11.- ACCESO A LA JUSTICIA Bolivia

Suppose we are in the BB on the turn in a nearly-PvBC spot. For now, we will bet-bet our value hands, although we’ll find some reasons to get more creative later. However, we face a decision when we hold a potential bluff. SBs who check back the flop with bluff-catchers commonly follow up with exploitable turn play as well. In particular, they do not fold enough to a lead. First, the flop was not very scary, and Villain checked back because he had “something” that he wanted to take to showdown. Then, the king on the turn seemed like the perfect card for any weak showdown-value hand. It could not have improved any hands that were not already ahead, and it even made them less likely because of card removal. (Keep in mind that a reasonable BB pre-flop defending range has a seven or better here over a quarter of the time. I found this number surprisingly high.)

Since the SB is ahead of most of his opponent’s range, folding to a single bet seems weak-tight. Thus, many players adopt a plan like, “start out with a call and then re-evaluate river”. Call-and-re-evaluate is a common approach in many spots where players know their range is narrowly defined as bluff- catchers and also know that their opponent holds many potential bluffs. They call at least once without giving the decision much thought and intend to take a more balanced approach if they face additional action.

How can we exploit this tendency? It figures to make betting our value hands even better if anything, but how should we play our air? From an equilibration exercise perspective, folding on the turn keeps the BB indif- ferent between giving up with bluffs and single barrelling. The more the SB folds, the more the BB is incentivized to take a stab with his bluffs. However, if the SB never folds to one bet, then Hero should never bluff once and then give up. We can save our B chips by simply giving up on the turn, although it could still be best to bluff twice. So, if our opponent tends to check back the flop with bluff-catchers and call a turn bet (at least on certain cards), then the most important thing is that we never bluff once and then give up – we should only bluff with the intention of following through on the river.

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Now, we still need to decide between check-check and bet-bet. This choice depends on how often Villain calls a river bet. If he calls too frequently on both streets, check-check is clearly best. We just have to give up with our bluffs. However, suppose he calls the second bet with a more moderate frequency – suppose he calls just enough to make our bluffs indifferent on the river. What should we do then? The EV of check-check is just S, while our choice to bet-bet means always seeing a river and being made indif- ferent to betting there. On the river, our EV of checking is S−B, and our EV of continuing the bluff must be the same. Since we always get to that spot if we decide to bet the turn, that is the EV of bet-betting on the turn. Since

S−B is much less than S, check-check is still clearly best.

Notice what happened here. If Villain is calling too much on the turn and then “playing well” on the river, we still do best by far to just give up with bluffs on the turn. Even if he is folding a bit “too much” on the river so that we prefer bluffing with any air we do happen to bring to the river after betting the turn, it is still better to not get to the river that way in the first place. We prefer an EV of S to one of a bit more than S−B. This is an impor- tant symptom of multi-street play and shows that it is important to keep the big picture in mind when developing a strategy. If we are in the SB, we cannot decide to start with a call on the turn and then appeal to the single- street river game solutions for help when we face a decision on the river. The BB can exploit this by giving up with his bluffs on the turn, making all our river calls very unprofitable.

For completeness, when can we actually bet-bet our bluffs in the BB? Es- sentially, Villain has to fold enough to make up for our having put B in the pot on the turn with no equity and no chance of winning immediately. We are indifferent between bet-betting and check-checking with a bluff if:

(10.2) So, we prefer to give up with bluffs on the turn unless Villain folds the river at least S/(S+B+P) of the time. This is about 71%, given the sizings in the example, quite a high frequency. By the way, this result should not be sur- prising. We know from single-street situations that the general form of a folding frequency necessary to keep complete air indifferent to bluffing is

Turn Play: Polar Versus Bluff-catchers Redux risk/(risk+reward), where risk is how much we lose when called, and reward

is how much we stand to win. In the single-street case, we find a folding fraction of B/(B+P). Here, we essentially have to risk our whole stack (by way of a turn bet and river jam) to try to win the pot plus Villain’s turn bet, and so risk/(risk+reward)=S/(S+B+P).

Here’s another way to think about this spot. Players often constrain their own play because of their stylistic preferences or ideas about what consti- tutes generally good play. This effectively removes options from the game tree. For example, many opponents constrain themselves to playing min- raise-or-fold pre-flop from the SB 50-BB deep. Additionally, we described above why a SB might always call when facing a turn lead on the K♣-7♥- 3♦-K♦ board. Regardless of whether these are good strategies, it is impor- tant to study situations under these constraints so that we can effectively strategize in these contexts. We can model our two-street, static-PvBC situation where the bluff-catching SB refuses to fold the turn with the al- tered decision tree shown in Figure 10.1. This is effectively the same as Figure 10.1 except that the SB is forced to call when facing a turn bet.

Figure 10.4: Altered decision tree for the PvBC situation on the turn with static hand values in the case where the bluff-catching SB constrains his own play with a “call and re-evaluate” strategy.

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What is the equilibrium of this modified game? Well, Hero bet-bets his nuts, and he bets only bluffs that he can continue with on the river. His betting range on the river makes the SB indifferent to bluff-catching. If he bluffed any less than this, the SB would always fold the river, which would incentivize him to bluff more, and vice versa. Thus, the BB’s strategy is es- sentially the same as in the single-street PvBC game, except that he now bets twice with all of his betting hands.

How about the SB? He only has a river decision here, and his folding fre- quency at this point is chosen in order to keep Hero indifferent between his two-street pure strategies: bet-bet and check-check. This frequency is the one we just found (Equation 10.2) and is significantly larger than in the single-street river PvBC situation. If the SB folds any less, Hero will not bluff at all. In this two-street game, the SB’s river play must make Hero in- different between the two-street lines give-up and bet-bet with bluffs. It is not enough for the SB to just make Hero indifferent to bluffing the river

after he has already put in B on the turn. He needs to fold more to incentiv-

ize Hero to risk that first bet as well.

In other words, if you consider just the river play alone, you notice that the SB is not folding enough to make the BB indifferent to bluffing. It appears that the BB can “automatically profit” by betting bluffs on the river, since he strictly prefers bluffing to giving up, given the SB’s high folding frequency. However, the SB is not actually exploitable in the context of this game, since the BB had to put in the turn bet to get to this river spot in the first place.

Consider the altered PvBC turn game shown in Figure 10.4. The SB is forced to call a turn bet but can play more intelli- gently on the river. Suppose the players are playing the GTO strategies we have just discussed. Is this a bad situation for the SB? If we re-introduced his option to fold the turn, could he use it to increase the EV of his bluff-catchers?

This spot was a bit contrived, since we forced the SB to always call the turn. However, we will see similar effects in more realistic situations. For exam- ple, “floating” is a move where a player calls a bet on an early street with little or no equity, simply because he believes he can take the pot away with a bluff later. Suppose we face a later street bet that we suspect might

Turn Play: Polar Versus Bluff-catchers Redux

be a float. To discourage such a move, we do not have to call with so high a frequency as to make Villain indifferent to the later street bluff after he has already floated, but only enough to make him indifferent to his whole line, which involved risking an earlier bet also. In general, the frequencies suggested by the bluffing and bluff-catching indifferences in multi-street situations can be different than in the single-street case, and some care must be taken when trying to analyze the river in a vacuum. We can often get a better view of a strategic situation by taking a more holistic view. Now, this does not mean all our previous work on the river in a vacuum was wrong. Let’s think about why. The BB arrives at the river spot at the end of Figure 10.4 with the perfect range such that he can bet all of it and make the SB exactly indifferent to calling. This is a rather unique situation! If he arrived at the river with infinitesimally less air, then he could bet his whole range, and the SB would still strictly prefer to fold. This is a break- down of indifference that we covered in Chapter 7. On the other hand, if he held infinitesimally more air, then betting his whole range would make the SB strictly prefer calling. Go through the equilibration exercise, and you can convince yourself that the equilibrium will indeed be the normal one from the single-street models we have seen previously.

The special thing about the solution in our altered game that lets the SB call sometimes but less than in the single-street solution is that the BB gets to the river with exactly the right range to bet all of it and still make the SB indifferent. Essentially, the SB gets away with folding too much (from a single-street perspective), since the BB cannot easily bluff more, because he is already bluffing all the air he brought to the river, and he cannot eas- ily arrive with more, since he has to call the turn bet to do it. The BB’s river starting range in this unique case lies right on the boundary between starting ranges corresponding to those that lead to breakdown of indiffer- ence and those that do not. It turns out that, if we solved this river spot by itself, the SB’s calling frequency would be completely unconstrained – any frequency between 0 and 100 percent would be co-optimal. It’s the addi- tional constraint from the turn spot that fixes things. Thus, our previous river analysis is correct. It’s just that in special cases, play on the river is not completely determined by the river situation alone, and thinking about the larger context can provide more guidance and also inform how the river

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starting distributions should look in the first place.

Furthermore, this situation is unlikely to arise in reality, even if the BB tries hard to choose such a range on the turn. Hand values change some from one street to the next, and particular river cards will make him end up with slightly more or fewer value hands. On cards that complete draws or are particularly good for his range, he may have too many value hands, so that the SB can happily fold with bluff-catchers. On blanks, the BB may have an abundance of potentials bluffs, so that the single-street solution applies. It is even harder to deviate from the single-street solution when the distributions are not exactly PvBC. If the SB is folding too much for rea- sons such as we have seen here, then perhaps the BB will be incentivized to start bluffing with some weak made hands that, under normal circum- stances, would try to show down and not contribute much to either his value or bluffing ranges. This additional source of BB bluffs would moti- vate the SB to begin calling more.

Lastly, suppose Hero plays the role of the SB in this K♣-7♥-3♦-K♦ hand. In practice, opponents’ turn and river bluff-leading frequencies from the BB vary widely – take note of them since many will have very exploitable fre- quencies after we check back the flop. That said, many experienced players will identify the particular turn situation in this example as a poor one in which to make a bluff. They expect the SB to call at least once and often twice with significantly too high a frequency for all the reasons we men- tioned previously. As a result, even players with good average turn bluff- leading frequencies will often actually have too low a frequency on a board such as this one, while perhaps bluffing too often on others. Against such an opponent, Hero can make exploitative, tight folds with his bluff- catchers when he faces a turn lead. It is important to pay attention not only to opponents’ probing frequencies versus missed c-bets, but also to how these frequencies depend on the board.