This is indeed an interesting scenario. Something is particularly weak about those B>C preferences. It could be one of two things:<div><br></div><div>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.</div>
<div><br></div><div>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. </div>
<div><br></div><div>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.</div>
<div><div><br></div><div>I wouldn't be surprised if there is an alternate equilibrium where the A voters approve B, and a more traditional chicken dilemma ensues.</div><div><br></div><div>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.</div>
<div><br></div><div>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.</div>
<div><br></div><div>Jameson<br><br><div class="gmail_quote">2012/2/28 Kevin Venzke <span dir="ltr"><<a href="mailto:stepjak@yahoo.fr">stepjak@yahoo.fr</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Hello,<br>
<br>
I have been adding some code to help investigate cases where Approval<br>
shows greater "perception of spoiler" than, say, IRV. To make the<br>
scenarios easier to visualize I just allocated six voting factions<br>
proportionately along 1D, positions ranging from -1 to 1.<br>
<br>
I found an interesting case with the candidate positions:<br>
.939, 0.333, -.06 (call them A, B, C)<br>
<br>
Approval showed perception of spoiler as 27%, whereas IRV, TTR, and FPP<br>
showed none. So I checked to see if it was consistent and what was<br>
happening.<br>
<br>
With six blocs the scenario looks roughly like this (with the pipe<br>
indicating the location of average utility for the bloc):<br>
~3 C>B | A<br>
~1 B>C | A<br>
~1 B>A | C<br>
~1 A | B>C<br>
<br>
Under IRV, all votes were sincere. Under FPP and TTR, the lone A bloc<br>
was compromising and voting for B. The result was that the sincere CW<br>
(either C or B) was always winning and no one perceived spoilers.<br>
<br>
Under Approval, the C>B voters bullet-voted, the two B blocs voted for<br>
their top two candidates, and the A bloc bullet-voted.<br>
<br>
(A much rarer outcome had the B>C faction bullet-voting, with the B>A<br>
and A factions voting for both A and B, giving the same result as the<br>
other three methods. I think it's clear that this outcome was rarer<br>
because the B>C voters are happier with settling for C than the A<br>
voters are with settling for B.)<br>
<br>
The result of this is that Approval was only electing the sincere CW<br>
half the time. Instead of alternating between C and B winning, C won by<br>
far the most often. B or A won rarely (and, I'd say, largely thanks<br>
to the AI confusion that results from one candidate winning most of<br>
the time).<br>
<br>
Note that C is easily the closest candidate to the median. Even when<br>
B has a majority win over C, B is still not likely to be the utility<br>
maximizer. Approval's success rate at electing the utility maximizer<br>
was thus nearly perfect (instead of 50%).<br>
<br>
I'm not sure what I think of this personally. I'm sure this scenario<br>
isn't any kind of general rule for Approval, but suppose that it was?<br>
Would it be a viable trade-off, to elect the utility maximizer more<br>
often, in exchange for more complaints about spoiled elections?<br>
<span class="HOEnZb"><font color="#888888"> <br>
Kevin Venzke<br>
----<br>
Election-Methods mailing list - see <a href="http://electorama.com/em" target="_blank">http://electorama.com/em</a> for list info<br>
</font></span></blockquote></div><br></div></div>