Sincere Voting and Information Aggregation with Voting Costs

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1 Sincee Voting and Infomation Aggegation with Voting Costs Vijay Kishna y and John Mogan z August 007 Abstact We study the oeties of euilibium voting in two-altenative elections unde the majoity ule. Votes ae ivately infomed about who is the bette candidate and have ivate costs of going to the olls. We show that sincee voting is always an euilibium. Futhemoe, even though the euilibium ecentage tunout goes to zeo in the limit, infomation fully aggegates unde the sincee voting euilibium. The Model Thee ae two candidates, named A and B, who ae cometing in an election. The winne needs a simle majoity of the votes cast. In the event of a tied vote, the winning candidate is chosen by a fai coin toss. Thee ae two eually likely states of natue, and : Candidate A is the bette candidate in state while candidate B is the bette candidate in state : Seci cally, in state the ayo of any vote is if A is elected and 0 if B is elected. In state ; the oles of A and B ae evesed. The size of the electoate is a andom vaiable which is distibuted accoding to a Poisson distibution with mean n: Thus the obability that thee ae exactly m eligible votes (o citizens) is g n (m) = e n n m m! Pio to voting, evey citizen eceives a ivate signal S i egading the tue state of natue. The signal can take on one of two values, a o b: The obability of eceiving a aticula signal deends on the tue state of natue. Seci cally, P [a j ] = and P [b j ] = We suose that both and ae geate ; so that the signals ae infomative and also that both ae less than ; so that they ae noisy. Thus, signal a is associated with state while the signal b is associated with : Peliminay daft. Comments welcome. y Penn State Univesity. vkishna@su.edu z Univesity of Califonia, Bekeley. mogan@haas.bekeley.edu

2 Conditional on the state of natue, the signals of the votes ae ealized indeendently. The osteio obabilities of the states afte eceiving signals ae ( j a) = + ( ) and ( j b) = + ( ) We assume, without loss of geneality, that : It may be vei ed that ( j a) ( j b) Thus the osteio obability of state is given the signal a is smalle than the osteio obability of state given the signal b even though the coect signal is moe likely in state. Of aticula inteest ae cicumstances whee voting is sincee whee, in euilibium, all those who eceive a signal of a vote fo A and those who eceive a signal of b vote fo B:.0. Comulsoy Voting Fist, conside the case whee all those in the electoate must vote fo eithe A o B; that is, no abstentions ae allowed. When the signal ecisions ae eual (that is, = ), it is easy to see that sincee voting is indeed an euilibium. Conside a vote with signal a and who votes sinceely. He ayo s ae only a ected when he vote is ivotal (that is, changes the outcome of the election). When the election is in a tie, thee ae an eual numbe of a and b signals ecevied by the othe votes and, since signal ecisions ae eual, these signals cancel each othe out and thee is no net infomational e ect on the vote s osteio beliefs. Since he own signal favos state ; it ays to vote sinceely. Next, conside the case whee A tails by a single vote. In that case, thee is one moe b signal than a signal fom the othe votes, and the infomational e ect of this signal is exactly cancelled by the vote s own a signal. In othe wods, the osteio beliefs of the vote in this cicumstance ae eual to he ios and hence she is indi eent between voting fo A and voting fo B: Thus, it ays to vote sinceely if the signal ecisions ae eual. When the signal ecisions ae not eual, say >, sincee voting does not constitute an euilibium. Once again, to detemine the otimality of voting sinceely, a vote needs to conside only cicumstances in which he o she is ivotal. Seci cally, let P iv A denote the set of cicumstances in which a single vote fo A will change the outcome of the election in favo of A: The obability of the event P iv A in state is P [P iv A j ] = X e n (n ) k (n ( k!k! k=0 )) k + X k=0 e n (n ) k (n ( )) k+ k!k +! The st exession etains to situation in which thee is a tie (so that A is winning with obability ) and a single vote fo A will esult in A winning fo sue. The second exession etains to situations in which A is losing by exactly one vote and a single vote fo A will thow the election into a tie. Myeson (998) has shown that,

3 when n is lage, the ivot obabilities fo candidate A in the two states and can be aoximated by the fomulae ( ) P [P iv A j ] en 4 n ( ) P [P iv A j ] en 4 n ( ) ( ) + + Similaly, let P iv B denote the set of events in which a single vote fo B will change the outcome of the election in favo of B: Using the same aoximation fomulae, we obtain ( ) P [P iv B j ] en + 4 n ( ) ( ) +! P [P iv B j ] en 4 n ( ) Fo sincee voting to be otimal, it must be the case that, afte eceiving signal a; a vote s exected ayo fom voting fo A exceeds he ayo fom voting fo B: Fomally, this amounts to the condition ( j a) P [P iv A j ] ( j a) P [P iv A j ] ( j a) P [P iv B j ] ( j a) P [P iv B j ] Reaanging tems, the incentive comatibility constaint becomes ( j a) (P [P iv A j ] + P [P iv B j ]) ( j a) (P [P iv A j ] + P [P iv B j ]) Next, we comae the obability of being ivotal in state comaed to state : When n is lage, the atio of ivot obabilities is P [P iv A j ] + P [P iv B j ] P [P iv A j ] + P [P iv B j ] en ( ) ( ) k ( ; ) whee k ( ; ) is ositive and is indeendent of n: Since > > ; ( ) > ( ) and so P [P iv A j ] + P [P iv B j ] lim n! P [P iv A j ] + P [P iv B j ] = 0 This imlies that, when n is lage and a vote is ivotal, state is in nitely moe likely than is state : Thus, the euied ineuality fo the incentive comatibility condition does not hold and hence a vote with an a signal will not wish to vote sinceely in a lage election. To summaize, we have shown, Poosition Unde comulsoy voting, sincee voting is not an euilibium in lage elections with di eent signal ecisions ( 6= ). 3

4 . Costly Voting U until now, we have assumed evey vote cast a vote fo eithe A o B; that is, abstention was not allowed. In eality, of couse, abstention is always ossible. A second asect is that votes incu some costs of going to the olls to cast thei votes. Fo the emainde of this ae, we add simultaneously intoduce the ossiblity of abstention and voting costs. We suose that voting costs vay acoss votes. Seci cally, the cost of voting fo each vote is ivate infomation and detemined by a ealization fom a continuous obability distibution F with suot [0; c] : We suose that c and that F admits a density f that is stictly ositive on (0; c) : Finally, we assume that voting costs ae indeendently distibuted acoss votes and indeendent of the signal as to who is the bette candidate. Thus io to the voting decision, each vote has two ieces of ivate infomation his cost of voting and a signal egading the state. Euilibium with Costly Voting We will show that unde majoity ule, thee exists an euilibium of the voting game with the following featues.. Thee exists a ai of cut-o cost levels, c a and c b, such that a citizen with a cost ealization c and who eceives a signal i = a; b votes if and only if c c i :. All those who vote do so sinceely that is, all those with a signal of a vote fo A and those with a signal of b vote fo B: Let us denote by a the obability that a vote with signal a will actually vote, so that a = F (c a ) : Similaly, b = F (c b ) : Conside an event whee (othe than vote ) the ealized electoate is of size m and thee ae k votes in favo of A and l votes in favo of B. The numbe of abstentions is thus m k l: The obability of this event in state is P [hk; l; mi j ] = e n n m m k + l m! k + l k ( a ( ) b ) m k l ( a ) k (( ) b ) l Notice that in the fomula above, the obability of voting fo A deends jointly on eceiving signal a (which occus with obability ) and a su ciently low cost ealization (which occus with obability a ). The obability of voting fo b deends jointly on eceiving signal b (which occus with obability ) and a su ciently low cost ealization (which occus with obability b ). The comlement of these two is the obability of abstention. It is useful to eaange P [hk; l; mi j ] as follows: P [hk; l; mi j ] = e n( a ( ) b) (m k l)! e na k! (n a ) k e n( 4 (n ( a ( ) b )) m k l l! ) b (n ( ) b ) l

5 Of couse, the size of the electoate is unknown to the vote at the time of the vote. The obability of the event that the vote totals ae k and l, witten hk; li, iesective of the size of the electoate, is P [hk; li j ] = X m=k+l P [hk; l; mi j ] = e na (n a ) k k! wheeas the obability of hk; li in state is e n( ) b (n ( ) b ) l l! () P [hk; li j ] = e n( ) a (n ( ) a ) k e n b (n b ) l k! l! If we de ne P iv A to be the set of events in which, by voting fo A as oosed to staying home, a citizen can change the outcome of the election, then we have P [P iv A j ] = = X P [hk; ki j ] + k=0 X k=0 X P [hk; k + i j ] k=0 e n(a+( ) b) n ( ) a b k k!k! ()! + n ( ) b k + and P [P iv A j ] is detemined similaly. Similaly, de ne P iv B to be the set of events whee voting fo B athe than staying at home changes the election outcome. Paticiation Decisions Using this notation, the exected ayo to a vote with signal a and cost c who votes sinceely is E a (c) = ( j a) P [P iv A j ] ( j a) P [P iv A j ] c and similaly, the exected ayo to a vote with signal b and cost c who votes sinceely is E b (c) = ( j b) P [P iv B j ] ( j b) P [P iv B j ] c We may use these exessions to chaacteize the aticiation behavio in the election. Seci cally, in any euilibium, it must be the case that the cost thesholds c a ; c b (o euivalently, the obability thesholds a ; b ) solve ( j a) P [P iv A j ] ( j a) P [P iv A j ] = F ( a ) (3) ( j b) P [P iv B j ] ( j b) P [P iv B j ] = F ( b ) (4) Lemma Thee exists a solution ( a; b ) to euations (3) and (4) such that 0 < a < and 0 < b < : 5

6 Poof. Notice that, at any oint (0; b ) P [P iv A j ] = e n( ) b ( + n ( ) b ) P [P iv A j ] = e n b ( + n b ) We claim that P [P iv A j ] > P [P iv A j ] : This follows fom that fact that the function g (x) = e x ( + x) is stictly deceasing fo x > 0 and that > : Hence, at (0; b ) ( j a) P [P iv A j ] ( j a) P [P iv A j ] > 0 = F (0) since ( j a) > : Next, notice that at any oint (; b ) ( j a) P [P iv A j ] ( j a) P [P iv A j ] < F () Thus, fo any b ; thee exists a value of a (0; ) solving euation (3) : An identical agument shows that, fo any a ; thee exists a value of b (0; ) solving euation (4) : Theefoe, an inteio solution to euations (3) and (4) exists. Voting Decisions Next, we show that fo lage n; it is otimal fo a single vote to vote sinceely given the aticiation thesholds detemined by euations (3) and (4) and given that all othe votes ae voting sinceely. Conside a vote with signal a who decides to cast a vote. If all othes aticiate accoding to the theshold costs detemined above and if voting, vote sinceely, it is otimal fo a vote with signal a to vote fo A if and only if ( j a) (P [P iv A j ] + P [P iv B j ]) ( j a) (P [P iv A j ] + P [P iv B j ]) (5) Similaly, given that all othes vote sinceely, it is otimal fo a vote with signal b to vote fo B if and only if ( j b) (P [P iv A j ] + P [P iv B j ]) ( j b) (P [P iv A j ] + P [P iv B j ]) (6) Myeson (998) has shown that fo lage n; the ivotal obabilities can be aoximated by P [P iv A j ] en P [P iv A j ] en a( ) b a ( ) b 4 n a ( ) b a( ) b ( ) a b 4 n a ( 6 ) b a + ( ) b (7) a ( ) a + b ( ) a (8)

7 of P [P iv B j ] en P [P iv B j ] en a( ) b a ( ) b 4 n a ( ) b a( ) b ( ) a b 4 n a ( ) b a + ( ) b (9) ( ) b ( ) a + b b (0) Recall fom euation (3) that any individual voting fo A obtains a goss bene t ( j a) P [P iv A j ] ( j a) P [P iv A j ] > 0 Hence, at any a and b that satis es (3) ; P [P iv A j ] ( j a) > P [P iv A j ] ( j a) We will show that, fo lage n, it is also the case at any a and b that satis es (3), P [P iv B j ] ( j a) > P [P iv B j ] ( j a) and togethe, these inualities imly that the incentive comatibility condition fo signal a, (5); is satis ed. Lemma When n is lage, fo all a and b ; P [P iv B j ] P [P iv B j ] > P [P iv A j ] P [P iv A j ] Poof. Using Myeson s aoximation fomulae (7) to (0), this is the same as which educes to o euivalently a+ ( ) b a+ ( ) b ( )b a > ( ) a+ b ( ) a+ b b ( ) a > > ( ) ( ) which clealy holds. We have thus shown that at any a and b such that the aticiation theshold euation fo signal a, (3), is satis ed, the incentive comatibility condition fo signal a is also satis ed. 7

8 The agument fo incentive comatibility fo signal b is analogous. We need to show that if a and b ae such that then it is also the case that P [P iv B j ] ( j b) > P [P iv B j ] ( j b) P [P iv A j ] ( j b) > P [P iv A j ] ( j b) So it is enough to show that fo all a and b ; P [P iv A j ] P [P iv A j ] > P [P iv B j ] P [P iv B j ] which, by Lemma, holds when n is lage. To summaize, we have established that Theoem In the costly voting model, when n is lage, thee is an euilibium in which (i) thee is a ai of ositive theshold costs; and (ii) voting is sincee. Having identi ed an euilibium in which voting is sincee, it is useful to examine the e ciency of majoity voting in tems of selecting the coect candidate that is, A is elected in state and B is elected in state : Notice that sincee voting alone is not enough to guaantee e ciency. Two tyes of things can still go wong. Fist, if too few individuals (in exectation) come to the oll, then even with sincee voting, infomation will not aggegate. Second, if votes come to the olls in the wong atios (based on di eences in signal ecisions), then again even if thee ae an in nite numbe of notes in exectation, infomation will still not fully aggegate. In the next two sections, we conside each of these otential oblems seaately. Section 3 examines the limit oeties of vote aticiation decisions whilst Section 4 examines the e ciency of election outcomes. In the oof of the theoem above we used the assumtion that the exected numbe of votes is lage. This enabled us to exloit the aoximation fomulae fo the ivotal obabilities deived by Myeson (998). As the following examle shows, howeve, a sincee voting euilibium may exist even when n is small (in fact, we know of no examle in which such an euilibium does not exist). Examle Suose that the exected size of the electoate n = 0; the signal ecisions = 3 4 ; = 3, and cost distibution F (c) = c on [0; ]: Then the (uniue) sincee voting euilibium has aticiation obabilities a = 0:68 and b = 0:3: The elevant lines deicting the incentive and aticiation constaints fo each tye of vote ae shown in the gue below. 8

9 b IC b E a = c a E b = c b b IC a a a Figue : Euilibium with Sincee Voting 3 Paticiation in Lage Elections We begin by consideing the euilibium aticiation thesholds in lage elections. Lemma 3 In any seuence of sincee voting euilibia, the theshold costs tend to zeo; that is, lim su n! c a (n) = lim su n! c b (n) = 0: Poof. Suose to the contay that, fo some seuence lim n! c a (n) > 0: In that case, the goss bene ts (excluding the costs of voting) to votes with a signals fom voting must be ositive. That is; lim ( ( j a) P [P iv A j ] ( j a) P [P iv A j ]) > 0 n! whee it is undestood that the obabilities deend on n. We know that along the given seuence, lim a (n) > 0: By taking convegent subseuences, if necessay, suose that lim b (n) also exists. If lim b (n) > 0; then 9

10 fom (7) and (8) we have that lim P [P iv A j ] = 0 = lim P [P iv A j ] : If lim b = 0; then lim P [P iv A j ] = lim e na = 0 Likewise, lim P [P iv A j ] = lim e n( ) a = 0 Thus if thee is a seuence such that lim a (n) > 0 then along some subseuence lim P [P iv A j ] = 0 = lim P [P iv A j ] : But this means that along this seuence, the goss bene t of voting fo A when the signal is a tends to zeo. This contadicts the assumtion that lim n! c a (n) > 0: Next, we show that, desite the fact that the theshold cost fo voting goes to zeo in the limit, in any sincee voting euilibium, the exected numbe of votes with signal s fa; bg is unbounded in lage elections. Thee is a ace between the seed at which the aticiation thesholds aoach zeo elative to the size of the electoate. A common intuition one may have is that the winne of this ace deends on the shae of the cost distibution aticulaly in the neighbohood of 0. As we show below, howeve, sincee voting euilibia have the oety that, in lage elections, the numbe of votes becomes unbounded egadless of the shae of this distibution. That is, in the sincee voting model, the oblem of too little aticiation to achieve infomation aggegation does not aise in the limit. Lemma 4 In any seuence of sincee voting euilibia, eithe lim n a (n) = o lim n b (n) = : Poof. Suose to the contay that lim n a (n) < and lim n b (n) < : In that case, thee is a subseuence of theshold obabilities such that the exected numbe of votes with each signal is nite in the limit. Along such a seuence (o a convegent subseuence, if necessay), it is clea that lim ( ( j a) P [P iv A j ] ( j a) P [P iv A j ]) > 0 This, howeve, contadicts Lemma 3. To establish this oety of sincee voting euilibia in lage elections, we st ove a technical lemma about the atio of aticiation ates in the limit. Lemma 5 (i) lim inf n! a(n) b (n) > 0; and (ii) lim inf n! b(n) a(n) > 0: Poof. To ove at (i), suose to the contay that lim inf na(n) n b (n) 4, it follows that lim inf n b (n) =. Using the fomulae = 0: By Lemma 0

11 P [P iv B j ] en P [P iv B j ] en a( ) b a ( ) b 4 n a ( ) b a( ) b ( ) a b 4 n a ( ) b a + ( ) b () ( ) b ( ) a + b b () it may be vei ed that P [P iv B j ] n P [P iv B j ] K b ( ) a ( ) b ne! a b whee ( ) K n = ( ) and 0 < lim K n < : Taking limits a b + ( ) ( ) a b + ( ) lim P [P iv B j ] P [P iv B j ] = But this contadicts the individual ationality constaint which euies that (jb) P [P iv B j ] (jb) P [P iv B j ] 0 o The oof of at (ii) is analogous. P [P iv B j ] P [P iv B j ] (jb) (jb) < We ae now in a osition to show Lemma 6 In any seuence of sincee voting euilibia, the exected numbe of votes tends to in nity; that is, lim inf n! n a (n) = = lim inf n! n b (n) : Poof. The oof follows as a conseuence of Lemmas 4 and 5. To summaize, we have shown that, even though the cost thesholds fo aticiation go to zeo in the limit in the sincee voting model, they do so su ciently slowly that the exected numbe of votes with a and with b signals is unbounded as n gets abitaily lage.

12 4 E ciency in Lage Elections We now tun to the uestion of whethe majoity ule is e cient unde costly voting. In othe wods, is it the case that in lage elections, the ight candidate is elected? Lemma 7 In any seuence of sincee voting euilibia, Futhemoe, a ( ) b a ( ) b lim n! = a ( ) b ( ) a b lim n! a ( ) ( ) = a b + b (3) and hence a b fo n su ciently lage. Poof. Fist, note fom Lemma 5 that a + ( lim inf a ) b < and b + ( ) a lim inf < b and hence, in the exessions fo the ivot obabilities, the exonential tems dominate in the limit. Suose, contay to the statement of the Lemma, that the atio di eed fom in the limit. In aticula, suose that the atio conveged to a numbe stictly smalle than. In that case lim P [P iv A j ] P [P iv A j ] = 0 and it would then follow that state is in nitely moe likely in a P iv A event than is state :This, howeve, would imly that the goss bene t to a vote with signal a fom voting is negative, which contadicts Lemma 3. Similaly, if the atio conveged to a numbe stictly geate than, then lim P [P iv B j ] P [P iv B j ] = 0 and, it would then follow that state is in nitely moe likely in a P iv B event than is state : This, howeve, would then imly that the goss bene t to a vote with signal b fom voting is negative, which also contadicts Lemma 3. Thus the atio must eual in the limit. If = ; then a = b : If > > ; eaanging tems in the atio exession imlies lim b a ( ) ( ) = a b +

13 The ight-hand side of the above exession is ositive. Hence, a < b : We ae now in a osition to show that infomation fully aggegates in lage elections. Poosition In any seuence of sincee voting euilibia, infomation fully aggegates. Fomally, in any seuence of sincee voting euilibia, the obability that candidate A is elected in state goes to one while the obability that candidate B is elected in state also goes to one. Poof. We will show that when n is lage, both and a ( ) b > 0 (4) b ( ) a > 0 (5) since, togethe with Lemma 6, this imlies a majoity of A votes in state with obability one and a majoity of B votes in state with obability one. Since a b fom Lemma 7, the euied ineuality fo euation (5) is always satis ed. We now establish the euied ineuality fo euation (4) : Since > ; we have and so fom (3), fo lage n, ( ) ( ) + < ( ) o euivalently, b b a a b < a a b < and since a < b ; we have that fo all lage n ( ) ( ) b a < ( ) n, Now suose that the ineuality in euation (4) is false; that is, fo some lage which is euivalent to a ( ) b 0 b a 3

14 Substituting this into the ineuality deived above we have ( ) < Reaanging this yields < ( ) ( ) < ( ) ( ) < 4 ( ) (3 ) < (3 ) < which is clealy imossible. Refeences [] Austen-Smith, D. and J. Banks (996): Infomation Aggegation, Rationality, and the Condocet Juy Theoem, Ameican Political Science Review, 90, [] Böges, T. (004): Costly Voting, Ameican Economic Review, 94, [3] Feddesen T. and W. Pesendofe (996): The Swing Vote s Cuse, Ameican Economic Review, 86, [4] Feddesen T. and W. Pesendofe (998): Convicting the Innocent: The Infeioity of Unanimous Juy Vedicts unde Stategic Voting, Ameican Political Science Review, 9, [5] Ledyad, J. (98): The Paadox of Voting and Candidate Cometition: A Geneal Euilibium Analysis, in Essays in Contemoay Fields of Economics, G. Howich and J. Quik (eds.), Pudue Univesity Pess. [6] Ledyad, J. (984): The Pue Theoy of Lage Two-Candidate Elections, Public Choice, 44, 7 4. [7] Myeson, R. (998): Extended Poisson Games and the Condocet Juy Theoem, Games and Economic Behavio, 5, 3. [8] Myeson, R. (000): Lage Poisson Games, Jounal of Economic Theoy, 94, [9] Palfey, T., and H. Rosenthal (983): A Stategic Calculus of Voting, Public Choice, 4,

15 [0] Palfey, T., and H. Rosenthal (985): Vote Paticiation and Stategic Uncetainty, Ameican Political Science Review, 79, [] Taylo, C. and H. Yildiim (005): A Chaacteization of Vote Tunout in Lage Elections, mimeo, Duke Univesity, Octobe. 5

556: MATHEMATICAL STATISTICS I

556: MATHEMATICAL STATISTICS I CHAPTER 5: STOCHASTIC CONVERGENCE The following efinitions ae state in tems of scala anom vaiables, but exten natually to vecto anom vaiables efine on the same obability