Backgammon Cube Handling in Races

Cube Handling in Races

Forum Archive : Cube Handling in Races

Pip-count formulas

From:   Tom Keith
Address:   tom@bkgm.com
Date:   21 June 2004
Subject:   Cube handling in noncontact positions
Forum:   GammOnLine

I have just written an article on cube handling in noncontact positions. http://www.bkgm.com/articles/CubeHandlingInRaces It compares several pip-counting methods to see which ones work best. To compare the methods, I collected positions from real games and rolled them out using GnuBG. Five pip-counting methods were evaluated: Thorp count, Keeler/Gillogly count, Ward count, Lamford count, and my own "Keith" count. The formulas are judged on their ability to account for wastage and on how good they are at making accurate cube decisions. There is lots of interesting information there, and several pretty graphs. Please let me know what you think. Tom

Tom Keith writes:

Chuck Bower asks: > 1) What are the strict requirements to make the database: > a) any non-contact position? Yes. > b) include positions with players owning their own midpoints (nearly > non-contact)? No, though that would be interesting to do too. > c) was there any requirements/restrictions on how many checkers were yet > to be borne in? Already borne off? No. > d) I assume that a single game typically generated many entries to the > database, since after each play a new valid situation results. True? True. --- HappyJuggler0 asks: > I am also intrigued that the colored line graph of the various methods > at different pip counts seem to show all the counting methods except the > Keith Count to be inferior than a straight, unadjusted pip count once > you get to the 80+ range. Or did I read that wrong? Yes, I assume the reason for this is that some adjustments such as "subtract 1 pip for each occupied home-board point" really aren't useful in longer races.

Michael Sullivan writes:

I'd like to see this compared to a heuristic using effective pips a la Trice and Zare, but it's hard to see how to get that into an algorithm and a function that's easily coded. In practice there's a lot of handwaving, but I find I get results (without being especially good at it yet) that are closer to database values that way than with any other count I've tried, especially for positions with <25 pips, and I note that *all* of Tom's heuristics tested, including the Keith Count have a fairly large RMS error at these small pip counts. A .05 error in assessing cwc can easily translate into a whopper of a double or take decision. It looks to me like Keith count works about as well for midrange races as it's competition, but it's the only one that is as good or better than plain pipcount in long races. OTOH, in short races, it looks still to be worthwhile to do a full analysis from a decent mental database of known positions. This isn't all that surprising really. Edge effects average out better the longer the race. In really long race, the pip count is a pretty good estimate, in medium races, taking care of the large scale effects (simple heuristics) is good enough to get a good estimate. In really short races, the details pay. Michael

Rob Adams writes:

24 23 22 21 20 19 18 17 16 15 14 13 54 pips +---+---+---+---+---+---+---+---+---+---+---+---+---+ O | O O O O O | | | O | O O O O O | | | | O O | | | | O | | | | | | | | X | | | | X | | | | X | | | | X X | | | | X X X X | | | | X X X X X X | | | [2] +---+---+---+---+---+---+---+---+---+---+---+---+---+ 1 2 3 4 5 6 7 8 9 10 11 12 39 pips I had this position yesterday online. So please let me know if this is how the Keith Count should work or not. X has 39 plus 10 for the acepoint checkers plus 1 for the 2pt is 50 plus 1/7 is about 57. O has 54 plus 2 for the acepoint checker is 56. Nothing for having a gap on the 2pt or having 2 off right? So ahead 57-56 would be a redouble/drop as X isn't ahead by 2.

Tom Keith writes:

Yes, that's how to do it. Now I hope a rollout doesn't show this to be too far wrong! ... A rollout shows dropping is a 0.089 error here. Even though the formula was wrong, in a way this is a good example because it reminds us that the formula gives only approximate answers and will make its share of errors. Cubeful equities: 1. Double, take +0.911 2. Double, pass +1.000 (+0.089) 3. No double +0.844 (-0.067) Proper cube action: Redouble, take

Douglas Zare writes:

Very well done. I hope you also publish this in backgammon periodicals. If you don't mind, I'd like to include a hyperlink to your page in my next GammonVillage column, but I think you could write an article or series of articles based on your work. I like the simplicity of the Keith count, but I suspect its accuracy in relevant positions could be improved by tweaking some of the parameters and adding a couple of items (e.g., reduce the penalty for checkers on the 3 point, but penalize the difference between the number of checkers on the lower three points and the number on higher points).

Bob Koca writes:

Good article Tom. It should be mentioned though that some of your recommendations apply only to longer races. You state that if cubeless chances are between 30% and 70% then a pip is worth about 2.5%. The length of the race is important here as well. If the pipcount is 104 to 100 then gaining one pip is worth about 2% whereas if the pipcount is 50 to 54 then a gaining one pip is worth nearly 5%. Similarly the 68%, 71% 78% rule is for longer races. As the race gets shorter all those numbres decrease. ,Bob Koca

Tom Keith writes:

Thanks for the comments, Bob. You make some good points. For the value of a pip, you are right that the length of the race matters, though I don't get as big a difference as you do. The following chart shows the probability of winning for the player-on-roll at different length races and various leads. The chart is an average of the positions I rolled out. According to the chart, the average value of a pip in a 100-pip race (pip count of player-on-roll) is 1.9%, which is about the same as you said. For a 50-pip race, the chart shows gaining a pip is worth about 2.5%, larger than a 100-pip race but not as much as your 5%. ---------------- Length of Race ------------------- Lead 20 30 40 50 60 70 80 90 100 -14: .101 .154 .193 .222 .244 .258 .270 .285 .297 -13: .114 .177 .214 .238 .265 .280 .290 .305 .317 -12: .149 .211 .246 .273 .295 .307 .315 .325 .334 -11: .177 .247 .279 .298 .319 .329 .338 .349 .357 -10: .210 .283 .309 .328 .347 .354 .363 .371 .375 -9: .250 .312 .338 .356 .369 .377 .382 .389 .395 -8: .288 .340 .359 .377 .387 .397 .408 .413 .415 -7: .333 .380 .410 .423 .422 .425 .430 .432 .435 -6: .384 .426 .442 .449 .450 .449 .452 .456 .459 -5: .430 .470 .480 .478 .475 .474 .477 .480 .482 -4: .494 .505 .503 .504 .508 .503 .501 .501 .500 -3: .530 .537 .540 .531 .529 .529 .526 .524 .521 -2: .569 .555 .553 .557 .559 .555 .549 .547 .547 -1: .616 .614 .600 .588 .577 .574 .573 .568 .565 0: .648 .643 .635 .619 .610 .604 .594 .588 .583 1: .691 .672 .654 .643 .634 .624 .616 .610 .602 2: .727 .707 .693 .671 .655 .645 .636 .630 .622 3: .762 .735 .712 .693 .676 .667 .657 .648 .641 4: .782 .760 .741 .718 .700 .690 .677 .667 .661 5: .806 .780 .764 .741 .726 .712 .698 .689 .680 6: .835 .805 .786 .767 .748 .732 .717 .708 .698 7: .856 .828 .800 .776 .765 .753 .734 .720 .712 8: .871 .850 .828 .800 .780 .767 .753 .739 .728 9: .890 .866 .842 .816 .799 .786 .770 .755 .744 10: .907 .883 .856 .831 .815 .802 .786 .775 .763 AvgPip: .033 .029 .027 .025 .024 .022 .021 .020 .019 Tom

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