[EM] How to find the voters' honest preferences

[Forest Simmons]

Kristofer,

Thanks for your insights and considerations regarding the method and my
questions.

It is true as you noted that there is no game theoretic incentive to vote
insincerely on the second ballot,(unless this election is considered as
part of an ongoing game including future elections as well), it is also
true, as you point out, that there is no incentive to vote complete
preferences on the second ballot.

You then say ...

"If you just want to find a winner, then an ordinary runoff might work as
well: select the finalists as above, then have a majority-rule election in
the second round."

The trouble with this ordinary runoff idea is that the runoff stage
(potentially) over-rides the strategic pairwise preferences implicit in the
three slot ballots.  In other words it throws out our burial disincentive.

Concerning the burden of two ballots:

In practice, voters that consider their three slot ballots to be sincere
could opt out of the second ballot, or they could opt to replicate their
favorite's preference order, etc.

My Best,

Forest




On Sun, Sep 8, 2013 at 3:27 AM, Kristofer Munsterhjelm <km_elmet at t-online.de
> wrote:

> On 09/08/2013 02:50 AM, Forest Simmons wrote:
>
>> The following method makes use of two ballots for each voter.  The first
>> ballot is a three slot ballot with allowed ratings of 0, 1, and 2.  The
>> second ballot is an ordinal preference ballot that allows equal rankings
>> and truncations as options.
>>
>> The three slot ballot is used to select two finalists: one of them is
>> the candidate rated at two on the greatest number of ballots.  The other
>> one is the candidate rated zero on the fewest ballots.
>>
>> The runoff between them is decided by the voters' pairwise preferences
>> as expressed on the three slot ballots (when these finalists are not
>> rated equally thereon), or (otherwise) on the ordinal ballots when the
>> three slot ballot makes no distinction between them.
>>
>> [Giving priority to the three slot pairwise preference over the ordinal
>> ballot preferences is necessary to remove the burial incentive.]
>>
>> Note that there is no strategic advantage for insincere rankings on the
>> ordinal ballots.
>>
>
> This sounds like an automated runoff. You use the first ballot to perform
> an initial election, then you use the second ballot to determine the
> outcome of a runoff between the two you picked from the initial election.
> Because majority rule is strategy-proof with n=2, there is no incentive to
> be insincere on the second ballot.
>
> But some might say that in certain situations there's no incentive to
> fully rank the second ballot either. Say that you're pretty sure the runoff
> will come to X vs Y. Then you only need to fill in X vs Y.
>
> Now, if the voters are basically honest and strategy concerns only come in
> second place (overriding their honesty if the pressure towards strategy is
> strong enough), then the voters will submit a full honest ranking anyway.
> But if they're not, then they might not give you all their preferences.
> (This question is related to other discussion as well. I've sometimes
> argued that if you have a criterion X that says "there is no incentive to
> be insincere in way Y unless enough people do it", that will keep the
> voters from doing so, because their primary concern is honesty; while
> others think that the voters will be insincere in way Y anyway because
> there's no disincentive either. The difference is whether we need "there
> must be an incentive to being honest" or whether "there is no incentive to
> being dishonest" is enough.)
>
> Anyway, to get back on topic, the method you mention seems to work in a
> game-theoretical sense. The ordinal ballots will be sincere. However, I
> think real world voters would be confused by it. "Why do I have to submit
> two ballots?", and so on. I suspect that strategic voters will be strategic
> on both ballots, while honest voters will be honest on both unless they
> feel like they have to use strategy on the first.
>
> If you just want to find a winner, then an ordinary runoff might work as
> well: select the finalists as above, then have a majority-rule election in
> the second round. If the purpose is determining the honest preferences,
> then your method would have an advantage since it requires the voters to
> state their preferences ahead of time (before they know who the finalists
> are going to be) and so will have to submit more information than in a
> runoff.
>
>
>  Questions.
>>
>> (1) What are some near optimal strategies for voters to convert their
>> complete cardinal ratings into three slot ratings in this context?
>>
>
> A strategic voter would like his preferred candidate to be matched with
> someone that can't win against him. So at a cursory glance, it would seem a
> good (zero-order) strategy is an exaggeration tactic: give the favorites 2
> points to get one of them into the runoff. Then put the no-hopes (compared
> to your favorites) at rank 1 to get one of them into the runoff as well.
> Finally, give the dangerous competitors zero points to keep them out of the
> runoff.
>
> There's also a converse strategy: 2 points to the weak competitors and 1
> point to the favorites. But giving the favorites 2 points doubly secures
> the voter: the favorites are in the running for both spots, while the weak
> candidates are only in the running for the second spot. So this helps
> increase the chances that the runoff will be favorite vs favorite (in which
> case it doesn't much matter which favorite wins).
>
> It may pay to push one of the favorites into the 1-point slot if the
> strategic voter has enough information, though. Consider a case where the
> voter prefers A > B > C > D, and if he votes honestly, then the contest
> will be A vs B and B will win. But if the voter puts B in the zero-points
> slot, then it will be A vs C and A will win. Then there's an incentive to
> do so -- because B will win, he's more a "dangerous competitor" to A than
> he is a favorite. One could probably calculate expected value to determine
> whether to give B a single point or none.
>
> But of course no strategy survives contact with the enemy. So n-th order
> strategy would be considerably more difficult. The principle would be the
> same, however: a voter desires to make one of his favorites enter the
> second round against someone whose chance to win is much less than the
> disutility should he win.
>
> (N-th order strategies also have to take into consideration the problems
> of a burial spree: if everybody puts no-hopes second, then these no-hopes
> win the runoff by pairwise preference.)
>
>
>  (2) We have a "sincere approval" method of converting cardinal ratings
>> into two slot ballots.  What is the analogous "sincere three slot" method?
>>
>> [Sincere approval works by topping off the upper ratings with the lower
>> ratings;  think of the ratings as full or partially full cups of rating
>> fluid next to each candidate's name.  If you rate a candidate at 35%,
>> then that candidate's cup is 35% full of rating fluid.  Empty all of the
>> rating fluid into one big pitcher and use it to completely fill as many
>> cups as possible from highest rated candidate down.  Approve the
>> candidates that end up with full cups. This is called "sincere approval"
>> because generically (and statistically) the total approval (over all
>> voters) for each candidate turns out to be the same as the total rating
>> would have been.]
>>
>
> The answer depends on what you'd like to reproduce. Even if you'd like to
> reproduce the cardinal score by giving candidates two points for top rank
> and one point for middle, then it's not obvious which of the many solutions
> to pick. For instance, you could completely disregard the top slot and just
> do your rating fluid solution, considering the middle and lower slot as
> "approved" and "not approved".
>
> So there would have to be additional constraints. One might be that if the
> voter rates any candidate above minimum, then at least one candidate has to
> be put in a top slot. Another might be that the error (difference between
> actual cardinal sums and quantized ones) should be minimized for all
> possible ways of filling in cardinal ballots.
>
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