Yes, with X voters and Y voters voting co-operatively, rating eachother's candidate barely short of max, X will win in Score and (probabilistically-voted) Approval.
Something similar can be done in MJ too, and I'll take Jameson's word for it that MJ's tiebreaking bylaws will keep the result from being a tie. Jameson said that, with 89% support needed, it would be easy for defectors to spoil that. But would they want to? Defection by Y voters will hand the win over to Z if Y doesn't do as well as X. The example's premise is that the Y voters intensely dislike Z. They perceive a _much_ greater merit difference between X and Z than between Y and X. So, how rational would it be for any significant number of the Y voters to defect? About MJ's tiebreakers, I just feel they amount to an elegant, artificial Frankensteinian graft. Mike Ossipoff On Mon, Sep 10, 2012 at 6:46 PM, Jameson Quinn <[email protected]> wrote: > Michael Ossipoff recently claimed to have shown that MJ had a bigger problem > with the chicken dilemma than approval or score. Though he'd made a basic > mistake (apparently he thought that MJ used a coinflip as a tiebreaker), > it's still an interesting question, and worthy of sensible discussion. > > So, let's take a realistic chicken dilemma scenario (using XYZ because I'll > use ABCDF for grades in MJ): > > 30: X>Y>>Z > 25: Y>X>>Z > 45: Z>>X=Y > > Why do I use 30, 25, 45 instead of 27,24,49? Because the tight margins in > the latter case means that even 1 or 2 percent of voters who prefer X>Z>Y or > Y>Z>X would upset the whole result. Since in reality there will always be > such people (for instance, the existence of Nader>Bush>Gore voters is > well-documented), I think it's best not to set a scenario too close to the > edge. On the other hand, I don't want to be accused of making the scenario > too easy, so I'm not using my usual numbers of 35, 25, 40. > > Obviously, under Score, or MJ, with honest voters, X will win. Under > approval, if all voters approve everything to the left of >>, then X and Y > will tie; but if some small, constant fraction approve only their favorite > instead, then X will win. > > It's worth looking a little harder at what happens with MJ. Say the grades > are A/B/C/D/F and the X voters all give Y a B, while the Y voters all give X > a B. Now the median for both X and Y will be B. But the tiebreaker will suss > out the difference and X will win. > > That will happen using either the MJ tiebreaker, which involves removing > median votes from both tied candidates simultaneously until they have a > different median; and the CMJ tiebreaker, which involves adjusting each > median by the ratio of [the excess of votes above versus below the median] > to [twice the votes at the median]. The former is like¹ using the smallest > trimmed mean possible which shows a difference; the latter is like² using > the smallest trimmed mean possible which encompasses all the votes at > median. > > So, what happens if a few voters are strategic? In approval, since the > "honest" result is a tie, it only takes a tiny number of strategic Y voters > to swing the result from Y to X. Similarly, in range, if the "honest" vote > for the X and Y factions is to vote the second choice relatively high — say, > 90/100 — then this is almost the same as the situation in approval; it takes > just a tiny 0.6% fraction of strategic Y voters to overcome X's advantage. > > In MJ, the median points for X and Y (50%) are closer to the 60% border with > C than they are to the 30% or 25% (respectively) borders between factions. > Thus, even 1 strategic voter who lowers the other candidate from an honest B > to a semi-strategic C will succeed. It would appear, then, that MJ is at > least as bad as approval, and certainly worse than range. > > However, I think this appearance is false, an artifact of the oversimplistic > scenario. In real life, the X faction (for instance) will never be perfectly > homogeneous. Instead of all of them grading Y at B, they will split between > giving Y a B or a C, with perhaps a few of them giving Y an (honest) A or D > or F. Thus the grades for Y will be something more like: > 25%: A (from the Y faction) > 15%: B (from 50% of X supporters) > 10%: C (33% of X supporters) > 5%: D (17% of X supporters) > 45%: F (Z supporters) > > Meanwhile, X's grades will be: > 30%: A > 12.5%: B > 8.33%: C > 4.17%: D > 45%: F > > With the numbers above, if the Y faction starts to use strategy, starting > with the strongest anti-X partisans among them, it would take a full 5% of > strategic voters to shift the result in MJ or CMJ. > > Obviously, it is possible to create a scenario which is not quite so > perfectly monolithic as the original scenario, where MJ still fails. But you > can see from the above why I believe that in realistic situations, strategy > will NOT be as tempting for the marginal X or Y voter. > > However, in this situation, it's not only the voters who can act > strategically. Consider the situation of candidate Y, deciding whether to > run an "honest" negative ad about Z, or a "backstabbing" negative ad about > X. While there is certainly a risk of it backfiring into voter disgust, from > a purely game-theoretic perspective the "backstabbing" ad is more > attractive. Starting from the original scenario, that holds true in all > three systems: Approval, Score, and MJ. > > If voters are purely rational, they can use Strategic Fractional Rating > (Mike Ossipoff's SFR) to guard against a Z win. For instance, in this > situation, the polls say that Z has 45% support, +/- 2%. For safety's sake, > let's call it 47%. So X and Y together have 53%. If the X and Y factions > were equally matched (26.5% each), then under score, each faction would have > to give the other at least 23.5/26.5 = 89/100 on average to ensure that at > least one of the factions (the larger) would win. Similarly, under approval, > each faction would have to approve the other with at least 89% probability. > In my opinion, 89% is quite a high level of cooperation; the kind of thing > that even a few backstabbing ads would almost certainly overcome. > > However, in MJ, it only takes an 89% chance of giving the other faction's > candidate anything above an F. It's much easier to cooperate when > cooperating means giving your favorite an A and the other faction a D, than > when it means giving the other candidate an 89/100 or even a full approval > on the same level as your own candidate. In fact, the good thing about MJ is > that, as explained above, if everyone is honest, you already have up to a 5% > buffer against strategy — the maximum possible. And if voters want to be > strategic, but not to use any probabilistic strategies that would risk a Z > victory, they can simply give the other faction's candidate a D. (And > generally, those who are most inclined to be strategic will be exactly those > who would honestly rate the other faction's candidate lowest; as long as > this group is smaller than the half the combined margin of victory over Z, > their strategy will have little or no impact). > > OK, that post is long enough for now. If anyone has any questions, I'd be > happy to clarify. > > Jameson > > ¹"Like" in this case means "the same, except in cases which are vanishingly > improbable with realistic numbers of voters" > > ²"Like" in this case means "gives the same ordering of candidates, except in > some cases if the trimmed mean in question would include more than two > grades; which is to say, if there are more votes at the median than at the > grades next door; which is to say, if the cumulative grade distribution is > unusually bendy around the median." > > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info > ---- Election-Methods mailing list - see http://electorama.com/em for list info
