Stake sizing, Part 1

In the last post we used Python code to take a look at a classic gambling situation, the coin flip, to make a point about the importance of choosing the highest odds available to bet at. Today, we’ll again use the coin flipping example to investigate another fundamental principal of successful gambling: stake sizing.

Now, imagine we’re one of the lucky punters from the last post who were allowed to bet on a fair coin flip at odds of 2.03. As I stated then, this is pretty much like a license to print money – but how much of your bankroll should you bet on each flip of the coin? Knowing that the coin was indeed fair and you would be getting the best of it, a natural instinct could be to bet as much as you could possibly cough up, steal and borrow in order to maximize your profit. This is a poor strategy though, as we’ll soon come to see.

The reason for this is that even if we do have come across a profitable proposition, our edge when betting at a (I’ll empasize it again: fair) coin flip at 2.03 odds is only 1.5% – meaning that for each 1 unit bet we are expected to net 0.015 units on average. This conclusion should be absolute basics for anyone interested in serious gambling, but to make sure we’re all on the same page I’ll throw some maths at you:

The Expected Value, or EV, of any bet is, simply put, the sum of all outcomes multiplied by their respective probabilities – indicating the punter’s average profit or loss on each bet. So with our coin flip, we’ll win a net of 1.03 units 50% of the time and lose 1 unit 50% of the time; our EV is therefore 1.03 * 0.5 + (-1 * 0.5) = 0.015, for a positive edge of 1.5% and an average profit of 0.015 units per bet. For these simple types of bets though, an easier way to calculate EV is to divide the given odds by the true odds and subtract 1: 2.03 / 2.0 – 1 = 0.015.

An edge of only 1.5% is nothing to scoff at though, empires has been built on less, so we’ll definitely want to bet something – but how much?

Stake sizing is much down to personal preferences about risk aversion and tolerance of the variance innately involved in gambling, but with some Python code we can at least have a look at some different strategies before we set out to chase riches and glory flipping coins. Just like in the last post I’ll just give you the code with some comments in it, which will hopefully guide you along what’s happening  before I briefly explain it.

Here we go:

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

def coin_flips(n=10000,odds=1.97,bankroll=100,stake=1,bankrupt=False):
    '''
    Simulates 10000 coinflips for a single punter, betting at 1.97 odds,
    also calculates net winnings

    NEW: default bankroll and stake set at 100 and 1, respectively
    now also calculates if player went bankrupt or not
    '''

    # create a pandas dataframe for storing coin flip results
    # and calculate net winnings
    df = pd.DataFrame()
    # insert n number of coinflips, 0=loss, 1=win
    df['result'] = np.random.randint(2,size=n)
    # calculate net winnings
    df['net'] = np.where(df['result']==1,stake*odds-stake,-stake)
    # calculate cumulative net winnings
    df['cum_net'] = df['net'].cumsum()

    # calculate total bankroll
    df['bankroll'] = df['cum_net'] + bankroll

    # if bankroll goes below the default stake, punter will stop betting
    # count times bankroll < stake
    df['bankrupt'] = np.where(df['bankroll']<stake,1,0)
    # count cumulative bankruptcies, with column shifted one step down
    df['bankruptcies'] = df['bankrupt'].cumsum().shift(1)
    # in case first flip is a loss, bankruptcies will be NaN, replace with 0
    df.fillna(0,inplace=True)
    # drop all flips after first bankruptcy
    if bankrupt:
        df = df[df['bankruptcies']==0]

    return df

First off, we’ll modify our original coin_flips function to take our punter’s bankroll and stake size into consideration, setting the bankrupt threshold at the point where a default sized bet can no longer be made. By default, our punter will have an endless stream of 100 unit bankrolls, but if we set the parameter bankrupt to True, the function will cut away any coin flips after his first bankruptcy.

def many_coin_flips(punters=100,n=10000,odds=1.97,bankroll=100,stake=1,color='r',plot=False,bankrupt=False):
    '''
    Simulates 10000 coinflips for 100 different punters,
    all betting at 1.97 odds,
    also calculates and plots net winnings for each punter

    NEW: now also saves punter bankruptcies
    '''

    # create pandas dataframe for storing punter results
    punter_df = pd.DataFrame()
    # loop through all punters
    for i in np.arange(punters):
        # simulate coin flips
        df = coin_flips(n,odds,bankroll,stake,bankrupt)
        # calculate net
        net = df['net'].sum()
        # check for bankruptcy
        bankruptcy = df['bankrupt'].sum()

        # append to our punter dataframe
        punter_df = punter_df.append({'odds':odds,
                                      'net':net,
                                      'bankrupt':bankruptcy},ignore_index=True)

        if plot:
            # plot the cumulative winnings over time
            df['cum_net'].plot(color=color,alpha=0.1)

    # check if punters ended up in profit
    punter_df['winning'] = np.where(punter_df['net']>0,1,0)

    return punter_df

We also want to modify the many_coin_flips function so that it’ll also take bankroll and stake size into consideration, counting up how many of our punters went bankrupt.

We won’t use the compare_odds function here, instead we’ll write a new one to compare stake sizing – but if we ever want to use it again sometime in the future a few minor changes will be needed here as well:

def compare_odds(punters=100,n=10000,odds=[1.97,2.00,2.03]):
    '''
    Simulates and compare coin flip net winnings
    after 10000 flips for 3 groups of punters,
    betting at odds of 1.97, 2.00 and 2.03, respectively.
    Also plots every punters net winnings
    '''

    # create figure and ax objects to plot on
    fig, ax = plt.subplots()

    # set y coordinates for annotating text for each group of punters
    ys = [0.25,0.5,0.75]
    # assign colors to each group of punters
    cs = ['r','y','g']

    # loop through the groups of punters, with their respective odds,
    # chosen color and y for annotating text
    for odd, color, y in zip(odds,cs,ys):
        # run coin flip simulation with given odds, plot with chosen color
        df = many_coin_flips(punters,n,odd,color=color,plot=True)
        # calculate how many punters in the group ended up in profit
        winning_punters = df['winning'].mean()
        # set a text to annotate
        win_text = '%.2f: %.0f%%' %(odd,winning_punters * 100)
        # annotate odds and chance of profit for each group of punters
        ax.annotate(win_text,xy=(1.02,y),
                    xycoords='axes fraction', color=color,va='center')

    # set title
    ax.set_title('Chances of ending up in profit after %s coin flips' %n)
    # set x and y axis labels
    ax.set_xlabel('Number of flips')
    ax.set_ylabel('Net profit')
    # add annotation 'legend'
    ax.annotate('odds: chance',xy=(1.02,1.0),
                xycoords=('axes fraction'),fontsize=10,va='center')
    # add horizontal line at breakeven point
    plt.axhline(color='k',alpha=0.5)
    # set y axis range at some nice number
    ax.set_ylim(-450,450)

    # show plot
    plt.show()

Now, with all our previous coin flip functions taking bankroll and stake size into consideration, we can go ahead and evaluate a few stake sizing strategies with a new function:

def compare_stakes(punters=200,n=10000,odds=2.03,stakes=[100,50,25,10,5,2,1,0.5],bankroll=100):
    '''
    Similar to compare_odds, but here we instead want to compare different
    staking sizes for our coin flips betting at 2.03 odds

    Increased number of punters in each group, from 100 to 200

    Also prints out the results
    '''

    # pandas df to store results
    results_df = pd.DataFrame(columns=['stake','win','lose','bankrupt'])

    # colors to use in plot later, green=1=win, yellow=4=lost, red=2=bankrupt
    colors = [sns.color_palette()[i] for i in (1,4,2)]

    # loop through the groups of punters, with their respective odds
    for stake in stakes:
        # run coin flip simulation with given stake
        df = many_coin_flips(punters,n,odds,stake=stake,bankrupt=True)
        # calculate how many punters in the group ended up in profit
        winning_punters = df['winning'].mean()
        # ...and how many went bankrupt
        bankrupt_punters = df['bankrupt'].mean()
        # lost money but not bankrupt
        lose = 1 - winning_punters - bankrupt_punters

        # append to dataframe
        results_df = results_df.append({'stake':stake,
                                        'win':winning_punters,
                                        'lose':lose,
                                        'bankrupt':bankrupt_punters},ignore_index=True)

    # set stake as index
    results_df.set_index('stake',inplace=True)

    # plot
    fig = plt.figure()
    # create ax object
    ax = results_df.plot(kind='bar',stacked=True,color=colors,alpha=0.8)
    # fix title, axis labels etc
    ax.set_title('Simulation results: betting %s coin flips at %s odds, starting bankroll %s' %(n,odds,bankroll))
    ax.set_ylabel('%')
    # set legend outside plot
    ax.legend(bbox_to_anchor=(1.2,0.5))

    # add percentage annotation for both win and bankrupt
    for x, w, l, b in zip(np.arange(len(results_df)),results_df['win'],results_df['lose'],results_df['bankrupt']):
        # calculate y coordinates
        win_y = w/2
        lost_y = w + l/2
        bankr_y = w + l + b/2

        # annotate win, lose and bankrupt %, only if >=2%
        if w >= 0.04:
            ax.annotate('%.0f%%' %(w * 100),xy=(x,win_y),va='center',ha='center')
        if l >= 0.04:
            ax.annotate('%.0f%%' %(l * 100),xy=(x,lost_y),va='center',ha='center')
        if b >= 0.04:
            ax.annotate('%.0f%%' %(b * 100),xy=(x,bankr_y),va='center',ha='center')

    plt.show()

By default, our new compare_stakes function creates a number of punter groups, all betting on fair coin flips at 2.03 odds with a starting bankroll of a 100 units. For each group and their different staking plan, the function takes note of how many ended up in profit, how many lost and how many went bankrupt.

As we can see on the plot below, the results differ substantially:

01

Just like last time, I want to remind you that any numbers here are only rough estimates, and increasing the size of each punter group as well as the number of coin flips will get us closer to the true values.

So what can we learn from the above plot? Well, the main lesson is that even if you have a theoretically profitable bet, your edge will account for nearly nothing if you are too bold with your staking. Putting your whole bankroll at risk will see you go bankrupt around 96% of the time, and even if you bet as small as 2 units, you’ll still face a considerable risk of screwing up a lucrative proposition. The truth is that with such a small edge, keeping your bet small as well is the way to go if you want to make it in the long run.

But what if some fool offered us even higher odds, let’s say 2.20? First off, we would have to check if the person was A: mentally stable, and B: rich enough to pay us if (or rather, when) we win, before we go ahead and bet. Here our edge would be 10% (2.2 / 2.0 – 1), nearly 10 times as large as in the 2.03 situation, so we’ll likely be able to bet more – but how much? Well, the functions are written with this in mind, enabling us to play around with different situations and strategies. Specifying the odds parameter of our new function as 2.20, here’s what betting at a fair coin flip at 2.20 odds would look like:

02

As can be seen from the new plot, with a larger edge we can go ahead and raise our stake size considerably, hopefully boosting our winnings as well. So the main take-away from this small exercise is that even if you have an edge, if you want to make it in the long run you’ll have to be careful with your staking to avoid blowing up your bankroll – but also that the larger your edge, the larger you can afford to bet.

That’s it for now, but I’ll hopefully be back soon with a Part 2 about stake sizing, looking at a staking plan that actually takes your (perceived) edge into account when calculating the optimal stake size: The Kelly Criterion.

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Stake sizing, Part 1

Flipping coins, and the importance of betting at the highest odds

As I stated in the previous post, this blog will now focus more on gambling, using Python code to investigate whatever comes to my mind around the subject.

Today I’ll have a look at a classic gambling example – the flip of a coin – but before I go ahead and talk you through the code I want to state a few things that I know some of you will be wondering. Though R seems to be the language preferred by most in the football analytics scene, I have chosen Python simply because I feel it is so much more intuitive and easier to learn. RStudio seems to be the tool of choice for the R folks, but I don’t know of any real dominant counterpart for Python. I use Spyder, available through downloading Anaconda, mainly because it’s easy to use and comes with a lot of useful stuff pre-installed. If you’re thinking about testing it out yourself, I would suggest switching the color scheme of the editor to Zenburn for that dark and cool programming look that really make your code look super important, and run your scripts in the included IPython console.

One final, very important thing: I am not in any way an expert programmer, statistician, mathematician or anything like that. I am simply a gambler looking to use these fields to get an edge. It’s totally OK to simply copy and paste any code I publish here to use yourself and play around with it however you may wish. If you notice any mistakes or if something doesn’t add up, please comment. I’m happy to learn new stuff.

Flipping coins, and the importance of betting at the highest odds

The inspiration for this post came the other day when I noticed that a few hours prior to kick-off in this year’s Super Bowl, the bookmaker Pinnacle offered 1.97 odds on the opening coin flip. A sucker bet, I thought to myself, knowing the true odds of a fair coin to be 2.00. The coin flip is a very popular Super Bowl prop bet though and as it was pointed out to me on Twitter, a few books actually offered the fair odds of 2.00. Choosing the highest odds available is crucial if you want to make money gambling in the long run, so I decided to write up a nice little Python script to visualise my point.

The layout of these blog posts will be that I simply throw a piece of code at you, before explaining it. The comments in the code itself should also help you out, and for those of you who already know Python much will be simple basics, while those who’s completely new to coding or Python will hopefully learn a few things.

Here we go:

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

def coin_flips(n=10000,odds=1.97):
    '''
    Simulates 10000 coinflips for a single punter,
    betting at 1.97 odds,
    also calculates net winnings
    '''

    # create a pandas dataframe for storing coin flip results
    # and calculate net winnings
    df = pd.DataFrame()
    # insert n number of coinflips, 0=loss, 1=win
    df['result'] = np.random.randint(2,size=n)
    # calculate net winnings
    df['net'] = np.where(df['result']==1,odds-1,-1)
    # calculate cumulative net winnings
    df['cum_net'] = df['net'].cumsum()

    return df

Allright, so after importing all the needed modules for this piece, we go ahead and define our first function, coin_flips, which will be used to simulate the coin flips and calculate the net winnings of a single punter. I’ve chosen 10,000 flips and Pinnacle’s odds of 1.97 as our default values here.

Creating a pandas dataframe, we can easily store the result of each coin flip. Now, as we assume that the coin is fair, there’s no need to even consider which side our punter would call each time, instead we can simply go ahead and use numpy to simulate a series of ones and zeros, representing either a win or a loss. Calculating the net result of each flip is also very straightforward as when he wins, our punter will pocket the net end of the offered odds, 0.97, while losing will see his pocket lightened by 1 unit. Calculating the cumulative net winnings is also very easy using pandas’ built-in cumsum function.

For coding reasons, the function is set to return the dataframe so calling it will simply make a lot of numbers pop up, but running the coin_flips()[‘cum_net’].plot() command in the IPython console will let you simulate a punter’s coin flips, and also plot his cumulative net winnings like this:

01

Every time you run the command another simulation will run with a new, different result. Doing this a couple of times, you’ll likely understand why I described this as a sucker bet. Sure, you can get lucky and win, even a couple of times in a row – but betting with the odds against you, you’ll find it very hard to make a profit long term.

But that single punter flipping coins 10,000 times actually doesn’t say that much, maybe he just got unlucky? To dig deeper we want to know just how likely you are to end up with a profit after 10,000 coin flips. So we write another function, using the previous one to simulate the results of many more punters betting on 10,000 coin flips. How many do you think will end up in profit?

def many_coin_flips(punters=100,n=10000,odds=1.97,color='r'):
    '''
    Simulates 10000 coinflips for 100 different punters,
    all betting at 1.97 odds,
    also calculates and plots net winnings for each punter
    '''

    # create pandas dataframe for storing punter results
    punter_df = pd.DataFrame()
    # loop through all punters
    for i in np.arange(punters):
        # simulate coin flips
        df = coin_flips(n,odds)
        # calculate net
        net = df['net'].sum()
        # append to our punter dataframe
        punter_df = punter_df.append({'odds':odds,
                                      'net':net},ignore_index=True)

        # plot the cumulative winnings over time
        df['cum_net'].plot(color=color,alpha=0.1)

    # check if punters ended up in profit
    punter_df['winning'] = np.where(punter_df['net']>0,1,0)

    return punter_df

The slightly more complicated many_coin_flips function uses the earlier coin_flips to loop through a group of punters, 100 by default, and save their results into a new pandas dataframe, punter_df, where we’ll assign a 1 to all punters who ended up in profit while all the losers get a 0. We also plot each punters cumulative net winnings with a nice red color to symbolise their (very) likely bankruptcy.

This function also returns a dataframe so running it will again make a lot of numbers pop up in the console, but it also plots out the financial fate of each punter, like this:

02.png

As we can see, there actually are a few of our 100 punters who got lucky enough to end up winning after 10,000 coin flips. But most of them ended up way below the break-even point, losing a lof of money. If this was a real group of punters we can only hope that even if they were stupid enough to set out betting on 10,000 coin flips at these odds, they’ll at least at some point realise their mistake and quit.

But how about if we change the offered odds? As I mentioned earlier, some books actually put up the fair odds of 2.00. How would 100 punters do after 10,000 coin flips betting at those odds? Well, we’ll have to write a new function for that. Also, just for fun (or to make a point) I’ve included an additional group of 100 punters lucky enough to be allowed to bet on the coin flips at odds of 2.03 – literally a license to print money.

def compare_odds(punters=100,n=10000,odds=[1.97,2.00,2.03]):
    '''
    Simulates and compare coin flip net winnings
    after 10000 flips for 3 groups of punters,
    betting at odds of 1.97, 2.00 and 2.03, respectively.
    Also plots every punters net winnings
    '''

    # create figure and ax objects to plot on
    fig, ax = plt.subplots()

    # set y coordinates for annotating text for each group of punters
    ys = [0.25,0.5,0.75]
    # assign colors to each group of punters
    cs = ['r','y','g']

    # loop through the groups of punters, with their respective odds,
    # chosen color and y for annotating text
    for odd, color, y in zip(odds,cs,ys):
        # run coin flip simulation with given odds, plot with chosen color
        df = many_coin_flips(punters,n,odd,color)
        # calculate how many punters in the group ended up in profit
        winning_punters = df['winning'].mean()
        # set a text to annotate
        win_text = '%.2f: %.0f%%' %(odd,winning_punters * 100)
        # annotate odds and chance of profit for each group of punters
        ax.annotate(win_text,xy=(1.02,y),
                    xycoords='axes fraction', color=color,va='center')

    # set title
    ax.set_title('Chances of ending up in profit after %s coin flips' %n)
    # set x and y axis labels
    ax.set_xlabel('Number of flips')
    ax.set_ylabel('Net profit')
    # add annotation 'legend'
    ax.annotate('odds: chance',xy=(1.02,1.0),
                xycoords=('axes fraction'),fontsize=10,va='center')
    # add horizontal line at breakeven point
    plt.axhline(color='k',alpha=0.5)
    # set y axis range at some nice number
    ax.set_ylim(-450,450)

    # show plot
    plt.show()

This last function makes use of the two previous ones to simulate the coin flips of our three groups of punters, plotting their total net winnings all on the same ax object, which we later make use of to add a title and some nice labels to the axes. We also add a horizontal line to be able to better compare the punters’ winnings with the break-even point, as well as some text annotation to explain the colors of the three groups.

Now, running the compare_odds() function in the IPython console will hopefully result in something like this:

03

Here we clearly see just how important betting at the highest odds really is. Have in mind though that the numbers to the right are only rough estimates. As you can see, the yellow group of punters who bet at the fair odds of 2.00 did not win exactly 50% of the time, but close enough. I actually had to re-run the function a few times to get this close. But it’s only natural since we only had 100 punters, a very small number in this context, in each of our groups. The more punters and coin flips we use in our simulations, the closer we’ll come to the real win percentages – but here speed is more important than super accuracy.

So as we clearly see in the above plot, betting on the coin flip at Pinnacle’s 1.97 odds really is a sucker bet, albeit an entertaining one if you were planning to watch the Super Bowl. But if you hope to make a profit from your betting, finding the highest available odds to bet on is crucial, as is shown by the green group of punters who were allowed to bet at odds of 2.03. It’s only a difference of 0.06, but it makes all the difference in the long run. The margins in betting are tiny, but they add up over time.

The lessons learned here can easily be transferred to sports betting in general and football betting in particular, were the Asian Handicaps and Over/Under markets focus on odds around even money. The coin flip example is special though as we knew the true odds of the bet beforehand, something you’ll never be able to know betting on football. But as shown in the last plot, by consistently betting at the highest available odds, you at least give yourself a much better chance of ending up in profit.

Flipping coins, and the importance of betting at the highest odds

With a new season approaching, the blog changes course: Gambling, probability and programming!

As you may have noticed, I haven’t written anything in months. There’s two reasons for this, one being of course that the Swedish football season I’ve primarily focused on ended in November, but it’s also because I’ve taken on a new job. Working full time for the first time in my life has simply left me with little time to do any writing. (Yes, I did use the word time three times in that short sentence.)

But now, having settled in at the new job I’m anxious to get back to writing again. There’s one thing though: as I now work with compiling odds on Swedish football I wouldn’t feel comfortable publishing football analytics about Allsvenskan, telling you which teams are underrated and who’ll win the league title. And knowing I set the odds, and potentially profit from your mistakes, why would you believe anything I said?

So this blog will take on a slightly new focus: gambling. I originally set up the blog intending to write about this topic as well as football analytics, using maths, statistics, probability and psychology to discuss interesting things related to gambling, but the football part soon took over completely.

As I’ve published my football work on the blog I’ve now and then gotten some questions about programming, so I’ve taken the decision to include Python code whenever applicable. Learning to code has made a huge difference for me both in my gambling and football analytics endeavours, and though the blog won’t turn into a Python tutorial per se, if any of you who are new to programming should learn a new thing or two through my writing, I’d be glad.

I’m still hoping to write the occasional football analytics piece though, and if I do it’ll likely be for StrataBet, using their data as I did when I had a look at headers in Allsvenskan and Norway’s Tippeligaen.

That’s it for now, but I already have a new post in the works, coming up very shortly!

With a new season approaching, the blog changes course: Gambling, probability and programming!