[EM] How to find the voters' honest preferences

[Kristofer Munsterhjelm]

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.