[Aisthesis]

Well, even without having the math double-checked and more critique on the formulae, I couldn't resist running some more hands through. The results are sufficiently internally consistent and plausible that I can't help but think that the math is pretty close to correct.

At this point, I've done all pairs down to 66 as well as AKs and AKo.

First, one methodological note: As favorites, I have been taking any hand that is a favorite AT ALL heads-up to the given hand UNLESS it's only a favorite by less than 50.5-49.5. The reason for this is that I've just rounded the percentages of the individual hands to the nearest percentage (and hence would get exactly 50% if, for example, I included 22 against AKs when one of the 2s has the same suit as the AK). But for the aggregate percentage on all hands, I'm using 2 decimal places after the percentage. For example, there are 74 hands that beat AKo by this rule, and their average win rate is 57.15% by my calculations. I thought the increased accuracy was probably fairly important once we were dealing with moderately large numbers of hands.

I find some confirmation to this method in an experiment I did with 77. JTs is a 51% favorite over 77.

If one includes JTs in the list of favorites, the stack-size result for 9 players if you have 77 is: 5.39 (24.03 for 3 players). But if you exclude JTs from the favorites list I get stack-sizes of 5.48 and 24.22 for the respective numbers of players. These results are sufficiently close that I don't think it matters very much whether or not one includes the hands that are extremely close to true coinflips. Hence, the method outlined above in terms of inclusion or exclusion of coinflip hands (the "favorites" list is to include only those hands that are at least 50.5% favorites).

A few highlights of results on this method:

66 gets a maximum stack-size of 4.48 with 9 players and goes up to 20.69 with 3. With 7 players, it's at 6.27.

AKs is actually always well within the range of possible all-in hands, getting a maximum stack-size of 20.88 even with 9 players (98.01 with 3 players).

Surprizing to me is how far the result goes down for AKo, on which just a few percentage points are added on every hand: For 9 players, I come up with a max. stack size of only 11.85 (about half of the result for AKs!!) and for 3 players it goes up only to 56.89

I haven't run hands like AJs or AJo yet. I do think that should be pretty revealing with regard to "good candidates" in EP--it would be nice to find a hand which gets a stack-size somewhere around 6 for 9 players.

Then also a good MP hand, where I'm guessing something like KQs might end up getting around a 6 (it should be a dog to any AX, right?).

Finally, a hand or 2 in the neighborhood of 6 with only 3 players. But it's just a lot more work when there get to be large numbers of favorites over the given hand (at least using pokerstove as the main odds calculation tool at my disposal).

I guess the last question is: Is anyone except me particularly interested in the results of all this calculation? I'm not planning on going through anywhere near all possible hand matchups, but I'd like to find some representative hands for orientation here. Once these are done, I basically have a spreadsheet format for sharing it with anyone who's interested. But it includes a lot of formulae in its current form and certainly won't just paste into this message board.