Three robots, the 1st, 2nd and 3rd seeds, are in the final round of a weightlifting championship.
How the round works.
- The robots publicly declare the weight they will attempt, one after another: first the 3rd seed, then the 2nd seed, then the 1st seed. A weight can be any nonnegative real number. No robot may declare exactly the same weight as a robot that declared before it.
- After all three weights are declared, every robot attempts its lift. The robot that successfully lifts the heaviest weight wins. If all three robots fail, they repeat the same three lifts, as many times as needed, until at least one succeeds.
- Every robot lifts a weight w successfully with the same probability p(w), independently of everything else. The function p is known exactly to every robot. It is continuous and strictly decreasing, with p(0) = 1 and p(w) → 0 as w → ∞.
- Each robot plays to maximise its own probability of winning. Every robot knows that the robots declaring after it will do the same.
Limit convention. Sometimes a robot's best prospects can only be approached, not reached, for example by declaring "a tiny bit more" than a weight already declared. In that case, treat the robot as getting that limiting win probability, and take every robot's win probability in the same limit. However, if a choice that can actually be made does exactly as well as such a limit, the robot makes that actual choice.
The situation. The 3rd seed has already declared its weight w₃, and p(w₃) = 15/64. The 2nd seed now declares its weight w₂, and the 1st seed declares last.
Question. What is p(w₂) for the weight w₂ that the 2nd seed should declare?
Answer format. Give p(w₂) as an exact fraction in lowest terms.