There is an interesting paper called The Truel. It is about 3 players A , B and C shooting under some rules.The two snippets of the pages relevant to my Question are given below, the full paper is given in the above link. My question is about the first tree, Length 1, Outcome AB. I do not follow how have arisen the outcomes in the first 5 edges in that tree 2 of them thin and next two thick and the last thin again. For the first edge A kills B and C kills A is clear. But now for the second edge A kills C why there is an outcome B; why B kills A as A is not his antagonist ? Also if A doesn't shoot (~) and B kills A why the remaining is C; i.e. the fifth edge. Which rule determines the remaining player if the shooter shoots as given ? I think that there are some concealed procedure behind the one who shoots whom which is displayed? I do not understand the rules of this.
The players’ preferences are defined in Section $4$, “Perfect Marksmanship and Additional Goals”. The answer to both of your questions is that killing the antagonist is merely the tertiary goal – the primary goal is to survive, and the secondary goal is to kill as many opponents as possible. Thus, if a player’s antagonist is already dead but the other opponent is still alive, the player will kill the other opponent.