Smash Vegas

There isn’t a gambling forum on the internet without hundreds of posts about “unbeatable” betting systems based on the martingale.  They are usually either fraudulent systems for sale or naive newcomers who have just discovered a foolproof betting system - and they can show it’s mathematically sound!

In fact the martingale has been around for hundreds of years.  The system is simple - by increasing the stake with each loss, when the gambler eventually makes a win it will cover all the previous losses and make a profit.

For example if we are flipping a fair coin with 1:1 payout if we guess correctly, our starting bet would be $1.  After each loss we double our stake and when we win the stake is reset to $1.  So if we lose once before winning we gain $2 - $1 = $1.  If we lose twice before winning we gain $4 - $2 - $1 = $1.  Losing three times before winning nets us $8 - $4 - $2 - $1 = $1.  And so on.

As you can see the win always nets us $1 - so mathematically it is unbeatable!

Of course there is a problem which I hope you can see.  The size of the bets grows exponentially and so the stakes quickly get very large indeed.  This means that sooner or later we will either run out of money to make the next bet, or hit the limit of what the casino will let us wager.

This happens sooner than you’d think.  In the example above after 10 losses in a row we would be betting over $1000 to win $1.

The martingale is only an unbeatable strategy if:

1. You have infinite wealth
2. There is no limit to the size of bet you may make

It tricks a lot of people because if you try it for a short period of time you are likely to finish ahead.  Unfortunately if you keep playing you will eventually hit a big loss which will wipe you out.

What the martingale really does is trade a lot of small wins now for a big loss later.

No betting system can change the expected value of a game.

You’ve tossed a fair coin 9 times and it’s come up heads every time.  The odds of a fair coin coming up heads 10 times in a row are 0.5¹⁰ = 1/1024.  So if you were asked to bet on the outcome of the tenth toss the odds would favour you choosing tails, right?

Of course the answer is no.  No matter how many times you toss a fair coin it is still 50/50 that it will land heads or tails.

That’s because the coin has no memory and each throw is independent of the last.  Not recognizing this and thinking that a result is “due” to “balance out” the probabilities is the gambler’s fallacy.

It’s an easy mistake to make because we assume that short term trends should be representative of long term trends.  We know that in the long term that coin will come up heads about 50% of the time so we expect deviations from the average to even out.  We fail to understand that in the short term a run of coin tosses is unlikely to “look random” and clustering of results is the norm.

The key to avoiding this fallacy is recognizing when a game has memory and when each round is independent of the last.

Throwing a coin obviously has no memory.  While playing texas hold’em getting dealt pocket aces twice in a row does not change the odds of you being dealt them again in the next hand (provided the deck is shuffled between hands).

On the other hand since the deck in blackjack is not shuffled every hand, a high occurrence of low cards does mean that more high cards are “due”.  This is because the hands are not independent, and is the basis of card counting.

The gambler’s fallacy is at the root of many fraudulent betting systems, don’t get caught out by it.

Small Stakes Hold’em

David Sklansky, Ed Miller, Mason Malmuth

This book on small stakes should not be mistaken for a beginners book.  Although it focusses on the smaller stakes where beginners tend to dwell the concepts and analysis are applicable to any game with loose players - this makes it one of the best books available for improving your online play.

Content

Small Stakes Hold’em (SSH) begins with a discussion of gambling concepts such as expectation, pot odds and pot equity.  It’s a well written introduction which I thoroughly approve of because I (like the authors) believe that every poker decision boils down to “which choice maximises my expected value?”.  As you’d expect with Sklansky on board the authors do a great job of explaining these concepts in simple terms and detailing why they are important.

The next chapters cover the theory behind playing each of the streets (preflop, flop, turn, river) with an emphasis on recognizing the type of hand you hold and comparing your odds to the pot odds.  Hand types and the strategy for playing them are discussed for each street - e.g. play speculative preflop hands like low suited connectors or pocket pairs if they can see the flop cheaply, try to win straight away on the flop by protecting made hands that are good but not great, etc…

The advice is good and general, emphasising the critical importance of pot odds.  There is also some discussion of awareness of the game structure.  For instance the authors argue that because of the minimum bet increase in many games on the turn and river it is sometimes useful to forego betting on the flop to reduce the pot odds being offered to an opponent on the turn.  This is thought provoking stuff and the many sections like this (along with the lack of a basic “these are the rules of poker” chapter) ensure this is definitely not a beginners book and would be useful to a wide range of players.

The last part of the book is an extensive test yourself chapter, with questions covering all the topics discussed in the book.  It is one of the largest question and answer sections I’ve seen in a poker book and is very comprehensive.

If I had to sum up the theme of the book it would be to play tight and aggressive in a loose game, being constantly aware of the pot odds.  The authors argue (correctly in my view) that in a loose game where players will call with rags the correct attitude is not to become defensive because of “bad beats” but instead to punish the opponents for their constant weak calls.

Presentation

The book is generally well laid out, although some of the tables could be a little clearer.

There is some discussion of poker mathematics which does not go into too much depth but may put off some readers with the display of equations.

The writing style is good and the book is a quick read despite its length (over 350 pages).

The test yourself sections are a useful way of absorbing the theory by practical examples.

Summary

I like SSH and regularly recommend it to friends wanting to improve their online play.  I think it does a good job of teaching players to rely on playing a solid “by the numbers” game of poker instead of fancy hollywood poker plays.  There’s a lot here even for advanced players who are perhaps not as familiar with the lower stakes and the book does a good job of explaining not only the differences in strategy at this level but the reasons for the differences.

With all this talk about pokerbots I’ve mentioned monte carlo simulations quite a lot.  But what are they?

They’re a method of computation which use repeated random sampling to produce results.  A simpler way of putting it would be “you try the thing lots of times to see what happens”.

A simple example would be calculating the probability of flipping two coins and getting two heads.  We know the probability is 1/2 * 1/2 = 1/4.  What if we couldn’t work that out?  A monte carlo method to calculate the probability would be to try the experiment many times, record the results and see what proportion of the time both were heads.  Obviously the more trials we perform, the more accurate a result we will get.  We will never get a perfectly accurate result but as the number of attempts increases, the result will get closer to 1/4.

This example probably seems silly because the answer is so obvious.  Where monte carlo methods come into their own are in problems which are too complex to calculate exactly.  This is why they are so crucial to pokerbots.

Consider a game of heads up holdem.  If we want to calculate our probability of winning a particular hand then we must compare it to all possible opponent hands.  Suppose we are preflop -

• there are 50×49/2 = 1,225 possible opponent hands
• there are 48×47x46×45x44/120 = 1,712,304 combinations of board card
• that leaves us with 1,225×1,712,304 = 2,097,572,400 possibilities to consider

That’s not too bad for a modern computer to calculate, but this is the simplest case.  Things quickly get out of hand as the number of opponents increases - e.g. two opponents have 690,900 possible hand combinations while nine opponents have 6.2211×10²⁰ (622 quintillion).

Much better would be to simulate the outcome of the hand many (10,000 - 1,000,000) times.  It is less computationally intensive and so will take less time.  100,000 trials will give us a winning percentage to an accuracy of several decimal places and we can adjust the number of trials to meet our desired accuracy.

Another advantage is that we can distribute the cards in the simulations in a non-random way.  Suppose we know that an opponent only goes to the flop 20% of the time, we could include this information in the simulation and only deal him the top 20% of hole cards.

Probably best to continue that train of thought in another post though.

The move from wordpress.com is complete.  It was a good way to begin but was holding me back from what I really wanted to do with the site.

I hope everyone likes the new design.  There are still a few bugs (the archives button is playing up for instance) but overall it wasn’t too painful a move and I think it’s quite an improvement.

We’re now hosted by DreamHost and I’ve just found out that all their hosting is carbon neutral, a nice bonus to what seems like a great company.  No complaints so far.

All that remains is to apologise again for the disruption and let you know it’s back to business as usual.