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leveling in poker

Leveling: How Overthinking Costs You Money at the Poker Table

In 1999 (and then later in 2006) David Sklansky in his celebrated book The Theory of Poker introduced the world to the various levels of thinking in poker.

For those unfamiliar with the concept, they can be summarized as follows (slightly renumbered from their original version to include purely irrational actions):

  • Level 0: I do not even care about my own cards.
  • Level 1: How strong is my hand?
  • Level 2: How strong is their hand?
  • Level 3: How strong do they think my hand is?
  • Level 4: How strong do they think I think my hand is?
  • Level 5: How strong do they think I think they think my hand is?
  • Level 6+: And so on.

Since the above is a bit hard to read, I often like to replace the rather complex third level statement: “How strong do they think my hand is?” with the much more intuitive: “What do I represent?“, which is a more concise way to express what is essentially our opponent’s perception of our hand.

The above sequence then simplifies to the following:

  • Level 0: I do not even care about my own cards.
  • Level 1: How strong is my hand?
  • Level 2: How strong is their hand?
  • Level 3: What do I represent?
  • Level 4: What do they represent?
  • Level 5: What do they think I represent?
  • Level 6+: And so on.

Let’s give a few behavioral examples of actions that fall within these categories to get a feel for them:

  • Level 0: Someone who goes all-in blind every hand.
  • Level 1: A play based on one being “married” to their own hand (good or bad).
  • Level 2: A fit or fold line which “believes” the opponent’s story.
  • Level 3: A sophisticated bluff which takes advantage of a scary texture.
  • Level 4: A play that “reads between the lines” and catches a bluff successfully.
  • Level 5: When Alex thinks that Bob will expect her to bluff, so she value-bets instead.

For example, when Alex bluffs by taking advantage of a scary card which is more likely to have improved her own range, rather than her opponent’s, she is thinking at Level 3, because she is using the board texture to represent strength.

Similarly, when Bobbie is trying to bluff the river simply because he cannot win at showdown (i.e. his hand is not strong enough) he is only thinking at Level 1. Interestingly enough, to the untrained eye they both seem to be doing the same thing (bluffing) although their justification varies greatly.

As we will see, this is not at all coincidental as these levels have an innate cyclicality to them. As a matter of fact, this “cyclicality” is exactly what makes poker so complicated and exciting.

To make matters even worse, it turns out that facing 1st Level thinking is the worst nightmare of 3rd Level thinking. It may sound paradoxical, but in a head to head battle, Level 3 loses to Level 1 every time. You have to be exactly one level above your opponent for this type of thinking to be effective.

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Spiral Leveling in Games

At this point, it should be rather unsurprising that Spiral Leveling plays a huge part in games. Perhaps the most classic example is with Rock-Paper-Scissors (RPS).

rock paper scissors leveling graphic

Rock Paper Scissors (By Enzoklop – Own work, CC BY-SA 3.0)

In that game, it is a rather well known fact that most beginners lead with Rock. This is known as “Rookie’s Rock” (Level 1). This fact alone creates a very interesting dynamic. For example what should Alex play in each of the following scenarios?

  • Level 2: Alex expects Bob to lead with Rock
  • Level 3: Alex expects Bob expecting her to lead with Rock
  • Level 4: Alex expects Bob expecting her to expect him to lead with Rock
  • Level 5: Alex expects Bob expecting her, to expect him, to expect us to lead Rock

A little bit of thinking reveals the following answers:

  • Level 2: Paper
  • Level 3: Scissors
  • Level 4: Rock
  • Level 5: Paper

For example, at Level 2, Alex simply expects Bobbie to lead with Rock, making Paper the obvious strategy for her. Next, at Level 3 Alice assumes that Bob is aware of the Rookie’s Rock strategy so she thinks he will try to take advantage of it by leading with paper. Thus, the proper strategy for her would be Scissors.

Similarly, at Level 4, Alex thinks that Bob is expecting an attack from Alex against his expected Rookie’s Rock. In other words, Bobbie is expecting Paper from Alex and thus he will likely lead with Scissors. If that’s the case, Alex should lead with Rock, effectively completing a full circle! And so on…

So what happens if say Alex thinks at Level 3 but Bobbie is not a sophisticated RPS player who simply leads with Rock (Level 1), like most other people? The answer is simple, Alex’s Scissors will lose to Bobbie’s Rookie Rock. Effectively, Alex has overthought the situation to her own detriment.

Once again, Level 1 beats Level 3! Notice, that if she simply thought at just one level higher than Bobbie (namely Level 2) she would have chosen Paper instead and she would have won.

Exactly One Level Ahead, No More, No Less…

The key here is that it does not matter how high one is in the Leveling Spiral of thinking. All that matters is their relative position on the circle that we get from the top view of the spiral. Rock beats Scissors no matter how advanced Alex’s thinking process is.

That being said, having knowledge of her opponents’ strategy is useful because she can then outsmart them by thinking at exactly one Level higher than they do. She does less than that and she immediately falls behind, she does more than that and she is now overthinking it. There is a very fine line between the two.

Relation to Poker

This brings us back to poker.

Why does any of that matter to winning players like Alex? There are several reasons.

For starters, the Leveling Spiral defines the game of poker in a fundamental way, very much like it defines a simpler game like Rock-Paper-Scissors.

Moreover, it sits at the core of what makes poker a game of skill. Anticipating how her opponents are likely to behave is exactly what makes Alex a winning player. In other words, the Spiral is the cornerstone of most winning strategies. Understanding how it works and how to take advantage of it, is what differentiates casual from professional players.

I find this idea so incredibly fundamental to the game that I have a really hard time choosing a practical example from poker which would be representative enough to do it justice. The issue is that the Leveling Spiral is technically applied to every poker hand imaginable. Regardless, here is a rather simplified example to send the point home.

Hand Example

Say the game is a live $5/$10 with 100 BB’s effective stacks. Alex opens from the Button to $30 with 7 6 and her very loose and passive opponent Bobbie calls from the BB.

Like most loose passive opponents, Bob is more or less thinking at Level 1, namely he primarily cares about his holding and nobody else’s. Alex on the other hand, is a thinking player who is both positionally aware and cognizant of board textures.

In any case, the flop comes A 5♠ 4♠ and Bob checks. Alex fires $45 into a $65 pot with her open-ended straight draw and her backdoor flush draw. Bobbie calls rather quickly. The turn is the J♠ making a spade flush possible and Bob checks again. Alex this time bets $120 to rep the flush and fold any weak pairs. Bob tank-calls this time.

Finally, the river is the 7♣ and action checks to Alex yet again. Without skipping a beat, fearless Alex bets $300 into an almost $400 pot. Bobbie now tanks before he eventually makes the call with A♣ 2♣ and wins the hand with top pair, no kicker!

What exactly happened there? On the outside, it may seem that — given the action — Bobbie made a rather terrible call with his pure bluff-catcher.

With her bluff, Alex was trying to “sell” what she believes to be a rather convincing story of strength. In other words, Alex is thinking on Level 3 (What do I represent?). Problem is, Bob is not Alex and he does not necessarily know what Alex does, nor does he pay attention to the story that unfolds before his eyes.

Bobbie, may very well be thinking: “I have a pair of Aces!” as if this is the only thing that matters in the hand. This is the typical Level 1 thinking. And as we have seen before, this meager level demolishes Level 3 every single time!

Ok, but what could have Alex done differently against that type of player?

The answer is rather simple: She could have thought and acted on Level 2. In other words, she could have tried to establish whether Bob likes his hand or not. If he does like it even remotely, she should avoid most bluffs and favor thin value betting instead.

If Bobbie is loose enough to give Alex action on a texture that wet with only top pair and no kicker, there is a ton of value to be had for the better part of Alex’s range.

Of course, that is only if Alex is smart enough to wait to hit that higher part of her range and then do the necessary betting! This is easier said than done of course.

There is a rather crude joke that pokes on the inability of newer player to beat the lower stakes by making them sound like this:

I want to move up in stakes where they will respect my raises!

What this claim really says is that:

I am unable to adjust

In other words, any discomfort to perform against thoughtlessly “sticky” opponents, reveals an inability for thin value betting and also painful big folds (which are equally necessary when the roles are reversed).

An Important Caveat

Now some of you may think: Wait isn’t the Leveling Spiral an exploitative approach to poker? And if so, doesn’t that neglect Game Theory Optimal (GTO) strategies, which are so popular lately? Yes and yes!

Except that the silent assumption here is that I perhaps neglected to consider them or simply forgot about them. GTO has been carefully considered but consciously not included in the basic picture of profitability of winning poker.

As someone who has studied and taught the topic for years, I feel that it is often misrepresented in the industry as being more relevant to poker success than it actually is.

GTO strategies are like martial arts. They are great to know and they have a lot of benefits, but we are better off not having to use them. This reminds me of Sun Tzu’s famous quote:

The best way to win a fight is to avoid it.

Similarly, GTO strategies should be viewed as something we don’t want to have to use. Let us not forget their defensive and conservative nature. They allow for someone like Alex to “lock” a specific piece of the pot for herself, no matter what her opponents do.

This means that unless her opponents make a mistake, that piece will be zero (or less if we take into account the rake). For example, one can clearly see that the GTO strategy for RPS is to randomize one’s choices so that the expected payoff is always zero.

However, if Alex knows the types of mistakes her opponents are making, why not trying to fully take advantage of those mistakes by using exploitative strategies?

Admittedly, on occasion Alex is up against some tough competition, so perhaps some GTO ideas may bail her out, in the sense that she may apply them to either lose the minimum or get her fair share at best. Or, she can use GTO strategies when she doesn’t have any reliable information on which to base exploitative strategies.

Luckily, other than very few exceptions where she has little to no choice on who she is going to compete against, game and table selection can go a long way in terms of creating a profitable environment for her. Perhaps Sun Tzu was up to something after all…

Important Editor’s Note: It’s crucial to not mix up exploitative adjustments based on reliable information with circular thinking based on assumptions.

If you’ve ever heard someone say something like “you should fold because no one would ever bluff in that spot,” you are already familiar with circular thinking. Watch the 4-minute video segment below for an explanation on the dangers of circular thinking.

The author of this article is Dr. Duncan Palamourdas, whose new book on developing a fundamentally sound approach to poker is now available for preorder. Preorder Why Alex Beats Bobbie At Poker here!

Want to read more from Dr. Duncan? Read his fantastic article series This Is How You Should Study Poker If You Want To Win In 2020.

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About the Author

Konstantinos "Duncan" Palamourdas

Duncan is a math professor from UCLA who specializes in the mathematics of poker, as well as in poker education. He currently teaches poker classes at UCLA extension that always fill up early and have long waitlists. In his book Why Alex Beats Bobbie At Poker, he uses simple language to scientifically explain how and why money flows from poker amateurs to professionals. Preorder the book here!

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