Monday, November 06, 2006

I've thought a lot about alternative voting systems over the course of the past decade, and a recent conversation with a coworker followed by a recent weblink provided by my brother have me thinking about the topic again. We have a system, by and large, in which we elect a single winner from each of several districts; this lends itself to elections in which one has, at best, a choice between two plausible candidates, where additional candidates are more likely to "split" votes of candidates most similar to them than to create a plausible third alternative that might better represent the average voter than either of the other two candidates.

I'm among those who wish we had a system that were more amenable to more alternatives, and one that resulted in greater representation of minority viewpoints. The state of Illinois, up until 1980, actually had such a system for the lower house of the legislature. Each district elected three members to the lower house, and each voter cast up to three votes; the three candidates with the most votes were elected. This may sound like the election system used for the lower house of the New Jersey legislature, in which each voter casts up to two votes for different candidates for two positions per district, but New Jersey requires that the votes go to different candidates, whereas Illinois did not — a voter could cast all three votes for the same person. In New Jersey, then, a district with 55% Democrats, voting along party lines, is likely to send two Democrats to the legislature; in Illinois, if one third of a district was Democrat and two thirds Republican, the Democrats could give all their votes to the same candidate, and could therefore guarantee that that candidate would be one of the three winners. This is called cumulative voting. In fact, one wouldn't need to let people cast three votes; one could effect the same outcome by giving each person one vote. Each Democrat would then vote for the same person, and each Republican could just vote for one candidate; the same results obtains.

At this point one still typically has a primary or some such, and in practice one still has limited options come the general election. If the list isn't limited, one still gets vote-splitting; a second Democrat running, for example, could take about 1/6 of the vote, leaving 1/6 for the other Democrat; each of three Republicans could get 2/9 of the vote, allowing a third Republican to slip through. Coordination is required to achieve proportional representation. The system itself, though, can be modified to handle this automatically, as long as each voter declares a second choice. The candidate with the fewest votes can be dropped, and his voters transfered to their second choices; in the example I've given, with (again) straight-party voting, the votes from the Democrat getting fewer votes transfer to the candidate getting more, assuring the latter's election. If people vote based on things other than party, though, this system can accommodate that, too; formal parties aren't required. Meanwhile, within formal parties, the candidates who win are those selected by the voters. This is a system called "single transferable vote" — one has a single vote, but it may transfer from one candidate to another if the initial choice can't make any good use of it. It ends up producing what I like to call a "Latin square effect", where voters who base their votes on a number of factors will end up being represented by a council that is proportional in all of those factors — it will tend to have the right proportions by party, the right proportions by culture, by community, and so on.

This system isn't necessarily the best way of selecting a single winner, though, as when we might want to elect a single person to serve as an executive. In that case, the system reduces to an "instant run-off" system, in which the candidate with the fewest first-place votes is removed at each stage and the stages progress until a single winner is chosen. This removes the vote-splitting problem, but it often selects winners who may not represent the bulk of the population well; if about 40% of the voters vote for one candidate, about 40% for an opposite candidate, and 20% for one in the middle, it may be that the 60% of the people prefer the middle candidate to the first one, and 60% prefer him to the second. They would prefer to compromise, but the compromise candidate is dropped first, and one of the other two will be elected. A different system is simply to note who wins pair-wise elections, and, if one candidate would beat each of the others head-to-head, designate that person the winner. This is Condorcet's method, and there are different ways of dealing with ties.*

With Condorcet's method, the median voter theorem applies; because it consists of a number of two-candidate races, it tends to result in that candidate nearest the median (again, along every dimension in which voters vote) winning. (If some other mechanism is used to reduce the race to only two candidates, the candidate closer to the median will be favored, but the candidate (of the original set) that was closest to the median may well have been eliminated, depending on the mechanism.) This is essentially the result I was arguing for in the preceding paragraph; we would rather have a system that tends to choose a candidate near the median than one near the extremes. It seems to me, though, that this isn't quite right; in particular, it may make sense for a minority that feels strongly about one issue to dominate over a majority that opposes them on that issue, but doesn't care as much. In particular, one can suppose that each voter has some sort of utility function the sum of which we are willing to accept as a social-welfare function; in English, we posit that there is some measure of individual happiness/welfare/preference such that 1) a voting system should maximize the total of this for each voter, and 2) each voter will take whatever steps he can on his own to maximize his own amount. (This second stipulation is essentially the principle of revealed preference, which is what economists call the maxim that "actions speak louder than words"; we ascertain what a voter prefers from how he votes, and it is these preferences revealed by voting behavior that we wish to maximize globally.) Given a simple model in which a voter's utility is determined by specifying a weight and a viewpoint on each possible issue, and adding up those weights on which the winning candidate agrees with the voter and subtracting those on which he disagrees, the best candidate is that nearest the mean voter. Thus, to allow people more say on issues about which they feel more strongly, we want something that will effect some sort of mean-voter principle.

If the pool of candidates is more or less like the pool of voters, the natural thing to try is to add up, for each candidate, the number of other candidates they beat on each ballot. If a candidate were to switch positions on an issue nobody cares about, they would pass few other candidates (going either up or down) in many ballots; on a more polarizing issue, they would move more on more ballots; and if they switch to a position that is less popular, but on which the minority feels more strongly than the majority, they may well gain votes. The problem with the Borda count is that, in a Borda count system, it is not in a voter's interest to actually rank his preferences honestly. Most likely there would be a few candidates with a real chance of winning, and voters would have a tendency to put their least favorite likely candidate at the bottom, below many candidates who are simply unqualified, in order to improve the relative performance of their preferred candidate. While our first-past-the-post system suffers a little bit from such "gaming" — one tends not to vote for one's favorite candidate, so much as one's favorite candidate who has a reasonable chance of winning — the Borda count is far worse in incenting a vote far less reflective of what the voter actually wants, and tends to lead to a system in which the connection between the outcome of the election and the voters' preferences are weakly coupled.

(One system that is almost equally famous for not being susceptible to gaming is "approval voting", in which each voter is allowed to vote or not vote for each candidate standing for election. The winner is then the candidate with the most votes. This system is simple to understand, easy to administer with a show of hands for small groups, and will often discover a popular "third-party" candidate where voters prefered a candidate whom they didn't expect to win to those they did expect to win. There is no reason, with approval voting, to vote for any candidate you like less than a candidate you are not voting for, and so any strategy comes down only to drawing a line between the candidates you will vote for and those you won't.)

Essentially what is happening in the Borda count is that voters are pushing one candidate above another not because they prefer that candidate, but because it gives them a way of affecting the candidate's standing relative to a third candidate. One way to eliminate this factor is to force the voter only to choose within pairs; if candidate A and candidate B are only competing with each other, one might as well rank them honestly, since neither's performance affects another candidate. Simply providing each voter with a random pair of candidates won't do; the voter, seeing an unlikely candidate against a candidate the voter wants to prevent from winning, faces the same incentive to vote against the latter candidate as he does with a simple Borda count. What is needed is to pair up all of the candidates in a fixed way, such that each candidate is only facing one other candidate.

Ideally, then, to construct a legislative body that reflects the electorate well, we want a system that, in practice, gives each voter a choice between two candidates. Empirically, we know an easy way to get this: first-past-the-post voting in single member districts.

Happy Election Day.

*The Smith set is the smallest nonempty set of candidates such that each candidate within the Smith set beats each candidate outside the Smith set in a head-to-head race. The set may contain exactly two members only if they exactly tie against each other; larger numbers can be produced without ties, if, for example, 1/3 of the voters vote A/B/C, 1/3 vote B/C/A, and 1/3 vote C/A/B, such that A beats B beats C beats A. I tend to use the term "Condorcet's method" to refer to any method that, by construction, selects as its winner a member of the Smith set; it may be that there is a conventional mechanism for selecting among its members that is officially considered the Condorcet method, but I don't know for sure, and, if there is, I probably don't much care for it.

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