[EM] Efforts to improve on CR's strategy

[Bart Ingles]

Ken Johnson wrote:
> >
> >From: Bart Ingles
> >
> >...  In your 10 candidate, 1 issue trial, are you able to account
> >for why sincere CR, exaggerated CR, Condorcet, Borda, IRV, and Plurality
> >all yield exactly the same average across 100,000 elections?  It looks
> >like top-two Runoff is within 0.1% of the same score.
> >
> I think it's simply the case that with 1 issue, all voters' CR profiles
> are precisely correlated (i.e., any two profiles differ only by a
> multiplicative scale factor), so all these methods become equivalent.

Here's what I think of as a single-issue trial:

Assume a single (unnamed) issue, in which any voter or candidate can
take a position pro or con.  You have 3 candidates (A, B, and C) who are
at positions +.7, +.2, -.5.  You also have a number of voters who each
can be placed at some point along the same scale.  Voters' utilities can
be determined by distance from each candidate.  For example, a voter at
+.4 would have absolute ratings (distance) for the three candidates of
-.3, -.2, -.9.

Normalizing these distances (what you call exaggerating) would give
utilities of A=.86, B=1.0, C=0.0 for this particular voter on a scale of
[0..1], with a mean of (.86 + 1.0)/3 = 0.62.  You can normalize to
[-1..1] instead if you want, but most of the literature uses [0..1]

Note that if each voter's position is a random number between 0 and 1
(or any other endpoints), then the voters' utility profiles will not be
simple vector multiples of one another.  In fact I'm having trouble
imagining a plausible model where they would be.

...

As for optimal strategy based on the above example:

Since both A and B are above the mean of 0.62, the optimal strategy in
Approval for this voter is to approve of both A and B.  The optimal
strategy under CR is to give A and B maximum points, and C minimum
points.


> >Stranger still, exaggerated CR should be equivalent to Approval, but the
> >scores here are wildly dissimilar.
> >
> What I call "ExaggerateCR" is not actually the optimal zero-info CR
> strategy, which would be equivalent to Approval.

I see that now.  What you call "ExaggerateCR" is what I would call
"sincere CR", since the voter should at least give maximum points to his
favorite, and minimum to his least favorite.  Otherwise the voters
aren't sincere, they're just being stupid, in effect partially
abstaining.

It's even worse with what you call "SincereAV", since a voter with all
positive or all negative unnormalized ratings would either approve all
or not vote at all.  This would be senseless behavior on the part of a
voter.


> "Sincere strategy"? From my perspective all strategies are insincere.

Nevertheless, you use the term to describe some variants of Approval and
CR strategy in your simulation.


> Based on the optimum zero-info strategy, should the approval cutoff be
> at the mean CR of ALL candidates, or of just the highest- and
> lowest-rated candidates? (I assumed the latter for "ExaggerateAV".)

All candidates.


> Here's a conceptual example that I think better illustrates the problem
> that I observed. Suppose you vote in an election in which there are 6
> candidates and you have no idea how anyone else votes. Your sincere CR
> profile for candidates A ... F is
>     SincereCR: A(0.7), B(0.5), C(0.3), D(0.1), E(-0.1), F(-0.3)
> (This assumes signed CR's, with an approval cutoff of zero.) What I call
> "ExaggerateCR" simply applies a linear transformation so that the max
> and min CR's are +1 and -1:
>     ExaggerateCR: A(1.0), B(0.6), C(0.2), D(-0.2), E(-0.6), F(-1.0)
> Sincere Approval is based on the Sincere CR profile:
>     SincereAV: A(1), B(1), C(1), D(1), E(0), F(0)
> Strategic Approval is based on the ExaggerateCR profile:
>     ExaggerateAV: A(1), B(1), C(1), D(0), E(0), F(0)
> All the rank methods sort the candidates by CR:
>     A > B > C > D > E > F

Ok, what you call SincereCR and SincereAV are not real strategies from
my point of view.  It would be meaningless to compare them with other
methods.

What you call "ExaggerateCR" is what I would call "SincereCR", since you
are basically just translating a raw rating info into a useful utility
range.

What you call "ExaggerateAV" is sort of a mediocre "quick-and-dirty"
approval strategy which might be useful, if for no other purpose but to
see how close it measures up to an optimal AV strategy.

What I (and most of the literature) call "sincere AV" (described above)
is simply making best use of the voter's known preferences, and not
trying to make use of polling info, etc.


> Now suppose the ballots get counted and it turns out that the total
> ballot count is 1. No one else bothered to vote, so your ballot
> determines the election. With the exception of Approval, all methods
> give A as the winner, whose CR is 0.7. SincereAV gives a 4-way tie
> between A, B, C and D, and ExaggerateAV gives a 3-way tie between A, B,
> and C. In computing the winner's CR, I assumed that a tie is broken by
> random choice, and I report the probability-weighted CR as the result.
> For example, with SincereAV A, B, C and D each has a 25% chance of
> winning, so I report the winner's CR as the average of A, B, C and D -
> which is 0.4. Similarly, for ExaggerateAV the winner's CR is reported as
> 0.5.
> 
> The 1-ballot assumption is obviously unrealistic, but I think this
> example conveys the essence of what was happening with my simulations.

If your simulations are behaving like 1-ballot elections, you had better
check your random number generator.

Bart