15. The Prisoner's Choice

Three prisoners are given one final chance to survive. Each prisoner must simultaneously choose a positive integer.

The prisoner who chooses the smallest integer that is chosen by exactly one prisoner survives. If no such integer exists, all three prisoners die.

Assume that all three prisoners use the same mixed strategy and that each chooses their strategy to maximize their probability of survival. Determine the symmetric Nash equilibrium strategy. Derive the probability of choosing each positive integer.