Let's say for whatever reason we decide we want to open/raise with a certain linear range, and let's say we decide 10% of opening hands is about right.

Using equilab we get this range of hands.

10.26%
99+, ATs+, KTs+, QTs+, JTs, ATo+, KQo

The reason those hands are considered 10.26% is due to the combinatorics.
For pkt AA there are 6 combo's out of a total of 1326 = 0.045%

The problem I am having is that even though Aces are ordered at the top, they are still equally scaled in proportion of combinations for PF ranges, and I feel that they shouldn't be. I feel that AA should be weighted much more than any other pair, then KK etc., and that they should be based more on a logarithmic scale rather than a linear one.

In other words, because aces are so much stronger than other hands, they might make up something like the top .3% of total strength instead of only 0.045%. Hands at the bottom of the scale would be reduced so that 100% of hands by actual strength are still represented.

So if we want to use a 10% range that actually reflects the top 10% of hand strength, I am just guessing, but for example it might look maybe something like...

JJ+, AQs+, AKo

That looks way tighter (3.32% based on usual combinatoric weighting) than our original 10.26% range, and that original range 10.25% combination might actually represent something like the top 25% of hands based on actual hand strength.

Comments appreciated. Has this been studied before? Is this consideration already built into existing suggested PF starting hand ranges, etc.?

You only get AA 0.045% of the time, so there's no logic in putting more weight than that on it.

The thing that matters here is the frequency at which you're doing something, not which hands represent what % of your range's strength. Combinatoric weighting is what matters. The effect that AA has on your range is represented in the equity of the range.

33 occurs exactly with the same frequency as AA so, yes, it is the equity (among other factors) that differentiates between the two hands. But, which equity? Showdown against 1, 3, or 9 opponents. What about realized equity, which factors in fold probability? Maybe equity combined with profitability.

I think the question is valid. In evaluating hands, there often is reference to the top or bottom of one's range so there is a implied value-weight metric, which is the rank order that depended on how the hands were ordered -e.g., Equilab/PokerStove, ProPokerTools, Sklansky, Billings,et. al, etc.

Whether a numeric weight can be assigned and be useful is something to think about and which I don't recall seeing.

NOTE: I couldn't figure out how to edit the post, but probability of a single pair like AA is of course 0.45% not 0.045%


Hal, you are the master. Will need to study it a bit more and give an update if I come across it in the literature anywhere.

there's more to rfi ranges than absolute eq in a gii vs any2 scenario, which is what equilab is outputting
in higher rake enviroments esp, blockers to getting 3b will matter quite a bit and also there's actual eq realization vs defending ranges