Scribe Notes for Mossel’s CS294-063/Stat206A, October 7th

We will now focus on the instance when k = 3, and we will denote the three alternatives as a, b, and c. For ease of notation, we will map each voter’s ranking, σi to a tuple (xi, yi, zi) ∈ {−1, 1}3, where xi = 1 if σi(a) > σi(b), and −1 otherwise. Similarly, yi = 1 if σi(b) > σi(c) and −1 otherwise, and zi = 1 if σi(c) > σi(a), and −1 otherwise. Finally, we will let x := (x1, . . . , xn), y := (y1, . . . , yn), and z := (z1, . . . , zn). Remark 2. Note that (xi, yi, zi) corresponds to a σi if, and only if, (xi, yi, zi) ∈ {−1, 1}3 \ {(1, 1, 1), (−1,−1,−1)}.