Q
9.You're down to only a handful of chips. Barely enough to post your next blind. You have to win a pot by the time that blind hits you, or you're out of the game. And that button is making its way around the table alarmingly fast. You look down and see a piece of cheese like Q9.
We can't answer that exactly -- we'd have to know just how willing your opponents are to call your small all-in bet with questionable hands, among other things. But here's an idealized question we can answer:
Suppose that a) I am going to have to go all-in on one of the next n hands to survive; b) when I do go all-in, one player is going to call me on any two cards, and everyone else is going to fold any two cards. What hands should I play from what seat, and what is my chance of survival?
We can solve this question for any n by working backwards from the simplest case:
You've run out of time. This is your last hand. You have to play it, no matter what you are dealt. Since you and your opponent are both playing any two cards, your chance of survival is exactly 50%. (We're not losing any sleep over what happens if we tie -- we survive to see another hand, which is good, but immediately posting the small blind, which means we're worse off than before.)
Therefore....
If you fold this hand, on the next deal you will be forced to play any two cards, for a 50% chance of survival. Therefore, on this hand, you should play any hand which does better than 50% against a random hand, and fold anything worse.
Which hands are these? You can scan through a heads-up equity table, and find them for yourself, or you can take my word for it: any pair; any ace; any king; Q3s, Q5o or better; J5s, J8o, or better; T7s, T8o, or better; and 98s. Those add up to just less than half of the possible starting hands -- 650 out of 1326 of them, to be exact. The other 676 you will fold and take your 50-50 shot next hand. You can add up the chances over all those hands, and get our total chance of success for the n=2 case:
6/1326 x 85.3%(AA) + 4/1326 x 67.0%(AKs) + 12/1326 x 65.4%(AKo) + ... 676/1326 x 50% = 54.03%You can also, if you like, first average the first 650 hands together, finding you have a 58.23% chance on the occasions you play your second-to-last hand, and arrive at that same overall chance of success via
650/1326 * 58.23% + 676/1326 * 50% = 54.03%So -- now you have a strategy for what you'll do on your last two hands, and we can tackle what to do with three.
We continue to follow this same line of reasoning. If you choose to fold this hand, you'll have two hands left to make a stand, and we just calculated that your aggregate chance of success on those two hands is 54.03%. Therefore, on this hand, you should play only those hands that have more than a 54.03% chance to win against a random hand, and fold the rest.
The list of hands good enough to play is: 44 or better; any ace; K4s, K6o, or better; Q7s, Q9o, or better; J8s or better; JTo; T9s, a total of 462 of the 1326 possible hands. We've quit playing hands like pocket threes, which have only a 53.7% chance of success,because we can do better by hoping for luckier cards on our last two hands.
We can go through the same calculation as above to find out what our new overall chance of success is. The average success rate of the 462 playable hands is 60.78%. So, with three hands to choose from, our chances are
462/1326 * 60.78% + 864/1326 * 54.03% = 56.38%.
Do you see the pattern? If we fold this hand, we have a combined 56.38% chance of success on our remaining three hands, so we should play only cards that have more than 56.38% chance of winning against a random opponent.
The list of hands good enough to meet this criterion is: 44 or better; any ace axcept A3o and A2o; K6s, K9o, or better; Q9s, QTo, or better; and JTs, 350 hands. The average success rate over these 350 hands, should we be lucky enough to be dealt one of them, is 62.56%, so our overcall chances with four hands to go are
350/1326 * 62.56% + 976/1326 * 56.38% = 58.01%.
Since we have a 58.01% chance of success if we fold, we play only those hands that give us more than a 58.01% chance of success now. These are 55 or better; A4s, A7o, or better; K8s, KTo, or better; QJs, QTs, and QJo, 250 hands. The average success rate of these hands is 64.32%, so our overcall chances with five hands to go are
250/1326 * 64.32% + 1076/1326 * 58.01% = 59.20%.
Since we have a 59.2% chance of success if we fold, we play only those hands that give us more than a 59.2% chance of success now:55 or better; A5s, A8o, or better; KTs, KJ o, or better; and QJs, 228 hands. The average success rate of these hands is 65.12%, so our overall chances with six hands to go are
228/1326 * 65.12% + 1076/1326 * 59.20% = 60.22%.At a shorthanded table that may already be more hands than you get between blinds, but we'll continue on up to 9 hands, the largest number you can have between the time you survive your previous small blind and the time you go all-in by posting your next big blind.
Play only hands with more than a 60.22% chance: 55 or better, A7s, ATo, or better; KTs, KJo, or better; and QJs, 188 hands. The average success rate of these is 66.12%, and our overall chances with seven hands left are
188/1326 * 66.12% + 1138/1326 * 60.22% = 61.07%.
Play only hands with more than a 61.07% chance: 66 or better; A7s, ATo, or better; KTs or better; and KQo, 154 hands. The average success rate of these is 67.46%, and our overall chances with eight hands left are
154/1326 * 67.46% + 1172/1326 * 61.07% = 61.80%.
Play only hands with more than a 61.80% chance: 66 or better, A8s, ATo or better, and KTs or better, 138 hands. The average success rate of these is 68.16%, so our overall chance of survival past our next big blind when we have a full orbit left at a full table is
138/1326 * 68.16% + 1188/1326 * 61.80% = 62.46%.
Are you surprised that the list of playable hands shrinks so rapidly, or that your chance of surviving another orbit creeps up so slowly? The latter shouldn't surprise you: you already know from bitter experience that you basically have to either catch a miracle hand or survive an all-in coinflip if you are to last through your next orbit. But I admit I was surprised at how long you could afford to wait before you were truly in extremis and forced to play any hand with any redeeming feature at all to it. As a useful mnemonic, we are pushing about half of our hands with 2 chances left; our best 1/3 with 3 hands left, our best 1/4 with 4 hands left, and so on.
In real life, of course, your callers will have slightly better than average hands -- there are certainly plenty of players who WILL call the short stack's all-in with any two if nobody else calls first, in an effort to make sure he busts, but often one or two people at the table will have normal playable hands. In light of this, your real-life chances of survival are substantially lower than 50% when you are down to one hand.
In real life you may also find yourself presented with an opportunity to triple or quadruple your stack if you survive, if you're in mid or late position and there are limpers in front of you. In this case you should be pushing your best multiway hands, rather than your best heads-up hands (less of the Axo and Kxo hands, more of the suited aces and suited connectors).
Still, I think I took a couple valuable lessons away from this heavily idealized calculation. I have been too willing to push "any ace and any pair from any seat" on the last orbit. Even with only one big blind left, I needed to remember to wait for something better than 22 or K2 to make my last stand, unless I am down to my very last couple of hands before my big blind of doom.