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Phil Rocquemore

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Change your Game Facing a Linear Range

“The snake which cannot cast its skin has to die. As well the minds which are prevented from changing their opinions; they cease to be mind.
Friedrich Nietzsche

The modern PLO games are saturated with players who have a basic understanding of hand value and a strong aversion to complex decisions. A fundamental component of these players’ strategy is a tendency to bet the flop with a Linear Range and not concern themselves with protecting their checking ranges.

This approach enables them to continue comfortably against check-raises, and to barrel frequently on the turn with a strong range. It crushes loose-passive fish, which explains why so many players rise through micro and small stakes playing this way.

Such opponents can be frustrating to play against, since it is not always clear which marginal hands we should continue past the flop. Sometimes we feel like we are playing too fit/fold and at other times we feel that we call the flop only to fold to the inevitable second barrel.

My analysis below will show you how to adjust your continuing ranges facing Auto C-bet and Linear C-bet strategies respectively on a flop texture which slightly favours the aggressor.

Scenario 1: Facing an Auto C-bet

In a 6-max game we call a 26% CO raise from the BB with 65% of starting hands, having excluded a small 3-bet range. We go heads-up to the flop, which comes down as:

K85

On this flop texture the CO has a small Equity advantage against us (56-44) and a minor polarity advantage (around 1.5 times as many sets, strong two pairs and assorted strong top pairs/overpairs).  The CO had a 56-44 Equity advantage against us Pre-flop so this flop is a typical example of the range interaction we can expect when we call from the BB.

Let’s take a closer look at how well we hit the board; shown below is the Hand vs Range Equity curve:

Hand vs Range Equity on K♣8♦5♥ for BB calling a CO open

If our opponent were to Continuation Bet his entire range, we can see from the graph that we would have little trouble finding hands with sufficient equity to continue.

On first analysis, since even a hand like QJT3 has 27% equity against our entire defending range, our opponent has a +EV check back for even the weakest hands in his range. Let’s say that our opponent uses a half-pot sized bet on this flop with his entire range. If we seek to prevent hit from C-betting any 4 cards more profitably than he might check back, we would not even need to continue to the Folding Threshold of 67%.

Yet from the graph it seems we can do substantially better than this if he C-bets his entire range. More than 50% of our hands on the flop exceed 40% equity against his range. The weaker hands are supported by a significant 10% of hands which dominate our opponent’s range with at least 70% equity. At the 67% Folding Threshold we find hands with only 33% equity against our opponent’s range!

Facing this opponent our main challenge is recognizing which weaker hands can profitably continue due to our opponent’s excessive C-betting. 30% of our hands fall between 30 and 40% equity and we want to select the best of these to continue with. Please note that this is an exploitative strategy; our opponent could restrict his flop C-betting range and prevent us from continuing with as many hands profitably as we shall explore later in this analysis.

Below is a table detailing the Equity for a selection of hand types, most of which fall inside the third quartile of our range:

Hand Equity vs 100% C-bet Frequency
AJ84♠ 40.6%
AQ65♠ 36.0%
KQ33♠ 52.7%
TTA6 33.4%
JT8♠6 38.9%
AJJ♠2♠ 39.8%
Q3Q♠2♠ 41.5%
AT96♠ 30.8%
J654♠ 42.9%
KT9♠8♠ 73.9%

The core of our continuing range is every {K,8,pair + gutter,OESD}. It’s also worth noting the steep drop in value from bare QQ to ATT. With underpairs the absolute rank matters as well as supporting side-cards. Check-calling a bare gut-shot to check the turn is a losing play unless our opponent likes to one-and-done. However, our opponent’s range is so wide that most Two pairs have become comfortable value XR. We can support these by check-raising some gut-shot straight draws on the flop.

The lesson from this investigation is that an opponent who C-bets too much of his air on the flop in position is in fact far less threatening than an opponent who restricts his range by checking back some of his weaker hands. Such a strategy is highly visible, a concept which I first introduced in ‘Predictability’, because it is easy to recognize our favourable equity curve against an opponent who automatically C-bets a flop texture.

Scenario 2: Facing a Linear C-bet

Most of the tight opponents you face will use a Linear C-bet construction on this flop texture. For the purposes of this example I shall use this 63% betting range:

{K,8,55,76,(5,TT,JJ,QQ):(96,97,64,74),AA}

I have not built the range strictly according to equity but rather from the perspective of a straightforward tight player. Such players prefer easy decisions and like to build pots whenever they have the potential to make the nuts by the river. Accordingly I have included within the betting range bet those hands which either already hit the board or have potential nut outs. I have assumed that this player is not concerned with protecting his flop check-back range.

Let’s examine how these changes to our opponent’s betting range affect our Hand vs Range Equity curve at the flop decision point:

Hand vs Range Equity on K♣8♦5♥ facing a Linear C-bet

Now only 32% of our range passes the 40% barrier. It is very clear that we do not want to defend every hand up to the 67% Folding Threshold, since hands at the threshold have only 27% Equity against our opponent’s C-bet range. With only 5% of our hands now exceeding 70% Equity we should refrain from check-raising, and allow the top of our range to protect the weaker hands in our check-call range.

Most of the hands we looked at earlier have dropped in Equity significantly:

Hand Equity vs 100% C-bet Frequency Equity vs Linear C-bet
AJ84♠ 40.6% 32.1%
AQ65♠ 36.0% 27.9%
KQ33♠ 52.7% 38.4%
TTA6 33.4% 24.9%
JT8♠6 38.9% 30.2%
AJJ♠2♠ 39.8% 27.6%
Q3Q♠2♠ 41.5% 26.4%
AT96♠ 30.8% 26.0%
J654♠ 42.9% 37.1%
KT9♠8♠ 73.9% 67.2%

It’s worth noting which hands drop the most in value against the linear betting range. The KQ33♠ and underpairs like Q3Q♠2♠ plummet in value because they have high equity against air and our opponent is checking back most of his air. But even bottom pair plus a low gut-shot remains strong enough to continue.

The following hands form the core of our continuing range:
{88,55,K,76,85,(8,5):(97,96,74,64)}

If we only continued with these hands we would be playing 42% of our range, with a 51% Equity Edge against the Linear C-bet range. Whilst we don’t want to continue up to the Folding Threshold, we still want to find some additional hands that can continue profitably. Evaluating the Range vs Range Equity alone is not sufficient to tell us which hands are the best candidates because the differences are marginal; we can see hands at the 45% mark with only 32% Equity and hands at the 60% mark with 28% Equity.

The method I suggest involves taking a closer look at the relative strength of the transitions for each player. Presently, the Equity Breakdown by Turn card for our core continuing range against our opponent’s Linear C-bet range looks like this:

Core range vs Linear C-bet

The favourable transitions for us are those which complete the straights: {9,7,6,4}. Keep this in mind as it will be important in the discussion which follows.

We know that we are going to be continuing some hands which either pair the 8 or the 5, have an interior pair already, or contain a gut-shot. The question is, “Which side-cards are sufficiently valuable to warrant the flop continuation?”

The best way to answer this question is to evaluate the Hand Equity Breakdown by Turn card in the specific context of the Range Equity Breakdown by Turn card. This method yields results which are frequently surprising. For example, look at the comparison between A973♠ (Left/Top) and A643♠ (Right/Bottom) below:

A context-agnostic approach, conditioned by many experiences of drawing to non-nut hands in multiway pots, and by years of hearing, “PLO is a game of the nuts” would prefer the A973♠ to A643♠ because it contains a nut gutter. In fact it is the A643♠ which has more playable continuations and the drop in equity when we spike our gutter does not mitigate this advantage.

The question that you should ask yourself when selecting these marginal hands is, “When I pair my side-cards which additional draws do I pick up?” That would enable you to recognize the advantage of the ‘weaker’ gutter here, since spiking any of the side-cards with the A643♠ brings additional straight draws, whilst spiking the side-cards with A973♠ does not. If you can pick up an additional straight draw when you pair the turn you will improve to a hand with multiple nut or near-nut river transitions.

You should also learn to recognize the range vs range advantage conferred on your pairing transitions. Consider the Equity Breakdown for QJ54♠  (Left/Top) and AQ53♠ (Right/Bottom), show below.

QJ54♠ might seem like a reasonable hand to continue since it can pick up an open-ender on the turn with the Ten, as well as spiking two pair or trips. Yet if we look more closely at the Equity Breakdown we see that the Ten is a very weak transition against a linear C-betting range. Our opponent will hold {Two Pair+} on the Ten transition 50% of the time and so we are usually unable to capitalize on our straight draw.

In contrast, the hand AQ53♠, despite having no transitions which bring an open-ender, is playable when it hits its side-cards.

Notice that the weakness of AQ53♠ on the straight transitions actually works in its favour. Recall that our range is already a favourite on these transitions and so we can expect to get free river cards with a high frequency. The AQ53♠ will have a profitable path through the hand when our opponent does check back, either by improving on the river to a hand with showdown value, or by bluffing the river to balance the many strong hands which we have on such run-outs.

This contextual transitivity of hand strength is what makes PLO such a complex, interesting game. Against linear betting ranges we place less of an emphasis on raw equity, and care instead about the frequency with which we improve to playable hands on transitions which do not heavily favour either player.

How to identify a player using a linear flop betting range


Now that we recognize which adjustments to make to our continuing range when we face an opponent C-betting with a linear range the challenge is to identify such opponents.

A linear flop C-betting range in a situation where a player only has a marginal flop Equity edge necessarily implies a weak flop Check-back range. The principal markers of a weak flop Check-back range are:

  1. High fold to Turn Lead after checking back.
  2. Low Delayed C-bet frequency

These two statistics are critical to most flop decisions when you are defending from the Big Blind and so should be in the main panel of your HUD. As a guideline for you to check your own performance a ‘Fold to Turn Lead’ over 60% is usually a serious problem. You can only fold this often if you are playing in very juicy loose games where you are consistently being paid off whilst playing ABC poker.

QUESTION(S) OF THE WEEK: Which situations do you consistently run into trouble against tight players? Comment below and your question could be the subject of my next article!

For my blog readers I am offering £250 off of the price of my comprehensive course, “Poker Math 2020: Pre-flop Principles” until midnight UK time on Friday 20th October.

Click here to view the Pre-flop Principles Course and reviews from the High Stakes Winners who bought it before you!

By | 2017-10-04T20:15:12+00:00 October 4th, 2017|

Why Equity Share is Bullsh*t

In this post I’m sharing a video from the first module of my acclaimed Poker Math 2020 course. I explain why ‘Equity Share’ is a terrible model for making pre-flop decisions and outline a new framework to give you a better understanding of strategic choices Pot-Limit Omaha.

Press play below.

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This is a great opportunity for ambitious low and mid-stakes players who would like to improve their understanding of the game and develop a working understanding of business development.

If you are familiar with my products, can work quickly and effectively with Odds Oracle and Camtasia, and would like to get early access to new material then I invite you to apply.

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By | 2017-04-10T13:22:45+00:00 March 22nd, 2017|

College is a Trap: The Advantages of a Poker Education

“Complacency is a state of mind that exists only in retrospective: it has to be shattered before being ascertained.”
Vladimir Nabokov

Professional online poker is a vocation unique to the early part of the 21st century. It didn’t exist prior to 1998 when Planet Poker opened it’s virtual card room to players with dial-up internet access.

As a child of the new century, online poker feels transient, impermanent, always on the brink of fading away. Whether it be due to the threat of impending government regulation, the perceived dwindling supply of fish or the saturation of knowledge (Sorry guys!) every year players opine that the games are soon going to die.

It is rare to find a player who expects poker to be his primary source of income in even 10 years’ time, let alone 20.

This inherent instability means that the online poker community is consistently concerned with what to do for a living when online poker ‘finally ends’. Many successful poker professionals chose to drop out of college to pursue their chosen lifestyle and, as a consequence, fear that there is something missing from their education.

It is true that there are well-documented benefits to going to college that a college dropout risks missing out on. However, few online players appreciate quite how unique the skill-set required to beat online poker is. Fewer still realize that it is precisely the skills honed by online poker that are vital to succeed in our modern ‘Digital Age’.

Part of my mission here at Cardquant is to enlighten those poker players who are in the dark about the wider utility of the poker skill-set. I want to show my readers exactly what advantages you have over your conventionally educated peers. I also want to open the minds of people outside of the poker community to the benefits of working with a current or former poker pro.

To fully appreciate the benefits of the brutal training that online poker subjects one to, one needs to understand the drawbacks of conventional education. With a couple of degrees from two of the world’s best universities on my CV before I started playing poker professionally, I’m pretty well placed to give a balanced perspective on this issue.

The poker skill-set combines probabilistic and inferential thinking with empiricism and psychology. In this article we will focus on the first of these skills- probabilistic thinking- and contrast it with the binary thinking instilled by a conventional education.

Binary Payoffs and Authority

Conventional education is a two decade period which spans the formative years of your life where you are trained that it is ‘bad to be wrong’. This training is implicit in the teaching methodology; ostensibly you are merely studying the subject matter, be it Mathematics, Geography or Physics. But, long after the facts and methods learned during your education fade, the manner in which they are evaluated does not.

Throughout your schooling you face countless quizzes, examinations and standardized tests, the overwhelming majority of which are graded within a Binary Payoff framework: you get 1 point for being ‘right’ and 0 points for being ‘wrong’.

Furthermore, every point in this Binary Payoff framework is allocated based on whether you got the correct (i.e. same) answer to a question that had already been solved. Even in that erstwhile bastion of tolerance to uncertainty- The Arts- the drive towards testing and standardization has led to students seeking to replicate the perceived ‘model answer’ rather than risk losing points by presenting their own interpretation.

A Binary Payoff framework has at its heart the ever-present figure of AUTHORITY so dear to state-sanctioned, standardized education. I speak here not of the individual professor himself, but rather of the concept that there exists some persistent, knowable objective truth which it is possible to check your responses against.

This educational structure is a Procrustean bed; the same blade that fashions unruly children into compliant cogs ready for The Corporate Machine hacks away at the legs of an independent and adventurous mind.

There are two problems with an approach that venerates educational authority and which evaluates responses by how closely they correspond with those prescribed by that authority.

The first problem is that in life the most interesting and valuable questions are the ones that nobody has solved

1. When we train people merely to recall facts and replicate existing procedures we insulate their minds from experiencing the uncertainty inherent to novel problems.

The standardized educational process generates minds with a strong aversion to uncertainty, unwilling to accept the temporary and contingent nature of all human ‘knowledge’. Such minds are easy to influence, keen to embrace rules and regulations that offer them ready answers to neatly packaged problems.

But this illusion of certainty comes at the cost of imagination and creativity, and stifles the vitality of both the ‘educated’ individual and the wider society in which he participates.

I would like to expand on the problem of authority and its relationship to uncertainty in future articles. For now we’ll direct our attention to the problem generated by rewarding students for giving answers which correspond with an existing model.

When a person is trained within a Binary Payoff framework he tends to overvalue ‘being correct’ in the Fluid Domains of life outside the classroom.

For isolated decisions where payoffs can take extreme positive and/or negative values Binary Thinking is catastrophic. The Binary thinker inhibits his ability to experience unusually high returns whilst simultaneously exposing himself to excessive downside risk. For iterated decisions the Binary thinker suffers from poor performance despite (or rather because of) his tendency to be consistently accurate. We explore this problem further in the example below.

The Power of Probabilistic Thinking

To illustrate how conventional education trains people to make poor decisions in Fluid Domains, I present a simple poker scenario:

Consider a one-street poker game where our opponent makes a pot-sized bet with a range composed of either the nuts or a pure bluff (perfectly polarized). We may choose either to call his bet and go to showdown, or fold and surrender the pot to our opponent.

We face this same decision 10 times and our opponent constructs his range with a mix of 60% value hands and 40% bluffs.

If we use the Binary Payoff model so dear to an educational system that emphasizes how bad it is to be wrong then we will fold every time. Our result? We get ‘the correct answer’ 6/10 times and make $0.

The professional poker player incorporates pay-offs into his model and so calls every time, not flinching at being ‘wrong’ more often than not. His result? By thinking probabilistically he gets ‘the correct answer’ only 4/10 times and yet profits 6*-1+4*+2 = $2.

Our lesson is this: The Correct Answer is NOT always the Best Response.

COLLEGE CONSTRUCT: Get the right answer to as many questions as you can. Payoffs are binary.
POKER PERSPECTIVE: The best response depends on the associated payoffs; there are worse things in life than being wrong.

Players who fold too much in these situations (nits) do so because their Binary Payoff worldview renders them overly attached to being ‘right’. This worldview leads to a miscalibrated, loss-averse mentality because the emotional payoff associated with being ‘wrong’ is excessive.

In those Fluid Domains where there are very large potential payoffs, both positive and negative, a bias towards getting ‘the correct answer’ means that the Binary individual makes bad decisions consistently and with increasing frustration due to his poor returns.

Since the Domains essential to life are almost all Fluid this Binary Payoff bias cripples a man everywhere outside of his academic or corporate cocoon. This bias is so pervasive that it deserves its own name; I call it “The Meta-stupidity of Geeks”.

The Meta-stupidity of Geeks

In his classic personal development book, ‘The Magic of Thinking Big’, David Schwartz relates an anecdote about Henry Ford, Founder of the Ford Motor Company:

“One time Henry Ford was involved in a libel suit with the Chicago Tribune. The Tribune had called Ford an ignoramus, and Ford said, in effect, ‘Prove it’.
The
Tribune asked him scores of simple questions such as ‘Who was Benedict Arnold?’ ‘When was the Revolutionary War fought?” and others, most of which Ford, who had little formal education, could not answer.

Finally he became quite exasperated and said, ‘I don’t know the answers to those questions, but I could find a man in five minutes who does.'”

In an era where we almost always have access to Google, Ford’s perspective becomes even more relevant: those who use their mind as a garage for facts will find success in artificial ‘examination’ environments but little in the real world.

We all knew fellow students in school with an encyclopaedic knowledge of trivia who were apparently incapable of getting to grips (for want of a better term) with members of the opposite sex.

Over the course of any given day, these students would have been faced with perhaps 100 questions on some academic matter or other and perhaps 1-2 interactions with a potential mate. Day-in, day-out they would score perhaps 95% on their academics and a big, fat zero in their sexual life.

Under the Binary Payoff model they were doing very well- still scoring over 90% on aggregate each day- yet a few years down the line they would regret not focusing their attention on the weightier questions of life2. Eventually their complacency catches up with them.

The Meta-stupidity of Geeks is this:
They are consistently right about everything that doesn’t matter.

 

QUESTION(S) OF THE WEEK: Where in your own life has using a Binary Payoff model in a Fluid Domain held you back? If you are comfortable sharing I would love to hear your stories in the comments below. On a lighter note, what opportunities has playing online poker given you that you would have missed out on if you had stayed in college?

For my blog readers I am offering £100 off of the price of my new course, “Poker Math 2020: Pre-flop Principles” until midnight UK time on Friday 9th December.

Click here to view the Pre-flop Principles Course and testimonials from High Stakes Winners!

Show 2 footnotes

  1. I originally wrote “The problem is that in life the most interesting and valuable questions are the ones that nobody has solved yet.” For now I’ll leave it to the reader to consider the change in meaning that comes from truncating this sentence.
  2. I have used the question of finding a mate as an example and yet the same argument can be applied to other weighty domains. The majority of people also delay the task of building a meaningful life until crisis forces it upon them. A young man who experiences crisis early holds a tremendous advantage over those peers of his who enjoy a balmier Spring. It just takes a while for him to realise it.
By | 2017-04-21T18:38:47+00:00 December 5th, 2016|