Long read: The Levelling Spiral and how to grow by running in circles

May 12, 2021inPoker

In this long read, poker author and educator Duncan Palamourdas expands on the theories that start with Sklansky’s poker levels and take you through to a way of thinking that might just help you level up the next time you play.

Duncan is the Author of Why Alex beats Bobbie at Poker: Developing a Fundamentally Sound Approach to Poker where he explores concepts like the one below, responsible for profit flowing from amateurs to professionals. For more details of that, and of Duncan’s other work, read on below. 

In 1999 David Sklansky introduced the world to the various poker levels of thinking in his celebrated work The Theory of Poker.

For those unfamiliar with the concept, they can be summarized as follows:

  • 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

I often like to replace the rather complex third level statement with the more intuitive: “What do I represent?”, which is a more concise way to express what is essentially villain’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

Sklansky’s levels of thinking…

Let’s give a few behavioural 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 opponents
  • 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
  • Level 6+: A world class play design to beat another world class play.

For example, when Alex bluffs by taking advantage of a scary card which is more likely to have improved her own range 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 (namely 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.

This “cyclicality” is exactly what makes poker so complicated and exciting.

To make matters even worse, it turns out that first level thinking is third level’s thinking worst nightmare. It may sound paradoxical, but in a head to head battle, Level 3 loses to Level 1 every time.

But how is that even possible?

Paths and cycles

To understand why this is, let’s talk about “progression” in a general setting.

This is what philosophers also call “motion”, a debated topic with history spanning thousands of years.

Let’s say for the sake of the argument that the way we perceive “progression” is simply by going from point A to point B.

For example, a trip from Los Angeles to London, or cooking a meal, or the passage from say childhood to adulthood.

Notice that points A and B are not always spatial, they can be temporal (relating to time).

The only tricky bit here is not to assume points A and B are different from one another. They often coincide! The classic example is a round-trip from say Las Vegas to somewhere else and then back to Vegas.

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This is what mathematicians often call a “cycle“.

This is to distinguish from the “paths” which are simply all the other progressions where the “destination” does not match the “start”.

Path or a Cycle?

So far so good. But before we fit this back into poker, what if we think of a concept that combines these two labels?

For example, let’s focus on one of the four seasons. In particular, let’s imagine that point A is the summer of last year and point B is the summer of this year.

Is that a path or a cycle?

On the one hand the arrow of time only moves forward, so no two moments in time can coincide with one another. That would make our thought experiment a path.

But it’s reasonable to admit a certain cyclicality due to both starting AND ending at the same season, namely summer. This would then clearly be a cycle, no?

What gives?

Formally speaking, the summer-to-summer progression is technically a path.

However, it is not just any path. It is the sort of path that loops around itself in a way that it creates the illusion of circularity.

Very much like a metallic spring or spiral. Looking at it from the side we can clearly see a path moving up/down. However, when we look at it from the top, it looks like a circle.

To expand on the metaphor, we can imagine that every ring of the spiral represents a single year, while we can think of the seasons as being vertically aligned and always in the same position of each ring (see picture below).

Vertically aligned

Once again, when we see the above spring from the top it looks simply like a circle (see figure 2). This is enough to create the illusion of a cycle, even though it is clearly a coiled path.

Levelling spiral from above

Spirals are more common than we think

The spring-like visual above is more common than we may expect.

For example, think of a hot stove. Let’s apply some of that poker thinking. Would you touch it with a bare hand? It’s rather unlikely (Level 3).

How about a kid that does not know any better? The chances increase dramatically (Level 2).

Now, how about a toddler who is only a few months old and can barely move, let alone walk all the way to the kitchen to touch the stove?

The chances drop again! (Level 1)

At first glance, it seems that both the toddler and the adult are equally safe from the stove.

Are they though?

Because it seems that the toddler is “safe” due to limited capacity, while adults are safe due to a conscious decision to stay clear of the stove, aware of the dangers if they do not.

That difference in causality matters a great deal.

Sklansky’s levels work in all sorts of situations…

For example, when the time comes to clean the stove, the adult can scrub it (and therefore touch it) whereas the toddler is still unable to do so.

The basic principle of thinking at a higher level is that one can always operate at a lower level if need be, but not vice versa.

That’s significant when you’re at the poker table. But first, take another example.

I remember once I was sitting at a dinner table, recovering from a sore throat. Unfortunately for me, the water pitcher was full of ice cubes.

Naturally, I attempted to pour the water carefully to avoid dropping any into my glass. I did this painfully slowly.

One of my friends tried to “help” me by grabbing the pitcher and tilting it towards my glass. Needless to say they “succeeded”, so my glass was now full of much needed water… and very much unwanted ice cubes.

My compassionate friend perceived me as being unable to perform the simple task of pouring water in the glass. That would have put me on Level 1 (sloppy pouring).

They were clearly able to perform that simple task, so they tried to help (Level 2 – skilful pouring).

Level 4 water pouring?

I was instead attempting to perform a rather different and harder task, namely selective pouring. That technically puts me on Level 3 (sloppy selective pouring).

My evident struggle made me look as if I was unable to perform on Level 2 although I was struggling to perform on Level 4 (skilful selective pouring).

From their perspective, the two situations look identical because there is struggle involved in both.

They were thinking on Level 2 because they perceived my action as Level 1. This is after all, Level 3’s biggest nemesis!

However, if they really wanted to help me, all they had to do was to operate on a Level 4 and skilfully help me pour the water but without dropping any ice cubes. Instead, they accidentally and inadvertently treated an adult like a toddler.

There are several other common examples of spiral-like processes, like studying an important poker concept again and again.

Spiral Levelling in Games

At this point, it should be rather unsurprising that Spiral Levelling plays a huge part in games as well.

Perhaps the most classic example is with Rock-Paper-Scissors (RPS).

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…

Start levelling up your RPS game…

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.

Just exactly one Level ahead, no more, no less…

The key here is that it does not matter how high one is in the Levelling Spiral of thinking. All that matters is their relative position on the circle that we get from the top view of the spiral.

This year’s Summer was hotter than last year’s Winter, and it will also be hotter than next year’s Winter.

Similarly, Rock beats Scissors no matter how advanced Alex thinking process is.

That 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 over-thinking it.

There is a very fine line between the two and Alex has to find it!

How this relates 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 Levelling 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 Levelling Spiral is technically applied to every poker hand imaginable.

Regardless, here is a rather simplified example to send the point home.

“Why Alex Beats Bobbie at Poker” by Duncan Palamourdas

A poker 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 big blind.

Like most loose passive opponent, 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 back-door 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 for two minutes 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.

Most profitable live players know that this line is rarely a bluff in a live setting.

Apparently, Alex knows that too, so she is 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.

“I have aces!”

This is the typical Level 1 thinking. And as we have seen before, this meagre level demolishes Level 3 every single time!

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

The answer is rather simple: She should have thought and acted on Level 2.

In other words, she should have tried to establish whether Bob likes his hand or not.

If he does like it even remotely, she should avoid most bluffs and favour thin value betting instead.

She should consider betting three streets with hands as weak as AT or possibly even worse.

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 a few more words, it says: “I am unable to think and perform on Level 2, in order to beat the predominantly Level 1 thinkers that dominate those stakes“.

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

An important Caveat: GTO

Now some of you may think: Wait isn’t the Levelling Spiral an exploitative approach to poker?

And if so, doesn’t that ignore 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.

Very much like the water pitcher story above, 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 constantly misrepresented in the industry as being more relevant to poker success than it is.

GTO strategies are like martial arts. They are nice to know, and they have a lot of side benefits, but we are better off if we are never forced 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 desperate measures of last resort.

Let us not forget their purely 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.

For example, one can clearly see that the GTO strategy for rock-paper-scissors is to randomize one’s choices so that the expected payoff is always zero. However, if Alex thinks that her opponents are capable of mistakes, 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.

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…

You can follow Duncan on twitter @AskTheMathDr

You can order the book directly from the publisher here or directly from amazon.

If you enjoyed the article, you can find similar articles like this by the same author on PokerStars, Upswingpoker, PokerNews, and Card Player.

 

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Author

Duncan Palamourdas

Duncan Palamourdas specializes in the mathematics of poker and poker education. His poker classes at UCLA always fill up early and have long waiting lists. He is also an author at Upswing Poker, Poker News, Card Player and Winning Network. Despite his impressive academic credentials, Duncan’s popularity is a product of his love for simple language and metaphors. Profitable poker play essentially revolves around correct risk-reward optimization. This is a complex topic and Duncan’s great strength is that he explains it in simple terms without resorting to technical jargon.

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