No, this isn’t about the recent US presidential elections.

In one of those “Yay I get to combine all my weird obsessions” moments, I’ve been playing around with using linear programming to construct provably minimal examples of specific election results. It actually works pretty well – it’s easy to to do for things like plurality voting, borda count and Condorcet winners, and it turns out to be almost possible to do for instant runoff voting (I haven’t figured out how to specify the instant runoff winner in a single linear problem, but you can specify an exact instant runoff drop-off order and then brute force by just considering all the relevant permutations).

Anyway, in the course of doing this I came up with the following really nasty small election example:

- 6 votes of A, C, B, D
- 3 votes of B, D, C, A
- 3 votes of D, B, C, A
- 1 vote of C, D, B, A

Why is this election so nasty? Basically the electorate is divided into two groups: A near majority who think A is great, and a bare majority who can’t agree on much of anything except that they hate A. If you want an interpretation of how these results came about, you could imagine that A is a right wing candidate in a slightly majority left-wing constituency, C is a centrist candidate, and B and D are two left wing candidates who are running against each other and have split the vote.

But regardless of interpretation, what results is an election that is *very* sensitive to which voting system you use. In particular four fairly common choices of system each elect a different one of the four candidates:

- The plurality winner is A, with 6 votes.
- The Borda count winner is B: The Borda scores are A at 31, B at 35, C at 34, and D at 30.
- The Condorcet winner is C: 7 voters (the first and last group) prefer it to B and D and 7 voters (everyone but the first group) prefer it to A.
- The Instant Runoff Voting winner is D: In the first round C drops out with only 1 vs 3, 3 and 6 first choice preferences. This transfers a vote to D, causing B to drop out in the next round at 3 vs 4 and 6 first choice preferences. B then transfers their first choice votes to D as well, and finally D has a majority at 7 out of 13 votes, so D wins.

This is the smallest election with these winners and this specific drop-off order, in the sense that it has the fewest number of distinct votes cast (caveat: Where no individual vote has more than 100 voters casting it. This probably doesn’t matter), and amongst those elections with four distinct votes having this property it also has the smallest number of total votes (it also does some minimizing amongst those, but that’s more for aesthetics and doesn’t correspond to an obviously interesting property).

I originally thought all of the voting systems were doing badly on this election, but actually on further reflection I think they’re all behaving in an *extremely* typical fashion for them.

To frame this in terms of the interpretation above:

- Plurality voting elects the right wing candidate because the left wing vote got split.
- IRV collapses the two left wing candidates together and elects the one with slightly larger support due to the centrist tie breaker vote.
- Borda vote does similarly but breaks the tie in the opposite direction because it counts the fact that the A voters prefer B over D for something.
- The Condorcet candidate is a centrist compromise between the two sides that everybody can tolerate but nobody is thrilled about.

So which candidate you think is correct really depends on what you’re going for – there’s a legitimate question to be asked about how centrist you really want your elected leader to be, and about how justified that is in this case. If you decide that because you’ve got a bare majority the you should elect a more left-wing leader, the question of B vs D seems to come down to whether you want to go for a candidate the other side hates or one the other side really hates.

I don’t know about you, but right now having leaders the other side only hates sound pretty good to be honest. So although I didn’t intend this to be an argument for any of these systems, maybe Borda count comes out of this looking pretty good.

Given that, I feel the title is rather over promising matters. So here’s an even worse election:

- 6 votes A, B, D, C
- 5 votes C, A, B, D
- 5 votes D, C, B, A
- 2 votes B, C, A, D
- 2 votes B, D, C, A

This has more or less the same properties as the above one: A is the Plurality winner, B is the Borda winner, D is the IRV winner.

However, C is no longer the Condorcet winner because there no longer *is* a Condorcet winner. Instead C is merely the Kemeny-Young winner, a system which always elects the Condorcet winner when there is one and has some claim to being the “best” such electoral system (in the sense of being the closest to capturing the same spirit as the Condorcet criterion), but is horribly impractical for any significantly large number of candidates.

To see why this happens, lets look at the majority preferences amongst the candidates:

- 11 voters prefer A to B
- 13 voters prefer A to D
- B and C are tied with 10 votes each
- 15 voters prefer B to D
- 14 voters prefer C to A
- 13 voters prefer D to C

So we have a Condorcet cycle where A is strictly preferred to B, B is strictly preferred to D, D is strictly preferred to C, and C is strictly preferred to A.

The Kemeny Young rule works by looking at *all* rankings of the candidates and giving them a cost associated by how much they disagree with this ordering. For each voter and each pair of candidates, the ordering incurs a cost of one if the voter disagrees with that ordering of that pair.

There are better algorithms for this than just considering all orderings of the candidates, but fortunately we only have four candidates so we can just brute force it to see why this works out this way.

The full list isn’t very informative, but here’s a partial list:

This gives us the following scores:

- “C, A, B, D” scores 50
- “A, B, D, C”, “B, C, A, D”, “B, D, C, A” and “C, B, A, D” each score 52
- “B, A, D, C “scores 54
- Everything else scores 58 or higher
- “D, B, A, C” scores 70

(The example was specifically constructed to give “C, A, B, D” as the unique Kemeny-Young best ordering: In much the same way as with IRV, it’s easy to specify the Kemeny-Young ordering but hard to specify just the Kemeny-Young winner)

Because it’s the leader of the unique minimum cost ordering, C wins under the Kemeny-Young rule.

As you can see it’s quite a close fight – there isn’t a huge amount of difference between the lowest cost and the runner up cost, and in the second lowest cost orderings don’t really agree on who should win – the only thing that seems to be consistent is that it shouldn’t be D. This sort of close race tends to be why Kemeny-Young calculations become difficult very quickly.

I’m finding it hard to come up with an interpretation of this election, so lets run the instant runoff vote and see if it’s enlightening (I’m ambivalent about IRV as a system, but watching how votes transfer can help elaborate on coalitions).

Initially B drops out, having only four first choice votes. These split equally between C and D, leaving us with the following configuration:

- 6 votes A, D, C
- 7 votes C, A, D
- 7 votes D, C, A

Now A, the previous plurality winner, has the lowest first choice votes so drops out and transfers votes to D:

- 7 votes C, D
- 13 votes D, C

So D wins the runoff over C.

So I think maybe the way to look at this is that A is an independent candidate of some sort that is distracting from the main stream set of candidates B, C and D. Lets see what happens to the election if we start by taking A out:

- 8 votes B, D, C
- 5 votes C, B, D
- 5 votes D, C, B
- 2 votes B, C, D

So B becomes the plurality candidate, with fully half the first choice vote. The majority preferences remain as they were (because they are not affected by the removal of A): C is still majority tied with B, B is still strictly preferred to D, D is still strictly preferred to C. So there’s still no Condorcet winner.

Removing has made instant runoff voting result in ambiguity though: C and D are tied as to who will drop out. If C drops out then it transfers votes to B, which wins. If D drops out, it transfers votes to C, which is now tied with B. In neither case does D, the original IRV winner, win! This is particularly interesting because the voters who put A first all strictly prefer B to D, and C to D, so removing their first choice candidate arguably results in a better result for them (if you break ties by flipping a coin, they get a better option 3/4 of the time).

The Borda winner (which can change by dropping a candidate) is still B.

I don’t know how enlightening that was, but it seems to reinforce the interpretation that B, C and D are an extremely fractious lot who are probably not going to get along, and unless we elect RON instead (the only candidate who can promise change!) we’re probably going to see the continuation of voting through other means.

### Addendum

If you enjoy these small election examples, I’m putting together a small ebook collating some of them. Let me know if you’d like to be a reader of an early draft.

Also (you’re going to be seeing messages like this a lot over the next year), if you like reading about this sort of thing and haven’t already, do consider voting with your wallet and sending some money to my Patreon for supporting my blogging here.

Daniel ReevesNice analysis! I’m a big fan of approval voting and am curious how it would compare here. I realize that means more assumptions about voters’ preferences (where their cutoffs are, acceptable vs not acceptable).

I guess to do this really right it has to be considered game theoretically. For example, in plurality voting you get less vote splitting than raw preferences predict because people have incentive not to waste their vote on the anticipated less popular of their acceptable choices.

davidPost authorThe answer is “It depends”. If you consider all feasible approval votes (that is, assume that each voter casts a vote for their top n candidates where n >= 1, but allow each voter to choose n arbitrarily) then any of the four candidates can be strict approval winners. There turns out to be a nice linear program you can write to prove this! (This is for the second one, I haven’t checked the first one due to being too lazy to convert it back to the right format).

In fact, with this election, every candidate is approvable even if every voter only picks either their top candidate or their top two candidates.

If we consider the more restrictive models “Every candidate votes for their top n for a fixed value of n” models, for n=1 it’s of course the same as the plurality winner, for n=2 C wins and for n=3 B wins.

A model where D wins the approval vote is this: Everyone but the “B, D, C, A” voters vote only for their first choice candidate, but the “B, D, C, A” voters also vote for their second choice candidate.

ETA: I stopped being lazy and converted it. The original election also has the property that depending on how people choose the numbers to approve, every candidate is electable under approval voting.

Daniel ReevesThanks David! Of course in all of these voting methods people will vote strategically and that really changes the analysis. My vague intuition for why approval voting is best is that the strategizing seems the most transparent and least perverse. Instead of, say, ranking a less preferred but more popular candidate above your true favorite, not “throwing away your vote”, etc, you just have to strategize about where to draw the line between acceptable and unacceptable. The preferences you’re expressing are still true, unlike with strategic voting in other systems.

(Hypothetical voter under approval voting: “I vastly prefer Bernie to Hillary, but given that Trump has a real chance of winning, I (quite truthfully) prefer anyone but him.” Compare to any other system where your incentives may be to untruthfully express a preference of Hillary over Bernie in order to maximize the chance of defeating Trump.)

Ok, I just pulled that rationalization out of my butt and won’t be too surprised (but will be very interested) to hear a counterexample.

What I really like about approval voting is that it’s so simple that everyone can immediately understand it, and it’s a trivial change from plurality voting: all the ballots and existing infrastructure work the same way — you’re just allowed to vote for as many candidates as you like. The winner is still whoever gets the most votes.

davidPost author> why approval voting is best is that the strategizing seems the most transparent and least perverse

I actually find the tactical voting of approval voting *really* hard. It’s difficult to know where you should put the threshold. Do you select everyone who is mostly OK? Do you pick only your top candidate and hope other people. I mean sure it’s just a single parameter and your vote is mostly “honest” within that, but I don’t feel like I’ve got a good handle on how to pick that parameter.

(And if you’re just going to pick it at random then you might as well just use a Borda vote)

> Ok, I just pulled that rationalization out of my butt and won’t be too surprised (but will be very interested) to hear a counterexample.

No, there’s no counter example. Approval voting is indeed monotonic (which I think is the property you want): The only thing to do if you want a candidate to have a better chance of winning is move them from a no vote to a yes vote.

Of course depending on the use case this might just move tactics to candidate selection. e.g. I like approval voting for a feed in to a runoff, but then when deciding who your favourite candidate is you need to not just think about which candidate you like most but also which candidate is most likely to beat candidates you don’t like in the runoff.

> What I really like about approval voting is that it’s so simple that everyone can immediately understand it, and it’s a trivial change from plurality voting: all the ballots and existing infrastructure work the same way — you’re just allowed to vote for as many candidates as you like. The winner is still whoever gets the most votes.

That’s true, but I’m not very keen on how little nuance you get to express your opinions as a result.

My favourite thing about Approval voting is really that it means that there’s never a good excuse for using plurality voting. I do like it for e.g. low-impact things where it doesn’t matter that much what you choose as long as people are mostly OK with it. (Even then I like it better with a runoff round at the end between the top two candidates afterwards)

davidPost authorApparently I’m actually wrong: “We construct another

example of a strategy combination which is a strategically stable equilibrium and where an insincere strategy is the unique best-response for a voter“

Michael ChermsideI just wanted to chime in and say that I think examples like this are *particularly* useful in describing voting systems to those who are not experts on the subject. Particularly when you weave a story line in that helps to set the stage (“Why is this election so nasty? Basically…”) it really shows people why voting matters and when you have constructed the situation so it produces different results depending on the method you use it helps to explain why the choice of method matters.