Why Do Random & Silly Bluffs Work?

Have you ever wondered why pure bluffs work when they do?
By pure, I mean situations where the player who is betting has absolutely nothing (not even a draw) while the player who ends up folding has something decent, say at least a pair.
Let’s consider the following situation in a live $5/$10 cash game with $1000 effective stacks in California. Action folds to Alice who raises to $30 with Q♥ J♥ from middle position and gets called by Bob from the BB. Alice has played with Bob and knows he is a positionally unaware loose-passive older gentleman who likes to play a lot of pots.
The flop comes (pot: $65): Q♠ 8♠ 5♦
Bob checks and Alice fires a $40 bet to get some value with her top pair. To her surprise, Bob comes over the top with a hefty $200 raise!
The action is back to Alice who thinks this is a very uncharacteristic line for Bob. He rarely raises without a strong hand like a set (ex: 55), two pair (ex: Q8s) or at the very least top pair with a strong kicker or a slow-played overpair (ex: AQ or AA).
Of course, the fact that Bob made a gigantic raise (five times bigger than Alice’s initial wager) is not lost on her either. This could mean that Bob is either trying to force her out of the pot or that he is afraid that another spade may come and thus his monster holding could lose to a flush. Incidentally, Alice does not think that Bob has a draw because, if he did, he probably would have called and waited to complete it before putting more money in the middle.
All in all, it appears that Bob is polarized, that is he is either bluffing or he has a very big hand. However, since in her estimation, a bluff seems unlikely in this spot, she decides to calmly fold her top pair and live to fight another day.
Without skipping a beat, Bob — having won the pot — jumps out of his chair and slams his holding on the table face up even though he knows very well that he didn’t have to. T♦ 2♦!
“I had the Brunson! I could not resist!”
Alice immediately smiles and politely taps the table saying “Very nice hand sir!” Of course, Alice knows that she got bluffed out of the best hand. But how could she possibly have known that ahead of time without seeing Bob’s cards?
The truth of the matter is, she could not have, and there is probably no way she ever would in a spot like this. Bob was likely to do the same thing with all the strong hands described above (sets, two pairs, etc.). Since there are many more legitimate hands than bluffs that Bob could have in this spot, Alice knows that she made the right decision in the long run. This means that if a similar situation occurs in the future, she will correctly fold again.
Having said that, Bob showing his cards is very useful to her. Alice now knows that T2 (suited, at least) is a hand that Bob likes to play, so she will certainly include it to his potential holdings next time she dissects his range!
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Thursday January 01, 1970

Okay, so far so good. But let us go back to our original question:
Why did Bob’s big bluff work so well even against a great player like Alice?
The answer is because it is rare!
In other words, Bob’s bluff is the exception, not the rule. If Alice had observed that Bob bluffs too much, she would never have folded her hand. The only reason Alice folded is because she knows that Bob does not bluff enough in that spot! Let’s do some quick math to confirm this, assuming that Alice’s assumptions are correct. According to her, Bob could have had one of the following hands: AQ, Q8s, 88, 55, or T2s.
Let’s also do a combo analysis on those. Taking into account that Alice already sees the Q♥ in her hand and the Q♠, 8♠, 5♦ on the board, the remaining combinations should be:


Thursday January 01, 1970

AQ: 8 combos
Q8s: 2 combos
88: 3 combos
55: 3 combos
T2: 4 combos


Thursday January 01, 1970

All in all, Alice loses to 8+2+3+3=16 combos while she beats only 4. That is exactly 4 to 1 against her. Since we established that the pot odds were close to 2-1, it should be clear that her failure rate is much higher than her reward. The math says she should fold!
It is also worth noting that Alice does not need to know what hands Bob is bluffing with. All she needs to estimate is how frequently he bluffs. So as long as Bob bluffs with less than 8 pure bluff combos (which is the threshold where her failure rate would match the pot odds: 2-1), she should fold every time! Exactly as she did.
Looking deeper
Technically speaking the above ratio (4 to 1) is slightly inaccurate as it does not take account Alice’s equity in the pot. In actuality Alice’s hand (Q♥ J♥) has around 25% equity against Bob’s range described above. This means that her failure rate is only 3 to 1. That being said, this is not exactly accurate either, as Bob may not always let Alice see both the turn and river cards in order to realize her full equity. All in all the ratio 4 to 1 is a pretty good estimate of the situation.
Incidentally, even if Alice thinks that Bob plays some of his better draws the same way, her decision to fold should not change. This is due to the fact that combo draws have a ton of equity against one pair hands and in some cases, they can be favorites.
For instance if we were to add the following hands to Bob’s range {J♠T♠, J♠9♠, T♠9♠, 9♠7♠, 7♠6♠, 7♠5♠, 6♠5♠, 6♠4♠, 5♠4♠, 5♠3♠}, Alice’s equity would only improve to 35%, which means that she will barely break-even at best, and that’s only if Bob let’s her see both of the remaining cards!
Population Tendencies
To conclude, Alice should fold because Bob does not bluff enough in situations like these. Despite what televised poker may indicate, experience shows that low-frequency bluffers, like Bob, are not the exception. Rather, they are the rule in the vast majority of poker rooms across the country and even more so at lower stakes.
Typically, players do not bluff nearly as much as they should. As a result, their betting actions are on average much closer to being “honest” (i.e., value hands) versus being “dishonest” (i.e., bluffs). This is what I like to call the Honesty Principle. An abbreviated version of it would be:
The Honesty Principle: On average and in the long run, Poker is an honest game.
By “honest,” I mean more honest than dishonest, in the sense that bets and raises are closer to being value oriented, rather than bluff oriented, as described above. An equivalent way of stating the principle is the following:
The Honesty Principle: As a whole, the poker community bluffs much less than it should.
(Note: The “poker community” includes every single person who plays poker, from the most inexperienced to the best player in the world.)
By “should,” I mean as dictated by what is called the game theory optimal strategy of the game (or GTO for short). That is a foolproof strategy which guarantees (at a minimum) a fair share of return for the player who employs it regardless of how their opponents play. The caveat is that we do not know what this optimal strategy is, although we know that it exists thanks to the theoretical work of the great mathematician John Nash. That said, we can approximate this strategy “locally”, by focusing on certain specific segments of the game, like the river for example. What we see from experience is that humans on average bluff much less than they should. In other words, humans are imbalanced in a way that favors value bets over bluff bets, with the gap being wider with less experienced players.
That’s all for part 1 of this series. Part 2 will be out next week.
If you want to learn the adjustments you should make given that the poker community under-bluffs, check out this article.
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