If you made random moves in a game of chess or checkers you'd get beat by a four year old with the barest grasp of the rules. In fact I would love to see a chess-playing robot that actually does that, it would thrill the crap out of my nephews to beat the big bad robot in a game of wits.
I'm unaware of the name of the property that RPS exhibits that makes it ungameable by a random opponent (zero-sum?) but ordinary RPS played by humans certainly doesn't exhibit it, only in the magic world of computers where things like simultaneity of play can be guaranteed does RPS exhibit that property.
If played in the real world like real humans traditionally played it, ongoing RPS matches between a human and an AI would soon see the AI dominating. I'd be very curious to see what real-world RPS would look like between two AIs that can 'explain their model'.
Imagine making a rule that gives a Xms window for making plays. Then strategy can start to emerge.