When can king and pawn win against another king and pawn? Assume such a situation: Both players (Black and White) only have two pieces, that is a King and a Pawn. They have the liberty of placing them according to their wishes on the first row of their side of board (White on row 1, Black on row 8) and then the game starts. (Yes, pawn start from the same row as kings).

Which starting positions guarantee a win to one side? Which will be drawn?

My intention: I wonder if such a chess variant would make sense. Since players have the liberty of placing their king and pawn on any column they please, are there any positions that are clearly best? Or (as I'd hope for) would it reduce to rock-paper-scissors, that is regardless of which position a player starts with their chances to win/lose/draw the game depend on which position their opponent starts with? Or would all games be necessarily drawn, which would make such a variant pointless?

So I guess I need a method of determining if any given position is drawn or won, or rather - general rules that govern this problem, so that I could grasp all possible positions without having to enumerate them.
As far as I can tell, in your proposed variant a player could always force a draw by placing the king in a reasonably centralized position, regardless of what the other side chooses. Even if he allows the opponent a seemingly superior king position, it still winds up as a draw with correct play.

Let's say Black chooses the king on the D file and the pawn on the C file. If the pawns start on the same file, Black could allow a position like this and still get a draw:

[FEN "3k4/8/8/2pK4/2P5/8/8/8 w - - 0 1"]

1.Kxc5 Kc7


With the pawns one file apart, it's a similar pattern:

[FEN "3k4/8/8/2p5/2KP4/8/8/8 w - - 0 1"]

1.Kxc5 Kc7


With the pawns two files apart, Black can take the diagonal opposition:

[FEN "3k4/8/8/2p5/3KP3/8/8/8 w - - 0 1"]

1.Kxc5 Ke7 Kd5 Kd7


Three or more files apart? No problem. White must abandon his pawn to eliminate Black's pawn.

[FEN "3k4/8/8/2p1K3/5P2/8/8/8 w - - 0 1"]

1.f5 c4 2.Kd4 Ke7


And, of course, Black can and probably will get his king to a better position than that. This is just to demonstrate that even if he doesn't, he can still draw.

As far as when a king and pawn can win against a king and pawn, there's one position that comes to my mind, outside of the obvious pawn race scenarios. This position is a loss for whoever has the move - they must abandon their pawn and will be out of position to block the opponent's pawn:

[FEN "8/8/8/1Kp5/2Pk4/8/8/8 w - - 0 1"]