# Ask the audience or Condorcet goes to Washington

by on October 19, 2004

What is the US Presidential election about? Well, one possible answer is that it is about which of George W. Bush and John Kerry would make the best President of the United States. Now there’s certainly room for disagreement about the relevant qualities to be best President, but much of the media and blogospheric discourse is couched in such a way as to appear to be discussing a matter of fact: best translates as “most competent”, “wisest” etc. I’m going to assume—for the purposes of this post alone and contrary to my saner instincts—that a matter of fact is indeed involved. Given that simplifying assumption, the matter of determining who would be the best President by a democratic vote is something we might justify by invoking Condorcet ’s jury theorem. According to the jury theorem (which I cite in Zev Trachtenberg’s formulation [1])

the probability that majority is correct ( Pm ) is given by the formula
v h-k/(v h-ke h-k
), where number of voters = n = hk , where h is the number of voters in the majority, v is the probability that each voter will give the correct answer, and e is the probability that each voter will give the wrong answer.

This has the remarkable consquence that just in case we expect each voter’s competence slighly to exceed the tossing of a fair coin (say we expect each voter to be right 50.1 per cent of the time), and just in case we can interpret “each voter” to mean “the average voter”, then with an electorate of, say, 100 million, the probability of a majority getting the result right approaches one. Of course, there’s a flip side to this: if the each voter has a < .5 probability of getting the right result, the majority will almost certainly be wrong!

So what should we think, ex ante , about the competence of the average American voter? The votemaster at the excellent electoral-vote.com opines:

Are the voters stupid? It is not considered politically correct to point out that an awful lot of voters don’t have a clue what they are talking about. A recent poll from Middle Tennessee State University sheds some light on the subject. For example, when asked which candidate wants to roll back the tax cuts for people making over \$200,000 a year, a quarter thought it was Bush and a quarter didn’t know. And it goes down hill from there. When asked which candidate supports specific positions on various issues, the results were no better than chance. While this poll was in Tennessee, I strongly suspect a similar poll in other states would get similar results. I find it dismaying that many people will vote for Bush because they want to tax the rich (which he opposes) or vote for Kerry because they want school vouchers for religious schools (which he opposes).

(Lest Carol Gould or her apologists think that this post reflects the anti-Americanism of a sneering Brit, let me say (a) I’m quoting an American and (b) that I’m far from convinced that citizens of the UK would fare much better than the people of Tennessee were their competence to be evaluated in a similar poll.) [2]

fn1. Trachtenberg, Making Citizens p. 281 n. 6

fn2. A commenter to a recent post of mine asked, sarcastically, whether the I thought flipping a coin would have been superior to having the Supreme Court decide on the outcome in 2000. Actually, I do think flipping a coin would have been a better method then. Whether it would be a better method than having the US electorate decide is questionable, although if voter-competence is such that individuals are more likely to get the wrong answer than the right one, it would yield a better chance of choosing the best President. Observant and thoughtful readers will also notice that, since Al Gore won a majority of the popular vote in 2000, I ought to believe that either Bush was the right answer then or that average voter competence has declined below the .5 level in the past four years….or perhaps I should believe that voter competence then as now exceeds .5 and that Kerry will inevitably triumph, or …..you do the permutations.

{ 36 comments }

1

lemuel pitkin 10.19.04 at 5:46 pm

Your argument and your evidence are weirdly out of sync here.

First you say that individual voters’ perception need to be only slightly better than random for the election outcome to be correct (given voters’ preferences), then you present as a problem evidence that voters’ perceptions, while not as acurate as we might prefer, are much better than random.

2

Chris Bertram 10.19.04 at 5:55 pm

I don’t say anything about “perception”, rather I wrote about competence. And I didn’t say whether I thought it was _actually_ above the threshold or not.

But on a matter of record, you refer to

bq. evidence that voters’ perceptions, while not as acurate as we might prefer, are much better than random.

Whereas the quote from electoral-vote.com was

bq. When asked which candidate supports specific positions on various issues, the results were no better than chance.

3

Jim Harrison 10.19.04 at 5:57 pm

The assumption here is that the voters are deciding a matter of fact rather than attempting to further their own interests or that all the voters are assesssing the same matter of fact instead of drastically different matters of fact as appears to be the case in the current election.

Rather like the equations in the beginning of Capital, the Condercet equation is useless in practice—ideological mathematics.

4

Chris Lawrence 10.19.04 at 5:59 pm

I’m far from convinced that citizens of the UK would fare much better than the people of Tennessee were their competence to be evaluated in a similar poll.

I suspect you are right. See, e.g., Gordon, Stacy B. and Gary M. Segura. 1997. “Cross-National Variation in the Political Sophistication of Individuals: Capability or Choice?” Journal of Politics 59:126-­47.

5

Chris Lawrence 10.19.04 at 6:01 pm

I’m far from convinced that citizens of the UK would fare much better than the people of Tennessee were their competence to be evaluated in a similar poll.

I suspect you are right. See, e.g., Gordon, Stacy B. and Gary M. Segura. 1997. “Cross-National Variation in the Political Sophistication of Individuals: Capability or Choice?” Journal of Politics 59:126-­47.

6

Giles 10.19.04 at 6:02 pm

“For example, when asked which candidate wants to roll back the tax cuts for people making over \$200,000 a year, a quarter thought it was Bush and a quarter didn’t know”

so by a margin of 60:30 the voterrs got it right that this was Kerry. By your formulation this is going to giveconvergence to the the right answer with a fairly small population.

I think you’re asking the wrong question – like Galtons Ox – the question is not whether the voters are stupid, the question is why do academic’s consistently think they are more intelligent than a crowd of idiots. Even when theory tells them otherwise.

7

Chris Lawrence 10.19.04 at 6:03 pm

Ick, apologies for the double-post; I got a 500 Internal Server Error message on the first one, so assumed it didn’t go through.

8

Giles 10.19.04 at 6:10 pm

“For example, when asked which candidate wants to roll back the tax cuts for people making over \$200,000 a year, a quarter thought it was Bush and a quarter didn’t know”

so by a margin of 60:30 the voterrs got it right that this was Kerry. By your formulation this is going to giveconvergence to the the right answer with a fairly small population.

I think you’re asking the wrong question – like Galtons Ox – the question is not whether the voters are stupid, the question is why do academic’s consistently think they are more intelligent than a crowd of idiots. Even when theory tells them otherwise.

9

Hogan 10.19.04 at 6:16 pm

After the uprising of the 17th June
The Secretary of the Writers Union
Had leaflets distributed in the Stalinallee
Stating that the people
Had forfeited the confidence of the government
And could win it back only
By redoubled efforts. Would it not be easier
In that case for the government
To dissolve the people
And elect another?

–Bertolt Brecht

(Someone had to do it.)

10

Chris Bertram 10.19.04 at 6:45 pm

Very good!

————

BTW, Jim Harrison, the Condorcet equation can actually be quite useful in some circumstances. I think there are even some applications in evolutionary biology. But I’m only deploying it here as a bit of fun.

11

Brett Bellmore 10.19.04 at 6:50 pm

Of course, it’s not like the people are voting without talking to each other; Perhaps we ought to be applying the Condorcet theorem to local opinion leaders, and not individual voters?

But I think Jim has it right: Voters aren’t all trying to answer the same question, so it’s simply inapplicable.

12

Nicholas Weininger 10.19.04 at 7:19 pm

Jim’s problem isn’t the only one. Condorcet’s theorem assumes that the voters’ decisions are independent, right? (That is, that each voter is flipping a different coin with a probability v of heads or “right” and a probability e of tails or “wrong”).

That’s a really bad assumption to make about a large voting populace, given the herd mentality/popularity contest dimensions of all real large-scale elections.

13

Sebastian Holsclaw 10.19.04 at 7:19 pm

“For example, when asked which candidate wants to roll back the tax cuts for people making over \$200,000 a year, a quarter thought it was Bush and a quarter didn’t know.”

Perhaps the problem is that most voters don’t think that particular issue is important one way or another.

14

lemuel pitkin 10.19.04 at 7:36 pm

Y’know, as soon as I posted that comment, I thought, why am I posting this? I don’t know from Condorcet and have no stake in this argument. I’ll shut up now….

15

Lukas 10.19.04 at 7:49 pm

But the Tennessee survey wasn’t a survey of voters; it was a survey of people over 18. I’m guessing people who care enough to become well-informed are more likely to vote, so voters are as a group more likely to be well-informed.

16

Phil Hunt 10.19.04 at 7:49 pm

I’m far from convinced that citizens of the UK would fare much better

Certainly not the 11% of the British population who think Hitler was fictional.

17

Lisa 10.19.04 at 8:55 pm

Showing that the majority of people only knew the answers on a couple of the candidate’s positions, or that they support a candidate whose position differs from theirs on an issue, doesn’t mean what I think you think it does (That voters are voting for a candidate out of a mistaken belief that he agrees with them, or for no rational reason whatsoever).

It’s very possible that a lot of people only CARE about one or two issues, and either a) don’t bother to inform themselves on the others, and/or b) don’t care if their preferred candidate disagrees with them on the others. For example, one might vote for a candidate that would provide one with the biggest tax cuts, even if one opposes his stances on every other issue. In fact, one might not even bother to find out his stance on things like vouchers/abortion/pollution.

Most people have a preference on most issues, but there are degrees of preference, and if you don’t assign much importance to an issue, then its perfectly rational to not inform yourself on it, or to ignore your candidates opposing stance.

18

Michael Otsuka 10.19.04 at 9:19 pm

Since so much is riding on this election, let’s play it safe and make _n_ (the number of voters) larger by adding the votes of the citizens of, say, Canada, France, the UK, Spain, Russia, Japan, Australia, Mexico, Israel, and South Korea, in which case it’s Kerry by a landslide:

I’m happy to add voters from other countries too. The more the merrier.

19

Michael Otsuka 10.19.04 at 9:25 pm

PS: Now I think I’ve figured out what Kerry means by his ‘global test’.

20

james 10.19.04 at 9:59 pm

It will only anger the American voter to suggest that foreign nationals should be involved in electing the US President.

21

g 10.19.04 at 10:46 pm

Lisa: In my opinion a voter who has one or two key issues and ignores everything else to the extent of not knowing (say) anything about the taxation policies of the candidates, is ipso facto incompetent to judge which candidate is better. We do not live in a single-issue world.

22

john b 10.19.04 at 11:07 pm

It will only anger the American voter to suggest that foreign nationals should be involved in electing the US President.

There’s only one left? Wow, I knew there was a downside to those disenfranchisement dirty tricks.

23

james 10.19.04 at 11:42 pm

john b – Are you certain the usage is incorrect? The phrase “The American voter” was used in a manner similar to “John Q Public”.

24

Janis Canon 10.19.04 at 11:58 pm

At times such as this, I think back to my sister’s response to a comment I made that began, “Anyone with average intelligence . . .” She just looked over her glasses at me and said, “Remember, average intelligence is 100.”

25

Janis Canon 10.20.04 at 12:13 am

The original post and all of the comments have been great fun for the mind. However, it may be that we need to simpify a little. I think back to my sister’s response to a comment I made that began, “Anyone with average intelligence . . .” She looked over her glasses at me and said, “Remember, average intelligence is 100.” Throw in a bit of cognitive dissonance, and you might have the answer to what is going on in the minds of many of the voters.

26

Janis Canon 10.20.04 at 12:14 am

The original post and all of the comments have been great fun for the mind. However, it may be that we need to simpify a little. I think back to my sister’s response to a comment I made that began, “Anyone with average intelligence . . .” She looked over her glasses at me and said, “Remember, average intelligence is 100.” Throw in a bit of cognitive dissonance, and you might have the answer to what is going on in the minds of many of the voters.

27

Tom T. 10.20.04 at 12:24 am

These considerations help to illuminate the old notion of “the worst form of government, but for all the others.”

28

Janis Canon 10.20.04 at 12:30 am

The original post and all of the comments have been great fun for the mind. However, it may be that we need to simpify a little. I think back to my sister’s response to a comment I made that began, “Anyone with average intelligence . . .” She looked over her glasses at me and said, “Remember, average intelligence is 100.” Throw in a bit of cognitive dissonance, and you might have the answer to what is going on in the minds of many of the voters.

29

Tom 10.20.04 at 1:03 am

As an empirical matter, is there any reason to think people who vote for Kerry because they think he has Bush’s policies outnumber those who vote for Bush because they think he holds Kerry’s positions? If we think mistaken votes are roughly equally between the two candidates, then I’m not sure that it matters, distressing though we may find it.

30

vernaculo 10.20.04 at 1:07 am

The best servants don’t do what they’re told, they anticipate the needs of those they serve. It’s the same with public servants.
It’s the issues we don’t know about yet that are going to be the most important.
Having a leader in place who can respond effectively, rather than just obediently, should be the goal of the electoral process. That it isn’t is an idictment of the mercantile bondage of the public. We make consumer choices now about everything, even matters of the spirit and the heart.
Except for the odd diet now and again, and budget constraints, consumers never sacrifice; but it’s a time for sacrifice. We need leadership in that, because most of us have forgotten how.

31

Brett Bellmore 10.20.04 at 11:39 am

But, G, a collection of voters who are interested in single issues, inform themselves about those issues, and vote on them, while being utterly indifferent to other issues, should be just dandy according to Condorcet, right? They’re contributing information on the one issue, and their randomly distributed votes on the other issues null out… ;)

32

David M 10.20.04 at 11:50 am

Interesting, but hopelessly complicated by the electoral college. It’s not a question of aggregating the wisdom of the American population, but the wisdom of various sub-communities that have different information consumption habits (aminly b/c of different newspapers, news programs, and media buys).

n is much smaller at the state level, and v likely varies considerably from state to state and region to region.

33

Mark Snyder 10.20.04 at 2:03 pm

Objections have been raised on the grounds that a multinomial distribution would be a better model than a binomial distribution:

“but the wisdom of various sub-communities that have different information consumption habits (aminly b/c of different newspapers, news programs, and media buys).”

and on the grounds that the independence assumption is unwarranted:

“Jim’s problem isn’t the only one. Condorcet’s theorem assumes that the voters’ decisions are independent, right?..That’s a really bad assumption to make…given the herd mentality/popularity contest dimensions of all real large-scale elections.”

To handle the former objection, you would have to partition the voters into N sub-groups, where
each sub-group i has their own set of probability e_i (where e_i=1-v_i). If we take the average of all of the e_i’s, i.e., e=(e_1+e_2+…+e_N)/N and use e for the original model, then the asymptotic behavior–the behavior when the number of voters gets huge–will be the same for both models. They both go to the same limit in the almost sure sense. So it’s reasonable to talk about the e for a typical voter, nationwide.

As for independence, sure they don’t make up there minds independently, so e may be thought of as a random variable and thus the votes cast are dependent until e is chosen–meaning until election day–but conditional on e equal to some number the voter preferences are all independent.

34

Mark Snyder 10.20.04 at 2:04 pm

Objections have been raised on the grounds that a multinomial distribution would be a better model than a binomial distribution:

“but the wisdom of various sub-communities that have different information consumption habits (aminly b/c of different newspapers, news programs, and media buys).”

and on the grounds that the independence assumption is unwarranted:

“Jim’s problem isn’t the only one. Condorcet’s theorem assumes that the voters’ decisions are independent, right?..That’s a really bad assumption to make…given the herd mentality/popularity contest dimensions of all real large-scale elections.”

To handle the former objection, you would have to partition the voters into N sub-groups, where
each sub-group i has their own set of probability e_i (where e_i=1-v_i). If we take the average of all of the e_i’s, i.e., e=(e_1+e_2+…+e_N)/N and use e for the original model, then the asymptotic behavior–the behavior when the number of voters gets huge–will be the same for both models. They both go to the same limit in the almost sure sense. So it’s reasonable to talk about the e for a typical voter, nationwide.

As for independence, sure they don’t make up their minds independently, so e may be thought of as a random variable and thus the votes cast are dependent until e is chosen–meaning until election day–but conditional on e equal to some number the voter preferences are all independent.

35

Mark Snyder 10.20.04 at 2:13 pm

In the above, “set of probability” should just read “probability”.

36

fling93 10.21.04 at 10:48 pm

The goal of an election is not to select the “correct” candidate. The goal is to select a candidate that is most representative of the voters.

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