Assume a player's RFI on the button is 40% at a 9-handed table. Is it correct to assume they are opening 40% of all 1326 hands, or is there any validity to weighting their range slightly stronger, to account for card distribution considering that 6 players had 'unplayable' hands?

Weird question, and almost certainly unimportant from a strategic standpoint. I've been getting into probabilities a lot recently, and can't quite wrap my head around if, or how, this (and similar problems) might be calculated.

Thanks!

i doubt it would be worth the effort to figure this out.Since you need a stronger hand to open from EP means that a lot of hands that would be playable from LP get folded.I doubt there is enough of a difference to make it worth the effort of finding out.

Yes.

You should do this everytime cuz players criteria on what is a top X%of hands varies but only at the middle-bottom section of those ranges, towards the top there´s seems to be some generic agreement (obv).

This is something you should always considered too, its called "card removal effect".

Thanks for the responses guys, but I didn't word my question very well. What I'm really looking for is a mathematical way to calculate the effect of unknown card removal, or more precisely, a formula (or many) that could be used to account for partial/imperfect information.

The button RFI thing was only meant to be an example of the kind of situation that this might be used for, but it was kind of a crappy one. I'm not sure how much math can be used to quantify imperfect information, but I can't be the first mathematically inclined person who has wanted to, so I imagine people have devised many ways to do so.

For example, I know how to calculate flop probabilities based on my hole cards, and I can quantify ranges using combinatorics, but how would I adjust the flop probabilities based on my assumptions about the ranges of 3 particular villains? I know it only makes minute differences in this instance, but I expect that, once I know how to solve these types of problems, I can find more impactful questions to ask.

I don't really know where to learn about the finer points of statistics and probabilities, I guess I might need to pick up a textbook or track down a maths student, but if anyone wants to give me a head start or point me in the right direction I would greatly appreciate it.

There's no formula. Spadebidder did a study years ago that showed that if it folds to you in late position then the deck is very slightly heavier in big cards, but it really is a very small effect.

http://www.spadebidder.com/flop-analysis/part2/

Thanks Rusty! That link is exactly the kind of stuff I'm looking for. Nerdgasm incoming.

This has been looked into over the years. Spadebidder looked at a ton of empirical data (the link that Rusty provided above plus others). Others tried to analyze it analytically.

IIRC the findings were that the two effects are both "small" but "significant".

In your example, due to card removal, by opening the ex ante top 40% hands the button actually will open slightly more than 40% of the time. Also, as you mention above, his range is slightly skewed to the top of his range.

Both of these effects can be clearly seen in toy examples. I believe that the math is essentially intractable for real 9-player poker, but simulations can give some insights.

I had some time tonight and coded up a simulation to help study the effect of card removal that OP raised. I used a hand-ranking from 10-handed (no folds) though I imagine that the results are robust to the hand-ranking.

On a 9-handed table I assumed that the players have the following ex-ante call pcts:

10% UTG
15% UTG+1
20% UTG+2
25% UTG+3
30% UTG+4 (hijack)
35% UTG+5 (cutoff)
40% UTG+6 (button)

In 10,000,000 deals I find:

- it is folded to the button around 19.46% of the time

- when folded to, the button opens around 41.63% of the time

- when folded to, the button's range is slightly skewed upward from his ex-ante top 40% range (e.g., the median opening hand now shifts one or two "hands" higher, from 55 to KTo or K7s).

Say you are playing a 6 max game where players are opening with any ace and it folds to the button. You would know that 6 cards are not aces so the button would have a
1-(42/46 *41/45) = 17% chance of holding an ace. Compared to the 1- (48/52*47/51)= 15% chance if the first 3 players card were unknown. If you are in the BB with out an ace the odds become 17.5% or 15.5%.

Since some A hands fold and some non-A hands raise your ability to adjust the hand strength is going to be less than the 2% difference above.

Upped the simulation to 50 Million deals with similar results to those posted above.

On a 9-handed table I assumed that the players have the following ex-ante call pcts:

10% UTG
15% UTG+1
20% UTG+2
25% UTG+3
30% UTG+4 (hijack)
35% UTG+5 (cutoff)
40% UTG+6 (button)

In 50,000,000 deals I find:

- it is folded to the button around 19.44% of the time

- when folded to, the button opens around 41.68% of the time

- when folded to, the button's range is slightly skewed upward from his ex-ante top 40% range (e.g., the median opening hand now shifts one or two "hands" higher, from 55 to KTo or K7s).

For another data point on the skewed range, when folded to, AA makes up around 1.29% of the button's opening range compared to around 1.13% of his ex-ante Top 40% opening range. When folded to, AKs makes up around 0.85% of the button's opening range compared to around 0.75% of his ex-ante Top 40% opening range, and when folded to, AKo makes up around 2.53% of the button's opening range compared to around 2.26% of his ex-ante Top 40% opening range.