https://apieceofthepi.substack.com/p/anti-inductive-dice

AntinductiveDice.py

# https://arxiv.org/abs/2503.16306

from collections import Counter, defaultdict
from itertools import product, repeat
from math import prod
a = [1,1,4,4,5,6]
b = [0,1,2,6,6,6]

def p(n):
    # prob that P(sum_a < sum_b) > P(sum_a > sum_b) after n times

    # (value, prob) for each player
    pa = [(x,cnt/6) for (x,cnt) in Counter(a).items()]
    pb = [(x,cnt/6) for (x,cnt) in Counter(b).items()]

    # O(n^2)
    # (sum_of_value, prob) after n times 
    sa = [(0,1)]
    for _ in range(n):
        d = defaultdict(float)
        for (x1,p1) in sa:
            for (x2,p2) in pa:
                d[x1+x2] += p1*p2
        sa = [*d.items()]

    sb = [(0,1)]
    for _ in range(n):
        d = defaultdict(float)
        for (x1,p1) in sb:
            for (x2,p2) in pb:
                d[x1+x2] += p1*p2
        sb = [*d.items()]

    sa.sort()
    sb.sort()

    # O(6*n)
    win_a,i,acc=0,0,0
    for j in range(len(sa)):
        while i < len(sb) and sb[i][0] < sa[j][0]:
            acc += sb[i][1]
            i += 1
        win_a += acc * sa[j][1] 

    win_b,i,acc=0,0,0
    for j in range(len(sb)):
        while i < len(sa) and sa[i][0] < sb[j][0]:
            acc += sa[i][1]
            i += 1
        win_b += acc * sb[j][1] 

    return (win_a, win_b)

for n in range(11):
    wa,wb = p(n)
    rel = "<" if wa < wb else "=" if wa == wb else ">"
    print(f"n={n}: P(a_win)={wa:.8f} {rel} P(b_win)={wb:.8f} ")