Monday, October 04, 2004


Short-Stacked - A Mathematical Approach

Let's get straight into an example. No-Limit Hold-Em, blinds 100-200. Everyone passes to us on the button, with 1400 chips, where we find Ah 2d. What will happen if we raise all in ? The worst case scenario is that our opponents will fold if they can't beat A2, and call if they can (See Note 1). This is what would happen if our opponents could read us perfectly (basically if they could see our cards). Computer simulation of the various outcomes tells us the following :

69% of the time, neither of our 2 opponents can beat A2. They pass, and we win 300.

The remaining 31% of the time, at least one of our opponents can beat A2 (note 2). There will be a showdown, which breaks down as follows :

18% of the time, we lose the hand. We lose 1400
5% of the time, we split the pot. We win 75
8% of the time, we win the pot. We win 1550

Note 5/10/04 : The split and win pot sizes now take into account the fact that if the SB calls, the pot is 3000, and if the BB calls the pot is 2900.

Adding all these outcomes up we get a total EV (expected value) as follows :

69% x (+300) = +207
18% x (-1400) = - 252
5% x (+75) = +4
8% x (+1550) = +124

TOTAL = +83

So even if our opponents play perfectly, this play still nets us, on average, 83 chips. Of course, in real life, our opponents can't see our cards and they will sometimes call with a worse hand than A2 (like KT) or fold a better hand (like A5 or 33). So our expectation will be better still.

Now, I know what you're saying right now. You're saying "but why should I risk getting knocked out of the tournament for 83 chips ? Are you nuts ?". The whole point is that your risk of being knocked out is included in the calculations above. When you lose, your equity drops to zero. If you double up, when you're short stacked, your equity doubles - as close as makes no difference (note 3). It's the two ways to win (nicking the blinds and winning anyway if called) that combine to make the play profitable.

It would be pretty tedious to do this calculation here for every hand that might come up. It would also, of course, be impossible to do it in the heat of the moment at the table. This is where the computer comes into its own. Being a clever programmer, I can write a clever program to do all these simulations and tell us the results. For example, if we do have A2, then how big does our stack have to be on the button before an all-in move becomes unprofitable ? The answer is 22 small blinds. Now we know that when we have 22 small blinds or less in our stack, if we find an Ace on the button and everyone folds before us, going all in is a profitable play.

One word of warning here - just because a play is profitable, that doesn't mean you have to make it. There may be alternative ways of playing the hand which are more profitable. It may also be worth passing up marginally profitable situations because, if we are eliminated, we will miss out on more profitable plays later on - but I stress don't overdo this. A bird in the hand and all that !

Next time I'll post the full simulation results, showing what hand is required in each position with various stack sizes.


(1) Since we are all in, we don't care whether they call or raise. Effectively they are calling.

(2) About 2% of the time, both opponents can beat A2. In many of these cases, one of them will fold anyway. This is such a small factor it can be omitted from the calculations. 5/10/04 How small ? Quick fag packet calculation, 2% of the time, they can both beat A2. A conservative estimate would be that at in least 1/4 of these, the BB would have to fold because of the SB's raise. So 1.5 % of the time, you're about 16% (that's a guess) to win 2800, 84% to lose 1400. EV = 0.015 * (.16*2800 - .84*1400) = - 10 chips. But the way we counted it before (just one person calling) would have been -5 chips, so it's only a 5 chip difference (compared to our result of 83 chips). I will make sure my simulation includes this factor before publishing full results (which is going to delay me by a day or two I'm afraid)

(3) Except when you are already in or close to the money, there are other players who are liable to be blinded away before you and waiting for them to go broke will earn you significant extra money.

Andy, A brief question rather than a detailed analysis. You typed:

"69% of the time, neither of our 2 opponents can beat A2. They pass, and we win 300."

Taking the hypothesis that they can see your cards, would it not be in their interests to call your all-in in certain cases because they are getting pot odds, even if they cannot beat A2?

Not sure how this changes the mathematics, and it probably does not in this particular case refute the argument. It would make it something like:

65% x (+300)= +195
20% x (-1400) = -280
5% x (+150) = + 7.5
10% x (+1700) = +170

which reduces the positive expectation slightly.

You argument also assumes (as you state) that chips = equity. Although it may be true that your equity "as near as damnit" doubles if you double through on a short stack, that is not, as far as I am aware, a term that would be acceptable in a Mathematics paper. Indeed, it's the complexity of how equity relates to chips that causes so many problems.

What I really want to see is an empirical study of chip holdings to real value. It would be an interesting piece of research, if I had the time.

Pete Birks
Quick query - how can we win 1700 if we win the pot in a showdown situation? If the SB calls 1300 more and the BB passes this will create a total pot of 3000 and we stand to win 1600. If SB passes and BB calls total pot = 2900 - we win 1500. You have discounted the additional complication of winning a showdown against both SB and BB, which would be +2800. Not a big difference but something would need to be accounted for in a computer program.
Thank you Pete and whoever :-). You are both right.

Pete's point does make a difference but a very small one. If my opponent is behind but getting pot odds then his call is not going to make very much difference in EV terms. By the Theory of Poker it must reduce my EV very slightly but I believe there is no point going down the Samuel road and working this stuff out to 4 significant figures because there is a much bigger inherent error than that in my assumptions : ie our opponents will not, in practice, play perfectly. The same goes for the point about winning 2800 with two callers - it's a low probability case and don't forget that the first caller (or raiser) will prevent the second player from coming in with many of his hands that beat me.

However the point about my figure of 1700 being wrong is totally correct, thank you. I will make sure that my simulation is doing this properly before posting any further results.

As for the variation in chip equity, you'll just have to take my word for it that increasing your chips from, say, 2% of the chips in play to 4% is to all intents and purposes doubling your expectation (short of the money places). This isn't a rigorous mathematical paper - it's my blog :-)

I have to say my Maths stretches to CSE grade 2, aka 1979, so when a discussion starts to highlight EV's and are applied to an expected outcome I tend to switch off.
However, for me, the best way to apply a short-stacked strategy is sometimes taking on a big stack to double through.
Put simply, I need to get some action. I am happy to call with any Ace, and most hands where I am getting 3-1+ for my remaining chips. Overall this works well for me and have recovered many times to swing the game more favourably in my direction.
Thanks for the comment. That's not the strategy I recommend, but there's more than one way to skin a cat, especially in poker. If I can borrow another saying from the The Buddha, when people asked him "But why should I do what you're saying" he would shrug and say "Just try it and see". Once I have enough of the strategy on here, you could always try it on some cheap sit and goes, and see how you like it.

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