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cused on play within the larger of the two trees in Figure 11.2. This game tree and our discussion apply to the case where one player takes the bet- ting initiative early in the hand, and by doing so, defines his range as rela- tively polar. His opponent just plays “check and guess”, and we have ar- gued that this is reasonable given the distributions. Of course, we know that raises and other changes of betting initiative are important parts of real play, and we will begin to consider them in the next chapter.
Here we saw how the presence of draws affects players’ range-splitting decisions at equilibrium. Given the decision tree, the structure of the SB’s turn strategy looks a lot like the analogous case on the river, the SB bet-or- check game. It is just that the betting thresholds are affected by the possi- bility of changing hand values. The BB’s calling range can be looser or tighter than the naive case depending on the SB’s bluffing cut-off. If the SB’s borderline bluff is a made hand that loses its showdown value by bluffing, the BB calls tighter to make up for it, but if the SB’s indifferent bluff is a draw, then the BB must call the turn more frequently. As for the SB, he value-bets hands corresponding to something like the top half of the BB’s calling range, but on early streets, he can bet somewhat wider for pro- tection-related reasons. Finally, coming full circle, the SB will have to in- clude enough bluffs in his betting range to make the BB’s bluff-catchers indifferent to calling. We saw that naive applications of indifference equa- tions lead to valuable intuition about unexploitable play but can lead to untrustworthy results in practice, and we had to account for some details to obtain accurate strategies.
Along the way, we found a rule for choosing our value bets on the turn, and we saw how the presence of draws modifies the rule of thumb from river play. On the river, with some simplifying assumptions, we can value- bet from the SB if we have at least 50% equity versus the BB’s calling range. On the turn, we found that this requirement is lessened because of draws – essentially, we are motivated to bet because we gain by pushing Villain off some of his equity.
Can we find an analogous rule for exploitative bluffing as well? How will it compare to the river case? Let us try to find an approximate bluffing crite- rion for the turn by comparing bluffing to shutting down. We looked at the
Nearly Static: Nearly PvBC Turn Play
bluffing indifference above, but now we want to find a quick rule of thumb for evaluating the tradeoff between bluffing and checking, à la Sec- tion 10.6.
For convenience, let Es, Ec, and Ef be the equity of the SB’s potential bluff versus the BB’s starting, calling, and folding ranges, respectively. Then, as- suming we just go to showdown once we make it to the river, we have a profitable single-barrel bluff if
Plugging in Equation 11.3 for Ec and rearranging, we find that a bluff is best if
(11.6)
where FT is the BB’s turn folding frequency, as usual. This looks messy, but it is actually easy to understand.
First of all, if we are on the river with pure air, both Ef and Ec are 0, and this equation tells us that we need FT>B/(B+P) to have a profitable bluff. No surprise there. If we are on the river with a hand with some showdown value, Ec is still 0, since we only get called by better, but Ef is not, and we find a profitable bluff when
(11.7)
What is this fraction on the right? Consider the numerator and denomina- tor separately at first. Since Ec=0, Equation 11.3 tells us that FTEf=Es, and so the numerator here equals FT−Es. FT is the total fraction of hands Villain folds to a bet, and Es is the total fraction of his range we are ahead of at the beginning of the street, so FT−Es is the fraction of his turn starting range consisting of hands better than ours that fold to a bet. And the denomina- tor, 1−Es is just the amount of Villain’s hands that are better than ours at the beginning of the street. So, the entire fraction on the right side of Equation 11.7 is just the proportion of Villain’s hands that are better than ours and that fold to a bet.
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to fold at least B/(B+P) of hands better than ours to profitably turn our hand into a bluff. As a quick example, suppose we are considering making a pot- sized bluff in position on the river. If our hand has no showdown value, we have a profitable bluff if Villain folds at least 1/2 of his turn starting range. If our hand has some showdown value, however, he needs to fold at least half of hands that are better than ours so that we can profitably bluff, and this requires him to fold more overall. So, this gives us a good way to think about bluffing on the river that takes into account our loss of showdown value without explicitly comparing EV(check) to EV(bet) 3.
Finally, on the turn, both Ec and Ef in our bluffing criterion might be non- zero – Ef because we have some showdown value if we check down, and Ec because we have some chance to improve if we bluff and get called. So we need to consider the full right side of Equation 11.6. The only difference, in comparison to the river case, which has a very simple interpretation, is the bolded term in the denominator. This term depends only on Ec and necessar- ily makes the right side of the inequality bigger. So, the more equity we have versus a calling range, i.e., the more likely we improve on the river after our bluff gets called, the more we prefer a bluff to a check. No surprise there. In order to see exactly how much bigger, it would be really great if we could write Equation 11.6 in the form of the easily-understood river result plus some extra contribution that depends on Ec, just like we wrote the minimum equity needed to value-bet as the river value ( 1/2 ) minus the protection term. It turns out that we can do that, approximately, using a mathematical technique known as a Taylor expansion. The details of the technique are not important – suffice it to say that, for fairly small values of Ec, we have the approximate bluffing criterion
Again, we see that the more equity retained by our bluffs, the less Villain needs to fold for us to profitably bluff.
There are a number of other interesting, practical questions to be an- swered here, but they are mostly small extensions of ideas we have already
Nearly Static: Nearly PvBC Turn Play
spent plenty of time on, so we will leave the exploration to you. Consider the following questions.
♠
The extra term here is due to any chance our bluffs have to improve if they get called – how big an effect does this have on our bluffing ranges in real spots?♠
How does this bluffing criterion look for the ideal made hand (Ec=0) and the ideal drawing hand (Ec=Es=Ef)?♠
Comparing check-check to bet-check gave us a quick way to get an idea of the effect of draws on our turn bluffing decision. However, to play most exploitatively, we know we need to consider the possibility that bluffing twice is actually best. What is a good rule of thumb for comparing check-check to bet-bet on the turn?♠
How might the possibility of other river action change our bluffing decision?11.6 EV Distributions
Finally, let’s look at some EV distributions to solidify what we have seen. We will focus on the K♥-8♠-3♦-2♣ case. Consider a game tree like the larger one in Figure 11.2, except that all the rivers are explicitly included. In other words, whenever the hand goes to the river, the tree includes river sub- trees corresponding to each of the 48 possible river cards, and the players’ ranges and strategies are potentially different for each of them. We have solved this game computationally, and the slight differences in river play affect EVs of the players’ turn decisions only slightly as compared to our earlier discussion. The primary difference from our earlier estimate is that the SB value-bets a little wider on the turn, since he can win some extra value by playing well on specific rivers. The other issues mentioned at the end of the last section do not come into play because of the fixed starting ranges and decision tree.
Figure 11.3 shows EV distributions for both players’ hands on the turn when players use their GTO strategies. As in an equity distribution, we
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have ranked hands from weakest to strongest according to their equity versus the opponent’s range. Unlike in an equity distribution, we show the hands’ EVs instead of equities at different points in the tree. We have drawn these distributions to include all hands with which the players get to the turn, not just those with which they actually take each action. For reference, the corresponding turn starting equity distributions are shown as insets. These are the same plots that were drawn larger in the second row and third column of Figure 11.1. We have indicated some correspond- ing holdings and numbered them for easy reference.
Figure 11.3: EV distributions for hands in players’ turn starting ranges in
the K♥-8♠-3♦-2♣ spot. Equity distributions are shown as insets for comparison.
The plot on the left illustrates the EV of betting and of checking the turn with all hands in SB’s turn starting range. These are his only two options at his first decision point, so of course, his actual play with any given hand corresponds to whichever EV is higher. That is, his actual EV curve at that decision point can be thought of as a third that is the maximum of the two drawn. Starting on the left side of the plot with the SB’s best hands, betting is clearly best. The top of his value region consists of 2-pair and better hands. The EVs here appear a bit erratic due to card-removal effects. For example, K-8o actually has significantly lower EV than K-3o, since it blocks more of the BB’s calling range. The first plateau in his EV (see arrow 1) cor- responds to his kings as well as nines through jacks. These are all nearly equivalent, since we assumed the BB’s range is capped at second pair. The
Nearly Static: Nearly PvBC Turn Play
SB’s eights, however, are certainly not all equivalent. These fall into the downward sloping section of the EV curves indicated by arrow 2, and the SB’s kicker makes a big difference here.
The two EV curves meet, so that the SB is indifferent between his options, right around the bottom of his 8-x hands. The small group of hands indi- cated by arrow 3 is the SB’s deuces. They have significantly more equity than the SB’s weakest hands, but not all that much more EV. They are also more or less indifferent between betting and checking at equilibrium. Ar- row 4 points to the SB’s draws. The best of these are low straight draws, but the bulk of them are hands with two overcards to the 8. These all have clearly higher EV(bet) than EV(check) (Why is this?) Finally, much of the rest of the SB’s range (see arrow 5) has one overcard to the 8 and is made indif- ferent, while his worst hands have no overcard or straight outs and strictly prefer to just give up on the turn.
Which parts of the SB’s betting and checking EV distributions should we expect to overlap in the three other turn situations featured in this chapter?
Now, how would these EV curves change if the BB were exploitable? Gen- erally speaking, if the BB called too frequently, the EV of checking would stay the same, while the EV of betting would increase for the SB’s better hands and decrease for his worse ones. If he called too little, the opposite trend would generally be true. What effect would these changes have on the SB’s range splitting, i.e., when would it affect which of the EV curves was higher at each point?
The second panel in Figure 11.3 shows the BB’s EV distributions, both at the beginning of turn play (solid lines) and after he actually faces a bet (dotted lines). The first pair of curves looks a lot like the second, just shifted upwards and slightly tilted. Why is this? Focus on the BB’s EVs after he ac- tually faces a bet, since that is when he actually has a decision to make. His strongest hands are eights. These have EV(calling) strictly higher than EV(folding) by virtue of the fact that they are actually ahead of some of the SB’s value-betting range. Most of the rest of his range is made indifferent by the SB’s polar betting range.
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We can get a sense of scale for the EVs here by keeping in mind the stack size at the beginning of turn play, S=21BB, and stack plus pot size,
S+P=29BB, that one player would end up with if his opponent just gave up
on the turn with his entire range. Notice that SB’s value hands are able to capture a lot of equity above and beyond the pot. For the BB, having a bluff catcher is not much better at all than having air once he faces a bet on the turn, and he usually faces a bet.
Compare the EV curves in Figure 11.3 to the corresponding equity distributions. If the players just checked down the turn and river, their EVs would be directly related to their equity. How does the betting affect the players’ ability to capture eq- uity? Which hands end up winning more than the entire pot at the beginning of turn play, on average? Do any expect to end up with less than their turn starting stack?
Which hands is it most important to play correctly to maxi- mize overall EV in these spots? For which hands does the player’s choice not significantly affect his EV?
In this section, we assumed the SB c-bet a polar range and the BB’s check-calling distribution was capped. If the players play these strategies, we can also find the SB’s flop checking range and the BB’s turn starting range after the SB checks. What do the equity distributions look like at the beginning of the turn after the SB checks the flop? Use the methods of this chapter to estimate the players’ equilibrium turn play, assuming the BB does all the betting.