[EM] (3) MJ -- The easiest method to 'tolerate'
Kevin Venzke
stepjak at yahoo.fr
Wed Sep 7 18:37:23 PDT 2016
Hi Jameson,
I wonder if you might remember that I created some simulations that attempted to determine methods' strategies in an organic way, using voters that don't have a concept of sincerity and just pull the levers (initially randomly) in repeated polls to learn what they do. Maybe you can see, from this concept, why I might find it unnatural to consider whether the strategic voter would make a decision to deviate from honesty in MJ. I tend to think the public dialogue preceding the election would color voters' thinking to the point that they wouldn't even perceive a difference between how they think they should vote, and what the ballot language would suggest is sincere. What I imagine is an occasional letter to the editor commenting on how things would be different if people actually did vote as they were asked to. I'm not sure if the tone would be humorous or critical.
In the below post you both suggest a weakening of IIA, and dismiss actual IIA failures as something that will happen only in bizarre scenarios. I wonder to what extent you feel that IIA is an advantage of MJ worth pointing out. Is it an issue you have to raise because other people are raising it?
Your concept of calculating the likelihood that a sincere ballot is suboptimal is an interesting one. You seemingly end up with incredibly high numbers, no doubt because you don't require that the voter can actually change the outcome. But that makes me wonder, in theory, how one would accurately measure this likelihood.
Kevin
De : Jameson Quinn <jameson.quinn at gmail.com>
À : Kevin Venzke <stepjak at yahoo.fr>
Cc : Kristofer Munsterhjelm <km_elmet at t-online.de>; "election-methods at lists.electorama.com" <election-methods at lists.electorama.com>
Envoyé le : Mercredi 7 septembre 2016 9h32
Objet : Re: [EM] (3) MJ -- The easiest method to 'tolerate'
I think I should be a bit more explicit with my mental models here.
In order to evaluate the quality of a voting system, you need a few things:
1. A probability model over electorates. This should be able to generate monte-carlo scenarios involving voters, candidates, and utilities for each voter-candidate pair. Or, in the reverse direction, it should be able to take a class of scenarios, and assign it some kind of plausibility score (that is, non-normalized probability). The process of figuring out how plausible a given class of scenarios is might be computationally difficult, but ideally we should be able to approximate it using some short-cuts.
2. A model for voter information; what does each voter believe about the rest of the electorate.
3. A model for voter strategy. This is a function that takes a voter's candidate utilities, information about other voters, and understanding of the voting system, and outputs a ballot, "strategic" or "honest".
Once you have the above, you can run a monte-carlo simulation and figure out VSE, voter satisfaction efficiency; that is, you can understand each system's expected utility.
Right now, we're not being that rigorous; we're just arguing verbally. But my arguments are intended to be educated guesses about what the above procedure would find.
So, in order to be able to make such guesses, I must have some rough idea of what I'd fill in at steps 1, 2, and 3. The first of those, the probability model over electorates, is the most crucial.
I'm imagining a model where each candidate has a point location in some n-dimensional space of issues and inherent qualities. Each voter has a single-peaked utility function over that space; we can approximate that by saying that the voter is "at" the peak of that utility function.
Of the n dimensions, one ideological dimension dominates; that is, it accounts for a majority of the variance of candidates, voters, and candidate/voter utilities. We can assume that candidates and voters are each distributed unimodally along this primary dimension. (I would not go so far as to assume that distributions are unimodal or even multivariate normal in the full-dimensional space, as that would essentially rule out Condorcet cycles. But I think that unimodal distributions over the primary ideological dimension are an unobjectionable assumption.)
Let's imagine that we have 3 candidates (P,Q,R), located at -100, 0 (center), and +70 on the main ideological dimension; and we have voters (p,q,r) located at -70, 0, and 199. Ignoring for the moment the other dimensions, we can say that U(x,X) is the negative of the ideological distance between candidate X and voter x; so U(p,P)=-30, U(r,P)=-300, etc.
What does this mean for the "honest" MJ ballots? Well, if by "honest" you mean "using a single canonical mapping from utility to grades", it's a mess. Say the cutoffs for A, B, C, D, and F are -40, -80, -120, -160, and -200. That would mean the "honest" ballots would be:p: P:A; Q:B; R:Dq: P:B; Q:A; R:Cr: P:C; Q:D; R:F
To me, that's a broken definition of "honesty". None of our three voters uses the full range of grades; and for voters q and r, that failure to use the full range was entirely predictable, and will probably continue in future elections.
For me, a better definition of "honesty" would be: calibrate an absolute scale such that the historical and/or plausible winners of similar elections would cover the range of grades from B to F, and any candidate better than 95% of historical winners gets an A. If the historical winners come from the ideological range {-80, 80}, that would mean the "honest" ballots would be:p: P:B, Q:C, R:Fq: P:F, Q:A, R:Dr: P:F, Q:C, R:B
Note that these "honest" ballots still don't use the full range of grades, but they come a lot closer than the "honest" ballots above. And for voter p, a candidate at -72 would indeed get an A; as would one at 90 for voter r, even though the absolute utility would still be -109, which would be worse than an F for voter q.
Note also that this definition of "honesty" still preserves IIA within any given election.
As this last example shows, one consequence of the model above is that voters near the center are likely to be much pickier. They're used to being more or less catered to ideologically, so they can afford to exaggerate their differences with both sides. That's what I meant when I said that it's likely that a centrist would give both sides an honest F, which was one of the factors of 1/3 in my rough calculation that an arbitrary honest ballot has less than a 1/27 chance of being strategically suboptimal.
And that, to me, is the key number. My information and strategic models (steps 2 and 3 above) are based on regret; that is, each voter looks at historical elections they've voted in and perhaps a noisy poll or two of the current election, and sees if they have any strategic regret. If such regret happens only 1/27 of the time, and even when it happens it means that the winner is only slightly worse than the strategically optimal winner... I think most voters won't bother attempting strategy, except in the trivial sense of rating at least one favorite candidate at A and at least one despised one at F. (Note that this strategy will almost certainly not affect the medians, and thus will not change the winner. Though technically it breaks IIA, it only does so in bizarre cases where both voter and candidate distributions differ severely from historical norms.)
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