# [EM] MJ, Approval, Score, and Chicken Dilemma

Michael Ossipoff email9648742 at gmail.com
Tue Sep 11 09:17:48 PDT 2012

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 <jameson.quinn at gmail.com> 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
> 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."
>
>
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