# [EM] FBLE probabilities - amazing but true

Warren Smith wds at math.temple.edu
Mon Nov 28 16:44:30 PST 2005

```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

```