This is an interesting variant. Like in checkers, the idea is to capture all the other player's pieces except for the king, (K v. K endgames O_o) and if a caupture is possible, it MUST be made. One last rule to avoid absolute manslaughters: A piece that has not moved can NOT be captured. :)

I have played a similar game as this before, and White definitely has the advantage.  It is a fun exercise. Although my friend and I just thought of playing this way off the top of our heads with no rules other than normal chess rules. I have heard of a similar variation, only it is called Dead-Drawn Game. But in that game there is no rule about not being able to capture a piece that has not been moved.

Have you heard of Prohibition Chess?  I have never tried it.  Anyway, the only rule that is different from normal chess is: no check is allowed except the giving of a checkmate. 

The "not moved - not captured" rule is basically there to prevent the queens, and a rook, knight, and bishop of each color from being captured in the first 5 moves :P