[EM] (3) MJ -- The easiest method to 'tolerate'

[Jameson Quinn]

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:D
q: P:B; Q:A; R:C
r: 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:F
q: P:F, Q:A, R:D
r: 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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