I have a round-based Multiplayer game for Android that has been working over XMPP before, and I want to switch to Google Play Game Services. In the old version, there was an
This is a standard issue of selecting a leader from a bunch of unorganized nodes. There's some fairly advanced ways to do it, I'm sure. If you want to check out: Automatic selection of a leader in a cluster of nodes and hte Paxos algorithm.
Now...that being said, since you have a limit of 4 people, theres easier ways to do it. You could just treat their ID's as a number, and who ever has the highest ID is player 1, and descending in order. That would probably be the simplest way.
You could also have all the players submit a random roll and broad cast it. In the extremely rare case of a duplicate, you can have a reroll. Once everyone has made their rolls, and everyone has the same data, each individual client can determine the player order based on the value of the rolls.
I wouldn't do anything related to the order in the array however. I don't believe that's guaranteed in any circumstance.