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I think that a first-player token will do what you suggested, but to be really fair, each player has to have the same amount of rounds being the 1st player? (e.g. with 4 players, you need to have 4n rounds).
There’s a math concept (fair division) that can be used, but I think you will need to have more dice than just 1 per player, but depending of what your game is about, there is a good idea. I learned about it in this video: https://www.youtube.com/watch?v=kaMKInkV7Vs
