If a player chose an aggressive action (a bet or a raise, as opposed to a check or a call) in the most recent spot where he had the chance to do so, then we say that he holds the betting initiative. For example, if the SB raises pre-flop, c-bets the flop, and the BB calls, the SB has held the initia- tive throughout the hand, up to the beginning of turn play. Traditional wisdom says that the player with the initiative either thinks he has a strong hand or is betting to represent such a hand. Players without the ini- tiative may respect that and check to the player with the initiative at the beginning of each street. Continuing with our example, the BB will usually check to the SB on the turn. It would be somewhat unusual if he decided instead to steal the initiative by leading the turn. On the other hand, if he checks but then the SB declines to follow up and also checks, nobody really has the initiative at the beginning of river play.
Initiative and Less Common Turn LInes
Betting initiative can be understood in terms of the players’ equity distribu- tions in common spots. We have seen that it often makes sense to play ag- gressively with strong hands and bluffs but not with intermediate holdings. So, aggressive play tends to polarize the aggressor’s range at the same time as it earns him the betting initiative. Then, by virtue of polarization, he keeps the initiative – we have seen that his bluff-catching opponent will usually play passively. In some sense, betting initiative is illusory. Many players place significant constraints on their strategies to respect initiative, despite the fact that these constraints are not enforced by the rules of the game. However, it does describe a pattern that emerges out of the solutions to very complex games. GTO strategies often respect initiative.
The multi-street decision trees we have used so far are ones where the owner of the initiative does not change (i.e., only one player bets). This was justified by early-street action that polarized one player’s distribution. In Chapter 11, we saw how draws and varied hand strengths can affect the range splitting within the context of this decision tree, but the set of lines we allowed was limited to check-check, bet-check, and bet-bet for the polar player, and fold, call once, and call twice, for his opponent.
These are the most common lines in real turn and river play because of the initiative-seizing, range-polarizing dynamic we just described. It is fair to think of any hand that breaks outside of this pattern as unusual. If the bet- ting initiative switches from one player to the other, something at least moderately uncommon must have occurred. Perhaps a player slow-played an earlier street and is thus able to find a raise later, or perhaps a card came which significantly affected relative hand strengths. More mundane reasons for initiative switches boil down to hand protection (which also has to do with the possibility of changing relative hand strengths). There are two primary ways for the initiative to change owners, and it is useful to separate them, since they give rise to very different spots. We will refer to these as type P and type M initiative switches. In the type P case, one player seizes the initiative. The BB can do this by leading a street when the SB holds the initiative, and both players can do this by raising a bet. A type M initiative switch is much less dramatic: one player voluntarily gives up the initiative. For example, if the BB 3-bets pre-flop, and his flop c-bet is
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called, but then he checks on the turn, he has given up the initiative. Note that when the SB gives up his initiative, we see another street immedi- ately, but this is not so for the BB.
Find a friend and play a session where you agree that neither of you can steal the initiative on the turn or river. How does this affect your turn and river strategy? How does proper early street play change?
Type P and M switches differ in the sorts of equity distributions involved. Suppose the BB holds the initiative at the beginning of turn play (as in the K♣-7♥-3♦-K♦ hand to which we will return shortly). Consider first a type P switch wherein the SB aggressively seizes initiative. Since he is in position, he can only do this with a raise. We saw what this looks like on the river in Section 7.3.3. In particular, Figure 7.19 shows the distributions after a BB bet and SB raise. The BB’s initial betting range was fairly polar, and the SB’s rais- ing range was even more so. Thus, we have an absolutely polar range run- ning into another absolutely polar range. In the model situation, the players essentially swapped places in a nearly-PvBC dynamic. Now, the situation is not always so dramatic on earlier streets. When draws are possible, there are some protection-related reasons to raise without a polarized range, as we will see. And, of course, if a raise is called, new cards can come and they can mix up hand values. In any case, a type P initiative switch involves two abso- lutely polar distributions running into each other – thus the label, P. A Type M switch involves ranges composed primarily of mediocre hands running into each other, and M stands for mediocre. This tends to happen when one player voluntarily gives up the initiative. For example, suppose the SB checks back the flop and calls a turn lead, and then the BB checks the river. In this case, the originally-bluff-catching SB still has a range of mediocre hands, and the BB’s failure to continue betting indicates that he is also unlikely to have a strong hand. Perhaps the BB holds one of the few medium-strength hands in his range – these might overlap significantly with the SB’s bluff-catchers. Alternately, perhaps the BB just has air that is giving up on its bluff. Thus, the BB’s equity distribution here might look a lot like that in Figure 7.10(c). The top part of it is a lot like the symmetric case, but he has some air in his range as well.
Initiative and Less Common Turn LInes
This discussion gives us a good framework for organizing the various possi- ble later street lines. First, we have those that involve no initiative switching and are contained in the PvBC tree. Then, there are lines that indicate that something unusual happened and the initiative changes hands. Both types of initiative switches frequently involve relatively similar ranges running into each other: polar vs polar or mediocre vs mediocre. When players hold similar ranges, we’ll tend to see changes of initiative, i.e., raises and re- raises, as in the symmetric distribution river games. In nearly-PvBC spots, especially on the river, we can expect one player to do most of the betting. In the next section, we will examine the exact solution to a two-street, nearly-static, nearly-PvBC situation, and we will see that the players do break out of the PvBC game tree with significant frequency. We will need to figure out why the (bluff-catching) SB might bet when checked to on the turn and how the BB should adjust his play in response. We will consider some other “outside the box” lines in this chapter as well. As you read over a description of the computational solution, pay special attention to spots where the initiative changes and how the subsequent action proceeds af- ter each type of switch.
12.2 K♣-7♥-3♦-K♦ Part Deux:
The Computational Solution
Let’s look at unexploitable turn-onward play in our K♣-7♥-3♦-K♦ example. We first considered this spot in the context of the ideal PvBC game. The players begin with 30-BB effective stacks, the SB opens to 2.5BB pre-flop, and the BB calls. The flop checks through, and we find ourselves at the be- ginning of turn play with a pot of 5BB and the turn starting distributions given in Section 10.4.
What does our approximate game tree look like? In our study of river play, we were able to tune the decision trees by hand to include the options that made the most sense depending on the players’ early-street play, river starting distributions, etc. However, most of our computational models
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describing play from the turn onward will contain hundreds of different river spots, and hand-tuning each of them is inconvenient. Thus, we will use some general rules for generating decision trees. The following para- graph describes this approach.
Our turn approximate games include both turn and river play. Whenever turn play ends without a fold or all-in, 48 different rivers are possible. Each of these is included in our game trees and is dealt with the correct fre- quency. The bet and raise sizing options are generated as follows. When a player has the chance to bet, he can always go all-in. He can also use each of the following sizes as long as it is at least a min-bet and at most 6/10 of an all-in: 1/2 P, 3/4 P, P, and (3/2 )iP for any positive integer i, where P is the
size of the pot. In other words, the over-bet sizings are 1.5P, 2.25P, 3.375P, etc. Additionally, we give the BB the option to block-bet 1/5 P at the begin- ning of each street and the SB the option to min-raise when facing a bet. Thus, the players have a wide range of strategic options available to them. We will use these bet-sizing options in all the computational calculations we present in this chapter and the next unless otherwise noted.
The turn starting distributions in our K♣-7♥-3♦-K♦ hand are shown in Fig- ure 10.3. The BB is fairly polar. He has some near-nut hands (mostly kings), some other high-equity holdings that are ahead of all of the SB’s range but which are not quite as invincible (his sevens). Overall, about a quarter of his range is a value hand. Then, he holds some hands with mediocre eq- uity, some of which are draws, and finally a sizeable portion of his hands need to improve to have much chance of winning. What do you expect equilibrium play to look like here?
Let’s see some results. First, consider the BB’s first turn decision point. He bets out about 80% of the time. This is more than we predicted when we applied the ideal two-street PvBC solutions to this spot. He also uses bigger bet sizings. In particular, the BB almost never uses his options to bet the size of the pot or smaller. Instead, he goes 3/2-pot with about three- quarters of his betting range and 2.25 times the pot with the rest. Recall that the GGOP sizing, although a bit larger than the pot, is smaller than both of these choices. To see why he bets this way, it is helpful to look at the holdings he plays with each sizing.
Initiative and Less Common Turn LInes
The BB’s strongest hands, and many other value hands as well, use the smaller sizing which is closer to the GGOP choice. This is desirable, since GGOP sizing essentially lets the bettor include as many bluffs as possible in his betting range while still keeping bluff-catchers indifferent. On the other hand, his range for using the larger sizing is capped – the strongest hand is a seven. These weaker value hands are also ahead of all of the SB’s range on the turn, but they only have about 90% equity. In other words, there is a small but significant chance they lose on the river. These hands have some- thing of a tradeoff. It benefits them to leave enough behind to make a sig- nificant river bet when called in order to bluff as often as possible. However, unlike the BB’s very strong holdings, they also gain from going a bit larger on the turn in order to prevent some of the SB’s hands from realizing their equity. We have seen that GGOP sizing is the multi-street analog to the river over-bet. On the river, nut hands can over-bet jam with impunity, while thin value hands often do better by betting smaller to ensure they get called by worse. Here, protection effects can make more vulnerable hands bet larger for protection, while the nuts take a closer-to-GGOP approach.
Why can the BB get away with capping his second value-betting range? On the river, if our value range’s best hand has 90% equity, then 10% of Villain’s range is effectively the nuts and can punish us with re-raises (neglecting card removal and chopping). Here, however, the SB’s 10% comes from the chance to improve on the river, not from any nut holdings that can happily re-raise on the turn. However, his capped betting range does result in his getting pretty well crushed by some rivers. For example, suppose the SB calls a 2.25P turn bet, and then an ace comes on the end. The BB’s previously po- larized range becomes about half air and half mediocre showdown value, while the SB now holds almost all pairs, many of which are rivered top pairs. In this case, the BB must almost always check the river, and he very often faces a bet when he does. Nonetheless, this is a relatively rare contingency, and it turns out that it does not happen often enough to motivate the BB to include any stronger hands in his 2.25P turn betting range.
The BB’s turned flush draws are almost all included in his smaller-sized betting range. Why does he bet them, and why does he use the smaller sizing to do so?
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How should the SB react when facing a bet? His considerations here are of the sort we focused on in the last chapter. He has to continue enough to make some of the BB’s low-equity air hands indifferent to checking. When the BB checks one of those hands, it gets to see a free river about half the time, and it improves on about 12% of rivers. So, when these hands try to check and get to showdown, they might expect to capture about 6% of the pot. However, the river action gains them a bit extra, so let us say they ex- pect 8% of the pot when they check. On the other hand, when they bet and get called, they mostly lose that bet, but again they have a bit of equity (in a larger pot) and a bit extra EV due to the river betting where they will be part of a relatively polar range.
How often should the SB continue when facing the BB’s 1.5P and 2.25P bets here? 4
Now, back to the BB’s first decision point – his checking range contains some nuts and some air (i.e., weak draws), but the majority of it is made up of those relatively few hands that do not fall into the nuts-or-air cate- gory, i.e., his weak showdown value. These include most A-x hands, which are both ahead of and behind significant portions of the SB’s range. The kicker has a big effect on these hands’ equities. The BB’s A-2 has about 20% equity while his A-10 has well over 40%. There are two reasons for this. First, the higher kickers are more likely to improve the BB to a winning hand if they pair up on the river. More significant, however, is the fact that A-x makes up a large portion of the SB’s range as well, and on most rivers, kickers will play.
How does the SB respond to a turn check? Of course, it is helpful to know what the distributions look like following the BB’s check – these are shown in Figure 12.1. The SB has about 66% equity on average. He holds no strong value hands nor any air, but some of his holdings are certainly better than others. The SB’s range at the end of flop play was composed of nearly equivalent bluff-catchers, but the BB starts the turn by checking primarily with his own mediocre hands, and this makes the distributions closer to symmetric than they were previously.
Initiative and Less Common Turn LInes
Figure 12.1: Equity distributions after the BB checks the turn in our K♣-7♥-3♦-K♦ example.
Now, it turns out that the SB bets a bit over half the time and checks back the rest when facing the BB’s turn check. Most of his hands here play with a mixed strategy – sometimes betting and sometimes checking. However, he does bet a somewhat polar range, considering the constraints of his starting distribution. Hands stronger than A-J and weaker than A-6 tend to bet, while hands from A-8 to A-10 tend to check.
We saw that, in a pure static PvBC situation, the bluff-catching player will never bet. After checking the turn, the BB always held air that was giving up on the pot, the players checked to showdown, and the SB won the pot with his bluff-catchers. Here, unlike in the model, the BB’s weak hands do have some chance of improving to win. This motivates the SB to bet to just take down the pot – a plan that works out great if the BB holds only bluffs that plan to check-fold. This is a protection-type effect reminiscent of the one that affects the polar player’s value betting decision as well. Of course, the possibility of a SB bet can tempt the BB to check value hands on the turn. We will discuss this dynamic in more detail in the next section. The SB always uses his smallest sizing option – 1/2 pot – when he bets here. It turns out that if we give him some smaller sizing options, 1/2 pot is still his favorite, although he uses a slightly smaller bet sometimes. If he
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goes much smaller than 1/2 pot, holdings with two overcards to the seven, such as 10-8o, begin to have the correct implied odds to call, which is non- ideal. In the next section, we will talk about some of the reasons behind these bets, and we will see why a small sizing works well.
Facing a bet, the BB folds over half his checking range, and the rest of his ac-