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Gambling to Optimize Expected Median Bankroll

Lunchtime Sports Science: Cracking a New Sport

This is the first and what will be a series of relatively short pieces on sports analytics. I'll be using a variety of sports for examples, including both team sports and single-player sports, and I'll also make my code available through my GitHub account.

Here are my recommended tools. If you're unfamiliar with some of these, don't worry. You'll pick them up as you go along, and they form a powerful suite that will keep you on the cutting edge even as a professional data scientist.
  1. Hardware - Ideally you want at least 4GB of RAM for larger data sets, but you'll be able to do high-level analysis with almost any modern computing hardware.
  2. Linux operating system - You can certainly do top-notch data analysis using any operating system, but Linux is an excellent (and free) working environment. There are a variety of ways to install and use Linux, but I'd recommend Ubuntu's Windows installer. This will allow you to easily install Ubuntu alongside Windows, and it also makes it easy to uninstall Ubuntu later (if you choose). Ubuntu is just one of the (many) Linux distributions available, but it's very popular and well supported.
  3. R programming language - R is a powerful statistical programming language, and it has thousands of available packages available. If you're using Ubuntu, installing R is simple - sudo apt-get install r-base. That's it!
  4. Python programming language -Python is a powerful and relatively easy to use programming language. One of the most common tasks for sports analytics is web scraping, and Python is an excellent choice thanks to libraries such as Mechanize, Beautiful Soup and lxml. It's also a great language for data cleansing. Installing Python under Ubuntu - sudo apt-get install python.
  5. PostgreSQL database server - There are many ways to store and analyze data sets, but a dedicated relational database server is necessary tool for high-level analytics. PostgreSQL is my personal recommendation, but there are other good choices (such as MySQL). PostgreSQL is free, fast, powerful and has a huge variety of procedural languages available (including R and Python). Installing PostgreSQL under Ubuntu - sudo apt-get install postgresql-9.2.
  6. GitHub account - Setting up a GitHub account is free and will allow you to automatically following any changes to various sports analytics GitHub projects (such as mine). Later, you can set up your own repositories if you'd like to share your own work with other people. Don't forget to install git under Ubuntu - sudo apt-get install git.
  7. We'll start with analyzing hockey. If you'd like to take a look at some of my hockey code and data that I've scraped, you can find them in my hockey GitHub repository. If you've set up Ubuntu and have installed git you can execute the command git clone to make a local copy of my repository.
Here's a basic outline for tackling a new sport.
  1. Understand how teams win - Build a model to project the likely outcome for a team when facing a particular opponent. Example - Krach for hockey (which is based on the Bradley-Terry model).
  2. Understand how teams score - Build a model to project how many goals/points/runs teams are likely to score or allow when facing a particular opponent. Exampe - Poisson distribution and hockey.
  3. Relate the two - Characterize the relationship between scoring and winning or losing. Example - Pythagorean win expectation.
  4. Understand how players score/prevent scoring - Determine which aspects of player performance impact team offense and defense and by how much.
  5. Understand player contribution to winning/losing - This is nearly automatic once you understand the relationship between team offense/defense and team winning/losing.
In my next article we'll build a basic power ranking model for hockey to predict likely game outcomes.


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Gambling to optimize your expected bankroll mean is extremely risky, as you wager your entire bankroll for any favorable gamble, making ruin almost inevitable. But what if, instead, we gambled not to maximize the expected bankroll mean, but the expected bankroll median?

Let the probability of winning a favorable bet be \(p\), and the net odds be \(b\). That is, if we wager \(1\) unit and win, we get back \(b\) units (in addition to our wager). Assume our betting strategy is to wager some fraction \(f\) of our bankroll, hence \(0 \leq f \leq 1\). By our assumption, our betting strategy is invariant with respect to the actual size of our bankroll, and so if we were to repeat this gamble \(n\) times with the same \(p\) and \(b\), the strategy wouldn't change. It follows we may assume an initial bankroll of size \(1\).

Let \( q = 1-p \). Now, after \(n\)  such gambles our bankroll would have a binomial distribution with probability mass function \[ \Pr(k,n,p) = \binom{n}{k} p^k q^{n-k…