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EPC Explanation

Posted By: Taper_Mike
Date: Tuesday, 9 August 2011, at 6:56 a.m.

In Response To: EPC Explanation (Max Levenstein)

Previous posters have already given some excellent citations for you. You may already have the answer you need. I would like to add just one other concept.

The programs that calculate EPC in a non-contact race do not work with pips. They calculate instead the expected (i.e, average) number of rolls remaining until the end of the game. Once the number of rolls is known, multiplication by 8 and 1/6 (the average number of pips in a roll) gives the EPC.

Understanding how EPC programs works is simple, at least, in concept. The trick is to work backwards, starting from the end of the game. With one checker on the one point, it is trivial to observe that there is one roll remaining in the game. So, move the checker back one pip, to the two point, and observe once again that there is one roll remaining. For a single checker on the three point, the result is the same. The fun begins when the checker in on the four point. Now, any roll except 21 will bear him off, but a 21 will only move him down to the one point. The expected number of rolls is:

E( 1 checker on 1 point ) = 1
E( 1 checker on 2 point ) = 1
E( 1 checker on 3 point ) = 1
E( 1 checker on 4 point ) = (34/36) + (2/36) * E( 1 checker on 1 point)

With one checker on the five point, we have:

31 rolls (any roll except 11, 12, 13) - Checker is borne off.
2 rolls (12) - Checker moves to 2 point.
3 rolls (11, 13) - Checker moves to 1 point.

E( 1 checker on 5 point ) = (31/36) +
(2/36) * E( 1 checker on 2 point) +
(3/36) * E( 1 checker on 1 point) +

If you are careful to sequence the computations just right, you will find that for each new postion you examine, a prior calculation will exist for the position that results after any roll and any move. That's why we have to work backwards. The necessary sequence of computation is given in the paper cited below.

Enumerating Backgammon Positions: The Perfect Hash
Arthur Benjamin and Andrew M. Ross

The general method is this:

  1. For any position, examine all possible rolls.
  2. For each roll, determine all possible (i.e, legal) moves, and select the best. The best move is found trying out each candidate move, and looking up the resulting position among the prior computations. The one with the lowest expected number of rolls remaining is the best.
  3. Form a weighted average (where non-doubles are weighted twice as much as doubles) of the rolls remaining after the best move has been made for each the 21 different rolls. This average gives the expected number of rolls remaining after a move has been made in the current position.
  4. Add 1 to get the expected number of rolls remaining before the dice are rolled in the current position.

Mike

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