The 1/n rule? (1 Viewer)

[Taghkanic]

I was reading Schoenberg’s Introduction to Probability with Texas Hold ’Em Examples (link to 2nd Edition)—which is mainly meant for statistics students, but contains some pretty useful analysis for players—and ran across this passage:

The author treats this idea as a standard “rule of thumb,” akin to the 2-and-4 guideline for estimating the probability of hitting your outs. But I have never encountered this concept for deciding how many hands to to open from various positions—at least not expressed in this way, as a fraction with the number of players left behind you as the denominator.

Of course any such rule must be applied with knowledge of the table dynamic, habits of those left to act, relative stack sizes, etc., so this is a very inexact guide. (In comments, I’ll try to compare the resulting percentages to other standard ideas about opening preflop.)

But more than debating the validity of the concept, I’m mainly asking whether people have heard this 1/n version before. Does it come, say, from some old poker book by Sklansky or Malmuth?

 

[pharaohone]

I don’t recall ever reading the concept

 

[Taghkanic]

So, nine-handed, this would yield percentages of hands to open as follows:

 

[Taghkanic]

If one accepts this as a general rule—a starting point from which to deviate based on the situation—then of course there is the question of “what hands do you include in that percentage”? Some people are going to mix in more suited combos, others more connectors, others more broadway combos, others more/fewer small pairs... But again, mainly curious why the author treats this as some common, broadly-known guideline.

 

[Taghkanic]

And of course, it can’t be correct to raise 100% of hands when it’s folded to you in the small blind, at least not against all opponents...

 

[pharaohone]

For what it’s worth, I think the rule generally works. I just thought through my opening ranges, and I’m opening about 20 hands in early position (about 12%) and closer to 100 hands when on the button (about 60%).

so as a directional rule, I think it works. But I had not heard it before. And certainly everyone’s opening ranges will differ slightly and differ based on the circumstances of the table.

Interesting tidbit - thanks for the post... like the rule!

 

[moose]

That is neglecting the probability that your opponents after you have a better hand than you. A better starting point of all "possible" hands would be the computer hand and up. So not 100% of all hands.

 

[pharaohone]

I think the rule is directionally correct for all but the small blind as you called out. I think if you are opening much more than half the time, let alone close to 100%, against the big blind when folded to you, you being out of position rest of the hand will put you in some tough spots.

 

[Jimulacrum]

Never heard of this "rule." Of course, in general, hand selection should vary inversely with the number of players left in the hand, and aggression should be dialed up the fewer players there are. Maybe not quite as precisely as 1/n, but that's not a bad starting point for thinking about it.

 

[upNdown]

For what it’s worth, I think the rule generally works. I just thought through my opening ranges, and I’m opening about 20 hands in early position (about 12%) and closer to 100 hands when on the button (about 60%).

so as a directional rule, I think it works. But I had not heard it before. And certainly everyone’s opening ranges will differ slightly and differ based on the circumstances of the table.

Interesting tidbit - thanks for the post... like the rule!

Yeah, I agree. Maybe it’s intended as a rule of thumb for beginners who haven’t yet become comfortable with how position affects their opening ranges?

 

[Kain8]

This is just another way of saying that your opening range should be wider in respect to the lower number of players are in the hand with you preflop.

 

[Goldfish]

They said there would be no math...

 

[Taghkanic]

I did a quick search on sites which recommend standard % opens from each position... The first one I found was an article on My Poker Coaching (MPC), whose recommendations I added to the table... I noticed that their numbers were closer to 1/(n+1), or the number of players left to act including yourself:

MPC’s percentages are very close to 1/(n+1), except for the button, where theirs is substantially (7%) higher.

 

[Jimulacrum]

I think the rule is directionally correct for all but the small blind as you called out. I think if you are opening much more than half the time, let alone close to 100%, against the big blind when folded to you, you being out of position rest of the hand will put you in some tough spots.

In the case of the SB, the problem is that you're up against specifically one opponent, he's up against specifically you, and you're in permanent worst position, so this one-size-fits-all mathematical approach fails.

Against some players in the BB, I raise 100% or nearly so. Against some, I raise like I'm UTG. But even against an unknown BB, I'm certainly not raising as much as this formula indicates—unless I were in position, like heads-up.

 

[Taghkanic]

This is just another way of saying that your opening range should be wider in respect to the lower number of players are in the hand with you preflop.

Yes, but it is positing a fairly specific way to think about that. I would tend to use a more “curved” distribution (even fewer in EP, slightly fewer in MP, and a lot more late). Depending of course on the table dynamic.

Note also that the book’s author does not discuss what to do about limpers. (I would downgrade my open % and increase my bet size, per limper, as a very general rule.)

 

[ChaosRock]

Note also that the book’s author does not discuss what to do about limpers. (I would downgrade my open % and increase my bet size, per limper, as a very general rule.)

Nor betting size it seems. He assumes a standard opening I guess.

 

[Taghkanic]

In the case of the SB, the problem is that you're up against specifically one opponent, he's up against specifically you, and you're in permanent worst position, so this one-size-fits-all mathematical approach fails.

Against some players in the BB, I raise 100% or nearly so. Against some, I raise like I'm UTG. But even against an unknown BB, I'm certainly not raising as much as this formula indicates—unless I were in position, like heads-up.

Agreed. Also, the fact that 7 players folded behind you would tend to “subtract” a lot of weak hands from the deck. I know subtraction is a topic which sometimes elicits strong feelings... But in the lower-stakes games I play, it is fair to assume that people are at least limping (and often opening) all Ax combos and pretty much all Broadway combos. So if the whole table has folded around to me in the SB, the number of aces and Broadway combos in the BBs range is a lot higher than usual. So I am not going to open anything like 100%. Maybe raising 50%, limping 25%, and folding the 25% of garbage I don’t want to play OOP.

 

[CrazyEddie]

It's (I presume) not a rule expressed by any poker players or authors, but it's not an unreasonable "basic rule of thumb" that a mathematician would devise using a simple model of the game, especially in a book trying to use poker as an example to teach probability to students.

My guess at the derivation of that rule: If no one raises ahead of you, they must not have good hands, and your top 1/N hand beats them. The question, though, is does your hand beat the people after you? You can't know this based on their behavior, because they haven't acted yet, but you can make the assumption that they are statistically uniformly distributed across all hands, and thus the best hand among them is in the top 1/N hands (in the modal case). If your hand is also in the top 1/N hands, then you'll be ahead of all of them but one, and on average will be "tied" (ahead or behind with 50% probability) with the best hand among them. Thus you should raise.

There's plenty that this model omits, but it's not a completely unreasonable model, especially for teaching purposes (as long as you're teaching math and not poker).

 

[CrazyEddie]

... I misread "players left to act behind you". So the model doesn't even rely on the players in front of you having weak hands just because they didn't raise; the model assumes anyone still in the hand (including limpers in front of you!) could potentially have a good hand, but still assumes that any hands left in are uniformly distributed.

A flawed model to be sure but again, not bad as a teaching tool.

 

[Prenders17]

Directionally, ok in concept but needs some massaging. This would have you raising 34% of hands, less the times it is opened in front of you. Might be a little stout.

 

[Taghkanic]

As a shorthand/starting place, I would go with the 1/(n+1) idea, in a vacuum, with no limpers, vs unknowns. Except make the button more like 40%. Then adjust to suit the situation.

I have a decent idea of hand rankings up to about 50%, so this seems easier to deploy than memorizing charts.