It's funny that you should mention that game. Here in Denmark we've played that game for at least 30 years. The parents of my generation claim to have played it with knives on grass.
The one that came to my mind was one we played I think around grade 2 that we simply called War. (I hope I haven't rambled on about this one before).
The idea was that you would draw a circle representing the world, divide it up equally amongst the players, name each section after a country, and take turns declaring war on each other. With everybody standing outside the circle, the kid whose turn it was to attack would say, "I declare war on..." [whatever country he'd choose to attack], and he'd would drop a stick on that section of the circle and take off running as fast as he could. I think if the stick didn't land in the country being attacked, the turn would end. But assuming the stick landed in the right section, the defender would pick up the stick and as soon as he had it, yell "stop," at which point the attacker had to freeze where he was. The defender would then throw it at the attacker and if he got hit then his attack failed. But if the defender missed the attacker got to redraw his borders taking a portion of the defender's land.
If we had any sense we probably should've used a ball of some sort instead.
Today we use a tennis ball. It's called "Jeg melder krig mod"(I declare war on):