# Mathematical Derivation of the Chen Formula

I was wondering if anyone knew how the Chen formula was mathematically derived and moreover what it is exactly that it is calculating?

I always find it very strange that it gives +2 for suited hands seeing as being suited really makes little difference percentage wise. So it gives things like:

(this is being used as an example to show that the Chen formula does not simply measure starting hand strength in a simple way)

Thanks for any help

Note: I am not looking for opinions on whither the Chen formula is useful or why it's not applicable to every situation and should only be used as a guide to for new players to get a feel for the strength of hands and so on. I am looking for a derivation of this formula or a source where I can find the derivation of this formula, or if this is not possible (I couldn't find one, but I could have easily missed it) some insight into its derivation.

Second attempt at clarification: I am asking if anyone knows or knows of a place where I can find a mathematical derivation of the Chen formula. I am asking for something that will involve quite a lot of maths so I was really hoping more for a source.

Third Attempt at Clarification: This question is not about people thoughts on the Chen formula (as that is outside the scope of a .se account) I am specifically looking for a derivation of the Chen formula. Chen has a phd is stats so I know he did not just make this up. I am not confused by my example it is just a trivial example of why the Chen formula is not a direct hand ranking system.

• possible duplicate of Thoughts on Chen's formula Feb 23 '13 at 13:53
• @SoboLAN I really don't think that this is a duplicate and if you think that it is I have probably worded my question incorrectly, I did look at that question before posting this one. I am asking how the Chen formula is derived and what exactly it is saying (which would come directly from it's derivation), I am not looking some opinions of where it is useful and why its not always applicable and so on. Feb 23 '13 at 14:35
• @hmmmm If you could clarify the specifics of the question you're asking and then use that as the title, that would help define the usefulness of this thread. Feb 23 '13 at 15:55
• @TobyBooth I think my title is pretty specific, I am asking for a derivation of the Chen formula. I know that this is a poker forum not a maths forum but there is definitely a strong enough connection. I'm not hopeful of getting an answer here now, but if I find one somewhere else I will post it here Feb 26 '13 at 14:13
• @TobyBooth yeah I think the word derivation is causing some problems. I am not asking how to calculate the Chen number of a specific hand, I am asking "how Chen came up with the formula" which would be some sort of game theoretic analysis (something similar to what is in his book). It is a question that is very mathematical in nature so maybe not best suited to this forum at this time (hopefully in the future). Thanks for the help anyway. If I find an answer I will come back and submit it but I appreciate that it's probably not appropriate at this time. Thanks Mar 1 '13 at 22:39

Here's one way that it might have been derived - this won't be the exact method, but it might have been a jumping off point for Chen.

First, take the Sklansky hand groupings - take it as given that this is what the Chen formula is trying to reproduce. Then produce a table of grouping for each hand, against a set of features that that hand might have (e.g. whether it is a pair, whether it is suited, whether it is a 1-gap or a 2-gap, what the high card is etc).

Then come with a way of converting the Sklansky hand groupings into scores. I took the simple formula

Score = 15.5 - GroupNumber

The subtraction is to ensure that higher groups have higher scores, and the constant 15.5 is to avoid the need for a constant in the regression I'm about to perform (just a technical detail, doesn't really matter).

Then, perform a regression of the score for each hand on each of the features you identified earlier. This gives each feature a contribution to the total hand score. The contributions for each separate feature look like this (roughly - I rounded each one to the nearest 0.5). Using this table, a hand like AA would get a score of 16.5 under this system (8.5 for being paired, 8 because the high card is an A). A hand like 98o would get a score of 8.5 (5 for being a connector, 3.5 for having high card 9) and 76s would get a score of 10 (5 for being a connector, 2.5 for being suited, and 2.5 for having high card 7).

A few interesting points to note - you can immediately see that having an A is way better than having any other card, and having K, Q or J is way better than 2-T. Being suited turns out not to be that important - it's about as important as having a 2-gap, but less important than having a 1-gap or a connector.

The neat thing about this is that the 'Score' can be anything. I derived a score from the Sklansky hand groupings, but you could use probability of winning the pot, or EV from a real-world hand history. It is quite plausible that Chen started with something like this, and and refined the formula to make it simpler and to correspond with his intuition.

• I'm going to except this as an answer because I think it is the closest I'm going to get (that is, I don't think anybody has the precise process that Chen used). But I in no sense mean that this is not a great answer (and was pretty much what I was looking for) Having had a few off topic answers/comments I didn't think anyone was going to have a proper answer. So Thanks this is great- definite +1! Mar 10 '14 at 20:22

It looks like the main reason behind using the Chen formula for different starting hands was so that you can categorize them based on the Sklansky and Malmuth hand groups table.

Have a look at:

http://www.thepokerbank.com/strategy/basic/starting-hand-selection/chen-formula/

There might not be heavy math behind it, after all, not the math from his book anyway. What kind of hand makes an experienced player happy or unhappy cannot always be translated into clear mathematical principles (or if it can, then that would be a long story). Experience is the key.

Here are a few thoughts that might help you get a feel for why / how the Chen formula is developed.

First, you have to realize that it's an approximation. Any simple set of rules for ranking hands is bound to have exceptions,a nd you will be able to find cases where it doesn't work.

Second, you have to ealize that it's not trying to rank hands against each other. Just because hand A ranks above hand B using the Chen formula, that doesn't mean that A has greater equity than B. Instead, it is supposed to be a measure of how playable a hand is pre-flop vs all other hands. It's trying to approximate something like the Sklansky hand groupings, so that in general, hands in higher Sklansky group will be ranked higher according to the Chen formula. To understand the Chen formula, it really helps to understand Sklansky groupings.

Looking at the Sklansky hand groupings, there are a couple of things that immediately jump out. For example, TT has greater chance of making the best hand than AKs, even though TT is in group 2 and AKs is in group 1. So why is TT ranked lower?

The key is to realize that chance of making the best hand is not the same as expected value, and expected value is what you really care about. To get a high expected value, it helps if a hand can (a) win you lots of pots, (b) win big pots, and (c) not lose too many big pots. The problem with a hand like TT is that it frequently makes the second best hand, losing out to any higher pair. If there's a single A-J on the table, then anyone with an A-J in their hand can beat you. So it fails test (c) - you often lose big pots with TT by having the second best hand. On the other hand, a hand like AKs will tend to make a very good or great hand (pairing the A or K or getting a flush) but rarely makes a hand that turns out to be second best. If it looks like the best hand, it frequently is. If it's not a good hand, it's obvious.

This also explains the example of 98o vs 76s. Although 98o has better equity (the chance of pairing the 8 or 9 more than makes up for the lower chance of a flush) it doesn't tend to play well against other hands, as the only way it can make the best hand vs a lot of aces, kings and queens is to hit a decent straight. The 76s can hit either a straight or a flush, and when you hit those hands, you tend to win big pots because you have great implied odds.

• I'd be interested to know why this was downvoted - I think that it explicitly answers the part of the question "I am looking for a derivation of this formula, or if this is not possible, some insight into its derivation." Downvotes without a comment to explain them are not helpful if you want the site to grow. Mar 10 '14 at 18:37

In hot/cold equity terms 98o is ahead of 67s, so why would you pick one over the other?

It's a question about post-flop playability, the types of hands you're attempting to make and ability to apply pressure. Put simply, with the suited hand you will see many more boards where you can apply pressure through semi-bluffing than with the 98o. The suited connector has the ability to make more nutted type hands than the offsuit connector.

With both these hands, if you hit showdown then you're trying to make some sort of nutted hand like a straight or a flush. A pair of nines is hardly better than a pair of sixes versus a lot opponent ranges, meaning the "high card" factor can be heavily discounted - they both lose to any paint pair and beat any high card.

• This answer wasn't really what I was along for I you re-read my question thanks Dec 23 '13 at 21:07
• There is no "mathematical derivation". Like I said, choosing starting hands is not about hot/cold equity. That's an incredibly simplistic way of looking at poker.
– tom
Dec 24 '13 at 14:19
• It's also about frequencies but I'll leave that for you to work out.
– tom
Dec 24 '13 at 14:27
• I think choosing starting hands is all about equity, in fact poker is all about equity and attempting to maximise it. I just think that you are using a very narrow idea of equity there. But again all I was saying was that the answer you gave really isn't answering my question Dec 31 '13 at 14:34