# How Voting Systems Work 1, Exercise 2: how often do three voters produce a cycle?
# Three voters each pick one of the 6 possible strict rankings of A, B, C.
# There are 6 x 6 x 6 = 216 equally likely electorates (ordered). Count the
# ones with a cycle, then estimate the rate for a larger electorate by
# simulation (seeded, so the run is repeatable).
import random
from itertools import permutations, product

rankings = list(permutations("ABC"))

def has_cycle(profile):
    def beats(x, y):
        return sum(1 for b in profile if b.index(x) < b.index(y)) > len(profile) / 2
    return (beats("A", "B") and beats("B", "C") and beats("C", "A")) or \
           (beats("B", "A") and beats("C", "B") and beats("A", "C"))

cycles = sum(1 for p in product(rankings, repeat=3) if has_cycle(p))
print(f"3 voters: {cycles} of {6 ** 3} electorates have a cycle ({cycles / 6 ** 3:.1%})")

random.seed(31)
trials = 2000
for voters in (3, 11, 101):
    hits = sum(
        1 for _ in range(trials)
        if has_cycle([random.choice(rankings) for _ in range(voters)])
    )
    print(f"{voters:3d} random voters: {hits / trials:.1%} of {trials} simulated electorates had a cycle")

# Output when run:
# 3 voters: 12 of 216 electorates have a cycle (5.6%)
#   3 random voters: 5.4% of 2000 simulated electorates had a cycle
#  11 random voters: 7.0% of 2000 simulated electorates had a cycle
# 101 random voters: 7.6% of 2000 simulated electorates had a cycle