[EM] Copeland Done Right

[Forest Simmons]

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!
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