[EM] Hybrid Beats-All/Approval v. Straight Approval

Forest Simmons fsimmons at pcc.edu
Mon Oct 29 16:50:34 PST 2001

Good point!

If voter X is almost sure that his ballot will make the difference between
a hated (by X) Condorcet Winner and a Condorcet tie (to be settled by
chance), voter X might be tempted to deliver up the election to chance
even if that required him to vote his favorite last and his most despised

Of course, if all like minded voters gave in to the same temptation, the
tie breaker odds of favorite to most despised would be adversely affected.

It's not clear that this effect wouldn't always over compensate for the
other effect, thereby removing the temptation for insincerity. But I
suspect that you are right.

Thanks for pointing this out.

So is random ballot the only method with reasonable expected utility that
is immune to manipulation?

It is OK in expected utility, but low in Banzhaf Power:

Under random ballot the probability that your vote will be pivotal is one
divided by the number of voters, whereas in lone mark plurality and other
common methods your power is more like one over the square root of the
number of voters, a dramatic improvement.

Low Banzhaf Power means lack of responsiveness to the electorate.

How about Cranor's method? Is it immune to manipulation?

Cranor takes everbody's (supposedly) sincere utilities for all of the
candidates and comes up with (supposedly) optimal recommendations for
their Approval votes.

Let's say that if everybody takes Cranor's recommendations as their
approval votes, that the resulting approval winner is the "Cranor Winner" 
of the election. 

Can a voter move the Cranor Winner higher up his true preference list by
faking the utilities that he submits to Cranor? 

If so, then what is Cranor supposed to offer over Instant Approval Runoff?

I suspect that the Cranor Winner is the same as the iterated Approval
runoff winner (iterated to convergence).

If so, then I don't see how Cranor's method could be manipulated.

But my intuition could be wrong here, too.

In my opinion Cranor and instant approval runoff share their main
problems: (1) lack of summability and (2) too complicated for the voting
public to use. 

If the candidates, as proxies for the voters, submitted to Cranor's method
or to (non-instant) Approval Runoff, these two problems would dissolve. 

If candidates were required to post their Cardinal Ratings of all of the
other candidates prior to the voters' designation of proxies, AND the
proxie method were non-manipulable (Cranor?), then it would make sense to
bind the proxies to their posted ratings. 

If the proxie method were (non-instant) Approval Runoff, it would make
sense to bind the proxies to the order of the posted ratings, if not to
the "utilities" themselves in the successive approval cutoff calculations
(for each stage of the runoff).

Demorep's vision of candidates being locked up without food and water
until they (as proxies) decide on a consensus winner seems (to me)
roughly equivalent to settling on the Cranor winner, except the weaker
willed proxies or those lacking in endurance might prefer Cranor, while
the mentally or physically intimidating candidates might prefer the locked


On Sun, 28 Oct 2001, Bart Ingles wrote:

> Forest Simmons wrote:
> >
> > Consider the case of a beats-all check followed by your random ballot
> > suggestion:
> > 
> > Voters are to submit ranked ballots with truncations allowed internally as
> > well as at the extremes (i.e. where there is no preference equal ranks are
> > allowed).
> > 
> > Suppose that the winner of the election is to be the beats-all candidate
> > if there is one, and otherwise the highest ranked candidate on a randomly
> > drawn ballot.
> > 
> > How would polling information make any difference in the way anybody
> > voted?
> I'm not sure how we defined "beats-all", so I'm taking it to mean "sole
> Condorcet winner".
> If the polling data shows your candidate to be drawing 35% of first
> preferences, you can expect your candidate to have a 35% chance of
> winning the random completion method.  If you view the likely beats-all
> candidate's utility as .30, then your best strategy is to vote in such a
> way as to prevent the ballots from showing any beats-all winner (try to
> force a cycle, in other words), assuming you can't otherwise make your
> own candidate the beats-all winner.
> If, on the other hand, false polling data convinces you that your
> candidate is only drawing 25% of the first-choice vote, you would want
> to help the beats-all candidate as much as possible.
> > After the election, no matter who the winner turned out to be, and no
> > matter how all of the other voters actually voted, how could any sincere
> > voter regret his/her voted ballot?
> >
> > So beats-all with random completion is strategy free and non-manipulable.
> The sincere voter would regret voting sincerely if the .30 beats-all
> candidate won, but the results showed that the voter's first choice
> would have had > 35% chance of winning the random drawing.
> I don't think I agree with claims in the rest of this message either,
> but they seem to be based on the earlier arguments, so I think I'll stop
> here until I'm sure I've understood correctly so far.
> Bart
> > If some other completion is used, then the election can be manipulated to
> > some degree, depending on the completion method.
> > 
> > It seems to me that the beats-all check on the front end of the method
> > can soften the effect of strategy mistakes on the back end "completion" as
> > long as the completion method isn't too blatantly prone to order reversal
> > strategy like Borda.
> >
> > Suppose that Approval is the completion method, for example.  It takes
> > polling information for voters to make a good choice of approval cutoff,
> > but no polling information to decide on the order of the candidates.
> >
> > If the polling information is totally wrong, the approval cutoff choices
> > may be bad, but with a little bit of luck, the election is decided before
> > the approval stage.
> > 
> > To minimize the potential for manipulation, Condorcet with Approval
> > Completion would be carried out with two trips to the polls.
> > 
> > If there is a CW, only one trip is necessary.
> > 
> > If not, then the results of the first trip reliably inform the choices for
> > the second (Approval) trip to the polls.
> > 
> > Of course (as Demorep is sure to point out) this method is too complicated
> > and inconvenient for public consumption, but it may have private
> > application in situations where it is important to minimize the potential
> > for manipulation.
> > 
> > More importantly, as far as I'm concerned, it helps us see the practical
> > limitations of strategy free methods.
> > 
> > Forest

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