LO8 - When is it correct to chase runner-runner for the nut low
 

A situation came up several times in Vegas that got me thinking.

The game was $4/$8 Omaha8 with numerous bad players that played too many hands and chased too many long shots. I can literally remember playing against two different players who did not fold a single hand, not one, preflop in separate 3 hour sessions. So I got to thinking…., <font color="red">how big does the pot need to be on the flop to consider chasing runner-runner low cards in hopes of getting ½ (or even ¼) of the pot? </font> Please check my logic and let me know where I screwed this up.

The Situation:
You hold A 2 9 K rainbow. The board reads J [img]/images/graemlins/heart.gif[/img] J [img]/images/graemlins/club.gif[/img] 8 [img]/images/graemlins/heart.gif[/img]. You assign a 0% chance of holding any high equity.

The Math:
There are 25 cards that could come on the turn that are helpful (4 each of 7s, 6s, 5s, 4s, 3s). The probability of catching a good card on the turn is 20 cards out of 45 unseen cards - 0.444. Then, you’ll need to catch 16 good cards on the river out of 44 unseen. 16/44 = 0.364. So the probability of catching good on both the turn and the river is 0.444 x 0.364 or 0.162. [img]/images/graemlins/confused.gif[/img]

Split Pot:
If I catch good on both the turn and the river, I’ll get a portion of the pot – either ½ the pot or ¼ of the pot (or maybe 1/6 of the pot – yuk). Since chasing runner-runner low to get a portion of the pot is a long shot (0.162), the pot will need to be pretty big. To create a big pot, it is necessary to have many contestants. With a large number of contestants in an Omaha8 game, the chance of another player holding the nut low as well must be considered. I used Alspach’s Poker math to estimate this probability at 0.3 (&lt;http://www.math.sfu.ca/~alspach/mag8/&gt;)
Therefore there is a 0.7 chance of winning ½ the pot and a 0.3 chance of wining a ¼ of the pot. So, if I catch runner runner low, I expect to get ((0.7 x ½) + (0.3 x ¼)) or 0.425 of the pot.

The Flop::
There is a 0.162 chance of catching runner runner by the river for me to win a 0.425 portion of the pot
0.162 x 0.425 means I have pot equity of about 7%. Therefore, I’d need the pot to be “huge” to make a call chasing runner runner low. I’d need 15 small bets to be in the pot to justify a 1 bet call. To get this number I took the number of bets (1) divided by the probability of winning (0.162) x the proportion of the pot that you’d win (0.425).
To get 15 bets in a pot preflop, I’d need 5 people to put in 3 bets or 8 people to put in 2 bets or 3 people to put in 5 bets (which is a cap in Vegas limit poker).
It gets much worse if you are facing a raise (or worse yet, 3 bets cold on the turn)
If you are facing 2 bets on the flop and still have 7% pot equity, you’d need there to be 29 bets (6 people in a capped preflop pot). &lt;&lt;2 bets divided by the probability of winning (0.162) * the proportion of the pot that you’d win (0.425)&gt;&gt;
If you are facing 3 bets cold, you’d need there to be around 45 bets (9 people in a capped preflop pot). &lt;&lt;3 bets divided by the probability of winning (0.162) * the proportion of the pot that you’d win (0.425)&gt;&gt;
<u>The Bottom Line: </u> you can call on the flop chasing runner runner nut low, if the pot is huge (15 or more small bets) and the flop betting is not raised and you don’t expect it to get raised (i.e., you are in last position).

The Turn:
Now, I have the same problem on the turn when I catch good. Do I continue to chase a low card to the river to get ½ or ¼ of the pot? The math is about the same. There are now 16 good cards out of 44 or 0.364. So, 0.364 – probability of making a hand x .425 – portion of the means I have about 16% pot equity So, we’d need 7 big bets in the pot to call 1 big bet on the turn &lt;&lt;1 bet divided by the probability of winning (0.364) * the proportion of the pot that you’d win (0.425)&gt;&gt;
From the flop discussion above, we must have at least 15 small bets preflop to call a flop bet. You can only call on the flop if it is checked or there is 1 bet; therefore, there are a least 7 big bets on the turn, so the 1 bet call on the turn is automatic if you catch good.
But what if you are facing a bet and a raise on the turn? With 16% equity, you’d need about 13 big bets to justify calling 2 bets on the turn. &lt;&lt;2 bets divided by the probability of winning (0.364) * the proportion of the pot that you’d win (0.425)&gt;&gt;
If you are facing 3 bets cold on the turn, you’d need 19 big bets to justify a call with 15% equity. &lt;&lt;3 bets divided by the probability of winning (0.364) * the proportion of the pot that you’d win (0.425)&gt;&gt; This would be a gigantic pot ($150 pot in a $4/$8 game).
<u>The bottom line:</u> You can call 1 bet on the turn if the pot is unraised. You can call 2 bets on the turn if there are 19 bets in the pot. To get 19 big bets in a pot and still have the correct calling odds on the flop and the turn, you’d need about 8 players in the hand each having put in 2 small bets preflop, 1 small flop bet, and 1 big turn bet…..not likely. You can’t really play for a raise on turn.

The River:
The river plays itself. If you catch good, call (bets and raises). If not, fold. You have about .425% pot equity &lt;&lt;1 bet divided by the probability of winning (1.0) * the proportion of the pot that you’d win (0.425)&gt;&gt; The pot has at least 7 big bets (the criteria for calling on the turn), so you are paying 1 bet to get 42.5% of at least 7 big bets.
What if 2 jackasses cap it on the river? You have 42.5% equity in a pot that has at least 7 big bets. If 2 jackasses cap it, there will be 21 big bets in the pot when you are faced with your last call (5 bets from jackass #1, 5 bets from jackass #2, 4 called bets from you, plus 7 bets to start with). Since you “own” 42.5% of this pot (21 bets x 42.5% = 9 bets that you own) and you only have to call 1 last bet, the call is automatic.

<u>The bottom line:</u> Call all bets and raises on the river when you hold the nut low. Duh! [img]/images/graemlins/tongue.gif[/img]

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

Situation is much different when you have counterfeit protection (i.e. pot size doesnt need to be as big as you have a much greater chance at catching good 2x in a row)

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

I'll usually treat a runner runner nut low draw like a gutshot--the pot has to be around 10 small bets on the flop. But I rarely chase it unless I:

1. have some sort of miracle high draw
2. have counterfeit protection
3. am pretty sure no one else has the nut lo draw

It's important to note that if the pot is big enough to draw to a backdoor low, there's a very good chance at least one other player has a backdoor low, in which case you are drawing to get 1/4 or 1/6...

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

I've been using the oversimplistic maxim of "in general, don't." (All the normal disclaimers apply if I have a good chance at the high; the maxim applies to low-only runner-runner draws.)

Nice to see the math that more or less supports that maxim.

Thanks for posting all that work, husker.

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

Thanks for this valuable info,cornhuskerfan.Like jjgoldy5 said,your chances are much better when you have counterfit protection.Could we possibly get you to run the numbers on this,too?Thanks a lot.Ben
(Buzz,Do you agree with these numbers?)

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

I used the same logic to determine the number of small bets required in a pot to chase runner-runner low on highish flop (2 high cards). Here's what I came up with. I had to do some rounding (i.e., # of bets is a whole number) and estimate some winning percentages with different holdings. Let me know if you agree/not.

On a highish board (2 high cards), you will have the correct pot odds to call one small bet, chasing runner-runner low for 1/2 (or less) of the pot, on the flop if you have the following hands and the pot contains <font color="red">n </font> bets:

You hold an "AB" hand (A2xx) with 0% high equity - <font color="red">15 </font> small bets (for example, you hold A [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/diamond.gif[/img] 9 [img]/images/graemlins/heart.gif[/img] 9 [img]/images/graemlins/club.gif[/img] and the flop is K [img]/images/graemlins/club.gif[/img] J [img]/images/graemlins/club.gif[/img] 8 [img]/images/graemlins/club.gif[/img])

You hold an "ABC" hand (A23x) with 0% high equity - <font color="red">9 </font> small bets (for example, you hold A [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/diamond.gif[/img] 3 [img]/images/graemlins/heart.gif[/img] 10 [img]/images/graemlins/club.gif[/img] and the flop is K [img]/images/graemlins/club.gif[/img] J [img]/images/graemlins/club.gif[/img] 8 [img]/images/graemlins/club.gif[/img])

You hold an "ABD" hand (A24x) with 0% high equity - <font color="red">10 </font> small bets (for example, you hold A [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/diamond.gif[/img] 4 [img]/images/graemlins/heart.gif[/img] 10 [img]/images/graemlins/club.gif[/img] and the flop is K [img]/images/graemlins/club.gif[/img] J [img]/images/graemlins/club.gif[/img] 8 [img]/images/graemlins/club.gif[/img])

You hold an "ABZ" hand (A27x) with 0% high equity - <font color="red">11 </font> small bets (for example, you hold A [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/diamond.gif[/img] 7 [img]/images/graemlins/heart.gif[/img] 10 [img]/images/graemlins/club.gif[/img] and the flop is K [img]/images/graemlins/club.gif[/img] J [img]/images/graemlins/club.gif[/img] 3 [img]/images/graemlins/club.gif[/img])

Now, lets say you hold "some" high equity (like a gut shot).
You hold an "AB" hand (A2xx) with "some" high equity - <font color="red">10 </font> small bets (for example, you hold A [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/diamond.gif[/img] 10 [img]/images/graemlins/heart.gif[/img] K [img]/images/graemlins/club.gif[/img] and the flop is J [img]/images/graemlins/club.gif[/img] 9 [img]/images/graemlins/club.gif[/img] 7 [img]/images/graemlins/spade.gif[/img])

You hold an "ABC" hand (A23x) with "some" high equity - <font color="red">7 </font> small bets

You hold an "ABD" hand (A24x) with "some" high equity - <font color="red">7 </font> small bets

You hold an "ABZ" hand (A27x) with "some" high equity - <font color="red">8 </font> small bets

It's difficult to memorize all of this, so I sort of summarized it in my mind as follows:

AB hand with no high - 15 outs - never call a raise
ABC,ABD, or ABZ hand with no high - 10 outs - never call a raise

AB hand with long-shot high - 10 outs - never call a raise
ABC,ABD, or ABZ hand with long-shot high - 8 outs - never for a raise.

Or maybe a simpler rule would be "don't chase runner-runner low unless you have an ABZ hand or better and there is 10 bets or more in the pot - with one exception, if you have an AB hand and there is 15 bets in the pot". I like this one. I'm going to call this the rule of "10/15"

In almost all situations, the pot will need to be multi-way with 1 or 2 preflop raises.

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

Good job,cornhuskerfan.Valuable info.You're the nuts.
Ben

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

[ QUOTE ]
You assign a 0% chance of holding any high equity

[/ QUOTE ]

I dont' consider myself a good omaha player at all really (especially as good as Cornhuskerfan, since I play live with him alot) - but when I am chasing to hopefully win half the pot - especially with A2 that everyone plays, and might get quartered, I lose intrest in the hand once the flop is 2 high cards.... Why put money in that you *might* get back, or not get at all, or get some of it back if 2 others have A2 and it comes runner runner 5-6.

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

[ QUOTE ]
You hold A 2 9 K rainbow.

[/ QUOTE ]

This is your first problem.If you had say A2sK9s then this hand is a bit more playable;but the 9 is basically the worst card you can hold so if you are offsuit you are essentially playing a dangler (in fact even suited the 9 would still be a bit off).

I would fold the flop and look for a better place to get my money.

Re:WhooFleuryScores
 

Quote:"I would fold the flop and look for a better place to get my money."

This is too tight,imo.It's great if all your cards are working,but the presence of a 9 in your hand shold not prevent you from playing an A2 hand.I personally play all A2 hands if the pot is not raised,but some other players who play better than me will pass some dry A2 hands depending on the situation(strength &amp; tightness of the game,position,etc).However,this is not a dry A2.The K is a strong card.I don't believe that I know any good players who would not normally play this hand.

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

[ QUOTE ]
Buzz,Do you agree with these numbers?

[/ QUOTE ]Hi Ben and Corn Husker Fan - I've been in Las Vegas for the past ten days, away from my computer, and not able to answer your question until now.
[ QUOTE ]
So the probability of catching good on both the turn and the river is 0.444 x 0.364 or 0.162.

[/ QUOTE ]Yes. That is correct.

[ QUOTE ]
I used Alspach’s Poker math to estimate this probability at 0.3 (&lt;http://www.math.sfu.ca/~alspach/mag8/&gt;)

[/ QUOTE ]Alspach is the mathematician, not me. However, that estimation (for getting quartered and sixthed) is incorrect (unless Corn Husker Fan is referring to a game where only seven players are dealt cards).
[ QUOTE ]
Therefore there is a 0.7 chance of winning ½ the pot and a 0.3 chance of wining a ¼ of the pot. So, if I catch runner runner low, I expect to get ((0.7 x ½) + (0.3 x ¼)) or 0.425 of the pot.

[/ QUOTE ]That's incorrect for a full game.

If (1) Hero is dealt a hand with exactly one ace and exactly one deuce, and with no back-up for low, and (2) when the flop has only one low card (not an ace or a deuce), and (3) if nobody folds the runner-runner nut low draw, and if the turn and river are both cards such that Hero's runner-runner low draw is successful, then<ul type="square">when ten players are dealt cards,
~57% Hero does not split low,
~37% Hero gets quartered for low,
~5% Hero gets sixthed for low, and
~1/1000 Hero gets eighthed for low.

when nine players are dealt cards,
~61% Hero does not split low,
~34% Hero gets quartered for low,
~4% Hero gets sixthed for low, and
~1/1000 Hero gets eighthed for low.

when eight players are dealt cards,
~65% Hero does not split low,
~31% Hero gets quartered for low,
~3% Hero gets sixthed for low, and
~1/1000 Hero gets eighthed for low.[/list]Something like that.

However, it's a moot point because strong players will probably not be drawing exclusively to runner-runner low. Thus if some fool draws for runner-runner low, he'll probably not have strong players sticking around to possibly quarter him. And that makes it impossible to compute how often Hero will get quartered or sixthed when drawing for runner-runner low.

Hypothetically we can create a situation where enough players call three or four bets on the first betting round and where there subsequently is no more raising on the second and third betting rounds, so that Hero has favorable odds to draw to a runner-runner low. However, my experience is after lots of action on the first betting round there is generally raising on the third betting round, if not on the second.

It's not absolutely impossible that Hero could have odds to continue to a bet - but after an action packed first betting round, even if there's no raising on the second betting round, how could Hero ever be sure there would not be raising before the river? And Hero would not have favorable odds to continue if there was more raising before the river.

Figure that if the probability of Hero making the runner-runner low draw is 0.162, then roughly Hero makes the draw one time out of six and misses the other five times out of six. When Hero misses, Hero misses roughly half the time on the turn (cost one more small bet, assuming there are no raises) and half the time on the river (cost one more small bet plus one large bet, assuming there are no raises). (It's worse for Hero with raises). Roughly (assuming there are no raises on the second or third betting rounds) Hero's cost averages two small bets = one big bet when he misses. And he misses five times out of six.

Hero has to make up the cost of those five misses when he doesn't miss - but that's unlikely. Hero's share of the pot will simply not average enough to make up for his costs when he misses. In other words, paying to draw exclusively for runner-runner low is simply not a good bet.

Assuming no raises, even without getting quartered or sixthed, Hero has to win at least five big bets (ten small bets) when he makes his runner-runner draw. That's realistic for scooping the pot, but not for splitting the pot. It's not impossible - just not realistic.

If I get a free draw, whoopee! One time out of six it might work out well for me - and even better than just getting half the pot. Sometimes you can turn a low winner into a scooper by just betting the low.

However, even taking into account that I might be able to convert a low hand into a scooper by aggressive betting, totally missing high on the flop and needing a runner-runner for low is a fairly standard check/fold for me. (I generally check, hoping for a free card, but fold when there is a bet).

I'm not saying that drawing to runner-runner low is something one should absolutely never do, but I think drawing to runner-runner low with no realistic chance for high is a fairly common mistake.

Buzz

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

This one of the best threads we have had in while, many thanks to cornhusker and Buzz!

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

[ QUOTE ]
From the flop discussion above, we must have at least 15 small bets preflop to call a flop bet. You can only call on the flop if it is checked or there is 1 bet; therefore, there are a least 7 big bets on the turn, so the 1 bet call on the turn is automatic if you catch good.

[/ QUOTE ]Hi Corn Husker Fan - How many players are still in the pot????? People don't only contribute on the first betting round! The number of people still improperly hanging around on the third and fourth betting rounds is generally much more important in terms of how well you do on a hand than the number of pre-flop raises you can get in with a nice starting hand.

Use one chip to represent each small bet. I'm using three colors of chips (red, white, and blue). I'm using white chips to represent Hero's contributions after the first betting round, red chips to represent what Hero has to win, and blue chips to represent more contributions from Villains.

If we figure Hero needs to average winning at least ten small bets in his portion of the pot to justify continuing after a flop where Hero’s only chance is to win is with a runner-runner bare ace-deuce low draw, then assuming no subsequent raising before the river but a bet every round, there would have to be fifteen chips in Hero’s portion of the pot, five white chips to represent Hero's subsequent contributions to the pot plus ten red chips to represent what Hero needs to win to justify continuing.

Hero needs to get awarded fifteen chips, five of which will be his own contributions after the first betting round. If Hero was certain he wouldn’t get quartered or sixthed, then there would need to be thirty chips total in the pot.

To get the thirty chips, take five white, ten red, and fifteen blue. Now subdivide these into smaller piles, each pile representing someone’s total contributions to the pot.

Hero’s pile will have one red chip plus five white chips. The one red chip represents hero’s contribution on the first betting round, and the five white chips represent Hero’s contributions on the second, third, and fourth betting rounds when the turn and river are both favorable.

(This is a no raise scenario. We could also do a raise scenario, but then Hero’s contributions would be more).

At any rate, the pile representing Hero’s contributions has six chips and we also can make four equal piles of chips to represent the total contributions of four opponents. Hero then only needs four opponents to stay in the hand all the way in this non-raising scenario. Will that happen? I don’t know. It’s not impossible.

More likely, in my games, five or six opponents will see the flop with me, but then there’ll be a drop off after the flop and then a gradual attrition.

I’m trying to figure a way for Hero to have favorable odds if there’s no raising but a bet every round. I’ve made one stack consisting of one red chip and five white chips to represent Hero’s contribution to the pot – and I still have twenty four chips left, nine red and fifteen blue. For the case where Hero has five opponents who also see the flop, I’m laying out (next to Hero’s stack of six chips) five chips to start each opponent’s chip stack. After the flop, let’s say four continue. And then if these four continue after the turn and if only three of them continue on the river, I can arrange the chips so that Hero has favorable odds for continuing but only if he’s reasonably certain he won’t get quartered or sixthed.

Trouble is if that many opponents continue, there’s a good chance one of them also has the nut low, same as Hero.

If Hero knew he’d get quartered, then instead of thirty chips total in the pot, there’d have to be sixty chips total in the pot. And that’s with no raising. If Hero knew he’d get sixthed, then there’d have to be ninety chips total in the pot.

I think your 30% estimation of how often Hero gets quartered is low, and it doesn’t account for getting sixthed. But let’s use your 30% estimation.

0.30*60 + 0.70*30 = 18+21 = 39.

If Hero gets quartered only 30% of the time and never gets sixthed, then when Hero makes low, the pot size when Hero makes the nut low has to average 39 chips in the pot instead of 30. That’s a bunch, but even so, if Hero has six opponents sticking around to the river for one bet each round, Hero may have favorable odds to continue.

But in a game with so many opponents continuing all the way to the river after a flop of J[img]/images/graemlins/heart.gif[/img], J[img]/images/graemlins/club.gif[/img], 8[img]/images/graemlins/heart.gif[/img], we had better use more realistic numbers for the probability of getting quartered or sixthed.

Considering the possibility of getting sixthed or eighthed in a loose game, I think you’d need 45 chips in the pot on the river to continue after the flop, and that’s with no raising on the second or third betting rounds.

Here’s my math:
In a ten handed Las Vegas game,
0.0012*120+0.053*90 + 0.377*60 + 0.569*30 =
0.12+4.77+22.62+17.07 = 44.58
Round that to 45.

You could get that (at least 45 chips in the pot, each chip representing one small bet or one half of a big bet) with at least seven opponents who regularly chased all the way to the showdown. But I don’t think that’s realistic.

If there’s a raise on the third betting round, then Hero’s contribution would be higher and consequently the number of chips needed in the pot on the river would be greater.

Often the betting will get checked around after a flop of
J[img]/images/graemlins/heart.gif[/img], J[img]/images/graemlins/club.gif[/img], 8[img]/images/graemlins/heart.gif[/img], and you get a free card.

The other alternative is to bluff a jack after a flop of
J[img]/images/graemlins/heart.gif[/img], J[img]/images/graemlins/club.gif[/img], 8[img]/images/graemlins/heart.gif[/img]. Most of the time (~72%) none of the five opponents who has seen the flop actually has a jack, and that seems especially true if they check to you. Trouble is, if five opponents see the flop, the probability of none of the five having at least one jack, assuming neither any preference for nor any distinct dislike of jacks, is only
1-C(41,20)/C(43,20) = about 28% - and some opponents check/call with a jack. And even if nobody has a jack, you have to tentatively put anybody who calls on a jack, although they might just be calling your bluff and intending to make a move on a later round. And there’s also the (remote) possibility of running into flopped eights full of jacks or two overpairs, either of which might simply call your bluff. So it’s pretty hard to follow through with another bluff or semi-bluff after the turn when you have a caller or two.

But sometimes a bluff works after this flop, especially if the pot is not too big and if your opponents view you as a solid and not overly tricky player.

However, when someone else bets this flop, I think you generally should give them the benefit of the doubt and fold, saving your chips for situations where you have more favorable odds.

I got into a dream game last week (alas, not located in Southern California) that was Omaha-8 heaven - an average of eight out of ten players seeing the flop in a 4/8, half-kill game! But even in that Omaha-8 paradise there were not enough chips in the pot at the showdown to justify playing runner-runner bare ace-deuce nut low draws after flops where my hand completely missed for high.

Buzz

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

[ QUOTE ]
Assuming no raises, even without getting quartered or sixthed, Hero has to win at least five big bets (ten small bets) when he makes his runner-runner draw. That's realistic for scooping the pot, but not for splitting the pot. It's not impossible - just not realistic.


[/ QUOTE ]

Hi Buzz, I hope you had fun in Vegas. There during the WSOP, did you go by the Rio at all? BARGE is next week, are you coming?

Anyways, that is the key, an unraised pot. There are several situations with enough players in a raised pot that a runner runner low draw is profitable.

Don

Re: LO8 - When is it correct to chase runner-runner for the nut low
 

Geez, I haven't worked out this stuff in a long time, but we had some real great threads on this a few years ago. With a bare A2, you need a lot more bets in the pot than an A23X or A234, I don't know that 15 bets is enough, but it should be more than adequate for A23.