# [EM] Dealing with ties and truncated ballots

Michael Rouse mrouse at cdsnet.net
Tue Jun 19 23:13:13 PDT 2001

```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.

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

```