[EM] Unranked-IRV, Cumulative, and Normalized Ratings

[Richard Moore]

Bart Ingles wrote:

> Richard Moore wrote:
> > Cumulative voting allows the voter to give different-sized votes to each
> > candidate, providing that the total of all his votes is less than or equal
> > to some constant. What Tom is proposing, I think, is that the voter has
> > a single vote that may be divided equally among the candidates he likes.
> > So if you want to approve 3 candidates, A gets 1/3, B gets 1/3, and C
> > gets 1/3.
>
> I think that's still considered cumulative voting.  Although
> successively eliminating the weakest candidates and recounting the
> ballots would not be part of cumulative voting.

There's an additional constraint there that isn't present in what we usually
refer to as cumulative voting. That constraint makes Tom's procedure
somewhat more like approval voting. But it's easier in Tom's method
than in approval to make a strategical error by spreading your vote too
thin. However, my gut feeling is that the subsequent elimination and
redistribution counteracts that problem somewhat. I just don't know if
it's enough of a counteraction.

> I think in most cases it's probably a strategy mistake to give different
> weights to your choices in cumulative voting anyway, so this is probably
> a good system.  In the rare exceptions, voters can give "statistical"
> partial votes simply by tossing a coin or rolling a die to determine
> whether to include a candidate.

It's a strategy mistake in cumulative voting, single-winner, to vote for
more than one candidate. But if you're going to make that mistake, then
you'd be compounding it if you gave the same weight to all your choices.
Cumulative is not approval; it's plurality in disguise.

> > Although I'm not satisified that this is the case, it's possible that this
> > constraint may keep the voter from giving all his vote to one candidate.
> > The strategy would have to be evaluated in the context of an elimination
> > procedure, so it's not obvious to me whether this is the case.
>
> Without elimination it's equivalent to FPP (simply vote for only one
> candidate).

True. Give your entire vote to the most strategically valuable candidate.
Don't give half to the most valuable and the other half to someone less
valuable.

> With elimination, maybe not.  For all I know the strategy
> might be equivalent to approval voting, but the performance of the
> system will not be as good (in single-winner elections).  A lot of work
> for a worse result.

I'm not clear how you've determined that the result is worse. Although
since approval performs so well, it's not likely that Tom's UIRV will
outperform it, except perhaps in a few cases.

> > I've been thinking of another alternative to cumulative, in which the voter
> > gives his ratings for each candidate. The voter's ratings are then scaled
> > by an amount which is equal to the square root of the sum of the squares
> > of his or her ratings. I don't know if this method has been discussed
> > before, but I call it Normalized Ratings.
> >
> > In Normalized Ratings, the voter's optimum strategy is to vote for each
> > candidate a rating that is proportional to the voter's strategic value for
> > that candidate. Thus, it avoids the drawback of Single-Winner
> > Cumulative, but it obviously fails to elicit a sincere vote, so it leaves a
> > lot to be desired. However, while an equally-divided approval vote
> > might encourage single-candidate voting in a non-eliminating counting
> > procedure, when combined with an elimination procedure it just might
> > avoid both the problem of Single-Winner Cumulative and the
> > problem of Normalized Ratings.
>
> I think you may be mistaken about the optimum strategy for this system.
> My guess is that for some voters, the best strategy will be the same as
> FPP, and for others it will be more like approval voting.  This may be a
> restatement of the previous paragraph, though -- as when the strategic
> value of the candidates are either one or zero.

Perhaps an example will clarify the optimum Normalized Ratings
strategy. Let's say you have strategic values for each candidate of
4, 3, 0, -3, and -4. Now look at the expected total strategic value
under Normalized Ratings for three different strategies:

(Note first that the normalized vote for any candidate ranges from
-1.0 to +1.0, but if any one candidate receives +1 or -1 from a voter,
then all other candidates must get 0. That is because the sum of the
squares of the normalized votes must equal 1.)

Plurality/cumulative strategy: Vote [1 0 0 0 0], which is already
normalized, and the expected strategic value is 4. Using -1 for
the second through fifth values means that the negative vote
for the second candidate hurts our strategic expectation, while
the normalization step would reduce the magnitude of the vote
going to each candidate, so the 1s and 0s seem like the optimal
expression of a binary plurality strategy.

Approval strategy: Vote [1 1 0 -1 -1], which is normalized to
[0.5 0.5 0 -0.5 -0.5], and the expected strategic value is 7.
I set the middle candidate to 0 because voting +1 or -1 for
this candidate will reduce the amount of votes going for
or against our favorites and least favorites. This ternary
approval strategy also gives a better result than the binary
[1 1 0 0 0] (normalized as [.707 .707 0 0 0] approval
strategy.

Proportional strategy: Vote [4 3 0 -3 -4], which is normalized to
[.566 .424 0 -.424 -.566]. Expected strategic value is 7.072.

The proportional strategy works best for Normalized Ratings.
Nobody is encouraged to throw his entire vote to one
candidate, or to rate two candidates the same unless the
strategic values of the two candidates are the same to begin
with.

I think Normalized Ratings would be a better system for
single-winner elections than Plurality, Cumulative, Borda, or
IRV. But as I said before, Normalized Ratings won't encourage
sincere voting, and that is a serious flaw, so it's not as good
as Condorcet or Approval. I think it might hold some
mathematical interest, though.

 -- Richard