It depends on the genre and the specific game. RTSs can do it this way and some fighting games do, although some fighting games use a variation that involves pr

It depends on the genre and the specific game. RTSs can do it this way and some fighting games do, although some fighting games use a variation that involves predicting player input and rewinding the game state on misprediction in order to reduce perceived latency. FPSs tend to use a totally different model that involves sending only the most recent relevant information to each player and trying to balance server authoritative cheat-prevention with allowing clients a little leeway in order to make the twitchy gameplay feel better. There's a great paper on Tribes' networking model that is still largely relevant for FPSs.