Condorcet's Jury Theorem and the Optimum Number of Voters
| Author | Jason Brennan |
| Published date | 01 June 2011 |
| Date | 01 June 2011 |
| DOI | 10.1111/j.1467-9256.2011.01403.x |
| Subject Matter | Research and Analysis |
P O L I T I C S : 2 0 1 1 V O L 3 1 ( 2 ) , 5 5 – 6 2
Research and Analysis
Condorcet’s Jury Theorem and
the Optimum Number of Votersponl_140355..62
Jason Brennan
Brown University
Many political theorists and philosophers use Condorcet’s Jury Theorem to defend democracy. This
article illustrates an uncomfortable implication of Condorcet’s Jury Theorem. Realistically, when
the conditions of Condorcet’s Jury Theorem hold, even in very high stakes elections, having more
than 100,000 citizens vote does no significant good in securing good political outcomes. On the
Condorcet model, unless voters enjoy voting, or unless they produce some other value by voting,
then the cost to most voters of voting exceeds the expected epistemic benefits to the common good
of their casting a vote. Anyone who is committed to democracy on the basis of the Jury Theorem
ought also to hold that widespread voting is wasteful, at least unless he or she can provide some
further justification of mass democratic participation.
Introduction
Condorcet’s Jury Theorem claims that:
‘on a dichotomous choice, individuals who all have the same level of competence
(or probability of being correct) above 0.5, can make collective decisions under
majority rule with a competence that approaches 1 (infallibility) as either the size
of the group or individual competence goes up’ (Estlund, 1994, p. 131).
If voters (1) are deciding between two candidates or policies under majority rule
procedure, (2) are on average more likely than not to make the right choice, then
(3) as the number of voters increases, the electorate is all but certain to make the
right choice.1 Many people use Condorcet’s Jury Theorem to defend democracy on
instrumental grounds (e.g. Barry, 1965; Dagger, 1997, pp. 96–97; Goodin, 2003;
Grofman and Feld, 1988; List and Goodin, 2001). They claim the Jury Theorem
shows that democracies are an excellent way to produce good political outcomes.
This article illustrates an uncomfortable implication of the Jury Theorem. Realisti-
cally, when the conditions of Condorcet’s Jury Theorem hold, even in very high
stakes elections, having more than 100,000 citizens vote does no significant good in
securing good political outcomes. On the Condorcet model, unless voters enjoy
voting, or unless they produce some other value by voting, then the cost to most
voters of voting exceeds the epistemic benefits to the common good of their casting
a vote. Anyone who is committed to democracy on the basis of the Jury Theorem
ought also to hold that widespread voting is wasteful, at least unless he or she can
provide some further justification of mass democratic participation.
© 2011 The Author. Politics © 2011 Political Studies Association
56
J A S O N B R E N N A N
Political theorists and others dispute whether Condorcet’s Jury Theorem, or some
version of it, can be used to defend democracy. Many theorists believe that the
conditions of the theorem do not hold in real life, and so democracies cannot be
modelled by the theorem. Others think that real democracies can be modelled by
the theorem. I do not take a stand on this issue here. Instead, I just want to inquire
what the optimum number of voters would be if Condorcet’s Jury Theorem does
apply. In high stakes elections, is there any real epistemic value in having a million
voters vote as opposed to 100,000? Or, alternatively, on the Condorcetian model, is
mass democracy a kind of epistemic overkill?
The marginal epistemic value of votes
It is well known that on Condorcet’s Jury Theorem even small electorates can have
a near certain chance of choosing the right answer. However, democratic outcomes
can be high stakes. Even if adding another competent voter only makes the
collective slightly more competent, perhaps the competent voter’s marginal contri-
bution is still significant enough to warrant him or her voting. Or, perhaps not. To
find out, we need to do some illustrative calculations.
According to Condorcet’s Jury Theorem, the probability (PN) that an electorate of
size N will make a correct decision is given by equation 1 (Estlund, 1994, p. 136;
Goodin and Estlund, 2004, p. 142; Mueller, 2003, p. 129):2
N
N
P
i
N i
= ∑
!
(1
) −
N
pc
pc
(1)
= (N − i)
( ) −
!i!
i m
where m = number of votes need to win: (N + 1)/2 for a simply majority,
pc = the probability that an individual voter will make the correct choice. If
pc > 0.5, PN rapidly approaches 1. When pc > 0.5, PN behaves like an exponential
rise function of N, with a limit of 1. For instance, suppose pc = 0.51; that is,
suppose that each voter has merely a 51 per cent chance of making the correct
choice on his or her own. If so, then when N = 101, PN = 0.58. That is, when
there are 101 voters in the electorate, there is a 58 per cent chance the electorate
will make the correct choice. When N = 501, PN = 0.673. When N = 1,001,
PN = 0.737. When N = 5,001, PN = 0.921. When N = 10,001, PN = 0.977. As N → •,
PN → 1. Thus, even when pc is just barely above 0.5, Condorcet’s Jury Theorem
shows that an electorate of only 10,000 voters is close to certain to make the
correct choice.
Let us assume, on behalf of the Condorcetian,3 that every voter has a greater than
0.5 chance of being correct. If so, it follows that every additional voter increases, by
some amount, the probability that the electorate will make the correct choice.
Let us call the amount of competence added to the electorate by the Nth voter the
Nth voter’s marginal contribution to group competence. The Nth voter’s marginal con-
tribution to group competence (DPN) in helping the electorate make the correct
choice between two candidates or policies is given by equation 2:
ΔP =
−
N
PN
PN−1
(2)
© 2011 The Author. Politics © 2011 Political Studies Association
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