Here is a game for 2 players.
Needed: An n-by-n grid drawn on paper (with an n of at least 8, I suggest).
Two markers (such as coins), one for each player. The markers should be small enough to both fit in one square of the grid (possibly by stacking them).
Two pencils/pens of different colors.
The players start the game by each placing their marker in any square of the grid different from where their opponent placed her/his marker.
The players then mark the squares they start in with an X (or with different symbols, especially if the colors of their pens/pencils are similar).
A "move" consists of both players moving one square each.
Players alternate, taking turns who is the "directioner" and who is the "decider".
On a move, first the decider decides if the players will move in opposition or in synch, and announces this decision. Then the directioner decides which of the 8 directions (up, down, left, right, or diagonally) the directioner will then move, then he/she moves his/her marker.
If the decider decided that the players move in synch, then the decider must move his/her marker one square in the same direction (from his/her current position) that the directioner moved her/his piece (from the directioner's previous position).
If the decider decided that the players move in opposition, then the decider must move his/her marker one square in the exact opposite direction (from his/her current position) that the directioner moved her/his piece (from the directioner's previous position).
Based on where the players are at the beginning of a move, the directioner must pick a direction where both players stay on the board.
If any player moves his/her marker onto an empty square, then that player draws his/her symbol in the empty square. But if both players move onto the same empty square, or a player moves onto a square with a symbol already in it, then no symbol is drawn by that player on that move.
When all squares are filled up with symbols, or when one player has more symbols than the number of symbols his/her opponent has + the number of empty squares, then the game ends. The player with the most squares with his/her symbol is the winner.
A variation: (Is this more or less fun than the first version?)
The directioner picks the direction first, then the decider decides if she/he will move his/her piece in synch or in opposition. (The directioner, in this version, only moves based on whether he/she can stay on the board, then the decider must move to stay on the board.)
Note: In either variation, if for some reason the decider can't move on a move, the he/she just stays put that move.