterized and used to determine the extent of strategic manipulation. In this section predictions about timing vote are discussed for such binary rules. Section 4 shows that strategic manipulation in quorum rules with observable participation can be reduced by holding multiple referenda at once, if voters submit all their ballots at once. Section 5 discusses how results are a ected if a transferable good is introduced in the game. Indeed, if partecipation is obsevable and contractible, abstention buying could then arise since some individuals bene t from others abstaining. Section 6 discusses some descriptive evidence on 5% quorum rules coming from Italian single and multi-topic referenda held from 974 to 25. Section 7 concludes. Proofs can be found in appendix 8. 2 Quorum Rules This section describes the committees and the binary rules with endogenous timing of vote considered in the study. First, notation is introduced, then the model and properties of quorum rules are discussed. 2. Notation I() denotes the indicator function, (X) denotes the simplex of a nite set X and for x 2 X, let [x] = [Pr(x) = ] 2 (X) denote a degenerate distribution. 2 For any vector y 2 X n, let #(xjy) denote the number of components of y taking value x: 2 I(L) = if the predicate L is true and I(L) = if it is false. (X) = np 2 [; ] jxj P o x2x p(x) = for any nite set X. #(xjy) = P n i= I(y i = x) () 3

4 F. Nava Quorum Rules Then, for x 2 X, let #(y) = f#(xjy)g x2x. The following conventions are adopted to shorten notation throughout the paper: for i 2 N, let i = Nn fig and for any given collection of sets fh i g i2n and M N, let H M = i2m H i. Finally, when no confusion arises, sets are used in a superscripts to denote their cardinality. 2.2 Mechanisms: Commitment and Information Structures Consider the problem faced by a committee N of voters with heterogeneous and unknown preferences having to choose whether to implement a reform. Let S denote the set of possible types for each voter in the committee and p the common prior on the type space S N. Assume that outcomes are binary Y = f; g. Interpret outcome as the reform and outcome as the status quo. Preferences for each voter are described by a common map u : Y S! R. Each voter s payo does not depend on the types of the others. Types can however be correlated across players since p(s i ) 6= p i (s i js i ). In this setup any voter s type completely determines his preferences and beliefs about prefernces in the committee. The Bayesian collective choice problem fn; Y; S; p; ug is symmetric, except for the prior which needs not to be. In this section utility is non-transferable and no costs are incurred when voting. Such stance is chosen to highlight the e ects of di erent voting rules on outcomes in absence of complicating factors. Preferences in this setup depend on outcomes and types alone, not on voting choices. For any voter of type s 2 S, let d(s) = u(js) u(js) denote the utility di erence of the two policies. The set of types can be partitioned into those favoring a reform S = fs 2 Sjd(s) > g, those favoring the status quo S = fs 2 Sjd(s) < g, and those who are indi erent S a = fs 2 Sjd(s) = g. For any given pro le of types s 2 S N in the committee, the number of supporters of each option is given by: n (s) = P i2n I(d(s i) > ) & n (s) = P i2n I(d(s i) < ) Also let n a (s) = N n (s) n (s) denote the number of indi erents and n(s) = [n (s); n (s); n a (s)]. Because in quorum rules supporters of the status quo can take advantage of voters that are indi erent, results are more pronounced in committees in which there is an ex-ante positive measure of undecided voters. Therefore, assume that n a = P S n a(s)p(s) > [A]. 3 For any given voter i type of type s i the probability distribution over the number of voters of each type in the committee #(s) isgiven by: g i (njs i ) = P S i I(#(s) = n)p i (s i js i ), for 8n 2 N S Di erent types may well display the same preference order, but di erent information about the preferences in the committee, so long as such di erence is consistent with the common prior assumption. The last assumption invoked on the committee is that p displays su cient symmetry to have that g i (js i ) = g(js i ) for s i 2 S and any i 2 N [A2]. This condition holds when the prior is symmetric. Indeed it holds for the independent case. But in such setup all uncertainty about preferences in the committee vanishes as the size of the committee increases. Thus, the weaker assumption. A simultaneous "-quorum rule is an anonymous single-stage mechanism m = fv; wg. Let V denote the choice set for any voter i belonging to the committee and let w : V N! Y describes how pro les of votes are mapped into outcomes. The quorum rules studied in this section require agents to choose between a vote in favor of either policy and abstention 3 This implies that for some type i s i 2 N S it must be that P S i n a(s)p(s ijs i) >. The more restrictive assumption, P S i n a(s)p(s ijs i) >. for all i s i 2 N S, strengthens conclusions. 4

5 Quorum Rules F. Nava V = f; ; ag. Outcomes are implemented so that the reform is enforced when at least an " fraction of the electorate votes and if the simple majority of those voting is in favor of the reform. Formally, for any vote pro le v 2 V N : w (v) = I (#(jv) > #(jv)) I(N #(ajv) > N") The parameter " 2 [; ] characterizes the fraction of votes necessary to validate the consultation. It takes such mechanisms from simple majority, for " =, to unanimity, for " =. 4 Simultaneous mechanisms best describe environments in which all committee members have submit their ballots either at once or at di erent times, but secretly so that no information on choices can ever be inferred before the ballots close. Often however quorum rules are implemented without such stringent informational requirements. An alternative rule could allow members of the committee to choose when to vote and could have turnout to be observable. Consider an anonymous voting mechanism in which members choose endogenously in which of T time periods to submit their ballot. In such setting for V = f; ; ag choice sets at any of the t 2 T stages would be determined for any voter i 2 N by: V if vi:t = fa; ;g (v i:t ) = v i:t if v i:t 2 f; g Therefore if a vote is ever submitted, there is no option to reverse that choice at later stages. The set of feasible choice histories up to stage t 2 T for any voter i 2 N is given by V t = v t i jv i:r 2 (v i:r ); 8r t. History pro les belong to VN t = (V t ) N. Outcomes are still determined by an "-quorum rule, but applied to votes received before ballots close. That is for any v T 2 VN T allowing for an arbitrary number of stages: w T v T = lim t!t w (v t ) In principle such mechanisms can at each stage disclose some information about ballots submitted at earlier stages. Indeed, suppose that at any stage t 2 T a signal about early ballots is publicly observed. Denote the signaling map by: t : V t N The function t describes how at stage t histories of vote pro les in V t N are mapped into signals in Z. 5 Throughout the analysis of the multi-period mechanism perfect recall is assumed by all voters. Thus, at any given stage t, the information available to voter i consists of his type, of his private history of choices and of the public history of signals:! Z s i:t = (s i ; v i: ; :::; v i:t ; z 2 ; :::; z t ) 2 S i:t where set of possible information stages is denoted by S i:t = S V t Z t. Such a multiperiod mechanism with disclosure will be denoted by M = ft; V; ; ; w T g. Throughout the paper time is discrete. T is either nite or countable. A countable T can be used to approximate a strategic environment in which ballots have to be submitted before a speci ed point in time. Though having T be countable can in principle threaten existence of equilibria, it is shown that such quorum rules always possess at least one perfect symmetric equilibrium. Di erent information structures are considered: (i) a secret information structure 4 The mechanism and the collective choice problem induce the Bayesian game of incomplete information fn; V; S; p; ug, for u(js i) = u(w()js i). 5 The signaling structures considered are deterministic, but the setup can easily allow for random signals. 5

6 F. Nava Quorum Rules that for any v t 2 V t N sets t (v t ) =, (ii) an information structure with observable turnout that for any vote pro le v t 2 V t N sets: t (v t ) = N #(ajv t ) (iii) One with observable abstentions that conditional on v t 2 V t N has voters observing t (v t ) = fi(v it = a)g i2n ; and (iv) an information structure that discloses all predetermined votes t (v t ) = v t for any v t 2 V t N. 2.3 Sincere Voting & Strategic Abstention Before turning to equilibrium analysis some qualitative properties of an "-quorum rule are discussed. Quorum rules have two margins on which voters can be pivotal in determining the outcome of the consultation: majority and quorum. On both such margins one may be pivotal in favor of the reform by voting in favor of it. However in order to be pivotal against the reform the action to be taken depends on the margin considered: a vote against on the majority, abstention on the quorum. The equilibrium analysis shows how di erent information structures may a ect the perception by status quo supporters of which is the relevant margin to act upon. n n FA. Sincere Voting: acceptance & rejection region.5.75 N na N na FB. Strategic Abstention: acceptance & rejection region When voting is sincere, those supporting the reform vote in favor of it, those opposing it vote against it and indi erents abstain. In any quorum rule, if voting is sincere, there are committees approving the reform, even when its share of supporters is smaller than the quorum threshold, n (s) < N". In such committees status quo supporters, n (s), could prevent the reform from passing by collectively abstaining. Indeed, for a vote pro le v 2 V in which only reformists vote, #(jv) = n (s) and #(ajv) = N n (s), one would get that w(v) =. Thus, quorum rules provide incentives for coordination amongst status quo supporters either on abstention or on a vote in against the reform depending on the preferences in the committee and the quorum level. In fact, whenever " :5, collective abstention by status quo supporters always leads to outcomes that are weakly preferred by them. Whenever " =, a vote against is their only weakly dominant action. When preferences in committee are uncertain, members opposing the reform face a trade-o between coordinating on voting sincerely and loosing if the quorum passes because of their votes, and strategically abstaining and loosing whenever they were a relative majority and the quorum passed. 6

7 Quorum Rules F. Nava 3 Equilibria with Information Disclosure This section discusses the e ects of information disclosure on the outcome of a quorum rule with endogenous timing of vote. Whenever voters can submit their ballot at di erent time periods and turnout observable, strategic behavior in quorum rules can bias a consultation in favor of the status quo outcome. The highest ex-ante equilibrium probability of rejection for the reform is used to measure the extent to which mechanisms with di erent disclosure can be manipulated by voters opposing the policy change. In a simultaneous quorum rule a behavioral strategy pro le consists of a collection of maps i : S! (V ) one for each voter i 2 N. In a rule with multiple time periods, instead, a pro le of behavioral strategy T consists of a collection of maps i:t : S i:t! ((v i:t )) one for each voter i 2 N at each time period i 2 T. Thus, T characterizes how voters at each stage act upon known information s i:t 2 S i:t. Throughout the analysis it is assumed that indi erents always abstain. Such assumption is invoked because, even though voting costs are ruled out of the model, the possibility that some members of the committee prefer to abstain seems compelling and bears strategic consequences on behavior of the other committee members. Speci cally I assume that d(s i ) = implies that i (s i ) = [a] for any s i 2 S in the simultaneous case and that it implies i:t (s i:t ) = [a] for any s i:t 2 S i:t, i 2 N and t 2 T in the sequential case.[a] A symmetric perfect re nement is chosen to re ne amongst the multiplicity of equilibria. Let E SP denote the equilibrium correspondence. Mixed strategy symmetric perfect equilibria are not likely to arise in this setup, because the probability of being pivotal with equal probability on both the reform and the quorum margin will be very small for any prior and any symmetric strategy pro le. Thus, only pure strategy equilibria will be discussed. If all uncertainty about preferences in the committee is fully disclosed to voters at the interim stage before actions are taken,status quo supporters can by acting cohesively maximize the ex-ante probability of rejection for the reform. 6 Such equilibrium rejection probability is obtained by having status quo supporters vote against the reform only when reformists alone exceed the quorum threshold and abstain otherwise. When all reformists vote in favor of the change, following such strategy is indeed a perfect symmetric equilibrium of the full disclosure simultaneous move game. 7 The rejection region for the reform in such an equilibrium is depicted in gure 2 and consists of the union of the rejection regions discussed of the previous section. Indeed, for: P P () = s2s I(n (s) > N" _ n (s))p(s) the following holds true: Claim If information is fully disclosed at the interim stage there is a perfect symmetric equilibrium in which the ex-ante probability of rejection of the reform is equal to P () In no perfect equilibrium of the simultaneous rule could the ex-ante probability of rejection ever exceed such value, since those favoring a policy change would never be convinced not to 6 Full disclosure implies that the interim distributions are degenerate: p i(s ijs i) = for some s i 2 S N and 8i s i 2 N S. 7 The following strategy for a voter of type s i 2 S is a SPE of the full disclosure game, p(s ijs i) = : 8 < [] if s i 2 S F I (s i) = [] if s i 2 S \ n (s) > "N : [a] if otherwise 7

8 F. Nava Quorum Rules vote in favor of the reform. Moreover, such result does not hinge on the assumption that no information is disclosed before ballots close. In fact: Proposition 2 If utility is non-transferable, the ex-ante probability of rejection of the reform cannot exceed P () in any symmetric perfect Bayesian equilibrium of any multistage quorum rule. This follows again because in any equilibrium committee members favoring a policy change always vote in favor of it. n N na F2. Full Disclosure: acceptance & rejection Mechanisms with di erent information structures are compared by the maximal equilibrium value that the ex-ante probability of rejection for the reform can attain. This comparison is suggestive of the extent to which equilibrium outcomes can be manipulated in favor of the status quo by the resolution of uncertainty. For any behavioral strategy pro le de ne the ex-ante probability of rejection for the reform in the simultaneous rule by: P (j) = P P s2s v2v w (v) Q i2n i(v i js i ) p(s) The interim expected utility of committee member i when of type s i, is given by: U i (js i ) = u(js i ) + d(s i ) Pr i (j; s i ) Pr i (j; s i ) = P h P s i 2S i v2v w (v) Q i j2n j(v j js j ) p(s i js i ) Therefore, all that matters to voters is how their choices a ect the interim probability of the reform taking place. Since types fully describe individual preferences and beliefs about preferences of others, in any symmetric equilibrium it must be that if s i = s j, then j (s j ) = i (s i ) = (s i ). 8 Therefore identities cannot a ect beliefs about the likelihood of the reform being enacted. Indeed, for any s 2 S : Pr i (j; s i = s) = P l2n S h P v2v w (v) Q s 2S (v j js ) l(s ) i g(ljs) = Pr(j; s) 8 When s i = s j, by symmetry d(s i) = d(s j) and by assumption [A2] g(s i) = g(s j). 8

9 Quorum Rules F. Nava Denote a behavioral strategy in the interior of the simplex by o i : S! int((v )). When ballots are submitted simultaneously, perfection requires that for any voter preferring the reform votes in favor of it. Indeed, for s 2 S and any o i : SN! int((v )) N : 9 [] = arg max i 2(V ) U( i; o ijs ) since Pr(j ; o i; s ) Pr(j[v]; o i; s ) for 8v 2 V That is, because a reformist can be pivotal in favor of the reform only by voting for it both on the quorum margin N #(ajv i ) = "N, and on the majority margin #(jv i ) = #(jv i ), he better do so in any perfect Bayesian equilibrium of the simultaneous game. For the simultaneous rule such behavior guarantees that the ex-ante probability of rejection cannot exceed P F I (). A similar perfection argument shows that if s 2 S and any o i : S N! int((v )) N : (s ) 2 arg max U( i; o ijs ), [ (js ) = ] i 2(V ) since Pr(j ; o i; s ) > Pr(j i ; o i; s ) for 8 i 2 (V ) s.t. i (js ) > Hence, perfection requires no voter opposing the reform never submit a ballot supporting it. The only dimension left out of the characterization of the symmetric perfect equilibria of the simultaneous quorum rule is the probability of abstention for all types s 2 S. Such probability depends on the strategy followed by the other committee members that oppose the reform and on the interim beliefs about the distribution of types in the population. For any prior distribution on the type space there is a symmetric perfect Bayesian equilibrium in which all agents types s 2 S strategically abstain: [a] if si =2 S SA (s i ) = [] if s i 2 S This follows because if only reformist were to vote a non-reformist would never have an incentive to vote against the reform since the probability of being pivotal on the abstention margin would by far exceed the one of being pivotal on the majority margin. In fact, for any type s 2 S there exists > : [a] = arg max U( i; o ijs) for 8 o i 2 B ( SA ) N i 2(V ) B ( SA ) = o : S! int((v )) k o SA k + Since for any o i 2 B ( SA ) N and for bc denoting the oor operator, it must that: Pr(N #(ajv i ) = bn"c \ #(jv i ) > #(jv i )j o i; s ) > (2) Pr(#(jv i ) = #(jv i ) + \ N #(ajv i ) > N"j o i; s ) A complete argument is reported in appendix. Strategic abstention can be the unique symmetric perfect equilibrium of the simultaneous game with incomplete information. Whether 9 For Pr(j ; o i; s i) = P h P s i 2S i v i 2V i w (; v Q i i) j2 i o (v jjs j) p(s ijs i) and w (; v i) = w (v i = ; v i). The norm used is o j = sup + s2s sup o v2v j (vjs) (vjs). For x 2 R let bxc = max q2n [q s.t. q x] denote the biggest integer not exceeding x. 9

10 Reformists F. Nava Quorum Rules Not indifferent this is the case will depend on the prior and the type space. strategy pro le de ned by: 8 < [] if s i 2 S SV (s i ) = [] if s i 2 S : [a] if s i =2 S [ S Consider the sincere voting Such strategy gives rise to symmetric perfect equilibrium SV 2 E SP if and only if for any >, there exists o i 2 B (b ) N such that condition [2] holds with the reversed and possibly weak inequality. 2 Let de, bc denote the ceiling and the oor 3 operator respectively and n = d(bn"c + )=2e. If types were independently distributed according to p 2 (S ) the su cient condition for sincere voting to be a perfect symmetric equilibrium would entail: P bn=2c N! n=n n!(n )!(N 2n)! qn q n ( q q ) N 2n (3) P bn"c n=n N! n!(bn"c n)!(n bn"c )! qn q bn"c n ( q q ) N bn"c for q v = p (S v ) for 8v 2 V. Intuitively, the condition requires a status quo supporter to be more likely to be pivotal on the majority margin, than on the quorum margin given the equilibrium behavior of others. For a committee with 7 members and a quorum of =3, gure 3 depicts in red the values of p for which sincere voting is not an equilibrium of the simultaneous move game.when the quorum is high or when voters are su ciently informed about preferences in the committee, the maximal ex-ante rejection probability P () can be obtained even if information is incomplete and voting is secret. The condition on the quorum threshold requires that: Claim 3 If indi erents abstain, when " 2 [:5; ], the perfect equilibrium maximizing the exante probability of rejection of the simultaneous rule is strategic abstention, SA. Thus, Pr(j SA ) = P (). When " =, sincere voting, SV, is the unique perfect equilibrium of the simultaneous rule. Thus, Pr(j SV ) = P (). 2 If n a(s) < N 2N" is known and N 2N" 2 (; N], then sincere voting is always an equilibrium. 3 For x 2 R let dxe = min q2n [q s.t. q x] denote the smallest integer not below x.

11 Quorum Rules F. Nava On the other hand even when " 2 (; :5) it is possible to approximately attain such bound. If types were independent and the population is large, any uncertainty about the number of voters of each type would limited by the law of large numbers. 4 In this scenario as the population size increases, the following strategy converges to the maximal rejection probability and is an equilibrium of the simultaneous quorum game for any value of " > and N > : 8 < [] if d(s i ) > I (s i ) = [] if d(s i ) < & N"=(N ) < p (S ) p (S ) : [a] if otherwise Such strategy prescribes that those opposing the reform vote against it only when they believe that the quorum is likely to be met even without them. As claimed lim N" Pr(j I ) = P (), since all the relevant information is revealed in the limit. By independence such strategy is always an equilibrium, since status quo supporters always pool their actions given the common prior and since they vote against only when the chance of hitting the majority margin is grater than the one of hitting the quorum margin. 5 Additionally, note that in a simultaneous game with independent types and p (S [ S ) <, the upper-bound on the ex-ante probability of rejection, P (), never exactly attained. Indeed, there is always a positive probability that the reform passes even when it could be blocked in full disclosure regime, since voters opposing it can condition their choice only upon their imprecise knowledge of preferences in the population. Hence: Claim 4 If indi erents abstain, " 2 (; :5), types are independent and p (S v ) > for any v 2 V, there exists no perfect Bayesian equilibrium leading to the maximal the ex-ante probability of rejection, Pr() = max 2E SP Pr(j): < P (). If types are correlated and " 2 (; :5), a similar strategy may be constructed by replacing the second line with: I (s i ) = [] if d(s i ) < and N"=(N ) < E(n (s)js i ) E(n (s)js i ) for some. This strategy constitutes for almost every type space an equilibrium of the simultaneous rule. In this equilibrium those that oppose the reform submit a ballot only if they believe that the quorum will be met. By the common prior assumption and by assumption [A2], the strategy will lead to a substantial pooling amongst status quo supporters. The constant,, will in general exceed unity to take into consideration the extent to which miscoordination can arise within such group of voters. Mixed strategy symmetric perfect equilibria are not likely to arise in this setup, because the probability of being pivotal with equal probability on both vote margins is very small for any prior and any symmetric strategy pro le. If the interim probability distributions on types are su ciently correlated and if revelation is almost complete for all types, opponents can still approximately attain the maximal ex-ante rejection probability by conditioning their vote upon preferences in the committee. [ADD: su cient conditions for Pr() < P () when correlated!!!] Now consider multi-period rules. Recall that by assumption [A] indi erents always abstain. If no information can be disclosed while ballots are open, the previous results still hold. In fact, the secrecy of such mechanism ensures that no information other than one s type can be used to condition the vote at any stage. Thus, all equilibria of a secret quorum rule with endogenous timing of vote can be represented as equilibria of the simultaneous rule 4 By independence, the fraction of agents of any given type converges to the prior probality of that type. 5 Since condition [3] always holds, for q v = P s i 2S I (vjs i)p (s i) for 8v 2 V whenever p (S ) 2 (N"=N ; p (S )].

12 F. Nava Quorum Rules by considering the induced distribution on votes. Speci cally note that, if the rule is secret, utility and beliefs are de ned for any s i:t = ( t ; vi t ; s i ) 2 S i:t by: U i:t ( T js i:t ) = u(js i ) + d(s i ) Pr i:t (j T ; s i:t ) Pr i:t (j T ; s i:t ) = P s i 2S i Pv T 2V T w (v T ) Q Q rt j2n j:r(v j:r js j:r ) i:t (v t ; s i js i:t ) i:t (v t ; s i js i:t ) = I(v t i = v t ) Q Q r<t j2nni j:r(v j:r js j:r )p(s i js i ) i Therefore, the probability of the reform taking place is independent of the timing and information. Indeed, for any i 2 N, t 2 T and s i:t 2 S i:t : Pr i: (j T ; s i ) = Pr i:t (j T ; s i:t ) = Pr i (j; s i ) i (v i js i ) = P v T i 2V T I(v i:t = v i ) Q t2t i:t(v i:t js i:t ) for 8v i 2 V Consequently, the distribution on nal votes of any player is independent of others strategies. Additionally, only strategies T such that the corresponding is a perfect equilibrium of the simultaneous game, can be perfect equilibria set of the multi-period secret game. Thus, so long as rules are secret, it is without loss to assume that players vote simultaneously. In such mechanisms the timing of votes remains undetermined in equilibrium. For any multi-period quorum rule, it is easily veri ed that both abstention at the nal [limiting] stage and or a vote against the reform at any stage are weakly dominated strategies for anyone favoring the reform. Therefore, in all perfect equilibria all pro-reform types vote in favor of the reform at some stages with probability one. Similarly, whoever opposes the reform never votes in favor of the reform by perfection. Hence, in all symmetric perfect equilibria the only variables to be determined are the timing of vote for the reformist and the timing and probability of vote for non reformist. If the set of time periods is countably in nite and turnout observable, independently of the prior distribution there exists an symmetric perfect equilibrium in which behavior of those opposing the reform discloses all relevant uncertainty about population preferences. Indeed, if z t = N #(ajv t ) such outcome obtains if voters choose for any s i:t 2 S i:t according to: 8 < [] if s i 2 S \ t t(s i ) b t (s i:t ) = [] if s i 2 S \ z t > N( ") : [a] if otherwise Where t(s i ) denotes the nite stage at which voter type s i 2 S submits his ballot. This strategy pro le completely reveals n (s) in a nite number of periods with probability one. Therefore, strategy b T leads to the same ex-ante probability of rejection of a simultaneous full information rule. Therefore, for Pr T () = max T 2E SP Pr(jT ) and: 6 Pr(j T ) = the following claim holds. P P s2s v T 2V T w (v T ) Q Q t2t j2n j:t(v j:t js j:t (s; v t ))p(s) Claim 5 A countable time period quorum rule with observable turnout, z t = N #(ajv t ), always possesses sequential equilibrium in which the ex-ante probability of rejection of the policy is maximized, Pr T () = P (). 6 Since the information structure is deterministic, otherwise one would replace j:t (v j:tjs j:t) with P z2z j:t (vj:tjsj:t ; vj:t ; z)t(zjvt ). (4) 2

13 Quorum Rules F. Nava A similar result holds if one assumes that the game lasts a random number of periods and ends in a nite number of time periods with probability one. Indeed, since reformists fear not to be able to submit their ballot at the next stage, they may be led to an early vote and, hence, to reveal all relevant information. No additional information disclosure can be exploited from those opposing the reform to attain higher ex-ante rejection probability in any symmetric perfect equilibrium. In fact, b T remains one of equilibria leading to the maximal ex-ante rejection probability even when more disclosure takes place. However, having more information be disclosed may drastically a ect results when utility is transferable, since the set of contractable actions increases from fag to V. In rules with a nite number of time periods, backward induction e ects and the existence of simultaneous terminal stage, may prevent or reduce the extent of strategic manipulation. In fact those favoring the policy change may decide to all vote at the last stage to prevent any information disclosure in the game. However, such assumption is a suitable description only of environments in which all voter can submit their ballots simultaneously in a single instance. Many relevant empirical examples, though, violate such condition. [[More: Adding quorum reduces turnout... & nite time period rules]] 4 Multiple Quorum Rules This section discusses the extent to which strategic manipulation can be reduced by holding several consultations at once. If a single consultation was held at once and turnout was observable, voters opposing the reform could exploit the mechanism because they could condition their vote upon the knowledge that the quorum had passed. But if agents have to vote on several propositions at once and can only observe turnout (aggregate abstention), manipulation can in principle be reduced. Indeed, whether such expedient solves the manipulation problem, depends on the correlation among preferences for di erent policies. Intuitively, if preferences on di erent policies are not perfectly and positively correlated across voters, one would expect holding several consultations at once to increase turnout and reduce the extent of strategic abstention. Though results generalize, the analysis focuses on case in which two consultations are held at once. For simplicity, assume preferences on di erent policies to be separable [A3], but possibly correlated. Hence, u(yjs i ) = u A (y A js i ) + u B (y B js i ) for any y = (y A ; y B ) 2 f; g 2. Again, de ne the utility di erences for each policy for any type s i 2 S by d A (s i ) = u A (js i ) u A (js i ) and by d B (s i ) = u B (js i ) u B (js i ). For notational convenience, de ne the preference type sets by S uv = (Su A \ Sv B ), for u; v 2 V. For this setup, a possible generalization of the single-topic simultaneous rule consists of a rule (V ; w): v = (v A ; v B ) 2 V 2 y = (w(v A ); w(v B )) Such rule assumes that both reforms are faced with the same outcome map w and a common quorum threshold ". It, also, assumes that players can a ect the outcome of both topics separately by choosing amongst votes in V for both of them. Note that, if voterest have to submit ballots simultaneously, the separability assumption guarantees that voting on each topic according to some equilibrium of the single-topic rule is an equilibrium of the double topic rule. Therefore, equilibrium behavior would not be a ected in such scenarioo. Again, if the full disclosure at the interim stage is assumed in the simultaneous rule, it is possible 3

14 F. Nava Quorum Rules to construct equilibria in which the upper-bounds for the ex-ante probabilities of rejection of the two policies, P A () and P B (), attain. 7 Indeed, such upper-bounds obtain by voting on each topic separately according to [??]. 8 The corresponding multi-stage rule is de ned by: (v i:t ) = V 2 if v i:t 2 f(a; a);?g v i:t if otherwise w v T = w (v T ) for 8v T 2 V T (5) The mapping requires voter to either to abstain or to vote on all topics at once. Such assumption is central to the de nition of the multi-stage rule and carries consequences on equilibrium behavior. Again consider a rule with observable turnout and a countable number of time periods: t (v t ) = N #((a; a)jv t ) (6) This information structure displays the publicly number, but not the nature of the ballots submitted. Hence, voters at time t + know how many of them chose to abstain on both topics up to time t. Note that, for information structure, even if only reformists vote on any given topic and #((a; a)jv t ) < N", it can be the case that no reform passes the quorum. This was impossible in the single quorum rule. But as in single-topic quorums it must be that, if #((a; a)jv t ) N", agents know that none of the reforms has yet passed. Again, it must be the case that there exists no perfect symmetric equilibrium strategy of that can lead to a bigger rejection region for both reforms in an endogenous timing mechanism. [[We are working on a claim that guarantees that both probabilities of rejection for the reforms do not exceed the upper-bounds in any SPBE of a mechanism satisfying conditions [5] & [6].]] Before we turn to general result for the multi-stage let us characterize two extreme cases. Again let us assume that voters that are indi erent on any given policy abstain. In the rst, suppose that preferences are perfectly and positively correlated across policies. For instance, suppose u A (yjs i ) = u B (yjs i ) for any (y; s i ) 2 Y S. Given the assumption, preferences at any stage are: U i:t (js i:t ) = d A (s i ) [Pr i:t ((; )j; s i:t ) Pr i:t ((; )j; s i:t )] + [u (js i ) + u (js i )] If this were the case, the multi-topic rule could be reduced to and played as a single-topic one, given the correlation among preferences. In fact, the following symmetric perfect Bayesian equilibrium strategy would, trivially, attain the full disclosure rejection region, for s i:t 2 S i:t and z t = N #((a; a)jv t ): 8 < [; ] if s i 2 S \ t > t + (s i ) + t (s i:t) = [; ] if s i 2 S \ z t > N" : [a; a] if otherwise Hence, if preferences are positively and perfectly correlated, the maximal extent of manipulation cannot be reduced by holding the two rules jointly. In the second example, preferences are still assumed to be perfectly correlated, but negatively. For instance, u A (yjs i ) = u B (yjs i ) for any (y; s i ) 2 Y S. In this scenario, all 7 For P A P () = s2s I(nA (s) > N" _ n A (s))p(s) and P B P () = s2s I(nB (s) > N" _ n B (s))p(s) when n A (s) = P i2n I(dA (s i) > ) and n B (s) = P i2n I(dB (s i) > ) and accordingly. 8 By playing according to a : S! (V 2 ), de ned in appendix, consistently with the full disclosure strategy of the single quorum, opponents of each policy strategically enlarge its rejection region. 4

15 Quorum Rules F. Nava agents, except for the indi erents, care for one and only one reform to pass and preferences at any stage are given by: U i:t (js i:t ) = d A (s i ) [Pr i:t ((; )j; s i:t ) Pr i:t ((; )j; s i:t )] In this committee, for all reformists not committing to an action pro le with probability one before the game ends is a weakly dominated strategy. The multi-stage quorum rule for this committee possesses the following symmetric perfect equilibrium independently of the prior, for s i:t 2 S i:t : 9 8 < T (s i:t ) = : [; a] if s i 2 S \ t > t (s i ) [a; ] if s i 2 S \ t > t (s i ) [a; a] if otherwise Additionally, the above described strategic abstention strategy may be the unique equilibrium. In fact, remark that, in this example, both factions of reformists would bene t from the knowledge of the number of indi erents, n aa (s), because members could, by cohesively conditioning their actions upon that information, attain the maximal rejection region for the undesired policy. But if agents were to act upon the disclosed information by voting against depending on the number of voters, delaying commitments in order to receive better information may be optimal for all non-indi erent agents and prevent any other equilibrium from arising. In fact, because for all agents that are not indi erent, the equilibrium probability of not committing in a nite number of time periods is zero by perfection, any reformist would bene t from a deviation that delays his commitment. Also, note that sincere voting without conditioning actions on information may not be equilibrium, in part because of the reasons discussed in the single topic section. Hence, for preferences that are perfectly negatively correlated it may be the case that the double topic multi-stage rule with observable commitment totals never attains the maximal ex-ante probability of rejection for the policy. But this would not the case if n aa (s) = n aa (s i ) for any i:s i 2 N:S, because all reformist may, by only being aware of the number of indi erents, attain the maximal rejection probability for the respective policies. In general we have that: Remark 6 If p(s 2 [S [ S [ S aa ] N ) =, the upper-bounds, P () and P 2 (), always attain in a symmetric sequential equilibrium of a quorum rule with countable time period and observable participation. And similarly if p(s 2 [S a [ S a [ S aa ] N ) = or p(s 2 [S [ S [ S a ] N ) =. 2 Indeed for any prior satisfying such conditions the problem is equivalent to a single-topic quorum rule. Hence, the previous considerations on the upper-bounds and manipulation still hold. Now, consider an slightly di erent outcome map in which the quorum is determined by participation to any of the consultations. Such rule is strategically equivalent to the one discussed in the empirical section. For any v A ; v B 2 V let: w v A ; v B = I #(jv A ) > #(jv A ) I(N #((a; a)j(v A ; v B )) > N") The map di ers from the pervious one, only to the extent that the quorum is determined for both policies from the number of commitments rather from the number of votes on the given topic. Consequently, we rede ne the double-topic simultaneous and multi-stage maps as: y = (w v A ; v B ; w v B ; v A ) for 8v; v 2 V w v T = w (v T ) 9 The equilibrium is unstable to joint deviations. 2 Clearly, the same result holds for the reversed policy order. for 8v T 2 V T 5

16 F. Nava Quorum Rules For this outcome map in all perfect equilibria whenever agents choose to commit they will do so sincerely on both topics, since once the quorum margin is a ected, the majority margin is all that matters to them. This phenomenon depends on the speci c nature of the mechanism that prevents any policy from not passing the quorum margin if the other one does. Again this rule is equivalent to a single topic rule if conditions in are met. Let us assume as we did for the other rules that s i 2 S aa ) t (s i:t ) = [a; a] and that s i 2 S xa ) t (s i:t ) = [y; a] for some x; y 2 V. Perfection of all equilibria considered in the multi-stage rule requires that in any SPB equilibrium: 8 >< t (s i:t ) = >: [; ] if s i 2 S \ t > t (s i ) [; a] if s i 2 S a \ t > t (s i ) [a; a] if s i 2 (S [ S a [ S a ) \ z t N" [; ] if s i 2 S \ z t > N" [; a] if s i 2 S a \ z t > N" [; ] if s i 2 S \ z t > N" Lines two, ve and six of the above matrix hold respectively for reversed preference types. Hence, the only part of the equilibrium strategy not determined by these restriction is the behavioral strategy for types in S and S whenever the quorum is not met. The behavior of these preference types when the quorum is not met depends on the speci c nature of the committee But remark, that this mechanism, per se, tends to increase the correlation among the number of voters on the two policies, with respect to the previous rule, because strategic abstention cannot be exercised independently on both reforms. [[More: examples & results]] 5 Abstention Buying This section investigates the extent to which vote buying can a ect a consultation s outcome under di erent disclosure regimes. A good is introduced in the strategic environment to transfer utility amongst voters. The good enters linearly and separably the utility function of all members, 8i 2 N: u(y; mjs) = u(yjs i ) + m i All agents are endowed with a nite and non-negative amount of the good m. For any type pro le s 2 S, let m(s) 2 R N + denote the pro le of endowments for the committee. Each voter knows his own endowment, but not necessarily the one of others. Such goods will be used by voters to write contracts upon observable voting choices, namely abstention. Voters however will not be allowed to write contracts upon the outcome of the consultation. This assumption is invoked to highlight the di erence between rules that are secret and the ones that are not, since contracts on nal outcomes do not depend on information structure implied by the voting rule. In such setup if all voting choices were secret, no incentive compatible contract would ever be signed since there would be no action to be contracted upon. However, in rules with observable abstention incentives could arise for those favoring the status quo to pay voters in order to get them to abstain. To compare voting rules, bounds on the equilibrium rejection probability for the reform are constructed. The bounds will be de ned so to hold for any distribution of endowment in the committee. Such bounds will, then, be used to measure the extent to which di erent information structures can be manipulated through vote buying. To construct the upper-bound suppose that all information on preferences and endowments were fully disclosed at the interim stage. Consider the information structure with individually 6

17 Quorum Rules F. Nava observable turnout de ned in case (iii) of section 2. Under these assumptions there are pro les of endowments for which a reform is passed only if nobody opposes it and more than the quorum favor it. Indeed, if all the wealth belonged to a determined status quo supporter he could pay su ciently many voters to abstain to make sure that the quorum would never be met. Therefore the upper-bound for the ex-ante rejection probability for the reform is given by: P P () = s2s I(n (s) > N" \ n (s) = )p(s) Figure 4A depicts the rejection region associated to such probability. In this context, so long as having an additional voter abstain increases the interim rejection probability for the reform, there are incentives for abstention buying to occur. The full disclosure assumption allows directs payments to the correct individuals and reduces their value. Payments made to players will be small when the probability of any one agent being pivotal is small. For speci c type spaces the upper-bound P () can obtain with a single voting stage so long as abstention can be contracted. Thus, absolute majorities of reformists could be overturned by abstention buying in a mechanism with individually observable turnout. [DO EXAMPLE?] In a rule with individual public abstention also reformists have incentives to buy votes, but less so. In fact, reformists can only pro t from writing contracts requiring others not to abstain, since any contract that requires a vote in favor of the reform is unfeasible. To construct the upper-bounds on the ex-ante probability of acceptance again assume full disclosure at the interim stage. Di erent cases that had identical behavioral implications when committee member could not be paid to vote need to be separated to construct such bound. In the rst scenario the vote of indi erents can be bought. In the second case the vote of indi erents cannot be bought. In the last scenario voters are allowed to abstain even in the poll and thus voting is not contractable. When indi erents can be paid to submit a ballot, the bound on the probability of acceptance depends on how such players choose to vote when paid to show up. If indi erents vote in favor of the policy the smallest rejection region is depicted in red in gure 4B. If, instead, they vote against the reform the rejection region is the union of the green and red regions. Given the behavior of indi erents, such regions consist of all pro les of preferences for which no possible vote split amongst those favoring the status quo can ever be exploited to pass the reform. To construct the bounds consider the case in which all wealth belongs to a su - ciently determined reformist. Such voter could implement the depicted bounds in equilibrium by buying abstention and turnout from the appropriate number of opponents and indi erents. Indeed, for the two cases respectively the bounds on the ex-ante acceptance probability are given by: P + () = P s2s I(n (s) > N"=2)p(s) P () = P s2s I(n (s) > maxf; N"=2 n a (s)g)p(s) In the second scenario indi erents cannot be bought. This assumption can be interpreted as requiring that there be a fraction of the electorate that simply cannot go to the ballots. To construct the bound again consider the case in which all wealth belongs to a determined reformist. Such voter can guarantee that the reform be passed by buying abstention and voting from the appropriate number of opponents whenever the number of reformists exceeds half the quorum and su ciently many votes can be bought. Therefore, in this scenario the highest ex-ante probability of acceptance for the policy would be given by: P 2 () = P s2s I(n (s) > N"=2 \ n a (s) < N Figure 4C depicts the rejection region associated to such probability. N")p(s) 7