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New for 2014: Small Stakes Solutions!

The mathematics of in-position play in Pot-Limit Omaha from pre-flop to river explained in 5 interactive lectures.
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How to do yourself Justice in the Battle of the Blinds

“Justice is a certain rectitude of mind whereby a man does what he ought to do in the circumstances confronting him.”
Thomas Aquinas

Aquinas’s concept of justice, whilst interesting, creates several problems in its practical application. The first problem is how to identify the circumstances that we find ourselves in. If we cannot recognize the reference class that the situation belongs to then Aquinas’ maxim gives us no guidance as to the correct course of action. The second problem is that it is impossible in all but the most trivial of situations to know what we ought to do without prior consideration.

As acting human beings we solve these problems in the moment by bundling any given situation into a class that appears to be sufficiently similar. Even very novel situations, which we cannot have considered fully before, have features that will enable us to associate them with some broad reference class

1.

It is excusable for us to spend very little time considering in advance what we ought to do in trivial and unlikely situations. However in situations of high value which we expect to encounter regularly it is our duty to be prepared. In today’s article I will help you do yourself justice in one such situation.

The question that we concern ourselves with here is,

“What is the optimal pre-flop strategy in a 6-max game when the action is folded to us in the Small Blind?”

What follows is a comprehensive analysis of the essential mathematics behind Small Blind versus Big Blind play. Since the correct choice of action for the Small Blind is contingent on the behavior of the Big Blind we shall, in the process of this investigation, discover some boundaries on the Big Blind’s play.

When the action is folded to us in the Small Blind we may choose one of three strategies:

  1. Limp our entire opening range. Do not include a raising range.
  2. Raise our entire opening range. Do not use a limping range.
  3. Use both a raising range and a limping range.

Strategies ‘1’ and ‘2’ offer the advantage of simplicity as we only need to determine the width of the playable range. We never encounter the doubt that a hand which is unprofitable as a raise may be profitable as a limp or vice-versa. Yet, as we shall illustrate here, simple solutions to complex problems are attractive but not necessarily correct.

We shall start our investigation into the relative merits of these three strategies by analyzing the advantages of a pre-flop raise.

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  1. We all, consciously or unconsciously, use a hierarchy of reference classes and prefer to utilize the one that appears to us as most relevant.
By | 2017-04-10T13:22:47+00:00 January 8th, 2016|

Folding Thresholds in 3-bet Pots: A Closer Look at your Opponents’ Strategy

“If you know the enemy and know yourself you need not fear the results of a hundred battles.”
Sun Tzu, The Art of War

Our primary considerations when choosing a line of play in a poker hand are the particular exploitable tendencies of a given opponent. Both ‘math’ and ‘feel’ players alike will agree that if a player folds ‘too much’ on a given street we can bluff him more often. But exactly how much folding is too much?

Away from the table we can do some valuable analytical work to calculate folding thresholds beyond which a player is vulnerable to exploitative bluffing. In so doing we also discover how our own exploitative lines expose us to being exploited ourselves. Thus the calculation of folding thresholds helps us both to more accurately target imbalances in our opponent’s strategy and to recognize when they are adjusting correctly to our strategy.

3bet pots provide a veritable haven of diverse exploitative lines for the observant player. Given their bloated size relative to single-raised or limped pots they also account for a disproportionate amount of the money per hand which you win/lose at PLO. In this article, we consider the mathematics behind bet/calling, bet/folding and other decisions post-flop in 3-bet pots.

My writing in this post is not meant to be an exhaustive examination of 3bet pots so we shall not go into much detail regarding our opponents’ ranges. The purpose of this analysis is to discover the threshold frequencies at which bet/folding the flop leaves a player open to being exploited. A knowledge of these fundamental frequencies will improve our awareness in 3bet situations at the table.
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By | 2017-04-10T13:22:52+00:00 June 4th, 2015|

The Equity Delusion: How much does chasing cost you?

“The future influences the present just as much as the past.”
Friedrich Nietzsche

Take a look at the picture above and ask yourself, “Why are these men running?”

We can’t be sure, but the chances are that they aren’t running from some rabid beast. In fact they likely aren’t running from anything at all. We can infer this from the fact that not only are these gentlemen wearing inappropriate beast-fleeing attire but that the lead character is looking at his watch. All rather mundane, so why, dear reader, have I bothered bringing this to your attention?

For those of you hoping for a surprise beast-chomps-man denouement complete with a graphic YouTube video this blog post will be something of a disappointment. My readers with a poker-focused disposition will have a lot more fun and should read on.

The point is that these gentlemen are roused to action by anticipatory anxiety: the threat of something bad happening in the future. In this case our heroes are presumably at risk of missing a connection or event. We routinely account for future events in our life decisions yet it seems harder to do in poker. Why? (more…)

By | 2017-04-10T13:22:53+00:00 April 20th, 2015|