This question is posted 4 times, probably not on purpose but be careful how many times you click the submit button in future please.

If the 2nd base runner reached or touched 3rd base, then UNLESS the 3rd base runner touched home and scored, he would be out at/or before 3rd because it would be occupied by the runner that came from 2nd. SO, Runner on 2nd advances to 3rd, runner at 3rd unless he scores, OUT.

"Goose chase" - that's a term I've never heard applied to a rundown. I often hear a rundown called a "pickle" but I dislike that term. It's simply a rundown... Anyway, no one is out unless someone is tagged (unless the runner originally on second base passes the runner originally on third base, then the runner who passed the preceding runner is out by rule). Unless forced (and no one is forced in this situation), the base belongs to the preceding runner until he legally acquires the next base. The following runner has no "right" to third base in this situation. If R3 (the runner originally on third base) does not advance to home plate, third base still belongs to him. R3 has the right to return to third base and if he does, it's his. If R3 safely retreats to third, R2 needs to go somewhere so he needs to attempt to retreat to second. If R2 attempts to advance and passes R3, R2. So R2's only course of action is to go back to second base. If R2 and R3 both end up on third base at the same time, R2 is out if tagged - not R3. But R2 must be tagged. If not tagged, R2 can legally attempt to return to second. Rule 7.03 (a) Two runners may not occupy a base, but if, while the ball is alive, two runners are touching a base, the following runner shall be out when tagged and the preceding runner is entitled to the base, unless Rule 7.03(b) applies. If R2 touched and went beyond third, he must retouch third base if he attempts to return to second base. If not, he is in jeopardy of being called out on appeal (but only if the defense executes a proper appeal) for failing to retouch third base on his way back to second. Rule 7.02 In advancing, a runner shall touch first, second, third and home base in order. If forced to return, he shall retouch all bases in reverse order...

As long as neither of the runners are tagged while off a base, or while they are both standing on 3rd base, the runner that was originally on 2nd can go back to second.