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.

Like I said, not going to make many journal-type posts, but here's one:

Spent the last two weeks of July grinding the headsup turbos during the 1.5x vpp promo and playing a bit of PLO. Neither went very well - some shots at higher PLO crahsed and burned. The hu turbo regs went from chastising me for getting in their games (apparently game-selection is a big deal to them and regs are supposed to avoid each other) to calling me a fish (thanks sharkscope!). I'd like to think I ran bad, but who knows. I've definitely got some work to do on the shorter-stacked open-shove and 3bet-shove game. Some game theory stuff I'm doing on raise/reraise ranges with 20ish BB may turn into an article at some point. Next couple articles will be on PLO; something on AAxx play is probably next.

But I racked up some vpps (~50k since getting home from Vegas) and am finishing July at 375k vpp, which is close enough to where I wanted to be to feel ok. I need ~125k/month from here out, or ~4k/day. August plan is to play ~100k hands of 2/4 PLO, mixing in some other stuff here and there. 2/4 PLO gets ~1 vpp/hand.

Also took a baby step toward getting my Stud High game stronger by playing a bit of 10/20. Really need to work on handreading there.

Final note, I just agreed to do a plo 3-video guest series for pokersavvy. First one should up fairly soon. If all goes well, I'll probably be working more with them.

Pot-limit omaha is a game of small edges. A lot of the time that two hands hit a flop hard enough to play for stacks, the underdog has at least 40% equity, and it is relatively rare for a hand to be an 70%+ favorite. The most common case of postflop domination in holdem is kicker-domination, such as AK v AQ on AT5, where the AK has 87% equity. Analogous situations involving one-pair and two-pair hands do exist in PLO, but they are less relevant because the underdogs in those spots rarely have a playable hand. For example, AQxx dominates A987 on an AQx board (~90/10), but a competent player will never put much money in with the A987. Even two pair against a worse two pair, which is a very bad spot for the lower two pair (~85/15), will rarely result in a big pot. Although there are a few situations (especially in aggressive shorthanded games) where it can be correct to play a big pot with less than top two, the underdog in those matchups will usually have a lot of outs. For example, with three overcard kickers and a gutshot, AKQT is only a 53/47 dog to dry top two on an A98r board and is rarely in terrible shape against wraps or other pair+draw combos.

The main cases where one made hand is a huge favorite over another and a lot of money often gets in are set vs top two, set vs set, straight vs straight, and flush versus flush. Dry top two has only ~25% equity against bottom set, is in worse shape (15%) versus middle set, and is totally dominated by top set (3%). Similarly, dry bottom set, with a single out to quads, has only 5% equity against a higher set. Non-nut straights and flushes are clearly drawing nearly dead against the nuts. In all cases, the underdog hand can improve its chances by having additional draws (which not only boost its equity when dominated but also can serve as blockers when facing a big draw), but in general it is easy to see that these are spots to avoid.

There is a second class of situation where one hand is a huge favorite which is less obvious but at least as important - draw domination. We are used to thinking of the classic PLO matchup as (the usually 60/40ish) big made hand versus big draw, but some draws are bigger than others and sometimes two draws will play for stacks. Fairly often, the stronger draw is a dominating favorite. One case occurs between two wraps where neither hand has a flush draw; the higher wrap or the highest pair is a large favorite. For example, on a JT5r board, KQ9x is an 75/25 favorite against Q98x. The combination of KQ9x having a higher high-card when both hands miss and having Q98x’s draws covered (only the 7 is a scoop card and the 9 and Q are losers) is deadly for the lower draw. Similarly, KQJ9 is a large favorite (70/30) over KQ9x and AQ98 has 71% equity against Q98x.

The situation can get worse for the dominated hand if a flush draw is involved. Even with the dangler, Q984 looks like a monster hand on JT5 (and it’s even favored over top set), but when it runs into KQ98 (better wrap, better flush draw) it has only 16% equity. Without the flush draw blockers, it’s 14%. Even KQ98 with diamonds versus KQ98 without diamonds, a simple freeroll scenario, is a 70/30 matchup.

What is the practical benefit of this knowledge? When holding a potentially dominated draw, we can’t know if we are facing the hand that has us crushed or the hand we’re flipping with, and when we compare our hand to a range that includes sets, two pair, and so on, we usually have something like 40-45% equity, which is easily enough to get it in once we’ve (correctly) made a semi-bluff bet or raise and gotten action. But consistently getting it in on the 45% end of a 55/45 (versus our opponent’s range) is a sure way to go broke playing PLO. Sometimes these situations are unavoidable, but we can minimize their frequency with a couple basic strategies. One, avoid playing the hands that get in these spots most often. Hands with danglers are the most obvious danger hands – if you consistently run three-card hands into four-card hands you’ll be dominated often (KQJ9 v KQ9x). Also dangerous are gapped runs with the gap near the top. As we saw above when KQ9x dominated Q98x, you are more likely to flop nut draws when holding gapped runs with the gap at the bottom (i.e. KQ9x, KQTx, 9875, rather than Q98x, KJTx and 9765). Two, when you do get stuck with a potentially dominated draw, look for ways to increase your postflop flexibility, especially in position. While it is usually fine to just jam and get it in when you have a 13-out wrap or better, keeping the pot smaller by calling rather than raising may give you ways to steal the pot from a better hand. Remember that a lot of the time two draws get it in on the flop, the dominating draw wins with a hand such as one pair that can’t take a lot of heat on later streets with deep money.

So far, I’ve focused on situations that occur in heads-up pots, but dominated-draws can also run into trouble in pots where three hands flop strong. A three-handed pot introduces an additional danger, which I’ll call “draw duplication.” A lot of the equity you have heads-up versus a set is dependent on all your cards being live. When the set has a couple blockers, your equity goes down a little. The situation is much worse when you are in a three-way pot with another similar draw and a set. Even if your draw isn’t dominated, you lose a lot of equity both because your outs are dead and because you’ll chop the pot a lot of the times that you make your hand. Even worse is when your draw can be dominated by the other draw, which brings me to a hand played by a student of mine recently:

Seat 1: CO ($1178.95 in chips)
Seat 2: BN ($892.90 in chips)
Seat 3: SB ($280.15 in chips)
Seat 4: BB ($497.90 in chips)
Seat 5: UTG ($2051.60 in chips)
Seat 6: Hero ($400 in chips)
SB: posts small blind $2
BB: posts big blind $4
*** HOLE CARDS ***
Dealt to Hero K T 8 J
UTG: raises $8 to $12
Hero: calls $12
CO: calls $12
BN: calls $12
SB: folds
BB: folds
*** FLOP *** 4 9 Q
UTG: checks
Hero: bets $48
BN: raises $147.30 to $195.30
UTG: raises $489.90 to $685.20.

Hero has a 12-out wrap and the fourth-nut flush draw. He makes a straight by pairing any of his cards, and all of his straight outs but the Ks are to the nuts (he has the T and 8 and the J gives him a straight flush), so he has 11 nut-outs, although most of them have to dodge a redraw if he’s facing a higher flush draw. He even has a backdoor flush draw. In most situations, this is a huge hand. Obviously you would prefer a higher flush draw, but having blockers and a straight-flush draw is the best situation you can be in if matched up against a better flush draw. This is certainly a hand that is susceptible to being dominated, or at least freerolled, but in a heads-up pot it is an automatic stack-off for 100bb because most of the time most of its outs are live/unduplicated and it’s on one side or the other of a 55/45.

Here, however, Hero should be facing two very strong hands. In particular, the UTG player’s range should not be much wider than top set or a monster draw such as AKJT with nut spades. To re-re-pot 200bb deep with bare middle set, JT99 no spades, or a dry KJTx would be really bad against most opponents. And although the first raiser’s range can be a bit wider, he almost always should have either a draw similar to ours, the nut flush draw with a pair/gutshot/open-ender, or a set. With two opponents, it is fairly likely that either one of them has a hand that dominates us (such as KJTx with higher spades, or a set and the nut flush draw) or that the two of them combined dominate us; that is, that between them we’re facing the same straight draw we have, a higher flush draw, and a set. The worst case might actually be facing two other draws, because it would be very likely that one or both opponent’s would have a higher flush draw or a pair, putting us in very bad shape.

In none of the above cases do we have much more than 30% equity, and in some we have a lot less. Against a wrap with better flush draw and a set of queens we have 19% equity. Even against a random KJTx and a random QQxx, we have only 34% equity. The only good case is to be against two sets, but even then 99xx will often have blockers and duplicate some of our straight outs. Altogether, if we assume both opponents are decent, we likely only have ~30% equity versus their ranges; what appeared to be an easy shove is a very close decision. Factor in that the first raiser may fold, giving us overlay and cleaning up our outs, and we probably should call. But it is much closer than it looks at first glance. Even without getting deeper into the math, it should be clear how much better off we’d be with the K-high flush draw (due to dominating the other draw rather than being dominated) and how much worse off we would be if we had a lower wrap such as JT8x.

Overall, while the strength of a draw is somewhat important when facing a set (it is always better to have more outs or blockers), it is even more important when facing another draw. Consistently being the one holding dominating draws in these matchups is an important element in winning PLO play.

Hello all, I joined tworags and started this blog today. The purpose of this post is to introduce myself and describe what I intend to do with the blog. My name is Tom Chambers and I've been playing poker full-time for a little over two years, mostly online. I use the name Entropy xx on Pokerstars, where I play the majority of the time. I use the name LearnedfromTV on most other sites, as well as here and on two plus two.

I play a variety of games, including 2/4-5/10 PLO, 10/20-30/60 HORSE, and all forms of nlhe. For the past few months I've put in more hours at PLO than anywhere else, and I expect that to continue.

I do not plan to do much journal-style blogging; my primary purpose is to write strategy articles, mostly on general poker theory and specific topics in games other than no limit holdem. A couple small exceptions: One, I am chasing supernova elite on Stars and will periodically update my progress. Two, I am planning to play the $50k HORSE and most/all of the Championship Events at next year's WSOP and will be putting in study and play hours each week toward preparing for that goal. Particularly, I'll be working at stud high and limit holdem, which are easily my worst two games. Again, I'll use the blog to update progress.