The intended answer for this problem is the Frequentist Heresy in which ROB’s decisions are treated as nonrandom even though they are unknown, while the output of our own RNG is treated as random, even though we know exactly what it is, because it was the output of some ‘random process’.
Instead, use the Bayesian method. Let P({a,b}) be your prior for ROB’s choice of numbers. Let x be the number TABI gives you. Compute P({a,b}|x) using Bayes’ Theorem. From this you can calculate P(x=max(a,b)|x). Say that you have the highest number if this is over 1⁄2, and the lowest number otherwise. This is guaranteed to succeed more than 1⁄2 of the time, and to be optimal given your state of knowledge about ROB.
This might be “optimal given your state of knowledge about ROB” but it isn’t “guaranteed to succeed more than 1⁄2 of the time”. For example:
My prior is that Rob always picks a positive and a negative number.
Actually Rob always picks two positive numbers.
Thus when we play the game I always observe a positive number, guess that it is the larger number and have a 0.5 probability of winning. 0.5 is not greater than 0.5
The intended answer for this problem is the Frequentist Heresy in which ROB’s decisions are treated as nonrandom even though they are unknown, while the output of our own RNG is treated as random, even though we know exactly what it is, because it was the output of some ‘random process’.
Instead, use the Bayesian method. Let P({a,b}) be your prior for ROB’s choice of numbers. Let x be the number TABI gives you. Compute P({a,b}|x) using Bayes’ Theorem. From this you can calculate P(x=max(a,b)|x). Say that you have the highest number if this is over 1⁄2, and the lowest number otherwise. This is guaranteed to succeed more than 1⁄2 of the time, and to be optimal given your state of knowledge about ROB.
This might be “optimal given your state of knowledge about ROB” but it isn’t “guaranteed to succeed more than 1⁄2 of the time”. For example:
My prior is that Rob always picks a positive and a negative number.
Actually Rob always picks two positive numbers.
Thus when we play the game I always observe a positive number, guess that it is the larger number and have a 0.5 probability of winning. 0.5 is not greater than 0.5
Right, I should have chosen a more Bayesian way to say it, like ‘suceeds with probability greater than 1/2’.