MTG Rules Question

[Androgeus]

I think this is the applicable rule

800.4a When a player leaves the game, all objects (see rule 109) owned by that player leave the
game, any effects which give that player control of any objects or players end, and all spells and
abilities controlled by that player on the stack cease to exist. Then, if there are any objects still
controlled by that player, those objects are exiled. This is not a state-based action. It happens as
soon as the player leaves the game. If the player who left the game had priority at the time he or
she left, priority passes to the next player in turn order who’s still in the game.

So Keiga's control effect ends as soon as Michael loses, and as Adam is the Arbiter's default controller and the only control changing effect affecting it is Adam's Reins of Power, the Arbiter will stay in play and under Adam's control after the end of turn.