It also completely depends on what kind of multiplayer game it is. If you're doing a simple turn-based card game, obviously you can run the entire game on the server, and have the clients just do input and display of the game state. If you're doing a fast-paced first-person-shooter, you need to run more logic on each game client and trust them more. Add a real-time physics engine and it gets even trickier, because now you need to make sure it's deterministic regardless of client hardware.

If you can solve the latter problem in a general way, you can easily build a company around it. There's already a bunch that do nothing but this. It's just a fundamentally hard problem because the speed of light is what it is, the distribution of computing hardware is what it is, and the distribution of players is what it is.