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.