Diagon -- The Game

Here is a game for any plural number of players.

Start with an n-by-n grid drawn on paper, where n is an odd positive integer.

Fill in the 4 corner squares of the grid.

Players take turns filling in grid-squares, one square per move. On a move a player fills in any empty square that is adjacent to and immediately above, below, right-of, or left-of any filled-in square.

A player gets a point when that player fills in a square such that that square is in a 2-by-2 group of squares where two diagonally adjacent squares are filled in (including the square just filled in) and the other two diagonally-adjacent squares are empty (empty just after the move when the point is scored).

For instance, a point is scored if we have a 2-by-2 group of squares that looks like this:
* o
o *

or this:
o *
* o

(o = empty square, * = filled in square, where one of the filled in squares is the square just filled in by the scoring player.)

The game continues until there is no possibility that any more points can be scored.

Highest score wins.

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If during the game we have a situation like so, say:
o *
* o
o *
(newly filled in square is the middle filled-in square), then how many points has the player scored? One or two?I leave how many points that can be scored on any single move, one at most or more, be up to the players to agree to among themselves.

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Question: Is there a simple way for one player, say the first or second to move in a 2-person game, to always win? (If there is, I probably should edit this game to eliminate the possibility of using the simple strategy.)

Thanks,
Leroy Quet