Suppose Israel does choose to switch to regional voting, with a single winner in each region. How should the winner be chosen?

Simple, you say. Just use the tried-and-true plurality method: let every voter choose their favorite candidate and let the candidate with the most votes win. This plurality method has the virtue of simplicity, but it suffers from certain well-established weaknesses. The main issue is the Ralph Nader effect. Suppose there are three candidates: the moderate Red candidate (R) gets 48% of the vote, the Blue candidate (B) gets 49% of the vote and the extreme Red candidate (R+) gets 3% of the vote. Under the plurality system, B wins. But, since the R+ voters obviously prefer R to B, the true majority preference (R) is thwarted by the plurality method. (Side comment: there was a lie in the previous sentence. As is clear from a cursory reading of the meshugosphere, R+ voters prefer anybody but R.) Formally, we say that R is a Condorcet winner, since in head-to-head, R would beat each of the other candidates. Thus, the plurality method might not choose a Condorcet winner, which is a bad thing. (A similar thing happens when, instead of one major Red candidate and one minor one, there are two major Red candidates that split the vote.)

Another, closely related, weakness of plurality voting is that voters are forced to vote strategically, rather than sincerely. To use the above example, if your real preference is R+, but you know that voting for R+ might tilt the results from R's favor to B's favor, you're likely to vote for R (your strategic choice), rather than R+ (your sincere choice). This is also a bad thing.

The weakness of plurality voting lies in the fact that voters give very little information. We know their preferred candidate (or at least, their strategic choice), but we don't know what they think about the other candidates. Thus, the fact that R+ voters prefer R to B is never given expression and is not taken into account in computing results.

A wide variety of solutions have been proposed and all involve eliciting more information from the voters. The simplest proposed improvement is approval voting: a voter can vote for all the candidates he approves of. If he has one clear favorite, he can vote just for that candidate. If he doesn't care much about most candidates but really despises one of the candidates, he can vote for all but the one he hates. (This should be immensely popular around these parts.) The candidate with most votes wins.

This method has much to recommend it. First of all, in our above example, R+ voters can simply vote for both R+ and R. This solves both problems mentioned above. First, voters are more likely to vote sincerely, not strategically (they really do prefer both R and R+ to B). Second, R is likely to beat B, thus giving the voters their true choice. In fact, it can be shown that approval voting generally (though, not always, as we'll see) results in victory for Condorcet winners.

But, there are problems. First of all, imagine that there a whole bunch of candidates running and a voter has a clear ranking of the candidates in his head. He still wouldn't quite know where to draw the line: between his most favorite and second favorite? between 5th favorite and 6th favorite?. This mushiness can result in two voters with identical preferences casting vastly differing ballots.

If that doesn't sound especially awful, consider another situation. There are three candidates. Our old friends, R and B, and a third candidate, whom we'll call P for Pareve. Let R be the favorite of 70% of the voters, B the favorite of 30%, and P the favorite of nobody. Now , it is perfectly possible that P is so perfectly inoffensive that 75% of the voters (whether they prefer R or B) also vote for P. Then despite the fact that R is a massive Condorcet winner and P is nobody's preference, P would win the election.

Well, if approval voting addresses the problems with plurality voting by eliciting more information from the voter, we can try to solve the problems with approval voting by eliciting even more information. Specifically, we can ask a voter to rank the candidates in order of preference -- most favorite, 2nd favorite, and so on. This, of course, tells us a lot more about a voter's preferences than just approve/don't approve. Of course, once we know every voter's preferences, it isn't exactly obvious how we ought to determine the winner. We need to define some appropriate aggregation function that takes all the votes and computes the winner.

Now, it's important to know that there's no perfect aggregation function. This follows from Arrow's Impossibility Theorem, which argues that some really simple set of properties (that you surely would hope for an aggregation function to have) are not simultaneously satisfied by any aggregation function. Nevertheless, there are some reasonable aggregation functions. The most famous of them is the Borda count, more familiar to Americans as MVP voting (sort of). If there are ten candidates, then a candidate gets 10 points for each first-place vote, 9 for each second-place vote, and so on down the line.

The extra information provided by ranking does indeed come in handy. Thus, in our example with the pareve candidate, P might indeed get lots of second-place votes, but R would still win, since second-place votes are worth less than first-place votes.

And yet, no surprise, there are problems with this too. First, it's really asking too much off voters to have them rank a whole gaggle of candidates. Second, it seems that using Borda count, Condorcet winners are actually less likely to win than they are using approval voting. Moreover, Borda count invites strategic voting. For example, suppose there are two real candidates, R and B, and eight bogus parties, which we'll call Pensioners1, Pensioners2, etc. Let's assume that nobody really wants to see any of the Pensioners candidates anywhere near the victory stand. Nevertheless, it's pretty clear what voters will do. (You know what you would do.) The R voters would bury the B candidate behind all the Pensioners candidates and the B voters would return the favor. One of those Pensioners candidates could even win as a result, to everybody's horror.

Of course, there are those who propose to remedy these flaws by asking voters to provide even more information. This can be done by having voters assign a grade, say from 0 to 100, to each candidate. This is clearly more general than ranking, since it imposes an implicit ranking and even facilitates indicating by how much, say, one's 4th favorite candidate is preferred over one's 5th favorite. This is a new idea and the literature on it is still a bit raw.

I personally think the simplest thing would be to use approval voting but limiting a voter to at most two approved candidates. The reasons I prefer this will have to wait for another occasion.