The goal of determining an optimal (in the game theory sense) bluffing frequency is to make our opponent indifferent to calling or folding.  This makes the frequency dependent on two factors, our hand and the pot odds (and in no limit games we can play with the latter).  In practice, our opponent won't usually call with the exact frequency he needs to in certain situations.  We can adjust our bluffing/valuebetting frequencies to account for this (an exploitative adjustment) and readjust if he seems to be calling/folding with different frequencies as result.

Let's take a look at a simple spot where we always either bet (for value and as a bluff) or check/fold (we never check/raise or check/call) against a hand we suspect is somewhere near the median of predraw pat hands, say something like a T8753 (but we don't know exactly what it is).  How often should we bluff?

Say we have the monster of all draws, 7432.   There are 5 ranks of cards that are good for us (so 20 cards max) and we likely on average have somewhere around 17 outs to improve.  If we bet half pot (giving Villain 3 to 1) we need to bet for value 3 times as often as we bluff to make Villain indifferent.  So with a 17 out draw, we'd want to bluff 5.66 cards.  It usually makes sense to do this with the worst of our catches (even though in this situation we're assuming we have no showdown value with anything worse than a T and could choose randomly as long as we were consistent).  So maybe we'd bluff on 7,7,7, the 4,4 not of the same color of the 4 in our hand all the time, then 2/3 of the time we could bluff the other 4 (and we could do this 2/3 randomization in various manners).  Say we notice after a few times in this spot (or similar spots) that this player never/rarely folds when we bet.  We can exploit this by bluffing less (if we don't want want it to be too noticeable) or not at all (and if we stop bluffing against this player we need to be vigilant about how this looks to other players in the game).

Now, if instead of betting half pot, we decide to overbet 2x pot.  Villain only gets 3:2 and we should be bluffing twice as many (11.333) cards (so maybe 77744433322 and 1/3 [via some randomization] of the 2 of the same color in our hand).  In this case we may find our opponent is never calling and we should increase the number of times we bluff until Villain catches on.

We can use these ideas in other circumstances as well and to think about how we should establish our various ranges in the different drawing spots, though they require more rigor.

Here's a table you can use to interpolate:

 

I'm not sure about the answer to this, but it's one I've posed in multiple forums without really a clear resolution.  It relates (mainly before the final draw) to a spot where we're 3 ways and the round went 1:1:2 (or some permutation).  Should we, if we don't think the 1cd that has position on us is good enough to always bet when we check (or even if he is) be betting to get value from the 2cd (and I guess also for some kind of balance since we'll lead our pat hands here).  About 55 minutes in Triple the Gold episode 3 DeathDonkey suggests we bet to get value from the 2cd -- is it really worth it?

There are a few possible outcomes based on how our opponents draw when we bet.

1. Both opponents get pat (I'm assuming pat here means 8 or better which is obviously tighter than they'll be in reality most likely) -- this will happen probablistically somewhere around 2-3% of the time assuming the 1cd will get there about 1/4 of the time and the 2cd will get there around 1/10 of the time.
2. One pats -- this will happen about 30% of the time -- 1/4 the 1cd will get there 9/10 the 2cd won't, and 1/10 the 2cd will, 3/4 the 1cd won't.
3. 1cd unimproved, 2cd improves to 1cd -- this will happen about 36% -- 3/4 when the 1cd doesn't improve, 5/10 the 2cd imrpoves to a 1cd.
4. Both unimproved -- this will happen about 32% of the time based on the fact the 1cd won't improve 3/4 and 2cd won't improve about 4/10.

So what happens in each of these cases?

1. Here we lose 1BB and any equity (which is going to be fairly small most of the time) we had in the pot most likely since it's going bet/raise/3bet back to us and we're not even closing the action.
2. Here we lose some fraction of 1 BB (over checking and calling) depending on the smoothness of our draw and the pot odds we're getting (plus how often he's doing this with things like breakable Ts/9s) or relinquish equity based on our average equity in this spot.  I'll assume against a pat hand we have something like 15% equity on average (sometimes we'll be splitting our equity with the other 1cd when it's the 2cd that pats).
3. We get marginal value at best depending on the smoothness of draw.
4. We fold out 2cd and the equity he folds (which I think is going to be around 18%) is redistributed to Ivey and us (so we pick up something around 10% of the current pot size in value).

So what's our EV from each situation look like?

1. -1BB + ~0 (we'll just assume we're drawing dead or would never draw here).

2. -0.85BB

3. ~0

4. 10%*Current pot size

Taking into account the relative occurence: EV = (-1BB)(2%)+(-0.85BB)(30%)+(0)(36%+(10%*POT)(32%)

So if the pot is 5BB, EV ~ -0.12BB, 10BB, EV ~ +0.05BB

So maybe we should only tend to be doing this when the pot is big?

How important do we think not always having a pat when leading is here?

I think I need to do a bit more thinking about the 1:0 and 0:1 as pat before writing any more to avoid providing terrible advice.  The 0:1 spot where we're patting OOP is especially tricky and opponent dependent.  So I've decided to write about another spot instead for the time being.

2:1 – “card ahead, come and get me”
This spot is less frequent to occur, particularly with tighter players and players who are familiar with the game.  We know drawing 2 is generally bad, especially OOP.  However, there are definitely times where we’ll consider defending 2cds (mostly shorthanded) and there will definitely be villains who do this more frequently than they should.  We’ll look at this from the 2cd and 1cd perspective.

1cd: Because we’re “card ahead” we’re more often than not going to end up with the best hand post draw.  It’s difficult to draw 2 cards to a really strong hand (7x2 will make an 8 or better like 9% of the time, and a T or better about 22% of the time if I've done my math correctly) and drawing 1 we’re more likely to hit our hand anyway.  Because our opponent misses so often, we’re going to do well to bet both strong value hands and hands with little to no showdown quite a bit of the time.  Because we anticipate our opponent to adjust to us polarizing our range (to something that likely includes more bluffs than value type hands) by calling us with a slightly wider range than normal, we can somewhat capitalize by betting for thinner value as well.  Hands that might be check backs in 1:1 or 1:0 spots can be good betting spots.  If our opponent adjusts to us doing this by check/raising or raising more frequently rather than calling we can readjust with 3b bluffs and changing the ratio of value to bluff hands in our range.  When we’re IP taking 1 card can be a profitable snow line with any 5 cards against some opponents (from time to time of course, unless our opponent is letting us get away with this constantly).

2cd: When OOP we should almost always be checking against competent Villains who are taking advantage of the above fact.  If the opponent is not capable of value betting wide then we’d do better to lead value hands and bluffs and check/calling bluff catchers and check/folding other nonshowdownable hands.  Better hands to bluff actively with include things that we might snow with or cause us to tend to snow with predraw (hitting more twos and lowcards, pairing small etc..).  Basically as the 2cd we want to be adjusting in a way that will make the 1cds life as miserable as possible.  Basically we're going to actively attempt to do everything we're attempting to guard against as the 1cd.  If we feel our opponent is betting too wide, we can adjust by bluffraising more or consider some kind of donkbetting strategy if he's tending to give us too much credit.

The two other spots I'll eventually write about will be the 1:1:1 and 1:1:0 spots.

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Last TDL session of the day +39BB in 0.5/1, +2BB at 1/2, so not a bad result :) for 80 hands.

Hand Distribution

Before discussing the pat side of the 1:0 and 0:1 spots, I just wanted to post this as I'm not sure it's enumerated this far elsewhere.  Impress your friends by telling them their #372 is no good. :)

Hand//ways to make//worst hand rank in group

75432//1//#1

76xxx//3//#4

Any 7: 0.16%

85432//1//#5

86xxx//4//#9

87xxx//9//#18

8 or better: 0.71%

95432//1//#19

96xxx//4///#23

97xxx//10//#33

98xxx//19//#52

9 or better: 2.04%

T5432//1//#53

T6xxx//4///#57

T7xxx//10//#67

T8xxx//20//#87

T9xxx//34//#121

T or better: 4.75%

J5432//1//#122

J6xxx//4//#126

J7xxx//10//#136

J8xxx//20//#156

J9xxx//35//#191

JTxxx//55//#246

J or better: 9.65%

Q5432//1//#247

Q6xxx//4//#251

Q7xxx//10//#261

Q8xxx//20//#281

Q9xxx//35//#316

QTxxx//56//#372

QJxxx//83//#455

Q or better: 17.86%

K5432//1//#456

K6xxx//4//#460

K7xxx//10//#470

K8xxx//20//#490

K9xxx//35//#525

KTxxx//56//#581

KJxxx//84//#665

KQxxx//119///784

K or better: 30.77%

A5xxx//1//#785

A6xxx//4//#789

A7xxx//10//#799

A8xxx//20//#819

A9xxx//35//#854

ATxxx//56//#910

AJxxx//84//#994

AQxxx//120//#1114

AKxxx//164//#1278

Any unpaired, non flush, non-straight hand (A or better): 50.16%

Some other info:

Each unique hand, 86532 for example has 1020 possible ways it can be made and represents 0.04% of total hands (There are 52c5 ways to make a 5 card hand = 2598960 combos).

1cds to 7s make up 1.77% of hands

1cds to 8s make up 4.73% of hands

2cds to 7s make up 9.70% of hands

When trying to determine what types of hands your opponents will take certain lines with, think about these combinations before making a decision.  Think about how many worse hands of the same rank he'll call you with and hands from other ranks.  You can also get an idea of the relative frequencies of each hand and if you wanted to create a target VPIP/PFR for each spot you could probably do it with a little more info, but I'll leave that to you.

1:0 – “pat hands skew rough”

1:0 is the convention used to describe the situation where one person is drawing one card and the other is patting.  1:0 would be where the 1cd is OOP, 0:1 is where he’d be IP.  This is going to be a really common scenario and it’s important to think about the situation from both sides of the coin.  We’ll look at it from four angles – having the 1cd in position, having the 1cd OOP, being pat in position, and being pat OOP.

1cd in position:  Assuming our opponent is competent the range of hands he has patting OOP will be stronger than those he pats IP.  If this is a 1:0 spot where are opponent (who is likely somewhat bad) calls and pats OOP or limp/call pats the majority of his range is going to be marginal hands in the neighborhood of T9xxx.  If they check postdraw I’d be even more likely to give them things like rough Ts and occasionally Js.  The closer we are to the button the rougher their range is likely to be, and they are usually turning their hand into a bluff catcher.  You should probably avoid bluffing them at first, but go for thinnish value with things like T8s to see how they react to you betting.  In my experience players like this don’t like to fold.  When they pat then lead their range is a bit wider, but includes more strong hands.  Some people will be making defensive bets, some think because they were ahead predraw that they have to bet now even though it wouldn’t be for value with things like T9s, and some will show up with somewhat wtf hands like 97s or better that they decided not to 3bet pre.  Try to get a feel for what type of bad player you’re against.  Depending on how weak they seem to be patting and the types of hands they are leading vs. checking, we may come after this guy with postdraw bombs both for value and as bluffs.

1cd OOP:  This is going to be a somewhat frequent occurrence when defending from the blinds.  We draw 1 to our T 9 8 or 7 and the villain pats behind us.   Really pay attention to what types of hands your opponents are doing with this.  Many will tend to play their hand for hot/cold equity and turning things like jacks and tens into snap checkbacks postdraw, essentially just hoping you didn’t outdraw them.  Against these opponents are going to want to be leading a pretty wide range of value hands and bluffs.  Test them out, see how they are responding to your leads.  If you are getting too many folds, expand your bluffing range.  If they are making suspicious calls, see what happens when you overbet, but in general look to go for lots of thin value with rough tens and smooth jacks and tend to avoid bluffing.  Against more aggressive opponents who will bet thinly for value and occasionally snow we should probably check a larger portion of our range so we can c/r for value and as bluffs more credibly.

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2nd tdl session of the day went pretty well.  19.5BB more at 1/2 in 119 hands.  I'm still leaking a few BB here and there, no real interesting spots though in this session.

---------------------------

3rd tdl session started out pretty poorly.  Got stuck 25BB pretty quickly losing a #2 vs #1 and losing a 16BB pot against a rivered wheel with a #6 (though I lose 0 bets on the river somehow).  Ended up getting unstuck and +6BB.  Was 3tabling 1/2 and 0.5/1, 396 hands.

Continuing with some of my general thoughts/initial impressions of sdl:

Concepts like light 3betting, cold 4bet bluffing, and squeezing for the most part are non-existent at the lower levels of this game and snowing is infrequent.  Before attempting any of these plays, try to gage first how the players you are using them against are likely to react.  If someone isn’t opening light, don’t 3bet light (unless perhaps they have a high f3b%, then we can 3bet things like our medium and good one card draws).  Good candidates for the light 3bet would also include those isolating weaker players limping.  Tend not to cold 4bet bluff/semibluff if the opener is tight or the 3bettor doesn’t 3bet frequently or isolate and do this sparingly in general.  As previously mentioned hand equities don’t run close together and we’re repping an extremely small range of hands.  If I am cold4betting (as a bluff/semibluff) , OOP I’d tend to do this with snows (provided of course that we're semi-deep such that I can bluff postdraw), while IP I’d tend to do this with good 1cds.  Don’t squeeze very wide (probably limit to medium-good 1cds or better and occasional snows) as the guy in the middle isn’t often folding and the guy opening isn’t usually opening wide.  We will very infrequently fold out people who have good one card draws.

Balance is perhaps more important in this game (and drawing games in general) than other games. Exploitative play is difficult (but obviously possible) over two streets (there's only so many things we can exploit), but to keep maximum pressure and uncertainty we want to have a pretty balanced game both pre and postdraw.  In practice I think it’s often best to balance your strong drawn hands with your weakest (when you pair your top card or second card top card) and turn the middle parts of your range into bluff catchers and check backs.  Sometimes we’ll merge middle parts of our range into our value and bluffing ranges respectively based on our opponents and how we’ve been playing.

_____________

Just played first tdl session of the day at 1/2.  Won about 22.5BB in 261 hands.  Ran hot against one guy, ran pretty poorly against 3 or 4 others.  I missed at least 2 bets on whiffed c/r attempts against generally aggro value bettors and played 2 or 3 hands pretty somewhat poorly including breaking incorrectly once or possibly twice and making a weird bet/fold first to act in a 2:2:1 when I decided to make an out of tempo lead (it's only a donk bet when it's my opponents doing it :)) with a 87543 (because of how frequently it's checked through at this limit, even though last to act is card ahead), got raised and 3bet before it was back to me.  I couldn't break and thought capping with the plan to lead into two likely pat hands would be spewy.  It ended up going 0:1 after the 2cd just called the 3bet.  Then bet call bet fold.

This is more or less a continuation of the last post:

Note: Everyone says this about their favorite games, but position in NL2-7 is more important than the vast majority of games.  The nature of it being a 2 street game with a single draw means we have very limited information to make decisions from.  Knowing how many cards our opponents are drawing and his action postdraw will give us a huge edge and allow us to play hands far more optimally.  For example, a pat Jack is a favorite against a one card draw (a Queen is a favorite against a two card draw).  If we were playing our hand hot and cold in position (which may be the case in tournament situations, when stacks are shallow in cash games (so we don't have much RIO), or when we don't want to break to bad/rough draws ) we can often choose to pat our hand if we see our opponent drawing, or “break” the jack and draw better if our opponent pats.  Being out of position with the same hand adds a lot of guesswork and is going to frequently get us into trouble.  

Opening Ranges: While the hands you open should definitely be a function of the players at the table and your relative position to them, flow, etc… the following provides a loose guideline to get started based on my experience:
UTG: Any 9 draw and any pat T97 or better.  The T97 gives us a reasonable two-way hand (by which we mean we can pat or break).
UTG+1: All UTG hands and smooth Ten draws (T7 or better).
MP: All UTG+1 hands, any pat T, any Ten draw except the rougher ones.
HJ: All MP hands, breakable jacks, any ten draw.
CO: All HJ hands, any pat jack, most jack draws, occasional good two card draws (like 742xx).
BTN: All CO hands, any pat QT, any jack draw.
SB: UTG range over limpers, we can raise far wider bvb, particularly against tight big blinds.
BB: UTG range, we can defend wider against minraises and obvious things like that, raise wider when it's bvb

Note: The above corresponds to unopened pots. Against looser players limping we can treat the pot as effectively being unopened, but against the tighter ones I’d tend to open tighter parts of the range (maybe open the recommended range from a seat or two in front of you and maybe only the top 70% or something of the UTG range from the blinds.  Watch out for things like how often someone is limp/folding and if someone is never limp/folding.

Note: In addition to these value-ish type hands that we’ll be patting or drawing to, we’ll also be incorporating “snows” (pure bluffs) in our range.  Hands that are great to snow with include 5 low card hands that include two pair or three of a kind (things like 77442 or 33386 and in games that are playing deeper, the more deuces you have the better since the best hand our opponent will be able to hold is an 86543 -- a #9).  Please avoid snowing with five high card type hands as our opponents are both more likely to hold better hands/draws and to get there.  Be aware of the implications to follow you getting caught snowing and how you should adjust to how they are adjusting to the fact you are snowing.  At 50NL2-7 there tends to be little snowing from what I've seen (although you'll the occasional random who'll drop in and play 10 hands before he finds out it's a lowball game -- thereby accidentally snowing fullhouses etc...) and people don't expect it much from you either. 

Snowing at microstakes nl2-7 in general is going to be pretty profitable if you're choosing your spots correctly, and some hands may start one place and turn into snows.  For example, when you raise and get flatted by someone who pats in front of you (this tends to happen quite a bit when we open in LP and one of the blinds calls) the villain almost always has something between a T8 or JT if they are in the general pool of newbie-ish players (from the really new players you'll see this with both stronger and weaker hands as well because they can't evaluate the strength of things like any 8 and good 9s [which are quite strong generally] or things like QJT53).  In any case, instead of drawing to rough hands we may have in LP (and sometimes even smooth hands depending on stack depth) quite a bit of the time we're better off patting and bombing the pot postdraw to put our opponents who we know have a marginalish hand to a tough decision and because people tend to think you're never snowing in the micros, they'll give you credit and fold an absurd amount of the time.  So instead of winning the pot say 35-45% (or worse)  of the time with our draw we can win the pot close to 100% of the time.  Another great spot is when our opponents call and draw 2 OOP.  We can pat behind pretty much anything and pick up the pot postdraw the 78% of the time they fail to make a ten or better.  Obviously we have to balance this decision with the implied odds we're foregoing in a given spot because wasting a 7432x as a snow just because a guy drew 2 or patted weakly in front of us won't always be a good idea, especially since we're still a pretty big equity favorite against the 2cd (but now have implied odds since he'll call us wider and because we can 'cooler' him).

That's it for now I guess, more to come.

I started playing nl2-7 a few months ago and thought I'd post some of my general and specific thoughts on the game -- I can't say for sure everything I think is correct, but it's worked reasonably well for me thus far.  I'm going to start out with a general introduction since some people who end up reading this (hopefully) may not know anything about the game.

No limit deuce to seven single draw lowball (SDL), also known as Kansas City Lowball (KCL), or just “deuce” is a 5 card draw lowball game with a two-street (predraw and postdraw) structure.  Aces are high only, straights and flushes count against you, so the best hand (which is often called a #1 or a wheel) is a 75432.

Note 1: In lowball games we always read the hand from the highest ranking card down.  Beginners often get confused by this concept.  But to give you an example, a 23459 (which lowball players will read as a 95432, a “smooth” 9, a ninety-five, a 9 perfect or a #19) is a worse hand than a 34578 (which we would read as 87543 and call a “rough” 8, or an eighty-seven or a #13).
Here are a few more examples:

8d9h2s3d4s vs. 5d7h9s6d4h -- 97654>98432
As2h3d4d5cs vs. 2d2c3s4h5c -- this one confuses lots of people at first, ACES are HIGH ONLY, you cannot make a traditional wheel, therefore A5432>22543
7d3d4d5d2d vs. KdQsJc9s2c -- the hand on the left is a wheel in this game, unfortunately it's all diamonds, KQJ92>75432sssss

The game is normally played somewhere around 7 handed and there is a small and big blind to the left of the dealer button just like in unlimited texas hold thems (UTG MP HJ CO BTN SB BB).  However, when this game is played live, the vast majority of the time there will also be antes, usually on the order of half a small blind to induce action.  Obviously, the size of this ante will impact how we play to some degree, as we need to play looser in high ante games and should be more reluctant to get in unnecessary spots in low and no ante games (this is true of any game involving antes).  On pokerstars (the only site I’ve ever played on that offers this game), no ante is used in the cash games (though it does appear in the tournaments and sngs they run).  Live, this game pretty much only runs at fairly high stakes (at the commerce with Billy Baxter they play something like 200/400/100, occasionally at the Bellagio during WSOP time and as part of mixed games).  There is currently a $1500 and a $10k bracelet event at the WSOP.

Note 2: Because NL2-7 is a two-street game, games tend to play shallower than NLHE.  In order to play 100bb pots in NL2-7 (particularly with no antes) there requires (or should require) a large degree of cold-decking (or higher level aggro play).  I’d say something on the order of 50-60bb is going to be a typical buy in size, with most medium sized pots being in the 10-20bb range.

Note 3: Unlike NLHE or PLO, hand equities in NL2-7 DO NOT run that closely together.  It’s not uncommon to be drawing dead or have your opponents drawing dead before the draw.  This both gives us disincentive to play marginal spots (when calling and more so when OOP) and sort of as a corollary to apply more pressure on those who are calling and are OOP.

My name is Jared W. I go by RabbiRounder on stars.  I decided to start a blog and post my thoughts about various things poker related.  I play all the games that are spread online with various degrees of aptitude.  My regular stakes online are something like:

Stakes (highest I've played online in that game)

25NLHE (100NLHE)

10PLO-25PLO (50PLO)

1/2 O8 (8/16 O8)

0.5/1-1/2 stud, stud8, razz (2/4)

50NL2-7-100NL2-7 (200NL2-7)

0.25/0.5-1/2 TDL (5/10 TDL)

0.25/0.5 Badugi

0.5/1-1/2 8game (4/8 8game)

Lately I've been playing far more TDL and SDL than anything else.

Since it's really the only thing I have tracked, here is the majority of my lifetime graph at 25NL (I'm sure I've lost hands prior to this db)

 and sngs....