We are now well on our way to estimating the players’ turn strategies. We know which of the SB’s hands will be made indifferent to bluffing, so we can now write down the bluffing indifference to find the BB’s calling fre-
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quency. The naive application gave a frequency of about 57%. What extra contributions to the EVs in the bluffing indifference apply to the cut-off hands we identified in the previous section?
11.3.1 K♥-Q♠-3♦-2♣
Take the K♥-Q♠-3♦-2♣ board first. The 9-8o was representative of the SB’s bluffing cut-off here. It has 6 pair outs versus about half the BB’s turn starting range and no outs versus the rest for a total of about 6% equity if it checks down. We need to account for this in our estimate of EV(check). When he does improve, it will not be to a hand strong enough to value-bet. However, he will have no trouble showing down and capturing that equity whenever he wants, since the BB cannot lead the river. Thus, if he checks the turn, he will just capture his small amount of equity in the pot – no more and no less. We can account for this effect by adding 0.06*P to the SB’s EV(check turn).
What about the EV of bluffing with 9-8o? We saw above that this hand gives up essentially all its equity in order to turn itself into a bluff. Thus, the esti- mate of this hand’s EV(bluff) from the naive estimate is still accurate. If it bluffs, the hand wins the pot the FT of the time it gets a fold, and loses its bet otherwise. It is always effectively air on the river, so it should break even on any river action as well, and no extra contribution is necessary to account for the river play. Applying the bluffing indifference, we see
so that the SB’s turn calling frequency here is 1−FT≈54%. This is a bit less than the naive estimate. As we expected, BB has to fold a bit more to en- courage a weak made hand to give up its showdown value and bluff.
11.3.2 K♥-8♠-3♦-2♣ and A♥-8♠-3♦-2♣
The situations on these two boards are again quite similar. Here, hands with one overcard to the eight were indifferent – take Q-6o. This hand has
Nearly Static: Nearly PvBC Turn Play
approximately 10% equity versus the BB’s turn starting range, so we will add 0.10P to the SB’s EV(check turn). If it bluffs and is called, the BB usually has an eight, so the one overcard is good for about 7% equity. We can es- timate the corresponding contribution to EV(bluff turn) by assuming he wins 7% of the larger pot (P+2B) the (1−FT) of the time we get called. Thus, we have
Here, it turns out the extra equity after checking and the smaller amount of extra equity in the larger pot cancel each other out, and we find a call- ing frequency of (1−FT)=57% as in the naive case.
Now, if we hit our overcard on the river, after betting or checking the turn, we have a hand strong enough to value-bet. This is in contrast to the case on the K♥-Q♠-3♦-2♣ board. Here, we effectively assume that we always check down on the river, but in fact, when we improve, the river betting action will increase our EV somewhat. However, this is a relatively small effect, and it applies to both sides of the equation (i.e., after betting and checking the turn), so it cancels out to some degree. We will consider this sort of effect further as we continue exploring the effects of multi-street play.
11.3.3 9♥-9♠-3♦-2♣
In this case, the SB needs to bluff a ton, so much so that his indifferent bluffing hand actually has unfavorable card-removal effects. Take 10-4o, for example. This hand has 6 outs versus much of the BB’s range if it checks down or its bluff is called, for a total of about 10% equity in either case. Making the same sort of adjustments to the bluffing indifference as on the previous boards (and again neglecting the possibility of river ac- tion), we find
so that FT=0.4 and the BB’s turn calling frequency is 60%. The bluffing hand here is essentially a pure draw in that it maintains all its equity when
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called. If the BB only called the 57% of the time that the naive estimate would suggest, this hand would strictly prefer to bluff.
There is an extra catch here that has to do with the card-removal effects. Keep in mind that the bluffing indifference applies to a particular cut-off hand of the SB’s, and so the folding frequency we solve for is the frequency when the SB holds that particular hand. Here the SB’s cut-off 10-4o was chosen for its negative card-removal properties, since the bluffing range needed to be so wide – it blocks some of the BB’s folding range. Thus, the BB is calling more versus this hand than versus the SB’s average turn bet. That is, the BB needs to call with somewhat less than 60% of his turn start- ing range in order to end up calling 60% versus the SB’s 10-4o. The oppo- site case is probably more common. If the bluffing range is relatively tight, the strongest bluffing hand might be chosen for its good card-removal ef- fects, and then the BB will need to fold more of his range on average than he does versus the SB’s cut-off. Anyway, the BB’s average calling frequency here is something close to 57% again.