[EM] Ranked Preferences

[Forest Simmons]

Consider the following non-instant runoff method:

Voters cast preference ballots where truncations and other collapses are
allowed.

Sample ballot:

P > A = Y > R = B = C > X = W

If no Beats All Pairwise winner is found the voters return to the polls.
Their second ballot must have at least one collapsed preference.

If there is still no Beats All winner on the basis of the new ballots, the
voters return to the polls again. 

This time all ballots must have at least two collapsed preferences.

Similarly, on the n_th trip to the polls, each ballot must have at least
n-1 collapsed preference.

If your first ballot already had four collapsed preferences like the
sample ballot above, you could resubmit it unchanged for the first four
return trips to the polls.  After that you would have to decide which
additional preferences you would be most willing to sacrifice.

To be clear, no candidate is eliminated before the winner is found
(unless you want to eliminate a lose-to-all candidate at the first
possible stage in order to ensure satisfaction of the order reversal
criterion).

Now suppose that we wanted an instant version of the above method.

Then in addition to ranking the candidates we would have to rank the
preferences.

The voter of the above ballot might cast something like this for the
instant runoff:

P >> A = Y >>> R = B = C > X = W 

Taking into account the four equal signs, the voting machine would
automatically beef it up to look like this:

P >>>>>> A = Y >>>>>>> R = B = C >>>>> X = W 

In each round of the runoff, each group of inequalities loses a ">".

After four simulated trips to the polls the ballot would be like the (non
beefed up) one cast by the voter.  Then after one more trip it would look
like this: 

P > A = Y >> R = B = C = X = W 

If no beats all winner is found at this stage, then the ballot becomes an
Approval ballot (as would all other ballots cast in this election):

P = A = Y > R = B = C = X = W 

This method satisfies the Condorcet Criterion as well as the Favorite
Betrayal Criterion.  If the "lose to all" loser is eliminated at the first
possible stage, then the method also satisfies the order reversal
criterion.

Now suppose that we wanted to speed up the process, and carry only n
squared log n pieces of information in our ballot summary instead of n
cubed.

Then we could use Dyadic Ballots and collapse (the weaker) half of the
preferences at each stage.

Suppose we really wanted to economize.

Then we could use Demorep ballots, and collapse directly to Approval in
the case of no Condorcet Winner.

Now suppose that the voter did no ranking of preferences, and submitted
her ballot just as in the non-instant runoff:

P > A = Y > R = B = C > X = W 

Then the three inequalities would be beefed up to their average allowable
strength.  In other words the beefed up version shown above would be
modified to become

P >>>>>> A = Y >>>>>> R = B = C >>>>>> X = W 

so that each string of inequalities would have a strength of six, instead
of strengths of five, six, and seven.

After six stages, all of the preferences on this ballot would be
collapsed.  If more rounds were needed to find a winner, this voter's
wishy washy ballot would have no further influence.

That's all for now, but later I will share my ideas of some ways of
converting medium and high resolution CR ballots into Ranked Preference
ballots.

Roy and I have already posted some ideas about converting CR ballots into
Dyadic (and Triadic) ballots, but it gets interesting when you try to
convert medium range CR ballots into (maximally) Ranked Preference
ballots. 

One last word on the philosophy of Ranked Preferences.  It has some of the
spirit of Ranked Pairs, but it requires more information. The voters
supply directly the information about the relative strength of preferences
in Ranked Preferences.  On the other hand Ranked Pairs attempts to infer
which preference omissions would least offend the voters on the basis of
the pairwise matrix summary of the ballots.

Ranked Preferences requires more information, but it satisfies the FBC.

The requirement of more information is not that big a deal if that
information can be encoded in medium resolution Cardinal Ratings ballots
in a natural way.

By "natural way" I mean a way that will not put the unsophisticated voter
at any serious disadvantage.

In other words, if the unsophisticated voter rates the candidates in an
order consistent with her preferences, and makes the biggest gaps in the
ratings consistent with the preferences of greatest strength, then that
should be close enough to optimal to almost nullify the sophisticated
voter's slight advantage, whether in the zero or perfect information case.

But, as I said, that is the subject of a future posting.

Forest