Game theory, as it applies to poker, is a complicated and confusing subject. It's easy to feel overwhelmed when reading an equation-filled book like The Mathematics of Poker or browsing the “solutions” generated by solvers like Pio and GTO Range Builder.
It's important to realize, though, that you don't have to master game theory, or – shudder – memorize the details of solutions to specific situations in order to use concepts from game theory to improve your results. The point of studying game theory is not to memorize specifics, but rather to learn and understand broad principles.
One of the most important lessons I've learned from game theory is that bluff-catching is overrated. That's not to say that it doesn't have a place in your strategy or even that it can't make you money, just that it should rarely be Plan A.
In “equilibrium solutions” - those were both players pursue an unexploitable strategy – bluff-catching tends to be a break-even proposition. That is, unless you can beat some portion of your opponent's value range (in which case I would argue the term bluff-catching no longer applies) or you have some blocker to that range, the only way to make money from bluff-catching is if your opponent bluffs, not just occasionally, but too much – out of proportion to his value betting. With the right hands, it's a freeroll to try, but it's not an especially profitable or desirable spot to end up in.
When facing a bet, the value of your pure bluff-catchers, hands that beat all bluffs but lose to all value bets, is entirely dependent on your opponent's bluffing frequency. Against an opponent with an optimal bluffing frequency, the value of calling with your pure bluff-catchers is $0, the same as it would be if you folded.
If calling and folding both have an EV of $0, you might wonder, why call at all? Remember that an opponent bluffing at a game theoretically optimal frequency is, well, theoretical. It's the assumption that we make in the absence of any better estimate of her bluffing strategy. If it turns out that she has a particularly poor strategy, bluffing with hands that are really too strong to turn into bluffs, then your bluff-catchers that break-even against an optimal bluffing strategy will actually make money. So, calling with those hands is a freeroll, as long as you don't call with so many of them that you end up incentivizing your opponent to stop bluffing entirely.
It's true that if you have reason to believe that your opponent will bluff too often in a given situation, then bluff-catching can show a profit. However, many people tend to misjudge these situations, which lead them to emphasize bluff-catching too much in their strategies.
For example, a “good spot to bluff” is not necessarily a situation where an opponent will be weighted towards bluffs in a way that makes excessive bluff-catching profitable. I hear many questionable calls justified on the grounds that “I thought he was just bluffing the scare card” or “It's a good spot for him to bluff.”
Generally, the reason that scare cards are scary is that they are legitimately good for an opponent's range. While it's true that they may induce an opponent to bluff, what's left out of this analysis is that they also open a lot of value betting opportunities. Because what matters in bluff-catching is not simply how many bluffs your opponent has but rather the ratio of bluffs to value bets, so-called “scare cards” are not intrinsically better or worse than any other card in a way that enables you to profit from excessive bluff-catching.
The inverse mistake is common as well. Many players will mistakenly conclude that an opponent who bets at an innocuous turn card is not bluffing because that card “doesn't change anything.” Because it is not a “scare card”, the logic seems to go, then the opponent would never bluff and therefore must have a value hand.
For many of the same reasons, that argument does not hold water. It may be the case that your opponent will bluff less often on a card that is difficult for her to value bet, but there is no a priori reason to assume that she will bluff never or even infrequently enough for you to exploit her with excessive folds.
Even in cases where you have either a blocker or read that enables you to make profitable bluff catches, these calls still tend to be only marginally +EV. In other words, you'd generally be better off not facing the bet.
When you take a bluff-catcher to showdown, your EV is equal to the chance that you have the best hand, which is generally a value much greater than $0. Once your opponent bets, the chance that you have the best hand generally decreases, and on top of that you must put additional money into the pot in order to see the showdown, which means that your Expected Value has declined. Turning your hand into a bluff-catcher is something your opponent tries to do to you, not generally something you should try to do yourself.
The major mistakes I have in mind when I say “turning your hand into a bluff-catcher” are cases where players take a hand that is strong enough to bet for value and instead check it, claiming that they want to give an opponent “a chance to bet at it”. What we have just seen, however, is that your hand generally does not gain value when facing a bet. If it does, this is usually the result of a really large betting mistake on your opponent's part, not the kind of thing that happens commonly.
Most opponents, in keeping with the general loose-passive profile of a “fish”, are more inclined to make loose (calling) mistakes than aggressive (betting) mistakes. By checking strong hands, you don't only give your opponent a chance to bet – you also give her a chance to check.
Often, both the bet and the check cost you money. When your opponent bets, your hand loses value because it is reduced to a bluff-catcher, even if it is a slightly profitable one. When she checks, it loses value because her checking range will consist mostly of hands weaker than yours, including many that have a non-zero chance of improving by the river. You lose value both by letting those hands realize some of their equity for free and by letting lesser hands that would have reluctantly called a bet get closer to showdown. In other words, you missed the opportunity to take value from your opponent by turning many of her hands into bluff-catchers.
This mistake is particularly egregious when made with extremely strong hands, which can be deceptive because “bluff-catching” with these hands also has the highest Expected Value. That's because they are ahead – often way ahead – even of hands in the opponent's value range and so aren't really bluff-catchers at all. It should not be surprising that these hands have more value than other bluff-catchers: nut and near-nut hands have high EV no matter how you play them (as long as you don't fold). The question is not whether they are profitable bluff-catchers, but rather whether bluff-catching is the mostprofitable way to play them.
It rarely is. Yes, if you check a strong hand, your opponent may mistakenly assume that she has you beat and value bet hands that are worse than yours. The important thing to realize is that these hands would also have called a bet had you made one. It is almost always the case that a hand must be stronger to value bet than to call a bet, so there really should not exist a set of hands that would bet for value if checked to but fold when facing a bet.
Think instead about two other sets of hands: those that will bet but not call a bet (bluffs), and those that will call a bet but not bet (bluff-catchers). If you try to get value from the former, you are at the mercy of your opponent's bluffing frequency. Even if she does bluff sometimes, she would have to bluff often enough to make up for all the value you're missing when she checks a hand that would have called a bet.
Betting your strong hands leaves less to chance. The only thing you have to worry about here is your opponent making excessively tight folds. That's a far less common mistake – again, most players err on the side of being loose and passive – and the sort of player who will fold too often isn't generally the sort who would have bluffed excessively when checked to anyway.
There are times for calling with pure bluff-catchers, but you shouldn't actively seek to create those situations. They will arise naturally, in the course of your pursuing more profitable opportunities.
Suppose you raise A5 in early position and get called by the button. The flop comes A 82. I suggest checking this flop, because you are unlikely to be called by worse. You're not exactly hoping for a bet, which would represent either a bluff or a hand that can beat yours, but calling one should be slightly profitable.
What you're really hoping for is a check, so that you can find a value bet on a future street. After both you and your opponent check the flop, many of the strongest possible hands become less likely for both of you. That means that if the flop checks through, your hand will be stronger, relative to both your range and your opponent's, than it was on the flop. This is true almost regardless of what the turn card is. In the process of trying to create this situation, you may end up bluff-catching, but it's not your primary goal.