published about 6 years ago

Michael (mikej) primarily plays mid stakes NLHE 6-max and recently graduated from Cal with a degree in MSE.

__I. The Situation__

400NL, 6-max, effective stacks of $400

Villain opens to $12 from the button, you 3-bet to $42 from the small blind with a 75s type hand, and he re-raises to $100. Should you 5-bet all-in?

__II. Analysis__

1) How do we begin?

Some players will solve for the fold rate that will make the 5-bet breakeven. This involves assuming a value range for villain, and from there itâ€™s straight math. In our example, if we assume a 4-bet/call value range of TT+/AK (3.5%), 75s has 27.46% equity when we get called according to pokerstove.

When our opponent folds, we win $146. And when our opponent calls, we lose $137.22.* (*For computations, see last section of article.)

This leads to a breakeven fold rate of 48.5%.*

What does this result mean? If our opponent folds 48.5% of the time or more, our 5-bet will be breakeven or better. The problem is itâ€™s difficult to have an intuitive sense for how often our opponent is actually folding to our 5-bet. This is because when we are the button in this example, we set a value range and a bluff range. Our fold rate to a 5-bet is incidental. Additionally, in calculating the breakeven fold rate, we didnâ€™t utilize all the information we have about our opponentâ€™s range.

2) Villainâ€™s range

In this spot, most buttons have a polarized 4-betting range (and they should). He will 4-bet/call with the strongest hands in his opening range. And he will call with the next tier of hands in his opening range which are not quite strong enough to 4-bet/call, but are good enough to continue to postflop. The rest of the hands in the buttonâ€™s opening range are not strong enough to 4-bet/call or flat, so they are his potential 4-bet bluff hands.

A button player might open with 45% of hands (40-60% is typical): => 4-bet/call with the top 3.5% {TT+/AK) (3-6% is typical) => Flat with the next 10.0% {suited broadway/A8s+/AQ/AJ/KQ/KJ/scâ€™s down to 76s/66-99} (7-14% is typical) => And either 4-bet/fold or fold to the 3-bet with the remaining 31.5% of hands

When considering whether to 5-bet, we need to think about the composition of our opponentâ€™s 4-bet range. It is completely composed of hands from two ranges of hands: (1) a â€œvalue rangeâ€ and (2) a â€œpotential bluff rangeâ€.

3) The potential bluff range and the bluff rate

Notice the relative sizes of the two ranges. The â€œvalue rangeâ€œ is 3.5% of hands and the â€œpotential bluff rangeâ€ is 31.5%, 9 times as large. This is a â€œsmall value range/big potential bluff rangeâ€ spot. In this spot, the button only needs to choose to bluff with a hand from his â€œpotential bluff rangeâ€ a relatively small % of the time to make a 5-bet profitable.

Remember, we needed Villain to fold 48.5% of the time to make a 5-bet breakeven. Since his value range was 3.5% (TT+/AK), Villain only needs to bluff at a rate of 10.4%* with a hand from the potential bluff range to make a 5-bet breakeven.

=> If Villain bluffs with a potential bluff hand higher than 10.4% of the time, his 4-bet range is going to be weak enough that your 5-bet will be profitable.

=> If Villain bluffs with a potential bluff hand lower than 10.4% of the time, his 4-bet range is going to be strong enough that your 5-bet will be unprofitable.

How often is Villain bluffing with a potential bluff hand or, more precisely, what is Villainâ€™s bluff rate? We have a more intuitive sense for this value.

We find that the profitability of the 5-bet hinges on the rate at which the button bluffs with a hand from the â€œpotential bluff rangeâ€. In the example given, this (breakeven) bluff rate is 10.4%.

4) Additional considerations

Once you fix the bet sizes, the breakeven fold rate is only dependent on your equity when your 5-bet gets called (Table 1). In Tables 2-4, the second row corresponds to the example case used in the article. From Table 2, notice that if our equity is 35% when called, we only need him to choose to bluff a very low 5.8%+ of the time with his â€œpotential bluff rangeâ€ to make a 5-bet breakeven or better. Table 3 shows how the breakeven bluff rate varies with the size of Villainâ€™s opening range. As his opening range increases from 40% to 60%, the breakeven bluff rate drops by nearly half from 12.4% to 7.1%. Table 4 shows that reasonable changes in Villainâ€™s flatting range do not impact the breakeven bluff rate very much.

If button is prone to flat AA, KK or with some other value hand, then the â€œvalue rangeâ€ shrinks relative to the â€œpotential bluffing rangeâ€. This lowers the breakeven bluff rate.

This analysis does not only apply to situations where our opponentâ€™s initial range is wide. For instance, say an UTG opponent opens 15% of hands and youâ€™ve noticed that vs. a 3-bet he almost always flats QQ+/AK making his 4-bet value range very slim. If he begins to have an appreciably high bluff rate, a 5-bet becomes very profitable for you.

__III. Discussion__

The analysis in this article is not limited to 5-betting. One basic example in which this analysis applies is a river spot where your opponent has a â€œpotential bluff rangeâ€ (maybe whiffed draws/pure air) and a â€œvalue rangeâ€ and you have to decide how often heâ€™s bluffing with that potential bluff range. You can use this analysis to look at the profitability of your initial 3-bet with 75s in this article (How often does Villain need to 4-bet bluff with his â€œpotential bluff rangeâ€ to make my 3-bet unprofitable?). It can be used to analyze the profitability of the 4-bet from Villainâ€™s perspective. You can also use this to analyze a flop c-bet spot on a dry board.

In â€œsmall value range/big potential bluff rangeâ€ spots specifically, itâ€™s easy for opponents to inadvertently create weak unbalanced ranges because the temptation to bluff with their potential bluff range is too high. Youâ€™ll see many players at mid and lower stakes with this specific leak in these spots.

**The key factors to consider are the relative size of the â€œvalue rangeâ€ to the â€œpotential bluff rangeâ€ and the bluff rate. **

__IV. *Computations__

1) Calculating the breakeven fold rate

When our opponent folds, we win $146 ($100 from Villain + our $42 reraise + $4 bb). EVfold = +$146

When our opponent calls, we have 27.46% equity.

EVcall = 0.2746($400+42+4) + (1-0.2746)(-$358) = -$137.22

F = fold rate, C = call rate, F + C = 1

F_be = breakeven fold rate

C_be = breakeven call rate

EV_be = breakeven EV = 0

EVfold*F + EVcall*C = EV

EVfold*F_be + EVcall*C_be = EV_be = 0

EVfold*F_be + EVcall*(1 â€“ F_be) = 0

EVfold*F_be + EVcall â€“ EVcall*F_be = 0

(EVfold â€“ EVcall)*F_be = -EVcall

F_be = breakeven fold rate = -EVcall / (EVfold â€“ EVcall) = 137.22 / (146 + 137.22) = 0.485, or 48.5%

2) Calculating the breakeven bluff rate

V = 3.5% = value hands in 4-bet range, B = bluff hands in 4-bet range

F_be = B / (V + B)

=> B = [F_be / (1 â€“ F_be)]*V = 3.29%

potential bluff range = 31.5%

b = breakeven bluff rate = B / (potential bluff range) = 3.29 / 31.5 = 0.104, or 10.4%

Is there a possibility with HEM to filter for specific opponents the ratio of 4bet/call and 4bet/fold in a steal-situation?

Maybe its a good way to get closer to the right decision in these spots.

Excellent article!!

Allermand, the idea is to make you aware of the ballpark of these %s in a spot that may not be obvious. Ideally, you should work out similar spots away from the table from time to time just as you do for other spots when you do your post-session reviews. Eventually you'll develop feel and it'll take you less time to make an in-game estimate.

Nice article!

Great work, sir!

I've been doing my own sums with smaller raises and trying to apply your calculations. I'm fine up till where you get the figure 3.29% can you explain that formula please? And also what does â€ mean?

nice article

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