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You consider the Randomized Yudkowsky-Kasparov system, but this is only one point on a continuum. There are stupider and smarter systems, generated by weaker and stronger chess players, respectively.

One interesting system is Randomized Kasparov-Kasparov, where we show arbitrary chess positions to Kasparov and ask him to assign probability distributions to what he'd do. Chess is a game of perfect information, so arbitrary chess positions can be shown without their history (of course, we have to count physically invisible state as known; a position consists of both the arrangement of the pieces and factoids like "the king has not moved, but the queen's rook has moved, so queenside castling is forbidden but kingside castling is allowed"). I assert without proof that RKK is an extremely smart system, and will turn both RYK and you yourself into chunky salsa.

Here's the question: is Kasparov smarter than RKK? We can imagine that first we develop RKK by properly incentivizing Kasparov (e.g. by telling him that we're going to use it to play against every human on the planet, and for every time that RKK comes out victorious, we'll donate a dollar to his favorite charity, or if RKK loses even once, we'll drown a kitten, or whatever), and then (after we've shown Kasparov the million zillion chess positions and recorded his assigned probability distributions) we turn the tables and properly incentivize him to beat RKK (charity donation, kitten drowning, whatever). Can Kasparov beat RKK? Does it matter if he knows he's playing against RKK?

Then the metaquestion: what if Kasparov was a weaker player? (For values of Kasparov that equal Yudkowsky...) What if he was a stronger player? One limiting case is easy. If Kasparov was very very smart, he could solve chess, and so RKK could solve chess, and they would be equally strong. (That is, if the solution says that White wins, then whoever plays White wins, and so forth.) There's a probability distribution that solves chess, after all (with 100% for the right move in any given situation and 0% for the others).