uNL HEADER: THIS IS LARGELY THEORETICAL. IF THAT PUTS YOU OFF, DON'T READ IT.
****, that was quick. Seems like only yesterday that I spent the summer writing my
Pooh-Bah post (on hand-reading) – and don't worry, I don't intend to drag this out to the same epic length that post reached. In fact, I already touched on this topic a little in that post – so I'm just going to reference that for the introductory material (before reading on, make sure you know what a capped range, an uncapped range, and a relatively capped range is - all of these are discussed in the link above – some basic knowledge of game theory in poker would also be useful).
It should be noted that I've posted this in the Beginners' Forum not because it's material which is aimed at beginners, but rather because it's my spiritual home on TwoPlusTwo. Some of the material in this post gets pretty advanced, and I wouldn't try and incorporate it into a 2NL strategy, for instance. Although the old maxim 'just go for fat value at the microstakes' is now completely outdated, there are still concept which will likely be detrimental to a strategy.
I'm going to address this in three parts. This is in no way intended to be exhaustive of all that can be said about capped ranges, nor is it immune to criticism – everyone can be wrong – although I don't see a way in which any of this material will be completely irrelevant at any stage. Firstly, the issue of basic exploitation of capped ranges will come up. Then, countering this exploitation will be discussed, leading on to a final discussion of the fundamental implications of capped ranges for the expected value of both players' ranges in a heads-up pot (pots with three or more players are somewhat different, but it can be said that capped ranges matter somewhat less in those pots as when one player is capped, everyone tends to capped to around the same extent). Unlike my Pooh-Bah, some points will be left open-ended. The reason for this varies: in some cases it is because it is more useful for the reader to think about it themselves; in others it's to avoid excessive deviation from the core of the matter; in others still it's because I simply don't know the answer.
As a general guide:
Absolute beginners: read something else, this is going to seem like mumbo-jumbo to you.
Beginners (2NL-5NL): Read the Pooh-Bah and the first section if you feel comfortable with the material on capped ranges therein.
Intermediate players (10NL-25NL): Read the first two sections and the third if you're comfortable with the second section.
Everyone else: all of this should be comprehensible, though this may not be too useful.
I: How to exploit a capped range
(players who beat the upper microstakes or higher may wish to skip this part)
In general, the issue with having a capped range is that your opponent can always have a better hand than the best hand in your range. This has two main implications. Firstly, it means that your opponent can value-bet a part of his range which is much wider than just the nuts as though it were the nuts (at least on the river) in addition to whatever thin value hands he may wish to bet. Secondly, it means that your opponent can bluff, knowing that you can never really have more than a bluff catcher. This has the effect of creating a requirement to make some uncomfortable calldowns (if hero is folding his entire range at any point, it is a mistake, or indicates that a mistake has been made elsewhere in the hand).
So, there are two main ways to exploit a capped range, which are hopefully obvious from the above. The first is particularly effective at the lower microstakes, or against other players who are bad enough to be unaware of their range. This is to bet and barrel very liberally with bluff hands. Often, the case will be that your opponent will end up folding their entire range by the river, and if not, they will likely fold too great a portion of it. This gives you the opportunity to bluff very profitably.
The second is true against anyone when their range is capped relative to yours – you can clearly value-bet with a wide range. So, for instance, in a 1-card poker game, if villain never has better than a J, we can value-bet
at least an ace through a jack, and likely also a ten and possibly a nine. Of course, in some cases, simply bet-bet-betting with all your value hands may not be optimal, but that's not a discussion for this point.
II: How to counter exploitative strategies
For this part, it will be assumed that hero's range consists of cards with value 1 and 0, with villain's range being mixed between cards with value 2 and value 0 (for those of you who are unaware, this is a very simple example of a polarised range). Where such polarisation exists, game theory dictates that based on an opponent's bet size, there is an optimal calling frequency to minimise our opponent's expectation (though if both players play optimally, the bettor's EV with his range is pot and the caller's EV is zero). This frequency is such that our opponent is indifferent to bluffing with any given bluff hand within his range (in our case, any 0 card) – in other words, a bluff has an expectation of zero. For instance, assuming a rakeless game, a pot-sized bluff needs to work 50% of the time to break-even, so we'd need to call 50% of the time.
Assuming we cannot yet identify any imbalance in villain's strategy when facing a capped range (this is often the case), it is suggested that preventing our strategy from being exploitable is a legitimate end. Thus, when villain bets pot, we would call 50% of the time, when villain bets half-pot, we would call two-thirds of the time etc. It should be noted that, in our example, this requires us to have a range which is at least n% 1, where n is the percentage of the time that we need to call. If our range is greater than (100-n)% 0, we have the problem of chopping with villain's bluffs some of the time when we call a sufficient portion of the time, which increases the amount of the time we need to call (presumably until that percentage surmounts 100, though I haven't done the maths and wouldn't know where to begin).
However, most of the time, when you cap your range, you will often have nearly exclusively bluff-catchers in your range (as people tend to bet and raise polarised ranges and cap their ranges by playing passively), so this should not be an issue. Thus, despite the potential problem of having too many air hands in one's range (at which point one either needs to reshuffle ranges to protect the capped range at an earlier point in the hand or make up the EV with other ranges in the hand), most of the time, the solution which obtains at least 0EV (and probably more as it is unlikely opponents will be perfectly balanced) is to be acutely aware of one's own range and to call down a sufficient percentage of it to avoid being exploitable.
III: But what about the EV of ranges?
So let's say we head into a one-street game. Player A has an uncapped range which is x% 2, y% 1 and z% 0, and we have a capped range which is 100% 1. It will be assumed that stack sizes are pot, and that villain is in position and will always check his hands with value 1. Hero checks to villain. How is the EV of each range affected by altering numbers x, y and z?
It is clear that y% of the time, both players have an EV of half pot. If both players are perfectly balanced, as y tends to zero, EV(hero) tends to 0 and EV(villain) tends to pot. Indeed, if both players are perfectly balanced, EV(hero) can be expressed as EV = y(pot/2), while EV(villain) can be expressed as EV = y(pot/2) + (100-y)(pot). However, depending on values x and z as a portion of (100-y), it may be impossible for both players to be perfectly balanced.
With stacks of pot, it is possible for villain to be perfectly balanced if 0 < z ≤ (x/2) (arguably it is possible if z = 0 but the bet size for perfect balance in that case is 0, and in reality nobody is ever checking back the river with their range of 100% nuts).
Thus, where villain has at least twice the number of value hands in his range as he does bluffs, his expectation is maximised (and ours minimised) by y tending to zero. In real terms, hands with a value of 1 in this spot are essentially showdown value hands which are not strong enough to value bet. As villain's range gets stronger relative to ours (i.e. the difference between the top of his range and the top of our range increases – it should be noted that this is not using the traditional equity-based definition of strength), y decreases (it should also be noted that in practice he can increase his expectation with stack sizes of pot where x/2 > z by turning some y into a bluff, but the maths for this is more complex), and thus the expected value of villain's range increases and the expected value of ours decreases.
Therefore, in terms of hypothetically optimal players, where such ranges exist from the beginning of the game, the greater the difference in strength, the greater the difference in EV. This should come as no surprise – I will be the first to admit it is a common-sense conclusion – but sometimes it is useful to remind ourselves of a couple of misconceptions. Firstly, some believe that having a capped range is necessarily 0, or more ridiculously, negative EV (it is impossible for a player to necessarily have <0EV with his range for obvious reasons when one considers the EV of folding). Secondly, that having a capped range is necessarily a bad thing and should be avoided. Although it decreases the EV of one part of hero's range during a hand, it must be noted that we have been considering one street in isolation, and in reality, having a capped range normally indicates that hero has had a strong betting range with a likely correspondingly high EV on a prior street. It must be remembered when considering game theory that the game theoretically optimal solution to poker is not necessarily to be perfectly balanced in every spot. Finally, of course, a note should be added that exploitative strategies should be taken where they are available.
Thanks for reading.
