I’m going to take a break from the algebra posts for a few hours and concentrate on a very applied area of math, electoral systems. The best-known one in the English-speaking world is, of course, plurality vote (or, if you’re a Brit, first-past-the-post): a few candidates run for office, and the one getting the most votes wins.
Electoral reforms, at least those that don’t focus on proportional representation, which is another matter entirely, generally advocate an alternative called IRV, or instant runoff vote (or alternative vote, or preference vote, again depending on the country). That’s a real shame, given that IRV is barely an improvement over plurality, and is of no use at all if you want a real multi-candidate election.
Because Anglophone countries are so used to two-party systems, in which additional parties are mere spoilers, the examples used to illustrate IRV in action tend to focus on a scenario with only two viable candidates, which is the only scenario in which IRV works.
You may have seen the following configuration: 45% of the voters are Republicans, 40% are Democrats, and 15% are Greens. In plurality, the 15% are sufficiently pissed off at the Democrats that they vote Green, guaranteeing the Republicans a victory. In IRV, they vote Green > Democratic > Republican, so their candidate is dropped and his support goes to the Democrats, guaranteeing them victory.
The number of bloggers who stop at that special case is astounding. To show that IRV helps, you need to pick one of two paths: the empirical path, which involves showing that IRV has changed politics for the better where it is used (which it hasn’t), and the theoretical path, which involves showing that IRV is sound in general.
In fact, IRV is not sound at all in general. First, the spoiler effect is fully revived when a third party becomes viable. If in the example above, the Greens siphon 10% from the Democrats and 3% from the Republicans, we get the same situation, with the Democratic candidate winning 58-42 in the second round.
Now, suppose that 3% of the Republican voters want to ensure that the Republican wins nonetheless, by voting G > R > D instead of R > D > G. The first round result then becomes 39 R, 30 D, 31 G, and the Democrat is dropped. Since in such a situation the remaining Democrats will likely be fairly centrist, it’s safe to assume at least half of them put R in second place, throwing the electing to the Republicans.
Plurality has its problems, but at least you can’t make a candidate win in it by voting against him.
Even without pernicious tampering like that, there’s a serious spoiler effect. Suppose that the above situation with the Republican winning is in fact the voters’ true preferences, that is that the Greens get 31% and their second-place votes are 28-3 Democratic. These 28% are in effect spoilers now, since their preference for the Democrat over the Republican no longer registers; to register it they must vote D > G > R instead, just like in plurality.
There are serious alternatives to plurality, but they are used by experts more than by reformers. Approval vote seems to be the mathematicians’ choice; it has a fairly good and intuitive strategy for deciding who to approve, and it never gives voters the incentive to approve a candidate but disapprove a candidate they like more.
Range voting gives the voters more choice than any other voting system; unfortunately, strategizing about where to place candidates except the most- and least favored could be the dictionary definition of “complicated.”
Condorcet is a good system that allows you to rank the candidates, and only starts breaking down in a very restricted case; not only must the election feature at least three viable candidates, but also there have to be at least two disjoint political axes with each candidate’s voters displaying specific perceptions of which is more important.
For an example of the above with Condorcet, consider Mexico, which has a center-right corrupt party (PRI), a less corrupt rightist party (PAN), and a less corrupt leftist party (PRD). In theory, if Mexico used Condorcet instead of plurality, it could break down if PRD and PRI voters’ second choices were PRI and PAN respectively based on ideology, and PAN voters’ was PRD based on non-corruption. In practice, what happens is that the PAN’s second choice is based on ideology as well; the voters’ choices are less clear, but I think PRD and PAN voters have equal proportions of PRI-second-choicers.