*“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:

- Limp our entire opening range. Do not include a raising range.
- Raise our entire opening range. Do not use a limping range.
- 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?

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:

**Expected Profit(SB Steal) =
f(Opponent Folds)*1.5 +f(Opponent 3-bets)*(-2.5) +
f(Opponent Calls)*((Expected Return)*((2B Pot Size) -2.5)
**

*Now…*

** f(Opponent Folds) = 1- f(Opponent Calls) -f(Opponent 3-bets) = F**

*Assume…*

**f(Opponent 3-bets) = 15%
(Expected Return) = 0.25**

**Expected Profit(SB Steal) = F*1.5 -0.15*2.5+ (0.85 – F)*(0.25*6 – 2.5)
**

*If the Expected Profit is greater than zero then we can profit by opening with our weakest playable hand from the Small Blind.*

*We solve for F…*

**F*1.5 -0.375 + (0.85 – F)*(-1)** **= 0
2.5F – 1.225 = 0**

**F = 0.49**

**We have found that if the Big Blind folds 49% of the time or more we can justify utilizing a raise-only strategy from the Small Blind.**

It follows that if we wished to use a mix of raises and limps from the Small Blind then the raising component should not exceed 50% in width and should likely be far narrower. *Why?*

Because we can assume that the Big Blind should be able to defend profitably with at least as many hands as we can open-raise profitably. If we wish to include some of our weakest playable hands in our raising range then those hands need to have the opportunity to profit from a steal and hence we must limit our raising frequency.

If the weak hands are sufficiently well protected by those strong hands which compose the rest of our raising range then they can raise their ‘vacuum expected return’ from 0.25 which will further increase the Expected Profit of these weak hands.

So introducing a limping range from the Small Blind could, in principle, increase the profit generated by our raising range. This is a necessary condition to justify using a limping range from the Small Blind but is not by itself sufficient.

Let us now evaluate the cost of a limp to get a better insight into the value of a strategy which includes a limping range.

## The Price of a Pre-flop Limp

Our opponent can respond to our limp from the Small Blind by either checking back, allowing us to see the flop for free, or by raising.

Hence when we limp pre-flop, if we assume that we never fold after limping, our average price to see the flop is given by the equation:

** Average Price = f(Opponent Checks)*0.5 + f(Opponent Raises)*2.5**

** ***Since…*

** f(Opponent Raises) = 1- f(Opponent Checks) = R**

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

** Average Price = (1-R)*0.5 + 2.5R = 2R + 0.5**

f(Raise) | Average Price to See Flop |
---|---|

20% | 0.9bb |

30% | 1.1bb |

40% | 1.3bb |

50% | 1.5bb |

60% | 1.7bb |

We see that when the Big Blind increases his raising frequency from 20% to 60% he almost doubles the Average Price that the Small Blind must expect to pay in order to see a flop.

We can now more clearly understand the value of open-folding a fraction of our range pre-flop: if we attempted to limp 100% of our range then our opponent could preserve a range advantage post-flop by raising with 60% of starting hands.

It is hard to imagine that open-limping an 80th percentile hand (such as A♣9♠5♦4♥ or J♠8♠7♣3♥) expecting to pay 2bb on average to see a flop is superior to open-folding and taking the fixed 0.5bb loss.

Let us compare our decision regarding whether to complete the Small Blind with the decision to call a pot-sized raise pre-flop in a Heads-up game. In the Heads-up game we are offered 2:1 on our pre-flop call. If our opponent were to use a 100% opening range it would be analogous to the ‘forced bet’ with a 100% frequency that the Big Blind posts in a 6max game.

*In the Heads-up scenario we would gladly accept that 90% of hands were playable, so why not here?*

If our opponent chose to never raise our limp then we would be getting 3:1 pre-flop and could indeed justify limping almost any 4 cards. Should our opponent veer to the other extreme and raise 100% of starting hands then we would always contribute 2.5bb to a final pot of 6bb and thus only be getting 1.4:1 on our limp. This is a vastly inferior situation to the heads-up game described and thus we cannot justify limping 100% of hands facing such a strategy.

In fact the Big Blind raising frequency that creates ‘Effective Odds’ of seeing the flop of 2:1 is only 16.7%.

*The Big Blind raising range facing a limp should never be less than 17% else he enables his opponent to profitably limp almost any 4 cards.*

## The Weapon of Choice

At the beginning of the article we set out to choose one from three possible strategies:

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

We have used some simple models and a series of assumptions to create some boundaries on an optimal blind battle strategy:

- The Big Blind must defend an absolute minimum of 50% of starting hands to a raise from the Small Blind.
- If the Small Blind’s expected return is less than 25% of the flop pot then he would prefer to fold pre-flop.
- If the Small Blind’s expected return exceeds 50% of the flop pot then he would prefer a raise to a limp.
- If the Big Blind raises a limp only 40% of the time then the average price for the Small Blind to see the flop when he limps is at most half the price of an open-raise.

**We will now use these boundaries to evaluate the relative merits of the three Small Blind strategies outlined above.**

The main problem with Strategy ‘1’– raising every hand in our opening range- is that it limits the total number of hands we can play profitably.

We already know that our opponent must continue an absolute minimum of 50% of starting hands. If we raise more than 50% ourselves then even our opponent’s weakest starting hands can continue profitably. Thus we reduce the probability that an opponent folds to our raise with any significant frequency pre-flop and limit our opportunities to steal the pot with a weak hand.

The main problem with Strategy ‘2’– only opening with a limp- is that it enables 100% of the Big Blind’s starting hands to see a flop for no additional cost.

We never get the opportunity to steal the pot pre-flop nor can we build an especially large pot post-flop: the flop pot size is only 2bb unless our opponent raises.

In addition both Strategies ‘1’ and ‘2’ share another inherent flaw- namely that to take only one action with some frequency enables our opponent to model our range perfectly and thus presents him with trivial pre-flop decisions. It turns our that, in the Battle of the Blinds, choice *is itself* a weapon.

Our analysis leads us to prefer Strategy ‘3’: use both a raising range and a limping range.

This is the framework that many players have converged on, switching to a raising strategy only when facing a player who over-folds in the Big Blind. Yet most of you will find, if you review your database, that your open-limping range has a mediocre win-rate.

This is no accident.

In the next article in this series we will examine how the exact composition of most players’ open-limping range gives rise to serious strategic problems post-flop.

For now I invite you to leave a comment on this article below. The sharp strategy minds among you can respond to this question…

**QUESTION(S) OF THE WEEK:** Under what conditions does a strategy that includes a limp-raise component effectively exploit your opponent’s pre-flop strategy in the Big Blind?

HaruJuly 22, 2016 at 3:17 amI will try to answer the question of the week.

Let’s say the bb is 3betting 15% when sb raises, and rasing limp p*100%.

If we have a good hand to limp-raise such as KQJTds, our equity is decent so I will just assume that expected return when seeing a flop is 0.5 to simplify the calculation (but this should vary a bit for sure.) Assuming this, we want to make the pot the largest.

1) Average pot when sb raise:

0.85*(6bb) + 0.15 (18)=7.8

2)When we limp raise

(1-p)(2bb)+p(18bb)= 2+16p

when we solve 1)=2), p is 0.3625. This looks surprisingly small. Maybe I oversliplified (Ignored the chance to get 4bet) but I guess if bb is raising limp 40%+ then limp-raise is better. I am little bit worried that we don’t include AA in our limp-raising range because we want to bet/4bet with AA, but maybe including some of Axxx ds into limp-raising range is enough.

jakeJanuary 11, 2016 at 8:28 amIn a case where the BB overly aggressive raises vs limps but doesn’t 3b enough to expect to or/4b our strong hands . Yet calls our raises with high frequency. This way we protect our limp/calling range and may archive that he reduces his raising frequency to see the flop cheaper with our mediocre holdings.

Sounds good?