All posts by Jérémie Smaga

Martingale Strategy simulation on Roulette

Good afternoon everybody,

I know that most people who have a slight background in probability theory try to apply it to casinos games. I even reckon that most teachers now choose this approach to interest their students to the field.

However, I have the chance to know people working in casinos, and I know too well that “the casino always wins” to be foolish enough to try and test a strategy for real.

It is interesting to distinguish the different kind of games you can find in the casinos. Mainly, I’d say that you have all kinds of electronic games, the blackjack and the roulette.

Ed Thorp famously showed in his book “Beat the dealer” the “counting strategy” which allows the player to have an advantage over the dealer in the long term. We now know that the casinos have very little sympathy for card counters, and I will let you try your luck if you fancy your chances and don’t care too much about your health.

Electronic games have been beaten as well by computer scientists who identified the pseudo-random sequences of the machines and hence were able to make significant profits, allegedly. Indeed, this is just a rumor, but I would be interested to have a concrete article about that if you know a real story.

Several tries have been made to beat the roulette. Ed Thorp, again, tried to have some portable computer trying to guess the expected landing zone of the ball, but didn’t do very well. The thing with roulette, is that it is one of the fairest game you will find in the casino. Indeed, the only thing that give the edge to the house is the presence of the “0” on the board (some even have a “00”). If it wasn’t for that, the house wouldn’t be able to make any money on the long-term, and the game would be fair.

Let’s consider a simple example, the expectation of the returns for a bet of 1 on red without the zero is:

$$E[X]=0.5 \cdot 1 + 0.5 \cdot -1=0$$

Yet with the zero (which pays 1/2 to any color bet) the expectation is:

$$E[X]=\frac{18}{37} (1-1) + \frac{1}{37} (-0.5) = \frac{1}{74} = -0.0135$$

This argument should be enough to discourage anybody from going further with this kind of color betting strategy without having an edge on where the ball will land.

However, you will often hear some people pretending to be able to make profit by playing the so-called “Martingale strategy”. The idea is to bet the amount that was lost so far, plus the minimum bet to make sure that, at the first win, you will end up with a profit of exactly the minimum bet.

In essence, this idea is not too bad. However there are 2 majors problems. The first one is that the casinos limit the maximum bets and the second one is that the player doesn’t have infinite wealth. Hence, with a long enough sequence of losses, the player would not have sufficient funds to play the next bet.

I think people tend to underestimate how quickly they would be out of money to apply such a strategy. Therefor, I decided to implement a little software which allows you to do a simulation with a defined initial wealth.

The strategy I follow is simple: I apply the martingale strategy as long as I lose, and I keep all the extra earning in my pocket. Hence, I stop playing when I am facing a run of losses exceeding my initial wealth. I then leave the casino with the sum of the excess earnings I won.

You can download the software here.

You will be able to do a manual simulation which will give you the exact evolution of the strategy as well as the history of the roulette.

You will also have the possibility to run several instances of the strategy in the background and to get a summary of your performance.

Assuming you use the default initial wealth of 100, and assuming the minimum bet is 5, you will see that the results converge with sufficient simulations.

It turns out that, with such parameters, you will end up leaving the casino having lost on average 25, that is, you lost your 100 but you earned 75 along the way.

It is also interesting to note that, if you increase the initial wealth, you will not perform better. This is due to the fact that you actually lower the damage limitation.

Note also that, in that strategy, losses sequences increase you bids exponentially whereas winning streaks keep a bid of 5.

Try it yourself, and let me know what you think of the app.

See you next time!

“The Quants”, a great book that everybody can read.

Good morning again,

I decided to create a new section in the blog dedicated to a selection of books. Indeed, I personally find it hard to find the right book to buy on a specific topic, and sometimes their are so many choices that I do not necessarily know how to pick the right one.

I start this section by talking about “The Quants” from Scott Patterson.

The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed It

First of all, this book is not looking to give any quantitative approach whatsoever. The idea is to tell the stories of the most famous quants having conquered Wall Street and the world of finance in general.

You will, for example, learn the story of Ed Thorp who can be considered as the first quant and who took part at Black & Scholes famous research. You will also find a long description of the creation process of Renaissance Technologies, probably the most impressive financial group ever. Who created it ? When? What was their background? How come nobody knows how they manage to do that well?

Patterson will give you sometimes a few quantitative concepts to explain what kind of strategies a fund was focused on, but you will not find any formula. You would, for example, get a qualitative definition of a stat-arb.

Anyway, the book is very easy to read and has a certain sense of humor. The author has the ability to put you in a situation so that you can really feel the atmosphere and hence fully appreciates the information he provides you with.

You can use the image above to get a link to the amazon page. I would recommend a hard-cover version because this book is that great that you wouldn’t want to damage it.

Let me know what you think of it!


Penalty Shots in Ice Hockey: a game-theoretic approach.

Good afternoon everybody,

As I promised a few days ago, I am now uploading the first research I made regarding game theory applications to Sports.

As you have probably guessed by now, the aim of this project was to model the penalty shot situation in Ice Hockey, and to find the equilibrium.

This research was made in collaboration with Geneva Servette Hockey Club (GSHC) and in particular with the help of Sébastien Beaulieu who I would like to thank again for his support.

You can download the file here, and the abstract is provided below:

This paper aims to analyze the penalty shot piece of play in Ice Hockey within a game-theoretic framework. A mathematical model will be provided for this situation in order to apply the Bayesian games setup. To make the model more realistic, measures taken with a professional ice hockey club will be used. The data will then be used to compute the mixed strategies equilibria and the meaning of these results will be discussed.

The results if this study provide a mixed strategy equilibrium with probabilities that could possibly be applied by players.

I think it would be a good idea to see if these strategies are actually being used by professional hockey players, maybe using videos. I would be glad if something conducted such research.

For those of you who do not wish to look at the mathematical aspects of the modeling, you can simply skip part 3 and focus on the introduction and the conclusion.

Please feel free to post comments to let me know your feedback.

See you soon for more.

What about Game Theory?

Hi everybody,

Just a little introductory post about the “Game Theory” section of this blog.

I like to say that game theory is the link between mathematics, economics and human behavior. It is a special field of mathematics aimed to analyze and understand conflicting situations. A comprehensive summary of the field is available on this page, and there is an interesting blog available here.

My aim in this blog is not to talk about the theoretical/purely mathematical aspects of Game Theory, but more to apply the concept to real world situations. As a matter of fact, there are a lot of classic, day to day situations where this field can be applied, and I would like to analyze a few of those and provide intuitive solutions. I am particularly interested in applications concerning the sports world.

I will soon post a research I made during my studies at the EPFL, in the Computational Game Theory class. The idea what to discuss a game-theoretic approach to penalty shots in Ice Hockey.

Until then, have a nice evening!

Here we go!

Hi everybody, welcome to my blog.

I am trying this experience in order to share thoughts, studies and research I’ve made recently.

Since most of the articles I will be publishing on this blog will be in English, I decided to adopt this language for this website.

Basically, the topics I will discuss in here will be Finance (news, research, books), Game Theory (research …) and Sport (comments, betting, …).

Please feel free to comment any post if you want to express your opinion or contribute to the subject.

I’ll post again soon to add content.

See ya,