# [EM] FBLE probabilities - amazing but true

Scott Ritchie scott at open-vote.org
Mon Nov 28 23:58:34 PST 2005

```I found this sentence when following the favorite betrayal link on the
website:

"Range voting, however, is immune to this problem in the sense that in
any 3-candidate election, it is never strategically desirable for any
voter to range-vote as though X>Y when his true opinion is Y>X. So in
the situation illustrated, the six C>A>B voters would not find it
strategically desirable to cast a range vote in which A>C>B. And the
word "never" above makes us believe that with range voting, 2-party
domination is not inevitable."

Here I am, the range voter, and I notice that A and B are the
frontrunners, yet I prefer C>B>A.  Under range voting, I correctly rate
C a 100% and A a 0%.  But now I can insincerely rate B at exactly equal
to 100%, making my ballot effectively C=B>A.  If I'm not allowed to do
an equal ranking, then I of course do have an incentive to rank C at 99%
and B at 100%, so I'm going to presume you're always allowing equal
ratings.

This got me thinking: can any deterministic election method other than
random candidate prevent "2-party" domination?  Suppose we have an
election that has two candidates with near majority support -
frontrunners.  Just because an election method can enable voters to
sincerely rank a third party candidate without hurting themselves, the
candidate will still lose if he doesn't command a similar near majority
support compared to the others under just about every fair election
method we can come up with.

It seems, then, we probably shouldn't be talking about cases of which
method is better for third parties when they don't have enough support
to win anyway, like the strategic C preferers in the examples given.  It
might be far more productive to analyze how voting systems perform when
stressed with more than two candidates that could conceivably win, ie a
true three-way race (incidentally, a sizable portion of the randomly
generated elections you have created have this property, not
coincidentally at the times when the C preferers don't have so much a
favorite betrayal incentive since C can actually win.)

Thanks,
Scott Ritchie

On Mon, 2005-11-28 at 19:44 -0500, Warren Smith wrote:
> Probabilities of "favorite-betrayal lesser-evil" (FBLE) situations in 3-candidate
> ranked-ballot elections
>
> Definition of "FBLE situation": Call the election-winner A.  An FBLE situation then
> occurs when some C>B>A voters, by switching to B>C>A ("betraying their favorite" C)
> can make B win (an outcome they prefer).
>
> Probabilistic model: All elections equally likely. That is, with V voters, there are 6^V
> possible elections since each voter can vote in 6 ways:
>    A>B>C, A>C>B, B>A>C, B>C>A, C>A>B, C>B>A.
> (We disallow "truncated ballots" and "ranking equalities.") We shall consider the large-V limit.
>
> Theorem 1:
> Basic Condorcet's FBLE probability is 25%.
>
> Theorem 2.
> Instant Runoff's FBLE probability is arctan(1/sqrt(2)/pi = 19.5913...%.
>
> Isn't that amazing? :)  I had previously analysed this latter
> incorrectly and thought 25%.
>
> Note.
> In both theorems the CBA voters are at least as well off switching to BCA
> IF they believe C's winning chances are well below 25% and 19.6% - thus
> the betrayal in some sense is strategically justified with 100% probability in
> both cases.
>
> The proofs are at http://math.temple.edu/~wds/crv/IRVStratPf.html .
>
> wds
>
>
> ----
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