This is why Doyle > Math

The usual after-game chat on thursday morning with Changs
set me thinking. For those of you who were there on
Wednesday night, you may recall the questionable hand where
I shoved pre-flop with AT and was called by Changs with
pocket nines.

Let's recap that hand.

The hand was min raised twice to me, and I min-raised it
again. Folded to Mike who calls the raise and Changs who
puts in 30 on top. I pushed him all in, and he reluctantly
calls, to flip over pockets nines. As mentioned before I
held AT of clubs.

Pot was about 25 dollars before Changs put in his 30, and I
had him covered for 83 and change more.

The results are irrelevant to this post, but I'll post it
anyway. I lucked out, spiking a ten on the first board and
an ace to secure the full pot on the second run.


What really set me thinking was my defence against Chang's
accusation of my apparent donkey shove.He feels that I
should not have shoved when he was apparently the favourite,
no matter how slight.(I'd told him I knew he had an
underpair before I shoved) I know many people would will agree with Changs here; shoving when you're behind, no matter how slight intuitively sounds like a -EV play.

Though I have poven it to be mathematically correct, this opinion of it being a donkey play bugged me for days. Why is it intuitively a bad move? After brushing up on my statistics and sorting through the 2+2 forums, I figured it out. It feels like a bad move because it IS a bad move.


So how can a move be mathematically correct, yet wrong at the same time? The answer lies in the underlying assumption that acts as a blindsight for most math based poker players. The advice "Make the right mathematical move every time and you'll end up winning in the long run" is flawed. What it doesn't mention is the risk of ruin and it's close cousin(which is relevent here), geometric cummulative variance.

Let's backtrack a little. What is bankroll management? It is money management in a sense so that you do not play with too much of your bankroll at any one time, exposing yourself to variance which can severely cripple, or ruin you. Most people can tell you bad management spells doom, no matter how perfect your poker math is. The reason this is a danger is a little something called risk of ruin. Risk of ruin predicts that the proportion which you risk in ratio with your bankroll is exponentially proportional to the chances of you going broke due to variance.The quirky part of it is, of course, that no matter how superb your management is, as long as you continue to risk a part(no matter how small) of your money playing games of variance, you end up with a possibility above 0 that you lose everything in time. Most people think casinos win because they have a slight edge in the long run(which is true). But in reality, it is this quirk of statistics that really help seperate the cash from the gamblers.

Why did i talk about the theory of risk of ruin? Well, it's an unwelcome opposing force to perfect mathamatical poker due to variance. I'm about to introduce you to another unwelcome variance concept: the geometrical cummulative.

I like what Mike offered as a compromise that night. He told me "It's not that it's a bad play. But you could have picked a better spot to put your money in". Wise Mike. He's absolutely right of course. When you're talking about putting all your money in, you hope to put it in when you're a clear favourite. But that's tourny Mike speaking. In tournaments, you only get one chance to go broke, so obviously waiting for a better spot is nicer. But in cash games, when the player can rebuy again(and the stakes are hopefully insignificant to his bankroll)...how does this work out to be right? After all, the right move in the long run always brings in the cash right?

Remember what I mentioned about how risk of ruin increases exponentially the more you expose yourself to variance? That was when variance was held as a constant. Now imagine if you will, the significance of the coeffecient in this equation, the variance. A small difference in variance will lead to a significantly larger difference in the probability of ruin.

Why does this matter? Well for starters, variance over the long run is not calculated by the normal averaging to find the mean of all variances(in all your hands). It's geometrically calculated. Meaning, instead of ADDING them together to find the total, you MULTIPLY. Big variance(big swings) mean the mean variance will end up much bigger.

Well that's the math part of it anyway. If you can't wrap your head around the math, here's another explanation. When you make a move based on math, the assumption that the long run will reap rewards only holds true if everything else remains on constant. This means, for example, that for any random play(let's take an example we see almost every game next), you must be assured that you will see that exact random play again in your lifetime. That is no easy feat. Let's see an example.

The pot is now 600 dollars. The flop is dealt, pending the river. Mike and Changs are both all in. Kelvin has 190 dollars in front of him with the nut flush draw. He puts neither on a set, meaning if his flush hits, it's very likely to scoop the pot. Mathematically this is a no brainer to call the all in right?

Here's what the math poker pros don't tell you. Mathematically, yes, it is correct. In the long run, you will always make money if you shove here. Here's the catch. For that long run to come true, you need to be sure that in your lifetime, you will encounter the exact situation two more times. That's what the long run means. That doesn't sound unlikely yet? Let's review this.

Let's say Kelvin holds AT of spades, flop is 4s8sKd. Changs holds KcAc and Mike holds 8d4d. Changs is in position of Mike who is in position of Kelvin. You will need these exact things to happen again twice more in your poker career. Let's list down the things that need to happen twice more:

1) Kelvin out of position to both Players A and B
2) Kelvin has to hold AT of a suit on a board of 48k, with 4 and 8 of the suit he holds
3) Player A in position of Player B, has to hold AK of another suit.
4) Player B in position of Kelvin, has to hold 84, with the 4 and 8 in the same suit as the king on the flop
5) both players have to go all in for exactly the total of 600 dollars
6) At this point, Kelvin must have exactly 190 dollars left in front of him

The likelyhood of the exact events replaying again the required number of times should influence how well you do in poker if you play with math. Imagine now, if Kelvin at another point in time encounters almost the exact situation, with 180 dollars left in front of him instead. If he shoves now, he is starting another cycle of hoping to catch another 2 exact situations, instead of fulfilling the above one. That's how sick it is. That's how unlikely it is.

I am reminded of the much overquoted quote:

"In the long run, we'll all be dead"


After all that ranting, what has this got to do with my AT vs 99 hand? Well, imagine my horror when I finally broke out of my bubble and realised what cummulative variance meant. How long do I have to wait for another hand as unlikely as an AT vs 99 all in pre flop cash game? Or to be more sadistic(since I won that one)....how long does CHANGS have to wait to balance out his variance call there? The upside of course, is that he only needs for it to happen once more because he was getting coinflip odds.

You're right Mike. We both could have chosen better spots.