[EM] Copeland Done Right (fatal flaw)
Forest Simmons
fsimmons at pcc.edu
Tue Dec 1 13:15:07 PST 2020
Fatal flaw in the monotonicity argument: it turns out that raising X to Top
on one ballot might increase the approval of Y on other ballots where
alternative Y is still ranked higher than X. So even though we have shown
that the approval of X does not decrease, there is a possibility that the
approval of alternative Y might surpass it.
On Monday, November 30, 2020, Forest Simmons <fsimmons at pcc.edu> wrote:
> A while back I made an attempt to de clone Copeland while preserving the
> property of electing uncovered alternatives. Although I got tantalizingly
> close I could not quite pull it off at the time. But recent discussions
> about the difficulty voters have deciding approval cut offs have led me to
> explore various ideas one of which gave me the key to success in our old
> Copeland sprucing up endeavor.
>
> Although it is tempting to completely remove he scaffolding and reveal the
> solution in its Stark Beauty with no trace of the method of discovery, in
> honor of Leonard Euler and with no disrespect for Carl Friedrich Gauss I
> would like to lead you through the successful line of thinking hoping that
> you will enjoy the journey as much as destination.
>
> As I mentioned above, pondering on approval strategy got me started on the
> right path. In particular, an idea Joe Weinstein suggested in the early
> days of the EM list: approve an alternative X if and only if it seems more
> likely for the winner to be someone you like less than X than for the
> winner to be someone you like more than X.
>
> Two immediate corollaries of this rule are to always approve your favorite
> and never approve your most despised alternative since there is no
> likelihood at all that the winner will be an alternative that you like more
> than your favorite nor is there any likelihood that the winner will be an
> alternative that you like less than your most despised.
>
> Another corollary, as Weinstein pointed out, is that when there are two
> clear front-runners, and you like one of them better than the other, you
> should have proved that one but not the other. How about the Alternatives
> in between? Approve them only if the front-runner that you approved is less
> likely to win then the one you did not approve.
>
> What if all of the candidates seem equally likely to win ... in other
> words what if we have zero information about winning probabilities? Then
> Weinstein's rule posits that we should approve every alternative above the
> median and disapprove every alternative below the median, and flip a coin
> to decide about the median alternative itself.
>
> This zero information case exposes two weaknesses of the rule: (1) unless
> the winning probabilities respect clone sets, the rule gives clone
> dependent advice, and (2) it cannot truly give optimal approval advice
> because it takes into account only ordinal as opposed to cardinal
> information beyond the likelihood estimates themselves.
>
> Compare for example, the optimal zero- info strategy that takes
> objectively quantifiable ratings (e.g. dollar costs/benefits) into account
> when they are available: approve every above mean rated alternative.
>
> So for now, with Weinstein we humbly settle for doing the best we can with
> rankings as opposed to ratings.
>
> So back to (1) ... how do we de-clone Weinstein's rule? Here we make use
> of a standard clone independent probability distribution as a plausible
> surrogate for "winning probabilities:" namely the random ballot probability
> distribution ... after all if the winner were chosen by random ballot (a
> clone independent method of election) the random ballot distribution would
> be by definition the distribution of winning probabilities. Note by way of
> contrast that the distribution we resorted to in the zero-info case above
> was the "random candidate" distribution. But why settle for that when we
> have access to the (clone independent and information rich) random (ballot)
> favorite probabilities as soon as the ballots are tallied?
>
> It was disappointing the first time I tried implementing Weinstein's rule
> with random ballot probabilities ... and reminiscent of our recent
> disappointment in our efforts to de-clone Copeland; the clone problem was
> solved, but at a cost of loss of monotonicity (mono raise).
>
> Weinstein's rule has a certain symmetry comparing winning probabilities
> above and below the alternative in question. As it happens in my most
> recent attempts I considered giving partial approval to the "cutoff
> alternative" i.e. the one which has a majority of the probability neither
> above nor below it. Something kept drawing me back to this idea... perhaps
> we could use something like Andy's mental coin flip estimate of whether the
> cutoff alternative was closer to Top or Bottom to decide whether to approve
> it or not ... I was willing to abandom purely ordinal ballots if necessary
> to get something useful out of this!
>
> The turning point came when I finally got the courage to give up on
> symmetry and always approve or always disapprove the cut off alternative.
> There did not seem to be any a priori way to decide between these two
> extremes because on the one hand always including could mean approving
> Bottom if Bottom had 51 percent of the probability or disapproving Top if
> Top had 51 percent of the probability. Which would be worse?
>
> If you think about it, the first of these two bad approval decisions is
> the one that is harmless ... why? Because if Bottom has 51 percent of the
> (random ballot) probability, then any decent rankings based deterministic
> method should elect Bottom ... so no harm done.
>
> So here is the DSV (designated strategy voting) method for automatically
> transforming ranked ballots into approval ballots:
>
> First tabulate the random ballot probabilities.
>
> Then on each ballot B, approve each alternative X such that the combined
> random ballot probability of the winner being ranked strictly ahead of
> (i.e. above) X on ballot B is at most fifty percent.
>
> In other words if there is an even chance or greater that the winner of a
> random ballot election would be ranked (by ballot B) below X or equal with
> X, then approve X.
>
> If you like, you could distnguish between truncation and being "ranked" at
> the bottom. So the above rule applies when X is ranked, and no truncated
> alternative is approved period!
>
> So let's seen how this asymmetry confers mono-raise compliance:
>
> Suppose that the only change is that X is raised on some ballot B. The
> only potential problem is if the probabilities change, and that can only
> happen if X is raised to equal first. That would would result in X being
> approved on ballot B ... so far so good.
>
> But what about on some other ballot B'? Could an increase in Prob(X)
> actually move the approval cutoff up so that on ballot B' alternative X
> goes from approved to disapproved?
>
> The answer is no, because whatever amount of probability is lost by the
> alternatives below X on B' is gained by X, so the amount of probability
> less than or equal to X is at least as great as before, so X does not lose
> approval on ballot B'.
>
> It turns out that the same asymmetry trick works to preserve monotonicity
> in de-cloned Copeland, as I will show in the next message!
>
>
>
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20201201/f9eedc85/attachment.html>
More information about the Election-Methods
mailing list