Fencer: my suggestion was very simply to replace the word "with" with the phrase "that employs" or "that uses" in the first sentence about the doubling cube. in other words, changing

"Backgammon (and its variants) is the only game which can be defined as a match with a doubling cube"

to

"Backgammon (and its variants) is the only game which can be defined as a match that employs a doubling cube."

thanks.