Suppose $f : \{-1,1\}^n \to \{-1,1\}$ is a voting rule for a $2$-candidate election. Making the impartial culture assumption, the $n$ voters independently and uniformly randomly choose their votes ${\boldsymbol{x}} = ({\boldsymbol{x}}_1, \dots, {\boldsymbol{x}}_n)$. Now imagine that when each voter goes to the ballot box there is some chance that their vote is misrecorded.

[...]

## Recent comments

Ohad Klein: Are the indexing in (the start of) 7 OK?Ohad Klein: "learning algorithm running in time in time"Amir: In the proof of Theorem 16, and in the equation immediately ...Ohad Klein: In example 6, should "of codimension less than n" be "of pos...Ohad Klein: In 15c (18c in the book), I think it should be $\cap_j{V_j}$...Ohad Klein: Bracket typo: In the proof of thm 10 (12 in the book), $sgn(...Ohad Klein: Oops, my bad.