I have a multiplayer browser based quiz game built on php, javascript and MySQL. If I place load balancer and autoscaler to scale my game then I am concerned about how below scenario will be handled:
Basically what will be good option to maintain game and player state across different servers.
You can have different HTTP servers (for PHP and Javascript) without problems, but you need to have a single MySQL database. Then, it doesn't matter which PHP server is connected to the player, all servers just need to get the data from the same database.
If you need to have different servers also for the database I suggest you to look at MySQL Clustering.