optimal_strategy.py

"""
https://brilliant.org/community-problem/loser-keeps-bieber-we-stood-no-chance/
?group=4cgSapeNTVXO&ref_id=115277
"""

import random


def play_defensive(them):
    if random.random() <= 0.04:
        them += 1
    return them


def play_offensive(us, them):
    r1 = random.random()
    r2 = random.random()
    if r1 <= 0.05:
        them += 1
    if r2 >= 0.95:
        us += 1
    return us, them


def simulate(us=1, them=0, defensive=20, minutes_left=20):
    """
    If leading with less than `defensive` minutes left,
    play_defensive, else play_offensive.
    """
    for minute in range(minutes_left, 0, -1):
        if defensive >= minute and (us > them):
            them = play_defensive(them)
        else:
            us, them = play_offensive(us, them)

    if us > them:
        return 1.0
    if them > us:
        return 0.0
    return 0.5


A = 0
B = 0
C = 0
D = 0
tries = 100000
for i in range(tries):
    A += simulate(defensive=10)
    B += simulate(defensive=20)
    C += simulate(defensive=15)
    D += simulate(defensive=5)

print
print A
print B
print C
print D


import IPython; IPython.embed()

output

$ python optimal_strategy.py

76397.5
72070.0
74839.5
76561.0