A Hand Analysis: EV Versus Variance

Here is the situation:

Pretty loose-aggressive 1-2NL table at the Extra Big Blind on Friday night, no obvious idiots, people (other than me) are straddling nearly every hand.

I've got 375 or so in front of me in the small blind, there are a mix of 50-250 shorter stacks around to the cutoff, who has the biggest stack with about 500 and has shown a seriously aggressive streak (and who has tried a move and lost a decent sized pot to me about a half hour before) and another, relatively tight medium stack with about 200 or so on the button.

Straddle is on at 5, as it has been for near every hand that I'm not UTG in for the past hour and it gets around to me with 3 limpers including the cutoff and the button when I look down to AQ of hearts.

In theory, I try not to go too crazy with AQ.

Ah hell, who am I fooling? AQ suited has a long history of giant pots with me; I have lost a few big pots (200+, once having been smooth called by AK and flopping Axx, another time 300+ having been smooth called by AK and flopping AQx and turning a K-also, rat bastard KATM once bluffed me off top two with this hand with a 99 on board), but no less than 3 times I have managed to stack or double up against someone with 1000 or more in the pot. Twice this happened at Playstation and once at the Rio during the summer blogger get together. The first time I flopped the nuts on the flop against bottom set, the second time my large flop bet was called and the turned flush was not believed, and at the Rio, I got it all in on the flop over the top of pocket 3s(!) who somehow called with a board of 24T, two of my suit (it was an amazing good read under the circumstances, but he was still behind 9:11-I would have said it was good EV too with the pot laying him 2 to win 3, except that the loss caused him to tilt off enough in short order to make his true odds more like 4 to win 3).

In a weird way this hand has sort have been the reverse Hilton's for me, winning me more than I should have rather than losing it.

So theory schmeery, AQ soooted gets me excited.

But 1st position in a potentially 6 way pot does not. So I raised it to 20. In retrospect (hell, I knew it at the time), this was too small to get heads up, but I also didn't want to make the old, you know 100% I don't want you to call me sized raise and figured I could lay her down if the flop wasn't good. I was nevertheless able however to shed all but the cutoff and the button.

Flop came a pulse raising 2h 3h 4s. I know immediately that if I get action on this hand, a whole lot of chips are likely to get in the middle. So part of me is hoping just to take it down quickly. I bet out 25 (this is a bit of weak lead, I'm trying to induce a bluff by the cutoff). The cutoff thought for a short while and then pushed a stack of 100 in the middle. Then the button called! What the hell is going on?

I need to get a grip on what they have. For the first player, I had him on anything from a set to crap trying a move on what looked to be a continuation bet, but my best read was a middle pair. I did not have the second player read for strength (I can't imagine him smooth calling in that situation with a set), but for a good draw or high middle pair and no respect for the first player's raise.

I definitely can't fold here for 75 into a 300 pot.

The question is, can I call?

Of course I can, it's a perfectly reasonable option.

But what about pushing? I have about 330 behind, and am fairly sure the button is pot committed since he only has 100 more behind. The question is the cutoff.

If he has a hand better than an overpair, he's calling, and I am in for a wild ride. If he has an overpair, he's either making a mathematically correct call (but paying me a decent price), or folding and giving up EV.

If he has no hand, I get him out of the pot, but I do take away his ability to bluff all in if I don't improve on turn (which I am fairly like to, even if I have 18 outs).

The analysis is pretty complicated since this pot is very likely to be multiway and the hand ranges remain quite wide. Pokerstove to the rescue!

Assuming I can put the first player on a range of TT-22, A5s, KhJh, 8h6h, 7h6h, 64s+, 54s, 43s, 32s, A5o, 65o, 54o, 43o, 32o (hold off questioning this range until you see what he had, I'm not even including total whiff bluffs) and the second player on
TT-99, 55-22, KhJh, KhTh, JhTh, Th9h, 7h6h, 65s, 54s (I've taken out some of the junkier holdings, assumed he'd fold the weaker pairs that hadn't tripped up or brought a strong draw with them, and threw in a lot of overcard flush draws to reflect the "drawish" feel I had for his call. The wideness of player 2's range reflects the fact he can still put me on a continuation bet with player 1 on a move. Still, I am sure he has something.

Assuming the second player is calling no matter what (my EV goes up if this assumption is wrong), let's assume player 1 will call about 5/7 of the time, folding only 66-TT (he might call occasionally with these, but he might also fold some of the other hands, the numbers actually don't change that much against my hand).

The win equity for the main pot for my hand 3 ways is 37.3% versus 33.8% for the cuttoff and 28.9% for player 3. That's 37.3% of a main pot of 615 (229.40) minus the 175 I'm putting in, or plus EV of +54.40, times 5/7 is +38.86.

For the side pot heads up I have slightly negative equity against player 1's range, 46.3%. So I'll win 46.3% of 350,or 162.50, minus 175 or -12.50, times 5/7 is -8.93.

If player 1 folds, my equity in the main pot jumps to 50.4% against player 2's range (with of course quite a bit of dead money sweetening the pot), with the main pot dropping to 500, so that's 252 minus 175 or +77 times 2/7, or +22.

Total EV of a push is thus +47.93.

Interesting thing is even if you assume 0% fold equity, the total EV for the push is still well positive, at 54.40 - 12.50, or +41.90.

How about calling?

Again, it's 75 into a 300 pot. If I hit a flush or a straight—8-12 outs depending on how many live cards I have, say 10 for the sake of simplicity—or something like 1 in 5, I am pushing but I may only get called by a weaker flush (and probably not both players in any case). So 20% of the time I will be able to push on the turn and say half the time player 1 calls a push (with on average 8 redraws--assume worse flush with 0 outs sometimes, set with 10 outs more often) and half the time player 2 does (with same redraws). For sake of simplicity, assume they never both call, (this is probably pessimistic but 50% for either calling is probably optimistic if a heart is on board). If player 1 calls, I'm getting + 85%(865) 735.25 - 75 - 255 or 405.25, roughly + 40.5 EV for when I hit and player 1 calls and I'm getting + 85%(515) 437.75 - 75 - 100 or 262.75 or + 131.38 EV for when I hit and player 2 calls.

When I miss, assume player 1 pushes 70% of time if I check to him and they both check through 30% of the time. I must fold if player 1 pushes, making 70%*80% or 56% a minus 75 proposition, and 30% of the time I get a free card. Assume I'll hit that free card the same 20% of the time, but only get paid off half as much, for +33.8.

Thus:
10% of the time I win 405.25 (+40.53) (all in on turn)
10% of the time I win 262.75 (+26.28) (all in on turn)
56% of the time I lose 75 (-42) (fold on turn)
2.4% of the time I win 202.6 (+5) (all in on river)
2.4% of the time I win 131.38 (+3.2) (all in on river)
19.2% of the time I lose 75 (-14) (fold or showdown loss on river).

Total EV of a call is +19.01.

(I realize there are several other permutations, like I hit they both fold, I improve with an ace which holds up on the river, and so on, but believe it or not I am running out of steam and despite the decimal points above, I am only seeking rough justice.)

The worst you are likely to lose by calling is 75 (though you can still get stacked on a redraw hand), whereas a significant portion of the time, you will get stacked by pushing.

If my analysis is roughly correct, taking that extra risk is worth +25-30 EV or so.

It is a clear case of variance dead set against EV.

I really don't think calling and seeing the turn here is a terrible move, but it simply does not maximize profits.

By now you have probably guessed what I did.

They both called.

The turn was a blank, and my gut ached as I saw my stack about to vanish.
But the river was a lovely, offsuit, five of all things. I wasn't 100% out the woods, as I could put someone on 6h 4h or 6h 7h, but the cutoff flipped over his 4c 3s and began castigating me for making a terrible, terrible move putting all my money in the pot so far behind. I don't know what the button had, but I doubt it was a set since he looked pretty resigned from the moment the money went in and he didn't utter a peep as he gathered his belongings on the way out.

I am a fish, what can I tell you.