Let's frame this in the simplest terms:
You can only 'win' if you know something about the mean of the probability distribution that produced the numbers. As we say in physics, this breaks the symmetry of the problem. You have more information that just the value of one number, and from there you can compare your number with the mean. If your number is higher than the mean, than you are likely to have the higher number. Otherwise it is likely to be the lower.
If you have a uniform probability distribution between + and  infinity, one could argue that the mean is zero. Therefore, if the number you viewed is positive, it is likely the higher one and vice versa.
However, I don't feel comfortable saying it is the mean  too many infinities are involved. It depends on your definition of zero and infinity. A good definition of zero is that it is the mean of integers, rational and/or irrational numbers.
A bell curve, although infinite, has a definite mean and finite integral. The probability is zero at +/ infinity, so there are fewer affronts to nature.

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