Pot Limit Omaha Poker Strategy for a New Generation

So-called ‘Artificial Intelligence’ approaches to poker do not play by using an intelligent understanding of the game. Instead they largely rely on using a self-play algorithm, iterated over vast numbers of hands, to build lookup tables that store near-equilibrium action frequency data for game trees that the algorithm has explored. They then use those lookup tables to generate an output action corresponding to the current state of the game as recognized by an algorithm designed for that purpose.

The problem is that you are not a machine, and so you have no chance of learning lookup tables for games which are not trivially small (Tic-tac-toe you can handle). So, in order for you- a human- to play strong poker, you’re going to need to rely on your understanding of the game and a different set of skills.

So what are the skills required to thrive in large, complex games?

[Note: I am aware that there is more to AI approaches to poker than just self-play. This article is concerned with the skills humans need to compete in PLO poker games, and is not intended as a prelude to a substantive discussion of AI.]

This article will explain the importance of having a strong, meaningful conception of PLO and demonstrate use of structural analysis to derive a fundamental strategic concept. We will see that the power of pattern-recognition heuristics is determined by the relevance of the insights used to develop them.

Poker is a game where history matters- both the ‘image’ that your opponents have of you, and the sequence of actions that led to you facing a particular situation should influence any non-trivial decision that you make. In Pot Limit Omaha you will face a far greater variety of preflop action histories to get you to the flop than you do in No-Limit Holdem.

Open-limping is still commonplace, especially in live games, and players tend to call (and overcall) raises wide- alert to the manifold opportunities to hit the flop inherent to holding four hole cards, which are most obvious in double suited starting hands. As a consequence you should expect to play many more multiway pots- which makes naïve solver-imitation approaches less useful than in simpler games like No-Limit Holdem.

The reason for this is simple: solver-imitation approaches rely on modeling the real game tree with a pruned game tree and using a self-play algorithm to derive near-equilibrium strategies for that reduced game tree. The imitator then attempts to substitute mimicry of the machine for a meaningful understanding of the game. This makes such approaches highly effective in games with small stack-to-pot ratios and few, if any multiway post-flop spots because it is possible to build pruned game trees which closely resemble the real game tree.

The table below shows how the number of flop decision nodes for SPR 10 grows as we increase the number of players in the pot on the flop:

For this evaluation I only included a single, pot-sized bet or raise as an alternative to checking or calling for each player. In practice, of course, strong play relies on utilising a variety of bet sizes, calibrated for that particular context.

As you can see, the sheer number of decision nodes makes a solver-imitation approach to understanding PLO extremely unwise for anybody seeking to play well in multiway pots at high SPRs- situations which characterize a large fraction of the contexts that you will find yourself in at the table. If you yourself have been starting your PLO analysis with a solver and have been disappointed by your results, then you have first-hand experience of the limitations of a computation-driven approach to a complex game.

That is not to say that solvers are not an extremely useful tool- the Cardquant Research Institute has its own in-house solver to assist my work on strategy development. However I start my analysis with a conceptual approach that places structure ahead of computation and this approach guides any solver work rather than the other way around. (You can read more about causal-conceptual versus statistical-computational approaches in my philosophical writing).

You see players making conceptual errors at the poker table every time that you sit down to play. One of the most common is inaccurate play with hands which block the nut hands for the current board state. In PLO poker, blockers to nut hands are highly significant in certain contexts- far more so than in No Limit Holdem. Nowhere is this more relevant than on flushing or monochrome boards, like this one:

T♥8♥2♥

When you only hold a flush 5.5% of the time on the flop, as you do in No-Limit Holdem, the fact that the A♥ blocks the nut flush has little significance to practical play at commonly contested SPRs. However in PLO, well-constructed ranges flop a flush on a monochrome flop 22-25% of the time, and a range of width 25% will flop the nut flush 12% of the time and the bare nut blocker 10% of the time. As a consequence, both the nut and the second nut flush blockers should regularly take aggressive lines on both flushing boards and on those boards which contain a flush draw.

Yet whilst blockers to nut hands are easy to identify, recognizing the contextual features which make it appropriate to play them aggressively is not. A weak concept of blockers and their relevance is the source of countless mistakes in practical play. I am particularly fond of this example: the common, yet astonishing ‘bluff-shove’ at an SPR of between 1 and 2 on a flop of:

A♠Q♥J♦

with bare Kings (say K♠K♥6♥4♣). Your opponent’s mistake here is a conceptual one- he has applied the idea of ‘nut straight blockers are good bluffing hands’ to the wrong context. It his failure to correctly identify the context that is the source of his error. If you have the good fortune to play against very aggressive, inexperienced players in a live game you will encounter many shoves of this nature.

The alternative to an attempt to play PLO by imitating a poker solver is to let a structural approach to the game guide our understanding. A structural approach to the game enables us to reduce the effective complexity of the game by choosing a strong metric to classify situations according to their most relevant features. In my theoretical work I refer to well-defined contexts as reference classes.

The art of generalizing strategies from a set of reference classes to unique situations without excessively sacrificing precision is central to my work on Pot Limit Omaha strategy. I then combine Pattern-recognition Heuristics and Action-generation Heuristics to teach you how to recognize these reference classes, and to think clearly about the most relevant considerations as you make decisions at the table.

We have established that for a human to learn a strategy to play effectively in a complex game, that strategy must be formulated conceptually.

If you have read or watched other introductions to the game you will most likely have been exposed to narrative descriptions which place an excessive emphasis on ‘Nut hands’ and feature a nebulous discussion on the importance of ‘Playability’. These descriptions are close to useless as a guide to practical strategic play in any game that is not populated by beginners. The problem is that these two phrases are introduced as concepts to guide your thinking, yet utilising these as starting elements for thinking about the game masks the fact that both of these ideas, to the extent that they are defined, are themselves contingent upon the context that the hand is played in.

This means that prior to analysis, even prior to defining the concepts that should guide our thinking, and certainly prior to computation we need to identify and categorize contexts into useful reference classes. A structural approach to the game, as developed here at the Cardquant Institute, begins by doing exactly this. What follows is a simple example of how a useful concept can be derived using methods of structural analysis.

In the bar graphs below we compare between two games the 5-card hand types that a random hand will make on an unpaired board by the river. The graph for No Limit Holdem is on the left, and the graph for Pot Limit Omaha is on the right.

Notice that a random hand in No Limit Holdem makes one pair or weaker almost 80% of the time on an unpaired board! In contrast a random PLO hand makes two pair or better more than half of the time by the river on such boards. However the most interesting statistic is that a random hand will make a straight or a flush almost one quarter of the time by the river on an unpaired board, which compares well with the 11% frequency that such hands are attained in a No Limit Holdem game.

The consequences of the increased frequency of straights and flushes are profound- and the first of these is that the value of a hand on the flop in Pot Limit Omaha is far less strongly coupled to its immediate made hand value.

To illustrate this point, let’s continue with a concrete example. Consider a flop of:

K♥8♠7♠

and compare the frequencies with which a player will hold certain hand types between No Limit Holdem and PLO poker.

From the tabulation of this example you can see that draws to flushes and straights play a much more prominent role on most boards in PLO poker than in No Limit Holdem. This means that the reference classes that players use to evaluate hand strength in No Limit Holdem are inappropriate for PLO poker. Any player that attempts to transfer the reference classes that are so familiar to him in No Limit Holdem to his play in PLO will make consistent errors that stem from his faulty conceptualization of the game. The irony is that this faulty conceptualization is likely derived from a solver-imitation approach to No Limit games!

In fact PLO necessitates regular use of an ability generally underrated in professional poker- imagination.

Whenever we see a flop in any community-card poker variant our hand’s showdown value is not its current showdown value, it is the showdown value of the set of hands we can expect to make on the river, weighted by their respective probabilities. (In fact the process of visualizing your future hands is relevant even preflop, but I’m keeping it simple here for what is intended as an introductory article.) What makes PLO a special game is not that ‘equities run closer’, but that future showdown value is less strongly coupled to flop made hand value and, as a consequence the relevance of different hand categories shifts as we proceed down the game tree.

This continual shifting of hand relevance makes constructing strong ranges far less trivial in PLO than in No Limit Holdem, which in turn opens up many more opportunities for exploiting imbalances in your opponent’s range.

In this article I have made the case for building a meaningful conception of PLO to guide your play rather than imitating a machine-generated solver solution (or imitating an imitator of such solutions). I then gave an example of how faulty concepts lead to inaccurate play and concluded the article with a demonstration of the power of a structural approach for building useful concepts.

If you made it through the entire article and understood it, even if it was after the second or third reading, then congratulations. This is not a bite-size article and I haven’t ‘explained it to you like you’re 5’. Yet the discipline and intelligence that enables you to understand the sophisticated approach that I present here will stand you in good stead for learning a more powerful framework for PLO and developing your own game.

If you want more Pot Limit Omaha Strategy, then read Beyond the Solvers: How to Evaluate Straight Draws in Multiway Pots next.