10K Post: Blind Steals on 4th Street

[MrWookie]

This is a post that started out being a submission to the 2+2 magazine that I didn't finish, and then it got way too long. I polished it off, though, and I think that at the end, this is quite worthy as my ten thousandth post. It's very long, however, and involves a fair bit of math. I'm glad I wrote it. I got me thinking hard about poker again, and I'd gotten pretty lazy. Grab a beer, and prepare to get your brain in gear.

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So you raise preflop on the button after it’s been folded to you with one of your typical stealing hands. You get called only by the big blind, and you autobet a flop that doesn’t pair you up. Unsurprisingly, the big blind calls, and then checks to you on a turn card that again fails to pair with you. What factors, now, determine whether you bet again, or take a free card?

This situation is an all-too-common occurrence amongst good poker players, especially 6 max players, and unfortunately, the choice is not always clear. It’s naturally a function of your cards, your opponent, the board, and to a lesser degree, your table image, but figuring out how these interact can be difficult. See, this situation represents an interplay between three important concepts: semi-bluffing (Theory of Poker p. 91), free cards (Theory of Poker p. 79), and inducing and stopping bluffs (Theory of Poker p. 191).

To review, a semi-bluff is a bet with a hand that’s probably not the best hand, but that may improve to the best hand. Thus, it can win either by folding the opponent now, or by improving to be the best hand on later streets: in this case, just from the turn to the river. Alternatively, you can take a free card. You offer yourself infinite odds on improving to the best hand, but you also offer them to your opponent. Now, for the purposes of this example, we hold no pair, but our hand may still be best. If it missed the big blind, too, then A high, K high, or even Q high may be the best hand. Unless we happen to have our opponent dominated, he likely has 6 or more outs to beat us, and he or she would really appreciate the free card. On the other hand, we may not be best, and betting could fold the big blind in this small pot.

On the other hand, getting check/raised here could be a disaster. If we’re confident that this check/raise indicates that the big blind has us beat, then we’re getting offered 6:1 on continuing. With those odds, we need to come up with 6 to 7 outs, depending on how this guy pays us off on the river, to call. Since our pair outs could well be dirty or not really outs (behind two pair or a set, for example), we got put in a position where we may have to fold a gutshot draw that would have loved a free card. This is why it’s said in Hold’em Poker for Advanced Players that one should be inclined to check hands with outs. Even if the big blind just calls instead of raising our bet, that call could well mean we’re drawing and wishing we had taken a free card.

The last point is that against many players, checking behind on the turn like this may induce a bluff on the river. If the player is aggressive, we may well have to call a river bet to catch a bluff with some pretty marginal holdings because the odds of him bluffing in this spot are more likely than the 4:1 the pot is offering on calling. Betting the turn, instead, will stop a river bluff. A river bet out of nowhere in that case would likely be with a made hand, in which case we can typically fold safely if unimproved. If the big blind is extremely eager to bluff, bordering on maniacal, then betting the turn in this spot may be inviting a check/raise bluff. If he is so eager to bluff that we’d have to call down a lot of these, we’d be getting 7:2 on it. These odds aren’t quite as good as the 4:1 we get from checking behind and calling a river bet.

With these points in mind, let’s take a look at some specific examples and work some math. At present, the pot contains three big bets, after approximating that the small blind has been removed due to rake. For the first example, lets say that our cards are A[img]/images/graemlins/diamond.gif[/img] J[img]/images/graemlins/diamond.gif[/img] on a board of 2[img]/images/graemlins/diamond.gif[/img] 5[img]/images/graemlins/diamond.gif[/img] 8[img]/images/graemlins/heart.gif[/img] T[img]/images/graemlins/club.gif[/img]. In this case, we are pretty likely to have the best hand. Many opponents will peel with any two overcards on a 258 flop, and we have a lot of those overcards dominated. Even if we don’t have the best hand, we have a minimum of 9 outs to the nut flush, and taking the standard deduction (half) for our overcards gives us three more. However, in situations like this, it’s usually excessive to discount our overcards quite that much. The big blind’s range of hands is so broad that we don’t have to be overly concerned with reverse domination. I’ll give us 4 outs worth for our overcards, but only 1 if we get check/raised.

Now, let’s assume that we’re behind. How much does betting out cost us? Well, if we bet and get called, then that’s the last money we’re putting in the pot unimproved. I’ll also assume that, on average, if we hit on the river, our EV will be +0.75 BB. This will roughly account for the times we bet and are called, and for the times we infrequently catch an A or a J, and get check/raised, for better or for worse. It also accounts for the times when he hit a flush and the big blind catches a second best hand he likes, and we win 3 or 4 big bets from him. Sometimes, though, he’ll just check and fold. This value naturally varies by players. A maniac will result in a higher river EV than a loose, passive opponent, who’ll in turn be more profitable than a rock. Since our draw will hit about 30% of the time, our EV if we bet and are called is 0.30 * (3.75 BB) + (1 – 0.30) * (-1 BB) = 0.425 BB. If we are check/raised, things are a little different. Our EV on the river will be higher, since we’ll likely be bet into, allowing us to raise. Since we often expect a call of our raise, I’ll say our river EV is +2.0 BB in this case. Thus, our overall EV is 0.22* (5 BB) + (1 – 0.22) * (-2 BB) = -0.46 BB, noting that we’re giving ourselves fewer outs this time. Our total EV, then, will depend on the probability of being check/raised, P(c/r) as P(c/r) * (-0.46 BB) + (1 – P(c/r)) * 0.425 BB. If P(c/r) = 10%, then we get 0.333 BB for our EV. Not too bad for when we know we have the worst hand, eh? This is obviously a profitable situation, having a strong draw in position, even if the pot isn’t all that large.

Instead, if we take the free card, our EV will be somewhat different. First of all, I’ll again set our river EV equal to 0.75. We are more likely to induce bluffs or value bets by worse hands by checking, but we’re less likely to have our raises called or reraised – all of our outs are pretty obvious outs, so reasonably alert opponents may fold even decent hands, such as Tx, to a raise. There’s still that possibility of him checking and folding, too. Thus, our EV is simply 0.30 * (3.75 BB) = 1.51 BB, giving us a difference in EV between betting and checking of 1.18 BB. Unsurprisingly, it’s quite preferable to take a free card if we have the worst hand.

The story of our EV by checking isn’t quite over yet, however. We may have the best hand, in which case we give our opponent a free card to beat us. For example, let’s give him 6 outs with, say K[img]/images/graemlins/diamond.gif[/img]7[img]/images/graemlins/diamond.gif[/img]. This is a pretty typical scenario. By giving him a free card, we give him a free chance at a draw that will come in roughly 13% of the time. Thus, the money we forfeit if we hold the best hand and give him a free card is 0.13 * (3 BB) = 0.39 BB. This is assuming, though, we play perfectly against him, and that he doesn’t bluff. We’ll get to bluffing later. Also, interestingly enough, when we put BB on a range of hands (below) that includes some hands that we dominate and some that have additional draws, it works out that the hands we beat average out to have just about 6 outs worth of equity against us. Keeping this calculation in mind is valuable.

One thing I’ve started to see lately is that “fold equity” is often tossed around as a good reason for making a bet here. This sounds appealing, like we’re really getting away with something when the BB folds. However, most of the folds are with worse hands. Indeed, with a hand like this, we’re ahead often enough that we’re often rooting for the BB to call with a 6 out draw when the pot offers 4:1 rather than for him to fold. In a game situation, a call here most often means we don’t have the best hand, but an ideally bad player would call here while drawing to beat us. Instead, a better way to look at this problem is from a more complicated function of the probability we have the best or worst hand, P(have best) and P(have worst), and the probabilities that he calls with worse hand, P(calls worse), or folds the best hand, P(folds best). We already worked out our EV if he calls or check/raises with a better hand to be WW, and we worked out the big blind’s equity in the pot if we’re ahead as 13%. I’m going to assume that there is no value in a river bet if we currently have the best hand, and that he doesn’t check/raise bluff to keep the equation somewhat under control.

EV of betting = P(have best)*[P(calls worse) * ((1-0.13) * (3.75 BB) – 0.13 * (1 BB)) + (1-P(calls worse)) * (3 BB)] + P(have worst) * [(1-P(folds best)) * (Betting EV when behind) + P(folds best) * (3 BB)] + (1 – P(have best) – P(have worst) * [P(fold split) * (1.5 BB) + (9 outs) * (4 BB)]

The last term is the chance that he has another AJ. This is quite favorable for us in this example, since he may fold, and if he doesn’t, we may hit a flush. It’s a little unwieldy, but with this equation now, we can substitute in the various probabilities and get a more complete picture of the EV our bet than simply the probability that the big blind folds to our bet times the size of the pot. So, let’s try it out. First, we need to put our opponent on a range of hands. For the purposes of this article, I’ll use a big blind who’s a little too loose and a little too passive, but who is straightforward enough to always 3bet his premium hands preflop. Thus, his range is approximately 22-99, 2Ts+, 2Ko+, 56s+, 5Qo+, 87o+, 86s+, T7o/s+ for the hands that beat us, the unpaired hands from A3-A9, K6-KQ, Q9, QJ, J9, 43s, 64s, 76, 97, and two [img]/images/graemlins/diamond.gif[/img]’s that we beat, and we have the chop with AJ. I’m not going to include all the nitty gritty math, but it pretty much works out to being just about equally likely to have either the best or the worst hand, and little under 3% of the time we are freerolling another AJ. Are there any better hands that he would consistently fold here? No. All the better hands are pairs or better, and people are pretty reticent to fold pairs in spots like this, especially on such a ragged board. It’s not impossible, though, so let’s say he folds a better hand 5% of the time. What about the hands we’re beating? How many of them will fold? Well, the better unpaired A’s are probably calling if they’ve made it this far, say A6 or better, and any gutshot or better draw. We’ll also toss in KQ and KJ making a bad call. That means he folds a worse hand about 1/3 of the time.

So, now we have a bunch of math to do, filling in the blanks from above:

EV of betting = 0.485*[0.67 * ((1-0.13) * (4 BB) – 0.13 * (1 BB)) + (0.33) * (3 BB)] + 0.485 * [0.95 * 0.333 BB + 0.05 * (3 BB)] + 0.03 * [(0.196 * (4 BB)]

EV = 0.485* ( 2.24 BB + 1 BB) + 0.485 * ( 0.316 BB + 0.15 BB) + 0.03 * (0.784 BB)

EV = 1.57 BB + 0.226 BB + 0.024 BB = 1.82 BB

Awesome! We’ve got it all worked out, and once we bet at this pot, we expect to win 1.82 BB. This is definitely larger than our hot-and-cold equity in the pot at the start of this street (52.3% of 3 BB = 1.57 BB -- behold the power of position), but we still may be able to do better. Let’s see what happens when we check, and to do that, we’ll start with another long, ugly, EV expression:

EV of checking = P(have best) * {[P(we improve) * EV(value bet)] – [P(he improves) * (do we call?) * 1 BB] + [(1 – P(we improve) – P(he improves)) * ( P(he bluffs) * (do we call?) * 4 BB + (1- P(he bluffs)) * 3 BB)]} + P(have worst) * {P(he bets) * [ P(we improve) * 4.5 BB + (1 - P(we improve)) * (do we call?) * (-1 BB)] + (1 – P(he bets)) * [P(we improve) * 3.5 BB]} + (1 – P(have best) – P(have worst) * [ 0.196 * (3.5 BB)]

So, this equation is a lot uglier. Most of my variables should be pretty self-explanatory. The EV(value bet) is for the times he calls with an unimproved A on the river. I’ll set this equal to 0.1 BB, a little less than the fraction of the time he’ll actually have A high here. The (do we call?) variable is going to be either 0 or 1, depending on what we do, for simplicities sake, although we could go on about optimal calling frequency in future discussions. The chances we each improve, as determined from previous sections, are 0.30 and 0.13, respectively. P(he bets) is the percent of the time he’ll value bet the river with his paired hands, and I’ve approximated him calling our river raise when we hit as 50% of the time.

So, let’s start getting specific. I’m going to run the numbers for a BB who never bluffs, and a BB who always bluffs. For the guy who never bluffs, I’m going to assume he’ll value bet the top 50% of the hands that were paired on the turn and everything that improves on the river. He’s pretty passive, after all, but not super passive. Since we’ll have a read that he never bluffs, we’ll never call UI.

EV of checking with no bluffs = 0.485 * {[0.30 *0.1 BB] – [0.13 * 0 * 1 BB] + [(1 – 0.30 – 0.13) * ((0 * 0 * 4 BB) + 3 BB)]} +0.485 * {0.5 * [ 0.30 * 4.5 BB + (0.70) * 0 * (-1 BB)] + (0.5) * [ 0.30 * 3.5 BB]} + 0.03 * [ 0.196 * (3.5 BB)]

EV = 0.485 * {0.03 BB + 0.57 * 3 BB} + 0.485 * {0.5 * [ 1.35 BB ] + 0.525 BB} + 0.02 BB

EV = 0.485 * {1.74 BB} + 0.485 * {1.2 BB} + 0.02 BB = 1.451 BB

Against an opponent who’ll never bluff, we come out ahead by betting the turn compared to checking. Apparently, we get enough value from the worse hands he calls to make up for skipping the free shot to improve against his better hands. This is very good to know.

Now, for the guy who always bluffs, we’ll need to always call. Also, he’s always value betting his good hands.

EV of checking versus bluffer = 0.485 * {[0.30 *0.1 BB] – [0.13 * 1 * 1 BB] + [(1 – 0.30 – 0.13) * ((1 * 1 * 4 BB) + 0 * 3 BB)]} +0.485 * {1 * [ 0.30 * 4.5 BB + (0.70) * 1 * (-1 BB)] + 0 * [ 0.30 * 3.5 BB]} + 0.03 * [ 0.196 * (3.5 BB)]

EV = 0.485 * {0.03 BB – 0.13 BB + 0.57 * 4 BB} + 0.485 * { [ 1.35 BB – 0.7 BB]} + 0.02 BB

EV = 0.485 * 2.18 BB + 0.485 * 0.65 BB + 0.02 BB = 1.392 BB

Now this result I find particularly surprising. People frequently advocate checking the turn behind against habitual bluffers to induce their mistake. However, by using their bluffing weapon against them, we’re getting hit harder by the most powerful weapon in limit hold’em: the value bet. We only improve on the river about 30% of the time, but we end up paying off every time he has something or drew into something because of the free card. Instead, by betting the turn, we’ve not only stopped villain’s bluffs, a powerful weapon in and of themselves, but we’ve also crippled the power of his value bets. We know much, much better to fold on the river UI if he bets out of nowhere. Now, granted, I did not account for a player who’ll frequently donkbet the river on a bluff 100% of the time after we bet the turn, but these characters are virtually nonexistent. Some players will bluff like this sometimes, but it is much less common. Players who’ll bluff the river frequently or always after checking on the turn are everywhere, especially as you move up in limits. And surprisingly enough, it’s a pretty decent strategy in spite of the number of 2+2ers who think they’re getting away with something by inducing players to do it! An expert player, though, would be better served, by only bluffing some of the time on the river (the game theory answer in this case is 25%) so as to induce getting paid off more or stealing the pot more.

Based on the above, is checking the turn in a related spot ever a good idea? Maybe. In order for it to be a good idea, checking behind on the turn can’t hone the edge of the river value bet. Thus, we need to have a hand where value betting on the river is already dead to us, or else we need a hand where some more river value bets are going to be with hands that are worse than ours. An example of the first would be a hand like J[img]/images/graemlins/heart.gif[/img]T[img]/images/graemlins/heart.gif[/img] on a board of A[img]/images/graemlins/heart.gif[/img] K[img]/images/graemlins/heart.gif[/img] 9[img]/images/graemlins/spade.gif[/img]. In this case, we’ll improve to a hand we can call with a similar percentage of the time, but we’re not calling anything unimproved. However, betting the turn in this case is somewhat more appealing, too, because we will actually fold better hands on the turn frequently, unlike the examples where we frequently had the best hand. An example of the second case would be having 77 on the above board of 2[img]/images/graemlins/diamond.gif[/img]5[img]/images/graemlins/diamond.gif[/img]8[img]/images/graemlins/heart.gif[/img]T[img]/images/graemlins/club.gif[/img]. In this case, a much larger number of his bets on the river we’ll beat. But it’s not clear-cut in this case, either. See, with 77, our turn bet will be more often with the best hand, and will be called by more worse hands like 2x and 5x incorrectly, which is quite profitable. The first case is I think more probable to be a check, especially as fold equity decreases against villain’s hand range (villain will have fewer unpaired hands that will call the flop and then fold the turn on that board). I considered working these examples out fully, too, but given the current length of this post, I think I’ll start wrapping things up and leave the EV calcs for the other examples for an astute reader or perhaps as a follow up.

Summary: Checking the turn behind when holding a draw that still has some showdown value in the name of inducing a bluff helps your opponents play better against you. By inducing bluffs, you become overly vulnerable to value bets. Checking is probably a much better idea with a hand that needn’t call a bet on the end unimproved, but I have not yet done the math to verify this.

Best of luck, guys, and we’ll see what I muster up for 20k.

[econophile]

Wookie,

Thanks

[ninenine_zoe]

[ QUOTE ]
Wookie,

Thanks

[/ QUOTE ]

[Peter Harris]

tl;dr

just kidding, will absorb this in the daytime when my brain works.

Congratulations for hitting 10k, let's hope by 20k you're a millionaire.

Pete

[RunDownHouse]

Wookie,

How confident are you in your calculation of outs? Specifically, I think giving us one out when check-raised is overly pessimistic.

Really good post.

[SlantNGo]

I'll have to look at it in more detail when I have time, but you result does indeed go against the advice I think a lot of us have been taught. Specifically, I'd like to experiment a bit with changing your assumptions in a certain way by (either over or under) by 25% and see how the result comes out.

[SlantNGo]

[ QUOTE ]
Wookie,

How confident are you in your calculation of outs? Specifically, I think giving us one out when check-raised is overly pessimistic.

Really good post.

[/ QUOTE ]

I agree. I've gotten check/raised here by not too aggressive players with TP and a decent kicker quite often. I'd definitely give myself at least 2.

[MrWookie]

My original draft actually gave 3 outs after we get c/r'd. Our EV given that we were c/r'd went from -0.4 to +0.1, but its effect on the final answer is pretty small, in the hundredth's place, because we're not check/raised all that often.

[Broom]

This is the best forum ever

[Mossberg]

You the man, man..

Thanks.