general equilibrium: campaigns and voting systems

Previously I extolled the virtues of approval voting, in which voters vote for as many candidates as they wish, with the highest vote total determining the winner; as I indicated, it is in many ways a substantial improvement on mandatory bullet voting, in particular in situations with large amounts of information (e.g. pre-election polls) and with rational agents, in which case it is in some sense optimal.

I've been thinking in the past day or two about the idea of allowing up to two votes (for a single winner), but not more than that; just as the single-vote system results in two major candidates, allowing two would by and large result in three or four major candidates, which might be less informationally demanding on voters than if there are a potentially unlimited number.  (An alternative is to have a primary of some sort to fix the number of general election candidates; in one of the environments I've mentioned, where one supposes that it is straightforward for voters to change their votes over a period of time with running totals publicly available, one way of doing this would be similar to a number of TV contests ("Survivor", "American Idol"), in which perhaps only the top 20 vote recipients on January 1 are allowed into the race, with the lowest vote getter eliminated every two weeks until you're down to, say, four.)

What I want to highlight this morning, in the general situation in which there are more than two serious (or reasonably potentially serious) candidates, is how this would affect campaigning; in particular, I would expect less "negative" campaigning.  If I'm candidate A running against candidate B with no other real opposition, it can make sense to devote a lot of resources to searching out the dumbest thing candidate B ever said, or even the thing candidate B ever said that is most amenable to being misconstrued; similarly, any unforced errors by candidate B — generically, anything that makes candidate B look worse — benefits me. To some extent this will still be true when candidates C and D are also reasonably in the mix, but it's not the case to nearly the same extent; even if I get a picture of candidate B paying a prostitute, if I don't have much to say for myself, there's a decent chance candidate C or D picks up his support.

I put "negative" in quotes because some of what is sometimes given that label is quite relevant to voters' decisions; some commentators seem inclined to label and ad that says "candidate B voted for bill X" as negative if run by candidate A and as positive if run by candidate B.  Voters are being asked to choose between options, differences between those options are relevant, whether they guide voters with different preferences in different directions, or whether they guide almost all voters in the same direction.  That said, some of the worst of campaigns — the seemingly deliberate misconstruals, or most brazen attempts to make the other guy seem alien, like "not one of us" — would become less relatively profitable if there were more candidates; indeed, on at least a rough basis, I imagine that having three opponents would make investing resources extolling your own virtues about three times as relatively valuable, compared to making the other candidate look bad, as only having one.