Miss. Stella's Revisit of the Ward Formula by Nigel Merrigan, © 2006

In this article **Nigel Merrigan**, Britisch Open Backgammon Champion 2006, discusses the influence of crossovers in race bear-off positions
in relation to wastage produced by gaps.

In the space of 24 hours, I had two meetings planned, one with a certain Miss. Stella Artois and the
other with a World Class thoroughbred; no, not the ante-post favourite at 2.45 Cheltenham, but rather
with the going is good Mr. Julian Fetterlein. Miss Artois's presentation went down well with all
the taste-buddies, although, there was a moment of interstellar hostility brought on by a near scare
of out-of-Stella experience. Miss Artois's justification for infringing on the use of common sense
was in accordance with the journal of Ferengi rules of acquisition ALC. Vol. 5.5: In the absence of
brain function, please feel free to continue to inebriate your fellow backgammon compatriots with
jabberwocky logic. In Miss Artois's absence, I was able to chat with Julian concerning the issue
of crossovers in race bear-off positions; in particular, how to calculate the value of crossovers.
With his assistance, I have refined the calculations, so here goes.

The focus of this article is race bear-off positions. Of particular interest is Wards racing
formula (see Marty Storer, Forum Archive, Jan 92). Compared to Thorp's 77% accuracy rate, the
Ward formula commands a chunky 90%. A 10% retainer is reserved for a few minor irritations in
accuracy, but otherwise it cuts the mustard in most positions. The majority of the step method
formulas share continuity when assigning penalties for extra outfield crossovers - ½ a pip for
each crossover. Assuming the least awkward types of bear-off positions with very little
variance in the way of extra crossovers, this additional ½ a pip works pretty well.

In the position below however, the picture is far from straightforward. An application of the Ward
formula predicts redouble - take, whereas extensive rollouts (Storer, 92) indicate redouble - pass.
Recent rollouts confirm redouble - pass with BLACK winning 79.1%. Marty's evaluation highlights a key
point to this position: the checkers in the outfield throw themselves at the mercy of the Ward
formula, which grants them leniency. In summation, each checker is fined the paltry sum of ½
a pip for the crime of vagrancy. Did they get off lightly?

Rollout Money equity: 0.582 0.0% 0.0% 79.1% 20.9% 0.0% 0.0% Evaluations 1. Double, pass 1.000 2. No double 0.841 (-0.159) 3. Double, take 1.012 (+0.012) Proper cube action:Double, pass

According to Kleinman, extra crossovers can in general be significantly harmful
(see Storer, 92). WHITE's position certainly qualifies for stiffer penalties for
the extra crossovers. The question is why and by how much. Positions of this
nature are interesting because of the dynamic interaction between wastage,
gaps and crossovers. The plausibility of crossovers being the main determinant
when considered alongside its counter-parts deserves further study.

The scope of this article postulates that crossovers as a causal factor produces a
negative effect on wastage greater than the causal effects produced by gaps on
crossovers. For instance, in the given position, Ward's assessment of the
gaps on the six and five-pts cost between 0-1 pips. The interaction between
wastage and extra crossovers cost 1.94.

The "Majestyk formula":P (W + EC's / PC * 100)W - Wastage obtained from the Ward formula EC's - number of extra outfield checkers PC - raw pip count P - Number of pips to the home board (adjusting for gaps) 11 (7 + 2 / 51 * 100)

The initial calculation (7+2/51*100) yields an interesting ratio worth
discussion. WHITE's position is not completely pipish. In fact, 17.6% of WHITE's
position is rollish - a product caused by the wastage to crossovers interaction.
Recent rollouts indicate WHITE wins 20.9% of the time. In turn, the probability of
O winning from a pipish position is 82.4 - 17.6 = 64.8% compared to her 35.2%
chances of winning from a rollish position.

In reality, WHITE's winning chances would largely depend on her rapid bear-off
formation. For this to take effect, she will need to roll Yatzee. Unfortunately,
for her, she will never get the chance to play the game out because of the inefficiency
produced by the pips (64.8%) to rolls (35.2%) ratio - assuming cube skill is not
synonymous with the Neanderthal characters from the Guinness advert. Opt for the
middle one; he may double late - 4000 years too late. On the other hand, if you
are playing someone useful you can expect the rock to be non-prehistoric and on
time 2006 Greenwich.

It gets worse when we complete the formula. Firstly, WHITE's winning chances from
a rollish position are 35.2 * 20.9 = 7.35% compared to 13.56% pipish.
Secondly, the 1.94 (17.6 * 11) pips she picks up in penalties for the pips
the extra crossovers produce will Pac-Man some of her rollish potential.

The Ward formula evaluates the position as a redouble, take with BLACK and WHITE's adjusted
counts yielding 58.3 and 59 respectively. With the extra penalties, WHITE's adjusted pip
count is 60.94 shifting a redouble, take (.7) to a redouble, pass (2.64) .64 shy of
a take from the moon. If you have omitted step six of the Ward formula, then applied
Kleinman's D ^ / ?, how much change do you get back from a 1296 rollout? Unfortunately,
the Chancellor of BLACK's checkers advocates a no respite policy for pip-dodgers. In a laboured
response, you argue that 78.9% is a fair price to pay for a rollout costing 79.1%. For god
sake, you cry; .2 is but a Blair, blur.

You may ask does the formula have any substance concerning validity. In response,
consider this direction of reasoning. Firstly, Ward assigns values for Wastage and
Gaps independently from each other; implying there is no direct correlation between the
two. Secondly, it is not clear whether the value of ½ a pip for each crossover is in
direct relation to the Wastage or Gaps, although, from the text, credence favours gaps.
This leaves Wastage and Crossovers as a potential significant relationship that has gone
partially undetected. You may think of wastage and crossovers situated on opposite extremes
of the continuum. A tug-of-war ensues between pips and rolls creating inefficiencies
in the position.

Regardless of the type of investigation undertaken, there is always the possibility
of extraneous variables producing the desired outcome, other than the operational
variables under measurement. In turn, any measurement may possess validity in the
absence of reliability. In view of these factors, consider the next problem. It is the
same position with one alteration, the third checker on WHITE's four-pt is now on her
eleven-pt. If you were to perform a Ward count and convert it to a percentage, you
would get a figure of 87%. A 1296 rollout gives 90.6% with the formula producing…wait
for it…90.4%.

Rollout Money equity: 0.812 0.0% 0.0% 90.6% 9.4% 0.0% 0.0% Evaluations 1. Double, pass 1.000 2. No double 0.961 (-0.039) 3. Double, take 1.555 (+0.555) Proper cube action:Double, pass

At this point, you may be screaming for a more detailed demonstration of how to calculate the
value of crossovers (+ WC's) in complex positions over the board. This the final position
popped up while playing John Harrison in the third round of the British Open. If I have the
position slightly out of place John, it will add flavour for what is to come. In addition,
the position is now for money with Snowie's evaluation omitted.

Would you double in this position? You are 16 pips up with a rapid bear-off formation!
If you were applying the Ward formula, you would have BLACK's adjusted count yielding 49+12+½ =
61.5 and WHITE's 65 - 2 = 63, double - take (according to interpretation). After the game, John
estimated my WC's in the region of 55-45, no double - take, impressive for a Chap who has been
playing for six weeks. I know I'm getting to it! An application of the Majestyk formula
gives the following:

19 (12 + 0 / 49 * 100) = 4.6

Wastage = 12 with 0 extra checkers since WHITE has 3 to BLACK's 2, so 12/49 *100 = 24.5% * 19,
the number of pips the crossovers produce to get to the 5 & 4 -pts which gives 4.6
penalty pips. Since WHITE has three crossovers (½ a pip for each), we can subtract 1.5
giving us 3 penalty pips to add to BLACK's raw pip count i.e. 49+3 = 52. Now we are
ready to apply the Ward formula.

BLACK's's adjusted pip count is: 52 + 12 for the wastage + a ½ for 1 extra crossover i.e.
BLACK has 4 crossovers to WHITE's 3, yielding 64.5 WHITE has 65 + 0 for crossovers since BLACK has
the greater number of crossovers + 2 ½ for potential wastage & gaps - 2 for each extra
home board pt = 65.5. Where have those 16 pips gone! An application of
Kleinman's D ^ / S gives a ratio of .198 or 63.4%. A 1296 rollout indicates
that BLACK wins 63.7%, no double - take.

In summary, does the Majestyk formula reflect the existence of a relationship
between wastage and crossovers? No, it does not! It does much more than that.
It takes in to account wastage, gaps, crossovers, and pips the crossovers produce,
indicating a greater cross interaction that produces a strong negative correlation.
Estimating short and long crossovers need not apply any further as the
Majestyk formula takes care of it accurately. You only have to compare BLACK's
penalty pips to WHITE's to tell you that. I suppose the only thing left is can
you apply it over the board? Yes, on the basis that Miss. Artois does not
catch your eye and vaporizes any intention you had of putting backgammon before
the lovely Stella.

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Last update: 27nd May 2006 |