Median Best Hand Part II

[PairTheBoard]

You know I'm not finding Sklansky in the Writers Section. I wonder what's up with that.

PairTheBoard

[Aisthesis]

That's strange. I'm almost sure that it was in cardplayer. If UPF is archived back that far, I think Lee was the one who originally referred me to it. But it was probably something like last December or January.

[PairTheBoard]

It was definitely in the CardPlayer. I don't know why Sklansky is not showing up in the Writers' Archive. I don't recall if he did a follow up article but in the one I remember David worked out the math for one example and discussed the principles behind the idea that you could compute a +EV for the play even assuming your opponent played perfectly against you. Since your opponent won't play perfectly the real EV will be even better.

David invited people to carry out the calculations for all hands. That discussion took place here at 2+2 where I believe K ended up doing the work. It would be nice if someone could find those threads and post a link to them that works. When I get the ambition I'll look for them myself.

PairTheBoard

[Aisthesis]

Well, fortunately I was able to get back to this sooner than I expected.

After seeing what the variables were, I figured it would be easiest to just throw in the little pairs as favorites against A9s and recalculate that one for all numbers of players 3-9. So, I set up a spreadsheet that will do most of the math for any hand, just filling in a few values. I wish someone could check my math here, as with the "improved" formula I do get a significant discrepancy for stack-size in comparison to my previous calculation: For 9 players, I come up with a stack of 3.14 times pot this time on A9s. I've double and triple-checked everything myself, though, on this, so I'm fairly confident it's right.

Anyhow, here are the results for maximum stack-size on A9s for various numbers of players:

9: 3.14
8: 3.80
7: 4.67
6: 5.91
5: 7.79
4: 10.94
3: 17.26

So, one might say that A9s actually starts more frequently becoming attractive for an all-in (in the given tournament shortstack situation)at button-3. It's more than adequate for shortstack situations when folded to you on the button.

I'll run through and post some results for further hands as I do them. The only real legwork now is just running through pokerstove all of the hands superior to the given hand.

[PairTheBoard]

Seems to be a descrepancy for A9s for it to become more attractive after you've thrown in more hands that can beat it. I would cross check the work between the two calculations to see what's going on. If the only change to the rankings is to throw in the small pairs then A9s is no longer Medain Best and probability of getting called goes up.

If you get tired of plugging Hand Matchups into PokerStove I'd suggest just using aproximations based on the Hand Type Matchup win frequencies.

I had a situation like this last night. I was in a multiway tourny and getting short stacked with K4o on the button - everyone folding to me. I probably should have gone All-In for the blind steal but decided to wait for a better shot. I ended up getting blinded down so bad that when I finally got a decent toss up I would still have been too short stacked to do much good had I won it. From the work you're doing I'm beginning to think it is a BIG MISTAKE to underestimate the value of everyone folding to you when you're on the button or cuttoff in these situations.

PairTheBoard

[Aisthesis]

Interesting result for JJ: I ran this hand in 2 different ways. On one scenario, I have AK and AQ also calling. On the other I have both of those (underdog) hands folding.

Maximum stack for 9 players with the call turns out to be 18.73 times the pot.

Maximum stack for 9 players with AK and AQ folding is only 13.64. Quite a difference there, too.

I really don't think there's much of an alternative to running the individual candidates through pokerstove, as the exact percentages will make quite a difference in the calculation. I already do have the imprecision of not considering the special situation (that you mentioned) with SB and BB--although the JJ calculation indicates that additional callers with inferior hands will (not surprizingly) actually improve your EV and hence make the maximum stack larger.

As to checking the math: Probably the easiest way would be to run an easy hand like JJ against the superior pairs and just see if you get the same results I do (at this point, I'm just copying all the formulae to the next line in my spreadsheet, so if it's right in one instance, it should be right everywhere). In running JJ against the superior pairs, I always picked a pair where exactly one of the cards was the same suit as one of the jacks (evening out the presumed difference with same suits vs. different suits, which probably isn't a big deal in this case anyway).

Here are the formulae I'm using:

B = number of superior hands (just counting through them), so for JJ (counting all the QQ-AA hands left in the deck), there should be 18 of them.

T = number of hands that are a tie (for any pair, there's only one other pair that's tied with it, but for A9s, T=3).

n = number of players to consider (for 9 players, n=9)

f = average percent favorite of the superior hands over the given hand (for JJ, I come up with 81.67% against QQ-AA).

With those variables, my formula for each column is this:

p = [(1325-B-T)/1325]~(n-1) This should be the probability that no one has a superior hand and hence all will fold.

s = 1 - [(1325-T)/1325]~(n-1) This is the probability that there is some hand that's a tie with yours (it's always pretty low).

Stack-size is then figured as:

X = p/[(2*f-1)*(1-p-s)]

I got that by figuring the break-even for stack-size as when

p = (1-p-s)[(f-(f-1)]X

on the one side of the equation are the expected winnings for the full table folding. On the other side are the expected losses when someone calls with a superior hand.

Do my formulae look right to you (again, with the simplification of ignoring the SB and BB specifics)?

If so, I'd be most interested to see if you get the same results on JJ. That's a fairly easy one to do since you only have to consider QQ-AA. If you get the same results as I do on that one (I have it figured all the way through to 3 players, just didn't post since it's clearly going to be good enough already with 9 players), I think everything else will be completely sound under the given assumptions. If not, then it would be a good idea to figure out where I might be off before running a bunch of other hands through the spreadsheet.

[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.