Flips a fair coin (under the same “this is a quick way to say ‘the ROB has a private random oracle’ assumptions as the arbiter has and game-theory in general requires”). If heads: A=0/B=1, otherwise A=1/B=0.
By a symmetry argument no agent can beat this ROB more than 50% of the time. (Any gains when the ROB coinflip is heads are balanced by an equivalent loss when the ROB coinflip is tails, and vice versa.)
(Regardless of what might happen in weird infinite distributions, it doesn’t really matter. This ROB should be a sufficient counterexample, and doesn’t do anything fancy.)
Consider a ROB that does the following:
Flips a fair coin (under the same “this is a quick way to say ‘the ROB has a private random oracle’ assumptions as the arbiter has and game-theory in general requires”). If heads: A=0/B=1, otherwise A=1/B=0.
By a symmetry argument no agent can beat this ROB more than 50% of the time. (Any gains when the ROB coinflip is heads are balanced by an equivalent loss when the ROB coinflip is tails, and vice versa.)
(Regardless of what might happen in weird infinite distributions, it doesn’t really matter. This ROB should be a sufficient counterexample, and doesn’t do anything fancy.)
If I am following, it seems like an agent which says “bet ‘higher’ if positive and ‘lower’ otherwise” does well
The goal is not to guess which of A and B is higher, but to guess whether the number TABI gives you is higher than the other one.
ROB’s choice of order for A and B is irrelevant.
“Please use spoiler tags liberally” Spoiler please?
And yes, I misinterpreted the question.