Guess The # Of Unfilled Squares

Here is asimple and unoriginal, but possibly fun, game.
Any number of players >= 2.
Start with an n-by-n grid drawn on paper.
(I suggest an n of at least 5.)
Before play each player *secretly* picks a positive integer.
Each player then writes down his/her number.
Starting in the upper left square, players take turns filling in
one square each move such that:
* the filled-in square is next to the square filled-in most recently
by the previous player, and the filled-in square is immediately either
above, below, left of, or right of the square most recently filled-in
by the previous player.
* the filled-in square was not already filled-in by any player
in an earlier move.
Play continues until there are no unfilled squares immediately adjacent
to the last square filled-in.
When the game is over (when every square next to last filled-in square
is already filled-in), the winner(s) is the player whose secretly
picked number (each of which is now revealed) is closest to the number
of squares of the grid which did *not* get filled-in.
So, players might try to guess, by observing which squares their opponents
fill-in, what their opponents secret integers are.
(For instance, is their opponent trying to end the game early, attempting
to continue the game, bluffing,...?)
Sample completed game:
(Numbers are filled-in squares. *'s are empty squares.)
1 2 * * *
4 3 101112
5 8 9 1413
6 7 1815*
* * 1716*
Player whose number is closest to 7 (the number of *'s) wins.
thanks,
Leroy Quet