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I am trying to identify the optimal bet of a parlor game in order to see if I can validate the Kelly Criterion's theoretical optimal bet size. Here is the game. You flip a coin N Times. If heads, you win 2x your bet. If tails, you lose your bet. According to the Kelly Criterion, the optimal bet size is .25 of your bank roll. My code is showing erratic values.

See slide 5 below: http://people.math.gatech.edu/~shenk/OptionsClub/kellyOptionTalk1.pdf

In order to solve the problem, I ran N simulations of N coin tosses each. It seems like the Central Limit Theorem is not helping me here. I ran 1000 simulations and 1000 coin tosses, and I am getting optimal bet sizes near 1 in some cases.

Is there a hidden issue with my logic?

import random

import matplotlib.pyplot as plt

import pandas as pd



#this program simulates a parlor coin game. You flip a coin N times. You 

#always pick Heads. If you bet heads and win your return is 2x your bet. If 

#you bet heads and lose, you lost your bet (not double your bet)



coin_value = ['H','T']

simulations = 10 #number of simulations of the N coin toss game

coin_tosses = 10 #number of times you flip a coin per simulation



game_result = 

for x in range(0,100):



    bet_size = (x/100) #set the bet percentage



    for sim in range(0, simulations): #we run N simulations, which play N 

    #coin tosses per simulation



    bank_roll = 100 #starting bank roll for each simulation



    for coin_toss in range(0, coin_tosses): #we toss the coin N times



        outcome = random.choice(coin_value)



        if outcome == 'H':

            profit_loss = 2 * (bet_size * bank_roll) #Double your bet and 

            #add that to bankroll

            result = 'W'

        elif outcome == 'T':

            profit_loss =  -(bet_size * bank_roll) #Loss your bet and add 

            #that to bankroll

            result = 'L'



        bank_roll = bank_roll + profit_loss 



        #below I create a dataframe which has columns below

        game_result.append([bet_size, result, bank_roll, sim, coin_toss])

     #this is your ending bankroll for each simulation



df = pd.DataFrame(game_result, columns = ['Bet_Size','result','Bankroll', 

'sim', 'coin_toss'])

df = df[df['coin_toss'] == (coin_tosses - 1)] #this selects the last  coin 

#toss for each simulation. Effectively, this is your ending balance after N 

#coin tosses.



x = df.groupby(['Bet_Size'])['Bankroll'].mean() #calculated E[V] by bet 

size by averaging the simulations based on bet size



x.plot(x= 'Bet Size', y = 'Expected Value')