A Universe of Sorts

Suppose A goes first at picking object from a collection of objects, then B.
B has an inherent disatvantage, since they went second.
So rather than repeating and allowing A to go third and B to go fourth (ie, we run ABAB), we should instead run AB BA, since giving B the third turn "evens out the disatvantage".
Now once we're done with 4 elements, what do we do? Do we re-run A B B A again? No, this would be argued as unfair by B. So we flip this to get the full sequence as ABBA BAAB.
What next? you guessed it... flip: ABBA BAAB|BAAB ABBA
And so on. Write the recurrence down :)
Reference

§ Thu Morse Sequence for Sharing