“If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts, he shall end in certainties.” - Francis Bacon, The Advancement of Learning

When your daily working life begins and ends at a computer screen it’s all to easy to be sucked into the maelstrom of clickbait, trends, and sensationalist Youtube videos. In the poker world the emphasis on solvers as some kind of Oracle is great for quickly generating content and has led to many poker players submitting to the idea that the best investment of their time is to become amateur data scientists.

And the solvers certainly are seductive! The rows of cards, numbers and percentages offer an apparent certainty that is comforting in the otherwise rocky, shifting game that is PLO. Yet the results that solvers give are very specific, and highly contingent, and meanwhile every day players commit errors at the PLO tables that would be easily avoided with a broader understanding of the structure of the game.

In the passages that follow, I present an analysis of straight draws in multi-way pots, which provides the justification for a number of action-generation heuristics that you can use to immediately improve your play at the tables. I am confident that many of you will learn more of practical use from this article than you would from several hours of self-study with a solver. The acute reader will understand that my choice to discuss a multi-way pot is no accident, but elaboration on that topic will have to wait for another article.

[Disclaimer: Not only have I worked extensively with solvers myself but I have, with my team, developed my own in-house solver. They are certainly wonderful tools and I applaud those developers of commercial solvers for their work. The purpose of this introduction is to draw attention to the reader’s investment of his time as he works to improve his PLO game.]

What is my Straight Draw Worth?

A regular opponent opens from the CO, and the BT and SB call. We elect to overcall with most of our continuing range here, rather than squeeze, and the flop comes down as:

The CO bets into the flop, the BT calls, and the SB folds. The questions that we are exploring in this article are,

How valuable are hands with straight draw components here?
What effect does the bet size have on our decision to continue in the hand?

First, some statistics detailing range interaction frequencies on this board. If we consider a range of {JJ,TT,66,JT,J6,T6,hh,KQ,Q9,98} a ‘hit’, then at least one of the four players in the pot will hit this board 97% of the time and at least two 77% of the time. (For the Hold’em players here only one player would hit the board with this range 53% of the time and two players 14% of the time in that game. This should give a player new to PLO some idea of quite how poor his intuition will be in this multiway situation.)

If we instead consider a tighter range of {JJ,TT,66,JT,J6,Ahh,Khh,(KQ,Q9,98):hh,KQ9,Q98,987,KQJ,KQT,AKQ} as a ‘hit’, then at least one of the four players in the pot will hit this board 86% of the time and at least two 46% of the time. If your instinct is making you feel queasy about the prospect of straight draws on this board given these numbers then your instinct is serving you well. For those of you who are thinking, “Three players to the turn is standard here,” the next part of my analysis will serve as a revelation.

Let’s start with the generous assumption that the CO initiates with this entire subset of his range and the BT continues with that same subset of hands in his range. The table below shows the 3-way equity of a selection of hands to illustrate the relative value of straight components.

Hand3h Equity facing a bet and a call

They do not fare well.

In fact, a similar analysis of the relative value of hands which already have two pair or a set will demonstrate that we should really consider even tighter ranges for our two opponents:

Hand3h Equity facing a bet and a call

Given the continuing ranges described bottom set will actually face either JJ or TT 40% of the time here! Even if the board pairs the J or the T on the turn bottom set will only have 39% equity 3h. There is therefore a strong case for folding bottom set on the flop.

Now we have a better picture of the relative value of hands even against two wider continuing ranges, let’s tighten one of the ranges to this:

At least one of the 4 players will hold this range 58% of the time and at least two 13% of the time. So you can be fairly certain that at least one of our opponents in this pot holds this range. Let’s give the CO this subset of hands to initiate with and permit the BT to play ‘fishy’ and call with the previous, wider range of:

For now we’ll assume that the BT is slow-playing his strongest hands as well, to avoid the complications of which hands to exclude as raises. We can examine the strength of some straight draws facing a competent CO betting a relatively linear range:

Hand3h Equity facing a solid bet and a fishy call

The true scale of the disaster that is calling the bare open-ender here is plain to see. Yet many a fish, and even weaker or tilted regulars, can be goaded into overcalling a half-pot bet here. What is perhaps harder to accept is that the combination flush and straight draws should still fold facing a half-pot bet and a call. Facing a pot-size bet and a call the naked nut wrap with no blocker is a clear fold, even holding a back-door flush draw.

The reason that the hands with straight components are so weak is that, even when they hit their outs on the turn, they only hover around 50% equity three-handed. As an example, the Equity Breakdown by turn card for KQJ♠5♠ is as shown:

This is one situation where simply counting our outs is a more reliable factor in our action-generation metric than evaluating our equity. There are at most 6 cards that we can hit where we wish to actively continue in the hand. With 45 unseen cards we have a 13% chance of spiking a major out by the turn. Of course there are occasions where some of our outs are blocked and/or counterfeited by one or both opponents.

It is also the case that, even when we hit the turn card, we are not guaranteed to win the entire pot nor to generate implied odds that are a significant multiple of the current pot. Set against this are those occasions where hitting the Jack or some of our minor outs will allow us to get to showdown when our opponents play passively. Weighing all of these factors is not an exact science but I assert that the point at which the calling decision is borderline is once the bet faced exceeds 1/3rd pot. This hand is a snap-fold facing a half-pot bet with a single call.

What makes PLO such a fascinating game is that the sheer variety of contexts, coupled with the large number of possible starting hands permits a broad variation in the relative starting hand strength given the game state. Treating your hands in multiway pots as simply slightly lower equity versions of those same hands in HU pots leads to severe classification errors and, as a consequence, costly post-flop mistakes. There are in fact material shifts in hand value in 3, 4, 5, and 6-handed pots that promote some components of a hand whilst relegating others to irrelevance.