On the redline

Most serious STT players obsess over their redline. This is a measure of how "lucky" they've been with allins. If you win an allin, you gain over your expectation; if you lose it, you lose what you expected. In case that's not clear, imagine you shove 63, and are called by KT. You now expect to win 33% of the time. If you do win, you won 66% more of the pot than you expected, and when you lose, you lose 33% more than you expected. The gap between what you expected and what you got is important to STT players, because they can tell themselves they were unlucky if expectation is a lot higher than results.

I will note straight away that for most typical nitty 2p2 types, particularly those who do not shove enough, expectation will very much be higher than results. That's partly because you so often get it in with the best hand, and obviously, when you lose in that case, what you lose is further below expectation than winning would be above. It's also, as I discuss below, perversely because you tend to do the opposite later in the game and get it in worse when the equity involved is higher.

But there are other components to luck. I was thinking about this the other day. I lost 9ish buyins and I had about the unluckiest day I could remember. But when I checked my expected results, I should have lost even more! I had actually run good in allin EV. How was that even possible?

Well, here are a few things that most players don't even think about.

Take that 63 hand. If you make KT fold, and you often do, you gain his equity in the hand for free. You don't consider yourself lucky, but think. If I had KK, shoved and he showed AA, I'm not going to be unlucky when he wins (he's a 4/1 favourite), but I sure am unlucky that he had the only hand ahead of me preflop! I don't have the maths to hand how often I can expect that, but it's not often. If it happens to more than a time or two in a set, you are running bad. But your allin EV doesn't consider that. The same goes for the times you threebet the LAGtard who is raising half his hands and this time he has a monster. He may have raised 20 times in a tournament, and the one time you went for it, he had you beat. He may well have not had a better hand every other time.

But I am focusing on how lucky you are to get away with shoves. HEM will credit you with a gain because you took the blinds and your equity increased, but it doesn't credit you for folding out better. But villain will often make a Sklansky mistake (should have called if he could see what you had).

We all know it's unlucky to shove AA, get called by JJ and then watch in horror when a J flops, or to get it in on the flop and watch JJ turn his miracle card. But when you raise AA, flop rags, get it in and are shown a set, you are not considered unlucky by HEM. Your expectation then is very low and you usually get nothing. But he flopped a set! 7/8 times that won't happen but this time it did. Effectively you were as unlucky as with the JJ hand where you got it in pre. But if he flops it for a raise, you weren't very unlucky at all according to your redline.

The same obviously goes for the times you raise AK, get called by A8, flop comes A83r and you get stacked. Again, you are not accounted unlucky preflop where you would be if you shoved.

Nor does it account for longer-term patterns that we would also consider "lucky". You're supposed to flop a set one in eight times, roughly. But you can call 20 raises with small pairs and hit nothing. Or you can hit your set and no one else has a piece, so you have had your "luck" in hitting, but you gain nothing from it. Of course, you can also hit and then get sucked out on. This should all even out in the long run, but the long run is not a day, a week or even a month. If I have a day where I play a hundred games and hit 30 sets in the early levels, getting paid off with every one, do I think I'm lucky? Well, it doesn't show in my redline, does it? (I don't know how often I get dealt pairs or hit flops, but that strikes me as hella lucky!)

Also, your redline is skewed by high-equity spots. If I shove the first hand of an 11 with JJ, and get called by AK and lose, I lose 5ish bucks; but if I shove that 63 on the bubble with 4K chips and get called by KT and lose, I lose more dollars (I'm not sure how many, but say I have $23 equity, I would lose 33% of that, which is about 7.50 or so). I am "unluckier" to get caught when shoving with a worse hand than I was to get caught with the better hand. Sure, I gain a ton when I win, but losing is worse than winning (you lose more equity when you lose than you win when you win because of the structure of STTs: note that I'm not saying that you lose more when you lose with 63 than you win! I am saying that the downside when you shove is always bigger than the upside unless you're heads up, so that proportionately winning twice when a 66% favourite does not make up for losing once). So we tend to lose out both ways! We get it in better when equity is low, and if we run bad, our losses take many similar spots to recoup; and we tend to have the worst hand at high equity, so we lose more in dollars even if we run as expected.