GROUP FORMATION AND VOTER PARTICIPATION

Transcription

1 GROUP FORMATION AND VOTER PARTICIPATION HELIOS HERRERA (ITAM) AND CESAR MARTINELLI (ITAM & UPENN) Abstrct. We present mobiliztion model of two prty winnertke-ll elections with endogenous voter group formtion: gents decide whether to be followers or become leders nd try to bring people to vote for their preferred prty-cndidte. The model gives closed form solution nd uniquely determines the number of leders in equilibrium. Expected turnout nd winning mrgin in the election re predicted s function of the equilibrium number of leders, their bility to mobilize voters nd the importnce of the election. Keywords: Voter s Prdox, Endogenous Leders. JEL Clssi ction: D7 Si nos hbitudes nissent de nos propres sentiments dns le retrite, elles nissent de l opinion d utrui dns l societe. Qund on ne vit ps en soi, mis dns les utres, ce sont leurs jugements qui reglent tout Jen-Jques Rousseu, Lettre M.d Alembert (758) Most people re other people. Their thoughts re someone else s opinions, their lives mimicry, their pssions quottion Oscr Wilde, De Profundis (95). Introduction Economists tend to view socil phenomen from two di erent perspectives. When deling with situtions tht re rife in strtegic interctions, such s mrkets with few prticipnts, or brgining mong smll group of people, we fvor gme theoretic models. When deling with situtions where individuls in uence in the environment is negligible, such s lrge, nonymous mrkets, we fvor competitive models. While this division serves us generlly well, there re socil phenomen which re not esily ssigned to either perspective. Consider, for instnce, elections with very mny potentil voters. A fully strtegic tretment of voters behvior in lrge elections is possible, but its implictions re disppointing. If voters re motivted only by the e ect of their ctions in the result of the election, nd there is but Dte: First Version: My th 4, This Version: 3th Jnury 5. Helios Herrer, ITAM-CIE, helios@itm.mx,

2 GROUP FORMATION AND VOTER PARTICIPATION slight cost involved in the ct of voting, gme-theoretic models predict dismlly low turnout, s long s voters re somewht uncertin bout the preferences of others. This is clerly t odds with mss prticiption in elections. A fully competitive tretment of voters behvior, emphsizing the privte gins from the ct of voting, is lso possible. From this perspective, voters vote motivted by the desire to express preferences or llegince to certin group, much s cheering crowds in sport event. But the competitive perspective seems to miss wht the strtegic perspective overemphsizes. Some individuls, such s politicl leders, prty ctivists, journlists, or other opinion mkers, devote time nd e ort to in uence the result of the election by mobilizing others to vote. These individuls do ct strtegiclly the time nd e ort they devote to the election depend to some extent on the time nd e ort they expect others to devote. Moreover, there is no cler line between voters nd opinion mkers. Under di erent circumstnces, n individul my decide to wtch the election s pssive specttor or to invest time nd other resources to in uence the result. Of course, it is true tht by nture of their professionl ctivity some individuls government functionries, editorilists, prominent economists, ctors nd other entertiners my hve n dvntge in in uencing others to vote. But it is lso true tht not everyone in ech of these ctegories becomes n opinion mker in every lrge election. For some, perhps mny, voters there is decision to be mde in n election regrding whether to try to in uence others or not. This mrgin of decision hs been neglected in the economic literture on elections, be it strtegic or competitive. Nonetheless, we believe nlyzing this mrgin of decision is crucil to understnd voter prticiption in lrge elections. In this pper, we propose model where voters decide whether to become opinion leders, nd extrct implictions in terms of the distribution of voter turnout nd the distribution of the winning mrgin in lrge elections. In our model, opinion leders rise endogenously out of lrge electorte of citizens with di erent politicl preferences. First, voters decide rst whether to be followers or leders tht try to in uence the outcome of the election by spending e ort to bring followers to vote. Being leder is costly s it is voting nd is driven by cost-bene t clcultion. Second, followers re rndomly ssigned to their leders nd must decide whether to vote independently or vote in complince with See e.g. Plfrey nd Rosenthl [6, 7], who build on erlier work by Ledyrd []. Another tretment of the subject is provided by Myerson [?]. See e.g. Brennn nd Buchnn [5] nd for very nice recent tretment, Schuessler [].

3 GROUP FORMATION AND VOTER PARTICIPATION 3 their ssigned leder in exchnge for rewrd. In the unique equivlence clss of equilibri, only xed number of voters for ech prty become leders: the in uence of ech leder, tht is, the leder s bility to swy the election one wy or the other is rndom but sttisticlly declines with the number of leders. More leders imply lower in- uence on followers of ech single leder. As in mobiliztion models, electorl turnout remins high becuse most voters re willing to vote in greement with their leder nd their group. Our model gives closed form solution tht llows us to derive the distribution of the electorl prticiption rte nd the winning mrgin in two-prty plurlity elections s functions of the importnce of the issue t stke in the election for voters, the cost of voting, nd the cost of becoming leder. Comprtive sttics on these prmeters of the model is ligned with stndrd stylized fcts. For instnce, expected turnout increses with the importnce of the elections nd the distribution of the winning mrgin generted by the model remins non-degenerte for the rbitrrily lrge electorte ssumed here. 3 Citizen-cndidte models l Osborne nd Slivinsky [5] or Besley nd Cote [] hve in common with this work the ide of endogenizing politicl ctivism. Nmely, out of popultion of citizens in equilibrium some citizens decide to become politiclly ctive cndidtes/leders. The gols nd the type of politicl ctivism re rdiclly di erent though. In the present pper leders rise not becuse they wnt to be elected to o ce themselves but becuse they cn ect the chnce tht their preferred prty wins the election (pivotlity). In other words, this pper tries to ddress the problem of the prdox of not voting with group-bsed mobiliztion model. The issue of turnout is not ddressed nd is not the objective of citizen-cndidte models which rther try to endogenize prty pltforms nd chrcterize prty formtion. By contrst in the present pper prty pltforms re given. Our work is lso relted to the socil interctions literture pioneered, inter li, by Gleser, Scerdote nd Scheinkmn [9]; some different pproches re presented by Becker nd Murphy [4] nd Durluf nd Young [7]. We borrow from the model of Gleser et l. the rrngement of gents in circle nd the ide tht some gents imitte their group behvior while others ct independently, depending on their voting cost. We devite in tht the group formtion, tht is the number of gents tht ct s leders is derived endogenously in the model, s well 3 This stnds in contrst with strtegic models of elections. For instnce, Plfrey nd Rosenthl [6] obtin high turnout rte in lrge election without uncertinty nd with no net bene ts of voting, but t the cost of predicting nerly tie.

4 GROUP FORMATION AND VOTER PARTICIPATION 4 s the decisions of followers to vote with their leder or independently. Lstly, voters re concerned not only by the decisions of their neighbors (group) s in locl interction models, but lso by the decision of the mjority (locl nd globl interctions). Six sections follow this introduction. The rst illustrtes the structure of the multi-stge gme, the second gives the prticulr circulr speci ction of how followers re ssigned to leders. The third section nds the unique clss of equilibri fter computing explicitly the pivotlity of leders. The fourth gives importnt comprtive sttics results, while the fth shows how ll the results re robust to vrious heterogeneities of potentil voters. The lst section summrizes nd concludes.. The Model We consider lrge election with two lterntives, A nd B, (e.g. two election cndidtes or two issues in referendum). There re is continuum of citizens (potentil voters) of mesure one of which strictly positive mesure re A-prtisns, nd B-prtisns, the rest re non-prtisns O. The only di erence in the preferences of these three types is tht prtisns enjoy gross gin of G > if their preferred prty wins the election nd zero otherwise, non-prtisns hve gin of zero whtever prty wins. A simple wy to describe this gme is to brek it down into three stges, in which citizens hve to mke sequentil decisions. In the rst stge every citizen chooses whether to become n ctive supporter of prty or not. We refer to n ctive supporter s n opinion leder (L), politicl entrepreneur committed to his prty, nd to n uncommitted voter s follower.(f): Nmely, ech plyer of ny type initilly hs one out of three choices to mke (L A ; L B ; F ), the rst two choices involve cost C > of becoming leder, the lst involves no cost. In the second stge once ll the leders re chosen, ny follower my (or my not) rndomly fll under the in uence of leder of one of the prties. Hence, ex-post there re three types of followers (F A ; F B ; F O ). All followers tht fll under the rnge of in uence of some leder of A or B, nmely F A nd F B, re o ered by their leder compenstion of (v ") in exchnge for committing to vote for the prty of tht leder (A or B). These in uentible followers hve the option of ccepting this compenstion, or rejecting it. In cse of cceptnce they will hve to vote for tht prty (V A or V B ). In cse of rejection (R) they receive no compenstion but re uncommitted nd free to bstin or vote for who they wnt, just like the independent followers

5 GROUP FORMATION AND VOTER PARTICIPATION 5 F O. In the third stge every citizen (leder or uncommitted follower) must choose whether to vote for one prty or bstin (V A ; V B ; A) the rst two choices involve cost of voting v >, bstining involves no cost. This lst third voting stge cn lso thought s simultneous to the previous second stge, the results do not chnge. The picture below tries to illustrte this gme. The rst stge leder-follower decision nd then the choice of nture, tht rndomly ssigns followers to leders, re depicted with thicker lines. The second nd third (voting stges) tke plce in correspondence to the dots in the picture. Recll tht the only ex-nte di erence mong citizen is their politicl view or type (A; B; O): In principle ny citizen cn end up in ny of the nl nodes of this gme. For ske of clrity we introduce only fter we present the min results of this pper other importnt initil di erences mong citizens such s voting costs, gins from the outcome of the election nd cost of becoming leder. All these dditionl heterogeneities do not genericlly chnge the results of this pper. We specify the move of nture fter we solve for the lst stge of the gme. The only property we need for now is tht leder cn in uence some strictly positive mesure of voters with probbility one... Lst Stges. Cse of No Leders. Note rst tht only A prtisn citizens my decide to be leders of prty A, likewise for B prtisns. No other type of citizen would wnt to incur the cost C of becoming leder if he cnnot get gin from doing so by ecting the election in his preferred direction. If no citizen chooses to be leder (which my hppen if the

6 GROUP FORMATION AND VOTER PARTICIPATION 6 cost C is high reltive to the gin G from hving the preferred prty win) then ll citizens end up in the node F nd then ply winner-tke ll election voting gme with continuum of potentil voters. For ny positive voting cost, no equilibrium of this subgme cn yield positive turnout, becuse tht would imply positive mesure of voters, ech with zero chnce of being pivotl. Zero turnout nd the prdox of not voting is nturl outcome of this subgme nd for this reson this subgme will not be reched in equilibrium s it will become cler lter. Cse of Some Leders. If t lest one citizen decides to become leder of one prty (sy prty A) then he exercises some in uence over rndom positive mesure of followers, nmely subset of the followers tht belong to the node F A : At this point the followers F A hve to decide whether to vote for the preferred prty A of their ssigned leder in exchnge for compenstion (v "), or reject the compenstion nd be free to vote for who they wnt or bstin. Proposition. In equilibrium ll followers of type F A (F B ) commit to vote V A (V B ) nd hve net gin of " >, ll followers of type F O bstin A nd obtin net gin of zero. Proof. The bove strtegy pro le n equilibrium becuse nobody wnts to devite from tht pro le: WLOG given tht pro le single F A follower would never be pivotl nd by choosing R he would give wy net gin of " to gin zero if he bstins or v if he votes. Likewise, ny F O follower bstins becuse he cnnot be pivotl. Furthermore there re no other equilibri. It is not n equilibrium tht ny positive number of followers my F A rejects R nd votes for A. A positive mesure of F A followers (ll the non prtizn O-type nd B-type followers, but lso most of the A-types) lwys ccept the o er of their A-leder nd mke voting for pivotl resons pointless for ny individul voter of ny type. In summry in equilibrium ll followers F A nd F B vote for whtever leder gives them some compenstion regrdless of their preferences. In ny equilibrium with some leders, ll followers know tht their probbility of chnging the outcome of the election is zero regrdless of whom they vote for. It is esy to see tht this outcome is similr with heterogeneity of voting costs. In this cse only F A followers with voting costs below the leder s compenstion (v ") (ssuming this compenstion is high enough so tht there is positive mesure of such followers) commit to vote V A, the rest reject nd then bstin together with ll the F O

7 GROUP FORMATION AND VOTER PARTICIPATION 7 followers. We elborte further on this issue in the section on heterogeneity. Now tht we re done with the voting decision of the followers, we turn to wht leders vote for. Leders know by there mere presence tht they cnnot ect the election with their vote, so they either bstin or vote for their preferred prty if we ssume tht they compenste themselves with (v ") for doing so. Either ssumption does not chnge the turnout, becuse there will only be discrete number of leders in ny equilibrium. Assuming they do vote without loss of generlity, the lst stge voting decisions of ll citizens re summrized by the rrows in the picture. We solved for the lst voting stges nd we re left with the initil leder-follower decision tht ll citizens fce initilly regrdless of their prty preferences. Solving bckwrds to nd the Subgme Perfect Nsh Equilibrium (SPNE) of this gme, we rewrite the pyo s resulting from the subsequent voting stges. The pyo s of F A nd F B followers become GPw A " A-prtisns GPw B " B-prtisns " O-prtisns where Pw A is the probbility tht prty A wins. The pyo s of the F O followers re the sme s bove without the ". If " is smll the pyo s of ll followers re GPw A A-prtisns GPw B B-prtisns O-prtisns All citizens tke into ccount these reservtion vlues when deciding whether to become leders or not. Given tht leders ect positive

8 GROUP FORMATION AND VOTER PARTICIPATION 8 mesure of followers then they consider their chnces of chnging the outcome of the election when deciding whether to py the cost C to become leder. If there re (A; B) leders lredy, citizen becomes n dditionl leder of prty A if nd only if G (Pw A (A ; B) Pw A (A; B)) > C A-prtisns G (Pw B (A; B ) Pw B (A; B)) > C B-prtisns > C O-prtisns Assuming (lter deriving) tht Pw A (A; B) is non-decresing in A, only A prtisns my consider to become leders L A ; likewise only B types my decide to become leders L B : We rewrite the initil lederfollower trde-o the A types re fcing s Pv A (A ; B) > C G where Pv A (A ; B) is pivotlity of the dditionl leder A, nmely by how much n A leder cn increse the likelihood tht the prty for whom he is leder wins the election. As we show in the extensions of the model, introducing heterogeneity on the cost nd/or bene t of becoming leder G, does not chnge the results either. C 3. Influence of Leders Being leder is costly (with xed cost C). The number of followers tht given leder gets is rndom nd it depends negtively on how mny other citizens become leders. This is relistic in the sense tht politicl entrepreneurs do not know how mny people they cn bring to vote but they do know tht their personl in uence over followers decreses the more entrepreneurs there re tht compete with them to bring citizens to vote. Nmely, the number of leders is su cient sttistic for the distribution of followers. We cpture this rndom dependence ssuming tht ll leders re dropped uniformly on circle of mesure one, which represents the popultion. Ech leder brings to vote for his prty n intervl of gents (to his right sy), until his intervl of in uence is interrupted. This cluster of in uence cn be interrupted by nother leder or my just die out exogenously. How likely it is to die out exogenously is mesure of the strength (or rther wekness) of the in uence of leder in the bsence of other leders. Ex post followers cn fll in the sphere of in uence of leder of prty A if the nerest leder to their left is of prty A nd the in uence hs not died out, in which cse in the SPNE they will vote for prty A regrdless of their preferences. Likewise they will vote for B regrdless of their preference if the nerest

9 GROUP FORMATION AND VOTER PARTICIPATION 9 leder to their left is of prty B nd the in uence hs not died out. Finlly they bstin if the in uence of the nerest leder to their left hs died out. If no citizens decide to be leders, citizens hve no externl in uence or rewrd from voting nd, s consequence, ll citizen ply regulr voting gme which gives the low turnout outcome described in the prdox of not voting. The lterntive with more votes wins the election, ties re zero probbility events. An equivlence clss of strtegy pro les of this one shot simultneous gme with continuum of plyers cn be summrized by (A; B) tht is, the number of leders for prty A nd for prty B, not who they re in prticulr. We only know tht in equilibrium they re prt of the A-prtisns nd B-prtisns respectively nd tht they re nite number or zero mesure reltive to ll voters in equilibrium. We ssume tht the popultion (e.g. politicl views) of the voters belonging to ny of the clusters in the picture bove is not di erent from the overll popultion. Followers re not selecting who is their leder in this model, rther some leder is ssigned to them (or emerges mong them rndomly). This leder my or my not hve the sme politicl views of the mjority of the voters in his cluster. 3.. Distribution without Abstiners. We wnt to nd how given number of leders (A; B) (considered s su cient sttistic) mps into the distribution of votes. As preliminry step for ske of exposition, we ssume tht ll followers fll under the in uence of some leder. Tht is, there re no F O followers or exogenously fding in uences. The in uence of leder cn be interrupted only by the presence of nother leder. To nd this distribution we rst need some sttisticl results. Theorem. The joint distribution of the spcings (x y ; ::; x k y k y k ; :::; x n y n )

10 GROUP FORMATION AND VOTER PARTICIPATION of the uniform order sttistic y < y < ::: < y n, i.e., the distribution g (x ; ::; x k ; ::x n ) is invrint under the permuttion of its components. Proof. See Reiss. p.4. This implies, in prticulr, tht Corollry 3. All mrginl distributions of (x ; ::; x k ; ::x n ) of equl dimension re equl. Assume tht there re no exogenous interruptions of the spheres of in uence of leders. There re A B leders in totl with A; B >. Proposition 4. The distribution of the number of votes for prty A hs the following pdf h (A;B) () Proof. See Appendix. (A B )! (A )! (B )! A ( ) (B ) < < The unconditionl pdf of the size of single cluster x, mong other n clusters, i.e. the in uence region of one leder mong other dditionl n leders is n ( x) n x As you cn see the in uence of ny single leder shrinks s n increses nd his in uence is crowded out by more other leders Distribution of Cluster Lenght The intersection with the y-xis of ny of the densities hppens to lso represent the number of dditionl leders n tht correspond to tht density.

11 GROUP FORMATION AND VOTER PARTICIPATION The turnout for prty A is rndom vrible tht hs expecttion E AB () A A B which is intuitive: the prty with more leders is expected to get more votes. 3.. Distribution with Abstiners. Assume tht the in uence of leders cn fde exogenously regrdless of the presence of other leders. This genertes F O followers tht then become bstiners in the SPNE. Abstention in this model is lck of ledership nd is obtined in the following wy. An exogenous number O of interruptions of the spheres of in uence fll uniformly on the circle. The smller this number O the stronger is the socil interction nd the stronger is the e ect of ledership on potentil voters. Now (A; B; O) becomes the su cient sttistic for the distribution of the turnout (votes) for ech prty (; b) : It is bivrite distribution de ned over the unit simplex. Proposition 5. The bivrite distribution of the number of votes (; b) for both prties given tht there re A; B; O leders is: h (A;B;O) (; b) (A B O )! (A )! (B )! (O )! (A ) b (B ) ( b) b for A; B; O (O ) Proof. See Appendix. If e.g. B we hve degenerte univrite density. If e.g. A B O the bivrte distribution is uniform on the simplex nd the

12 GROUP FORMATION AND VOTER PARTICIPATION mrginl distributions re liner e.g. h db ( ) The probbility tht prty A wins the election is Pw A Pr ( > b) Z hdbd hdbd The results should hold in ny circumstnce in which bstiners re introduced by lck of ledership through blind or indiscriminte interruptions of the in uence of ny given leder Probbility of Winning. We need to show tht the probbility tht prty A wins the election Pw A (A; B; O) Pr ( > b): Pw A (A; B; O) with h (A;B;O) (; b) is independent of O: Z h (A; B; O) dbd h (A; B; O) dbd (A B O )! (A )! (B )! (O )! A b B ( b) O Proposition 6. When O the probbility of winning is independent of O nd equl to BX AB k (A B k )! Pw A (A; B; O) (A )! (B k)! Proof. See Appendix. k Since the probbility of winning is independent of O for O grter or equl to one, we re left to prove tht the sme hods lso if O. Proposition 7. Pw A (A; B; ) Pw A (A; B; ) Proof. See Appendix. Hving estblished tht the probbility of winning does not depend on O we cn preform the clcultion with the simpler univrite distribution with O derived erlier in the no bstiner cse, which gives more trctble expression thn wht we originlly obtined for Pw(A; B; O) :

13 GROUP FORMATION AND VOTER PARTICIPATION 3 4. Pivotlity nd Equilibrium In the following we refer to Pv A s the probbility of ny A leder of being pivotl, the sme clcultions hold reversed for the probbility of being pivotl of B. We omit the vrible O since we hve shown it does not ect the probbility of winning nd therefore the pivotlity. WLOG referring to prty A, the probbility of being pivotl for the A- th potentil leder is the di erence between the probbility of winning with him nd winning without him keeping everything else constnt, tht is Pv A (A; B) Pw A (A; B) Pw A (A ; B) Proposition 8. Proof. See Appendix. 4 Pv A (A; B) (A B )! AB (A )! (B )! The expression for the pivotlity when A B is useful lter when we look for the equilibrium Pv A (A; A) A (A )! (A )! (A )! for A For instnce for A B, the probbility of being pivotl is the chnge leder cn mke from losing the election for sure to losing it with 5-5 chnce Pv A (A; A) The following monotonicity result is insightful nd will be useful lter. Proposition 9. For every k ; :::n Pv A (B k; B) < ::: < (Pv A (B ; B) Pv A (B; B)) > ::: > Pv A (B k; B) Proof. Pv A (A; B) Pv A (A ; B) Pv A (A; B) Pv A (A ; B) A B A A B A 7 () A 7 B () A B 4 I m very grteful to Aron Robertson for illustrting me the Wilf-Zeilberger method to solve hypergeometric identities.

14 GROUP FORMATION AND VOTER PARTICIPATION 4 So the probbility of chnging the outcome of the election is higher when the di erence between the number of leders of the two prties is smller. The other importnt monotonicity result is tht Pv A (A; A) decreses in A, becuse Pv A (A ; A ) Pv A (A; A) A < The pivotlity is highest when there re the sme number of leders for both prties, but this vlue decreses the more leders there re. The 3-D plot of the pivotlity function (done by using the gmm function rther thn the fctorils to be ble to plot over R ) shows how the pivotlity is highest in the digonl (A B) nd decreses s we move long the digonl. (A; B) (; 3) (; 3) The pivotlity function lso hs the following symmetry property. Proposition. () Pv A (A; B) Pv B (A; B) Proof. By de nition we must hve Pv A (A; B) Pv B (B; A) nd from proposition (8) we see tht Pv A (A; B) Pv A (B; A)

15 GROUP FORMATION AND VOTER PARTICIPATION 5 which mens tht for ny (A; B), how much n A-leder increses the chnce tht prty A wins (Pv A (A; B)) equls how much B-leder decreses tht chnce (Pv B (A; B)). We re now redy to nd the equilibrium of the gme. 4.. Equilibrium. If the cost of becoming leder is constnt then we hve tht n gent becomes leder of prty A if GPv A (A; B) C > We hve the following equilibrium result. Proposition. There exist unique clss of equilibri. They re of the form (A; A) Proof. See Appendix. Intuitively given the shpe of the pivotlity function symmetric equilibri re not possible. In ny con gurtion (A; B) with A 6 B whenever the prtisns of the winning prty hve no incentive devite incresing or decresing the number of their leders, then necessrily some prtisn of the losing prty hs incentives to become n dditionl leder. Note tht symptoticlly (using Stirling s formul) Pv A (A; A) ' p 4 (A ) which is slow decrese of the pivotlity long the digonl B A, so the equilibrium number of leders is not necessrily very smll. In ny equilibrium, the number of leders increses with the importnce of the election nd decreses with the cost of being leder. This drives the following comprtive sttics results. 5. Comprtive Sttics 5.. Turnout. The expected turnout in ny equilibrium (A; A) with level of bstention O is E (T b) ( b) h (A; A; O) dbd (A O )! (A )! (A )! (O )! (A O )! (A )! (A )! (O )! ( b) (A ) b (A ) ( b) (O ) dbd! A b (A ) (O ) ( b) (A ) b A (O ) ( b) dbd

16 GROUP FORMATION AND VOTER PARTICIPATION 6 From the normliztion (see ppendix) note tht A b B ( b) O dbd (ABO )! (A )!(B )!(O )! Hence E (T ) E (T ) (A O )! (A )! (A )! (O )! A A O O A (AO )! (A )!(A)!(O )! Expected turnout increses with the equilibrium number of leders nd decreses with the level of bstention. It is 5% when the number of leders of both prties equls the number of exogenous interruptions of leders in uence. 5.. Closeness. The expected closeness of the election, i.e. the expected winning mrgin of ny prty, is in ny equilibrium (A; A) equl to CL (A; O) E (j bj) E ( bj > b) ( b) h (A; A; O) dbd CL (A; O) (A O )! ( b) A b A ( b) O dbd (A )! (A )! (O )! A (Pw (A ; A) Pw (A; A )) A O A (Pv (A ; A )) A O (A)! A O A A! (A )! with more leders, tht is more importnt election, we hve the following result. Proposition. More leders mke closer election if nd only if there re enough of them A > O

17 GROUP FORMATION AND VOTER PARTICIPATION 7 Proof. CL (A ; O) (A )! A O A (A )! (A)! CL (A ; O) CL (A; O) A O A A O A () A > O A AO < We hve tht more importnt election is likely to hve smller winning mrgin if there re su cient leders reltive to bstiners tht is if the turnout is more thn 5%. If on the one hnd, the number of voters is smller thn the number of bstiners, then extr leders re more likely to bring bstiners to vote rther thn to stel voters from other leders. If on the other hnd, in expecttion there re more voters thn bstiners, then new leders tend more to stel voters from ech other rther thn bringing bstiners to vote. In the ltter cse, the number of votes for A or B, i.e. the rndom size of the sum of the cluster sizes of prty A or B tends to stbilize more (lower vrince). This decreses the di erence between the ggregte voter shres, i.e. increses the closeness of the election. 6. Extensions: Heterogeneity This model ssumed ex-nte identicl gents except for their prty preference. It cn be extended to llow for di erent voting costs v nd di erent gins from winning G (or equivlently di erent costs of being leder C), under some regulrity ssumptions. 6.. Di erent Voting Costs. Assume tht voting costs re heterogeneous nd distributed ccording some continuous pdf d (v) with v [v; v] with v >. In this cse ll F O followers still bstin, nd for given compenstion c, F A nd F B followers vote when their voting cost is lower thn the compenstion promised by leders nd bstin otherwise V A or V B if v < c A if v > c As result the pivotlity clcultion of leders is the sme nd so is their equilibrium number. The fct tht the results do not chnge is cler if you think tht followers re never pivotl when there is t lest one leder ecting the turnout. Only the turnout number for the

18 GROUP FORMATION AND VOTER PARTICIPATION 8 homogeneous cse needs to be scled down in the heterogeneous cse by the fctor D (c) (the CDF evluted t c), tht is, the frction of people tht vote in every leder s cluster. Note tht the citizens with low voting cost tht become F A or F B followers my obtin positive net bene t from voting for leder. This rises their expected reservtion vlue from being followers nd bises the rst stge leder-follower decision towrds being followers. Citizens with low voting costs re more likely to be followers thn leders (unless we ssume s we did tht leders rewrd themselves the sme wy too for voting). In either cse these di erent reservtion vlues do not chnge the results. They cn be seen s specil cse of the heterogeneity of gins from ledership tht we illustrte in the next section. 6.. Di erent Bene ts. Assume tht citizens my hve di erent bene ts from the outcome of the election (or di erent costs of ledership) distributed ccording to come continuous pdf z (G) with G G; G Recll tht the decision to become leder depends on the pivotlity cost-bene t clcultion Pv (A; A) > C G g with g g; g. As long s the bove is true for some vlue of g there will be dditionl citizens tht become leders. This process stops when the LHS, which does not depend on the prmeter g, crosses the threshold g, more precisely when Pv A (A; A) > g > Pv A (A ; A) The integer A tht stis es the bove condition, de nes the unique clss of equilibri in this heterogeneous cse. If the bove inequlity is not true, we cn lwys nd some citizens willing to become leder. So there is no di erence with the homogeneous cse nd ll the results nd comprtive sttics follow through. There my be some ine ciency though since the chosen leders not necessrily re the citizens with lower costs nd higher bene ts. This is due to the discrete nture of the leders, ll nd only the citizens with g g; Pv A (A; A) my become leders in this heterogeneous cse.

19 GROUP FORMATION AND VOTER PARTICIPATION Di erent E ectiveness of Leders. We ssumed so fr tht the region of in uence or cluster of leder ws rndom but the effectiveness of the leders of ech prty ws the sme. Nmely for ny given region of in uence we ssumed the leder could ttrct ll the gents in tht region to vote for his prty. More generlly we cn ssume tht leders of prty/issue A (B) cn ttrct only frction [; ] ( [; ]) of the potentil voters of his region of in uence. These setup is generl nd llows for proportions nd to depend on the originl preferences of the voters. For instnce, n A leder could ttrct ll A prtisns only within his cluster or ll A prtisns nd ll non-prtisns within his cluster or ll gents within his cluster like we ssumed in the benchmrk model, nd so forth. For [; ] the equilibrium number of leders of the model is the one we previously obtined: only the turnout should be scled down ccordingly if <. In generl if leders of di erent prties hve di erent e ectiveness 6, then the equilibrium is no longer necessrily symmetric nd the equilibrium number of leders for ech prty need not be the sme. The model extends s follows. Proposition 3. The pivotlity of leders is Pv A ( ) A B (A B )! (A )! (B )! Pv B ( ) ( ) A B (A B )! (A )! (B )! Proof. See Appendix. The constnt rtio of the pivotlities Pv B (A; B) Pv A (A; B) is mesure of the dvntge of B reltive to A. Nmely, if > then prty B hs reltive electorl dvntge. Then Pv A (A; B) ( ) B Pv A (A ; B) A Pv B (A; B) A Pv B (A; B ) B

20 GROUP FORMATION AND VOTER PARTICIPATION So Pv A (A; B)? Pv A (A ; B) () (A ) 7 (B ) Pv B (A; B)? Pv B (A; B ) () (A )? (B ) Tht is the pivotlity of A increses when A < (B ) nd similrly for B. Both pivotlities pek round the vlue where the number of leders for ech prty re in the reltion A B Note tht where both pivotlities pek the prty with the electorl disdvntge hs more leders then the prty with the dvntge, tht is pivotlities pek where the election is more likely to be close election. This cn be checked numericlly by looking t the probbilities of winning, mking sure they re the closest to / when A. In B equilibrium the prty with more e ective leders (e.g. prty A when > ) will hve less leders so tht the probbility of winning stys close to 5%. nd the pivotlity peks. The following picture shows the pivotlity when prty B hs n dvntge. ; (A; B) (; ) (; ) 5

21 GROUP FORMATION AND VOTER PARTICIPATION To hve pure strtegy interior equilibrium the two conditions Pv A (A; B) > C G > Pv A(A ; B) Pv B (A; B) > C G > Pv B(A; B ) must be stis ed simultneously, which implies tht pivotlities must pek: A. The equilibrium of course depends on the cost bene t B rtio nd on its heterogeneity, if ny. 7. Conclusions There is no cnonicl rtionl choice model of voting in elections with costs to vote. But, s Feddersen point out in his recent survey rticle [8], while cnonicl model does not yet exist, the literture ppers to be converging towrd group-bsed model of turnout, in which members prticipte in elections becuse they re directly coordinted nd rewrded by leders. This pper is contribution to this literture in two wys. First, it trets ll gents s ex nte identicl (except their politicl inclintion) nd hs leders self-select endogenously out of this homogeneous popultion, shedding some light on how these groups of voters cn be formed out of the voter popultion in the rst plce. How nd why people join or identify with their groups is (ccording to Feddersen) still mjor concern tht these group-bsed model hve not ddressed in stisfctory wy. The second contribution is more technicl. This model gives nice closed form solution, which is desirble becuse it llows to obtin immedite nd intuitive comprtive sttics results. Moreover, the solution of the problem gives lwys pure strtegy SPNE tht pins down uniquely the number of leders for ech prty, the expected turnout, nd the closeness of the election. Existence of equilibri (let lone uniqueness) is centrl problem in this literture, compounded with the fct tht the mixed strtegy often used present conceptul problems of interprettion in group-bsed models. This model solves ll these technicl spects. Of course, further reserch is needed to understnd better the inner mechnisms of voter group formtion. We consider this pper one step in tht direction.

22 8.. Proofs. GROUP FORMATION AND VOTER PARTICIPATION 8. Appendix Proof. 4 WLOG pick one of the (n ) leders tht re uniformly distributed over the unit circle. Just to strt counting from there cll tht leder nd cll tht point. From to the remining n leders re distributed uniformly. The size of cluster of the leder t zero (which hs the sme distribution s the cluster of ny other leder) is equl to the coordinte of the lowest of the remining leders, tht is, it is distributed s the rst order sttistic of (n) iid uniform drws on the unit line n ( ) n Similrly the cumultive cluster size of k djcent leders (WLOG the rst k leders including ) is equl to the coordinte of leder k or the k-th order sttistic. n! (k )! (n k)! k ( ) n k The cumultive cluster size of k non-djcent leders is distributed in the sme wy s bove becuse the underlying distribution is uniform (see Corollry 3). If the totl number of leders is (n ) A B then, the cumultive cluster size of k A of them is distributed s in the sttement of the theorem. Proof. 5 The joint pdf of two order sttistics of order n for uniform underlying distribution is f ( i ; j ) n! (i )! (j i )! (n j)! ( i) i ( j i ) j i ( j ) n j i < j In this cse, reordering the clusters (see Corollry 3) so tht there re rst A leders then B leders, the cluster sizes re i ; b j i A i; A B j; A B O n ) b (B ) ( b) (O Hence we hve (A B O )! h (A;B;O) (; b) (A )! (B )! (O )! (A )

23 GROUP FORMATION AND VOTER PARTICIPATION 3 Proof. 6 Clculte the inner integrls rst obtining Z Hence h (A;B;O) dbd h (A;B;O) dbd Pw A (A; B) k (ABO )! (A )!(OB )! A ( ) OB (ABO )! PB A B k ( ) O k (A )! k (B k)!(o k)! (A B O )! (A )! (O B )! A ( ) OB (A B O )! (A )! BX BX k AB k (A B k )! (A )! (B k)! Becuse integrting by prts itertively we obtin I Tht is AB k ( ) O k d (O k)! (A B k ) AB k (A B k )! (A B O )! AB k ( ) O k (B k)! (O k)!!!! AB k ( ) O k d (B k)! (O k)! AB k ( ) Ok d (O k)! ( ) OAB (O A B )! d AB k (A B k )! d AB (A B O )! k (A B k )! (B k)! Proof. 7 When O, the distribution is univrite nd the probbility of winning is the probbility tht > :5 Pw A (A; B; ) Pw A (A; B; ) (A B )! (A )! (B )! (A B)! (A )! (B )! A ( ) B d b b A b B ddb

24 Need to show tht: (A B) The LHS is (A B) GROUP FORMATION AND VOTER PARTICIPATION 4 b b Hence subtrcting the RHS A A A A B A b B b (A B) b A b B ddb A b B ddb A B A ( b) A b A db A ( ) B d B ( ) A d b B B ( ) A AB ( b) A b A db B ( ) A d A B ( ) A d (A B) B ( ) A A B ( ) A d B B ( ) A A B ( ) A d! AB Where in the lst step we integrting by prts the second term obtining AB A B ( ) A d B B ( ) A d Proof. 8 By de nition Pv A (A; B) Pw A (A; B) Pw A (A ; B) nd from proposition (6) we hve BX Pw A (A; B) Hence Pv A (A; B) BX k k AB k (A B k )! (A )! (B k)! AB k (A B k )! (A B k ) (A )! (B k)!! AB

25 k GROUP FORMATION AND VOTER PARTICIPATION 5 So we re left to prove the following identity BX AB k (A B k )! (A )! (B k)! (A B k ) (A B )! AB (A )! (B )! The identity is equivlent to S(A; B; k) () k (A B k )! (A B )! BX S(A; B; k) k De ne G(A; B; k) s R(A; B; k) A B k A B k G(A; B; k) R(A; B; k)s(a; B; k) (B )! (A B k ) (B k)! Then G(A; B; k) () k (A B k )! (B )! (A B )! (B k)! With simple lgebr you cn check tht S(A; B; k) G(A; B; k ) G(A; B; k) for k ; :::; B S(A; B; B) G(A; B; B) for k B Hence BX S(A; B; k) G(A; B; ) k Proof. Any interior equilibrium (A:B) must stisfy simultneously Pv A (A; B) > C G > Pv A(A ; B) Pv B (A; B) > C G > Pv B(A; B ) which implies tht the pivotlity functions must be decresing. By proposition (9) this implies both A B nd B A. Hence we must look for interior equilibri of the form (A; A). From proposition (8) it is lwys the cse tht Pv A (A; A) > Pv A (A ; A ) Pv A (A ; A) So for given xed cost we cn nd some A Pv A (A; A) > C G > Pv(A ; A ) Pv A(A ; A)

26 GROUP FORMATION AND VOTER PARTICIPATION 6 Given tht there re (A; A) leders nobody else wnts to be leder of prty A. And no leder of prty A wnts to be follower. The sme clcultions nd resoning holds for the pivotlity of B. We re left to check if there re corner equilibri, i.e. equilibri of the form (; B), or equivlently (A; ). For this to be the cse we need the following conditions to be stis ed Pv B (; B) > C G > Pv B(; B ) Pv A (; B) < C G From the property () the bove conditions re equivlent to Pv A (; B) > C G > Pv A(; B ) Pv A (; B) < C G From the monotonicity result (9) we hve Pv A (; B) > Pv A (; B) so the property cn never be stis ed. Note tht no leders is the unique equilibrium when C G becuse in tht cse the probbility of being pivotl is one hlf, ssuming the election is decided by fir coin toss in the cse of no leders. Hence (A; A) is the unique clss of Nsh Equilibri. Proof. 3 The proof is divided into four prts or lemms. In the rst two lemms we show the result Pv A (A; B) ( ) A B (A B )! (A )! (B )! Pv B (A; B) ( ) ( ) A B (A B )! (A )! (B )! for O : In the lst two lemms we show tht the result holds for ny O ; ; ; ::: by showing tht the probbility of winning of ny prty Pw does not depend on O:

27 GROUP FORMATION AND VOTER PARTICIPATION 7 Lemm 4. The Cse O : The probbility of winning of prty A when O is Pw A (A; B; ) Integrting by prts we obtin (A B )! (A )! (B )! A ( ) B d (A B )! (A )! (B )! (A B )! (A)! (B )! (A ) ( ) (B ) d A ( ) B d (A B )! (A)! (B )! A B nd integrting by prts itertively we obtin the sum Pw A (A; B) AX k (A B )! (A k)! (B k )! ( )A k Bk where Lemm 5. We need to show tht for O we hve Pv B (A; B) Pw B (A; B) Pw B (A; B ) ( ) ( ) A B (A B )! (A )! (B )! The probbility of winning is Pw A (A; B) H (B; k) BX H (B; k) Pw B (A; B) k (A B )! (A k )! (B k)! ( )Ak B k

28 GROUP FORMATION AND VOTER PARTICIPATION 8 Hence H (B; k) H (B ; k) (A B)! (A k )! (B k)! ( )Ak B k (A B )! (A k )! (B k)! ( )Ak B k (A B )! (A k )! (B k)! ( )Ak B k (A B )! (A k )! (B k )! ( )Ak B k D (B; k ) D (B; k) where D (B; k) (A B )! (A k )! (B k )! ( )Ak B k Note tht summing the terms we hve telescopic sum on the RHS BX H (B; k) k BX H (B ; k) k BX (D (B; k ) D (B; k)) Pw B (A; B) (Pw B (A; B ) H (B ; B )) D (B; B ) D (B; ) Tht is H (B ; B ) Pv B (A; B ) D (B; B ) D (B; ) ( ) AB Pv B (A; B ) ( ) AB (A B )! (A )! (B)! ( )A B Hence Pv B (A; B ) ( ) A B (A B )! (A )!B! To obtin Pv A (A; B) (or in generl Pw A from Pw B ) given the symmetry of the problem, you cn switch A with B nd with ( ) ( with ). Lemm 6. We show rst tht Pw A (A; B; O ) Pw A (A; B; O ) k

29 GROUP FORMATION AND VOTER PARTICIPATION 9 By strightforwrd clcultion we hve Pw A (A; B; ) Pw B (A; B; ) Z Z (A B)! (A ) b (B ) dbd (A ) (A )! (B )! Z B Z! (A B)! (A ) (A )! (B)! d (A ) ( ) B d B Z (A B)! (A )! (B)! B (AB ) d Pw A (A; B ; ) AB Pw A (A; B ; ) (A B )! (A )! (B)! (A B )! (A )! (B)! B ( ) A ( Pw B (A; B ; )) Thus, it must be the cse tht Pw B (A; B ; ) Pw B (A; B; ) But since by lemm () we hve Then (A B )! (A )! (B)! B ( ) A (A B )! (A )! (B)! B ( ) A Pw B (A; B ; ) Pw B (A; B; ) Pw B (A; B; ) Pw B (A; B; ) Lemm 7. It is left to show tht for O ; ; ::: Pw A (A; B; O) Pw A (A; B; O ) Integrting by prts we hve Pw A (A; B; O) Z Z (A B O )! b b B A ( b) O ddb (A )! (B )! (O )! b (ABO )! A!(B )!(O )! b R B b A ( b) O ddb b (ABO )! A R A A!(B )!(O )! b AB b O db Now the rst term is (A B O )! A! (B )! (O )! Pw A (A ; B; O ) Z b B b b A ( b) O ddb b (B ) dbd!

30 GROUP FORMATION AND VOTER PARTICIPATION 3 The second term fter iterted integrtion by prts is A Z O (A B O )! b AB A! (B )! (O )! b db A Z O (A B O )! b AB A! (B )! (O )! (A B) b A O Z (A B O )! b ABO A! (B )! (A B) ::: (A B O ) (A B )! ( ) A B A! (B )! Pv A (A ; B; ) Pw A (A ; B; ) Pw A (A; B; ) Thus for O ; ; ::: Pw A (A; B; O) Pw A (A ; B; O ) (Pw A (A ; B; ) Pw A (A; B; )) Now proceed by induction. For O, we hve for ll A nd B Pw A (A; B; ) Pw A (A; B; ) Hence substituting this reltion in the previous one we obtin Pw A (A; B; O) Pw A (A ; B; O ) (Pw A (A ; B; ) Pw A (A; B; )) So for O 3, we hve nd so forth for ll vlues of O. Pw A (A; B; 3) Pw A (A; B; ) 8.. Normliztion. Inner integrl. Integrlly on the b vrible: b B ( b) O db (B )! (OB )! (O )!! B ( b)o (B ) b O db B 3 ( b)o (B ) (B ) b db O (O ) ( b) OB (B )! (O B ) ::: (O ) O db Resulting in the Inner integrl ( b) OB db db db

31 GROUP FORMATION AND VOTER PARTICIPATION 3 b B ( b) O (B )! (O )! db (O B )! Outside Integrl. Finlly the integrl over : A ( ) OB d Resulting in: (A )! (OBA )! (OB )! ( ) OB ( ) OBA d A ( ) OB (A )! (O B )! d (O B A )! Hence, the integrl without the normliztion coe cient yields the inverse of the normliztion coe cient (B )! (O )! (A )! (O B )! (O B )! (O B A )! So the integrl is normlized to one. (ABO )! (A )!(B )!(O )!

32 GROUP FORMATION AND VOTER PARTICIPATION 3 References [] Aldrich, John Rtionl Choice nd Turnout, Americn Journl of Politicl Science 37- (993), [] Besley, Timothy nd Stephen Cote, An Economic Model of Representtive Democrcy, Qurterly Journl of Economics (997), [3] Blis, Andre To Vote or Not to Vote: The Merits nd Limits of Rtionl Choice Theory, Pittsburg University Press (). [4] Becker, Gry S. nd Kevin M. Murphy, Socil Economics (), Cmbridge, Mss. nd London: Hrvrd University Press. [5] Brennn, Geo ry nd Jmes Buchnn, Voter Choice: Evluting Politicl Alterntives, Americn Behviorl Scientist 8 (984), 85-. [6] Downs, Anthony, An Economic Theory of Democrcy (957), New York: Hrper nd Row. [7] Durluf, Steven N. nd H. Peyton Young (editors), Socil Dynmics (), Wshington, D.C.: Brookings Institution Press, nd Cmbridge, Mss. nd London: MIT Press. [8] Feddersen, Timothy J. Rtionl Choice Theory nd the Prdox of Not Voting, Journl of Economic Perspectives 8- (4), 99-. [9] Gleser, Edwrd L., Bruce Scerdote nd Jose A. Scheinkmn, Crime nd Socil Interctions, Qurterly Journl of Economics (996): [] Ledyrd, John, The Pure Theory of Lrge Two-Cndidte Elections, Public Choice 44 (984), 7-4. [] Morton, Rebecc, A Group Mjority Model of Voting, Socil Choice nd Welfre 4: (987), 7-3. [] Morton, Rebecc, Groups in Rtionl Turnout Models, Americn Journl of Politicl Science 35 (99), [3] Noelle-Neumnn, Elisbeth, Turbulences in the Climte of Opinion: Methodologicl Applictions of the Spirl of Silence Theory, Public Opinion Qurterly 4 (977), [4] Noelle-Neumnn, Elisbeth, Public Opinion nd the Clssicl Trdition: A Re-evlution, Public Opinion Qurterly 43 (979), [5] Osborne, Mrtin J. nd Al Slivinski, A Model of Politicl Competition with Citizen-Cndidtes, Qurterly Journl of Economics (996), [6] Plfrey, Thoms R. nd Howrd Rosenthl, A Strtegic Clculus of Voting, Public Choice 4 (983), [7] Plfrey, Thoms R. nd Howrd Rosenthl, Voter Prticiption nd Strtegic Uncertinty, Americn Politicl Science Review 79 (985), [8] Petkovsek, M., Wilf, H. nd D. Zeilberger AB, A K Peters, Ltd., of Wellesley, Mss. (997). [9] Reiss, R.-D., Approximte Distributions of Order Sttistics (989), Springer- Verlg New York Inc. [] Schuessler, Alexnder A., A Logic of Expressive Choice (), Princeton nd London: Princeton University Press. [] Shchr, Ron nd Brry Nlebu, Follow the Leder: Theory nd Evidence on Politicl Prticiption, Americn Economic Review 89 (999), [] Uhlner, Crol, Rtionl Turnout: The Neglected Role of Groups, Americn Journl of Politicl Science 33 (989), 39-4.