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#1
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Distribution of CP2-7 Hand Values
I put together a graph showing the hand values for 1M CP2-7 hands:
http://markgritter.livejournal.com/354410.html What I find interesting is that no hand in the sample has a value less than -3 points, while there are a few hands with values between +3 and +4. (quads/wheel/trips works well.) As a result, although the median value of the set is close to zero (0.007), as it should be, the mean value is slightly negative (-0.07). Can anybody explain this result? (I.e., prove that no hand scores less than -3 on average?) Does the same thing occur in normal CP? |
#2
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Re: Distribution of CP2-7 Hand Values
Beautiful!
Can anybody explain this result? I'll try to do some thinking on this. From what I understand, you generated a million hands and graphed the best setting for each hand. If you graphed the worst setting, would you find that they went from -4 to +3? How about if you graphed a random setting? Also, the nature of CP2-7, with cards bad for the ends being good for the middle migh have something to do with it. I wonder what a graph of straight CP would look like... |
#3
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Re: Distribution of CP2-7 Hand Values
[ QUOTE ]
I'll try to do some thinking on this. From what I understand, you generated a million hands and graphed the best setting for each hand. If you graphed the worst setting, would you find that they went from -4 to +3? How about if you graphed a random setting? [/ QUOTE ] Hmm... worst settings are generally pretty bad. For example, the best in the sample, TTTTQ 23457 KKK, has QT543 TTTKK Q72 as a legal setting, and this gets scooped by the worst hand in the set: A7652 T7652 TJQ. Random settings are pretty bad too--- the initial runs of my experiment, against a random setting, have scored in excess of +3 on average. I think it would might interesting to plot the second-best setting for every hand on the same graph--- that would be feasible (though I'd probably want to work with a smaller subset, as recalculating all 1M hands takes a while.) [ QUOTE ] Also, the nature of CP2-7, with cards bad for the ends being good for the middle might have something to do with it. I wonder what a graph of straight CP would look like... [/ QUOTE ] I've been thinking of the high hand 223345789TQKA, which seems to be about the worst I can come up with. The settings 22334 A9875 KQT or 33457 2289T AKQ both have some strength in one of the hands and could avoid enough scoops to make it to -3. I have been thinking of putting in knobs in my code to handle high-only and 1-6 scoring as well. |
#4
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Re: Distribution of CP2-7 Hand Values
[ QUOTE ]
I've been thinking of the high hand 223345789TQKA, which seems to be about the worst I can come up with. The settings 22334 A9875 KQT or 33457 2289T AKQ both have some strength in one of the hands and could avoid enough scoops to make it to -3. [/ QUOTE ] You've got a wheel in there, you know. |
#5
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Re: Distribution of CP2-7 Hand Values
[ QUOTE ]
[ QUOTE ] I've been thinking of the high hand 223345789TQKA, which seems to be about the worst I can come up with. The settings 22334 A9875 KQT or 33457 2289T AKQ both have some strength in one of the hands and could avoid enough scoops to make it to -3. [/ QUOTE ] You've got a wheel in there, you know. [/ QUOTE ] Crap! Too much 2-7 rots the brain! |
#6
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Re: Distribution of CP2-7 Hand Values
Interesting tidbit demonstrated: It is always incorrect to play an ace as the kicker to two pair in front. Do you see why?
Mark, can you explain this better? Thanks |
#7
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Re: Distribution of CP2-7 Hand Values
Also posted: preliminary results about hand strengths in front, middle, and back!
Unfortunately, I have now found three bugs in my code that have affected this data set, probably in small ways. (Straight flushes were not handled correctly. Pairs in back may have been treated like trips in some cases--- not predictably, possibly not at all in the context of this data set. Two pair kickers greater than either pair were given an incorrect strength.) Use with caution. http://www.lowballgurus.com/1M-backs.txt (219KB) http://www.lowballgurus.com/1M-middles.txt (284KB) http://www.lowballgurus.com/1M-fronts.txt (27KB) Each file is ordered with the weakest hands first. The first column describes the hand, the second column specifies how many occurrences of that hand in the 1,000,000 settings examined, and the third column specifies the percentage of the 1M hands less than or equal that hand in strength. (Note that the percentage is not exactly how often the hand "should win" because your opponent's distribution is altered based on the cards you hold--- it would consist only of a nonrandom subset of the possible hands.) Median hands from this sample: Back: KQJ42-flush Middle: 98754 low Front: QQK Interesting tidbit demonstrated: It is always incorrect to play an ace as the kicker to two pair in front. Do you see why? Back hands: Straight flushes: 0.2% Quads: 3% Full houses: 34% Flushes: 27% Straights: 11% Trips: 1.6% Two pair: 14% One pair: 7% High card: 0.2% Middle hands: 75432: 5% 7-low: 14% 8-low: 20% 9-low: 17% T-low: 13% etc. |
#8
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Re: Distribution of CP2-7 Hand Values
Wow, I was stunned by some of these results. I was expecting hand values in the front and back to be higher because of the low in the middle. Instead, in the back you have a KQ flush at 50 %ile and Smolen has a KJ flush. 70 %ile: Gritter(CP2-7) == 5s full, Smolen(CP) == 4s full. 90 %ile: Gritter (CP2-7) == Qs full, Smolen (CP) == Qs full.
The fronts are a different story. Whereas Smollen has 552 at 50%ile, you have QQK (552 is 26 %ile). 70 %ile: AA6 vs TT9. 90 %ile: 333 vs. KKQ. |
#9
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Re: Distribution of CP2-7 Hand Values
[ QUOTE ]
Wow, I was stunned by some of these results. I was expecting hand values in the front and back to be higher because of the low in the middle. Instead, in the back you have a KQ flush at 50 %ile and Smolen has a KJ flush. 70 %ile: Gritter(CP2-7) == 5s full, Smolen(CP) == 4s full. 90 %ile: Gritter (CP2-7) == Qs full, Smolen (CP) == Qs full. The fronts are a different story. Whereas Smollen has 552 at 50%ile, you have QQK (552 is 26 %ile). 70 %ile: AA6 vs TT9. 90 %ile: 333 vs. KKQ. [/ QUOTE ] This makes sense because the front is much less constrained. It might be more apropos to compare the front in CP2-7 with the middle in CP high. The only "constraint" on the back, however, is maximizing total value of the 13-card hand. Always making the best possible hand in back is wrong, but not too badly wrong--- correct to a first approximation. So it's not too surprising that the front hands in 2-7 are only a tiny bit stronger. (I would be interested to see things on the low end: is the 10th percentile significantly stronger or weaker in CP2-7 than in CP high? It may make a lot more sense in 2-7 to weaken the front in order bolster the middle.) |
#10
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Re: Distribution of CP2-7 Hand Values
[ QUOTE ]
(I would be interested to see things on the low end: is the 10th percentile significantly stronger or weaker in CP2-7 than in CP high? It may make a lot more sense in 2-7 to weaken the front in order bolster the middle.) [/ QUOTE ] CP high - front 10% K32 20 KQJ 30 AQ6 40 AKT 50 552 60 88T 70 TT9 80 QQ8 90 KKQ CP 2-7 - front 10 KQT 20 AK8 30 88J 40 JJ8 50 QQK 60 KKJ 70 AA6 80 AAQ 90 333 |
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