Here are a class of backgammon games in which the rules for the two sides are not the same. Since one player plays by different rules than his opponent, the rules must favor one side or the other. And that's part of the fun of playing these gamestrying to figure out which side has the advantage, and by how much.
Since the rules are not symmetric, it is only fair that players take turns playing the two sides so that neither has a long-term advantage.
Or another possibility, if playing for money, is to adjust stakes according which side seems have an advantage. Each player separately estimates the odds they believe are fair, and the game is played according to the average of the two estimates. For example, suppose Jack believes White has an advantage of 1.5-to-1, and Jill believes White's advantage is 1.2-to-1. Since Jack has a higher opinion of White's advantage, Jack plays White's side, betting $1.35 (the average of 1.5 and 1.2) against Jill's $1.00. Both players should be happy with this arrangement because each is playing at odds more favorable than their estimate of what is fair.
Players play a five-point match. White starts with four points; Black starts with zero. There is no doubling cube.
Both players follow the normal rules with the following exception:
Whenever White rolls a 1, he doesn't get to play it and his opponent plays it instead. So if White rolls 1-1, he loses his turn and Black gets to play the four 1's and then take his normal turn.
Is White's four-point lead enough to overcome his disadvantage in not being able to play any of his 1's?
Two Rolls versus Choice
White, on his turn, gets to play any roll he wants except for doubles. He merely announces his roll and then plays it.
Black, on his turn, plays twice. He rolls as normal, but he gets to roll and play twice every turn.
The strategies are quite interesting. The two-rolls-per-turn player has a huge racing advantage, not only because of the extra rolls but also because his doubles count.
The choice-of-roll player invariably hits any blot he chooses, and quickly builds blocks and primes.
White starts with five pieces on the opponent's one-point and two pieces on the opponent's twelve-point. (This was first played when someone set up the board wrong by accident.)
This is a more balanced game than you might expect and good practice in back games and holding games.
White starts with his fifteen checkers set up in the normal position. Red starts with nine checkers on the bar, and two checkers each on White's one-point, two-point, and three point.
The rules are the same as in backgammon, but the strategy is quite different because of the unusual nature of Red's position.
Snake is intended to give practice in how to play, and how to play against an extreme back game.
Two Rolls versus Choice