# Backgammon

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In the early days, the only difference between match and money play was that people were much tighter and it was quite rare to see a 4 cube. Gradually players realised that the score had a significant influencing factor on cube decisions. Ideas such as not doubling so readily when ahead in a match became standard. Soon, some of the keener minds decided to work out match equity tables.

A match equity table gives you the percentage chance of winning a match at any particular score. Trailing 5-4 in a match to 7 points, your chances are 41 per cent; leading 10-3 in a match to 15, they are 83 per cent. Three players, Robertie, Woolsey and Kleinman, derived tables based on a mixture of mathematical theory and empirical evidence. Their three tables, however, had some differences. Over the years methods have become refined and the empirical evidence of real matches has greatly increased, so there is now general agreement on the table values.

The problem is, how do you remember tables? A table for a 15-point match has 225 entries, a little too many to remember. Luckily help is at hand. I shall give two methods by which the match equity for any score in any length match can be calculated.

The first, and most commonly used, is the Janowski Formula, derived by Rick Janowski of Rochdale. If D is the difference between the two players' scores and T is the number of points the trailing player needs to win, then the match-winning probability of the leading player is 50 + (D x 85/(T+6)). If the Crawford game is being played this changes to 55 + (D x 55/(T+2)). This may seem daunting but after a little practice it becomes quite easy to use.

If you don't like multiplication then try the Underwood Formula, named after the late US player Fleet Underwood. If T is the trailer's score, L the leader's score and W the match length then the leader's match-winning probability is 50 + the first (L-T) numbers from the following sequence: 9 8 6 5 4 3 2 2 2 2 2 2. If W-L > 4 then subtract ((W-L) + (W- T))/4 from the number previously calculated. Again, this may seem complex but is surprisingly easy to use. For example, leading 5-2 in a match to 11 the leader's probability of winning is 50+9+8+6 - (6+9)/4 which is 69 per cent.

Of the two formulae the Janowski is slightly more accurate, particularly for longer matches where one player has a significant lead. We shall return to this topic in a few weeks' time to see how to apply it all.