Supposing (without loss of generality) that A > B, we have two cases:
TABI handed you A.
TABI handed you B.
Your job is to ensure that you answer “yes” more often in case 1 than in case 2. For this purpose it suffices to probabilistically answer “yes” according to some function that increases monotonically across the reals; this ensures that the probability of answering “yes” will always be higher for A than for B, given any two numbers A > B.
There are infinitely many functions with this property, of course, but without further knowledge of ROB’s algorithm, there is no way of knowing which of them performs best. However, since any such function will result in >50% accuracy, any one of them suffices as an answer to the question as posed.
I don’t think that works? Or at least I’m missing something.
I do not see how your “without loss of generality” holds. If B > A, your ‘correct’ response with a monotonically increasing function becomes an incorrect response with a monotonically decreasing function—and with two distinct random reals the probability of this happening is… 50%. (At least in cases where this is probability is even defined...)
It’s not clear to me what you’re trying to say. Do you have a concrete counterexample in mind, e.g. an algorithm for ROB to follow, alongside a monotonically increasing function of your choice, such that the math does not give a strictly >50% accuracy for any agent which replies “yes” with probability determined by the function in question?
Supposing (without loss of generality) that A > B, we have two cases:
TABI handed you A.
TABI handed you B.
Your job is to ensure that you answer “yes” more often in case 1 than in case 2. For this purpose it suffices to probabilistically answer “yes” according to some function that increases monotonically across the reals; this ensures that the probability of answering “yes” will always be higher for A than for B, given any two numbers A > B.
There are infinitely many functions with this property, of course, but without further knowledge of ROB’s algorithm, there is no way of knowing which of them performs best. However, since any such function will result in >50% accuracy, any one of them suffices as an answer to the question as posed.
I don’t think that works? Or at least I’m missing something.
I do not see how your “without loss of generality” holds. If B > A, your ‘correct’ response with a monotonically increasing function becomes an incorrect response with a monotonically decreasing function—and with two distinct random reals the probability of this happening is… 50%. (At least in cases where this is probability is even defined...)
It’s not clear to me what you’re trying to say. Do you have a concrete counterexample in mind, e.g. an algorithm for ROB to follow, alongside a monotonically increasing function of your choice, such that the math does not give a strictly >50% accuracy for any agent which replies “yes” with probability determined by the function in question?
Long story short, I misinterpreted the question. (I was thinking it was trying to predict if ROB chose A > B or A < B)
Long story short, I misinterpreted the question.