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Bankroll Management

Went to Burswood the other night with my oldest child, now 18, to play some poker. I didn't remember that the lowest stakes game Burswood spreads is $2/3, and so, even with the extra $50 his mother gave him, Primus (nicknames for my three kids are Primus, Secundus and Spare) was buying in short with $100. My advice to him was therefore to 'play tight' and be prepared to shove with a monster pre-flop.

To cut a long story short, after some early advances, the night didn't go well for either of us. Primus was up to $180 but then lost 2/3 of his stack with 10 10 to K K, and the rest with AK on a flop of A K 9 to trip 9's. I had a flopped nut flush cracked by 10 6, who flopped a pair of sixes, made two pair with a 10 on the turn to enable me to get all my money in, and a six on the river. Talk about a cold deck.

Anyway, moaning about that isn't the point of this post, rather, the next day I was talking about the session with Primus's mother, who was telling me that 'he should have just bought in for $50, and that way if he went all in, he would still have some money left and could buy in again'.

I tried to explain how, buying in that short could work, but was not generally a good idea because any pre-flop bet was likely to be $15, and a three bet would mean all-in, and even to get to that point, the blinds might eat significantly into the stack first. My example was to say you have a hand like AK, in position, with a raise in front to $15. To re-raise that hand would mean all-in, but lets say we just call, nothing hits on the flop, and the pre-flop bettor then makes a pot size bet of $30 - you have to let it go in most cases. Next hand we are in is A Q suited, this time in mid position, we make it $15 and get called by the button and the BB. Again the flop is nothing, the button bets out $50, and again we have to let it go. So now we are down to only $20, get rubbish through the blinds and suddenly have only $15 left.

I did a bad job of explaining it, because I don't think I managed to change her mind - or maybe I should have picked a better time than trying to explain it while she was hanging out the washing. In any event, I have been thinking for a while about bankroll management in general, not just for poker, but for any form of casino gambling. I read a while ago that the average Las Vegas casino 'take' for every dollar passed across the tables was 22%. When you consider that the house edge on even the gimmickiest table games is seldom more than 5%, and as low as 1.5 or 2% on blackjack and craps, where does the extra casino profit come from?

Sure, mug punters, alcohol and other Las Vegas distractions will all take their toll, but very few people walk into the casino for the express purpose of throwing their money away (at least intentionally). My theory is that the casino makes _most_ of its money through people sitting down at a table 'short' for the basic bet they are placing.

Just about every book you read about any form of gambling (or poker) tells you that bankroll management is vital, and you need at _least_ 50-100 bets in front of you, and 500 in your bankroll. But no book I have read so far seems to give other than a cursory explanation as to why that is. So I thought I would have a go at giving one.

First, I find it helps to imagine if your 'luck' is represented by a long string of random numbers that stretches through your life. There is a number from the moment you are conceived, and every microsecond after, until you die, and every one is completely random. Then, every time you partake in something that involves chance, you use the number from that particular instant in your life. Say you are crossing the road and are distracted and don't look both ways - there is some chance you will be knocked down by a car, let's say that chance is 3%. The 'number' on your random number line at that point in your life is 3.3426% - very close, but more than the 3% or less that would mean you were hit. So the car swerves, you step back, and live another day.

So too, imagine you are sitting at a blackjack table, you are playing perfect basic strategy, and the house edge is 2%. Each hand that is dealt 'taps into' your random number line, and if the number is 48 or above the house wins, and less than 48 you win.

It doesn't matter what the game is, roulette, craps, poker, the numbers are random, and you have absolutely no way of changing them or knowing what they will be in advance. Just like in poker where you have trapped your opponent with the nut flush and he has to hit runner-runner to make a full house. Sometimes the number is just going to more than 97% and you will lose.

Enough of my bad beats, back to the blackjack table....

Like a not un-typical casual casino patron, I sit down at the table and place my $100 in front of me to exchange for 20 red chips. Being fortunate to find a $10 min bet table, I make my first wager, and what to you know, I win, and now have $110. So too for the next four hands, plus one blackjack, I now have $165. This game is easy. But then the next ten hands lose, and with a couple of splits and double ups, I am down to just $30. Oh well, $30 isn't much good to anyone, so I keep playing, who knows, I may get lucky. And I do, for the next two hands. But then the five after that all lose, and I am out. In about 20 minutes, the casino has made $100.

But wait a minute, I was playing close to perfect basic strategy. I placed 23 bets, or $230 on the table, I knew my EV would be negative, but shouldn't it only be $230 x -2% = -$4.60? Where did the $100 loss come from?

Let's look at the random number line for that session (I take the sequence from this site, that generates true random numbers). Every number below 48 I win, every number 48 and above, the casino wins:

...95 66 56 22 62 23 42 14 81 84 33 10 02 21 17 38 28 73 53 94 80 48 97 99 62 84 68 16 14 62 84 74 91 96 24 64 26 18 10 97 65 94 76 74 11 81 89 76 18 43 58 28 41 02 63 74 98 45 50 49 34 60 02 81 60 03 04 43 53 96 14 36 85 38 37....

The bold numbers are my winning hands, the italics are the casino's, the first bold number is the point in time when I sit down, the last italic when I bust out and leave.

You can see that if I had one more bet, I would have won the next hand, then lost, then won three and lost the next five, and so on. But I didn't, I only had the $100 I started with, and when that was gone, that was it.

What happened was, I sat down with ten bets and that was not enough, even with some wins, to overcome a short term variance swing of several losses.

This is the FUNDAMENTAL KEY to bankroll management. You MUST have enough bets to overcome short term variance, and 'ride through' the inevitable downswings. Without that, your EV will always be far, far worse, by orders or magnitude, as per the above example.

There is one more thing though. Just like the string of random numbers goes on and on, there is nothing to stop me going back to the casino another day and hitting a sequence where I win eight out of the ten bets I make. That is only slightly less likely than a sequence where the casino wins, and a completely valid subset of the lifetime number string.

So here I am, having a great time, $200 in front of me from my $100 buy in. I'm on a roll, the night is young, the attractive cocktail waitress is on her way back with my complimentary beverage. I'm ahead for the night, and back to even in total. Time to pick up and leave - yeah right. As if that is going to happen.

50 hands later, up down, up down, what do you know, its all gone again. Hmmm. $200 down where EV says I should be only about $20.

Still, you will notice one thing; the difference between my EV and actual loss from the first to the second session is slightly less. In the first session it was $95.40, or about 95% worse than expected. After the second session, it was $183.40 or about 91% worse - effectively 4% closer to the expected value of -2%. (Does nothing to help with the $200 shortfall for this months rent though.)

Which, hopefully, illustrates the point that a bankroll of $200 is better than a bankroll of $100 for a bet unit of $10. Therefore a bankroll of $300 would be better again, as would $500 or $1000. Eventually, with a large enough bankroll, the actual win/loss is going to match the EV for the way I play. How large a bankroll would I need? Infinite is the mathematical answer. But only the casino has infinite money. The player is always going to be limited by some factor, namely, not being Kerry Packer or Bill Gates (and consequently, the casino is always going to be getting better than EV on every bet). Never the less, with a sufficient bank roll, I can _approach_ the true EV, and bigger is always going to be better.

Which still leaves the question of, for a given bet size, what is a reasonable bankroll. Using the very, very, rough example above, of improving 4% for every $100, and, roughly again, running that down to the EV of 2%, we end up with the amount of ((96/4) -1) x $100 = $2,300. Say $2,500 to be safe, should let your win/loss come close to the real EV with the 'given' of a 2% house edge.

Have I answered the question of how the casino makes more than 20% on each dollar bet? Simply, they rely on people playing with inadequate bankrolls, who, when they win, will stay at the table until variance causes them to lose all the money they are prepared to bet. The casino has the cash behind it to ride out any swing in the punters favour, the punter however can only survive to the limit of their bankroll, and is less likely to leave when they are ahead.

As far as setting a bankroll of whatever game/stakes you want to play at, I think that would be easy enough to do by getting a large enough sample of random numbers, knowing your EV, and than running a spreadsheet where every number above your EV is a win, every one below is a loss. Then run an average of the win/loss column, and when that number looks close enough to your EV for your liking, the number of rows equals the number of bets you need in your bankroll. (But don't be a ninny and only do it for 100 or less values, do it for at least 1,000, and then, several times).

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