Aggregate Accuracy under Majority Rule with
Heterogeneous Cost Functions
Minoru Kitahara

Yohei Sekiguchi

Graduate School of Economics, University of Tokyo

Graduate School of Economics, University of Tokyo

Abstract
We investigate an election model with costly accuracy improvement by allowing
heterogeneity in the cost functions. We find that the aggregate accuracy in large elections ischaracterized by the average value of the inverse of the second derivative at zero information.

We are grateful to Akihiko Matsui, Toshihiro Matsumura, and Daisuke Shimizu. We also thank Associate Editor Jordi Massoand two anonymous referees for their helpful advice. Financial assistance for this paper was provided by the JSPS Research
Fellowships for Young Scientists.
Citation: Kitahara, Minoru and Yohei Sekiguchi, (2006) "Aggregate Accuracy under Majority Rule with Heterogeneous Cost
Functions." Economics Bulletin, Vol. 4, No. 25 pp. 1-8
Submitted: February 14, 2006. Accepted: July 21, 2006.
URL: http://economicsbulletin.vanderbilt.edu/2006/volume4/EB-06D70001A.pdf

1. Introduction
Consider the following situation. A large society is using majority rule to try to choose the“correct” alternative of two choices. However, each member of the society, say i, must invest
the costs of Ci (qi ) so that he or she can vote for the correct alternative with probability qi .
The members’ utility from the correct choice by society is normalized to 1, and the incorrect
one to 0.
There are two opposite effects of the size of the society on the accuracy of society’s choice.
On the one hand, society can utilize the law of large numbers.1 On the other hand, large
size gives each member only a negligible incentive to improve his or her accuracy.2 If so, howdoes the aggregate accuracy, namely the probability that the majority of members vote forthe correct alternative, depend on the parameters of the cost functions?
In his pioneering work, Martinelli (2004) considered cases with homogeneous cost functions (i.e., Ci = C for all i), and found that, if C 0 (1/2) = 0, then the probability depends
only on C 00 (1/2). We allow heterogeneity here, and find that the generalized key parameter
is not the average of Ci00 (1/2) but the inverse of the average of 1/Ci00 (1/2).

2. The Model
2.1 Settings
For n ∈ N , the following normal form game is considered. There are 2n + 1 players, i.e.,“voters”. The strategy of each voter i = 1, ..., 2n + 1 is the accuracy of his or her vote,
qi,n ∈ [1/2, 1]. The corresponding payoff is given byÃ2n+1!
XPr
xj (qj,n ) ≥ n + 1 − Ci (qi,n ),(1)j=1where each xj (qj,n ) is independently drawn from its corresponding distribution as½
1 with probability
qj,n.
xj (qj,n ) =0 with probability 1 − qj,n1

At the extreme, this effect makes the accuracy converge to 1. This line dates back to Condorcet (1785).
For general results with exogenous accuracies in this line, see, e.g., Berend and Paroush (1998).
2
At the extreme, this effect makes all members give up any investment (i.e., qi = 1/2 for all i), even witha finite size of their society, which results in its choice being as if it were decided by a toss of a fair coin (i.e.,being correct only with probability 1/2). This line dates back to Downs (1957). For models with a binary
choice of accuracies in this line, see, e.g., Mukhopadhaya (2003).

1

The first term in (1) represents the expected utility from the chosen alternative, which, bynormalization, is equal to the probability that the correct alternative is chosen.3 The second
term represents the costs that yield the accuracy.
For simplicity, we focus on cases
P with a finite number of cost functions, {Ck }k=1,...,K ; forall k, mk,n voters have Ck , and k mk,n = 2n + 1. We assume that, for all k, mk,n /(2n +1) → αk∗ > 0 as n → ∞. For all k, Ck (q) is strictly increasing, strictly convex, and twice
differentiable in q. We also have Ck (1/2) = 0, Ck (1) > 1, and Ck0 (1/2) = 0.

2.2 Equilibrium and Aggregate Accuracy
The equilibrium concept is that of pure Nash. The first-order conditions imply that4Ã!
XPr
xj (qj,n ) = n = Ci0 (qi,n ) for i = 1, ..., 2n + 1.

(2)

j6=i

By the assumptions about cost functions (e.g., Ck (1) > 1 and Ck0 (1/2) = 0 for all k assure
the interiority), (a) (2) is the sufficient and necessary condition for {qi,n }i to constitutean equilibrium, and (b) there exists at least one strategy profile {qi,n }i satisfying (2) by
Kakutani’s fixed point theorem.5
The aggregate accuracy, namely the probability that the correct alternative is chosen in
the equilibrium, is given byÃ2n+1!
XPrxi (qi,n ) ≥ n + 1 .
i=1

3. The Result
We yield the following asymptotic property.
Theorem. Consider any equilibrium sequence {{qi,n }i }n . Then,Ã2n+1!
XPrxi (qi,n ) ≥ n + 1 → Φ(c∗ ) as n → ∞,(3)i=1
3Strategic voting is not explicitly considered here. For the importance of distinguishing between strategic
and sincere voting, see the seminal work of Austen-Smith and Banks (1996).
P
4Note that voters i can affect the “outcome” only when their vote is pivotal, i.e., j6=i xj (qj ) = n.
5
There may be multiple equilibria.

2where c∗ ∈ [0, ∞] solves
4φ(c∗ )
1
P=,1∗c∗k αk C 00 (1/2)(4)

kwhere Φ is the standard normal distribution function and φ the corresponding probability
density function.6
Thus, in a large society (i.e., asymptotically), the aggregate accuracy depends only on
the inverse of the average of 1/Ci00 (1/2).
PPi
Proof: For simplicity, denote Sn ≡ 2n+1
j6=i xj (qj,n ), and let cn be
i=1 xi (qi,n ) and Sn ≡defined byEn− 12√cn ≡ 2n+1≥ 0,Vnwhere En ≡ E[Sn ] =2n+1

Pi qi,n and Vn ≡ var(Sn ) =√Pi qi,n (1 − qi,n ). Then, (2) yields

Vn Pr (Sni = n) 1 φ(cn )Ci0 (qi,n )=.
VnEnφ(cn )cn− 122n+12n+1(5)

Pr(Sni = n) = qj,n Pr(Sni,j = n − 1) + (1 − qj,n ) Pr(Sni,j = n)(6)P
Denote Sni,j ≡ h6=i,j xh (qh,n ). Then, Pr(Sni,j = n − 1) ≤ Pr(Sni,j = n), since qh,n ≥ 1/2 for
all h 6= i, j. Thus,

is nonincreasing in qj,n . Therefore,Pr(Sni = n) ≤ PrÃX!
xj (1/2) = n

j6=iµ ¶
2n=(1/4)n ,n(7)which implies, by (2),lim max |qi,n − 1/2| = 0.

n→∞iNote that, by (6) for i and j,¡¢i,ji,j1
1 Pr(Sn =n−1)−Pr(Sn =n)i+q−i,ji,jj,n22 Pr(Sn =n−1)+Pr(SnPr(Sn = n)=n)=.
¡¢i,ji,jj
1
1 Pr(Sn =n−1)−Pr(Sn =n)Pr(Sn = n)+q−i,ji,ji,n22 Pr(Sn =n−1)+Pr(Sn =n)

Thus, by (8),¯¯¯¯¯¯¯ Pr(Sni = n)¯
1− 1¯¯ ≤ 4 max ¯¯qi,n − ¯¯ → 0 as n → ∞.
max ¯¯jii,j2Pr(Sn = n)
6We use conventions as 1/∞ = 0 and 1/0 = ∞.

3(8)By (2), it implies¯¯ 0¯¯ Ci (qi,n )lim max ¯¯ 0− 1¯¯ = 0.
n→∞ i,jCj (qj,n )Therefore, since αk∗ > 0 for all k,Ci0 (qi,n )Ci0 (qi,n )1
1=→P ∗ 1= P
as n → ∞.
P2n+10 (qC)
En1
q−1/2
j,nj,n2n+1 j
αk C 00 (1/2)
j=1 (qj,n −1/2)−2k02n+1j=1 C 0 (q) C (q)2n+1ii,njj,nk2n+1

Suppose that, along some subsequence, cn → ∞. Then, by (7), we have¶np µ2n¶ µ 1 ¶n µp
1i
Vn Pr(Sn = n) ≤Vn
1−22nrr¡2n¢Vn2n + 1 n
1=→√
as n → ∞,n42n + 1πn √πn2πpwhere the first term converges to 1/4 as we will see in (ii) below, and the third term
converges to 1 by Stirling’s formula.7 Thus, along such subsequences, the LHS of (5) converges
to 0, which equals 4φ(∞)/∞.
Suppose that along some subsequence, cn → c∗ < ∞. Since(i)2n+1X1lim
E[exp(xi (qi,n ))] ≤ exp(1) < ∞,n→∞ 2n + 1
i=12n+1X
11var(xi (qi,n )) = > 0, and(ii) lim
n→∞ 2n + 1
4
i=12n+1
X
11(iii) lim
min{Pr(xi (qi,n ) = 0), Pr(xi (qi,n ) = 1)} = > 0,n→∞ 2n + 1
2i=1

where (ii) and (iii) follow from (8), the premises (I), (II), and (III’) of Theorem 1 in McDonald(1979) are satisfied. Note that |(n − 1) − En | ≥ (2n + 1)(1/2) − (n − 1) = 3/2 > 1, and|(n − 1) − En ||(n + 1) − En |lim p
= lim p= c∗ .
n→∞n→∞(2n + 1)/4(2n + 1)/4
Therefore, the local limit theorem of McDonald (1979) implies that√Vn Pr (Sn = n − 1)
= 1,limn→∞
φ(cn )
7nNote that √4πn =√2π(2n)(2n)2n e−2n√.
( 2πnnn e−n )24(9)and that if c∗ > 0, then,√limn→∞Vn Pr (Sn = n + 1)
= 1.
φ(cn )(10)

As we will briefly see in the final paragraph,Pr(Sni = n) ≥ Pr(Sn = n − 1), and| Pr(Sni = n) − Pr(Sn = n + 1)| ≤ | Pr(Sn = n + 1) − Pr(Sn = n − 1)|.

(11)
(12)

Thus, by (9) and (11), if c∗ = 0, then, along such subsequences the LHS of (5) diverges toinfinity, which equals 4φ(0)/0. If c∗ > 0, then, by (9), (10), and (12),¯√¯¯ Vn Pr (Sni = n)¯¯lim max ¯− 1¯¯ = 0.
n→∞ iφ(cn )Therefore, since limn→∞ Vn /(2n + 1) = 1/4, as seen in (ii) above, along such subsequences
the LHS of (5) converges to 4φ(c∗ )/c∗ .
These imply that cn → c∗ solving (4) as n grows. (3) is then a simple consequence of thecentral limit theorem.8
Now, we briefly see (11) and (12). Since qi,n ≥ 1/2 for all i,Pr(Sni = n − 2) ≤ Pr(Sni = n − 1) ≤ Pr(Sni = n), andPr(Sni = n − 1) ≤ Pr(Sni = n + 1).

(13)
(14)

Note that for any m,Pr(Sn = m) = qi,n Pr(Sni = m − 1) + (1 − qi,n ) Pr(Sni = m).

(15)(13) and (15) imply (11). If Pr(Sni = n + 1) ≥ Pr(Sni = n), then,Pr(Sn = n + 1) ≥ Pr(Sni = n)
≥ qi,n Pr(Sni = n − 2) + (1 − qi,n ) Pr(Sni = n − 1) = Pr(Sn = n − 1),which implies (12). If Pr(Sni = n + 1) < Pr(Sni = n), then, by (13), (14), and (15),0 <≤≤=8Pr(Sni = n) − Pr(Sn = n + 1)
Pr(Sn = n + 1) − Pr(Sni = n + 1)
Pr(Sn = n + 1) − (qi,n Pr(Sni = n − 2) + (1 − qi,n ) Pr(Sni = n − 1))
Pr(Sn = n + 1) − Pr(Sn = n − 1),

See, e.g., Feller (1971).

5where the second inequality follows from qi,n ≥ 1/2. Thus, (12) holds.

2

Our theorem can derive the corresponding result of Martinelli (2004).
Corollary (Martinelli (2004)). Suppose that all voters have an identical cost function, C.
Then, the aggregate accuracy converges to Φ(c∗ ) as n grows, where c∗ solves
4φ(c∗ )
= C 00 (1/2) .
c∗(16)To be precise, only symmetric equilibria (i.e., qi,n = qn for all i) are considered in Martinelli(2004). Note that qi,n 6= qj,n with Ci = Cj can occur under our equilibrium concept. Thus,in allowing the possibility of asymmetric equilibria, we generalize his result even within caseswith homogeneous cost functions.

4. Discussions
Our result provides some answers to the topics below. Note that one could not find them if
one misinterpreted the result of (16) in the homogeneous cost cases and used the average ofCi00 (1/2) as the proxy.

4.1 Severity of Convergence to 100% Accuracy
Martinelli’s (2004) result of (16) implies that, if cost functions are homogeneous, then theaggregate accuracy cannot converge to 1 unless Ci00 (1/2) = 0 for all voters.9 One may
conclude, based on this, that the convergence to 100% accuracy is implausible. However, ourresult suggests that such a conclusion is too premature. Observe that, if Ck00 (1/2) = 0 forsome k, then, however small αk∗ > 0 is,Xkαk∗

1Ck00 (1/2)= +∞,which implies Φ(c∗ ) = 1. Thus, to attain convergence to 100% accuracy, extreme costfunctions satisfying Ci00 (1/2) = 0 are required only for an arbitrarily small fraction of voters.

4.2 Heterogeneous Utilities and the Effect of Pie Allocation
9Note that Φ(c∗ ) = 1 corresponds to the RHS being 0.

6

Consider that all the members have an identical cost function, C(·), but may differ in theirutilities, ri , from the correct choice by the society, i.e., the payoffs of (1) are replaced byÃ2n+1!
Xri Prxj (qj,n ) ≥ n + 1 − C(qi,n ).
j=1

We can deal with this situation within our framework by letting cost functions as Ck = C/rk .
Then, how does the change in the distribution of the utilities, {rk }k=1,...,K , affect theaggregate accuracy? Our result provides a clear-cut answer. In fact,P ∗X
α rk1∗αk 00= 00k k .
Ck (1/2)
C (1/2)kP
Thus, the aggregate accuracy depends only on k αk∗ rk , the average utility among the members.
Note that, if the average is unchanged, then the sum is also unchanged. Thus, we caninterpret changing the distribution of the utilities, but retaining its average, as changing theway of allocating the pie resulting from the society’s choice. Then, the above result means
that, unless we assume a difference in voting abilities (represented by cost functions) among
members, at least asymptotically, the aggregate accuracy is not affected by the manner of
allocating the pie; for example, we can allocate more to vulnerable people in the society
without causing any deterioration of the aggregate accuracy.

References
Austen-Smith, D. and Banks, J. S. (1996) “Information Aggregation, Rationality, and theCondorcet Jury Theorem” American Political Science Review 90, 34–45.
Berend, D. and Paroush, J. (1998) “When is Condorcet’s Jury Theorem Valid?” Social
Choice and Welfare 15, 481–88.
Condorcet, M. de (1785) Essai sur l’Application de l’Analyse à la Probabilité des Décisions
Rendue à la Pluralité des Voix: Paris.
Downs, A. (1957) An Economic Theory of Democracy, HarperCollins Publishers: New York.
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, John Wiley& Sons: New York.
Martinelli, C. (2004) “Would Rational Voters Acquire Costly Information?” forthcoming in
Journal of Economic Theory.
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McDonald, D. (1979) “A Local Limit Theorem for Large Deviations of Sums of Independent,
Nonidentically Distributed Random Variables” The Annals of Probability 7, 526–531.
Mukhopadhaya, K. (2003) “Jury Size and the Free Rider Problem” The Journal of Law,
Economics and Organization 19, 24–44.

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