jeopardy.py

#!/usr/bin/env python3
# Code to help answer the Riddler Express from 11/13/2020:
# https://fivethirtyeight.com/features/can-you-snatch-defeat-from-the-jaws-of-victory/

from fractions import Fraction
import random

TOTAL_ON_BOARD = (2 + 4 + 6 + 8 + 10) * 100 * 6  # Total amount a Jeopardy! board is worth


def get_exp_value(call_order):
    """
    Takes a list of 30 amounts corresponding to the values of answers chosen and
    returns the expected value (as a fraction) for the return if you answer every
    question correctly and bet the maximum amount on a daily double.
    """
    left_on_board = TOTAL_ON_BOARD
    money_on_hand = 0
    exp_value = Fraction(0, 1)
    for amount in call_order:
        # 1/30th of the time, we hit the daily double here
        bet_amount = max(money_on_hand, 1000)
        total_won = money_on_hand + bet_amount + (left_on_board - amount)
        exp_value += Fraction(total_won, 30)
        money_on_hand += amount
        left_on_board -= amount
    return exp_value


# Answer the original question: what happens if you go in order across the columns,
# from the lowest to the largest values:
amounts = []
for amount in range(200, 1001, 200):
    amounts.extend([amount] * 6)
exp_value = get_exp_value(amounts)
print("Across, top to bottom:", exp_value, float(exp_value))

# Not asked, but interesting: what happens if you go from top to bottom,
# and what happens if you go category by category, from top to bottom and bottom
# to top.
amounts = []
for amount in range(1000, 100, -200):
    amounts.extend([amount] * 6)
exp_value = get_exp_value(amounts)
print("Across, bottom to top:", exp_value, float(exp_value))

amounts = list(range(200, 1001, 200)) * 6
exp_value = get_exp_value(amounts)
print("Category, top to bottom:", exp_value, float(exp_value))

amounts = list(range(1000, 100, -200)) * 6
exp_value = get_exp_value(amounts)
print("Category, bottom to top:", exp_value, float(exp_value))

# Extra credit, what happens with a random order.
# Try 100000 random games
N_GAMES = 100000
exp_value = Fraction(0, 1)
amounts = list(range(200, 1001, 200)) * 6
for _ in range(N_GAMES):
    random.shuffle(amounts)
    exp_value += get_exp_value(amounts) / N_GAMES
print("Random Walk:", exp_value, float(exp_value))