Pawn wars

One of the training exercises I used today with the ACT Junior Development Squad was a tournament involving a subset of the starting pieces. Each round started with 8 pawns and a king each, but after round 1 certain pieces were added (or taken away). For example we had knights only (plus king and pawns), bishops only, 2B+2N, Rooks only etc
Interestingly while describing the rules at the start one player suggested the even simpler 8 pawns v 8 pawns (without any pieces at all) is a win for the player moving second. The argument is a 'classical' one, in that Black simply mirrors Whites moves, until White is forced to sacrifice a pawn and supposedly lose the game.
However I suspect there is a flaw in this reasoning, depending on the rules used. If you use the "both queen = draw" rule then 1.f3 f6 2.c3 c6 3.e4 e5 4.d4 d5 5.exd exd is a draw as both players then push to d8/d1.
But my question is: Is this game a win for one side or the other (ie has this game been solved)? If so, what about the case where kings are added?