No, not every multiplayer game needs an authoritive server. It depends largely where your priorities are. For myself, having a base of players with some cheaters would be a great problem to have.
Yeah but, the solution to your great problem is to write the authoritative server, so I would recommend you just do that in the first place. It's not really any extra work if you are starting a new project. (An authoritative server is just a headless client after all :) )
