[EM] Dealing with ties and truncated ballots

[Michael Rouse]

While I was working on my "Crosscut" method of voting, I realized early on 
that I needed a way of dealing with truncated ballots. If I simply ranked 
all truncated candidates in last place, I would be encouraging people to 
truncate everyone but their top choice, which kind of defeats the purpose 
of rank voting. At first I handled truncated ballots as if they were "tied" 
ballots. In other words, if there were candidates A, B, C, X, Y, Z, and 
someone only voted ABC, then X would receive 1/3 of a vote for fourth 
place, 1/3 of a vote for fifth place, and 1/3 of a vote for sixth place; 
the same for Y and Z. This doesn't encourage truncated ballots, but doesn't 
really discourage it either.

I then realized that if I assumed that truncated ballots indicate no 
preference rather than identical preference, I could re-order the truncated 
ballots so the electorate as a whole was slightly happier without lowering 
the happiness of the voter who truncated his or her ballot. As an analogy, 
if I disliked the colors beige, almond, and cream equally -- or if they all 
looked the same to me -- allowing someone to choose almond over beige and 
beige over cream would make him or her happier without harming me. To use 
the ballot above, if one person voted ABC and another voted AXBYCZ, then 
turning the truncated part of the first vote to XYZ -- in effect making his 
vote ABCXYZ -- would improve the overall satisfaction of the electorate.

The same can be done with tie votes. Say that instead of truncating my 
ballot I voted ABC(XYZ tie). Let's assume that the rest of the electorate 
has the following combined rankings for the XYZ triplet:
order XYZ: 35%
order YZX: 25%
order ZXY: 40%

We would then have the adjusted rank of ABC (X35, Y25, Z40)(X40, Y35, 
Z25)(X25, Y40, Z35) where each set of parenthesis encloses a rank, and the 
numbers indicate the percentage of the rank for each candidate. We could go 
even further and have the winner of (XYZ) in the remaining electorate have 
sole possession of the first available rank, the #2 candidate in (XYZ) the 
second available rank, and the last one the remaining rank. This has the 
added benefit of only handling "whole" votes. In the vast majority of 
single-seat elections this truncation/tie method will have very little 
influence, but in multi-seat elections it might help increase the overall 
happiness of the electorate by electing someone who would otherwise be 
narrowly defeated.

The cool thing about this is this gives a positive incentive to vote for 
every rank if you have *any* preference at all: not only does it prevent an 
order of candidates you don't like, but it gives you a bit more influence 
if someone *else* does not fill out his or her  ballot. If you *really* 
have no preference between two candidates, you can tie or truncate however 
you want --  you don't have to fill out the ballot randomly for your 
remaining votes to be counted. On the other hand, if you want to put a 
hated rival to your favorite candidate at the bottom of your ballot 
"strategically" or sincerely, it is in your best interest to rank every 
candidate in between.

Of course with my method this would have to be recalculated after every 
pass, but computer time is cheap and this would add (marginally of course) 
to the overall happiness of the electorate. In addition, if those truncated 
or tied formed a circular tie,  you could use a Condorcet completion method 
to choose a rank order to put into the list.

Mike Rouse
mrouse at cdsnet.net