# [pzhon] The Golden rule

• Bronze
Joined: 15.10.2008
When you are on SB on full table and everyone folded to you, do you and how adjust your pushing range to fact that BB dont have "normal" distribution of cards? (Probabilty of AA in BB is not 1/220 but something about 1/150.)

How to calculate equity of some Kx SB push, when BB have more Ax hands than "normal" and higher probabilty of AK than A2?

Thank you and welcome to Poker Strategy.
• 20 replies
• Bronze
Joined: 01.12.2008
I concluded from "killer poker by the numbers" that the calling frequency increases by about 25% when pushing from the sb. The author ran computer simulations that support this theory.

In other words if bb calls top 20% of hands, he will actually be calling about 25% of the time. In order to compensate for this you have to push 20% less hands than what is optimal if you don't take into account this change in distribution.

I hope I've made at least a little bit of sense.
• Bronze
Joined: 17.06.2010
It's a good point by ChoChikun that the BB's hand is not random after many players fold. The players who folded are more likely to have folded low cards than high cards, so the remaining deck is richer in high cards, particularly aces. This is called the "bunching" effect. This is ignored by programs like SNG Wizard, ICM Trainer, or the Nash calculator on HoldemResources.net.

The main reason the bunching effect is ignored is that it is easy to ignore it, and very hard to include it, particularly when you are trying to compute the push/fold Nash equilibrium. If you can ignore the bunching effect, then to compute the Nash equilibrium for 4 player, you can start by solving blind-versus-blind, then see when the button would be better off pushing, then add the CO. If you have to take into account the bunching effect, you can't build the solution this way, since you don't know yet what the blinds have when the CO and button fold.

Many people hope that the bunching effect is negligible in Hold'em, and there are reasons why it is smaller in tournaments than in some of the simulations for cash games. When there are fewer players at the table, then the bunching effect is reduced. Also, people may play fewer weak aces like A2s or A5o in early position when the stacks are shallow than when the stacks are deep.

If the chance to run into AA is 20% higher than if the BB had a uniformly random hand, the chance to run into AT might only be about 10% higher. AA is an extreme case.

As noclaninator mentions, the bunching effect suggests pushing a tighter range than you could against someone with a random hand. This is just one effect left out of the usual models. There are others, such as that players don't call with a consistent range: If you get called 15% of the time, it won't be by the top 15%. You will see AA, JJ, AQ... and some surprisingly weak hands like 98o. The inconsistency of your opponents lets you push wider.

While the bunching effect may discourage you from making some marginal pushes from the small blind, it may encourage you to push wider from earlier positions when you have a hand like Ax. The Nash calculator sees that when you push an ace, then there are only 3 aces left in the deck. However, when you fold an ace, it assumes the ace is shuffled back into the deck. One reason to fold is that you hope your later opponents will collide. However, if you have Ax, it is less likely that two of your opponents will have a pushing and a calling hand. So, folding A2 is not as good as folding 32, and you might be right to push A2 when you wish you could change your hand to 32 and then fold.

It will be a step forward when programs can incorporate the bunching effect in their analyses, and can tell us whether the bunching effect is significant or not.
• Bronze
Joined: 17.06.2010
Maniac asked, "How do you calculate the avg. loss?"

I believe this was about pushing 75s against someone calling with a 38.5% calling range at about the 8:00 mark in the video. When you get called, you have risked 9.5 bb, and you get back some share of the 20 bb pot on average. I used PokerStove to calculate that 75s has 37.94% equity against the calling range:

 code: ```839,028,960 games equity win tie pots won pots tied Hand 0: 62.056% 61.56% 00.50% 516510245 4157444.50 { 22+, A2s+, K2s+, Q6s+, J8s+, T9s, A2o+, K6o+, Q8o+, J9o+ } Hand 1: 37.944% 37.45% 00.50% 314203826 4157444.50 { 7c5c }```

37.94% x 20 bb is 7.59 bb. You invest 9.5 bb and get back 7.59 bb on average, for a loss of 1.91 bb when you get called.

I also used PokerStove to count combinations to see how often you get called because PokerStove reports the number of "games." Every calling combination means PokerStove evaluates 48 choose 5 = 1,712,304 boards. So, the number of calling combinations is 839,028,960/1,712,304 = 490 out of the 50 choose 2 = 1225 possible combinations. 490/1225 = exactly 40% calls.

The unconditional average gain from pushing is 60% x 1.5 bb - 40% x 1.91 bb = +0.14 bb.
• Bronze
Joined: 29.03.2006
hi,

is this 839,028,960 the numer of boards possible?

48 choose 5 do you mean deck +flop?

50 choose 2 obv. your hand and the rest of the deck

thanks and greez
• Bronze
Joined: 22.10.2007
Thanks for the great chart, going to print it out (I mean the one in the vid). I was looking for such a chart for a long while.
And the whole video is very good as well, I understood every important aspect after the first view although english is not my mother tongue. Seriously, its your first video and its already one of the best on this site, keep it up!
• Bronze
Joined: 17.06.2010
Originally posted by maniac
is this 839,028,960 the numer of boards possible?
839,028,960 is the number of boards analyzed by PokerStove when you calculate the equity of 7 5 against the range.

48 choose 5 do you mean deck +flop?
48 choose 5 means the number of ways of choosing 5 objects out of 48. Once you select a hand for yourself and your opponent, there are 48 more cards, and the number of possible flops+turns+rivers (when you don't care about the order) is 48 choose 5 = 1,712,304.

For each combination in the calling range, there are 48 choose 5 boards, so to count the combinations you can divide the 839,028,960 (reported by PokerStove) by 1,712,304 to get 490 combinations. This is a faster way to count the calling combinations than saying

6 combinations of AA
6 combinatiosn of KK
...
3 combinations of 77 since you have a 7
6 combinations of 66
3 combinations of 55 since you have a 5
...
12 combinations of A8o
9 combinations of A7o since you have a 7
etc.

You can verify that this method, dividing the number of reported games by 1,712,304, gives the same count of combinations in some simpler situations, like AK vs. QQ+.

I hope this helps.
• Coach
Coach
Joined: 27.07.2006
Phzon you undoubtely have a very deep understanding of SNG and MTT Theory, and you also have a very nice and clear way of making your points.

I just find it strange that people were flaming Jonathan Little a few months ago for misapplying the Nash HU chart in several instances and now praise the golden rule.

This 'Rule' is just a rule of thumb and therefore might minimize mistakes, but does not help to find optimal ranges, let alone foster a true understanding of why Ranges change the way they do (ok you do show some parameters that influence shoving/calling ranges, I admit).

It might help people to meorize ranges more quickly in the beginning but thats all.

Or am I totally missing your point? verwirrt
• Bronze
Joined: 17.06.2010
I'll copy the answer I gave in the German forum:

Original von JayGatsby
Phzon you undoubtely have a very deep understanding of SNG and MTT Theory, and you also have a very nice and clear way of making your points.
Danke.

This 'Rule' is just a rule of thumb and therefore might minimize mistakes, but does not help to find optimal ranges, let alone foster a true understanding of why Ranges change the way they do (ok you do show some parameters that influence shoving/calling ranges, I admit).

It might help people to meorize ranges more quickly in the beginning but thats all.

Or am I totally missing your point?
The Golden Rule is just a rule of thumb, and you are right that there is much more to finding optimally exploitive ranges, which depend on your reads on your opponents. The Golden Rule doesn't even tell you the Nash equilibrium strategy, since it just tells you to push at least as widely as you would HU when you aren't covered by a lot. Sometimes the Nash equilibrium is much wider than that even with equal stacks.

If the Golden Rule helps you to minimize your mistakes, that is very valuable! I believe the Golden Rule should help many beginners avoid gross mistakes by referring to the chart, and the HU chart is more detailed and more accurate than the SB columns of more general charts. In my experience, many solid winning players improve further by using the Golden Rule. Some of my students came to me after getting a star from the Sharkscope leader boards, and they still missed Golden Rule shoves, particularly for more than 10 bb. Since their opponents were calling tighter than the Nash equilibrium calling ranges in most situations, they were missing some profitable pushes.

There is more to blind-versus-blind play. However, a good exercise for many players is to make sure not to miss any Golden Rule pushes in a session by accident. Only fold a hand which is a Golden Rule push if you have a clear reason to override this useful heuristic.
• Bronze
Joined: 26.11.2007
Originally posted by JayGatsby
I just find it strange that people were flaming Jonathan Little a few months ago for misapplying the Nash HU chart in several instances and now praise the golden rule.
Jonathan used the chart for calling, too, and in bubble situations. That`s without doubt a huge mistake.
I will check out your highly praised vid, pzhon.
• Bronze
Joined: 19.02.2007
Not really feedback, but i think you said you are open to new suggestions for topics in future videos.
I really have trouble to adjust to the loose pushes from the regs in mp co and bu. I think its quite common for players have problems adjusting to that when they move up to the stakes were the regs are more frequently and better (and might even abuse your tightness).
And since your old video (on stox) about stealing in early and midgame really helped me to boost my roi in these stages* i think you are the right guy for that.

*(and made me a retarded spewmonkey , im now working on removing the retarded)
• Coach
Coach
Joined: 27.07.2006
Originally posted by 00Visor
Originally posted by JayGatsby
I just find it strange that people were flaming Jonathan Little a few months ago for misapplying the Nash HU chart in several instances and now praise the golden rule.
Jonathan used the chart for calling, too, and in bubble situations. That`s without doubt a huge mistake.
Yeah thats right.

What was even more shocking is that he used it in a 500\$ SNG on Stars, where no doubt you have to shove/call superexact in order to have even a tiny edge.

I also remember him saying that you can divide the hu equilibrium by 2 for the BU range in 3 handed play (50/30/20).

Thats probably why I was a bit skeptical about Phzons golden rule.

However, having watched the vid again, I feel kinda bad about my critique.

It was never your point to find exact ranges, as you said it is just a useful heuristic for analyzing bvb situations. So my critique was actually an ignoratio elenchi...
• Coach
Coach
Joined: 27.07.2006
Originally posted by NamenIos
Not really feedback, but i think you said you are open to new suggestions for topics in future videos.
I really have trouble to adjust to the loose pushes from the regs in mp co and bu. I think its quite common for players have problems adjusting to that when they move up to the stakes were the regs are more frequently and better (and might even abuse your tightness).
Why would you wanna see a video about that? Thats just basic Wizard Work!

Assign people shoving ranges, look up your calling range, that easy.
• Bronze
Joined: 19.02.2007
Thats true, maybe it was a bit late and i was lazy in my head. Time to get my ass up and play around with wiz.
• Bronze
Joined: 17.06.2010
Originally posted by JayGatsby
Why would you wanna see a video about that? Thats just basic Wizard Work!

Assign people shoving ranges, look up your calling range, that easy.
Actually, I plan to make a video on calling all-in soon. There is a lot to study.

Choosing the right range for your opponents is difficult, and is worth several percent ROI to determine how tightly your opponents are pushing. Further, SNG Wizard does not let you choose reasonable pushing ranges. SNG Wizard's hand ordering by the Sklansky-Chubukov rankings means it will push hands like J4o before 76s. J4o should not be in a 50% pushing range, but 76s should be in a 30% pushing range.

Second, it is useful to study situations to see how risk-averse you should be. How much equity do you need against the pusher's range to call? I created ICM Explorer to study questions like this.

Third, which hands have enough equity against the range you give your opponents? This is not trivial. I still see regs make calls which would be wrong against a 100% range. You can study the equities using PokerStove and PokerStrategy's Equilator.

SNG Wizard combines the second and third steps, which can be useful at times, but I think it is good to study them separately.
• Coach
Coach
Joined: 27.07.2006
Yeah the Wiz Handranking is really annoying.

Why is it actually so hard to program Wiz in a way that you can assign ranges freely like in the Equilator, where you can even exclude certain Suits and Stuff?
• Bronze
Joined: 02.03.2008
Originally posted by JayGatsby
Yeah the Wiz Handranking is really annoying.

Why is it actually so hard to program Wiz in a way that you can assign ranges freely like in the Equilator, where you can even exclude certain Suits and Stuff?
Hi!
That is annoying lots of users, and the dissatisfaction has reached the Wiz developer long ago. I copy the answer from the Wiz faq:

"...The problem with supporting user-defined hand ranges is speed, or lack thereof. By using a fixed hand ranking (like the KS [=Sklansky-Chubukov] rankings) I can pre-compute all the raw hand equity calculations. This permits the SitNGo Wizard to compute the hero’s hand range in a fraction of a second. The same calculation (the Compute All button) takes several seconds in SNTPT. Actually, the speed difference is even greater because SNGPT is only computing heads-up showdowns. SNGWIZ is computing 3-way showdowns in addition to heads-up showdowns.

If user-defined hand ranges were supported, the equity calculations would have to be computed in real-time. Imagine running hundreds of PokerStove solutions every time you open a hand. I fear the performance would be dreadful...."

At least these raw hand equity calculations should have been pre-computed for some more rankings. Probably Wiz would take up a few more MBs of hard disk space but no one would mind I think...
• Bronze
Joined: 22.11.2009
Does anyone have a reference to the technical details of how to do these computations? I'd love to look into the computational issues, and maybe even get a student to work on it.
• Bronze
Joined: 01.09.2005
As I really like the Nash- Chart you use in the video (Min 9:20) I want to ask you where I can download this chart or if you can upload it? I've made a Screenshot and printed it out but I'd also like to have the chart for Calling.

Secondly I recognized that there are some slight differences between the chart you show in the video and the chart on holdem-resources. What is the reason for that?
• Bronze
Joined: 17.06.2010
Originally posted by jbpatzer
Does anyone have a reference to the technical details of how to do these computations? I'd love to look into the computational issues, and maybe even get a student to work on it.
I'm not sure which computations you want.

There was a reference on another forum about how it is computationally difficult to take into account the card-removal effects. See "Shoving mathematics" in the PokerRoad Red Pro forums.

If you are just looking at SNG Wizard-style calculations for blind versus blind shoves with no card-removal effects from other players, then it should be a straightforward calculation once you have a table of how each hand does against each other, and account for the card removal effects of particular hands.

It is more complicated if you want to compute the Nash equilibrium. As long as you don't take into account card removal effects, this can be done recursively using fictitious play. There is a sense in which the Nash equilibrium is unstable: Exploitive responses to strategies near the Nash equilibrium may spiral out away from the Nash equilibrium. So, you may need to take smaller steps, averaging your last strategy with the play which maximally exploits your opponent.