Poker Math How-To: SB vs BB short-stack

It's folded to you in the small blind. Your villain has 20 big blinds left after posting the blinds. You look down at K5o. Can you show him your hand and shove over him and be assured that it's a profitable play?

Step 1: What range of hands can villain call with profitably if we shove K5o?

Villain will be calling 20bb in a pot of 21.5bb, so he can call profitably with any hand that has 48.2% equity or better against K5o. This range of hands is: 22+,A2s+,K5s+,JTs,T9s,A2o+,K5o+.

Step 2: If villain calls with exactly the range from Step 1, are we profitable?

Villain's optimal range found in step 1 is 30.6% of hands. It also has 66.0% equity against K5o. The value of our shove is found by adding up the following:

We shove and he folds (profit of 1.5bb): (1-0.306) * (1.5)
We shove, he calls, we win (profit of 21.5bb): (0.306) * (1-0.660) * (21.5)
We shove, he calls, we lose (profit of -20bb): (0.306) * (0.660) * (-20)

Which comes out to be (1.041 + 2.237 - 4.039) = -0.761. This means that if villain calls with the optimal range against our hand that we will lose on average 0.761 big blinds per hand.

Now a question for you to work out and answer: What is the range of hands you can just shove here and be guaranteed to profit, even if your opponent could see your hand?

Presentation by Lee Jones at 2005 WPPC
Copyright © 2005 Lee H. Jones
Note to organizer: each primary numbered entry should be the title of a single Power
Point slide. Lettered sub-entries should be bullets on that slide.
Heads-up Strategy for No-Limit Hold’em Tournaments
1. An Equilibrium Heads-up Strategy for No-limit Hold’em
a. Developed by James Kittock of Mission College, California, with
assistance from Lee Jones
b. All information here Copyright © 2005 James Kittock and Lee Jones
2. The Scenario
a. You are the last two players in a no-limit hold’em tournament.
b. The button has the small blind and acts first.
c. The ratio of stack sizes to the big blind has gotten relatively small (less
than 10:1).
3. Assumptions
a. Every chip has identical value. That is, there is no extra bonus for winning
your opponent’s last chip, and no extra penalty for losing your last chip.
b. The small blind is one-half of the big blind.
c. Because the blinds are large compared to the stacks, the only viable
strategy for the button is to move all-in (“jam”) or fold. This is true
almost always in sit-and-go’s and smaller tournaments. In major events,
the stack:blind ratios are often much bigger than 10:1 (at least initially).
d. If the button jams, then (by definition) the big blind can only call or fold.
4. What is an equilibrium strategy (ES)?
a. This strategy can not be exploited. If you play this strategy, your
opponent must play the equivalent strategy, or his EV will go down.
b. If either player deviates from the strategy, his results will suffer.
5. How was the strategy determined?
a. Take all 169 possible starting hold’em hands (e.g. KK, QJo, 85s) and rank
them (from AA down to 32o) by determining how each does against a
random hand, taking into account the relative likelihood of various
matchups (e.g., AA vs. AKs is relatively uncommon compared to 85s vs
32o).
b. Determine the EV of each against all the others.
c. Plug these numbers into a model that calculates the expected change in the
small blind’s stack size (“delta SB”).
d. Use game theory (specifically, the “minimax” technique) to determine the
ES for each player, for various stack-to-blind ratios.
6. The results – a qualitative view
a. Most players do not adjust correctly for changing hand values as the
number of players goes down.
b. This lack of adjustment becomes egregious at heads-up play.
c. Most players jam from the button/small-blind far less than they should.
d. Most players, in the big blind, call a jam by the button/small-blind far less
than they should.
e. The equilibrium strategy will clobber most opponents playing by the seat
of their pants.
7. The results – quantitative
Optimal Top % and Cutoff Hands
SB (jam) BB (call)
R
Top
%
Cutoff
Hand
Top
%
Cutoff
Hand
Delta SB (in BB
units)
1 89% 62s 100% 32o 0.010
2 79% 64s 89% 62s 0.051
3 74% 95o 70% 75s 0.061
4 71% T4o 60% J5o 0.047
5 68% 96o 53% 97s 0.026
6 64% 76s 48% Q2s 0.002
7 61% T4s 42% Q7o -0.018
8 58% J2s 39% K2s -0.042
9 55% 98o 36% 33 -0.063
8. The “Sit And Go Endgame” System (SAGE)
a. Copyright © 2005 by James Kittock and Lee Jones
b. Developed by James Kittock as an easy-to-remember approximation to the
equilibrium strategy (ES).
c. Gives up virtually no advantage to an ES player. And (like the ES)
crushes most players – even experts.
d. Far easier to compute than looking up than the top N percent of hands in a
table.
9. SAGE: Computing a Power Index (PI)
a. The “power number” of each card is its rank. J=11, Q=12, K=13, A=15
(don’t forget the ace is 15!)
b. Take the power number for your higher card and double it.
c. Add the power number of your lower card.
d. If it’s a pocket pair, add 22.
e. If they’re suited, add 2.
f. The sum is the Power Index (PI) of your hand.
10. SAGE: Using the PI
a. Compute the ratio (R) of the shortest stack to the big blind.
b. Look up the necessary PI for that value of R.
c. If the PI of your hand is greater than or equal to that value, then jam (if
you’re the button/small-blind) or call (if you’re the big blind).
11. The SAGE Numbers
12. SAGE Example 1
a. Blinds are 500/1000. After the blinds are taken, the SB has 5635 chips
and the BB has 2865 chips.
b. The SB has pocket 3’s. PI = (2 x 3) + 3 + 22 = 31
c. BB has J4s. PI = (2 x 11) + 4 + 2 = 28
d. The value of R is the smaller stack (2865) divided by 1000 ≈ 3.
e. Looking at the table, the SB should jam, and the BB should call, since
both have PIs higher than the respective entry in the table (22 for the SB
and 24 for the BB).
13. SAGE Example 2
a. Same circumstance, but the blinds are 200 and 400.
b. Now R is 2865 divided by 400 ≈ 7.
c. The SB should still jam (his PI of 31 is greater than the necessary 26).
d. If the SB jams, the BB should now fold, since his PI (28) is smaller than
the corresponding entry (30).
14. Why should we give this away?
a. It improves the game of poker. A field of study can only advance if its
participants discuss, argue, and (in this case) prove what is true and what
is not.
b. It’s not fair that just a few people have this. If you think we’re the first
people to do this, you’re wrong. If you find yourself heads-up with Greg
Raymer or Chris Ferguson, don’t think for a moment that they are without
this information.
c. Don’t think this will make the games terrible. Look what Brunson did
with Super/System. Look what Sklansky did with Theory of Poker. Look
R Jam (SB) Call (BB)
1 17 any
2 21 17
3 22 24
4 23 26
5 24 28
6 25 29
7 26 30
what Harrington is doing with the Harrington on Hold’em books. The
games continue to be great.
d. It’s fun to make a big splash in the swimming pool.
15. [entire slide should be this quote]
“In poker, as in every other area of human endeavor, the considered opinions of a
collection of the world’s best practitioners might be right, but it might also be quite
wrong. The consensus of what is considered true seems obvious and inevitable until
some brave soul comes along and says ‘No, the truth is really like this.’”
-Dan Harrington, Harrington on Hold’em, Volume II, p. 162

For cash games, there is a table called Chubukov Rankings
google it for more info, or look it up on No Limit Hold 'em: Theory and Practice by David Sklansky

The first post you made is in reference to the "Sage" system. Here is a link: http://www.cardplayer.com/magazine/article/15250

I think this was meant to get people to think themselves, not google someone elses answer .

Originally Posted by bjsaust

I think this was meant to get people to think themselves, not google someone elses answer .

Well, Sage is pretty useful information so no big deal.

For stacks 30bb and under, it's not hard to get a feel for it, or even to just make charts. I once did 5bb to 35bb push/fold charts for the same thing in around an hour.

20BBs and over I tend to use a b/c range than a push/fold range. You think push/fold is the best option between 20 and 30?

Originally Posted by bjsaust

20BBs and over I tend to use a b/c range than a push/fold range. You think push/fold is the best option between 20 and 30?

Against a 30bb or less stack I tend to just push/fold because it's easier. The stack sizes being small means position is less of a factor so you can't really argue that so much as straight-up convenience.

You occasionally get people to do some kind of stupid stuff too, like call your shove 25bb deep with Q8s.

But with over 30bb push/fold is pretty clearly not ideal, and with under 20bb it probably is ideal, so the 20-30bb range is probably a matter of personal taste.

Originally Posted by bjsaust

I think this was meant to get people to think themselves, not google someone elses answer .

Yeah, maybe i should've shut up, i still think, googling it and doing some research/reading will help people.
The topic is a bit complicated for someone to re-think the whole Sklansky-Chabukov ratings all over by doing the whole math himself, so pointing where to look may help.

Originally Posted by asdpikas

Originally Posted by bjsaust

I think this was meant to get people to think themselves, not google someone elses answer .

Yeah, maybe i should've shut up, i still think, googling it and doing some research/reading will help people.
The topic is a bit complicated for someone to re-think the whole Sklansky-Chabukov ratings all over by doing the whole math himself, so pointing where to look may help.

The implications of the thought process involved when making these calculations extend far beyond even the preflop aspect of the game, so it's an important concept to learn for some people.

[sarcasm]Why bother learn something for yourself when you can go to www.google.com and find it.[/sarcasm]

Also, note that the SAGE system doesn't necessarily apply to cash games:

When does the SAGE System apply?
Let's be sure that you understand exactly when you can apply SAGE:
1. You are heads up with a single opponent at the end of a no-limit hold'em tournament. The important point is that there is just a single prize left. Note that a one-table satellite into a larger event also has this feature (the single prize is a seat in the larger event).

And now the question that no one has bothered to respond to:

Originally Posted by spoonitnow

Now a question for you to work out and answer: What is the range of hands you can just shove here and be guaranteed to profit, even if your opponent could see your hand?

Originally Posted by spoonitnow

Also, note that the SAGE system doesn't necessarily apply to cash games:

Maybe that's why i posted twice...
once for tourneys, once for cash games (mentioned in my posts)

If all the thought processing is what's important, then the specific range for your example is not the important thing. As stated, anyone can look it up on a table.

The answer, in a few words would be to find the range that makes opponent fold X% of the time. Thus X% of the time we win the BB. This ammount of BBs won should compensate for the loss of EV we suffer when he calls with 100-X%
The larger our stacks (actually, only the smallest stack of both) compared to the BB, the more EV loss we suffer when he calls allin with 100-X. Thus, our range must be much stronger the bigger effective stacks are compared to BB.

EDIT

this X amount of folds that we need to achieve, is also known as folding equity, very important concept in semibluffing. The more folding equity we have with our range, the more we compensate for the good calls he makes.

All this assumes he will call correctly, if we include bad calls with worse hands, where we would be +EV when called, then the folding equity doesn't need to be that high.

Originally Posted by spoonitnow

The implications of the thought process involved when making these calculations extend far beyond even the preflop aspect of the game, so it's an important concept to learn for some people.

[sarcasm]Why bother learn something for yourself when you can go to www.google.com and find it.[/sarcasm]

The implication of your post (and the one about game theory and bluffing) is that you've read Sklansky's books. I hardly see how this post teaches anything beyond that. You don't do a very good job at explaining the concepts.

[sarcasm mode on]Why bother learning something for yourself and doing the reading/research if you can just read a simplistic post in a forum[sarcasm mode off]

Originally Posted by asdpikas

Originally Posted by spoonitnow

Also, note that the SAGE system doesn't necessarily apply to cash games:

Maybe that's why i posted twice...
once for tourneys, once for cash games (mentioned in my posts)

If all the thought processing is what's important, then the specific range for your example is not the important thing. As stated, anyone can look it up on a table.

Hopefully now that you have stated the obvious (in bold) people will understand that it's the process of going through the calculations that will teach you important things, not the actual results themselves.

Originally Posted by asdpikas

The answer, in a few words would be to find the range that makes opponent fold X% of the time. Thus X% of the time we win the BB. This ammount of BBs won should compensate for the loss of EV we suffer when he calls with 100-X%
The larger our stacks (actually, only the smallest stack of both) compared to the BB, the more EV loss we suffer when he calls allin with 100-X. Thus, our range must be much stronger the bigger effective stacks are compared to BB.

EDIT

this X amount of folds that we need to achieve, is also known as folding equity, very important concept in semibluffing. The more folding equity we have with our range, the more we compensate for the good calls he makes.

All this assumes he will call correctly, if we include bad calls with worse hands, where we would be +EV when called, then the folding equity doesn't need to be that high.

This is a very inefficient thought process when approaching this problem. The two-step process I outlined in the OP is the most efficient and most instructive way to find these results. A simple spreadsheet could speed the process up, though.

Originally Posted by asdpikas

Originally Posted by spoonitnow

The implications of the thought process involved when making these calculations extend far beyond even the preflop aspect of the game, so it's an important concept to learn for some people.

[sarcasm]Why bother learn something for yourself when you can go to www.google.com and find it.[/sarcasm]

The implication of your post (and the one about game theory and bluffing) is that you've read Sklansky's books. I hardly see how this post teaches anything beyond that. You don't do a very good job at explaining the concepts.

[sarcasm mode on]Why bother learning something for yourself and doing the reading/research if you can just read a simplistic post in a forum[sarcasm mode off]

The implication of my posts are that the mathematics behind what are sometimes considered non-beginner problems in no-limit hold'em is not as hard as it seems. The reason you aren't finding the perhaps less-than-obvious lessons in these posts are that you aren't trying to apply it to other aspects of the game, but this is intentional. My style of teaching in all things is to provide enough of a foundation in a subject for someone to learn the concept sufficiently to apply it in more areas than the example(s) given. The result is that it lacks a "holding your hand" approach that can often be confused as me not doing "a very good job at explaining the concepts."

There is a lot more to my posts than rehash of the same old material. You, however, seem content to copy and paste something you find on the Internet. The difference is that I take people who are willing to learn through a series of thoughts and discoveries that give them ideas to think about on their own. It is absolutely ridiculous that you would try to attack my posts when it's blatantly obvious that you have not thought critically about what has been presented. I say that it's blatantly obvious because if you had, you would have questions and ideas instead of a clipboard full of copied material from various places on the Internet.

I hope that this will be a lesson to anyone who happens to read this: the journey is far more important and more rewarding than the destination.

Originally Posted by spoonitnow

There is a lot more to my posts than rehash of the same old material. You, however, seem content to copy and paste something you find on the Internet. The difference is that I take people who are willing to learn through a series of thoughts and discoveries that give them ideas to think about on their own. It is absolutely ridiculous that you would try to attack my posts when it's blatantly obvious that you have not thought critically about what has been presented. I say that it's blatantly obvious because if you had, you would have questions and ideas instead of a clipboard full of copied material from various places on the Internet.

I hope that this will be a lesson to anyone who happens to read this: the journey is far more important and more rewarding than the destination.

LOL at that, all u've done, as i said before, is take a couple of topics from known books, and post a simplistic example of them.
Well, a simplistic post, deserves what it got, a simplistic response (i.e. copy/paste)

You are nobody to assess whether i think or not about what you present.
I am, and i'll tell you what i critically think of these posts of yours, it's useless crap.

Originally Posted by asdpikas

Originally Posted by spoonitnow

There is a lot more to my posts than rehash of the same old material. You, however, seem content to copy and paste something you find on the Internet. The difference is that I take people who are willing to learn through a series of thoughts and discoveries that give them ideas to think about on their own. It is absolutely ridiculous that you would try to attack my posts when it's blatantly obvious that you have not thought critically about what has been presented. I say that it's blatantly obvious because if you had, you would have questions and ideas instead of a clipboard full of copied material from various places on the Internet.

I hope that this will be a lesson to anyone who happens to read this: the journey is far more important and more rewarding than the destination.

LOL at that, all u've done, as i said before, is take a couple of topics from known books, and post a simplistic example of them.
Well, a simplistic post, deserves what it got, a simplistic response (i.e. copy/paste)

You are nobody to assess whether i think or not about what you present.
I am, and i'll tell you what i critically think of these posts of yours, it's useless crap.

It's blatantly obvious that you're completely oblivious to the intricacies of what's being explained here so I wouldn't expect you to think any different.

asdpikas: I have yet to see any thread/post by you even half as meaningful and thought-provoking as this thread is. It honestly doesn't matter whether this concept is posted on the damn FTR homepage, as it is still overlooked by beginners, and even more advanced players, all the time. The power of this thread doesn't come in the form of knowing which hand can be shoved profitably. That is just a byproduct and simply added to this thread. The real use of this thread gets players to begin thinking for themselvs on this topic, as well as many more related topics. They begin to understand how things work and, more importantly, WHY! We see that you can google something and copy/paste the information. But instead of spitting the information back up, let it digest and learn from it (fucking stupid comparison, but oh well!).

KTo+, 22+, K8s+, A3o+, A2s+
I haven't done the suited Qs or lower yet, but I haven't done math in 20 years either so I'm taking a break. I was surprised that JQo wasn't profitable. I would be shocked if JQs, TJs weren't.

Originally Posted by chardrian

KTo+, 22+, K8s+, A3o+, A2s+
I haven't done the suited Qs or lower yet, but I haven't done math in 20 years either so I'm taking a break. I was surprised that JQo wasn't profitable. I would be shocked if JQs, TJs weren't.

I wonder if it's because 20bbs is on the deeper side for what you normally see in tournaments as push/fold situations?

for QJs

step one.652 * 1.5 = 0.978
step two.348 * .436 * 21.5 = 3.26
step three.348 * .564 * (-20.0) = -3.92
step four ???
step five PROFIT

forgot to cut/paste range so check it independently
I think it was any ace, any king, QJs/QJo, any pair

Originally Posted by iopq

for QJs

step one.652 * 1.5 = 0.978
step two.348 * .436 * 21.5 = 3.26
step three.348 * .564 * (-20.0) = -3.92
step four ???
step five PROFIT

forgot to cut/paste range so check it independently
I think it was any ace, any king, QJs/QJo, any pair

No, it's not.

QJs is profitable. The range below is 33% of hands.

equity win tie pots won pots tied
Hand 0: 43.216% 42.54% 00.68% 1209040176 19343998.00 { QJs }
Hand 1: 56.784% 56.10% 00.68% 1594696468 19343998.00 { 44+, A2s+, K2s+, QJs, A2o+, K2o+ }

1 + .33((21.5*.432)+(-20*.568))= 1-.68= +.32

rofl i can't believe you still have that microsoft avatar

(after reading thread)


oh its because you are a total cock, now it all makes sense

chardrian's range is better

Originally Posted by spoonitnow

Originally Posted by iopq

for QJs

step one.652 * 1.5 = 0.978
step two.348 * .436 * 21.5 = 3.26
step three.348 * .564 * (-20.0) = -3.92
step four ???
step five PROFIT

forgot to cut/paste range so check it independently
I think it was any ace, any king, QJs/QJo, any pair

No, it's not.

Btw I was referring to the bold

oh - my bad.

QTs is the last one in.

I was again susrprised that JTs didn't make the cut.

And 20BBs is no longer becoming "deep" in MTTs. Good players are frequently openshoving 20BBs from the CO, button, and SB.

For the math geniuses out there - what is the formula (and please explain it step by step for us math retards out here) for figuring out unexploitable shoves from the button, the CO, etc.?

bump for answer to chard question
would like to set up at spreadsheet for this, but my brains too rusty