emily account problem

emily_problem.py

memoize_case_max_payoff = {}

def find_max_entropy_case(n: int):
    if n % 2 == 0:
        return n//2
    else:
        return n//2 + 1
    
def solve_case(n: int, majority: int):
    # we already saw this case!
    if (n, majority) in memoize_case_max_payoff:
        return memoize_case_max_payoff[(n, majority)]
    
    elif majority > n:
        raise "majority cannot be greater than n, you goofed somewhere"

    # base case: everything is one color, we win. nice! (contains the n=1 case, optimize out sidewalls of tree)
    elif majority == n:
        #print("all one color case")
        memoize_case_max_payoff[(n, majority)] = 2**n
    
     # flipped majority and minority case (optimize out half the tree)
    elif majority < n / 2:
        memoize_case_max_payoff[(n, majority)] = solve_case(n, n - majority)
    
    # actual strategy piece
    # always bet on the majority color
    # outcome A and outcome B are minimal outcomes for either outcome of first bet
    # A is where you guessed right, B is where you guessed wrong
    # after A, you get into case of (n-1, majority - 1)
    # after B, you get into case of (n-1, majority)
    # so the best outcome is to guess x where (1 + x) * A = (1 - x) * B
    # then, solving: always guess the majority color with bet x = (b - a)/(a + b)
    # your minimum payout will be 2ab/(a + b)
    else:
        # print("actual strategy piece for n", n, "majority", majority, "find_max_entropy_case(n-1)", find_max_entropy_case(n-1))
        # you got it right!
        a = solve_case(n-1, majority-1)
        #print("a", a)
        # you got it wrong!
        b = solve_case(n-1, majority)
        #print("b", b)
        max_payoff = 2*a*b/(a+b)
        #print("max_payoff", max_payoff)
        memoize_case_max_payoff[(n, majority)] = max_payoff

    return memoize_case_max_payoff[(n, majority)]

def solve_for_n(n: int):
    max_payoff = solve_case(n, find_max_entropy_case(n))
    return max_payoff

print(solve_for_n(52))

# output is 9.08132954942779