Kelly

The Kelly Criterion

The Kelly Criterion is a formula used for bankroll management in gambling and investing. It helps determine the optimal bet size to maximize long-term growth, while minimizing the risk of losing the entire bankroll.

In [367]:

def kelly_criterion(win_probability, odds):
    return (win_probability * (odds + 1) - 1) / odds

¶

In [368]:

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

# Example win probability and odds
win_probability = 0.6 
odds = 2.0  

# Calculate the recommended bet size
win_probabilities = np.linspace(0.01, 0.99, 100)
bet_sizes = [(kelly_criterion(p, odds) * 100) for p in win_probabilities]

# Plot the recommended bet size as a function of win probability
plt.figure(figsize=(10, 6))
plt.plot(win_probabilities, bet_sizes)
plt.xlabel('Win Probability')
plt.ylabel('Recommended Bet Size (%)')
plt.title('Kelly Criterion Recommended Bet Size vs. Win Probability')
plt.grid(True)
# Highlight the point for the example win probability
recommended_bet_size = kelly_criterion(win_probability, odds)
plt.scatter(win_probability, recommended_bet_size * 100, color='red', label='Example (60%)')
plt.legend()

# Display the graph
plt.show()

# show result
df_res = pd.DataFrame({
    'Win Probability':win_probabilities,
    'bet_sizes':bet_sizes
})
print(df_res)

    Win Probability  bet_sizes
0          0.010000 -48.500000
1          0.019899 -47.015152
2          0.029798 -45.530303
3          0.039697 -44.045455
4          0.049596 -42.560606
..              ...        ...
95         0.950404  92.560606
96         0.960303  94.045455
97         0.970202  95.530303
98         0.980101  97.015152
99         0.990000  98.500000

[100 rows x 2 columns]

¶

Create function to test different batchs of parameters with Kelly criterion¶

In [369]:

def run_simulations(num_simulations, num_rounds, initial_bankroll, win_probability=None, odds=None):
    # Arrays to store results
    bankroll_progressions = []
    final_bankrolls = []

    for _ in range(num_simulations):
        
        if win_probability == None:
            # Generate random win probability (between 0.01 and 0.99)
            win_probability = np.random.uniform(0.01, 0.99)
            fixed_probability = False
        else:
            fixed_probability = True

        if odds == None:
            # Generate random odds (between 1.0 and 3.0)
            odds = np.random.uniform(1.0, 3.0)

        # Initialize bankroll and progression for this simulation
        bankroll = initial_bankroll
        bankroll_progression = [bankroll]

        # Simulate betting and update bankroll progression
        for _ in range(num_rounds):
            if bankroll > 0:
                # Calculate recommended bet size
                recommended_bet_size = kelly_criterion(win_probability, odds)
                if recommended_bet_size <= 0:
                    # do nothing
                    bankroll_progression.append(bankroll)
                else:   
                    outcome = np.random.choice([-1, 1], p=[1 - win_probability, win_probability])
                    bankroll *= (1 + recommended_bet_size * outcome)
                    bankroll_progression.append(bankroll)
            else:
                bankroll_progressions.append(0)

        # Store bankroll progression for this simulation
        bankroll_progressions.append(bankroll_progression)
        
        # Store the final bankroll for this simulation
        final_bankrolls.append(bankroll)
        
    return bankroll_progressions, final_bankrolls, fixed_probability

¶

Run simulation with parameters¶

In [370]:

import random

# Number of simulations
num_simulations = 25
# Number of rounds or iterations in each simulation
num_rounds = 100
# Initial bankroll 
initial_bankroll = 100
# win_probability
win_probability = 0.6
# odds
odds = 2

bankroll_progressions, final_bankrolls, fixed_probability = run_simulations(num_simulations, num_rounds, initial_bankroll, win_probability=win_probability, odds=odds)

if not fixed_probability:
    # Plot the recommended bet sizes (optional, for variable win probabilities)
    plt.figure(figsize=(10, 6))
    plt.hist(bet_sizes, bins=20, alpha=0.7, color='blue')
    plt.xlabel('Recommended Bet Size')
    plt.ylabel('Frequency')
    plt.title('Distribution of Recommended Bet Sizes')
    plt.grid(True)


# Plot the min, max, average bankroll progression the batch of simulation
plt.figure(figsize=(10, 6))
mean_bankrolls = np.mean(bankroll_progressions, axis=0)
plt.plot(range(num_rounds + 1), mean_bankrolls, color='green', linewidth=2)
plt.xlabel('Number of Rounds {0}'.format(num_rounds))
plt.ylabel('Average Bankroll')
plt.title('Average Bankroll Progression for {0} Simulations with a win rate of {1}'.format(num_simulations, win_probability))
plt.grid(True)

# Plot the bankroll progression for a random simulation
plt.figure(figsize=(10, 6))
progression_selected = random.randint(0, len(bankroll_progressions)-1)
rand_progression = bankroll_progressions[progression_selected]
plt.plot(range(num_rounds + 1), rand_progression)
plt.xlabel('Simulation progression n°{0}'.format(progression_selected))
plt.ylabel('Bankroll')
plt.title('Bankroll Progression for simulation n°{0} with a win rate of {1}'.format(progression_selected, win_probability))
plt.grid(True)

plt.show()

¶

And check details...¶

In [371]:

# return 
def evaluate(data, initial_bankroll):
    df = pd.DataFrame(data)
    #(initial_bankroll - df['Final Bankroll'])*-1 
    df['Profit'] = df['Final Bankroll'].apply(lambda x: (initial_bankroll - x)*-1 if x > 0 else 0)
    df['Win/Loss'] = df['Profit'].apply(lambda x: 'win' if x > 0 else ('loss' if x < 0 else '#N/A'))
    # nb profit vs nb loss
    profitRate = (df['Profit'] > 0).sum() / len(df)
    bankrupts = (df['Final Bankroll'] == 0).sum()
    return df, profitRate, bankrupts

In [372]:

# Plot the final bankrolls
plt.figure(figsize=(10, 6))
plt.hist(final_bankrolls, bins=20, alpha=0.7, color='green')
plt.xlabel('Final Bankroll')
plt.ylabel('Frequency')
plt.title('Distribution of Final Bankrolls')
plt.grid(True)

plt.show()

# Create a DataFrame to store the final bankrolls
data = {'Final Bankroll': final_bankrolls}
df, profitRate,bankrupts = evaluate(data, initial_bankroll)
print(df.head(25))

    Final Bankroll         Profit Win/Loss
0         1.132005     -98.867995     loss
1         0.485145     -99.514855     loss
2       994.632193     894.632193      win
3    374541.554786  374441.554786      win
4         1.132005     -98.867995     loss
5      5415.219719    5315.219719      win
6       994.632193     894.632193      win
7         2.641344     -97.358656     loss
8     12635.512677   12535.512677      win
9         0.485145     -99.514855     loss
10       33.554855     -66.445145     loss
11       14.380652     -85.619348     loss
12      426.270940     326.270940      win
13    29482.862913   29382.862913      win
14       78.294662     -21.705338     loss
15       78.294662     -21.705338     loss
16       14.380652     -85.619348     loss
17       14.380652     -85.619348     loss
18   374541.554786  374441.554786      win
19      994.632193     894.632193      win
20        2.641344     -97.358656     loss
21        0.207919     -99.792081     loss
22     2320.808451    2220.808451      win
23       14.380652     -85.619348     loss
24       78.294662     -21.705338     loss

In [373]:

print("\n******************************************************************\n")
# Profits ?
print("Estimated profit rate : {0}\n".format(profitRate))
# Bankrupts ?
if bankrupts == 0:
    print("No bankrupts \n")
elif bankrupts/len(df) <= 0.05:
    print("Take care, bankrupt can occurs ({0} times with this simulation\n)".format(bankrupts))
else: 
    #Bankrupts/nbRecord > 0.05:
    print("Forget this, try something else...\n")
print("******************************************************************\n")

******************************************************************

Estimated profit rate : 0.4

No bankrupts 

******************************************************************

¶

Profit rate vs Win probability (without fees)¶

In [374]:

# List of win probabilities to test
win_probabilities_to_test = np.linspace(0.01, 0.99, 10)

all_results = []

for win_probability in win_probabilities_to_test:
    bankroll_progressions, final_bankrolls, _ = run_simulations(num_simulations, num_rounds, initial_bankroll, win_probability=win_probability, odds=odds)
    df, profitRate, bankrupts = evaluate({'Final Bankroll':final_bankrolls}, initial_bankroll)
    all_results.append([win_probability, final_bankrolls, bankroll_progressions, [df], profitRate, bankrupts])

# Plot the results
label = []
plt.figure(figsize=(12, 6))
for result in all_results:
    label.append("Win Probability: {:.2f}".format(result[0]))
    for progression in result[2]:
        plt.plot(range(num_rounds + 1), progression, label=label)

plt.xlabel('Number of Rounds')
plt.ylabel('Bankroll')
plt.title('Bankroll Progression regarding Win Probabilities')
plt.legend(label)
plt.grid(True)

In [375]:

df_result = pd.DataFrame(
    {'Win probability':[proba[0] for proba in all_results],
    'Profit rate':[profit[4] for profit in all_results],
    'Nb Bankrupt':[bankrupt[5] for bankrupt in all_results]}
    )
print(df_result)

   Win probability  Profit rate  Nb Bankrupt
0         0.010000         0.00            0
1         0.118889         0.00            0
2         0.227778         0.00            0
3         0.336667         0.00            0
4         0.445556         0.00            0
5         0.554444         0.28            0
6         0.663333         0.68            0
7         0.772222         1.00            0
8         0.881111         1.00            0
9         0.990000         1.00            0

In [376]:

# Plot the distribution of final bankrolls
plt.figure(figsize=(12, 6))
for final_bankrolls in all_results:
    result = final_bankrolls[1]
    plt.hist(result, bins=20, alpha=0.7)
    plt.xlabel('Final Bankroll')
    plt.ylabel('Frequency')
    plt.title('Distribution of Final Bankrolls\n Win Probabilities {:.2f}'.format(final_bankrolls[0]))
    plt.grid(True)
    plt.show()
    print(final_bankrolls[3])

[    Final Bankroll  Profit Win/Loss
0              100       0     #N/A
1              100       0     #N/A
2              100       0     #N/A
3              100       0     #N/A
4              100       0     #N/A
5              100       0     #N/A
6              100       0     #N/A
7              100       0     #N/A
8              100       0     #N/A
9              100       0     #N/A
10             100       0     #N/A
11             100       0     #N/A
12             100       0     #N/A
13             100       0     #N/A
14             100       0     #N/A
15             100       0     #N/A
16             100       0     #N/A
17             100       0     #N/A
18             100       0     #N/A
19             100       0     #N/A
20             100       0     #N/A
21             100       0     #N/A
22             100       0     #N/A
23             100       0     #N/A
24             100       0     #N/A]

[    Final Bankroll  Profit Win/Loss
0              100       0     #N/A
1              100       0     #N/A
2              100       0     #N/A
3              100       0     #N/A
4              100       0     #N/A
5              100       0     #N/A
6              100       0     #N/A
7              100       0     #N/A
8              100       0     #N/A
9              100       0     #N/A
10             100       0     #N/A
11             100       0     #N/A
12             100       0     #N/A
13             100       0     #N/A
14             100       0     #N/A
15             100       0     #N/A
16             100       0     #N/A
17             100       0     #N/A
18             100       0     #N/A
19             100       0     #N/A
20             100       0     #N/A
21             100       0     #N/A
22             100       0     #N/A
23             100       0     #N/A
24             100       0     #N/A]

[    Final Bankroll  Profit Win/Loss
0              100       0     #N/A
1              100       0     #N/A
2              100       0     #N/A
3              100       0     #N/A
4              100       0     #N/A
5              100       0     #N/A
6              100       0     #N/A
7              100       0     #N/A
8              100       0     #N/A
9              100       0     #N/A
10             100       0     #N/A
11             100       0     #N/A
12             100       0     #N/A
13             100       0     #N/A
14             100       0     #N/A
15             100       0     #N/A
16             100       0     #N/A
17             100       0     #N/A
18             100       0     #N/A
19             100       0     #N/A
20             100       0     #N/A
21             100       0     #N/A
22             100       0     #N/A
23             100       0     #N/A
24             100       0     #N/A]

[    Final Bankroll     Profit Win/Loss
0        88.581158 -11.418842     loss
1        83.422551 -16.577449     loss
2        81.770660 -18.229340     loss
3        85.963168 -14.036832     loss
4        86.827119 -13.172881     loss
5        86.827119 -13.172881     loss
6        77.782626 -22.217374     loss
7        95.004072  -4.995928     loss
8        88.581158 -11.418842     loss
9        85.963168 -14.036832     loss
10       85.107813 -14.892187     loss
11       80.957022 -19.042978     loss
12       85.107813 -14.892187     loss
13       82.592476 -17.407524     loss
14       85.107813 -14.892187     loss
15       87.699753 -12.300247     loss
16       81.770660 -18.229340     loss
17       87.699753 -12.300247     loss
18       88.581158 -11.418842     loss
19       85.963168 -14.036832     loss
20       88.581158 -11.418842     loss
21       81.770660 -18.229340     loss
22       85.107813 -14.892187     loss
23       86.827119 -13.172881     loss
24       85.963168 -14.036832     loss]

[    Final Bankroll     Profit Win/Loss
0        33.374949 -66.625051     loss
1         2.200295 -97.799705     loss
2         6.100048 -93.899952     loss
3         3.090995 -96.909005     loss
4         1.566258 -98.433742     loss
5         3.090995 -96.909005     loss
6        23.757631 -76.242369     loss
7         4.342260 -95.657740     loss
8         0.564951 -99.435049     loss
9         1.114926 -98.885074     loss
10        2.200295 -97.799705     loss
11        6.100048 -93.899952     loss
12       65.865131 -34.134869     loss
13       23.757631 -76.242369     loss
14       12.038384 -87.961616     loss
15        4.342260 -95.657740     loss
16        6.100048 -93.899952     loss
17        3.090995 -96.909005     loss
18       23.757631 -76.242369     loss
19        1.114926 -98.885074     loss
20        0.793649 -99.206351     loss
21        2.200295 -97.799705     loss
22       92.527969  -7.472031     loss
23        4.342260 -95.657740     loss
24       23.757631 -76.242369     loss]

[    Final Bankroll       Profit Win/Loss
0        36.749328   -63.250672     loss
1       145.899639    45.899639      win
2         0.294736   -99.705264     loss
3        73.223724   -26.776276     loss
4        18.443655   -81.556345     loss
5        36.749328   -63.250672     loss
6         0.147921   -99.852079     loss
7        73.223724   -26.776276     loss
8         0.147921   -99.852079     loss
9        18.443655   -81.556345     loss
10       18.443655   -81.556345     loss
11      290.707760   190.707760      win
12      290.707760   190.707760      win
13       18.443655   -81.556345     loss
14       36.749328   -63.250672     loss
15     2299.661134  2199.661134      win
16        4.645604   -95.354396     loss
17     2299.661134  2199.661134      win
18        0.004710   -99.995290     loss
19       73.223724   -26.776276     loss
20      145.899639    45.899639      win
21       18.443655   -81.556345     loss
22        0.587266   -99.412734     loss
23      579.240650   479.240650      win
24        0.294736   -99.705264     loss]

[    Final Bankroll         Profit Win/Loss
0        35.712998     -64.287002     loss
1       312.986743     212.986743      win
2    623698.420672  623598.420672      win
3         1.376502     -98.623498     loss
4       312.986743     212.986743      win
5     71166.402940   71066.402940      win
6        35.712998     -64.287002     loss
7      2742.998510    2642.998510      win
8         4.074991     -95.925009     loss
9    623698.420672  623598.420672      win
10     8120.361924    8020.361924      win
11      105.724619       5.724619      win
12    71166.402940   71066.402940      win
13      926.564714     826.564714      win
14       12.063588     -87.936412     loss
15     2742.998510    2642.998510      win
16        0.053055     -99.946945     loss
17    24039.487281   23939.487281      win
18     8120.361924    8020.361924      win
19    24039.487281   23939.487281      win
20    71166.402940   71066.402940      win
21        4.074991     -95.925009     loss
22       12.063588     -87.936412     loss
23    71166.402940   71066.402940      win
24      312.986743     212.986743      win]

[    Final Bankroll        Profit Win/Loss
0     2.775327e+05  2.774327e+05      win
1     9.774072e+12  9.774072e+12      win
2     8.548066e+10  8.548066e+10      win
3     4.148939e+11  4.148939e+11      win
4     5.718010e+04  5.708010e+04      win
5     7.475844e+08  7.475843e+08      win
6     7.475844e+08  7.475843e+08      win
7     5.718010e+04  5.708010e+04      win
8     3.628519e+09  3.628519e+09      win
9     1.347049e+06  1.346949e+06      win
10    3.173378e+07  3.173368e+07      win
11    2.775327e+05  2.774327e+05      win
12    7.475844e+08  7.475843e+08      win
13    9.774072e+12  9.774072e+12      win
14    3.173378e+07  3.173368e+07      win
15    1.178082e+04  1.168082e+04      win
16    3.628519e+09  3.628519e+09      win
17    1.761159e+10  1.761159e+10      win
18    3.628519e+09  3.628519e+09      win
19    1.540249e+08  1.540248e+08      win
20    2.775327e+05  2.774327e+05      win
21    2.775327e+05  2.774327e+05      win
22    3.173378e+07  3.173368e+07      win
23    2.013753e+12  2.013753e+12      win
24    2.632823e+16  2.632823e+16      win]

[    Final Bankroll        Profit Win/Loss
0     7.760195e+10  7.760195e+10      win
1     9.005904e+17  9.005904e+17      win
2     8.449250e+14  8.449250e+14      win
3     8.816393e+16  8.816393e+16      win
4     7.596897e+09  7.596897e+09      win
5     8.630869e+15  8.630869e+15      win
6     8.271452e+13  8.271452e+13      win
7     8.630869e+15  8.630869e+15      win
8     8.816393e+16  8.816393e+16      win
9     9.005904e+17  9.005904e+17      win
10    8.630869e+15  8.630869e+15      win
11    8.816393e+16  8.816393e+16      win
12    8.271452e+13  8.271452e+13      win
13    9.397235e+19  9.397235e+19      win
14    8.449250e+14  8.449250e+14      win
15    9.199489e+18  9.199489e+18      win
16    9.199489e+18  9.199489e+18      win
17    9.005904e+17  9.005904e+17      win
18    7.927003e+11  7.927003e+11      win
19    9.805570e+21  9.805570e+21      win
20    8.816393e+16  8.816393e+16      win
21    9.397235e+19  9.397235e+19      win
22    9.199489e+18  9.199489e+18      win
23    8.449250e+14  8.449250e+14      win
24    9.005904e+17  9.005904e+17      win]

[    Final Bankroll        Profit Win/Loss
0     5.971056e+31  5.971056e+31      win
1     2.576578e+25  2.576578e+25      win
2     4.512133e+29  4.512133e+29      win
3     3.409672e+27  3.409672e+27      win
4     3.409672e+27  3.409672e+27      win
5     5.971056e+31  5.971056e+31      win
6     4.512133e+29  4.512133e+29      win
7     4.512133e+29  4.512133e+29      win
8     5.971056e+31  5.971056e+31      win
9     1.947037e+23  1.947037e+23      win
10    3.409672e+27  3.409672e+27      win
11    3.409672e+27  3.409672e+27      win
12    4.512133e+29  4.512133e+29      win
13    3.409672e+27  3.409672e+27      win
14    2.576578e+25  2.576578e+25      win
15    3.409672e+27  3.409672e+27      win
16    4.512133e+29  4.512133e+29      win
17    4.512133e+29  4.512133e+29      win
18    5.971056e+31  5.971056e+31      win
19    5.971056e+31  5.971056e+31      win
20    2.576578e+25  2.576578e+25      win
21    4.512133e+29  4.512133e+29      win
22    4.512133e+29  4.512133e+29      win
23    3.409672e+27  3.409672e+27      win
24    4.512133e+29  4.512133e+29      win]