Question: If a game of backgammon were to be played between opponents, where one used standard dice and the other a pair of Sicherman dice, who if either would have the probabilistic advantage?

Assume that all other factors and rules were/remained the same.

References to literature, especially which explore this problem or an analogue, are also acceptable.

For those not familiar with either or both of these topics:

Background: Backgammon is a historic game in which two players try to move their pieces past each other. Noteworthy rules include

  • rolling two six-sided dice each turn, moving individual pieces by moves of length equal to one or both of the numbers shown. In particular, where a double is rolled, the number of moves is also doubled - rolling e.g. $22$ gives that player up to four movements of length 2

  • in a turn, a player must move pieces by the greatest possible total length. That is, turns which don't use all possible movement are not permissible; where no movement is possible, no movement occurs.

  • the ability to hit single opponent pieces, returning them to the bar, from which they must begin the movement process again. This is the primary method of gaining some advantage over the opponent.

  • the inability to land on, or hit, opponent pieces which have formed a point (wherein at least two pieces are stacked in the same spot). This is a significant part of strategy.

Background: Sicherman dice are a pair of non-identically-numbered six-sided dice $$(1,2,2,3,3,4)+(1,3,4,5,6,8)$$ with the property that the distribution of the sums of rolling them together, is the same as that of two standard six-sided dice, $$(1,2,3,4,5,6)+(1,2,3,4,5,6)$$ This can be shown by (among several methods) creating a table of sums, and counting frequencies, for each pair of dice.

  • $\begingroup$ My initial observations are that while the Sicherman player has only four possible doubles instead of six, and none are for high numbers, the Sicherman player's $8$ means the Standard player cannot trap them, even with a seven-block. These are notable parts of strategy that may affect the importance of different roll probabilities, and their usefulness in a game. $\endgroup$ – Nij Sep 2 '16 at 6:38
  • $\begingroup$ Because the moves depend on the specific pair rolled and not only on the sum, the play will likely be very different. If nothing else, a double is much more likely with the standard dice. $\endgroup$ – Aaron Sep 2 '16 at 6:39

Speaking as a backgammon player, the immediate question that comes to mind is how one would handle entering a checker on the bar if an $8$ is rolled. Would it be placed outside of the opponent's home row, on the eighth point? Or would it not count?

Another question is one of bearing off. Would all the checkers need to be in the home row?

As for not being able to form a blockade, actually, it is possible to create a $6$-prime, with a gap in the seventh position, and a point in the eighth position. Since a $7$ cannot be directly rolled, it is not possible to pass such a configuration with any roll. So the matter of entering from the bar becomes especially relevant here; if the opponent with Sicherman dice is only allowed to enter from points 1-6 in the home row, then rolling an $8$ does not help; whereas if they are allowed to enter in on the $8$-point, then a blockade cannot be made while allowing bearoff.

  • $\begingroup$ An excellent point about needing only seven points for an effective eight-block (!!). Note that the riles remain the same but for the dice used by one player, so I would think entering to the 17 is acceptable (distance from the bar), but bearing off from the 8 is not (must be in the home board). $\endgroup$ – Nij Sep 2 '16 at 6:58

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