I think you raise a very interesting question, and I like your initial solution. I feel like a game is fair if all players know that a die is biased, but no pl

I think you raise a very interesting question, and I like your initial solution. I feel like a game is fair if all players know that a die is biased, but no player has an advantage in predicting the outcome. To achieve that: for each die, for each roll have the players agree on a random permutation of the observed faces to actual faces for that roll. As a result even if all players know that 6 will land with probability 1 in the original throw, after permuting each side has probability 1/6. The agreement algorithm: have the parties each propose a permutation, and compose the permutations together in order to get a final "unbiased" permutation. I think if there is at least one honest party, this will work. Another approach is to let everyone sample the dies before playing until they are satisfied, and let everyone adjust their strategy accordingly.