A set of permutation fair dice work for any subset of dice and players. So a 4-dice perm-fair set allows any three dice to be used and is guaranteed to be perm-fair for 3 as well.

The “Go First” name is catchy for laypeople, but permutation fairness is the strongest and most interesting property.

There are sets we call “all subset place fair” which means any subset of the dice can be used and can fairly choose 1st, 2nd, and so forth, but this property is slightly weaker and doesn’t always make every ordering equally likely for every subset.

there seems to be no need to match number of dice with number of players - for example in 2 players the second player throwing second die (2,3) is really a noop as outcome is decided by the first player's die (1,4)