In a previous article (https://steemit.com/poker/@markgritter/hand-values-in-chinese-poker-with-2-7-in-the-middle), I described the distribution of hand values for 13-card or "Chinese" Poker with 2-7 in the middle. But those results were based on comparing hand arrangements against a 10 million hand reference sample, which isn’t something you can do in your head at the poker table. Now I will present a beginning strategy (for 2-4 scoring) that is feasible for a human to use in real-time and provides fairly good results. I will also try to explain the ways in which this strategy fails.
Scoring Arrangements Using Reference Hands
The key idea to finding the best arrangement of your thirteen cards is to identify how likely your front, middle, and back hands are to win. Then pick the arrangement which scores the most based on those estimates. The data presented in my previous article form the basis for an estimate: although it is not an exact match, a front hand which is better than 50% of the front hands your opponent will play should win its point about 50% of the time.
The strategy I present here is fairly simple. Memorize the table below which maps back, middle, and front hands to point values. Start with the highest-value hand and continue up the table until you find the hand that you can beat--- that is the row you should use. (You can get started by memorizing only every other hand, as I’ll discuss later.) Pick the best arrangement you can find which maximizes the sum of point values, and play that arrangement.
| Points | Back hand | Middle hand | Front hand |
|---|---|---|---|
| 1 | High card | Pair or worse | 432 |
| 2 | QQJT6 | 22KQT | QJT |
| 3 | 9933J | A9876 | KQ8 |
| 4 | JJ66Q | KT765 | AQ3 |
| 5 | QQJJ6 | QT976 | AK8 |
| 6 | 8-high straight | JT984 | AKQ |
| 7 | Q-high straight | J9852 | 88A |
| 8 | A-high straight | J6532 | TT4 |
| 9 | Q9754-flush | T9642 | JJ7 |
| 10 | KT863-flush | T7654 | QQ3 |
| 11 | KQJ92-flush | 98753 | QQA |
| 12 | AQ532-flush | 98542 | KK5 |
| 13 | AKJ63-flush | 97542 | KKJ |
| 14 | Threes full | 95432 | KKA |
| 15 | Fives full | 87632 | AA6 |
| 16 | Sevens full | 87432 | AAT |
| 17 | Nines full | 86543 | AAQ |
| 18 | Jacks full | 85432 | AAK |
| 19 | Queens full | 76532 | 444 |
| 20 | Aces full | 76432 | 777 |
It is important to consider all the arrangements which might be best. Does it make sense to abandon the middle and play a pair or worse? Should you play a flush in front, or instead play a pair in back, a pair in front, and a strong middle?
Example #1: 2h5c6h6s7d9cJcJsQcKhAdAhAs
We have no possible straights or flushes. The arrangements that might make sense are (back/middle/front):
| Arrangement back/middle/front | Back score | Middle score | Front score | Total score |
|---|---|---|---|---|
| AAAJJ 97652 KQ6 | 20 | 12 | 2 | 34 |
| AAA66 J9752 KQJ | 20 | 7 | 3 | 30 |
| AAA66 Q7952 JJK | 20 | 5 | 9 | 34 |
| AAA6Q 97652 JJK | 5 | 12 | 9 | 26 |
| JJ66A Q7952 AAK | 4 | 5 | 18 | 27 |
| AAKQ6 97652 JJA | 2 | 12 | 9 | 23 |
It is not worth considering, for example, AAA66 K9752 JJQ because that is strictly worse than the 3rd alternative. It also doesn’t make sense to put anything but the weakest kickers with trip aces, because obviously the kickers will never matter in comparison with another hand. The 4th and 6th alternatives look pretty bad (an experienced player would not consider them) but they are not, technically, dominated by any of the other options.
The scoring system says we should play either AAAJJ 97652 KQ6 or AAA66 Q7952 JJK. When there are ties, tend toward arrangements which put lower cards towards the back and bigger cards towards the front. (This is the bias built into the software I wrote to produce these results, and seems to work OK in practice.) For example, if a high-card hand in front and a small pair in front are tied, tend to go with the high-card hand. If using a big pair to make a full house in back, or using a smaller pair (but worse middle and better front) are the tied options, use the big pair in front.
That would suggest playing AAA66 Q7952 JJK, which does turn out to be the best option by about three-tenths of a point.
Example #2: 4d5d5c6h7s8c9cTcThJdQdQhAd
| Arrangement back/middle/front | Back score | Middle score | Front score | Total score |
|---|---|---|---|---|
| AQJ54-flush T8765 QT9 | 12 | 9 | 1 | 22 |
| AQJ54-flush Q8765 TT9 | 12 | 5 | 8 | 25 |
| AQJ54-flush 98765 TTQ | 12 | 1 | 8 | 21 |
| QQ55J 98764 TTA | 4 | 10 | 8 | 22 |
| TT55A 98764 QQJ | 3 | 10 | 10 | 23 |
| TT55J 98764 QQA | 3 | 10 | 11 | 24 |
In this example, the scoring system identifies playing a Q-high in the middle as the best play. It also identifies TT55J 98764 QQA as the second-best alternative. Again, both of these are accurate, although the flush is better by nearly four-tenths of a point.
Example #3: 2d2h2s3d3s5h5s6c7d8hTcJhQd
| Arrangement back/middle/front | Back score | Middle score | Front score | Total score |
|---|---|---|---|---|
| 22255 T8763 QJ3 | 13 | 9 | 1 | 23 |
| 22233 T8765 QJ5 | 13 | 9 | 1 | 23 |
| 22233 JT876 55Q | 13 | 6 | 6 | 25 |
| 22235 87653 TJQ | 5 | 14 | 2 | 21 |
| 22358 76532 TJQ | 1 | 19 | 2 | 22 |
The scoring system chooses 55Q in front as the correct play. A full analysis shows that the first alternative listed (QJ3 in front) may be slightly better, but the values differ by only about 3/100ths of a point.
Analyzing the Scoring Strategy
How well does the scoring strategy do? To answer this, I took a sample of 100,000 random hands and set them to maximally exploit the “best” arrangements from the original results, the results of the scoring strategy described above, and a few variants. This ensures that we are doing an “apples to apples” comparison--- the opponent’s advantage against a particular strategy comes from exploiting the strategy, rather than simply having better hands in some cases.
I examined using either 10 or 20 reference hands for each position. I also examined using a more complicated formula, based on calculating the expected value (EV) of each arrangement rather than the sum of scores. Finally, I examined multiplying scores together rather than adding them.
| Strategy | EV (points per hand) | % of hands altered |
|---|---|---|
| Original (based on 10M hand sample) | -0.0039 | 0% |
| 3x10 reference hands, scores added | -0.048 | 25% |
| 3x10 reference hands, EV formula | -0.043 | 24% |
| 3x10 reference hands, scores multiplied | -0.21 | 42% |
| 3x20 reference hands, scores added | -0.028 | 20.4% |
| 3x20 reference hands, EV formula | -0.027 | 20.0% |
There is a small benefit to using the EV formula rather than just adding points, but it is more complicated. First convert the points for each hand (call them “a”, “b”, and “c”) to percentages by subtracting one and dividing by 20. Combine the individual scores to a score for the total arrangement by plugging them into the formula:
a + b + c + ab + ac + bc - 2abc
(Subtracting two from this value and doubling the result gives a point estimate but is not necessary for comparing arrangements.) I wouldn’t recommend putting in the effort to use the more accurate formula unless you are already a lightning calculator. You can see that a+b+c is a good first approximation because the size of the multiplicative terms is likely to be small, and may cancel each other out.
Limits of the Scoring Strategy
There are three related sources of error in the scoring strategy: inaccuracy in scoring, card deletion effects, and hand correlation.
The scoring strategy could be improved further by memorizing more reference hands, at 2% or 2.5% intervals instead of 5% or 10%. This allows us to make finer distinctions between different arrangements. But, the payoff of the extra memorization is not very large; after all, we are already setting 80% of the hands correctly with just 60 reference hands.
For example, should you play 7h8d9dTdJc 3c5d6c7s8s 5hQsKs or 8d9dTdJcQs 3c5d6c7h8s 5h7sKs? They score identically--- one point less in back for the J-high straight but one point more in front for KQ. But K75 and KQ5 are much farther apart in value than the straights, so KQ5 is the correct play.
(An open question is whether you might be better off memorizing some other set of 20 hands and their values, rather than every 5th percentile--- does making fine distinctions in one part of the range of hands help more than in other spots?)
Some of the benefit of memorizing additional reference hands will be realized by just getting kickers correct. In example #3 above, 22255 T8763 QJ3 and 22233 T8765 QJ5 are given the same score, but T8763 in the middle is the better choice. This is an exception to the “bigger cards towards the front” rule I used. Some suggested rules of thumb for placing kickers are:
- When you don’t have a pair in the front, prioritize improving the middle, then front, then back, as in the 22255 T8763 QJ5 example above. Kickers are more important in mediocre middle hands than in poor front hands.
- If you play two pair in back, your best kicker should always go in front. Examples: 45577 469TJ QQK or 8899J 23457 QQA.
- If you only have a pair in back and a small pair in front; then it is usually correct to play your second-highest (or worse) kicker in front. Examples are 23QKK 23578 44J, 36JJA 23467 447, or 8JJQK 23567 449. With nines and higher, stick with the best kicker in back: 58TTQ 34578 99A.
- If you only have a pair in back and high-card in front, improve the high-card hand as much as possible. An example hand is 256TT 24568 9JK.
Unfortunately, even if you memorized scores for every possible hand, you would not always arrive at correct arrangements. This is because of card-deletion effects. The 13 cards you hold significantly impact the distribution of opponent cards and thus the strength of hands he or she will be able to make. No single table of scores can capture the true probability that a particular hand will win in front, middle, or back.
Consider (A) 2d5c5d5h5s 2s3d6c7dJc 7sQcKs versus (B) 2d5c5d5h5s 2s3d6cJcQc 7d7sKs. If we use the hand percentile tables for the full 10 million hands (i.e., if we had memorized scores for all of the possible ranks),
| Setting | Back | Middle | Front | Estimated EV using formula |
|---|---|---|---|---|
| (A) | 0.959 | 0.345 | 0.091 | -0.43 |
| (B) | 0.959 | 0.195 | 0.295 | -0.27 |
This analysis would suggest we choose (A). But comparing the two choices in detail shows that the two options are very close, with (B) a very slight favorite. This is because taking four fives out of the deck significantly changes the probability that our opponent will be able to play a strong hand in the middle. If we look at the hand percentile rankings after removing all hands from the 10 million that overlap with this hand, then a different picture emerges:
| Setting | Back | Middle | Front | Estimated EV using formula |
|---|---|---|---|---|
| (A) | 0.948 | 0.502 | 0.044 | -0.16 |
| (B) | 0.948 | 0.297 | 0.182 | -0.33 |
The new EVs are still not very accurate (the best values I have for the two arrangements are +0.019 and +0.034 respectively) but puts things in the correct order. Notice that the J7-high in the middle, which we previously thought would only win 34% of the time, is now expected to win 50% of the time. The pair we can make in back is revealed to be weaker as well.
I can only offer a hint as to how to adjust the strategy here: when you have many low cards, then (somewhat paradoxically) the value of a mediocre low holding such as a J or T improves substantially. You might be able to take this into account by fudging the middle table by a couple of points. However, my experiments about doing this in a systematic way by memorizing an alternate table for “low-card-rich” hands have not been successful.
Here is an example of how card deletion can affect hand ranking. Each column shows how the distribution of front hand strengths changes as we remove more and more low cards
| Percentile | Original Back | 2 dead low cards (2c2s) | 4 dead cards (4h4d) | 6 dead cards (7d7s) | 8 dead cards (5h6c) |
|---|---|---|---|---|---|
| 10 | 9933J | TT554 | TT883 | TT88Q | JJ22Q |
| 20 | QQJJ6 | KKJJ7 | 333xy | KKQQ6 | 333xy |
| 30 | QJT98 | KQJT9 | KQJT9 | KQJT9 | KQJT9 |
| 40 | Q9754 flush | QJ764f | QJT85f | KT643f | KJ853f |
| 50 | KQJ92 flush | AT843f | AJ975f | AQ853f | AQJ32f |
| 60 | AKJ63 flush | AKQJ9f | 333xx | 333xx | 333xx |
| 70 | 555xx | 666xx | 777xx | 888xx | 999xx |
| 80 | 999xx | TTTxx | TTTxx | JJJxx | JJJxx |
| 90 | QQQxx | QQQxx | KKKxx | KKKxx | KKKxx |
| Percentile | Original Middle | 2 dead low cards (2c2s) | 4 dead cards (4h4d) | 6 dead cards (7d7s) | 8 dead cards (5h6c) |
|---|---|---|---|---|---|
| 10 | A9876 | AJ976 | AQ765 | AQ986 | AQT83 |
| 20 | QT976 | QJ984 | K7654 | K9853 | KT763 |
| 30 | J9852 | JT875 | Q7643 | Q9532 | Q9854 |
| 40 | T9642 | J5432 | J8743 | J9643 | J9843 |
| 50 | 98753 | T8532 | T8762 | T9642 | T9752 |
| 60 | 97542 | 98632 | 98742 | 98754 | T6432 |
| 70 | 87632 | 96432 | 97532 | 97543 | 97652 |
| 80 | 86542 | 87532 | 87543 | 87632 | 87642 |
| 90 | 76542 | 85432 | 85432 | 86532 | 86532 |
Finally, front/middle/back hand strengths for a particular 13-card hand are not independent, but related in a complicated way. The rules of the game force the back hand to be no weaker than the front. But also, because it is possible to shift strength from one portion of the hand to another, a particularly strong component may be correlated with weakness elsewhere. For example, if we look at just hands from the 10M-hand sample which have 76432 or 75432 in the middle, then the back hands are considerably weaker than the original distribution:
| Percentile | Original Back | Back when 75432 or 76432 in the middle |
|---|---|---|
| 10 | 9933J | JJ632 |
| 20 | QQJJ6 | KKJ94 |
| 30 | QJT98 | TT664 |
| 40 | Q9754 flush | QQJJ3 |
| 50 | KQJ92 flush | J-high straight |
| 60 | AKJ63 flush | A-high straight |
| 70 | 555xx | KT754f |
| 80 | 999xx | AQT94f |
| 90 | QQQxx | TTTxx |
I believe that this anti-correlation explains why, even after taking card deletion into account, it is not possible to directly predict EV from hand distributions.
Conclusion
The strategy presented here is a good starting point for playing Chinese Poker with 2-7. A modest amount of memorization coupled with a systematic approach to arranging hands results in a strategy that is surprisingly strong even against perfect exploitive play. Losing $3 on average on every hand played for $100/point is not trivial, but the variance involved in CP and the magnitude of mistakes made by beginning players are much larger.
Expert play requires going beyond a simple strategy that assigns scores to hands in isolation. Some hands cannot be set correctly without taking into account card-deletion and anti-correlation effects. Computers can achieve this level of play, by coming at the problem via simulation instead of rule-based settings. In fact, the difficulty of describing an expert-level strategy suggests that even the best human players may have exploitable flaws in their game.
(Author's note, June 2018: Some CP2-7 games have added a qualifer in the middle, which changes the strategy significantly! Be sure you know the rules when sitting down to play. Originally published in Two Plus Two Magazine, October 2007.)