A, B and C are in a three way duel. Starting with A, they rotate in the order of A-B-C, each firing one shot at a time. They stand close to one another, so that each can kill one of the others or deliberately miss. They can also adopt probabilistic strategy, like shooting at an opponent with a certain miss probability, etc. A just referee will make sure the required probabilities are determined objectively by tossing dice.
If after many rounds there is still more than one players standing, the referee will randomly eliminate one player, until there's only one survivor.
Before the shooting begins, C can make a public statement, followed by B, and finally A. Specifically, a statement is a promise or threat which takes the form "if this happens, I will do this; if that happens, I will do that...". For example, C can state "if B makes no statement, I will kill them at my turn" or "if no one has being killed when it comes to my turn, I will shoot in the air" or "if A makes a statement to kill B, I will fire at A with a miss probability of 99%, else I will kill A at my turn", etc.
It is common knowledge that statements are binding and irrevocable. All players are selfish and intelligent, and will say and act to maximize their own surviving probability.
Question: What should C say to maximize their surviving probability?
More challenging questions: Suppose there're now 4 players dueling, firing in the order of ABCD and making statements in the order of DCBA. What should D say? What would E say if there're 5 players? Is there any way to tell how adding more players would affect the last person's surviving probability?