One nice reference that seeks to answer your question is the following paper,
The paper's main results are contained in the following theorem (Theorem I):
a. If both players have a dominant strategy in the basic game, the game is the same. (where the basic game is just the 2x2 bimatrix game)
b. If only one player has a dominant strategy in the basic game, then
the extended game has a unique equilibrium in un-dominated strategies.
It's different from the basic game outcome iff that outcome is Pareto dominated by another pure strategy outcome. The extended game equilibrium achieves
these Pareto dominating payoffs by having the player with the dominant
strategy play first and the other second.
Think of this as a way to achieve a Pareto superior vector of payoffs.
(C) If the unique simultaneous play equilibrium of the basic game is
in mixed strategies, then like in part (B) there won't be a change unless there's a Pareto superior P.S. vector. If Pareto dominance does
not exist, both players will wait until the second period and play
simultaneously. If Pareto dominance does exist, the unique extended
game equilibrium attains the Pareto dominating payoffs by selecting one
of the sequential move games.