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 ```

### Cube Handling in Races

Bower's modified Thorp count  (Chuck Bower, July 1997)
Calculating winning chances  (Raccoon, Jan 2007)
Calculating winning chances  (OpenWheel+, Nov 2005)
Doubling formulas  (Michael J. Zehr, Jan 1995)
Doubling in a long race  (Brian Sheppard, Feb 1998)
EPC example: stack and straggler  (neilkaz+, Jan 2009)
EPC examples: stack and straggler  (Carlo Melzi+, Dec 2008)
Effective pipcount  (Douglas Zare, Sept 2003)
Effective pipcount and type of position  (Douglas Zare, Jan 2004)
Kleinman count  (Øystein Johansen+, Feb 2001)
Kleinman count  (André Nicoulin, Sept 1998)
Kleinman count  (Chuck Bower, Mar 1998)
Lamford's race forumla  (Michael Schell, Aug 2001)
N-roll vs n-roll bearoff  (David Rubin+, July 2008)
N-roll vs n-roll bearoff  (Gregg Cattanach, Nov 2002)
N-roll vs n-roll bearoff  (Chuck Bower+, Dec 1997)
Near end of game  (Daniel Murphy, Mar 1997)
Near end of game  (David Montgomery, Feb 1997)
Near end of game  (Ron Karr, Feb 1997)
One checker model  (Kit Woolsey+, Feb 1998)
Pip count percentage  (Jeff Mogath+, Feb 2001)
Pip-count formulas  (Tom Keith+, June 2004)
Thorp count  (Chuck Bower, Jan 1997)
Thorp count  (Simon Woodhead, Sept 1991)
Thorp count questions  (Chuck Bower, Sept 1999)
Value of a pip  (Tom Keith, June 2004)
Ward's racing formula  (Marty Storer, Jan 1992)
What's your favorite formula?  (Timothy Chow+, Aug 2012)