Pick a Winner

by Brian on March 7, 2004

I rather liked the discussion that followed from John’s “earlier post”:https://www.crookedtimber.org/archives/001450.html on voting systems. So just for fun I thought I’d try a more complicated version of an example I brought up in the comments there, to see what people’s opinions are.

Imagine we have a race with four candidates, and 100 voters. The voters preferences, at least as expressed, are as follows:

46 vote ACDB (i.e. A first, C second, D third and B fourth)
45 vote BDCA
4 vote DBCA
3 vote CDAB
2 vote CDBA

Who should win? It’s a nice test case for voting systems, because the four voting systems discussed in the earlier thread say different things.

* A gets the most votes, so she wins on plurality or first-past-the-post.
* B wins on single transferable vote.
* C wins on Condorcet, however it is implemented.
* D wins on the Borda count.

But who __should__ win? I think B. It looks to me like there’s only two candidates with serious support, and of those B is preferred by the majority 51-49. So she wins. Now this is probably a consequence of my being brought up on STV, and always thinking about the 2-party-preferred vote, but however it came about that’s my intuition and I’m sticking to it.

By the way, despite the ‘block’ nature of the votes, the case isn’t entirely unrealistic. The overwhelming majority of voters for a party follow the party’s instructions about how to rank order the other candidates. I would guess this happens in 90 to 95% of cases, if not more. It would not surprise me to find a block of 100 votes in an election just like this, if A and B are the major party candidates and C and D minor parties.



Chris Bertram 03.07.04 at 8:30 pm

Brian, I get 52 preferring B > A on your figures, so 52:48 (I was trying to see who’d win on a 2-round French-style system).


Norm Jenson 03.07.04 at 10:01 pm

You don’t even consider Approval Voting why?


Brian Weatherson 03.07.04 at 10:23 pm

Chris, oops I mistranscribed the last two lines. I’ve corrected the example now.

I don’t really see the obvious benefits from approval voting, but in any case it doesn’t fit into the systems I’m considering because it doesn’t take the rank ordering of each individual as input and deliver a social order as output.

Approval voting strikes me as an odd compromise between different systems. One voting system would let each voter grade each candidate out of 100, and then the winner would be the candidate with the highest score, when we add up all the grades. Approval voting is like that, except it’s out of 1. Why be so restrictive?

Moreover, I’m really not sure how one is supposed to express fairly common opinions using approval voting. For example, imagine I like Labor, don’t mind the LibDems and hate the Tories. How am I supposed to vote? For Labor only? That would mean not distinguishing b/w the LibDems and the Tories. For the Tories only? That would mean giving the LibDems as much support as Labor. I don’t see how this is an improvement over plurality.


Norm Jenson 03.07.04 at 10:42 pm

You would vote both for Labor and LibDems but not for Tories. You vote for those you approve of. Did you follow the link and read about how approval voting works and the advantages it provides? Judging from your question it doesn’t look like you did.


Brian Weatherson 03.07.04 at 10:53 pm

I think it’s a bit simplistic to say I’d vote for Labor and the LibDems. I might suspect that (a) most Labor voters don’t hate the LibDems, and (b) if we all vote for them as well then the LibDems would win, whereas it is possible that if we all voted for just Labor, then Labor would win. And by hypothesis, I’d prefer a Labor win. There just seems like way too much need for tactical voting here, whereas STV provides a nice simple way of having my preferences recorded and counted.


Brian Weatherson 03.07.04 at 11:14 pm

The problem with approval voting I’m trying to get at there is that as a voter I have two separate desires that shouldn’t really come into conflict. The first is that I do my bit to maximise the probability of Labor winning. The second is that I do my bit to minimise the probability of the Tories winning. Under STV I can do both. Under approval voting, I have to choose (when I come to mark the LibDems box) between these two preferences. This seems like a bad thing – voters should not have to face choices like this.

Counting votes in approval voting would also be a nightmare. (Assuming we use paper ballots, which I think are necessary for security.) The tried and trusted method to count votes is to sort the ballots into stacks and count the stacks. Under STV you then have to resort the smaller stacks into 2 and count again, but you’re still sorting then counting stacks. Under approval voting, some ballots need to go in multiple stacks, and there’s no obvious easy way to do this. Maybe if you trust machine voting this isn’t a problem, but I don’t, and as I see it at the counting stage approval voting is much much more complicated than STV. Since the only advantage I can possibly see for approval voting over STV is that it’s simple, this is a fairly serious cost.


John Quiggin 03.07.04 at 11:32 pm

You can implement approval voting with preferential ballots, for example by counting 1st and 2nd preferences as 1 and 3rd and 4th as zero in the present example, which would make B the winner. For counting, it would then only be necessary to count the votes twice, once for first preferences and once for second, which is marginally less complex than STV (though the volume of the counting task would be larger).

My immediate take on this example, is that approval voting is a version of STV which asks for, and therefore produces, less information, and presumably therefore produces less accurate results on any reasonable criterion.

I suspect the appeal of approval voting is mainly to people who can see the inadequacies of plurality (first past the post) but are worried about the supposed complexity of preferential.


Alan 03.08.04 at 12:38 am

The precise problem with approval voting is that it forces you to assign equal values to different parties. It resembles the Borda silliness where the value of preferences is determined artifically.

Part of the ancient lore of STV is that it began with something like the Iowa caucuses. Voters stood beside the candidate they favoured. If your candidate had a quota you stayed. If not, you moved to stand beside another candidate.

Approval voting looks like a simple idea but it’s actually asking faintly weird questions like: ‘Do you support Labor 100% or 50%?’ where those numbers are determined by the number of options, not by the choice of the voter. If the object is a system that excludes the need for tactical voting, it fails.


Brian Weatherson 03.08.04 at 12:41 am

Right in this case it would only take twice as long to count, but in practice it could take an incredibly long time depending on how many voters mark multiple boxes.

I think the quickest way to count them would be to sort the votes first into those that mark one box only and the others. For the one box only group count them as normal (stack then count), and for the others go through candidate by candidate counting sorting them into those that voted for each candidate and those that didn’t, and count each of the stacks. I’d guess it will take roughly 2 to 3 times as long as a regular count (assuming not too many people mark multiple boxes) – but it will be horrendous to do recounts because the votes won’t be presorted. Recounts are always horrendous anyway, but this would make them even worse.


Phill Hallam-Baker 03.08.04 at 3:55 am

You ask who should win but you don’t give us the names of the candidates, their views on the issues or their experience. As Tina Turner said ‘What’s votes got to do with it?’

According to the principles favored by the GOP the winner is the person whose brother appointed the returning officer.

Another school of thought is that the winner should be E, who did not stand in the original election but will win a recall election caled a few months later.

Then there is the journalism school which does not count the votes, instead it counts the amounts the candidates have raised in contri-bribe-tions. This has the considerable advantage of being measurable before the count itself which from the journalists point of view is much more important than the result.

Applying these princples to the current presidential election John Kerry is completely ignored for four months because Dean has raised slightly more cash. And ‘w’ is considered a shoe in because he had $104 million in cash before the $10 million ad buy. The fact that spending $40 million seemed to do little for Dean despite having a much smaller geographic area to cover (the early primaries).

Of course W might not be as big a spendthrift as Dean.


TomD 03.08.04 at 12:36 pm

A little off-topic, but can anyone explain the Greek electoral system to me? (No-one here has managed yet.)

It is a list system with constituencies, and a proportional system in which the largest party doesn’t need a majority of votes to get a majority in Parliament. In other words, a complicated fudge. This may be why I haven’t found any good explanation of its mechanics.


Rock Howard 03.08.04 at 1:37 pm

Let’s say that I really don’t like C and D.
I don’t know which I dislike more, but I have
come up with some reason for preferring
one over the other in a ranked system. Ugh!

Then, depending on how the ranked votes are
counted, that distinction may not matter
anyway. (Most implementations of STV throws
away anything beyond the 1st and 2nd ranking
and so why ask for that info in the first place?)

Now extend this to a large field. The CA
governor’s race recently had 250 candidates.
How can any ranked system deal with that?

As for counting votes, ranked systems have one
horribly property. If new votes are “found” late
in the process, you have to restart the count from
the beginning. With Approval you simple add
the found votes to the existing totals. This
is a big deal in places where elections
allow mail-in voting.

See our website for more on Approval Voting.


Robert “Rock” Howard


arcseed 03.08.04 at 2:12 pm

“It looks to me like there’s only two candidates with serious support..”

I don’t know that you can say that. We only have the order of preferences, not the degree. A could be only a marginally better candidate than C to the ‘ACDB’ voters, in which case the preference of the last two groups for C should probably tip it to C. We can’t know if that’s the case just from the votes, but it’s certainly a possibility. And condorcet does make an awful lot of sense, in the abstract.

Approval voting: My understanding is that in the system where you give a candidate a 0-10 score, giving any candidate anything but a 0 or a 10 is wasting part of the power of your vote, making it more likely an unnacceptable candidate will win. Approval voting is the exact same system, constrained so that all voters have to vote strategically. Whether the LibDems are ‘unnacceptable’ or not is something the voter has to decide, but he should still give them either 0 or 10 in the ranking system.


Matt Weiner 03.08.04 at 4:25 pm

In STV do you have to rank all candidates? That would solve Rock’s problem, I guess–at least the first, and probably the second (you could just not list all the joke candidates).


Wayne Eastman 03.08.04 at 4:57 pm

The real issue isn’t voting systems but preferences.

To have voting make sense, you want to have people’s political preferences arrayed on a single spectrum. If voters support the candidate closest to them on a left-right spectrum, voting system paradoxes lose their sting. But you have to worry about a potential for destructive division if preferences are aligned along just one spectrum. And you also have to worry if that single spectrum avoids dealing with certain basic interests–that avoidance may moderate conflict in the short-term but leave the political system detached from reality and vulnerable to destructive long-term conflict.


Brian Weatherson 03.08.04 at 5:14 pm

In modern versions of STV, the point Matt makes is right – you don’t *have* to rank all the candidates. If all the candidates you give a preference for are eliminated, then you are treated as having not voted. This makes it easier in cases where there are 130-odd (not 250 as I recall) candidates.

And in practice you don’t have to redo the count every time. As long as it’s clear who the top 2 or 3 are, and it usually is, you can just count the first preferences and then throw them into whichver bundle they will finally end up in. Theoretically it could be complicated, but in practice it is really very quick. And having physical bundles of votes rather than just tallies (which is all you could have on approval) is *so* much better for recounts.


Daniel Lam 03.08.04 at 5:54 pm

Let’s ask a different question: who should lose?

The race for most-hated candidate is a simple two-horse race of A vs B. A loses 51-49.

After eliminating A, the race for most-hated candidate is still a two-horse race, B vs C. B loses 51-49.

In the remaining contest, D loses.

So C should win.


Kerim Friedman 03.08.04 at 6:21 pm

This is an interesting discussion. I have to admit that the mathematics of evaluating different voting systems is beyond me, but I find Brian’s dismissal of Approval Voting rather odd. Basically, he is saying that he doesn’t like it because voters’ can’t vote strategically – which turns out to be the main “benefit” of AV touted on their website!!!

If I remember correctly, Brian’s last post was concerned about how open the Borda voting system was to manipulation. On the AV site, they argue that Instant Runoff Voting as used in Australia has suffered from voter manipulation. Their argument, which I am not prepared to evaluate, seems to be that by eliminating any kind of ranking system, they force voters to vote more honestly in terms of their actual preferences. But Brian rejects this system because it is less easy for voters to manipulate the results to ensure the strategic outcome that they desire. Like I said, I can’t evaluate whether AV delivers what it promises, but it seems that this is a rather strange grounds on which to reject it! Especially considering the origins of this discussion.


Jason 03.08.04 at 8:42 pm

I thought there was little disagreement as to who should win when there is a Condorcet candidate?

C is preferred to any other candidate, so why shouldn’t C win?

Wouldn’t a better way of eliminating candidates in a STV type scheme be to eliminate Condorcet losers, rather than just the people who are not the most popular?


Jason 03.08.04 at 10:42 pm

Looking back through the archives, I see there really is some disagreement, and people think Condorcet winners are not desired. I’m a little confused as to the reasons why not. Maybe someone will explain (complexity arguements aside).

A field of 50 candidates, a few peoples first and everyone elses second favourite choice is A, but the extremists B and C take up most peoples first (and last) choices.

I know I’d like A to be picked …


Matt Brubeck 03.09.04 at 6:28 am

Jason makes a good point by turning Brian’s queestion around, and looking at who the voters want to lose.

Brian says that A and B are “only two candidates with serious support”, but neglects to mention that they are also the only candidates with serious votes against. He biases the discussion by treating first-choice votes as “serious”, while ignoring the last-choice votes.

I favor the Condorcet winner C over the STV winner B — and a majority of the voters agree with me.

(Assuming, of course, that their votes reflect their preferences. Different election systems would provide different incentives for voters and parties to prefer strategic votes.)

By the way, Condorcet ballots can be tallied in lots, like approval ballots.


novalis 03.10.04 at 1:33 am

How can you know the voters’ intent in those ballots unless you know the system they were voting under?

You’re assuming that people vote honestly — but there are very few systems under which this is advisable.

Comments on this entry are closed.