If we compare 2-person simultaneous game and corresponding sequential game (same payoff structure, but players make their decisions one after another) it often happens that:
If simultaneous game has several pure strategy Nash equilibria, corresponding sequential game has first player advantage (for example: Battle of sexes).
If simultaneous game has no pure strategy Nash equilibria, corresponding sequential game has second player advantage (for example: rock-scissors-paper).
Is it always the case? Are there any theorem with respect to the subject?