I don't know if I'm making your idea better or worse.
But player 1 is always just as you describe. Except player 1 is on the tablet. He's inside a city, as chaos reigns down. He's trying to survive.
Player 2 is in 1st person mode, playing as godzilla. The concept is basically like hungry hungry hippos. Most people think the winner of the game is the hippo that ate the most marbles. It's not. The point is to eat the one singular yellow marble.
That's what this game could be. Godzilla trying to kill player 1.....but there's no way to identify or find player 1. So you just have to destroy the whole city, and kill every single NPC.
Player 1 can find weapons, but that would identify him as the target, since all the other NPCs are running. So if you miss, or you have a weak weapon, you expose yourself as an easy target.
Player 1 can also call in decoys. NPCs that attack, but do minimal damage. But they expose themself as the "player".