# [EM] strategic voting

Forest Simmons fsimmons at pcc.edu
Thu Jan 18 15:02:50 PST 2001

```Mike, thanks for the valuable information.

in a version where the votes can be split into arbitrary positive reals
summing to a given value or less, the space of possible ballots is a
simplex, and the corners of the simplex are the points where the whole
amount is concentrated on one candidate.

As I mentioned before, in CR the space of possible ballots is a hypercube,
and the corners correspond to AV ballots.

I view Rankings as a way of forcing voters away from these extremes.

Since corners seem to be the reason for these extreme strategies, how
about making the space of possible ballots as round and smooth as
possible? The roundest such space would be a hypersphere.  The voters
would vote their ratings graphically between -1 and 1 .  As the voter
adjusts one lever the other levers would automatically move to keep the
sum of squares constant.  When the voter gets all the levers in the right
proportion, he/she submits the ballot.

Cumulative voting limits the L_one norm of the ballot.  Approval voting
limits the L_infinity norm.  This "round space" method would limit the
L_two norm of the ballot.

The L_two norm is widely considered to be the happy medium between the two
extreme norms.  In two dimensions, a constant L_two norm yields a circle
inscribed in a square representing the same constant value for the
L_infinity norm (of points in the plane.)  If the L_one norm is limited to
the same constant, it yields a diamond that touches the circle and the
square at the same points. The circle is the happy medium between the
square and the diamond.

In other words, the diamond  abs(x) + abs(y) = 100%, is inscribed inside
the circle given by  x^2 + y^2 = 100%, which, in turn, is inscribed inside
the square given by  max(abs(x),abs(y)) = 100% . Analogous geometry takes
place in higher dimensions.

Has anyone ever explored this constant variance idea before, as a method
of keeping the ratings from bunching up, but allowing more expressiveness
than rankings?

Forest

On Thu, 18 Jan 2001, MIKE OSSIPOFF wrote:

>
> My 1st reply didn't post, but my 2nd one did. I'm going to
> find my copy of the 1st reply & re-post it.
>
> Approval was first proposed by someone named Robert Weber, at
> Northwestern Univerisity, I believe, in Illinois. His article,
> "Approval Voting", in the Winter '95 issue of _Journal of Economic
> Perpective_ covers the derivation of the way of calculating
> strategy for Approval & Plurality.
>
> This subject is also covered by Samuel Merrill, in his book
> _Making Multicandidate Elections More Democratic_. It's out of print,
> I've been told, but of course it's available at some university
> libraries, and via interlibrary loan. Merrill also writes about
> strategy for other similiar methods, including Cardinal Ratings,
> and single-winner Cumulative (where we can divide some votes among
> the candidates however we wish). He shows that Cardinal Ratings is
> strategically equivalent to Approval, and that single-winner Cumulative
> is strategically equivalent to Plurality (One should give all his
> votes to the candidate for whom he'd vote in Plurality).
>
> I got the term "strategic value" from Merrill, if I remember correctly.
>
> An author named Cox, in one of his books on voting systems,
> writes about strategy for Runoff.
>
> For some discussion of one way of deriving those strategies for
> Approval & Plurality, and some discussion about estimating the Pij,
> Plurality strategy:
>
> http://www.barnsdle.demon.co.uk/vote/sing.html
>
> One thing I describe there is a suggestion by Tideman, for
> estimating the Pij from the various candidates' estimated probabilies
> of winning.
>
> At Laurie Crannor's website (I don't have the URL right here, but
> I believe that it can be reached via http://www.eskimo.com/~robla,
> a website belonging to the owner of EM), she describes Hoffman's way
> of estimating the Pij. It uses the same geometrical idea as Tideman's
> estimate, but instead of using estimated probabilities of winning,
> Hoffman uses a previous election result, with the same candidates or
> parties, for an effort at a more precise estimate.
>
> My article at the barnsdle website might have a link to Crannor's
> website, where her article can be found by selecting "Determining
> Pivotal Probabilities", or something to that effect, at her main menu.
>
> Her interest isn't Approval; she's interested in using Plurality with
> a system for collecting ratings from the voters and using those to
> calculate each voter's optimal strategy. She'd use each final Plurality
> result to calculate new Pij, and hope that the result converges.
> It sounds too complicated for a public proposal. But her frontrunner
> probability estimating article is interesting & maybe useful.
>
> Hoffman's approach requires more & more spatial dimensions, as we
> add candidates. So Crannor describes, in the same article, a suggestion
> for avoiding that need. I didn't understand her description of her
> procedure; it seemed to me that she didn't tell enough about it.
> If you find it & understand it, I'd be grateful for an explanation of it.
>
> I'd also expect that there's some way of statistically estimating
> the probability that a certain 2 parties' candidates will be the
> frontrunners, based on how well all the parties have done in various
> previous elections which don't necessarily have the same party lineup
> as the current one for which we want the Pij. I haven't pursued that,
> and wouldn't know how to, but I'd be interested to find out how
> someone else makes that approach work.
>
> Mike Ossipoff
>
> _________________________________________________________________