This is an extension of the well known "3 gunmen puzzle":
N gunmen with hitting probabilities in (0,1] take turns to shoot at each other. Firing order is fixed (gunman 1 shoots first, then gunman 2 shoots, etc, as long as they are still alive). They repeat this process until only one survives. A gunman's goal is to maximize his surviving probability. Hitting probabilities are common knowledge.
To be specific, consider 10 gunmen with hitting probabilities 0.1, 0.2, ..., 0.9, 1. Who are they going to shoot when all are still alive? How should we formulate the problem so it can be solved by programming?