Distribution of CP2-7 Hand Values

[MarkGritter]

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?

[Phat Mack]

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...

[MarkGritter]

[ 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.

[2461Badugi]

[ 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.

[MarkGritter]

[ 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!

[MarkGritter]

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.

[Phat Mack]

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.

[MarkGritter]

[ 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.)

[karpov]

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

[MarkGritter]

[ QUOTE ]
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

[/ QUOTE ]

I should have said "to two pair in back", sorry. I confirmed that this is still the case in the latest 1M-hand iteration (which has some bugs fixed.)

If you have two pair in back, the difference between a lower kicker and an ace kicker is less than 1%. Putting the ace in front instead always increases the strength of the front hand by more than this.

For example, 51.1% of fronts are QQK or worse, while 53.9% are QQA or worse. Now look at this example with two pair in back:

two pair Threes and Deuces Four kicker , 1 , 0.084274
two pair Threes and Deuces Five kicker , 4 , 0.084278
two pair Threes and Deuces Six kicker , 3 , 0.084281
two pair Threes and Deuces Seven kicker , 3 , 0.084284
two pair Threes and Deuces Nine kicker , 25 , 0.084309
two pair Threes and Deuces Ten kicker , 79 , 0.084388
two pair Threes and Deuces Jack kicker , 172 , 0.084560
two pair Threes and Deuces Queen kicker , 132 , 0.084692
two pair Threes and Deuces King kicker , 9 , 0.084701

Here the kicker makes < 0.1% difference.

two pair Kings and Jacks Deuce kicker , 421 , 0.217500
two pair Kings and Jacks Three kicker , 363 , 0.217863
two pair Kings and Jacks Four kicker , 366 , 0.218229
two pair Kings and Jacks Five kicker , 284 , 0.218513
two pair Kings and Jacks Six kicker , 317 , 0.218830
two pair Kings and Jacks Seven kicker , 254 , 0.219084
two pair Kings and Jacks Eight kicker , 304 , 0.219388
two pair Kings and Jacks Nine kicker , 278 , 0.219666
two pair Kings and Jacks Ten kicker , 221 , 0.219887

The difference in kickers is still less than 1% in hand strength. Note that there were no examples found where a Q kicker was played, either!

And of course it doesn't make any sense to play xxyyA ..... zzA, since you should pair the aces.

If you don't have a pair in front, then ace-high is again much better than any other high-card hand, so it outweights playing the aces as a kicker.

If you have trips in front, then you can't have two pair in back so the situation does not occur.

However, there are some hands where you must play two pair, ace kicker, in the middle.