# How Voting Systems Work 9, Exercise 3: how often do weight and power disagree?
# Assumed model: 200 random systems of 5 voters, weights 1-12, quota = smallest strict majority.
import random

def banzhaf(weights, quota):
    n = len(weights)
    critical = [0] * n
    for mask in range(1 << n):
        members = [i for i in range(n) if mask >> i & 1]
        total = sum(weights[i] for i in members)
        if total >= quota:
            for i in members:
                if total - weights[i] < quota:
                    critical[i] += 1
    s = sum(critical)
    return [c / s for c in critical]

random.seed(31)
dummy_systems = monotone_breaks = big_gap = 0
worst = (0, None)
for _ in range(200):
    w = [random.randint(1, 12) for _ in range(5)]
    q = sum(w) // 2 + 1
    p = banzhaf(w, q)
    if 0 in p:
        dummy_systems += 1
    for i in range(5):
        for j in range(5):
            if w[i] > w[j] and p[i] < p[j]:
                monotone_breaks += 1               # more weight but strictly less power
    gap = max(abs(p[i] - w[i] / sum(w)) for i in range(5))
    if gap > 0.10:
        big_gap += 1
    if gap > worst[0]:
        worst = (gap, (w, q, [round(x, 3) for x in p]))
print("systems with a dummy voter:", dummy_systems, "of 200")
print("cases of more weight but less power:", monotone_breaks)
print("systems where some voter's power differs from weight share by over 10 points:", big_gap)
print(f"largest gap {worst[0]:.3f} in weights {worst[1][0]}, quota {worst[1][1]}, Banzhaf shares {worst[1][2]}")

# Output when run:
# systems with a dummy voter: 46 of 200
# cases of more weight but less power: 0
# systems where some voter's power differs from weight share by over 10 points: 69
# largest gap 0.474 in weights [4, 10, 2, 2, 1], quota 10, Banzhaf shares [0.0, 1.0, 0.0, 0.0, 0.0]