Chinese Poker

From LowballWiki

Chinese Poker is played with two, three, or four players. Each player receives thirteen cards and arranges them into a "front" hand of three cards and "middle" and "back" hands of five cards. In normal Chinese poker, the hands must be arranged back to front in order of increasing hand strength. When all cards have been arranged, the hands are revealed and compared pairwise: front vs. front, middle vs. middle, and back vs. back. A player typically wins one point for each hand that beats his opponent's corresponding hands, plus bonuses for winning the majority, all three confrontations, or for making special hands.

1 Lowball Variants
2 Does a Pure Strategy Exist?
3 Example 2-7 Hands
4 2-7 Strategy/Theory Links

[edit] Lowball Variants

Here, of course, we are interested variants that include a lowball hand. A popular variation is to play with deuce-to-seven rankings in the middle hand, while the front and middle are played for high.

Also floating around in high-stakes games is a 17-card version of Chinese Poker that was developed by 2004 WSOP Main Event champion Greg Raymer. Played two or three-handed, players set four hands instead of the typical three. The back, middle, and front hands remain the same, but the fourth one is a four-card badugi hand. Ba-what? It's a four-card lowball game where the objective is to make your best possible low hand containing all four suits. Aces are low in this variant, and the best possible hand is an A-2-3-4 rainbow (all different suits).[1]

A somewhat sillier variant from Phat Mack on 2+2:

I thought of a 4-player, 13-card version: 5-card high in back; 4-card badugi in the middle; 4-card cribbage hand in front. Game is played with the bug, so a 53 card deck. After the hands are set, 53rd card is turned over and plays as part of the cribbage hands.[2]

[edit] Does a Pure Strategy Exist?

Is there an optimal strategy for Chinese Poker which specifies just one "best" setting for any 13 cards? Or is it necessary to mix up your play by switching between different settings for the same cards?

Originally we believed that a pure strategy exists for normal Chinese Poker, but not for CP with 2-7 in the middle. Timothy Cooper has argued that this is because the ordering constraint has been removed, and that normal CP without an ordering constraint would also have no pure strategy. (Suppose one player plays the normal CP orderings while the other is free to choose whether the middle or back hand is higher-rank. The second player would have a positive expectation from being able to take advantage of his opponent's fixed ordering--- it is obvious he would do at least as well, and it is easy to find at least one example where his greater freedom allows him to switch from losing all three set to losing just two.)

Here is an example demonstrating this idea in CP2-7:

A holds 2222344455578
B holds AAA334567789T

No flushes are possible. If A knows how B orders his cards, he can win at least 2 out of 3. Similarly, if B knows how A orders his cards, then B can win 2 out of 3. Because there is no dominant strategy for either A or B, a mixed strategy is necessary to play this particular example.

Some of A's choices:

A1: 55544 23478 222 (beaten by B1)
A2: 22244 23457 558 (beaten by B2)
A3: 22224 34578 554 (beaten by B3)
A4: 22224 34478 555 (beaten by B4)

Some of B's choices:

B1: AAA77 34568 T93 (beaten by A2 or A4)
B2: AAA33 45689 77T (beaten by A3 or A4)
B3: AAA93 34568 77T (beaten by A1 or A4)
B4: 6789T 33457 AAA (beaten by A1, A2, or A3)

However, recent work by Mark Gritter has demonstrated that if the "best" pure strategy is exploitable, it is almost certainly by less than 1/100th of a point. So, as a practical matter, CP2-7 can be played without recourse to game-theoretic concepts. (Jerrod Ankenman disagrees.) Some preliminary work suggests that exploitive play is a greater factor in CP high than in CP2-7.

Bonus factoid: The CP2-7 solver says:

1.3307  2c2d2h2s4d 3c4h4s7s8c 5c5d5h 
1.3307  2c2d2h2s5c 3c5d5h7s8c 4d4h4s 
1.82542 2c2d2h2s4d 3c4h5c7s8c 4s5d5h
1.82542 2c2d2h2s5c 3c4d5d7s8c 4h4s5h
2.21033 4d4h5c5d5h 2c3c4s7s8c 2d2h2s
2.22693 2c2d2h4d4h 2s3c4s7s8c 5c5d5h
2.22693 2c2d2h5c5d 2s3c5h7s8c 4d4h4s

So A2 and A1 are very close in value, as are B1 and B2. B3 and B4 are losing settings, however.

-0.879094 9sTcAcAdAh 3c4d5h6s8h 3s7c7d
-0.873139 3cTcAcAdAh 3s4d5h6s8h 7c7d9s
-0.871964 3c9sAcAdAh 3s4d5h6s8h 7c7dTc
-0.73758  6s7c8h9sTc 3c3s4d5h7d AcAdAh 
-0.149898 3c3s7c7dAc 4d5h6s8h9s TcAdAh
0.27002   3c4d5h6s7c 3s7d8h9sTc AcAdAh 
0.6622    7c7dAcAdAh 4d5h6s8h9s 3c3sTc  
0.802774  3c3sAcAdAh 4d5h6s8h9s 7c7dTc
0.810453  7c7dAcAdAh 3c4d5h6s8h 3s9sTc