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God cannot randomly select something.
He can't draw a piece of paper randomly from a hat, or pick a random card. Any time he picks up the dice, he knows what he will roll. If he gives the dice an extra shake, he still knows the outcome, and already knew that he would give the extra shake.
God is incapable of playing dice.
With this in mind, I'm not sure anyone omniscient could play a game, as they all rely on the unknown.
A game of tag where someone knows they can outrun everyone forever isn't fair. A game of snakes and ladders where someone decides what they will roll, and agrees to the game only after learning the outcome, seems like a breach of the implied rules.
God cannot play games.