Aggregate Accuracy under Majority Rule with Heterogeneous Cost Functions. Abstract

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1 Aggregate Accuracy uder Majority Rule with Heterogeeous Cost Fuctios Mioru Kitahara Graduate School of Ecoomics, Uiversity of Tokyo Yohei Sekiguchi Graduate School of Ecoomics, Uiversity of Tokyo Abstract We ivestigate a electio model with costly accuracy improvemet by allowig heterogeeity i the cost fuctios. We fid that the aggregate accuracy i large electios is characterized by the average value of the iverse of the secod derivative at zero iformatio. We are grateful to Akihiko Matsui, Toshihiro Matsumura, ad Daisuke Shimizu. We also thak Associate Editor Jordi Masso ad two aoymous referees for their helpful advice. Fiacial assistace for this paper was provided by the JSPS Research Fellowships for Youg Scietists. Citatio: Kitahara, Mioru ad Yohei Sekiguchi, (2006 "Aggregate Accuracy uder Majority Rule with Heterogeeous Cost Fuctios." Ecoomics Bulleti, Vol. 4, No. 25 pp. -8 Submitted: February 4, Accepted: July 2, URL:

2 . Itroductio Cosider the followig situatio. A large society is usig majority rule to try to choose the correct alterative of two choices. However, each member of the society, say i, must ivest the costs of C i (q i so that he or she ca vote for the correct alterative with probability q i. The members utility from the correct choice by society is ormalized to, ad the icorrect oe to 0. There are two opposite effects of the size of the society o the accuracy of society s choice. O the oe had, society ca utilize the law of large umbers. O the other had, large size gives each member oly a egligible icetive to improve his or her accuracy. 2 If so, how does the aggregate accuracy, amely the probability that the majority of members vote for the correct alterative, deped o the parameters of the cost fuctios? I his pioeerig work, Martielli (2004 cosidered cases with homogeeous cost fuctios (i.e., C i = C for all i, ad foud that, if C (/2 = 0, the the probability depeds oly o C (/2. We allow heterogeeity here, ad fid that the geeralized key parameter is ot the average of C i (/2 but the iverse of the average of /C i (/2. 2. The Model 2. Settigs For N, the followig ormal form game is cosidered. There are 2 + players, i.e., voters. The strategy of each voter i =,..., 2 + is the accuracy of his or her vote, q i, [/2, ]. The correspodig payoff is give by ( 2+ Pr x j (q j, + C i (q i,, ( j= where each x j (q j, is idepedetly draw from its correspodig distributio as { with probability qj, x j (q j, =. 0 with probability q j, At the extreme, this effect makes the accuracy coverge to. This lie dates back to Codorcet (785. For geeral results with exogeous accuracies i this lie, see, e.g., Bered ad Paroush ( At the extreme, this effect makes all members give up ay ivestmet (i.e., q i = /2 for all i, eve with a fiite size of their society, which results i its choice beig as if it were decided by a toss of a fair coi (i.e., beig correct oly with probability /2. This lie dates back to Dows (957. For models with a biary choice of accuracies i this lie, see, e.g., Mukhopadhaya (2003.

3 The first term i ( represets the expected utility from the chose alterative, which, by ormalizatio, is equal to the probability that the correct alterative is chose. 3 The secod term represets the costs that yield the accuracy. For simplicity, we focus o cases with a fiite umber of cost fuctios, {C k } k=,...,k ; for all k, m k, voters have C k, ad k m k, = 2 +. We assume that, for all k, m k, /(2 + αk > 0 as. For all k, C k(q is strictly icreasig, strictly covex, ad twice differetiable i q. We also have C k (/2 = 0, C k ( >, ad C k (/2 = Equilibrium ad Aggregate Accuracy The equilibrium cocept is that of pure Nash. The first-order coditios imply that 4 ( Pr x j (q j, = = C i(q i, for i =,..., 2 +. (2 j i By the assumptios about cost fuctios (e.g., C k ( > ad C k (/2 = 0 for all k assure the iteriority, (a (2 is the sufficiet ad ecessary coditio for {q i, } i to costitute a equilibrium, ad (b there exists at least oe strategy profile {q i, } i satisfyig (2 by Kakutai s fixed poit theorem. 5 The aggregate accuracy, amely the probability that the correct alterative is chose i the equilibrium, is give by ( 2+ Pr x i (q i, +. i= We yield the followig asymptotic property. 3. The Result Theorem. Cosider ay equilibrium sequece {{q i, } i }. The, ( 2+ Pr x i (q i, + Φ(c as, (3 i= 3 Strategic votig is ot explicitly cosidered here. For the importace of distiguishig betwee strategic ad sicere votig, see the semial work of Auste-Smith ad Baks ( Note that voters i ca affect the outcome oly whe their vote is pivotal, i.e., j i x j(q j =. 5 There may be multiple equilibria. 2

4 where c [0, ] solves 4 φ(c = c k α, (4 k C k (/2 where Φ is the stadard ormal distributio fuctio ad φ the correspodig probability desity fuctio. 6 Thus, i a large society (i.e., asymptotically, the aggregate accuracy depeds oly o the iverse of the average of /C i (/2. Proof: For simplicity, deote S 2+ i= x i (q i, ad S i j i x j(q j,, ad let c be defied by c E 2+ 2 V 2+ 0, where E E[S ] = i q i, ad V var(s = i q i,( q i,. The, (2 yields V Pr (S i = φ(c V 2+ Deote S i,j h i,j x h(q h,. The, Pr(S i,j all h i, j. Thus, Pr(S i = = q j, Pr(S i,j φ(c = C i(q i, E c. ( = Pr(S i,j is oicreasig i q j,. Therefore, ( Pr(S i = Pr x j (/2 = = j i =, sice q h, /2 for = + ( q j, Pr(S i,j = (6 ( 2 (/4, (7 which implies, by (2, Note that, by (6 for i ad j, max q i, /2 = 0. (8 i Thus, by (8, max i,j Pr(S i = Pr(S j = = 2 + ( q j, ( q i, 2 Pr(S i,j Pr(S i,j Pr(S i,j Pr(S i,j = Pr(S i,j = = +Pr(S i,j = = Pr(S i,j = = +Pr(S i,j = Pr(S i = Pr(S j = 4 max i q i, 2 0 as. 6 We use covetios as / = 0 ad /0 =.. 3

5 By (2, it implies Therefore, sice αk > 0 for all k, C i(q i, E 2+ 2 = C i(q i, P 2+ j= (q j, /2 2+ max C i(q i, i,j C j (q j, = 0. = P 2+ j= C j (q j, q j, /2 C i (q i, C j (q j, 2+ k α k C k (/2 as. Suppose that, alog some subsequece, c. The, by (7, we have V Pr(S i = ( ( ( 2 V 2 2 ( 2 V 2 + = 4 as, 2 + π π 2π where the first term coverges to /4 as we will see i (ii below, ad the third term coverges to by Stirlig s formula. 7 Thus, alog such subsequeces, the LHS of (5 coverges to 0, which equals 4φ( /. Suppose that alog some subsequece, c c <. Sice (i (ii (iii 2+ E[exp(x i (q i, ] exp( <, 2 + i= 2+ var(x i (q i, = i= > 0, ad 2+ mi{pr(x i (q i, = 0, Pr(x i (q i, = } = > 0, i= where (ii ad (iii follow from (8, the premises (I, (II, ad (III of Theorem i McDoald (979 are satisfied. Note that ( E (2 + (/2 ( = 3/2 >, ad ( E ( + E = = c. (2 + /4 (2 + /4 Therefore, the local it theorem of McDoald (979 implies that 7 4 Note that 2π(2(2 π = 2 e 2 (. 2π e 2 V Pr (S = φ(c =, (9 4

6 ad that if c > 0, the, As we will briefly see i the fial paragraph, V Pr (S = + φ(c =. (0 Pr(S i = Pr(S =, ad ( Pr(S i = Pr(S = + Pr(S = + Pr(S =. (2 Thus, by (9 ad (, if c = 0, the, alog such subsequeces the LHS of (5 diverges to ifiity, which equals 4φ(0/0. If c > 0, the, by (9, (0, ad (2, max V Pr (S i = i φ(c = 0. Therefore, sice V /(2 + = /4, as see i (ii above, alog such subsequeces the LHS of (5 coverges to 4φ(c /c. These imply that c c solvig (4 as grows. (3 is the a simple cosequece of the cetral it theorem. 8 Now, we briefly see ( ad (2. Sice q i, /2 for all i, Note that for ay m, Pr(S i = 2 Pr(S i = Pr(S i =, ad (3 Pr(S i = Pr(S i = +. (4 Pr(S = m = q i, Pr(S i = m + ( q i, Pr(S i = m. (5 (3 ad (5 imply (. If Pr(S i = + Pr(S i =, the, Pr(S = + Pr(S i = q i, Pr(S i = 2 + ( q i, Pr(S i = = Pr(S =, which implies (2. If Pr(S i = + < Pr(S i =, the, by (3, (4, ad (5, 0 < Pr(S i = Pr(S = + Pr(S = + Pr(S i = + Pr(S = + (q i, Pr(S i = 2 + ( q i, Pr(S i = 8 See, e.g., Feller (97. = Pr(S = + Pr(S =, 5

7 where the secod iequality follows from q i, /2. Thus, (2 holds. Our theorem ca derive the correspodig result of Martielli (2004. Corollary (Martielli (2004. Suppose that all voters have a idetical cost fuctio, C. The, the aggregate accuracy coverges to Φ(c as grows, where c solves 4 φ(c c = C (/2. (6 To be precise, oly symmetric equilibria (i.e., q i, = q for all i are cosidered i Martielli (2004. Note that q i, q j, with C i = C j ca occur uder our equilibrium cocept. Thus, i allowig the possibility of asymmetric equilibria, we geeralize his result eve withi cases with homogeeous cost fuctios. 4. Discussios Our result provides some aswers to the topics below. Note that oe could ot fid them if oe misiterpreted the result of (6 i the homogeeous cost cases ad used the average of C i (/2 as the proxy. 4. Severity of Covergece to 00% Accuracy Martielli s (2004 result of (6 implies that, if cost fuctios are homogeeous, the the aggregate accuracy caot coverge to uless C i (/2 = 0 for all voters. 9 Oe may coclude, based o this, that the covergece to 00% accuracy is implausible. However, our result suggests that such a coclusio is too premature. Observe that, if C k (/2 = 0 for some k, the, however small αk > 0 is, k α k C k = +, (/2 which implies Φ(c =. Thus, to attai covergece to 00% accuracy, extreme cost fuctios satisfyig C i (/2 = 0 are required oly for a arbitrarily small fractio of voters. 4.2 Heterogeeous Utilities ad the Effect of Pie Allocatio 9 Note that Φ(c = correspods to the RHS beig 0. 6

8 Cosider that all the members have a idetical cost fuctio, C(, but may differ i their utilities, r i, from the correct choice by the society, i.e., the payoffs of ( are replaced by ( 2+ r i Pr x j (q j, + C(q i,. k j= We ca deal with this situatio withi our framework by lettig cost fuctios as C k = C/r k. The, how does the chage i the distributio of the utilities, {r k } k=,...,k, affect the aggregate accuracy? Our result provides a clear-cut aswer. I fact, αk C k (/2 = k α k r k C (/2. Thus, the aggregate accuracy depeds oly o k α k r k, the average utility amog the members. Note that, if the average is uchaged, the the sum is also uchaged. Thus, we ca iterpret chagig the distributio of the utilities, but retaiig its average, as chagig the way of allocatig the pie resultig from the society s choice. The, the above result meas that, uless we assume a differece i votig abilities (represeted by cost fuctios amog members, at least asymptotically, the aggregate accuracy is ot affected by the maer of allocatig the pie; for example, we ca allocate more to vulerable people i the society without causig ay deterioratio of the aggregate accuracy. Refereces Auste-Smith, D. ad Baks, J. S. (996 Iformatio Aggregatio, Ratioality, ad the Codorcet Jury Theorem America Political Sciece Review 90, Bered, D. ad Paroush, J. (998 Whe is Codorcet s Jury Theorem Valid? Choice ad Welfare 5, Social Codorcet, M. de (785 Essai sur l Applicatio de l Aalyse à la Probabilité des Décisios Redue à la Pluralité des Voix: Paris. Dows, A. (957 A Ecoomic Theory of Democracy, HarperCollis Publishers: New York. Feller, W. (97 A Itroductio to Probability Theory ad Its Applicatios, Joh Wiley & Sos: New York. Martielli, C. (2004 Would Ratioal Voters Acquire Costly Iformatio? forthcomig i Joural of Ecoomic Theory. 7

9 McDoald, D. (979 A Local Limit Theorem for Large Deviatios of Sums of Idepedet, Noidetically Distributed Radom Variables The Aals of Probability 7, Mukhopadhaya, K. (2003 Jury Size ad the Free Rider Problem The Joural of Law, Ecoomics ad Orgaizatio 9,

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