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The Effective Pip Count

by Douglas Zare
25 April 2003

Douglas Zare

The pip count is a simple way to assess most races, including races to save the gammon. In a long race (50-120 pips), you can double with a lead of 10%-2 pips, redouble with a lead of 10%-1 pip, and take with a deficit of 10%+2 pips. In many situations, the raw pip count is unsatisfactory, and must be adjusted.

Which side do you like better?



Red trails 44-15 in the race.

In almost all circumstances, you should prefer the position with 44 pips over the position with 15. (In a head-to-head contest, Red would be a favourite even if White were on roll.) Here, you can't rely on the pip count. There are several methods available for adjusting the pip count to penalize the candlestick, and the effective pip count (epc) is the best that I have encountered. It not only allows me to assess most races to within a pip, but it provides a framework for learning. If I misunderstand a position, I can improve. It is also simpler than many inaccurate systems

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Subject: Re: The Effective Pip Count

From: Tad Bright

Date: 26 Apr 2003 00:46 EST

Great article. I've been trying to consolidate all of Walter's articles and this article helps. Walter said he tested Thorp and Ward over 180million positions and that Ward was better, about 80% accurate. I'm not quite 50% accurate with Quizmaster. Let's see how this article helps me.

For match cubes, can I take that 10% number and adjust them in match play. Is it true that each pip is roughly 3%? So if I'm trailing and need 65% for a double, can I be closer from the 10% by 3 pips to have a double?

Thanks again!

Subject: Re: The Effective Pip Count

From: zare

Date: 26 Apr 2003 13:25 EST

I was worried that this article was a bit too long, so I cut out the section on going from the epc to percentages. I'll elaborate more on that in the future. However, a basic idea is that you can interpolate between what you know is a borderline take/pass decision and what you know is an even race.

If you lead 70-73, your lead is 7 pips, counting the fact that you are on roll. At 70-66, you would win 50%. A lead of 70-79, 13 pips, would be a borderline take/pass decision by the 10%+2 pips rule, which corresponds to about 78% winning chances. So, 70-73 should be about 7/13 of the way from 50% to 78%, a little bit more than half way, or 65%. Each pip is worth a little bit more than 2%. The pips closer to the even races are worth a bit more, and those in more extreme races are worth a bit less, but this interpolation is fairly accurate, and can be used to extrapolate a few pips beyond the take point, too.

What happens if you lead 40-42? The 10%+2 pips rule is not accurate any more. 40-45 would be a borderline pass, rather than 40-46. That's a lead of 9 pips including the roll, and it would correspond to about 27% winning chances (the difference between 50-50 and 77-23). In that case, each pip is worth about 3%, so leading by 6 pips means you are about a 68-32 favorite.

The interpolation suggests that in races shorter than about 40 pips, each pip is worth more than 3%. In races longer than about 80 pips, each pip is worth less than 2%.

If you can estimate the epc within a pip, you can estimate the probability of winning within 2-3% in medium-length races. That still allows you to make 0.100 take/pass errors in money play. In order to avoid making errors of size 0.050, you need to estimate the race within a half pip. That's not too hard in a pips versus pips position, or pips versus a pure n-roll position.

This illustrates how important a pip is. Some of the other systems for adjusting the nominal pip count have adjustments that are off by 3 pips or more, e.g., they would penalize the optimal 7-5-3 position for having a stack on the 6 point and gaps on the 2 and 3 points, although they would recognize that it has fewer crossovers than most 79-pip positions. If you get numerical feedback on your estimates of the race, 3 pips is a completely unacceptable error (at least for a normal position), corresponding to passing a borderline redouble. Douglas Zare

Subject: Re: The Effective Pip Count

From: zare

Date: 26 Apr 2003 13:57 EST

By the way, I think I should be clearer in acknowledging Walter Trice as the inventor (as far as I know) of the epc method, and for coming up with the 7n+1 formula.

If you find that you are not doing as well on the Bearoff Quizmaster, it could be because you have set the equity range and position numbers to give you tough decisions. As Walter Trice pointed out, many of the cube decisions are trivial, where one side is far ahead, and you are probably skipping those. Also important are the magnitudes of your errors, which the Quizmaster doesn't count; calling an initial double a redouble is much better than calling it a beaver, and some errors are only by 0.001.

Another good way to use the Quizmaster is to set the equity range to something like 0.98-1.02, so that it shows you nothing but borderline take/pass decisions you can try to remember as reference positions. Don't worry about scoring 50% on such a quiz, of course.

Douglas Zare

Subject: Re: The Effective Pip Count

From: Tad Bright

Date: 27 Apr 2003 15:46 EST

This is great. Now I'll synthesize and make some bullet points for my pda. Also thanks for the quizmaster tips. I'm about 20 hours from putting it all together and 100 hours from being a master of racing cubes...maybe:) It seems that this is one aspect of the game that this intermediate can master at a higher level.

Subject: Re: The Effective Pip Count

From: 1296

Date: 28 Apr 2003 03:12 EST

excellent article. its back to racing kindergarden.

will look forward to seeing how the EPC for both sides is translated into double, redouble, and point of last take.

its hard to believe the rules are the same as those for nominal pips with low wastage.

Subject: Re: The Effective Pip Count

From: Slitherin

Date: 28 Apr 2003 13:23 EST

Great article which when combined with Walter's Boot camp articles gives a thorough view of epc.

Where can I get hold of Quizmaster please??

Subject: Re: The Effective Pip Count

From: zare

Date: 02 May 2003 10:33 EST

I think the only way to get the Bearoff Quizmaster is directly from Walter Trice. You can find his e-mail address in this web page: http://www.nebackgammon.org/clubinfo.html

Be careful that epcs don't follow the 10%+2 pips rule. If you subtract 7 pips for the wastage of an efficient position, the resulting quantities may satisfy the 10%+2 pips rule for pips versus pips positions. Another way of doing this is to assess the added wastage from having too many checkers on the low points plus the synergy, and add the added wastage to the nominal pip count.

Douglas Zare

Subject: Re: The Effective Pip Count

From: dorbel

Date: 05 May 2003 04:28 EST

The best DZ article that I have read, erudite and applicable. An understanding of EPC and the ability to adjust pip counts for gaps and wastage is essential to avoid expensive guessing games in live situations. This article should inspire all of us to do some work on these themes. Kudos as ever, to the inimitable W.Trice, who had all this stuff off pat long before the robot era. A couple of points. The average roll of 8.167 pips is not always a useful figure, as it is inflated by the huge 6-6, 5-5 and 4-4. As only 12 rolls are above the mean and 24 below it, it may well be correct to use a slightly lower figure in our calculations. I would also be interested to hear Doug's comments, perhaps in a later article, on the fascinating pip/roll hybrids discussed by Kleinman and Kazaross in (I think) "Only The Hogs..." The Zbot sounds interesting. I wonder who can be working on that?..........

Subject: Re: The Effective Pip Count

From: zare

Date: 05 May 2003 21:44 EST

In Position A, the question is how much wastage there is from the 10 checkers on the ace, deuce, and trey points. There is more wastage than in the reference position with a closed board and spares on the lower points, 13.71 pips, or 7 pips more than optimal. How much does this position waste? Zbot estimates that it wastes about 16 pips, about 9 pips more than optimal. I think the best possible 15 checker bearoff forward from White's position would have the shape 2-2-1-3-4-3, wasting about 15.5 pips. (In other words, from the reference position, moving a checker from the 4 point to the deuce point only saves 0.2 effective pips! 1.8 pips are wasted.)

If you start with the linear rules penalizing the second checker on the ace point 1 pip and each additional checker 2 pips, you only get a total of 5 pips added, 1+2 for the second and third checkers on the ace point and 1+1 for the third and fourth checkers on the deuce point. Having a large number of checkers on low points calls for a synergy adjustment, here about 4 pips.

Red's checkers are also not ideally placed for the pip count. High numbers will force checkers deep, and it is likely that there will be a combination of a thin 4 point and too many checkers on the deuce point. Zbot estimates that Red's position wastes about 9.5 pips, slightly more than in the second position in the section on gaps.

The combination says that the nominal pip count is misleading: White is not ahead by 6 pips, but behind by about a half pip. Red is on roll, worth 4 pips, but Red is not close to a redouble (or an initial double). On the other hand, it is also far from a beaver. In order to beaver in an even race, the pip count must be about 100, and here the race is shorter and favors Red slightly.

Position B is actually from the same game as position A, after the player holding the 4 cube rolled well including a great 2-2 in the bearoff. An efficient race of 36-29 would be a huge pass; to take in a race of that length you must be within about 10%+0.5 pip.

White's position is pretty efficient, although not ideal for a position with 36 pips, since it would be better to have no checkers on lower points and more checkers off. It wastes 7.11 pips, and it is easy to get this estimate almost exactly right over the board.

Red's position is ugly. The three checkers on the ace point get penalized 1+2=3 pips. In addition, there are too many checkers on low points, as well as a shortage of checkers on even points. That means that there will be a tendency to miss on 2s later in the bearoff. Red's position wastes 12.74 pips, so the synergy is worth about 3 pips. (Over the board, my estimate for the synergy was off by about a pip.)

Instead of leading by 7 pips plus the roll, the lead is more accurately described as a lead of about a pip and a half plus the roll. That makes it a small redouble. Redoubling is correct by 0.030 EMG: Not redoubling is worth 0.703 (times 4) while Redouble/Take is worth 0.733.

Position C is interesting because of the match score. Red gains much more than Red risks by redoubling, so Red should redouble as a substantial underdog in a last-roll position. Of course, the game will not end on the next exchange, so there are more things to think about than the doubling window.

Is Red the favorite? It is easy to estimate the effective pip count of White. An efficient position with 25 pips has an effective pip count of about 25+6=31. (31.14, here.) Red's position is slightly better than a 5 roll position. It is very likely that Red will roll 1 or 2 aces in the next few rolls, in which case the position will behave as a 5 roll position. It's more common that Red rolls no aces than that Red misses twice by rolling 3 or more aces. A true 5-roll position would have an effective pip count of 7*5+1=36, and this position has an epc of 34.83.

Red trails by just under 4 pips and is on roll, so Red is the cubeless favorite. (Red wins 51.0% of the time.) That is enough to redouble to 8 at this match score, according to live-cube rollouts using Snowie's match equity table. Of course, it is a huge take.

Redouble/Take is worth 0.281 EMG. No Redouble is worth ~0.257 EMG.

Douglas Zare

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