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Hit/Most/Six Rule — Is a "distance metric" needed?

Posted By: Taper_Mike
Date: Friday, 11 November 2011, at 2:52 a.m.

In Response To: "Hit/Most/6" rule spelled out (Nack Ballard)

I contrived the following position to demonstrate that a tie was possible (in theory) when evaluating the locations of owned points according to the 6pt Convention. Here's what I wrote in my initial post:

In the position below, the two best plays in the N (Near) family are 10/8 6/5 and 8/7 6/4. Both move one checker down and one checker inside. Both make two new points. Because no other play hits or makes two points, these two plays will be ranked first and second in the N (Near) family. But which should get top billing? Can the issue be decided on the basis of the locations of owned points? Or is it a tie by that measure, with the owned 5pt in one play matching the owned 7pt in the other? If it is a tie, of course, then the issue will have to be resolved by checking the locations of blots and spares.

What plays are nactated N and n?
Position ID: bD4HBwCoK/EBAw Match ID: cAkFAAAAAAAA

In his reply, Nack presented the updated Hit/Most/Six Rule. Its guidelines clarify that the tie is resolved in favor of the inner board point. There is no need to examine the locations of blots and spares. Thanks, once again, go out to Nack.

### Nactation Exercise

As an exercise in the application of the updated Hit/Most/Six Rule, I attempted to rank all the moves in the N (Near) family for the position above. There are 12 in all, most of them ugly. The result is summarized the following table. Several aspects of the Nactation system, some positive and some negative, were revealed. One proposed modification was generated.

The N (Near) Family
Play Far Down Jump Inside Hits on
These Points
(more is better)
(higher is better)
Owned Points Not Held
by All Members in Family
(more is better)
(closer to 6pt is better)
Blot and Spare
Destinations
(closer to 6pt is better)
(outer board: farther is better)
Nactation
10/8, 6/5 10/8 6/5 5pt* 8pt N
8/7, 6/4 8/7 6/4 4pt 7pt* n
10/8, 5/4 10/8 5/4 4pt* 8pt N
13/11, 6/5 13/11 6/5 5pt* 11pt n
8/7, 5/3 8/7 5/3 7pt 3pt* N
8/7, 4/2 8/7 4/2 7pt 2pt* n
13/11, 5/4 13/11 5/4 4pt 11pt* N
10/9, 6/4 10/9 6/4 4pt 9pt* n
10/8, 4/3 10/8 4/3 8pt* 3pt N
13/11, 4/3 13/11 4/3 3pt 11pt* n
10/9, 5/3 10/9 5/3 3pt* 9pt N
10/9, 4/2 10/9 4/2 2pt* 9pt n

*This is the deciding criterion.

### Accuracy and Precision

This study reveals the great precision that has been built into the Nactation system. The Hit/Most/Six Rule provides all the tools necessary to discriminate between the 12 moves in this family.

The rules of Nactation are also quite accurate. In this family, the moves ranked highest correspond to the plays that might be deemed best by expert players. That is, the best play is ranked first in the family. The second best ranks second, and so on, through at least the first quarter of the family.

Once I understood the rules, identifying (and ranking) all the members of the N (Near) family was a straightforward endeavor. You don't have to be a wizard to learn Nactation.

### Relative Nature of Nactation

No play by itself can be assigned a rank within a Nactation family. The rank of a play is relative. Hit/Most/Six does not define the rank of a play, rather it provides rules for determining the relative ranks of two plays. It allows one to say, "Play A outranks Play B," without giving the rank of either within the family under consideration.

This is not a problem when the plays being ranked are at the top of a family. But for the plays ranked lower, it can be an issue. No play can be accurately ranked unless every play of higher rank has been identified. Even for moves ranked as high as third and fourth in a family, a nactator must be careful that he hasn't overlooked any alternatives. It's easy to miss one.

This fact was driven home to me in this study. Initially, I stated that there were only two plays that would make two points, and therefore that these two would be ranked at the top of the N (Near) family. Exhaustive enumeration revealed a third that I had missed. Check the foregoing table. My only consolation is that the plays I claimed were the top two, in fact, were. Not quite pure luck, but close.

The introduction of underlined families for the Nactation of complex doublets should greatly reduce the number of plays that might be overlooked. By splitting the N (Near) family, for instance, into two families (a regular family and an underlined family), the size of each has been reduced. Smaller family size means fewer overlooked plays. (Note the distinction: when doublets are rolled, there are two N (Near) families, regular and underlined, but when a non-doublet is rolled, there is still only one.)

Otherwise, there's not much a nactator can do except to use great care when nactating one of the rare plays that present this sort of difficulty.

### Proposal: A Distance Metric for Blots and Spares

In another post I wrote, "The only remaining bit of fuzziness for me has to do with the apples-and-oranges comparison between a blot moved into the inner board and a blot moved to the outer board.". This exercise reinforced my belief that such a problem exists.

Its existance is revealed in the ranking of the three plays at the bottom of the family. Both pit the location of an inner board blot against the location of an outer board blot. In the outer board, a blot or spare that is farther from the 6pt is better than one that is closer. Without a quantitative measure of just what "better" means, however, ranking can be problematic.

Consider, for instance, the play 13/11 4/3. In the following table, I rank this play above the other two. I do this based on my notion of distance metrics. For a blot or spare in the outer board, the distance metric gives the distance to the midpoint. For all other locations, distance metric gives the distance to the 6pt.

Play Destination of
Inner Board Blot
(metric gives distance to 6pt)
Destination of
Outer Board Blot
(metric gives distance to 13pt)
Distance
Metrics
Relative
Rank
13/11, 4/3 3pt — (3 away from 6pt) 11pt — (2* away from 13pt) 2*, 3 1
10/9, 5/3 3pt — (3* away from 6pt) 9pt — (4 away from 13pt) 3*, 4 2
10/9, 4/2 2pt — (4* away from 6pt) 9pt — (4 away from 13pt) 4*, 4 3

*This is the deciding criterion.

Ranking plays then becomes a matter of minimizing distance metrics. The lowest metric of one play is compared with the lowest metric another. If they are different, then the play with the smaller metric outranks the other. If they are equal, then comparison proceeds with the second-smallest metric of each move. And so on.

Ties go to the inner board. As with the ranking of owned points, in case the distance metric of a blot or spare in the inner board is tied with the distance metric of a checker in the outer board, the tie is resolved in favor of the one in the inner board.

If the combined number of blots and spares created by one play is different from the number created by another play that is otherwise ranked equal, then the play that creates the fewest number of new blots and spares outranks the other. We don't need a new rule for this. Any play that creates a fewer number of blots/spares must have created a larger number of owned points or else borne a checker off.

Unless I have misunderstood something, the 6pt Convention needs some quantitave measure added to it so that the locations of blots and spares in the outer board can be compared with the locations of those in the inner board. The distance metric discussed here is one simple solution. It may not, however, be the only one. And, of course, in case I wrong entirely, perhaps no refinement is needed at all.

Mike

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