Does the complexity of poker increase unboundedly as effective stack sizes tend to infinity?

Question

You can assume this question to be about no limit texas hold'em, but answers about/including other variants are also interesting. I'm also not sure how to define complexity mathematically in a way that agrees with the following intuition, maybe someone will have an interesting answer anyway.

At small effective stack sizes poker is in practice much simpler than with deeper stacks. This makes sense: smaller stack sizes means less possible bet sizings so the game tree is smaller. However, if we had a game with no blinds, the game would be very simple in the sense that there's a very simple GTO strategy: for example fold every single hand or play only aces by shoving.

Does the game trivialize in some sense as the effective stack sizes tend to infinity? Or does the complexity keep increasing, and only trivializes at infinity?

Answer 1

I'm just an amateur here so take my thoughts with a grain of salt. Considering your question, I don't think the complexity will increase without bound as a result of infinite effective stack size.

What do you accomplish by making the stack size infinite? If you are thinking about able to make infinite different number of bet sizes, then you can already achieve infinite different number of bet sizes with finite effective stack size. Increasing the stack size only increase the number of BIG bet sizes. However, strategies are often similar when the bet size exceeds a certain number of BB, or a certain portion of the pot. Experimenting with GTO solvers will also yield similar conclusions as you include many different, ridiculously large bet sizing options. The solvers will either go with the largest one 100% of the time (or equivalently all-in) or simply ignore these large bet sizing options.

Technically there are infinite bet sizes to choose from, making poker "infinitely complex" if we were to consider every single one of them. However, by grouping similar sizing and even boards, we can narrow down strategies to a dozen scenarios that really matters. Hope that helps!

Answer 2

Defining complexity is very important here:

If you have ever used a solver, you will know that the game tree can grow very quickly if you add lots of different possible bet sizings on different streets.

If we are defining complexity as "the size of the game tree" then the nature of being able to bet any amount on any street means that at any stack size the complexity is extremely large. So large, that it practically wont make a difference above a stack size of ~10BB. Add in variables like exploitation and ICM and there is plenty of complexity to go around.

If we define complexity as "strategic complexity" then I think there is still a lot to go around at any stack size > 10BB. for example, if you plug in a pretty tight range into a solver and take a look at preflop bet sizings for different hands, solvers will often recommend min-raising with very strong hands (along with marginal hands for balance) because they can still play well post flop. There are always optimizations to be made strategically to both exploit different opponents and play in an unexploitable way.

There may be other definitions of complexity, these are the ones that came to mind for me.

As a TLDR: NLHE is very complex at any stack size > 10BB due to all of the different possibilities that practically speaking, there is no difference in complexity between stack sizes.

Answer 3

As explained in a comment, I'll stick to the definitions of "infinite stack poker" as a game where the blinds are zero and there are no restrictions to bet sizes (i.e: if we say a player has a stack of 1, any real number between 0 and 1 is a valid bet size).

In terms of the game tree, this is indeed a very complicated game. At depth 1 we already have an uncountably infinite range of possibilities. But this doesn't mean finding optimal solutions will be harder.

In fact, there are many situations in regular poker where adding more possible bet sizes would change nothing. Let's consdier a spot where the optimal bet is €1.25. If bets with additional decimals are allowed it may result that €1.248906734 is actually a better option, but you won't suddenly find out that there's a better bet somewhere between €1.73 and €1.74

In a no-blinds variant the game becomes even simpler from a strategic point of view as checking is always at least as good as betting (and in fact it's worse every time you don't hold the nuts).

A perhaps more interesting infinite stack poker game would be one where blinds are set 1/2 but players can make arbitrarely large bets without ever being all-in.

Answer 4

Pretty interesting question and a nice thought experiment. I strongly suspect that the complexity will converge at some point, because the number of hands, especially value hands in every spot is limited. When we bet, we want to make our opponent indiffernet between calling and folding (baisc game theory optimal concept). Given a fixed number of value hands, one can choose a bet size and right number of bluffs, such that the opponent will be indifferent. So the question is, can we benefit from an infinite stack aka infinite bet size? In my opinion no, because we can never add more bluffs than value hands to our betting range, else our opponent will be no longer indifferent and will profit from calling with any bluff-catcher.

So overall the game complexity should not change much. Nevertheless there might be some very weird situations arising, where both players have the nuts, e.g. the board is AAAAx and both players hold some K, and they raise each other over and over again... I am not sure, is there a limit of raises?

Answer 5

No. The fact that Aces can only be dealt one in every 216 hands is in some sense a fundamental constant of the game of poker. With enough stack depth, there exist preflop bets which are only actionable with Aces, and which target exactly Kings or other Aces on the razor's edge of a good price. Very deep-stack poker allows for a preflop 5 or 6-bet, where you start to see some of these dynamics come into play.

As the blinds become miniscule, it becomes increasingly important that you have a hand that can become the undisputed best hand (the nuts). There is an 'aces' for every board, and stack depths are such that an all-in will effectively only take place when one has the nuts.

The game is barren and boring at massive stack depth: huge raises and tons of folding, essentially no all-in bluffing on the river and chopped pots when players do go all-in. Since showdown happens so so rarely, hands turn into an auction where players are stratifying their hands, and as the nuts come into frame as a prominent possibility, all bluffs stop.

EDIT: I should say that bet sizes change and ranges will change up to a certain point, so the game definitely becomes different as stack sizes increase. But it does not become more complex per se, nor does it oscillate endlessly through various schema as the stack sizes increase. Eventually, the only thing changing is going to be bet sizing and nothing else.

Answer 6

The complexity of poker does not necessarily decrease as effective stack sizes increase, but it can be affected by a variety of factors such as the specific variant of poker being played, the betting structure, the number of players at the table, and the skill level of the players.

In no limit Texas hold'em, the complexity of the game can increase with deeper stack sizes because players have more flexibility in terms of bet sizing and can use a wider range of strategies to try to win pots. With deeper stack sizes, players can also make more complicated plays such as floating, bluffing, and check-raising, which can add an additional layer of complexity to the game.

However, it's important to note that the complexity of poker can also be influenced by other factors such as the specific variant of poker being played, the betting structure, and the number of players at the table. For example, a variant of poker with a more complicated betting structure, such as pot limit Omaha, may have a higher level of complexity than no limit hold'em, even with deep stack sizes.

Overall, the complexity of poker can vary depending on the specific circumstances of the game, and it is not necessarily trivialized at infinity.

In no limit Texas hold'em, the complexity of the game may increase with deeper stack sizes because players have more flexibility in terms of bet sizing and can use a wider range of strategies to try to win pots. With deeper stack sizes, players can also make more complicated plays such as floating, bluffing, and check-raising, which can add an additional layer of complexity to the game.

However, it's important to note that the complexity of no limit hold'em can also be influenced by other factors such as the skill level of the players and the number of players at the table. For example, a table with highly skilled players may have a higher level of complexity than a table with less skilled players, even with deep stack sizes.

Overall, the complexity of no limit hold'em can vary depending on the specific circumstances of the game, and it is not necessarily trivialized at infinity.