Secrets of the Amateurs

I ran into quite an interesting spot in the $50 Rebuy last night. Right on the bubble, 28 left, 27 paid, I had 15K and was the lowest stack. Two or three other players had less than 20. I was in the small blind with AQdd, it was passed around to the hijack who made it 5K (blinds were 1K-2K/200). So just for once I waited to see what would happen on the other tables, and saw that another player had busted. "Great, " my inner egg said, "locked up the cash (and the ranking points heh), let's gamble". I moved in, he called (correctly) and showed 97hh. The flop came all hearts, marv. But never mind, I've cashed for $320 anyway.

At this point, any of you who know Pokerstars' rules & procedures better than I do (or did yesterday) will be saying "Oh no you haven't !". Because, even though I busted out after the other guy in time, he was given 27th place because (I assume) he started the hand with more chips than me. Perhaps, while I was swearing and looking for a cat to kick, he was laughing at the total goon who had just handed him $320.

But was that the case ? Let's have a look. If I fold the hand, I'm in the money with 14K. If I call and win, I'm in the money with about 34K. And if I call and lose, not even my BFH as they used to say on Bullseye. To complete the equation, there are 1.458 million chips in play and the prize pool is $46,150 . I estimate the equities as follows :

Folding, $320 plus (the remainder of the prize pool after everyone gets their $320 * (my chips / total chips)) = 320 + (46150 - 27 * 320) * (14 / 1458) = $680

Calling and winning, similarly 320 + (46150 - 27 * 320) * (34 / 1458) = $1194

Calling and losing, not even BFH = $0

In the event, AQdd v 97hh is a 62.9% favourite, so calling $EV = 1194*.629 = $751

Against the tightest reasonable range I can create that includes 97s, AQs is 58.4%, so calling EV = 1194*.584 = $697

Compared to the folding equity of $680, it seems that the surprising answer to the question "Realising that I would be guaranteed 27th if I folded, but out in 28th if I played and lost, should I have played the hand ?" is yes. Providing ... my assumptions are valid. Firstly, that the big blind makes no difference. He had me covered by a few hundred. Typical players are, I would think, unlikely to move in this spot without at least [TT+, AK] which is only 4% of hands [1].

The other questionable assumption is the way I'm calculating my equity. This is reliant on the fact that even if I win the hand I'm only going to have 2.5% of the chips in play. It's pure guesswork, but even so, I'm guessing that with this small a proportion of the chips, Chip EV ~ $ EV, after we've taken out the $320 that everyone gets, which we've done. I'm now running some Monte Carlo simulations which seem to back this up. These are very interesting in general and I might post the results when I've finished (or I might not, they might be just too good !).

So in the end I can still swear and kick whatever soft thing is nearest to hand (or foot) but at least it's just because of a bad beat, not because I threw away equity because I didn't know the rules. In a way, I did hand the 27th guy $320, but only after first taking it from the geezer who raised the 97s, plus about $50 (depending on his exact raising range) for myself. Sklansky bucks this is, of course :-). I found this a bit surprising, which is always good because that means I've learned something. Without going through the whole thing again, if I had been on say 8K it might well have been correct to pass. 3K, definitely. But the chance of turning 14K into 34K, even as just a 60-40 favourite, was worth risking the elimination with this stack.

Finally, as we can't do these calculations with the timer ticking away, it's worth positing a rule of thumb. I would suggest that if your chip stack is greater than the chip equivalent of the bubble money increment ($320 = about 10K chips in the 50 rebuy), you should just go ahead and stick them in if you're fairly sure you're ahead (of his range that is, we don't have Hellmuthian super powers), even if someone has just gone busto and you can lock up the cash by folding. In the more normal circumstance, you'll still be waiting for someone to bust and I would say in that case, go for it even more. No one likes a bubble wuss :-)

[1] Note that because he has me covered (even if by 1 chip), he can call more loosely than if I have him covered (even by 1 chip). However, most players are oblivious to this distinction when overcalling (as you can see in Sit and Goes where this comes up quite a lot on the bubble)