Smash Vegas

A betting system is a method of bet sizing depending on predetermined factors.  The most common betting systems use a progression where you bet more or less depending on whether you have just won or lost.  There are also betting systems where you look for an event which is statistically “due” (e.g. you’re playing roulette and the last 4 numbers have been red so you bet black).  Obviously the latter are examples of the gambler’s fallacy.

The problem with all betting systems is that bet sizing does not affect the house edge.  It can increase your chance of winning in the short term, but that will always be balanced out by a large loss later (as in the martingale system).  The longer you play, the closer your loss will be to the house edge.

The reason that bet sizing cannot affect the house edge is simple - expected value is linear.  This means that:

expected value (A+B) = expected value (A) + expected value (B)

In other words it doesn’t matter what the individual amounts you wager are, overall your loss will equal the total you wager multiplied by the house edge.  There cannot be a clever way of sizing bets which would change this - otherwise expectation would not be linear.

Thus we find (although we already knew) that the equation for your loss at a gambling table is:

loss = (total amount wagered) * (house edge)

No betting system can alter that.

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.

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.

This piece of probability theory is probably [sic] best described in it’s form as a bar bet (or proposition wager).

It takes a little preparation but is worth it.  You need three cards.  Each is marked on both sides - one is red/red, one black/black, the last is red/black.

You ask your victim to choose a card blindly and place it on the table.  Let’s suppose the up side is red.  You bet your opponent the next round of drinks (for fun) that the other side is also red.

At this stage most people are thinking “It’s either the red/red or red/black card, so I have a 50% chance of being right”.  Consequently they will take the bet.  As with any bar bet it’s best to add a little theatre - a friend who used to use this would say “I know it’s a coin toss but I have the luck of the Irish” (hence the picture).

In fact the probability of the other side also being red is not 50/50.  We are observing not cards but card faces.  There are three possibilities:

1. This is the red side of the red/black card
2. This is the first red side of the red/red card
3. This is the second red side of the red/red card

Thus in 2/3 of the cases the other side is also red.  By betting on the hidden face being the same as the up face you will win two thirds of the time.

That works out as a fair few free drinks…