"1million simultaneous players"
- are those 1 mil req / sec or 1 mil req / minute?
"Please recommend the best hosting system for the API."
- without seeing the API, i don't think anybody can recommend the best one... maybe suggest a good provider for servers (location, budget?)
- couple of servers to handle that load with 2x HW load balancer in front
- standard mysql scalability issue: one master + multi slave (look into percona)
- multi php nodes
- optimize the api
- optimize the mysql indexes
- and don't forget: optimize the api :-)