Facing a bet on the turn, the BB can call, raise, or fold. The bluffing indif- ference, essentially, tells us that the BB’s bluff-catchers must sometimes be folding and sometimes not-folding at equilibrium. However, he can make the SB’s bluffs indifferent through either of his not-folding actions. When exploring equilibrium play in SB barreling spots previously, we assumed not-folding meant calling. However, under certain conditions, the BB pre- fers jamming to calling with his bluff-catchers facing a turn bet.
If the BB stops calling on the turn, and jams instead, what consequences does it have for the SB’s strategy? Certainly his value hands still do well to bet. And, at equilibrium, he will still bluff. However, his choice of bluffing hands changes. If the BB is playing jam-or-fold versus a bet, then the SB’s EV of bluff-folding is the same with draws and weak made hands – he
Initiative and Less Common Turn LInes
never sees a showdown, so his particular holding does not matter. The EV of checking, however, is better for draws, since they will be worth more on the river. Thus, versus a BB who is not flatting bets, the SB should fill out his bluffing range starting with those that capture the least of the pot on the river. This is different than what we found when the betting initiative was not allowed to change hands.
So, we can think of the turn-protection raise like this – generally, the polar SB will bet the turn with value and bluffs. He will fill out his bluffs starting with his draws, and the BB will play check-and-guess. However, if condi- tions are right for re-raising – stacks are short enough, bluffs have equity they can be pushed off of, the BB’s bluff-catchers have some outs versus the SB’s value, etc. – then the dynamic changes. The BB prefers to play jam- or-fold versus a turn bet. But, the SB will then be motivated to bluff with weak hands instead of draws. Then the BB’s jams lose their protection ef- fect. The SB is no longer pushed off draws, and the BB is incentivized to re- turn to playing call-or-fold versus a bet. And so on.
Thus, nearly-polar, nearly-static turn play will often proceed as before, without any initiative switches. However, if the SB’s using draws to fill his bluffing range makes the BB want to jam rather than call the turn, then the nature of play will change. The BB will be motivated to raise for protec- tion. (We can check when the BB might want to incorporate any jamming into his strategy using EV equations such as the ones in the previous sec- tion.) However, he can’t play strictly jam-or-fold, or else the SB’s response will motivate him to call rather than jam. Thus, at equilibrium, he will be indifferent between all three of his choices. Similarly, at equilibrium, both the SB’s draws and his weak made bluffs will be indifferent to bluffing. He cannot bluff all draws, since the BB’s counter-adjustment (playing jam-or- fold) will incentivize him to deviate, and similarly he cannot bluff only weak made hands.
Now that we have a few indifferences whose origin we understand, we can solve them to find some GTO frequencies. Because BB calling and jamming have the same effect on the SB’s weak made hands, while folding has the same effect on SB’s draws and weak hands, we can break down the process of finding the BB’s frequencies into two steps. First, he can make the weak
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made hands indifferent by choosing an appropriate folding frequency. Then, within his remaining not-folding hands, he can adjust the ratio of calls to jams to make the draws indifferent as well.
If we are in the BB, we can choose our folding frequency to make the SB’s air indifferent to bluffing and checking. How- ever, his EV of bluffing draws is equal to his EV of bluffing weak made holdings only when all of our not-folds are jams, i.e., when we do not call any. How can we still make his draws indifferent to bluffing when some of our not-folds are calls?
Now, we can solve for the BB’s GTO frequencies, and we will use numbers corresponding to our 9♥-2♠-9♦-A♠ example. In that hand, the SB has essen- tially two types of bluffs. He has weak, unpaired hands with pair draws, and he has flush draws that have significantly higher equity. The BB will call, fold, and raise so as to make both types of draws indifferent to bluffing. The SB EVs here will be similar to those at the beginning of this section, just with a more detailed accounting of the various possibilities corre- sponding to the deficiencies in the first model, which we listed above. A number of equities are necessary to write down the EVs here. Let:
♠
EQSBcheckdraw=0.34 be the equity of the SB’s flush draws if they check
back the turn (i.e., their equity versus the BB’s turn starting range).
♠
EQSBsemibluff=0.30 be the equity of the SB’s flush draws if they bet and
get called (i.e., their equity versus the BB’s check-calling range).
♠
EQSBdrawallin=0.32 be the equity of the SB’s flush draws if they bet the
turn and then call an all-in.
♠
EQSBcheckweak=0.16 be equity of the SB’s weaker bluffs if they check
back the turn.
♠
EQSBbluffweak=0.14 be equity of the SB’s weaker bluffs if they bluff and
get called.
♠
EQSBjam=0.96 be the equity of the SB’s value hands after they bet and
call a jam.
♠
EQSBcall=0.88 be the equity of the SB’s value hands if they bet and get
Initiative and Less Common Turn LInes
The first two equities, EQSB
checkdraw and EQSBsemibluff, are equal in the simple
case where the SB’s draws keep all their equity after bluffing. The fifth, EQSB
bluffweak, would be 0 if the SB’s weak made hand bluffs lost all their eq-
uity after bluffing. Here, however, they keep most of it. The last two equi- ties correspond to the SB’s value hands. They should be close to 1 but are somewhat less due to the BB’s chance to improve on the river. The SB has various value hands, and his draws are not all perfectly equivalent either, but these numbers are chosen to be representative. Notice that the SB’s value hands have more equity when facing a jam than a call, since the BB check-calls all of his slow-played three-of-a-kind.
Now for the EVs. How should we account for the river action? To good ap- proximation, we can write the EVs of any SB hands that are air on the end as if there were no river action, since they will be indifferent to bluffing. The SB’s value hands, however, gain from the river betting, so we need to find their capture factor. Strong value hands will expect to win the whole pot plus another bet whenever the BB calls a river bet. If the river bet is of size b into a pot of p, we know the BB’s calling frequency will be something like p/(b+p). So, a value bet will capture approximately p+[bp/(b+p)]. (Pre- viously, in neglecting the river action, we just awarded him the pot p.) Di- viding by p, we find that the fraction of the river pot captured by the SB’s value hands is
(12.3)
Note that this has the same numerical value whether the turn checks through and the bet is 6BB into 8BB on the river or whether a bet goes in on the turn leaving 15BB into a pot of 20BB – these two cases could have been different if we were not using GGOP sizings. So, the SB’s value hands win more than the whole pot on average, as expected. The closer the situa- tion is to the true static PvBC case, the better approximation this will be. In reality, at equilibrium here, the SB’s highest-value hand, A-A, has a capture factor (CF) of about 1.42 after the BB calls a turn bet. K-9o, a very strong hand that is still vulnerable to the flush draw, has a CF of 1.27. Finally, the SB’s A-x hands have a CF of only about 1.13, since they are vulnerable to some draws and lose to the BB’s slow-plays. So, our approximation looks to
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be right on the money for the pure nuts, but the CFs of non-nut value hands fall off fairly quickly.
As for the SB’s flush draws, we will assume that when they improve on the river, they will get 1.43 times the pot, just like the SB’s other value hands, and we will assume their equity is the chance they improve on the river. The same goes for the SB’s weak hands – it is just that they improve to value less frequently on the end. With this in mind, we can write the SB’s EVs. We have
In these two, the capture factor accounts for the value the SB gets when he improves on the river. Now, his EV of value-betting is just a case analysis – the amount of chips he ends up with after each of the BB’s possible re- sponses, weighted by the frequencies with which the BB takes each action. The additional value the SB achieves with a river bet is again accounted for by Rvalue here, and so is the fact that the SB does not always win the hand if
he gets called or jammed. Finally,
We assume here that bluffs fold if they face a turn all-in while semi-bluffs have the odds to call it off. Notice that we have neglected the BB’s slow- plays in writing down these EVs. We have seen before that if there are few of these and stacks are sufficiently shallow, they will affect profits but not players’ strategies.
We have accounted for each of the deficiencies in the original model listed above. The fact that the BB’s bluff-catchers have some chance to win on the river is incorporated into the SB’s various equities. The SB’s two differ- ent types of bluffing hands are modelled explicitly. Finally, we have ac- counted for river action, as well, using the capture factor.
Now, we want to find the BB strategy that makes the SB indifferent to bluffing with each of his holdings. That is, we want to find C, J, and F such
Initiative and Less Common Turn LInes
that
We have three equations and three unknown variables, so we can solve. Plugging in for the equities and sizings in this hand, we find J=0.14, C=0.42, and F=0.44. These numbers agree with the computationally-generated equilibrium to within a percent. The result is quite sensitive to the amount of equity had by the SB’s draws facing an all-in, EQSB
drawallin. The higher the
equity, the more the BB needs to jam to keep draws indifferent to semi- bluffing the turn.
In summary, the BB’s primary turn plays after check-calling the flop with bluff-catchers are fold, call-call, and call-fold. Generally, he is indifferent between all of these at equilibrium: when he checks the turn and faces a bet, they all have an EV of S. However, even in the ideal PvBC game, as stacks get shorter and shorter, his EV of jamming the turn gets closer and closer to his EV of calling. Several additional factors can push it over the edge: the equity of the SB’s bluffs, the BB’s equity versus the SB’s value, and poor playability of bluff-catchers on the river. Remember that in multi- street situations, a polar player can include many bluffs in his betting range. In the example above, he bet more than one bluff for each value hand on the turn, even with relatively modest bet sizing. If a jam makes the SB fold all of these bluffs, each of which had a significant amount of equity, then the result can be a large win for the BB.
Generally, a turn-jamming BB will motivate the SB to re-compose his bluff- ing range and begin using his weakest hands to bluff rather than his draws. Another option is for him to change his bet sizing. We have seen that GGOP sizing lets him include as many bluffs as possible in his betting range. By deviating and, say, going larger on the turn and smaller on the river, he can decrease his number of bluffs and thus make the BB’s jams less profitable. This is part of the reason we saw the polar player bet larger than GGOP with much of his range in the solution to the K♣-7♥-3♦-K♦ spot.
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The BB’s turn jamming range is composed of his best bluff- catchers, generally speaking: e.g., K-Qo in this section’s hand example. His slow-played nines do much better at equilibrium by continuing to call down on the turn. Why are his best bluff- catchers better than mediocre ones for raising, and why do the actual value hands prefer to call, given the BB’s EVs we wrote down above?
Suppose you are building a double-barrelling range in posi- tion. How would you choose your turn bluffs if the BB tends to play call-or-fold? What if he frequently check-raises?
This discussion foreshadows strategies we will encounter on earlier streets and in larger model games. We’ve seen that GTO strategies will often make Villain indifferent to certain decisions. We will often be able to do this more effectively when we allow ourselves more strategic options. We saw in the previous chapter that in SB barrelling situations where the BB can only call or fold the turn, he can generally only make one sort of SB hand indifferent to bluffing. Here, however, since the BB has two independent frequencies to adjust (his calling and jamming frequencies), he is able to make two sorts of holdings, weak hands and draws, indifferent to bluffing.
Now consider the real game, or at least a larger model. Suppose we’re making a flop c-bet. Not only can Villain adjust calling and raising fre- quencies on the flop, he can also adjust tons of subtle details of his strat- egy on all later streets. Maybe he can make our bluffs indifferent by fold- ing enough to the c-bet, while making our draws indifferent by changing how much he pays off on run-outs when draws come in, while making strong made hands indifferent by playing aggressively enough versus missed c-bets, etc. Thus, we can expect to see lots of indifferent hands (and thus very highly-mixed strategies) in early-street play.