Wednesday, November 29, 2006

More EV/Variance/Ruination Crap

Just got a very interesting comment on my post on the topic earlier this month, from someone named "David" (I play poker with several of those but they all go by their own nicknames so I wonder if I know this one).

I found it very interesting, so much so I start rambling off into a long comment. Long enough I just decided to promote it to its own post.

David said:
Your willingness to accept the consequences of statistical variance should depend on how deep your wallet is. If you are playing close to $0 EV in massive pots, then your risk of ruin is very high.

In this case, if you push, you risk $175 now to net $38 on average, giving
you about a 22% return (there's a number called the Sharp Ratio, which describes
your risk with respect to return and volatility, I have bastardized its use here to make a point). If you call, you risk $75 to net $19, giving you something like a 25% return. So it gives you a little more bankroll safety.

So, in the end, from a variance (and consequently, risk of
ruin) point of view, it is a wash, and is really up to your personal style
(which is fairly well documented).

My response:

Thanks for a very interesting comment. It made me think about things I haven't before.

I will need to read up on this Sharp ratio--I've heard of it in the most general of terms as applied to evaluating investment strategies. [after looking it up I see it's actually Sharpe and I see that as you said, you have indeed simplified it; because it's complicated beyond this discussion (and my present desire to comprehend it), I'll stick to the way I think you used it and drop the E.]

I am not fully following you however. What you have said at the beginning of the comment remains consistent with how I think about the problem in general--the willingness to take that risk has to be measured in terms of your bankroll.

Yet you go on to point to the similar adjusted returns of 22% and 25% and state that the implication is that it comes down in this case to a matter of style.

Don't you need to factor in bankroll and opportunity cost as well rather than just reward relative to amount risked? I mean as to bankroll, if this was your whole bankroll, risk of ruination would be extreme versus the push being for example 1/1000th of your bankroll where you'd really have to be pretty abysmally unlucky to go broke making this play in any iterated fashion.

As for opportunity cost, assuming risk of ruination of taking this risk relative to your bankroll is acceptably small (as it was for, though not quite 1/1000th to be sure), if I having nothing else to invest my bankroll in at that moment, isn't still a mistake not to take the higher EV?

To illustrate my attempted point, if you had $1000 and time for 100 coin flips and someone offered you the chance to win 11 cents against 10 cents per flip ("Sharp" 10%?) or 1.05 against 1.00 ("Sharp" 5%?) wouldn't you be silly to take the first bet over the second bet not withstanding the better ratio?

EV for the first choice is 1.00 with very low variance compared to $1000 and no chance of ruination, and EV of the second is $5.00 also with no chance of ruination, and a significantly higher variance relative to $1000 than the other bet (but still quite small relative to the bank roll).

More thought, the Sharpe Ratio also factors in a rate of riskless return, I wonder if that can be worked in as a missing piece to quantify what I am trying to get at with opportunity cost. Oh, this is starting to feel like I am still at work. Oh wait a minute...

Anyway, I'm glad you got me thinking about this.
See the flop...

Thursday, November 23, 2006

A Good Night (oh so close to a Great Night)

Objectively, the night went well for me. I exactly bubbled the final table in the 15k on Full Tilt which started at 8 and managed to take down the Mookie which I had played at simmultaneously from 10.

It was not quite as enjoyable as it sounds--funny how the way you feel about how things turn out jumps around with how high your expectations go.

I started playing the 15k guaranteed with Weak Player and the field seemed particularly awful--we lost 2/3 of the players by the first break.

I was hanging around about average for quite awhile without really get chipped up as we approached the cash bubble (sadly losing my wingman along the way). As I got near cash bubble, I would have been at least somewhat happy just have slipped in as my M had dropped to about 6.

I managed to go on a heater which started just before the bubble and lasted until I actually found myself 8th of 80 or so. I then chipped down, sliding down to the teens, until I had one huge hand where a jackass big stack decided to give me a lot of chips.

I hung out for awhile near the top of the leaderboard, while at the same time my situation in the Mookie looked bleak. At one point, I got down to 580 chips—leaving me to a desperation play.

I was feeling pretty good in the 15K, was playing very TAGish, much more so than in the Mook, where LAGSox came out.

I then got in a situation that I thought I might be able to use my stack to bully—something I hadn’t done a lot of so I felt it would work. I had presto, against a late position opener with a medium stack and figured he’d probably fold overs, and even some pairs from 66-TT. Problem was, he Aks, which is not the kind of hand most people will fold getting 3:1. When he called, I was happy to see I had a coinflip, but although I had been lucky with these in several places throughout the night, I was not here .

I dropped down to the middle of the pack, even plummeting to an M of less than 3, putting me in a position where I had to call an all-in from and even shorter player (if you can believe it) with 66. The results were not pretty as my M dropped to below two—Harrington’s “Dead Zone”.

I then when on dual heaters in both the Mookie and the 15K, doubling up several times in each, finding myself #6 of 12 in the 15K and in 2nd place, behind Waffles, in the Mook.

I played more aggressively in the Mook, in part because I was very willing to bust out to maximize my EV in the 15K. Given how things turned out, I’ll need to think about this more to see whether perhaps I needed to play more aggressively in the “real” tourney once we got down to the final two tables.

In the 15K, we got down to 10 players and I got moved to the other table for balancing. Unfortunately, I was the relative short stack on the new table, though as Bobby Bracelet pointed out, had no reason to hurry with an M of around 10-15. We lasted with 10 players all through a break and many, many hands, at least 20 minutes. I was chipping down a bit but had grabbed a couple of pots where I could to more or less tread water.

Then a hand came up where I had 99 in early position, a strong hand 5 with five players. I raised it up pot as I had twice before in the past two orbits, committing 1/3 of my stack. It folded around to the big blind, who was also the big stack and quite aggressive. He paused for a moment and min raised me.

Either he had monster and was trying to string me along, or he figured he could push me off my hand. I had to make a choice and I felt pretty strong it was the latter. I still considered folding—I would still have an M of around 4 if I folded, with a reasonable shot of out waiting someone else to the final table. But I wanted to get there with chips—true cliché: if you want to live you must be willing to die.

I pushed. He thought for a few more moments before deciding he was pot committed. My read was dead on, but my luck held out no more. It hurt, but at least it was over quickly, with an ace on the flop.

My consolation hand actually happened in the Mookie before this:


Aside from setting me up to take the whole thing down, I got revenge on Waffles for that awful beat he put on me with 24o cracking my cowboys way back when (not that I remember these things).

So all in all, it was a good night.

Thanks to Weak, Donkey Puncher and Bobby for sweating me at the 15K.
See the flop...

Monday, November 06, 2006

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.

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.
See the flop...