Hand Values in Chinese Poker with 2-7 in the Middle
A common variant of Chinese Poker is to play with the middle five-card hand scored using Deuce-to-Seven lowball ranks. This means that 23457 unsuited is the best hand, while straights and flushes are bad. This article presents some results about how hands should be arranged when playing under these rules. I will examine the game with 2-4 scoring and no royalties.
Here are five tricky hands. How would you set them?
- 2♣️3♦️4♣️5♣️6♣️6♥️7♦️7♥️7♠️8♣️T♥️J♠️K♣️
- 2♠️5♠️6♠️8♠️T♠️J♣️J♦️J♥️Q♠️K♣️K♥️A♣️A♠️
- 2♣️2♦️3♦️4♠️5♦️7♦️8♣️T♥️J♣️J♦️Q♥️K♣️K♦️
- 2♥️3♥️5♦️5♠️6♣️7♣️8♣️9♠️T♠️K♠️K♦️A♠️A♦️
- 3♦️4♣️4♦️5♣️6♥️7♣️8♥️9♣️9♠️T♥️Q♠️K♦️A♠️
Determining Hand Values
Just as in other forms of poker, how much a hand is worth depends on how your opponent plays his or her hands. (I have examined just heads-up play, although hand values in a multi-way pot should be similar.) In order to decide what arrangement is best, we have to have some idea of what the opposing hands will look like.
The way I approached this is to run an experiment using only a subset of hands. Player A was assigned 10 million hands chosen at random, and player B was assigned a different set of 10 million hands. Each player initially chooses a setting based on comparison with a ‘seed’ set of arrangements. Then, player A goes through his choices hand-by-hand and picks the best arrangement against the strategy chosen by player B. Player B then does the same thing using player A’s new arrangements. This iteration provides a strong set of settings for both A and B.
Of the 10M hands in player B’s set, only about 1.3% do not contain a card in common with any particular hand from A’s set, so the sample size for a single hand is about 130,000. The standard error (95% confidence interval) for evaluating a hand against this size sample is about +/- 0.012 points.
The set of arrangements does not converge to a pure strategy. There are hands that “cycle” back and forth between two different settings depending on the opponent’s strategy choices. That is, the pure strategies found by the iterative process described above are exploitable. In the sample of 10M hands, a bit more than 2% of hands cycle back and forth; however, the net value of this exploitive play is under 1/1000th of a point.
This “cyclic” behavior usually involves the decision about whether to put a pair in front at the expense of the middle and/or back. Here are some example hands where the best arrangement cycles between two alternatives, which I have named “P” and “NP”. Hands are shown in the order back/middle/front.
| More pairs in front (strategy “P”) | Fewer pairs in front (strategy “NP”) |
|---|---|
| 3cTdTsJcQh / 3s4d5d7c8c / 6d6hAs | 3c6d6hTdTs / 3s4d5d7c8c / JcQhAs |
| 2c2d5c5h5s / 3d4c7s8c9c / TdThJh | 5c5h5sTdTh / 2c3d4c7s8c / 2d9cJh |
| 2h5h6c6h9s / 2s4c5s7s8s / JhQcAd | 2s5s7s8s9s / 2h4c5h6cJh / 6hQcAd |
| 8c8d8h8sTd / 2c3d4h6s9c / 5c5sAc | 2c5c8c9cAc / 3d4h5s6sTd / 8d8h8s |
| 2d3s4d5d6d / 2h4s6h7h8h / TcJdJh | 2d4d5d6dJd / 2h3s4s6h7h / 8hTcJh |
Distribution of Hand Values
CP2-7 hands in the sample range in average value from -2.95 points to 3.89 points. Although the mean value is very close to zero (it is 0.00068 points due to exploitive play and sample error), this asymmetry means that the median hand is slightly negative (about -0.0655 points.) This means that although perfect play of CP will result in a long-term expectation of zero, you will still lose money on more hands than you win money!
The lower bound of -2.95 points suggests that it is always possible to avoid a scoop at least 50% of the time. Here are some sample hands (from strategy “NP”)
| Sample of worst hands | Average value |
|---|---|
| 2c3d4cJcJh 2d3h4d9dKh 9sThTs | -2.95 |
| 2c7c7s8c8d 2h3dJhQhAd 3hTdTs | -2.93 |
| 2c4d6d7cAc 2s4h6h7dTc ThJhQs | -2.93 |
| 4c5c5d9d9h 2c2h4dQsKd TcThAh | -2.90 |
| Sample of best hands | Average value |
|---|---|
| 6hJcJdJhJs 2c3c4c5d7d AcAdAh | +3.89 |
| 5s6s7s8s9s 2d3c4c5d7c JdJhJs | +3.88 |
| 7c7d7h7sKh 2d3d4h5s8h JcJdJh | +3.82 |
| 9cQcQdQhQs 2h3h4s6c7c TcTdTs | +3.71 |
| Sample of median and break-even hands | Average value |
|---|---|
| 4c6c7cKcAc 2h4s6s7s8s QsKhAh | -0.066 |
| 4d5d7d8dTd 5s7s8hTcQh 3c3d3s | -0.066 |
| 6d6h6s9h9s 3s4d5sTdJh QcQhAh | -0.066 |
| 3dTcTsJcJh 3s4c6d7h9c QcAcAh | 0 |
| 5d7d8d9dAd 2c2s5h7c8h QdQhQs | 0 |
| 8c8h8sQdAc 2c7c9cTsJh 6d6h6s | 0 |
Here is a graph showing 1 million hands ranked by value. Most hands are between -2 and +2 in value.
Front, Middle, and Back Percentiles
These tables show what hands lie at the 10th, 20th, etc. percentile for the front, middle, and back in the strategies arrived at through the iterative process described above. (Percentile is what percentage of the player’s fronts, middles, or backs are strictly less than the given hand--- this is not exactly the same as the percentage of the time that the specified hand will win the component, because holding 13 cards in your hand affects your opponent’s distribution. But it’s a decent first approximation.)
| Front Hand | P strategy percentile | NP strategy percentile |
|---|---|---|
| 432 | 0% | 0% |
| A32 | 10.3% | 10.6% |
| AKT | 20.2% | 20.5% |
| 223 | 25.1% | 25.3% |
| 884 | 30.1% | 29.8% |
| JJ4 | 40.0% | 39.2% |
| QQK | 50.0% | 50.9% |
| KKQ | 60.8% | 62.9% |
| AA6 | 70.3% | 69.9% |
| AAK | 81.3% | 80.9% |
| 444 | 90.1% | 89.9% |
| AAA | 99.7% | 99.7% |
| Unpaired hands | 25.1% of sample | 25.3% of sample |
| Paired hands | 62.2% of sample | 61.8% of sample |
| Trips | 12.7% of sample | 12.9% of sample |
Hands without big pairs in front are unlikely to scoop. The median front hand is QQK, so small pairs are sub-pair holdings.
The “nut low” in front, 432, appears in a variety of hands, usually paired with a strong middle and mediocre back, such as 7c8d9sTcJc 2d3c4h6d8s 2h3d4s (-1.29 points) or TcJcQcKsAh 2d3d4d5h9s 2h3s4s (-1.53 points.)
| Middle Hand | P strategy percentile | NP strategy percentile |
|---|---|---|
| Straight flush, Q-high | 0% | 0% |
| Flush, 75432 | 0.018% | 0.018% |
| A9865 | 10.0% | 10.0% |
| QT974 | 20.0% | 20.0% |
| J9842 | 30.0% | 30.1% |
| T9543 | 40.0% | 40.3% |
| 98743 | 50.3% | 50.7% |
| 97532 | 60.0% | 60.4% |
| 87543 | 70.3% | 70.6% |
| 86432 | 81.1% | 81.3% |
| 76432 | 91.6% | 91.7% |
| 75432 | 95.1% | 95.1% |
Nearly 5% of arrangements manage to place a wheel in the middle. The drop-off is rather steep, with the median hand a 98 low. However, it is still worth trying to avoid placing a pair--- many strong high hands will be forced to put ace-high or a pair in the middle, so having a J low may be sufficient to avoid a scoop.
| Back Hand | P strategy percentile | NP strategy percentile |
|---|---|---|
| T5432-high | 0% | 0% |
| AAK72 | 8.5% | 7.8% |
| 8855Q | 10.0% | 9.5% |
| QQJJ5 | 20.0% | 19.9% |
| K-high straight | 29.7% | 30.0% |
| 75432 flush | 34.8% | 35.1% |
| QT542 flush | 40.0% | 40.4% |
| A6543 flush | 50.0% | 50.4% |
| AKJ93 flush | 60.0% | 60.4% |
| Deuces full | 61.5% | 61.9% |
| Sevens full | 71.7% | 72.0% |
| Tens full | 79.8% | 79.9% |
| Aces full | 92.7% | 92.7% |
| Quad Threes | 95.9% | 95.9% |
| Straight flush 5-high | 98.7% | 98.7% |
| Royal flush | 99.9% | 99.9% |
Straights and two pair are not good holdings in CP2-7. The median holding here is an ace-high flush. Often it makes sense to abandon the back with low two pair in order to make a quality middle and big pair in front.
One interesting feature not displayed in the table is that it is never correct to play an ace kicker with two pair. The ace always has more benefit as a kicker in front. (And if you have trips in front you can’t be playing two pair in back.)
The “nut low” in back, T5432, appears in only one hand: 2345T 23457 789 unsuited. This hand is not among the worst, however, since it only scores about -2.01 points on average--- it can do no worse than losing two components and tying the third.
Breaking the Wheel
It is usually correct to break 75432 if you can still make a quality low (another 7, and 8, and maybe even a 9 or T) if doing so will let you significantly improve the front or back. Here are some example hands where a wheel was possible, broken into cases where the wheel was kept and where it was broken:
| Played 75432 in the middle | Broke the wheel |
|---|---|
| 3d5d6d7dTd 2s3h4s5s7c 8h8sQh (+0.08) | 3s4s5s6sTs 2h3h6d7h8h TcJdJs (-0.47) |
| 5hJhQsAcAd 2d3c4s5s7d 6d6hKh (-0.61) | 2c2h2s9dTs 3h4c5c7d8d JsAdAs (+1.08) |
| 6h6s9hQcQs 2d3h4c5h7d JcAdAs (+1.70) | 5c6c7cTcAc 2d3c4h5s8d 9c9sKs (+0.51) |
| ThJsQhKsAc 2d3c4s5c7c 3d3h5h (+0.12) | 2s3s4s5s6s 2c7h8s9dTc 2hQdAh (-0.02) |
| 9dThJdQsKc 2c3c4d6d7d 5d5s8c (-0.04) | |
| 2h3h4d5c6c 2s3s6d7cTs QcQhKs (-1.05) |
In a 10M-hand sample, the player held 23457 about 12.7% of the time (including a few suited 23457’s) but only made a wheel in 4.9% of hands, less than half.
AAx in front
It is a little surprising that the kicker matters when you have aces in front; my intuition was initially that the kicker will not usually play. But, the strategies developed here play AAK far more often than any other kicker. In the NP strategy, AAK is an 80.9% hand; AAQ is 77.3rd-percentile, while AA2 is 67.6th-percentile. Aces in front constitute 19.4% of the front hands; AAK is 6.2% of aces.
When you hold two aces your opponent will have the other two aces about 10.5% of the time, so aces vs. aces in front is not uncommon. When you hold exactly two aces, they should go in front more than 85% of the time! In contrast, when you hold exactly two kings they end up in front in just 41% of arrangements.
The same is true of any pair; it is usually best to put your best kicker in front, just as you would in Chinese Poker played for high. It may even be worth making a slightly worse flush in back in order to improve the kicker in front. For example, 3h5hJhQhKh 2c3s4c5s7c 6hAcAs and 3h5h6hQhKh 2c3s4c5s7c JhAcAs are extremely close in value, but the latter is better by about 0.02 points.
Answers to Hand Problems
The best arrangement is 4c5c6c8cKc 2c3d6hThJs 7d7h7s. This is worth about 0.56 points. A close second is 6c6h7d7h7s 2c3d4c5c8c ThJsKc, which wins 0.50 points. The best alternative that plays the wheel in the middle is 7d7h8cThJs 2c3d4c5c7s 6c6hKc, which is much worse, losing 1.21 points on average.
It is necessary to abandon the middle here and play a flush: JcJdJhKcKh 2s5s6s8sTs QsAcAs. This wins 0.92 points. 6s8sTsQsAs 2s5sKcKhAc JcJdJh is still a winning hand, but only worth 0.39 points.
Playing the wheel in the middle with 2c8cThKcKd 2d3d4s5d7d JcJdQh loses 0.33 points on average. It is slightly better to change the wheel to 87542 so that we can put kings in front 2c2dThJcJd 3d4s5d7d8c QhKcKd loses only 0.26 points.
This hand is a very close one. Playing the flush is appealing: 5s9sTsKsAs 2h3h5d6c7c 8cKdAd wins 0.74 points on average. However, it may be slightly better to play a pair in front with 5d8c9sAdAs 2h3h5s6c7c TsKdKs; this earns 0.76 points against the P strategy but only 0.72 points against NP.
You are just not going to be able to do very well with this hand. Playing the best middle with 4c7c9c9sTh 3d4d5c6h8h QsKdAs is your best bet, losing 1.71 points on average. Playing two pair or splitting pairs is worse: 4c4d9c9sQs 3d5c6h7c8h ThKdAs is at -1.85 and 9c9sThQsKd 3d5c6h7c8h 4c4dAs is worth -1.98 points.
(This article was previously published in the September 2007 issue of Two Plus Two Magazine.)
This also implies, I think, that you're never playing a pair on top that has a third card used in the middle, something that happens occasionally in the middle-qualifier form. But I guess I can see that it might always be better to throw the middle in favor of a full house in this form.
Is the code still in good enough shape that you could run similar simulations for 14-card, ten qualifier?
Whoa, that was eleven years ago? Time, it flies.