Results of a CP2-7 experiment

[MarkGritter]

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Everything I'm saying has to do with selecting a strategy facing a closed, totally random hand.

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Yes. The way you do that is by putting your opponent on a range of hands and figuring out what he'd do with them.

In my experiments, there are only 1,000 (or 10,000, or 100,000) hands he could possibly have instead of 52C13 = 365 million. You still don't know which one of those you're actually facing in any given situation. So what I do is pick the one that is best on average, given a complete listing of how my opponent would play those hands.

At some large-enough sample size you should get close enough to the same answer from a randomly chosen sample as from carrying out the same exercise with the entire set of hands.

The differences between real life and the experiment are:

1) In real life your opponent's hand can be one of 52 choose 13 = 365 million. In the experiment he's limited to some "small" random subset of that.

2) In real life you probably don't know your opponent's strategy exactly. (But if, say, somebody published a strategy for playing CP2-7, and your opponent followed it, you _would_ know.)

3) In real life you could not remember a specific play for millions of individual hands, you would have to have a more high-level strategy.

It is not a matter of playing "face up" (though the 2-hand example I gave earlier was--- the hand range was limited to just one hand.) The non-convergent behavior I describe doesn't arise because player A can see player B's cards and adjust, it's because player A knows player B's _strategy_ and adjusts.

[donger]

So any idea how we can establish a 0-100 win % range for the middle hand so my strategy can be tested among the others? I'm curious how it would be countered since it doesn't favor any one hand (back, middle, or front).

[Phat Mack]

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I've been trying to figure out how to determine the median 2-7 hand for this game. Any idea how to do this?

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If you pick up 13 cards, and try to set your best 2-7 hand from them, here are the percentiles:

10---T9532
20---98752
30---97653
40---87653
50---87543
60---86543
70---85432
80---76532
90---75432

Of course, when playing, certain cards may be needed to set other hands, which may in turn skew these results.

[MarkGritter]

I posted a list of what middle (2-7) hands were chosen in 20K hands here:

http://wiki.lowballgurus.com/article..._Probabilities

[Phat Mack]

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Here's an example from the current run: B's hand is 2s3c3d5h6h6s7c7s8d9cJcJhAs. The hands and arrangements in A's latest strategy that don't contain one of these cards are:

9h9sKcKhKs 2d3h4h6d7h ThQdAh
9d9hAcAdAh 2d4s5d6c9s TsJsKh
5c5dTcThTs 2h3s4c6c8h KdKsAc
8h9dTdJsQd 2d4d5d8sTs 4hAdAh
TdThQcQhQs 2h3h4c5c9d 3sKhKs
4d6dQdKdAd 2c4h5s6c7h 9sTsJs
5c5d5s9d9s 2c3h4c6d8h ThKhAc
QcQdKcKdKs 2d4d5dJsAc 9d9h9s
2c4c6cTcKc 2d3h4s5d8c 3sThKd
6cJdJsKcKh 2h4d5d6d9s QdAcAh
TcThQdQhQs 2c3s5d6c9d 6d9hKd
TcJdQcKhAc 2d3h4d7h8h 9dThTs
4d6dJdQdAd 2h5c6c8s9h 4h4sTh
TcThTsKcKd 3s5d8hJdQh 4d4h4s
4dTcTdThTs 2c2d4h8h9s QdQhAc
4c4d4sTdTh 2d3s8c9dQh 9sAcAh
4dJdQhQsKh 2d3s4s5c9h 2h8c8s
8hTsJdKcKd 2c3h5c6d7d QcQhAc
7dJdQdKdAd 3h5c8h9hTs 5sKcKh
etc. (113 possible hands out of 10000) All are treated as equally likely.

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As we proceed through iterations of A1,A2,A3... and B1,B2,B3..., does B keep playing this same hand against the same 113 hands being played by A?

[Phat Mack]

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What I'm aiming at here is a somewhat smaller goal, to see if CP2-7 actually has a "fixed" best arrangement for every hand, or if the game-theoretic solution must involve randomization (or at least nonlocal thinking.)

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I don't see a best fixed arrangement happening with two players, unless there was an anti-punting provision---it's too much like roshambo otherwise. But how about with four players and A1, B1, C1, and D1, etc.?

[MarkGritter]

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Here's an example from the current run: B's hand is 2s3c3d5h6h6s7c7s8d9cJcJhAs. The hands and arrangements in A's latest strategy that don't contain one of these cards are:

9h9sKcKhKs 2d3h4h6d7h ThQdAh
9d9hAcAdAh 2d4s5d6c9s TsJsKh
5c5dTcThTs 2h3s4c6c8h KdKsAc
8h9dTdJsQd 2d4d5d8sTs 4hAdAh
TdThQcQhQs 2h3h4c5c9d 3sKhKs
4d6dQdKdAd 2c4h5s6c7h 9sTsJs
5c5d5s9d9s 2c3h4c6d8h ThKhAc
QcQdKcKdKs 2d4d5dJsAc 9d9h9s
2c4c6cTcKc 2d3h4s5d8c 3sThKd
6cJdJsKcKh 2h4d5d6d9s QdAcAh
TcThQdQhQs 2c3s5d6c9d 6d9hKd
TcJdQcKhAc 2d3h4d7h8h 9dThTs
4d6dJdQdAd 2h5c6c8s9h 4h4sTh
TcThTsKcKd 3s5d8hJdQh 4d4h4s
4dTcTdThTs 2c2d4h8h9s QdQhAc
4c4d4sTdTh 2d3s8c9dQh 9sAcAh
4dJdQhQsKh 2d3s4s5c9h 2h8c8s
8hTsJdKcKd 2c3h5c6d7d QcQhAc
7dJdQdKdAd 3h5c8h9hTs 5sKcKh
etc. (113 possible hands out of 10000) All are treated as equally likely.

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As we proceed through iterations of A1,A2,A3... and B1,B2,B3..., does B keep playing this same hand against the same 113 hands being played by A?

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Yes, the set of 13-card hands being played does not change. Whnever B considers how to play 2s3c3d5h6h6s7c7s8d9cJcJhAs, he is up against the same 113 opposing hands. Only how they have been set up may have changed.

(I've actually thought about changing the experiment a bit to add a new new hands on every iteration but I concluded it wouldn't really add anything, compared with just doing a larger run.)

[MarkGritter]

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I don't see a best fixed arrangement happening with two players, unless there was an anti-punting provision---it's too much like roshambo otherwise. But how about with four players and A1, B1, C1, and D1, etc.?

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What, you've never played four-handed roshambo? [img]/images/graemlins/smile.gif[/img]

I don't know. I think if randomization is necessary for 2 players it is probably necessary for 4 players as well.

Running the same sort of experiment for 4 players would be very difficult because there would be so few valid hands if each player was assigned a moderately-sized random list. What would be possible is describing some algorithmic strategy for three players and then seeing if the 4th player can do better.

[Sweet]

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I posted a list of what middle (2-7) hands were chosen in 20K hands here:

http://wiki.lowballgurus.com/article..._Probabilities

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Mark, absolutely first rate stuff.
Could you post the same data for the top and bottom hands?

[MarkGritter]

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I posted a list of what middle (2-7) hands were chosen in 20K hands here:

http://wiki.lowballgurus.com/article..._Probabilities

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Mark, absolutely first rate stuff.
Could you post the same data for the top and bottom hands?

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I will do so after I get further with my 100K-hand experiment, which will produce a better data set.

(Early results suggest that even if there is cyclic behavior, it doesn't actually matter since the value is < 1/100th of a point. This was not the case at 1K or 10K hands. I'll have a better idea what's going on in 24 hours or so.)