# [EM] Truncation and Strong FBC

Alex Small asmall at physics.ucsb.edu
Sun Oct 6 22:53:53 PDT 2002

```I was at a boring retreat over the weekend and had the time to ponder two
ideas:

The most important one is that strong FBC becomes easier to satisfy if we
allow truncation.  The condition is no longer that the voters never derive
betrayal never give an outcome preferable to that obtained by sincere
truncation (listing favorite and no others in a 3-way race).  This might
not make strong FBC possible, but it seems to improve the odds somewhat.

Second, although this isn't particularly significant, I realized that
plurality voting can be derived from a symmetry condition when voters are
limited in the amount of info they can provide.  I started by thinking of
a situation where instead of 6 dimensions (for 6 types of ballots in a
3-way race, or 9 types with truncation) we work in some lower-dimensional
space, to try and make my geometric idea easier to visualize.  By
symmetry, such a space can only have 3 dimensions (or zero, in a
dictatorship).

Furthermore, by constraining the number of voters in each category to add
up to a predetermined number, we find ourselves working in a triangle.
Finally, the symmetry condition for making the election method unbiased
forces us to divide the triangle into three equal regions corresponding
exactly to plurality voting.

OK, so plurality voting is a no-brainer.  I just thought it was cute that
I could derive it as the only possible ranked method for a 3-way race when
we limit the amount of info voters can provide.  Anyway, it tells me that
the geometric idea might actually be useful for providing insight into
harder problems than pluralityl.

Just felt like sharing.

Alex

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