This is indeed an interesting scenario. Something is particularly weak about those B>C preferences. It could be one of two things:
1) Maybe you're using some kind of trimmed or decaying utility function, where the difference between a candidate who's 2/3 units away and one who's 1 unit away is negligible. Thus, your A voters are like Nader voters; so far out of the mainstream that the other two candidates appear more similar than they really are. So they bullet vote, holding out for a tiny chance of victory. The rest follows; the hapless B>A voters give A a vote, to prevent the likely C win; the B>C voters thus vote for C to ensure A doesn't win; and C's win is almost guaranteed. 2) Depending what you mean by "six factions proportionally from -1 to 1", the B>C>A voters could have tiny B>C preferences. They're either at 0.2 (if the factions are evenly-spaced points), which puts them .13333 from B and .26 from C; or they're at 0.16666 (if the factions are the center of evenly-spaced line segments) which puts them .16666 from B and .22666 from C, a difference of only 0.06. In the second case, the B=(>)C>A votes cause the A>B=(>)C votes and not vice versa. But in either case, the two blocs together form an equilibrium; neither has much motive to change until the other one does. I wouldn't be surprised if there is an alternate equilibrium where the A voters approve B, and a more traditional chicken dilemma ensues. The funny thing is that this is both a chicken dilemma, and precisely the opposite of a chicken dilemma, at the same time. A's bullet vote could be seen as trying to provoke a chicken dilemma between B and C, but since B voters are not unified on their second choices, the fight ends up being played out between B voters, not between B and C. Or you could say that C is trying to cause a chicken dilemma between B and A, and, with the help of some extremely weak-willed C>B voters, is succeeding brilliantly. Anyway: in real life, I think that the A voters would be able to see that if they changed, then the B>C voters would change, and so the A voters would only continue to bullet vote if they really were largely indifferent about B>C. Jameson 2012/2/28 Kevin Venzke <[email protected]> > Hello, > > I have been adding some code to help investigate cases where Approval > shows greater "perception of spoiler" than, say, IRV. To make the > scenarios easier to visualize I just allocated six voting factions > proportionately along 1D, positions ranging from -1 to 1. > > I found an interesting case with the candidate positions: > .939, 0.333, -.06 (call them A, B, C) > > Approval showed perception of spoiler as 27%, whereas IRV, TTR, and FPP > showed none. So I checked to see if it was consistent and what was > happening. > > With six blocs the scenario looks roughly like this (with the pipe > indicating the location of average utility for the bloc): > ~3 C>B | A > ~1 B>C | A > ~1 B>A | C > ~1 A | B>C > > Under IRV, all votes were sincere. Under FPP and TTR, the lone A bloc > was compromising and voting for B. The result was that the sincere CW > (either C or B) was always winning and no one perceived spoilers. > > Under Approval, the C>B voters bullet-voted, the two B blocs voted for > their top two candidates, and the A bloc bullet-voted. > > (A much rarer outcome had the B>C faction bullet-voting, with the B>A > and A factions voting for both A and B, giving the same result as the > other three methods. I think it's clear that this outcome was rarer > because the B>C voters are happier with settling for C than the A > voters are with settling for B.) > > The result of this is that Approval was only electing the sincere CW > half the time. Instead of alternating between C and B winning, C won by > far the most often. B or A won rarely (and, I'd say, largely thanks > to the AI confusion that results from one candidate winning most of > the time). > > Note that C is easily the closest candidate to the median. Even when > B has a majority win over C, B is still not likely to be the utility > maximizer. Approval's success rate at electing the utility maximizer > was thus nearly perfect (instead of 50%). > > I'm not sure what I think of this personally. I'm sure this scenario > isn't any kind of general rule for Approval, but suppose that it was? > Would it be a viable trade-off, to elect the utility maximizer more > often, in exchange for more complaints about spoiled elections? > > Kevin Venzke > ---- > Election-Methods mailing list - see http://electorama.com/em for list info >
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