Fully Managed Dedicated servers are a ideal choice to host games on it. If you do not have enough money to spend upfront, you can start with
Managed Cloud server. You can consider a Cloud server with atleast 1.5 Ghz of CPU and 1 GB of RAM initially and later on upgrade the resources as Cloud servers are highly scalable and its resources can be increased at any point of time. This will indeed turn out to be a better solution than VPS.