Pokerstars opened up 10/20 badugi a couple days ago and I'm addicted. I was playing around with starting hand strategy today, but I'm still sorting out relative strengths of 2-card, 3-card, and badugi starting hands. When no one is pat, it is so important to have the best three-card hand (unlike in triple draw, if you and your opponent draw 1 and both miss, whoever had the best draw always wins; so e.g. that final draw is like 80/20 instead of 55/45 or 60/40).

Getting there is by far the most important badugi skill, but 2a and 2b seem to be not missing bets when you have the best 3-card and avoiding getting stuck in pots with second-best (or worse) 3-card, especially in headsup pots. If people have good snowing frequencies, the pat vs draw situations get more interesting, but the Stars game doesn't seem to have much of that.

Only ~6.5% of hands are dealt badugis, and 3-card 6s or better are only another 6%, so while a full-ring EP strategy (I think Badugi is usually dealt 7-handed, though the Stars game is 8-handed) of folding everything but badugis and strong 3-cards makes sense, in late position and in shorthanded games it's pretty obvious that's too tight.

The question is how to compare good 2-card starting hands and rough 3-card starting hands. Clearly A23x > A27x > A2xx, but it isn't clear to me where, say, 654x should fall. It's a lot harder to turn a 3-card into a badugi than it is to turn a 2-card into a 3-card. Even though 765x will make more badugis than A2xx, A2xx will end up with a lot of better 3-cards, and 765x can't really promote itself to a better 3-card because it's so rough. On a single draw, A2xx has 10* outs twice to improve to beat 765x, and 765x only has 10* outs once to improve to a badugi, and a couple more to notch A27 (but none of the other three-cards A2xx can make); with three draws it has to be the equity favorite. It's not that hard to figure out a one-card draw's equity versus a pat badugi. Finding, for example, the weakest 3-card hand that is favored over A2xx would be a lot trickier, but the general problem of comparing 2-card hands to 3-card hands is something I want to work on.

* - I didn't account for card removal (blockers in the other hand) there, which affects the 2-card hands improvement chances a bit more than the 3-card's.

A bit of trivia: A2xx is the single most common starting hand. Any specific badugi is ~ 0.1% of the dealt hands, A23x is the most common 3-card at ~0.33%, and A2xx is ~1.3%. (The best two-card and three-card hands are more common than worse hands of the same type because you always discard the worst suited card e.g. As2c3d4d is played as A23 not A24.)

912k vpp and counting...

Flop texture is too big a topic to do it justice in a couple paragraphs, but if we ignore some nuance we can separate PLO flops into three big categories: dry flops, draw-heavy flops, and obvious-nuts flops. The archetypal dry flop is K82r; there's no way besides sets and pairs to hit it, and there are relatively few hands that hit it. There is a wide range of draw-heavy flops but a good example is Th9h5s; there are a ton of ways to hit this flop, including a variety of wraps, combo draws, and pair+draw hands, in addition to sets and top two. Obvious-nuts flops include monotone flops and super-connected flops like 654r. They look a lot like the draw-heavy flops, and in holdem they play essentially the same because in holdem the nuts isn't very likely and all of the obvious-nuts flops also contain a lot of pair+draw possibilities. But in PLO, the nuts is much more likely (and on monotone flops, there are no one-card flush draws), and as a result those flops share some characteristics with the K82-type flops. Specifically, both of those flop-types differ from the Th9h5s flop-type in that there are many fewer hands an opponent who doesn't have the nuts can easily be aggressive with. The strongest semibluffing hand on a monotone flop is a set, and it's a solid dog to a flush; the strongest combo draw on a 6s5s4c flop (flush draw+gutter) is an 11-outer versus 87.

What we're really talking about here is the distribution of equities among the hands that hit the flop reasonably hard. What separates Th9h5s is both that a wider range hits it, and that if you run equity calculations matching up individual hands in that range with the entire range, you get a narrower distribution of outcomes. In other words, there are a ton of hand vs range 55/45s on that kind of flop, and although there are individual hand versus hand domination situations (set over set, dominating draw) there really aren't any hands strong enough to dominate an entire range. At the other extreme is K82r, which doesn't have any semi-bluffing hands and where KKK has massive equity versus any range. Stretch from the driest flops to the slightly drawier dry flops, like AJ4r, and similarites emerge with flops like 654r and monotone flops; there are fewer combinations of hands with decent equity against the nuts, and there is more separation among relative hand strengths.

These categories are too restrictive and the dscussion above isn't sufficient to cover all important points. For example, a main difference between AJ4r and 654r is that there are 3 available two-card combinations of AA on the former and 16 of 87 on the latter). Also, flops like Th9h6s have a very vulnerable flopped nuts and are essentially a slightly different kind of draw-heavy flop. That said, just understanding this basic distinction will help with some flop decisions. Some of the same factors that may make K82r a good flop to cbet or to bluff raise apply to 654r. And you may find opponents are predictable in similar ways on AsJc4s and 6s5s4c; moderately passive opponents don't semibluff these flops much, and check-call a lot of decent hands that they will fold later.

Update on a few things, along with some strategy ideas that I haven't developed in writing and organized into an article. Unfortunately, writing articles can't be a priority right now because of Elite and putting my coaching program together. Regarding elite, I passed 800k yesterday so I'm in comfortable range but it won't be easy. 5k vpp/day the rest of the way. Since I started playing the 225-335 sngs I can do 1k+/hour so it isn't that tough. I wish I could play more plo though.

Between a marathon Sunday (from the warmup to the horse with tons of sngs all day) and a sick latenight 5/10 plo session Monday after I thought I was done for the day, I did 18k vpp in two days. So I've taken it relatively easy yesterday and today, made a hu plo video for Pokersavvy, and am working on my plo coaching program, which I am aiming to have ready for new students in January. I've been working through the first few lessons with current students and am pleased with the results, as well as the tweaks I'm able to make as a result of the sessions.


The rough sketch of the program as it is today:

Each Unit is 4 lessons long, comes with an outline for each lesson and other supplementary material; a starting hands spreadsheet, a flop textures spreadsheet, etc. The full program is 30 lessons and is designed as a whole, but the units can be stand-alone too so people can choose to skip parts or to just do one unit or to choose a few. I'm also planning supplementary units on headsup play and shortstacking/tournament play/cap game play, and students usuallysupplement the program with sweat sessions and hh reviews.

Intro Lesson - three goals; to introduce my approach to thinking about poker and plo, partly because it explains why the program is structured as it is, to go over the outline of the program, and to cover several core PLO concepts.

Unit 1 Preflop: sets/flushes/straights/big draws are the target hands, so we evaluate starting hands in terms of their connectedness, suitedness, and “high card strength” in order to organize sensible hand categories/rankings, discuss the conditions under which to play different hand types and the situations to encourage with different hand types, and conclude with a workbook-style lesson covering common situations.

Unit 2 Flop Texture: Organize flops first into connectedness groups (based on the number and type of wraps and straights possible) and then into more functional categories that account for suitedness, high-card strength, and paired boards, and conclude with a workbook-style practical lesson.

Unit 3 Math: Twofold purpose: One, to supply the combinatorics of the interaction of preflop hand and flop; that is, how often does stuff flop stuff? Two, to provide core equity math in a easy reference format, including basic hand versus hand and hand versus range matchups, as well as the stack-size-dependent math of reraised pots, including versus likely AAxx. Also includes a bit of implied odds math (e.g. calling a flush turn with a set to fillup, how often does he need to payoff?).

Unit 4 Game Theory: Lessons build from toy game examples to generalized plo situations (such as naked ace lines) to situational analyses using complex ranges.

Units 5-7 Postflop Each Unit “Flop” “Turn” and “River” splits into three pieces, an introductory lesson defining the general parameters of play on that street, two lessons covering the meat of the (pieces) of lines to be discussed, and one workbook-style lesson heavy with examples. Each unit contains some emphasis on transitional ideas (flop-turn combinations, turn-river combinations)

Wrap-up/Opponent Profiling – review core concepts, a bit of game theory (strategy/counterstrategy), use pokertracker stats and other information to review exploiting opponent tendencies.

Pricing starts at $250/lesson but there's volume discounts built in for people buying the whole program or multiple units. Exact details still uncertain.

Pieces of it are completed, but I'm about two months away from having the whole thing done, at which point I'll advertise on 2p2 coaching forum and start in with new students (post-PCA, probably) If things continue to go well with the coaching, I'll look into refining the material into an audiobook/ebook combo later next year.

It's a big project, which means it's basically the only project I have right now besides chasing elite, and making a video here and there, best case, one/week.
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And for anyone who does read here for strategy content (or at all!), I'll throw out a couple important ideas worth thinking about.

1. In PLO, the most important piece of information regarding how strong a starting hands is, and more importantly, how to play it, is the distribution of equities that it flops. Some hands, like dry AA/KK (KK73r) flop the very strong set occasionally, and the uncomfortably mediocre unimproved (and unimproving, due to crap sidecards) overpair. There's a huge separation; if the word didn't already mean something else, you might say KK73s equity over the range of flops is "polarized." QJT9ds, on the other hands, flops a very smooth distribution of equities. From flopped straight with redraws to monster combo draws to two-pair+openender to the medium draws to the variety of one pair+ hands. A QJT9ds yields a lot of postflop flexibility across a wide range of flops.

2. Whether to call a 4 bet preflop (presumably AAxx who will shove any flop) is a function of two factors related to the note above: how often do you flop good enough equity to stack off, and what is your average equity against AAxx when you do. Those two %s together are what matter, not preflop equity. The action will happen on the flop, so the flop equities are what matters. You are calling the 4bet not based on your odds of beating aces, nor on your odds of outflopping aces, in the strict sense. You are calling in order to be placed in the situation where face a shove you can call x% of the time, of which times you'll win the pot y%. If the value of that situation is greater than the cost of calling the 4 bet, call; if not fold. If this seems basic, good, because it is. But a lot of people don't play like they understand it.

This is a rough draft of an article I'm working on, the first in a handful of game theory articles. Some of them will deal with naked ace theory in PLO, and the scenario this one covers is similar. Important Disclaimer; the work here and in future articles is my own, but I should reference Chen and Ankenmann's Mathematics of Poker, which is the bible of poker game theory and covers, in some form, anything I'll write about.

This article will cover a simple contrived multi-street nuts-or-nothing betting scenario that can be solved using game theory.

The rules are simple. Pot-limit structure where you must bet pot. The game: two opponents Player A and Player B each hold a single spade. Pot contains 1 unit. Player A has and is known by B to have, the Ten of Spades. Player B's spade is unknown.


One street version


Since B has perfect information about A, the game begins with a checking to B, who then bets pot or checks behind.

So B has a 4/12 chance of beating the T, and an 8/12 chance of being beaten; if there were no betting, B's range has 33% showdown equity, A's has 66%.

In order to find the equilibrium solution for the play of this game, we must find the frequencies A can choose that will make B indifferent between his two options, and vice versa.

That is, B needs to bluff with a frequency that makes A indifferent to calling; and A needs to call with a frequency that makes B indifferent to bluffing.

We know B bets every time he has J,Q,K,A. To bet only those and check all losers would be expoitable; A would never call, and would claim the pot the 66% of the time B checks behind.

If we let B bluff x of the remaining 8 times, we find that if A calls he wins 2 units x/(x+4) of the time, and loses 1 unit 4/(x+4). So his total profit is determined by (2x-4)/(x+4), which equals 0 when x = 2

This makes sense. Since he's laying 2-1, he should have a vbet:bluff ratio of 2:1.

What is A's optimal response?

His goal is to make B indifferent to bluffing. The pot size is 1 unit and B's bluff costs 1 unit, so A makes him be indifferent to bluffing by calling 50% of A's bets.

What is the overall equilibrium strategy:

A bets all his winners and 1/4th his losers (say 9 and 8, it doesn't matter), and A calls half the bets.

12 cases:

1/2 the time B checks; A wins 1 unit, B 0 units
1/12 of the time, B bluffs and A calls; A wins 2 units, B loses 1
1/4 of the time, B bets and A folds; A wins 0 unit, B wins 1 units
1/6 of the time, B value bets and A calls; A loses 1 unit, B wins 2 units

So A wins 1/2 + 1/6 + 0 - 1/6 = 0.5 units
And B wins 0 - 1/12 + 1/4 + 1/3 = 0.5 units


A and B both have 50% equity. B gains 16.66% equity from the street of betting + the information advantage
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Two street version:

For simplicitly, call the first street the "turn" and the second street the "river".

Let's assume (for now)that B will always value bet the first street (ignoring that it is possible that the most profitable strategy would include checking the turn with the nuts some % of the time, in order to be more likely to gain a river bet)

Because B has perfect information, A's options are to check-fold the turn, check-call then check-fold, or check-call twice. B will always value bet his winners on both streets. When he has a loser he can check twice, bluff once and then check, or bluff twice (He can't check and then bluff, because he doesn't have a river valuebetting range after checking)


B's optimal bluff:value bet frequency on the first street is the strategy that will make A indifferent to calling or folding, given that the result of calling is being placed in the river situation where A will often face another bet.

If A check-folds he nets 0 units; if he check-calls he will be placed in the river situation described above. The pot will now be 3 units, but the exact same equilibrium strategy determined above will hold, provided that B has enough bluff hands in his range; that is, provided he bluffs at least 2 hands on the

turn. B will bet all his winners and 2 of his losers (2:1 value bet:bluff ratio), and A will call half the time.

B will bluff the turn with some number x of his 8 bluffing hands. If called, he will then check behind x-2 of those hands. So A's river outcomes are

(x-2)/(4+x) he wins 2 units (the 1 unit in the pot before the turn, and the unit B bet on the turn)
1/(4+x) he wins 5 units (the 1 unit in the pot before the turn, the unit B bet on the turn, and the 3 units B bets on the river)
2/(4+x) times he loses 4 units (the 1 unit he calls on the turn and the three units he calls on the river)
3/(4+x) he loses 1 unit (the 1 unit he called on the turn before folding the river)

So his total outcome is [2(x-2)+5-8-3]/(x+4). For him to be indifferent between calling and folding to the turn bet, we need to set this equal to 0 (the outcome if he folds the turn) and solve for x.
2x - 4 + 5 - 8 - 3 ----> 2x - 10 = 0 -----> x = 5

So B's strategy is to bluff 5 of his 8 losers on the turn, along with betting the 4 winners, and check behind the other 3.

A's goal is to make B indifferent to bluffing or checking the turn. If we let A's turn calling frequency = y, B's outcome for bluffing the turn is to win 1 unit (1-y)% (when A folds) and to enter the river game y% (when A calls).
In the river game, A calls half the time.

If B bluffs the turn and A folds, B gains 1 unit. If B bluffs the turn and A calls, the outcome is dependent on the river action. When he checks behind he loses 1 unit, when he bluffs and gets called (half the time) he loses 4 units, and when he bluffs and succeeds (half the time) he gains two units.

So if A calls turn y% of the time, the net outcome of B's bluff is

1-y +1 units
3y/5 -1 units
y/5 -4 units
y/5 +2 units

(1-y) - 3y/5 - 4y/5 + 2y/5 = (1-y) - y = 1 - 2y; y=1/2.


So, add it all up and we get the following outcomes:
1/4 of the time B checks turn; A +1 unit, B 0 units
3/8 of the time B bets and A folds the turn, A 0 units, B +1 unit
1/8 of the time A calls the turn, B checks the river, A +2 units, B -1 unit
1/12 of the time A calls twice and loses; A -4 units, B +5 units
1/24 of the time A calls twice and wins; A +5 units, B -4 units
1/8 of the time A calls the turn and folds the river; A -1 units, B +2 units

So A's total win is 1/4 + 0 + 1/4 - 1/3 + 5/24 - 1/8 = 0.25
B's total win is 0 + 3/8 - 1/8 + 5/12 - 1/6 + 1/4 = 0.75

In other words, A's share of the pot goes from 66% with no betting to 50% with one street to 25% with 2 streets.

I'll leave the 3-street version for another time.

I'm in the midst of a organzing a bunch of projects, and finding myself with only 24 hours in a day, so I decided to write a post in the hope that it will help me prioritize.

Job 1 right now, and until the end of the year is Supernova Elite. I have 607k vpp as of this moment, having done ~115k each of the past two months, which is solid but not enough to get me there if I repeat it the next few months. 4k/day is relatively easy, consistely doing 4500+ is a lot harder; and I need somewhere around 4500/day, depending on how many days off for holidays, to get it. 9 tabling plo gets 550/hour, 12 tabling around 700, and I've recently taken up 18ish tabling the 114 sit and goes, which gets ~1k/hour. When the PCA Steps kick in full speed that will give a boost which will be much needed in December. Right now I give myself about a 70% chance of getting elite; if I get close enough I'll play sick marathons if necessary to close it, but too many bad days (losing sessions early in day often leads to 3k vpp or worse days; family stuff left me getting 1k a couple days ago). Too much of that happens and ~mid November I'll decide it's hopeless. There's only 80 something days left, and I'll basically have to play ~8 hours a day to succeed.

Two priorities dueling for number 2 are the coaching videos I've been making for pokersavvy and the plo theory coaching program I'm putting together. There's a decent amount of work involved with most of the videos I want to do; selecting hands that fit a theme, incorporating powerpoint, and so on. The shell/outline of the coaching program is in place (viewable in the pokersavvy coaching forum; maybe I'll post it here too), but some of the supplemental work is still in progress. With new students on board in the next week or two, I need to get some more work done on a few projects, particularly in the short term the starting hands sheet and the flop textures sheet, which are mostly ready but appear early in the program.

There also are several articles I want to write, either to be posted here or at Pokersavvy. I've done a bunch of game theory work for plo, mainly focusing on the naked ace play but some other stuff as well, and I will turn that into a series of articles at some point. I also want to write articles on a handful of other topics, such as AAxx play, 3betting, and board texture. And way down the list are artcilaes on key horse concepts.

In the back of my mind is the possibility that all of this work will end up in a theory book I want to write next year. Students who do the full theory program are basically going to get a book's worth of supplementary content, and if the situation is right I'd clean that up and actually write the book. There's a weird market force at work here because the full twenty lesson program is definitely worth a few thousand to dedicated students who can use it to step into midstakes games and win ($3500 at that moment, likely $5k jan 1 or therabouts), but of course packaging that material in a book makes it worth $25 or whatever to the mass market. So I'm not sure yet what I want to do. There's another book possibility out there as well, but nothing concrete or that I can really consider until the year's over.

I'm also doing a handful of things regularly outside poker; working out, playing in tennis and basketball leagues, plus that whole social life thing. So it all adds up to probably not doing everything I'd like between now and the end of the year. Regardless, decent chance I'll be posting more over here in the near future. I'm not sure what might be done at pokersavvy with my articles, and I'll probably be debuting them here, at least the next couple, which I hope to write in the first half of October.