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Simlab: Tricks for Aces, Kings, Queens, and Jacks with Constant Total Points
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Some comments on my previous post (reporting some old simulation results) inspired me to do some new runs.  In these runs I held total HCP (or schmoints, the 6-4-2-1 scale) constant and tabulated the number of tricks obtained according to the number of aces, kings, queens, and jacks.  This post gives the raw numbers and some first-glance observations.  Maybe Yuan (or some other real mathematician) can use the data in some elegant way.

In the first case we have 25 total HCP; one hand has 15-17, and both hands are balanced with no 5-card suit.  This "should" be the situation where aces and kings are not so much undervalued in the 4-3-2-1 scale.  The sim reported:

Av tricks for 1 ace = 8.12

Av tricks for 2 ace = 8.39

Av tricks for 3 ace = 8.59

Av tricks for 4 ace = 8.76

Av tricks for 0 king = 8.96

Av tricks for 1 king = 8.70

Av tricks for 2 king = 8.53

Av tricks for 3 king = 8.49

Av tricks for 4 king = 8.48

Av tricks for 0 queen = 8.81

Av tricks for 1 queen = 8.66

Av tricks for 2 queen = 8.61

Av tricks for 3 queen = 8.42

Av tricks for 4 queen = 8.36

Av tricks for 0 jack = 8.53

Av tricks for 1 jack = 8.51

Av tricks for 2 jack = 8.53

Av tricks for 3 jack = 8.55

Av tricks for 4 jack = 8.60

(It would be nice to have a way to put these in a table...)  Some situations are very constrained.   We will always have at least one ace, to get to 25 HCP. When there are no kings there will always be four aces, for the same reason.  The high trick number for zero kings may reflect the advantage we get from all those aces, but it's puzzling that the tricks for no-kings are higher on average than the other situations with all the aces.   This happened also in the three other runs, and I do not claim to understand it.

In this situation it seems that aces are somewhat undervalued (even with everything balanced), but jacks are not as bad as we have heard.  Queens are the overrated cards here.

 

The second case keeps the 25-HCP total but requires that there be a five-card suit somewhere in the two hands.  We expect more tricks on average, and we might expect that aces and kings become more valuable relative to queens and jacks.  And that's pretty much what we get, at least for aces:

Av tricks for 1 ace = 8.19

Av tricks for 2 ace = 8.469

Av tricks for 3 ace = 8.80

Av tricks for 4 ace = 9.13

Av tricks for 0 king = 9.38

Av tricks for 1 king = 8.88

Av tricks for 2 king = 8.77

Av tricks for 3 king = 8.62

Av tricks for 4 king = 8.63

Av tricks for 0 queen = 9.34

Av tricks for 1 queen = 8.90

Av tricks for 2 queen = 8.80

Av tricks for 3 queen = 8.55

Av tricks for 4 queen = 8.47

Av tricks for 0 jack = 8.85

Av tricks for 1 jack = 8.81

Av tricks for 2 jack = 8.67

Av tricks for 3 jack = 8.69

Av tricks for 4 jack = 8.58

 

The last two runs held schmoints constant at a total of 33, which had seemed to be the number equivalent to 25 HCP.  We can expect aces to be less favored here, where the scale already gives them a heavy weight.  We might note that since the jack is the only honor with an odd weight, every deal with 33 schmoints in total will contain either one jack or three.  And again, a deal with no kings will have all four aces.  In the balanced case, with no 5-card suits:

Av tricks for 1 ace = 9.07

Av tricks for 2 ace = 8.64

Av tricks for 3 ace = 8.49

Av tricks for 4 ace = 8.21

Av tricks for 0 king = 8.85

Av tricks for 1 king = 8.45

Av tricks for 2 king = 8.53

Av tricks for 3 king = 8.45

Av tricks for 4 king = 8.63

Av tricks for 0 queen = 8.14

Av tricks for 1 queen = 8.27

Av tricks for 2 queen = 8.55

Av tricks for 3 queen = 8.56

Av tricks for 4 queen = 8.91

Av tricks for 1 jack = 8.41

Av tricks for 3 jack = 8.60

It does seem that the 6-4-2-1 scale underweights queens (and maybe jacks) in this  situation.  Aces are significantly overweighted.

Finally we can ask whether the presence of a five-card suit will bring the aces back into favor.

Av tricks for 1 ace = 8.73

Av tricks for 2 ace = 8.70

Av tricks for 3 ace = 8.69

Av tricks for 4 ace = 8.60

Av tricks for 0 king = 8.94

Av tricks for 1 king = 8.72

Av tricks for 2 king = 8.73

Av tricks for 3 king = 8.60

Av tricks for 4 king = 8.70

Av tricks for 0 queen = 8.46

Av tricks for 1 queen = 8.55

Av tricks for 2 queen = 8.69

Av tricks for 3 queen = 8.72

Av tricks for 4 queen = 8.92

Av tricks for 1 jack = 8.65

Av tricks for 3 jack = 8.72

Overall, only the queen seems to have any bias at all.

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