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# Backgammon tools in J

# 3: Two-sided bearoff probabilities

# by Howard A. Peelle (hapeelle@educ.umass.edu)

J programs are presented as analytical tools for expert backgammon [1]. Part 3 here develops a program to compute probability distributions for two-sided bearoff (without contact) and uses it to determine the probability of winning.

The inner board is represented as a list of six integers. For example, one piece on the 5-point and one piece on the 6-point:

board =: 0 0 0 0 1 1

Utility programs:

ELSE =: ` WHEN =: @.

Master program `PD`

produces a list of probabilities for bearing off in increasing numbers of rolls. Defined much like `Rolls`

in [1], it calls `PDBearoff`

or else returns 1 when all pieces are off the board:

PD =: PDBearoff ELSE 1: WHEN AllOff AllOff =: All@(Board = 0:) Board =: ] All=: And/ And =: *. PDBearoff =: 0: , Average@AllPDs Average =: +/ % # AllPDs =: PD"1@BestMoves BestMoves =: PDDoubles , 2: # Better@PDSingles doubles =: > 1 1 ; 2 2 ; 3 3 ; 4 4 ; 5 5 ; 6 6 singles =: > 1 2 ; 1 3 ; 1 4 ; 1 5 ; 1 6 ; 2 3 ; 2 4 ; 2 5 ; 2 6 ; 3 4 ; 3 5 ; 3 6 ; 4 5 ; 4 6 ; 5 6 singles =: singles ,: |."1 singles PDDoubles =: doubles Best@AllMoves"1 Board PDSingles =: singles Best@AllMoves"1 Board Better =: Best@,:"1/ Best =: {~ MinIn@:(N"1) MinIn =: i. <./

Note that program N (from [1]) is used in `Best`

to evaluate boards.

`AllMoves`

and its subprograms are the same as in [1]:

AllMoves =: (Die2 Moves Die1 Moves Board) ELSE ¬ (Die1 Moves ^:4 Board) WHEN (Die1=Die2) Die1 =: First =: {.@Dice Die2 =: Last =: {:@Dice Dice =: [ EACH =: &.> Moves =: ~.@;@(LegalMoves EACH <"1) LegalMoves =: (Possibles Move"1 Board) ELSE Board WHEN AllOff Possibles =: FromTo Where FromTo OK"1 Board FromTo =: On ,. On-Die1 Where =: #~ On =: Points Where Board > 0: OK =: Off Or Inside Or Highest Or =: +. Inside =: Last > 0: Off =: Last = 0: Highest =: First = Max@On Max =: >./ Move =: Board + To - From From =: Points e. First To =: Points e. Last Points =: 1: + i.@6:

For example, probabilities of bearing off the board above in 0, 1, 2, 3, and 4 rolls:

PD 0 0 0 0 1 1 0 0.166667 0.707562 0.124829 0.000943073

Such a probability distribution is related to expected number of rolls (N):

+/ pd * i.#pd =. PD (board) 1.96005 N (board) 1.96005

Further, `PD`

can be used to determine the probability of the first player winning:

Pwin =: X/@amp;,:&PD X =: +/ . * +/\.

`Pwin`

computes the `PD`

of both boards, then sums products of player 1’s probabilities times reverse cumulative sums of player 2’s probabilities – that is, the sum of probabilities of 1 bearing off times the respective probabilities of 2 not bearing off.

The result is the probability of player 1 winning the bearoff.

Example:

0 0 0 0 1 1 Pwin 0 0 0 0 2 0 0.766415

The first player is about 77% likely to win.

The programs above are extremely inefficient, so `PD`

should be re-defined to utilize a pre-established database of probability distributions for fast look up (as done for `N`

in Part 1).

Upon request, the author will provide a script with efficient programs.