# Random Point on Infinite Line Paradox

I've invented a paradox, or at least I think I have. Here is how it goes:

On an infinite line, a point is placed at random. You start at point 0 on the line, and your job is to find the point, but you can only recognize that you have found it by standing on it. You can only move left or right along the line.

Logically, you would want to go the greatest distance possible in either direction before turning around, as that would prevent backtracking the best. But you do intend to eventually find the point given infinite time, which would be impossible if you only go in one direction for infinity- turning around is a necessity. So you have guaranteed inefficiency, that is, you want to go as far as you possibly can before turning around but you can not go as far as you can, as that would strip you of the assurance of finding the point, cutting the odds down to 50%.

So the question is: have I really invented this paradox. More specifically, is this a paradox and has it been thought of before?

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Is the line discrete or continuous (i.e. is it $\mathbb Z$ or $\mathbb R$)? – Asaf Karagila Jan 31 at 0:40
Why would you want to prevent backtracking? Do you get paid more if you find the point faster? If so, what is the payoff as a function of the time you take? – Trevor Wilson Jan 31 at 0:42
On an infinite line, a point is placed at random... End of the road. – Did Jan 31 at 0:44
Even with the very naive formulation, what exactly is the paradox??? – Ittay Weiss Jan 31 at 0:53
If the distribution is unspecified, part of the problem does have a meaning: "An arbitrary point on the real line is chosen. Is there a strategy that is guaranteed to find the point?" But then you want to ask the question, "Which is the most efficient strategy?", for which you have to define efficiency, perhaps in terms of the expected time to find the point. And that has no meaning unless the distribution is specified. – Rahul Narain Jan 31 at 1:00

Let's consider the discrete version, because it is less complicated. It is only a paradox if you consider the following two assumptions to be intuitively correct:

(1) The distribution of the point's placement is uniform over all of $\mathbb{Z}$.

(2) Given a strategy (e.g. "always move left" or "always move right",) if for every finite time interval, that strategy maximizes the chance of finding the point within that time interval, then that strategy maximizes the chance of finding the point eventually.

Assume without loss of generality that you start at the origin, and begin by moving one step to the right. If you haven't found the point at time $n$, all you know is that it is somewhere outside the bounded interval that you've explored. If the distribution were uniform, you'd conclude that it was equally likely to be to the left or to the right. You'd rather find it sooner than later, so for any given $N$ to maximize your chance of finding it before time $N$ the best strategy is to keep moving to the right. By (2) this strategy would maximize your chance of finding the point eventually, but it doesn't because it makes the chance equal to 1/2 whereas the "zig-zag" strategy makes the chance equal to 1.

Therefore at least one of (1) and (2) must fail. In the usual formulation of "chance" as probability, (2) holds but (1) fails. One could imagine some other formulation of "chance" that allows (1), but then (2) or some unstated assumption would have to fail. Whether this is a paradox depends on your intuition regarding (1) and (2) and the other unstated assumptions.

I'm sure that people have considered problems like this before (I have) but I'm not sure whether anyone has thought of them as paradoxes.

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(1) is wrong right off the bat because there is no uniform distribution over $\mathbb Z$. – Rahul Narain Jan 31 at 1:04
Correct. That doesn't address the question of whether it is a paradox. – Trevor Wilson Jan 31 at 1:05
For example, the Banach--Tarski paradox is a paradox because many people start with the intuition that equi-decomposability preserves Lebesgue measure. This turns out to be false. But instead of just saying "Theorem: equi-decomposability does not preserve Lebesgue measure" we call it a paradox. – Trevor Wilson Jan 31 at 1:06