Poker, Math, and Theory (continued)

Okay, a quick update on the previous post: I consulted with my braintrust, two respected and feared poker players with healthy experience at the stakes I play. Here are there thoughts and my reactions:

(1) Jaconda felt I unfairly discounted the chance that Carl would flat-call my 4bet. She's right in that I should have given that possibility more weight; however, I think it's a weird play to flat a 4bet for a total of 25% of your stack. Also, we have to remember that Carl is out of position (OOP). Position is extremely important in poker and even more so in no-limit games like the one we're playing. Players out of position like to negate their disadvantage by getting the money in preflop if possible. Putting the money AIPF (all-in preflop) means that the player OOP won't be forced to make difficult decisions without the advantage of seeing how his opponent has acted.

Additionally, what is he gonna flat with here? I can see flatting AA and KK maybe but Carl has to think that we have a big hand and are gonna be c-betting virtually every flop (if only cuz we have such a strong image and the pot is so big). Would he flat AK? No way...he doesn't wanna play a flop with two big cards. What about TT, etc? Again, no way...he's not gonna wanna do this because he's gonna have such a hard time with a large number of flops. Unless Carl is setting up an elaborate and bizarre bluff, I can't imagine him flatting here. Even if he has QQ+, he's gonna wanna get it AIPF cuz my hand range loses so much strength after a flop and Carl wants to maximize value.

(2) Both Jaconda and Eltrain think that we should be folding to a 5bet shove. This disagreement comes from the fact that they're giving Carl credit for a much stronger range than I am. This is something that cannot be resolved, unfortunately, because only Carl knows what he would 5bet shove here. Assuming arguendo that his range is much stronger than I think (and that Carl is either gonna fold or shove to our 4bet), Jaconda and Eltrain are correct in that it doesn't matter what our cards are because we're folding to his shove. Ideally, we'd like to have an ace in our hand, only because that makes it less likely that Carl has an ace, which means that he's more likely to fold. Complicated, I know, but that's (online) poker.



By the way: I folded preflop, fwiw. No me gusta variance, haha.

Poker, Math, and Theory

This hand happened tonight, and I had no idea what to do. (This is rare for me, because I've played many, many hands...arguably too many, haha.) It turned into a pretty helpful mathematical exercise: hopefully the thought process is correct.

PokerStars Game #30473142887: Hold'em No Limit ($2/$4) - 2009/07/14 22:25:35 ET

Table 'Kalliope III' 9-max Seat #5 is the button

Seat 3: Billy ($400 in chips)

Seat 5: Carl ($400 in chips)

Seat 6: Dan ($400 in chips)

Dan: posts small blind $2

Billy: posts big blind $4

*** HOLE CARDS ***

Dealt to Billy [8s 8c]

Carl: raises $8 to $12

Dan raises $30 to $42

Billy calls, raises, or folds??!?

Okay, let’s take this from the top. Our goal, as it always should be in poker, is to play optimally. This means that we try to make the correct mathematical decision every time we have a choice of action (bet, fold, raise, check). En otras palabras, we wanna play like this guy:

In this case, of course, we only have three options. Let's go through them one by one:

Call: This one makes no sense for us. Past experience with Dan (we've played 2771 hands against each other) tells me that he's a very aggressive player. (For those who know what these numbers mean, he's 17.5/15.1 with an astounding 8.5 3bet% over the 2771 hand sample, which includes a lot of shorthanded play.) Since he's so aggressive, he has a relatively wide range here. (Range means "the various hands he can have.") He's almost certainly not going to have 72o (seven-deuce offsuit), but he could have something like T9s (ten-nine suited). If we call his bet, we're going to have a very hard time playing a flop against him because we have little idea of what hand he has. What if the flop is Q75? T93? AA2? What do we do? I have no idea. He's aggressive, so we know he's gonna bet out on the flop, which will force us to make a very hard decision. Hard decisions are bad--it's like lighting money on fire. We're only gonna be playing optimally when we flop an 8 (because we're never ever ever folding when we flop a set), but that happens about 12% of the time...not nearly enough to justify a call.

Raise: Fortune favors the bold, right? Certainly, raising has its advantages. We have an image of being tight, aggressive, and a solid player. Thus, a re-re-raise (a/k/a 4bet) from us here is scary as hell. It screams strength. If we scream strength, we have tremendous Fold Equity (FE). Fold Equity is an essential concept that many new players do not sufficiently grasp. If, for example, Dan folds to our 4bet any time he doesn't have AA, we should 4bet him at every opportunity because we're gonna win almost every pot. Yeah, there'll be that 1 time where he does in fact have AA and we lose a buy-in, but that'll happen so infrequently that the numerous small pots we rake in will more than make up for the one big loss. In other words, in that hypo the 4bet has a positive expected value. That's good for us: every time we make a +EV play, it's like we're printing money.

So, in deciding whether to raise, it's helpful to know how often Dan will fold to our raise. We can't know that, unfortunately, but we can guess. (Remember, since he himself raised, we can assume that his range is stronger than just any two random cards; as a result, he'll fold less often than if he just held random cards.) Of course, his fold percentage depends on the amount of our raise. If we raise to $72 (making the minimum raise), he'll fold maybe 20% of the time. Likewise, if we raise to $400 (going all-in), he'll fold maybe 80% of the time. So, why not just go all-in? Because we're risking a helluva lot of money to win what's currently in the pot. Theoretically, there's some magic number that gives us the most FE for the money we risk (the best bang for our buck, if you will), but that's incalculable without getting into Dan's head. Let's say that our magic number is around $120. (That's a pretty standard raise, so I feel good with saying that $120 is our best choice when deciding what amount to raise to.)

At this point, we need two pieces of information: what percentage of the time Dan folds to our 4bet to $120, and what is the range of hands he can have if he 5bets (re-re-re-raises) us. The range we can approximate: he probably will 5bet with AQ, AK, 77+ (77, 88, 99...). Now, time for some math:

We raise to $120. There's now $174 in the pot (12 + 42 + 120). Let's assume that Carl folds. Action swings back to Dan: he has to put in $78 to call ($120 - $42). He's either gonna fold or go all-in. (Trust me.) Let's see what happens if he goes all in:

Using pokerstove.com software, we have 39.5% equity against the range we've given him. That means that, on average, we're gonna win 39.5 * $812, or $321. That sounds great, except we started with $400.* So, we lose $79 over the long run.

Now, what if he folds? Well, easy: we win $58 (12 + 42 + the 4 we put in as big blind).

So, we have two possible results: winning $58 or losing $79. To determine whether a raise is a +EV play, we need to know the relative proportions that these two results will occur. (Again, if he folds 99% of the time, we're gonna win $58 ninety-nine times and lose $79 one time. That's +EV.) We can't know this exactly, but we can determine what the minimum FE we need to make the raise a +EV play. That's just seventh-grade algebra:

-79(1-x) + 58x = 0.

137x = 79

x = 57.7%

So, if he folds more than 57.7% of the time to our 4bet, we're making money in the long run. It's reasonable to expect him to fold at least 60% of the time, so raising is a +EV play.

Fold: Folding is not +EV in this case, for obvious reasons.

So, in conclusion, we should raise. In reaching this conclusion, we assumed a few things: that Dan folds to a 4bet >57.7% of the time, that Carl folds 100% of the time, that Dan will either fold or go all-in when confronted with our raise, and that we have 39.5% equity against Dan's 5bet range.

Now, here comes the interesting part. It should be clear that the more equity we have against Dan's 5bet range, the lower our FE can be to keep our raise +EV. For an obvious example, let's say we have AA. Then, our equity is about 83%. We win money (in the long run) if we get it all in, and we win money (in both the long and short runs) if he folds. Sweet. With AA, the best starting hand in poker, it's obvious that raising is +EV (the pertinent preflop question when we have AA is which action is most +EV, fwiw). With 72o, we have terrible equity, so we need some extremely high FE number to make the raise profitable. 88 fits in the middle.

But, let's mess around with our hand a bit. What if we have QJs? What if we have 22? What if we have A3o? Again, it's obvious that as long as we keep our equity above approximately 39%, raising is profitable.

QJs: 36.14% equity.

22: 31.62% equity.

A3o: 27.43% equity.

Yikes, not so hot for any of 'em. QJs is the most palatable, so let's do that calculation:

(.3614 * 812) -400 = -107

-107(1-x) + 58x = 0

165x = 107

x = 65%

So, we now need him to fold 65% of the time...hmm, that's right on the border. Good to know, though.

(btw: AQo: 37.41%; JTs: 36.39%)

Moral of the story: If anyone ever tells you that poker is a game of luck, just smile and nod. Then, challenge him to play you heads-up. Or, better yet, refer him to me and I'll play him heads-up.

_________________

*Technically, we started with $396 after we were forced to put in the big blind, but let's keep this simple.

11 Best Led Zep Songs

Operational Definition: these are the top 11 Led Zeppelin songs, in order. I have done my best to consider only the quality of the music (not the historical significance, popular opinion, cute backstory, etc.) for each song.

Also, this ranking is far different from the task of "make the ultimate Led Zeppelin album." En otras palabras, es posible que these songs do not go well together as a playlist.

My qualifications: On the "plus" side, I have every studio album, the BBC Sessions, How the West Was Won (DVD and CD's), and the important tracks from Coda. On the "minus" side, I
was born years after September 25, 1980; I have never heard Zep while high; and I don't have very many bootlegs.


11. Kashmir: Damn, this must be a helluva list if Kashmir can't crack the top 10. Hopefully my credibility hasn't gone out the window.

10. When the Levee Breaks: See, I look at Kashmir as PG's version of When the Levee Breaks. When I compare the two, I like Levee more. Close your eyes and I bet you can hear the opening drums. Then, the harmonica. The lyrics aren't too special, but this song is just so heavy and tight that it belongs on this list.

9. Night Flight: I'm a bit of a sucker for the happy Zeppelin songs, I have to admit. Also, when Plant sings, "Oh mama, well I think it's time I'm leavin'", I just smile and smile.

8. Fool in the Rain: Underrated because it's on one of their worst albums.

7 (tie). Going to California: Joni!
7 (tie). Ramble On

6. Bring It On Home: You're probably scratching your head at this one. I freely admit that I love this song more than the fabled Reasonable Person. Plant's vocals on the first verse don't cut it for me, but I love the harmonica and the curious crescendo. Also, the How The West Was Won version is tight.

5. Bron-Y-Aur Stomp: Jeez, more acoustic stuff? Yeah, sorry. You're lucky I didn't throw in "That's the Way."

4. Black Dog: I dare you to find a better opening to an album than the first 20 or so seconds of Black Dog.

3. Over The Hills and Far Away: Beautiful opening guitar riff, simple and solid lyrics, strong vocals. "I live for my dreams and a pocket full of gold." Is "gold" supposed to mean "money" or "marijuana"? We report, you decide!

2. Since I've Been Loving You: The guitar solo, the crescendo, Plant's vocals. "Do you remember mama when I knocked upon your door? I said you had the nerve to tell me you didn't want me no more. I open my front door, hear my back door slam...you know I must have one of them new-fangled backdoor men." Also, what good is a Zep list without acknowledging Zep's roots in blues?

1. Hey Hey What Can I Do (Street Corner Girl): I love how this song combines an upbeat melody (if that's the word) with somewhat depressing lyrics. (Cf. "Heavy Things" by Phish.) Also, I have a special affinity for songs that tell a bit of a story, as this one does. Plus, who among us hasn't dated a prostitute?


"Hey, why not..."

• "Achilles' Last Stand?" Admittedly, it's epic and one of the group's favorites. I just don't dig it that much. It starts so heavy that it has nowhere to go.

• "All My Love?" Incomprehensible lyrics, for one. Also, it's too pretty.

• "Ten Years Gone?" Eh, I guess I'm in too good a mood, haha.

• "Carouselambra?" Seriously?
  

• "Stairway to Heaven?" Because I like those other songs more.

I'd better publish this post now before I start second-guessing myself ("jeez, how can I not have 'Dazed and Confused' on this list?"). Peace out, Seacrest!