10 moves in a row for both players - advantage for whom?

[taatelikakku]

The whole premise of Thue-morse fair sharing sequence is that earlier moves have more value than later ones (beginning vs endgame) and that simply alternating b,w,b,w is not fair enough because black gets the biggest first move (or initiative?). Based on that idea it would be logical for the first player to get a very large advantage if he got the first 10 moves.