The (un)expected variance at poker
Enter your winrate and look at your risk of ruin.
Table of Contents
import time
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from pathlib import Path
from functools import lru_cache
from scipy import stats
%pylab inline
plt.style.use('fivethirtyeight')
@lru_cache(maxsize=512)
def calculate_variance(totalhands: int, winrate: float = 0.0, dev: float = 100.0, samples: int = 1000):
"""
Another Poker Variance Calculator like Primedope's one
Simulate randomly the amount of samples specified with winrate, totalhands and dev as parameters
Display 20 samples, worst & best samples, 95% and 68% confidence intervals, expected winnings,
as well as risk of ruin up to 20%
Print useful numbers about (un)expected variance
Inputs:
totalhands (int): number of hands e.g. desired length of the samples
winrate (float): observed (or true) winrate we want to calculate the variance
dev (float): standard deviation ("Std Dev bb per 100 hands" stat in HM2)
sample (int): number of samples we want to simulate
"""
#limit number of samples
if samples > 1_000_000:
samples = 1_000_000
start = time.time()
h = totalhands//100
EV = winrate*h
#standard error of the mean
std_dev = dev/(h**0.5)
#probability of loss with such a winrate and standard error
prob_loss = stats.norm.cdf(0,loc=winrate,scale=std_dev)
#x-axis
x = np.arange(0, totalhands + 100, 100)
#expected winnings
y = [i * winrate for i in range(h+1)]
#put ev values in a dataframe
df = pd.DataFrame(y, columns=["ev"], index=x)
#confidence intervals
df["conf1"]= [i * (winrate + std_dev) for i in range(h+1)]
df["conf2"]= [i * (winrate - std_dev) for i in range(h+1)]
df["conf3"]= [i * (winrate + 2*std_dev) for i in range(h+1)]
df["conf4"]= [i * (winrate - 2*std_dev) for i in range(h+1)]
#building ramdom samples with loc = winrate and scale = dev
arr = np.zeros(shape=(samples,h+1))
for i in range(samples):
arr[i,:]=np.cumsum(np.random.normal(winrate, dev, h+1))
#best and worst EV
bottom_ten = np.argsort(arr[:,-1])[:10]
top_ten = np.argsort(arr[:,-1])[-10:]
#bottom line for each sample
worst_downs = np.array([i for i in np.amin(arr, axis=1)])
#minimum bankroll required for each risk of ruin percentile
#if the i-th percentile returns a positive value, minimum bankroll required is set to zero
rr = range(0,101)
min_bkr = [int(-np.percentile(worst_downs,i)) if np.percentile(worst_downs,i) < 0 else 0 for i in rr]
rr_df = pd.DataFrame({"Risk of ruin": rr, "Minimum Bankroll": min_bkr})
rr_df = rr_df.set_index("Risk of ruin")
#best and worst samples
df["best"] = arr[top_ten[-1],:]
df["worst"] = arr[bottom_ten[-1],:]
#print computing duration
print(f'Duration: {time.time() - start}s')
#path to save image
IMAGE_DIR = Path.cwd().parent / "images"
plt.figure(figsize=(20,10))
#select randomly 20 samples to display excluding best and worst samples
idx_array = [i for i in range(samples) if i not in [top_ten[-1],bottom_ten[-1]]]
random_idx = np.random.choice(idx_array, 20, replace = False)
for i in random_idx:
plt.plot(x, arr[i,:],linewidth=1)
#display confidence intervals as well as best and worst samples
fmt=["b","g--","g--","g","g","c","y"]
labels=["EV","68% confidence interval","","95% confidence interval","","Best", "Worst"]
for idx, col in enumerate(df.columns):
df[col] = df[col].astype(int)
plt.plot(df[col],fmt[idx],linewidth=3,label=labels[idx])
plt.title("Samples over %d hands with confidence intervals" %totalhands)
plt.xlabel("total hands")
plt.ylabel("win/loss in big blinds")
plt.legend(bbox_to_anchor=(0.0, 1.0), loc='upper left')
plt.savefig(IMAGE_DIR / "variance.png")
plt.show()
#some numbers to print
ws = [winrate + std_dev, winrate - std_dev, winrate + (2*std_dev), winrate - (2*std_dev)]
bb = [df.conf1.iloc[-1],df.conf2.iloc[-1],df.conf3.iloc[-1],df.conf4.iloc[-1]]
std_dev_bb = int(std_dev*h)
print (f"Expected winnings: {winrate} bb/100 ({df.ev.iloc[-1]} bb)")
print (f"Standard deviation after {totalhands} hands: {std_dev: .2f} bb/100 ({std_dev_bb} bb)")
print("\n")
print (f"68% confidence interval: {ws[0]:.2f} bb/100 ({bb[0]} bb), {ws[1]:.2f} bb/100 ({bb[1]} bb)")
print (f"95% confidence interval: {ws[2]:.2f} bb/100 ({bb[2]} bb), {ws[3]:.2f} bb/100 ({bb[3]} bb)")
print("\n")
print (f"Top-10 over {samples} samples (bb/100):\n{np.around(arr[top_ten][:,-1]/h, 2)}")
print("\n")
print (f"Bottom-10 over {samples} samples (bb/100):\n{np.around(arr[bottom_ten][:,-1]/h, 2)}")
print("\n")
print (f"Probability of loss after {totalhands} hands: {prob_loss:.2%}")
print("\n")
#display the first 20th percentiles of risk of ruin dataframe
plt.figure(figsize=(20,10))
plt.plot(rr_df)
plt.title("Risk of ruin\n(q-th percentile over %s samples merged by their bottom line)" %samples)
plt.xlabel("risk of ruin %")
plt.xlim(0,20)
plt.xticks(range(0,22,2))
plt.ylabel("minimum bankroll in big blinds")
plt.savefig(IMAGE_DIR / "risk_of_ruin.png")
plt.show()
# return
calculate_variance(totalhands = 80000, winrate = 5.5, dev = 100, samples=1000)
