Thanks for all the comments on What’s Your Play? Facing an Overbet on River.
Time for Some Game Theory
It’s probably impossible to find a game theoretically optimal betting strategy for me on the flop, and likewise a calling strategy for Villain, as there are more than two players in the game. The river, however, is a different situation.
It’s quite likely that I’m playing a capped range after checking the turn (which may be sufficient reason not to check). If – and this won’t necessarily be the case, we’ll return to that question – Villain arrives at the river with a mix of nut hands (that beat everything in my range), air (that lose to everything in my range), and hands in between, then this is basically just the AKQ game from Mathematics of Poker, and Villain should probably bet all of his nut hands and all of his bluffs for a size that makes me indifferent between calling and folding with my bluff-catchers.
I hedge a little bit there because even in a straightforward scenario like the one I define above actual poker is a bit more complicated than the AKQ game. Blockers, for instance, can make the actual solutions more complex. Still, this is about as good an approximation as you’ll find for the AKQ game in an actual poker hand.
The biggest caveat is that Villain may not get to the river a lot of air. As several commenters point out, there are good reasons why he might prefer to bet hands without showdown value on the flop rather than check and call with them.
However, that’s not enough to decide that he’s weighted towards value here, for at least two reasons:
- Many of his strongest value hands might prefer to bet or check-raise the flop as well. Frankly, the flop check-call probably eliminates many of the hands that would otherwise be most likely to overbet the river, whether for value or as a bluff, including busted draws, sets, and even strong top pair hands. It’s a weird, uncommon line, which often means it’s likely to be unbalanced. However, if we can’t deduce or predict the imbalance, then game theory still provides a way to avoid playing into his hands.
- One option for “finding bluffs” on the river is to turn a hand with a small amount of showdown value into a bluff. If Villain played a hand like AXhh or bottom pair this way on the flop, he may well conclude that bluffing the river would be higher EV than checking, even though his showdown value was part of the reason he played the flop the way he did.
If I’m roughly indifferent between calling and folding at equilibrium, why should I ever call with a bluff-catcher? It’s an understandable question, but as I argue in my recent article To Catch a Bluff:
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.
There remains the question of whether my particular hand is a “pure” bluff-catcher or a better-than-average bluff-catcher. With only a vague idea of what Villain’s exact bluff or value hands might be, it’s hard to say with certainty which blockers could make my hand better-than-average. I’m inclined to think that having a heart would be bad and that blocking a set of 8s, top two pair, and a turned straight is good, but depending on the composition of his range for arriving at the river, these effects may be small or non-existent.
They Always Have It
My favorite response to this post came in a tweet that, sadly, I’m not unable to find. I mentioned posting a WYP about facing an overbet on the river, and someone responded to the effect of, “I didn’t read the post, but I fold.” This is reminiscent of a hashtag I’m fond of using, sometimes with tongue in cheek: #TheyAlwaysHaveIt.
There are lots of reasons why the average player bluffs at a lower-than-optimal frequency in many situations. One is just a simple fear of losing. Human brains tend to be loss-averse, which means that they tend to over-emphasize the consequences of losing a bluff. Mathematically, if a bluff of 150% of the pot will cause 70% of an opponent’s better hands to fold and will never be bluff-raised, then it will show a profit. However, if some part of your brain cares more about the money you lose when the bluff is called than the money you win when the bluff succeeds, then it may convince you that the bluff isn’t worth attempting.
On top of that, it can be hard to find bluff candidates. It’s usually obvious when you have a hand strong enough to value bet, but it’s not always clear which hands are best for bluffing. That’s especially true in a case like this one, where, as I argued above, bluffing at an optimal frequency may require the Villain to bet hands with some showdown value.
#TheyAlwaysHaveIt logic becomes less applicable the more sophisticated your opponent is. I didn’t have much experience with this opponent, but he seemed more than competent, and I didn’t want to pursue an excessively exploitive strategy against him.
My logic went something like this:
- The base rate for bluffing in this situation is probably too little rather than too much, so applying Bayes’ Theorem should lead me to fold pure bluff-catchers.
- This opponent seems better than most, and the above logic is exploitable, so I don’t want to deviate too drastically from a balanced calling strategy.
- My hand may be better than a pure bluff-catcher because of the blocker effect.
- I call.
Villain had Ah Jh, which makes perfect sense and, to my chagrin, was not a hand I considered. The call kind of feels like a mistake in retrospect, but that may just be results oriented thinking. Thanks for your input, everyone!