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:

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.