Hey, not sure I quite understand your question -- Best for what? What's to check?

So I'm using a program called piosolver which solves GTO betting frequencies (%to bet or check etc.) on no limit holdem flops for different type of pots ( e.g. btn v bb 3bet pot). I wanted a list of generic flops which can roughly cover the whole game (e.g. A high dry board etc.). I stumbled on your list and did some catergorizing and found quite a number of flaws: the list basically had no paired boards for low boards (highest card being 10), you had so many j high paired board (what's the difference between Jc4c4s and 2cJd2d really) and you had AcKhQs and KcAhQs which are the same board but different frequencies) I think ur list is buggy

Any reason you have 875r listed twice with different frequencies?

Suits are different in the examples of "duplicate" boards.

There are no "reasons" for any of the choices beyond the satisfaction of the constraints as described in op.

List is correct afaict.

Why would duplicate flops have different frequencies? And why do AKQ have a duplicate flop, it rarely appears...

So do you mean that this list only has different flops with different criterias (monotone, straight etc), but doesnt actually show which flops are actually important for studying (e.g. whats the point of studying every J high paired flops)

Gordons gold book spreads the flop types for nlh and more or less the same flop types are used in the mastering plo book. The strategy changes are about cbet size variations based on flop richness, that i dont exactly agree with as not to bet some rich or then bet hugh, cant be optimal all the time, and many poor flops i check as they are good flops for ops to check to you and i have little fear from free cards.

And more cbet bluffs are done the poorer and less on the range for the opponent, but the bet sizes are smaller as there is less to protect and get value from. These are the same in lhe, where i am thinking about poor flops, semipoor flops, rich flops, superrich flops additionally to high and low flops with a compromise cutoff on ten high flops, while in nlh it is in jack high flops because of the callers range generally containing many jacks and tens, and less higher cards.

The amount of gaps on the flop influence to the fact that the more gaps, the less good or any draws. There likely are more books that deal with flop types, but we have evolved to ranges there also, no matter what the flop is, if u have all the equity stats.

The next level comes with bots giving us the ev trees also to say if a flop check or bet is better, or a raise. The next it will be adjusted to opponents stats and the more the more stats we have and putting more weight to newer and that moments stats. And the bots already give lines vs. different bet sizes, and this is at 6 max level already, even if the bots dont play every situation well, the human learning can play better than he could without a bot, and better than he could with the current books, and new books are coming and they play stronger than the current books. Most already play well preflop and as cbet, even i am close of getting these all clear and i am just starting out.

Holy sht, just realized OP is W. Tipton

I'm such a donk haha.

But seriously, OP, I'm trying to understand what flops i should be focusing on my game types, and I'm having trouble finding good sets of flops to solve for lets say a 6m 100bb UTG v BTN flat etc.

Well again they're not duplicates -- they differ by suits. One of the properties we said we want our set of flops to have is the correct probability of any particular card coming. (Not rank, card). So suits matter for that ofc.

Anyhow as described in op, this set of (well, probability distribution over) flops was generated by solving a math problem in order to reproduce various properties of the full game. No hand picking of boards was involved. Also this set is not unique... this is just one choice.

As far as choosing some boards to study, staying with some of the high-probability boards in that list seems like a fine start, but "important" is sort of subjective.

Hi Will,

I've probably completely missed the point of this exercise, so apologies in advance. But out of curiosity I ran an EQ evaluation for all hand matchups with the subset of flops to see how accurate they would be.



count 812,175.0000
mean 0.0188
std 0.0141
min 0.0000
25% 0.0075
50% 0.0161
75% 0.0273
max 0.1117

It's actually not bad with just a few in the tails that have ~10% error. The hands with the biggest errors are.

AdTh Vs Kc5c 11.7%
3h2d vs Qh8d 11.09%
3h2d vs Qh8c 10.67%

----

I realize this wasn't really the point of this exercise, so apologies for the diversion but I was bored.

Hey, really cool. Just to verify, did you take into account the given probabilities of the flops being dealt? And you must have dealt with card removal too? I hadn't thought of doing this before, but a std deviation of ~1.4% seems surprisingly good to me.

Well yeah but as 1326 hands form only 169 strategically different preflop groups and 22100 flops form only 1755 strategically different flops, 875r boards are the same no matter what specific suits are (you could add the weigths and just have one assuming you also combine results for combos in the same group when summing up postflop to preflop results).
I don't fully understand the methodology about the choice of this subset but maybe there is something to improve there if we take into account above mentioned facts about preflop hands and strategically different flops.

By improvement I mean having smaller subset with the same (or better) properties.

Did I understand well:

For a pair of hands, XY ZT, if the 103 flops are representative,
eq(XY,ZT) should be weighted mean on all 103 flops chosen with right weight of eq(XY,ZT,flop i)? of course the flops should not have X,Y,Z or T in them and the other weights being changed accordingly?

and doing that only few pairs of hands have an error bigger than 10% of equity?
What SD should we get for a perfect set of flops, exactly zero I guess.

To answer about the equivalence of suited flops, I think you can group flops afterward and sum-up the weights and that is equivalent to taking the equivalence before doing the mean-square approximation, but is simpler.

Yeah I used the weights. In case of invalid flops (i.e. hand contains the same card) I skipped that flop.

At the end I renormalized with the sum of weights from the valid flops I did have. I think that's right.

I'm assuming the error actually decreases as well if you start working with ranges instead of hand v hand

Yup, sgtm

-- properties and frequencies of the full set of 22,100 flops vs. the subset of 103 flops:

unpaired (18,304 combos) -- 82.82% vs. 84.14% (sum of freq given condition)
paired (3,744) -- 16.94% vs. 15.64%
trips (52) -- 0.24% vs. 0.21%

rainbow (8,788) -- 39.76% vs. 39.76%
2-flush (12,168) -- 55.06% vs. 55.06%
3-flush (1,144) -- 5.18% vs. 5.18%

unpaired, rainbow (6,864) -- 31.06% vs. 36.58%
unpaired, 2-flush (10,296) -- 46.59% vs. 42.39%
unpaired, 3-flush (1,144) -- 5.18% vs. 5.18%

paired, rainbow (1,872) -- 8.47% vs. 2.97%
paired, 2-flush (1,872) -- 8.47% vs. 12.67%

high-card flops (w/o 3-straight) (16,440) -- 74.39% vs. 80.96%
3-straight rainbow (288) -- 1,30% vs. 2.02%
3-straight 2-flush (432) -- 1,95% vs. 1.16%
3-flush (w/o 3-str) (1,096) -- 4,96% vs. 5.18%
3-straight-flush (48) -- 0,22% vs. 0%

Axxr (w/o 3-str) (1,536) -- 6.95% vs. 10.80%
Kxxr ("") (1,296) -- 5.86% vs. 4.41%
Qxxr ("") (1,056) -- 4.78% vs. 11.93%

Axx2 ("") (2,304) -- 10.43% vs. 6.03%
Kxx2 ("") (1,944) -- 8.80% vs. 6.34%
Qxx2 ("") (1,584) -- 7.17% vs. 1.46%

Axx3 ("") (256) -- 1.16% vs. 1.04%
Kxx3 ("") (216) -- 0.98% vs. 0.62%
Qxx3 ("") (176) -- 0.80% vs. 0.12%


potential sources of errors are 1) the left handside is wrong; numbers are from http://forumserver.twoplustwo.com/25...holdem-300649/, post #7ff. and 2) i made a mistake in classifying flops; if anyone is interested in checking, you can pm me.

+1 to using the set (and frequencies) of strategically different flops to start with, it seems like this would drop a bunch of the original conditions and make room for strategically important conditions like the ones above. so far though, Will's list looks decentish.

Jxxr ("") (840) -- 3.80% vs. 0.87%
Jxx2 ("") (1,260) -- 5.70% vs. 6.45%
Jxx3 ("") (140) -- 0.63% vs. 0%

btw, does anyone know how many "duplicates" there are in total? one can simply use this set with arbitrary suits (ofc not breaking the flop type though) by adding up the frequencies. the only reason i can think of why someone is interested in such a subset to begin with is to figure out stuff pre, and pretty much the only info that matters is ev at flop dealt, which can be averaged over all combos for each hand type.

Not sure which % you think is right and which you claim is from my set, but for paired flops, both numbers are

1-(48/51*44/50)-3/51*2/50 = 0.16941

so I believe you've made a mistake somewhere.

ok, i'll check my list later today

ok, so i found one flop i misclassified as unpaired, the corrected numbers are:

great, so do you think any of the properties listed in OP are not satisfied?

(keep in mind that OP is mostly about the method, not the particular set found at the end -- other properties could have been specified, although it may have taken more flops to satisfy them all.)

i guess i indirectly checked for conditions 2-4 (2-flush, 3-flush, paired board) and the frequencies are correct (indirectly beause i did not account for specific suits or ranks but rather the total frequencies of flop types).

i'm a bit confused with regards to the frequency of a 3-straight board: i'm getting the true freqency of 3.48% (768/22,100) but getting a frequency of 3.19% in your subset. if the condition "a 3-straight board of any rank comes" is satisfied then this should mean that the total freqency of a 3-straight board coming should also be the same as in the full set, right?

any idea?

i have a question on methodology: how many conditions/frequencies are there, what is P? i'm not sure how you define the last two conditions, so 52 + 4 + 4 + 13 + 12(?) + ? + ?).

following approach: 1,755 distinct flops. we make a vector B containing our new P conditions -- example:

- any particular rank comes (13)
- a 2-flush comes (1)
- a 3-flush comes (1)
- a paired board with the pair being a particular rank comes (13)
- a 3-straight board of any rank comes (12)
- (other, strategically relevant conditions)

the frequencies in B are calculated as if every flop (out of the 1,755) has the same(!) probability of falling. for example, there are 24 strategically distinct flops which are AAx (which is one of the above conditions), so this means that the frequency for that condition is 24/1,755 = 1.37%.

we then make a Px1,755 matrix as you have explained, but instead of putting a 1 when a flop satisfies a condition, we put in: the true frequency of that flop (type) falling divided by 1/1,755. so there are 12/22,100 combos of the flop falling AA2r = 0.0542%, so whenever the flop AA2r satisfies a certain condition, we write 0.0542%/(1/1,755) = 0.9529 into the matrix.

is this correct? if yes, how "tough" is it to get a good x?

Interesting thread, I think the idea could be pretty useful. I've been using PioSolver (postflop equilibrium solver) to do some preflop calcs by running the solver over 400 random flops, see what hands are +EV/ -EV, etc. I think choosing a well-constructed subset should be faster and more accurate than just random flops.

I followed the methodology in the OP, but changed a couple things. I'm using suit isomorphisms and added the constraint "any particular 2 pair hits," which increases number of flops by quite a bit. I'm at 170 flops under these conditions.

Hey thanks for looking at this. I'll check it when I get a few mins.

Do the frequencies of these sample flops in this subset add up to 1? It should, right?