Plurality Quirks

How Voting Systems Work 1 ended with a nine-voter election in which the plurality winner was the last choice of a majority. This chapter stays with that rule, the simplest one in the course, and asks what else it does. Plurality is the rule behind the phrase “first past the post”, and it has three famous quirks: it can elect a candidate most voters dislike, it punishes candidates who share voters with a similar rival, and it nudges voters toward tactical choices. We will count each quirk in small Python programs, then simulate thousands of three-way races to see how often the rule disagrees with a head-to-head count. The maths stays at arithmetic and percentages.

The Rule: Most Votes Wins

Under plurality, each voter marks one candidate, and the candidate with the most marks wins, whether or not that is a majority. Nothing else on the ballot is counted. The Wikipedia article on Duverger’s law describes a tendency, which it is careful to call “a tendency rather than an invariant rule”, for plurality voting in single-member districts to favour two-party competition. It names two mechanisms, summarised in the table. The article attributes the law to Maurice Duverger but, as fetched, gave no date for it (check).

The same article lists exceptions, so treat the law as a pattern and not a certainty. In Canada, it says, the Green Party took about 5% of the national vote from 2004 to 2011 and won only one of the 308 seats in the House of Commons, which is the mechanical effect in numbers. It also says strong regional parties can win seats even when only two parties compete in each district, as in Canada, the UK and India.

Vote Splitting

Suppose Rae and Sam appeal to much the same voters and Tom stands apart. If those voters divide their first choices between Rae and Sam, neither may beat Tom, even though most voters would rather have either of them than Tom. Here are 100 voters (illustrative numbers):

Tom wins with 45%, but Rae beats Tom 55 to 45 head to head, and so does Sam. In the language of How Voting Systems Work 1, both Rae and Sam would beat Tom in a straight contest; the plurality count never asks. The next question is who is to blame, and for that we withdraw each candidate in turn and recount.

With all three standing Tom wins. If either Rae or Sam withdraws, the other takes the withdrawn candidate’s voters and wins. Rae and Sam are each a spoiler: the Wikipedia article on the spoiler effect describes it as a losing candidate affecting the result “simply by participating” (as fetched). Tom withdrawing changes the winner too, but Tom is the winner, so this is not a spoiler case. The article cites the 2000 US presidential election, where it says Ralph Nader’s campaign is widely cited as a spoiler, but the fetched text gave no vote totals, so none are quoted here (check). It also gives a 2009 mayoral election in Burlington, Vermont, with head-to-head figures of 4,064 to 3,476; that example involves instant-runoff counting, which How Voting Systems Work 3 covers, and I did not verify the figures beyond that one summary (check).

Simulating Three-Way Races

One hand-made example proves little, so we need a model that can produce thousands of elections. The simplest is a line from 0 (left) to 1 (right). Voters sit at random points on the line, candidates sit at positions, and each voter ranks the candidates by distance, nearest first. This one-dimensional line is an assumed model; real issues have many dimensions. With voters spread along a line, the candidate nearest the middle voter should win head to head (the median-voter idea; I did not confirm it from a source here, so treat it as a property of the model, (check)). The code searches for the Condorcet winner directly, so it does not rely on that idea.

Rae and Sam split the left and centre, Tom collects the right with 43 of 101 first places and wins, but Sam, who sits nearest the middle, beats every rival head to head. Now repeat this with random candidate positions and count how often plurality names someone other than the head-to-head winner:

With two candidates the rules cannot disagree. With three, they disagree in roughly a third of the simulated races, and the rate climbs as candidates are added: 35.8%, 54.0% and 65.3% for three, four and five. The sample is 2000 seeded races per row, so the exact figures belong to this seed and model; the trend, not the digits, is the finding. This is a simulation of an assumed model and says nothing about the frequency in real elections.

Votes Versus Seats

Plurality is usually applied district by district, and a party that is second or third everywhere can win few seats or none. Here are ten districts of 100 voters each with three parties, using invented numbers:

Green’s 255 votes elected nobody, which is what critics mean by wasted votes. A Green voter who understands this may vote for Blue or Red instead (the psychological effect in the Duverger table), and Exercise 3 asks what that does to the seats.