The previous model showed us how jamming could be reasonable under certain conditions but not how it could coexist with both calling and fold- ing at equilibrium. To see that, we need a model that more accurately cap- tures certain aspects of the game. Now, we could work harder to write down the exact EVs of various holdings, and we will. However, the crucial element turns out to be that the SB has multiple different kinds of bluffing hands – say, pair draws and flush draws as in the previous example, or draws and weak made hands, as we’ll consider here. By folding some and then picking the right balance between jamming and calling when he does continue, the BB can make both types of SB bluffs indifferent. Thus, our work here will not only have consequences for the BB’s protection raising, but also for the SB’s bluff selection. But we’re getting ahead of ourselves. In this section, we’ll look to make more accurate estimates of the relevant EVs and use them to look at the BB’s exploitative raise-vs-call decision. In the next section, we will take a more equilibrium-oriented approach. When exactly does he prefer jamming to calling? It depends on the SB’s
5 The book The Compleat Strategyst by J. D. Williams gives an elementary introduction to
two-player, zero-sum matrix games. Recordings of a Yale University course entitled Game Theory by Professor Ben Polak are highly recommended for anyone who prefers video to text. Both of these high-quality resources are freely available online.
Expert Heads Up No-Limit Hold ’em, Volume 2
betting range, which we will assume is composed of draws, weak made hands, and value hands. We will suppose that draws keep all of their eq- uity and weak made hands lose all their equity after bluffing. This last as- sumption is not great for the previous hand example, where the SB’s weak, unpaired holdings also act like draws. It fares much better on most other boards, where the BB’s ranges will not contain nearly as many unpaired hands, and the SB’s weakest holdings cannot count on pair outs after their flop and turn bluffs get called.
As far as the EVs, here are a few issues the previous model ignored:
1. The BB’s bluff-catchers have a significant chance to improve on the river, even if they go all-in and get called. This makes jamming and calling both better as compared to folding.
2. If there is money behind at the end of turn play, there is potential river action, and it matters. We have seen that bluff-catchers are not generally able to realize much of their equity in the pot when there is another round of betting. This makes calling significantly worse than in the previous model.
Both of these effects make jamming better for the BB as compared to his other options.
We need to write down the BB’s EVs on the turn. Let S, P, and B be the stack, pot, and bet sizes, as usual, and let D, W, V be the fractions of the SB’s bet-
ting range that are draws, weak made hands, and value hands, respec-
tively. In order to account for point 1 above, let EQBB{vs value} be the BB’s
equity with a bluff-catcher versus the SB’s jam-calling range (i.e., his value hands and a few strong draws).
Then, once he faces a turn bet, the BB’s EV of protection check-raising all-in is
Whenever the SB has a bluff, he folds to the jam, leaving the BB a stack of (S+P+B), but when he has value, he calls, and the BB gets his equity in the pot. On the other hand, if the BB calls, his EV is:
Initiative and Less Common Turn LInes
This deserves a bit of explanation, since we stuck something new in here. Generally, we can write the BB’s EV of calling as the amount of chips he has left after the call, (S−B), plus the size of the pot after he calls, (2B+P), times the fraction of that pot that he ends up capturing, which we have called R. Now, if the bet were all-in or if the river were guaranteed to check through, then R would just be the BB’s equity. However, due to the future street of betting, the BB’s bluff-catchers will not be able to realize all of their equity in the pot. It turns out that R≈0.30 – that is, he is able to capture about 30% of the pot at equilibrium. This is somewhat less than the 48.4% equity in the pot that he would win if the players were forced to check down on the river. We will call R the capture factor, because it describes the amount of the pot a player is able to capture. Now, this number might be bigger if the BB can play exploitatively on the river. Also, in general, it will not just be a single number – it will depend on the hand, just like equity depends on the hand – although for now, we can get away with a single number, since the BB holds mostly nearly-equivalent bluff-catchers here. Generally, a first good guess at a hand’s capture factor is its equity, and then we can adjust that estimate according to the type of hand and expected future action. We know that air hands tend to more or less break-even with respect to just giving up, so they will usually capture none of the pot on average – that is, they have R≈0. Nut hands will have R>1, since they win more than the whole pot if there is money left behind to bet. Bluff-catchers will gen- erally capture somewhat less than their equity, as is the case here (here, the BB’s bluff-catchers realize about 0.30/0.484≈62% of their equity). Notice that the chance that the SB’s drawing hands improve on the river has been subsumed into the capture factor. That is, R is not only a function of equity. It also depends on things like playability of all the hands in Vil- lain’s range. In fact, it accounts for lots of effects. In some sense, all of the multi-street effects that are difficult to account for are represented in that one number, so we have not really solved any problems – we’ve just shifted all the hard work into the problem of finding R. That said, there is a lot of practical utility here. More on this later.
The EVs we wrote above did not rely on any equilibrium-based arguments. We can see if jamming is better than calling by determining whether
Expert Heads Up No-Limit Hold ’em, Volume 2
EV(jam)>EV(call). The result does not simplify to anything especially nice, so we will not reproduce it here. Anyhow, it expresses a trade-off between the amount of equity the SB’s draws are folded off of by a jam and the fraction of the pot the BB’s bluff-catchers expect after calling. The more high-equity the draws that the SB folds, the more the BB likes a jam. The more of the pot he captures after calling, the more he likes to call, of course.
The result also depends on the composition of the SB’s betting range. The higher his value-hands-to-draws ratio, the more equity the BB needs to be folding out to find a profitable shove. On the other hand, the more equity the BB has when his jams do get called, the less equity he has to fold out to make shoving good. In our 9♥-2♠-9♦-A♠ example, the SB’s draws need only have modest equity before the BB begins to prefer jamming to calling. Of course, if the SB is bluffing with fewer draws and more weak made hands, the draws he is bluffing with will need to have more equity to make up for it.
What is the general requirement for EV(jam)>EV(call), given the above EVs? It involves a number of parameters. Can you think of a more useful way to express it than as a constraint on the equity of the SB’s draws?