The rationality of weakly symmetric functions

Towards the end of class I quickly said the following:

Let $f : \{-1,1\}^n \to \{-1,1\}$ be any social choice function, and suppose we try to use it to rank 3 candidates via 3 pairwise comparisons using $f$. The Rationality of $f$ is the probability (when the voters' preferences are i.i.d. and uniform) that this doesn't not result in a cyclic preference.

Then if $f$ is weakly symmetric, the Rationality of $f$ is at most $7/9 + 4/9\pi \approx .919$. Here is the proof. Since the Rationality of Majority is $3/4 - (3/2\pi) \arcsin(-1/3) \approx .912$, this is looking pretty good for Majority.

In fact, later in this course we will prove the Majority Is Stablest Theorem, which in particular implies that any weakly symmetric function has Rationality at most that of Majority (plus $O(\log \log n / \log n)$).