“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

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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.

Why Raise at all?

Consider a situation where we raise from the Small Blind to 3bb at 100bb starting stacks and are called by the Big Blind.

On the flop the pot will be 6bb and we will have 97bb behind. We can express our return from the hand as a fraction of the flop pot, which can take any value from -16.17 to +16.17 times the pot.

If we had instead open-folded the Small Blind we would have lost 0.5bb on the hand. The equivalent value from the raised pot would be us receiving 2.5bb from the 6bb pot; thus if we expect to return more than 2.5/6 = 0.417 of the flop pot when we raise then raising is superior to open-folding. For now we have neglected those times when our opponent will 3-bet, and we will discuss the implications of him sometimes folding to a raise below.

Alternatively we could open-limp the Small Blind and, if we did not face a raise from the Big Blind, we would need to win a smaller fraction of the flop pot in order to make our play more profitable than open-folding. The flop pot would now be 2bb and so as long as our return exceeded 0.5/2 = 0.25 of the pot, limping would be superior to open-folding.

We can now better understand the utility of ‘raising for value’. If our expected return from the flop pot is 0.5, then we expect to profit by 0.5bb irrespective of whether we open-limp or open-raise:

Expected Profit(Open-limp) = Expected Return – Pre-flop Investment
= 0.5*2 – 0.5 = 0.5bb

Expected Profit(Open-raise) = Expected Return – Pre-flop Investment
= 0.5*6 – 2.5 = 0.5bb

If our expected return exceeds 50% of the flop pot then we would prefer a raise to a limp. So, in a vacuum situation where our opponent will never fold or raise us pre-flop our preferred action is simply a function of our expected return from the flop pot:

If Expected Return < 0.25 then Fold
If 0.25< Expected Return < 0.5 then Limp
If Expected Return > 0.5 then Raise

Let us turn our attention to the other advantage that open-raising has over open-limping: the chance to immediately profit from our opponent’s pre-flop fold.

The Mathematics of the Small Blind Steal

How often do we need our opponent to fold for us to profit by raising any 4 cards from the Small Blind?

We raise to 3bb and thus risk 2.5bb to win the 1.5bb of money already posted between us and the Big Blind.

Minimum EV(SB Steal) = f(Opponent Folds)*1.5 + f(Opponent Continues)*(-2.5)

 Since…

 f(Opponent Folds) = 1- f(Opponent Continues) = F

  …we can express the Price in terms of one variable:

Minimum EV(SB Steal) = F*1.5 – 2.5*(1-F) = 4F – 2.5

 If the minimum EV is greater than zero then we can profit by opening with any 4 cards from the Small Blind.

 We solve for F…

4F – 2.5 = 0

F = 0.625

So we have found that the Small Blind has a profitable steal with any 4 cards if his opponent will fold to a steal more than 62.5% of the time.

Unfortunately the calculation in this form is not especially edifying since we have assumed in the model that the Small Blind loses 100% of the time whenever the Big Blind continues.

In practice, unless our opponent were to 3-bet 100% of his continuing range then even the bottom of our opening range (Which would fold to a 3-bet) would have some positive expectation post-flop in 2-bet pots. So to get a deeper insight into the value of a pre-flop steal from the Small Blind we need to ask a better question:

With your worst ‘playable hand’ for what opponent folding frequency are you better off raising than limping pre-flop?

A hand at the bottom of our raising range would presumably fold to a 3-bet pre-flop so we don’t have the complications of estimating it’s value in a 3-bet pot. We have already established that for a hand to qualify for a limp it must have an expected return of at least 0.25 times the pot in a 2-bet pot. So we’ll use a calculation which takes into account the possibility of playing a 2-bet pot post-flop to estimate the folding frequency at which we have no need of a limping range: