Dedicated servers for a Battle-Royale style Game

I have created a battle royale multiplayer game that uses the Battle Royale template from the marketplace, the game will be released on steam. Now I want all the players all over the globe to connect and play matches together, what do I need to do to make this work? Do I need to set up a dedicated server and install a server copy of the game all by myself? or will Steam do the entire job for me? and is there a price that I need to pay to Steam for the servers?

To all the other developers who are going to be releasing multiplayer titles how are you going to go about the Server part?