Thanks for all the comments on What’s Your Play? Worst Card to Bluff, and sorry that I’ve been slow in getting this results post up. It’s possibly a result of the fact that I’ve been driving back across the country this week and not online much, and possibly a matter of my not having fully worked out an optimal bluffing frequency for this situation yet. But more on that in a moment. For now I’m going to post what I do know as well as the results, and hopefully later in the week I’ll return with an update concerning the math behind bluffing optimally over multiple streets.
Villain’s Range
Ian encapsulates my thoughts on the flop: “Neither villain’s actions on the flop are particularly scary – the donk is usually weak in my experience, and the raise can therefore be quite weak, too – unpaired overs, an overpair possibly too, I suppose a 9. On the other hand, your overcall is really quite scary.” At every point where Villain has put money into the pot, he’s had a decent expectation of fold equity. I don’t have enough experience with him to say exactly how wide his range will be in a spot like this, and of course he could play strong hands like overpairs or even quads this way, but he could easily be wider than that.
This is the impetus for my wanting to play back at him in the first place. I have the rare hand that actually has some equity against the top of his range plus good reason to believe his range is wider than it “should” be.
Target
This raises the question of just ambitious we should be with our bluffing. The choice is whether to go after hands as strong as AA or simply to try to make him fold the portion of his range with which he himself was bluffing.
The latter can be done cheaply and is in my opinion not optional. Any player with half a clue is not going to expect you to be light here, and if he himself has nothing, you can expect him to give up easily. A bet of $150 – $200 offers you great odds and should drive him off the non-pair portion of his range.
The bigger question is whether to try to run Villain off of stronger hands. This is a more opponent-dependent decision, as some players simply aren’t going to fold big pairs no matter how plausibly you represent a narrow range.
Against the right player, though, you really can represent that narrow range here. There aren’t many hands that are going to cold-call a raise on this flop, and a big bet on either the turn or river should polarize you to the point where a good hand reader won’t expect you to bet worse than KK for value. At that point, he just has to decide whether you have quads or some sort of airball with which you called his flop raise. Many players won’t even call the flop raise with a straight draw, so it’s really very difficult to put Hero on a bluff here, with 66 being the most likely candidate.
Given the stacks, I think bombing the turn in order to set up a river shove is the best way to do this. Even if you don’t actually shove the river, you get the most leverage by sizing your turn bet in a way that suggests it’s coming. This should generate more fold equity per dollar than checking back and betting the river, even if you were to overbet shove the river. It’s also most consistent with how Villain probably expects you to play quads.
Balance
If you believe you know whether and when Villain will fold AA (or whatever you think the top of his folding range will be), then you can exploit that by taking the appropriate line. If you don’t know, then the best thing you can try to do is use the few bluffing hands in your range to balance the few combos of quads in your range.
We can reasonably suppose that the only hand with which you will pot the turn and shove the river for value is quads. This is an unbalanced range, since Villain could exploit it by folding all of his bluff-catchers when you pot the turn. Similarly, if you have bluffs in your range when you pot the turn but never when you shove the river, then Villain could exploit that by calling the turn with his bluff-catchers and folding if you shove the river.
Thus, you need to have some bluffs inyour range for both of these plays if you don’t know how Villain will respond to these bets with his bluff-catchers. Specifically, you need to bluff in proportion to the pot odds that your bet offers on each street.
Working this out over multiple streets is complicated, and David Sklansky’s article on the subject has been a great help to me. With a stack-to-pot ratio of roughly 4, there is room for a pot-sized bet on the turn followed by a pot-sized shove on the river.
The point I’m still mulling over is how the presence of the nuts in Villain’s range changes Hero’s optimal frequency, since Sklansky’s example assumes that Villain has a clearly-defined bluff-catcher (or “mediocre hand”, to use his terminology). I’m going to keep working on it myself, but any insight that you all may have would be appreciated. As I said, I’ll post an update when I think I have it figured out.
Results
The basic thrust of Sklansky’s article, and the fundamental value of leverage, is that a big turn bet puts tremendous pressure on Villains bluff-catching range. My plan, then, was to bet big on the turn and then give up on the river. This line could be exploited by a Villain who’s willing to call the turn with a lot of bluff-catchers, so I was gambling that my opponent was not that player. I bet $600, and he folded.
Nice bet Andrew. Did he tank at all before folding? In retrospect, do you think he did/would have folded a hand like AA or KK?
I ran into a similar situation. If you don’t mind, I’d like to get your take, as well as others, on my play here.
2/5 nl at the Hard Rock in Miami (well Hollywood, Fl to be precise). Before I get to this hand I must explain my image started off as a nit bc of the way I played my very first hand dealt in the bb w AKo and bc I bought in for the min $200. Half the table limps including utg. I just checked behind as I know nothing. Players are pretty deep. And they look and talk kinda aggro-ish. I thought utg might be trapping. Flop ace high. Checks around three streets and finally button bets $10. I raise to $30 he calls and shows ace. I show AK and say why don’t u bet????? He says he was waiting for high hand. Then the most loud and aggro looking player blasts me for not raising pre, “why don’t you raise pre????” He says. I asked him if that’s what I’m supposed to do as I’m new to the game. Anyhow, after a few orbits table talk states as fact that I’m a nit and I don’t have the balls to do some of the things I did unless I have a hand. For two hours I chipped up to $900 without showdown. About a half hour in, table decides to make it a deep 2/5 only $400/$1200 min/max buy in.
So, we get to this hand. I get 79cc utg+2 and open bet $15. Table is super aggro but I get mega respect when I open. I haven’t shown I’ve gotten out of line and I haven’t gotten too involved either. Folds around and button, who’s the one that said I don’t have balls, calls. Bb 3-bets to $45. Folds to me and I make it $110 to go. The table chatter was helping my image. Button snap folds and acts annoyed. I’m betting he called with ATC just to outplay me. They don’t like nits. I am not one but today I appeared to be one. Anyhow, villain flats in th bb (it’s heads up and we’re about $650 effective). Well that didn’t go as planned but that’s ok. I got position. My perception of the table was that seats 1-4 were on the nitty side and 5-9 were young aggro kids. I was in seat 2. Villain in seat 8. He was relatively new to the table for maybe an orbit or two. Flop comes 268r. Pretty good flop given i have 9 high in a 4-bet pot. Sb checks. I, naturally, cbet here. I don’t think checking is an option. My bet size is $175 into $245ish pot. He calls. Turn is an off-suit J. He checks. I check. I would do the same here with aces and kings. River 6. He checks and I make a small value looking bet of $150 (about 1/4 pot).
By the time we got to the river I ranged him on mid pair no set (77, 99, TT), AQs-AK, and QQ+ highly discounting KK+. He genuinely didn’t look comfortable check calling. I also discounted AQ. My target folds were the mid pairs and AK.
Thoughts?
Is it ok to post a hand here if its relevant to original post?
Very dry flop so you should probably be betting less, especially considering the SPR and the fact that you have position. I think I like something like ~135 on the flop (I’m assuming the 650 effective meant that’s how much you had after him calling the 4-bet).
The turn would then be ~510 and you’d have 510 in your stack, and at that point I think I’m just shipping it. This is especially important if you think that he is c-c OOP with AK/AQ as they will almost certainly fold the turn.
The main thing that confuses me about your post is your crazy nit image combined with the fact that you think you are getting 3-bet OOP (with a call of your 4-bet) by hands like 77. WTF? In any event, you got a great board to barrel off on, and I think that even if you might check behind AA on this turn (which you probably shouldn’t as it makes your barreling range too bluff heavy in this spot), villain doesn’t necessarily know that. Ultimately you want to take the line against this villain that will produce the most fold equity.
You state that checking the flop isn’t really an option, and while I understand the desire to fold out better hands (since you have 9-high), I think with the stack sizes here, you can shove over most turn bets that he makes (and since he is aggro, he is likely to make one). Most live players respect live turn aggression and they are also less likely to peel as there is only one card left. In the event that you get check-called, you can decide whether or not you want to bomb the river based on reads.
As Andrew mentions in his followup post (https://www.thinkingpoker.net/2013/08/optimal-bluffing-frequency-tentative-results/), there are two points in the particular hand that, I now believe, mostly preempt the need for taking full pen to paper:
1) If the Villain has too small a percentage of his range as the nuts, then the situation will be the same as the one that Sklansky described when the Villain holds only a bluff-catcher; the nuts are just the “best bluff-catchers” in this case
2) If the Hero has a high enough percentage of his range as the nuts, as he likely does in this case, then he gets to bluff all his non-nut hands and his opponent, in equilibrium, has to fold all non-nut hands.
I believe that Andrew’s analysis is accurate and complete. Having not considered it from this angle when recording the podcast (https://www.thinkingpoker.net/2013/08/episode-45-mike-stein-of-quantitative-poker/), I floundered around a bit thinking about the game theory of the less-specific situation of two-street nuts-or-bluff spots, so for anyone interested in the mathematics setup for such generalized approaches to a two-street problem, here are my algebra notes…
Assumptions:
– 1 unit in the pot
– 4 units remaining in the effective stacks
– Hero either gives up (never winning at showdown), pots the turn and gives up on the river, or pots the turn and pots the river
– Villain always checks to Hero and either calls or folds, never raising
– Hero’s range consists of only nuts (9x) and bluffs
– Villain’s range consists of nuts (9x) and some number of bluff-catchers (we can think of his flop bluffs or misses as just really poor bluff catchers; the weakest ones will fold to any Hero bet anyway and thus never reach showdown)
– Each player is fully aware of the other player’s ranges
All pretty reasonable in the specific situation that we’ve identified except the last one, but that’s how game theory optimal solutions go.
This sort of analysis is easiest when working backwards from the river in the case where the turn went bet-call. For the purposes of this street, we’ll assume that the ranges which reach the river are fixed parameters m and n, and we’ll solve for the bluff and call variables b and c.
Let m be the percentage of Hero’s river-reaching range of nuts, with 1-m being bluffs. Let n be Villain’s river-reaching range of nuts, with 1-n being bluff-catchers. The river strategy is defined by b, Hero’s bluff percentage (Hero bets all his nuts), and c, Villain’s call percentage among his bluff-catchers (Villain check-calls all nuts).
There are 3 units in the pot when we reach the river and 3 behind. Then Hero’s expected value from bluffing, as a function of Villain’s calling range, is:
(+3) * [P(Villain folds)] + (-3) * [P(Villain calls)]
= 3*[P(Villain has nuts)*0 + P(Villain has bluff-catcher)*(1-c)] – 3*[P(Villain has nuts)*1 + P(Villain has bluff-catcher)*(c)]
= 3*[(1-n)(1-c)] – 3*[n + (1-n)(c)]
= 3*(1-n)(1-2c) – 3n
And Villain’s expected value from calling, as a function of Hero’s bluffing range, is:
(+6) * [P(Hero was bluffing, given that he bet)] + (-3) * [P(Hero has nuts, given that he bet)]
= 6*[(1-m)b / (m + (1-m)b)] – 3*[m / (m + (1-m)b)]
= [(6(1-m)b – 3m) / (n + (1-m)b)]
The equilibrium occurs at the indifference points, that is, the optimal bluffing and calling frequencies will leave Hero to be breaking even on his bluffs and for Villain to be breaking even on his calls. Solving the above equations for b and c in terms of m and n yields:
c = (2n – 1) / (2(1-n))
b = m / (2(1-m))
Note that the equation for b is consistent with the shortcut from Andrew’s previous posts discussing these situations: if Hero bets m / (2(1-m)) of the time when he holds a bluff, since he holds a bluff (1-m) of the time, that means he’s betting m/2 of his bluff combos, i.e. having a bluff when he bets half as often as he has the nuts.
So Hero’s expectation for the river when he holds a bluff is:
[EV(Hero bluffs)*[P(Hero bluffs)] + EV(Hero gives up)*[P(Hero doesn’t bluff)]
= [(3*(1-n)(1-2c) – 3n)*(1-m)*b + 0*(1-m)*(1-b)]
= [(3*(1-n)(1-2c) – 3n)*(1-m)*b]
and Villain’s total expectation for the river when he holds a bluff-catcher is:
[EV(Hero gives up)*P(Hero gives up)] + [EV(Villain calls a Hero bet with a bluff-catcher)*P(Hero bets)]
= [3*(1-m)*(1-b)] + [(6(1-m)b – 3m) / (m + (1-m)b)]*(m + (1- m)b)
Now, moving back to the turn:
Let p be the percentage of Hero’s turn-reaching range of nuts, with 1-p being bluffs. Let q be Villain’s turn-reaching range of nuts, with 1-q being bluff-catchers. The turn strategy is defined by g, Hero’s bluff percentage (Hero bets all his nuts), and h Villain’s call percentage among his bluff-catchers (Villain check-calls all nuts).
Again, we set up indifference equations. So Hero’s expected value from bluffing the turn, as a function of Villain’s calling range and the EV of the river under those ranges, is:
(+1) * [P(Villain folds)] + (-1+[Hero’s EV on the river when he holds a bluff]) * [P(Villain calls turn)]
= (q*0 + (1-q)*(1-h)) + (-1+[(3*(1-n)(1-2c) – 3n)*(1-m)*b]) * (q*1 + (1-q)*h)
and Villain’s expected value from calling the turn with a bluff-catcher is
(-1 + [Villain’s EV on the river when he holds a bluff-catcher])
= -1 + [3*(1-m)*(1-b)] + [(6(1-m)b – 3m) / (n + (1-m)b)]*(m + (1-m)b)
We know what the river range parameters m and n must be as a function of the initial ranges p and q along with the turn strategy variables g and h:
m = p / (p + (1-p)*g)
n = q / (q + (1-q)*h)
This, along with the two turn indifference equations, allows us to solve for the turn strategies in terms of the initial range parameters. This part is ugly and time-consuming, but tractable. So I’ll stop here and leave the rest as an exercise for the curious/insomniac reader.
Thanks for your detailed calculation.
I hope I’m not the only one to have read it, there is a little typo for the first formula:
c = (1-2n)/(2(1-n))
plugging last formulas into this yields quite simple one:
c = ((q-1)h+q)/(2h(q-1))
to see what it looks like, a simple query to Wolfram alpha displays a plot:
http://www.wolframalpha.com/input/?i=%28%28q-1%29h%2Bq%29%2F%282h%28q-1%29%29
for b, we get an even simpler formula
b = p /(2g(1-p))
and a nice plot http://www.wolframalpha.com/input/?i=+p+%2F%282%281-p%29g%29
Now maybe a few realistic values for typical ranges in Andrew’s example would help to visualize the results as poker ranges…
JeffP,
Awesome, thanks! That was indeed a mistake. Nice catch on the simplification too, I had started those by hand but didn’t get very far. I forgot about Wolfram Alpha!
Another typo I had noticed, the n here should be an m:
“And Villain’s expected value from calling, as a function of Hero’s bluffing range, is:
(+6) * [P(Hero was bluffing, given that he bet)] + (-3) * [P(Hero has nuts, given that he bet)]
= 6*[(1-m)b / (m + (1-m)b)] – 3*[m / (m + (1-m)b)]
= [(6(1-m)b – 3m) / (n + (1-m)b)]”