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=0.25/B=0.75, otherwise A=0.75/B=0.25.
This is indistinguishable from a ROB that always returns those two numbers in a consistent order.
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=0.25/B=0.75, otherwise A=0.75/B=0.25.
This is indistinguishable from a ROB that always returns those two numbers in a consistent order.