n shooters numbered 1~n play a game. Their probability of hitting a target are 0<p1<p2<p3<...<pn<1.

Before the game, the referee arranges the n players in some order. When the game starts, each player takes turns to shoot each other according to that order (For example if the order is 4,3,6,1,... then shooter 4 fires first, then shooter 3 fires, then shooter 6, ect, if they are still alive). When the last alive person in the order finished shooting, the first alive person in the order again resumes, so on and so forth according to the order. The game ends when there's only one person left. We say that hesurvives.

After the referee arranged the order, each player mustsimultaneouslydetermine his shooting priority list for the rest n-1 people except himself. He must adhere to the list throughout the game, i.e., always shoot the first alive person on that list. We call this priority list hisstrategy. Obviously, each player has (n-1)! strategies to choose. Each player wants to maximize hissurviving probability.

Question 1: Suppose the referee arranges the shooters in the order: 1,2,3,...,n. Is there a dominant strategy for shooter 1? (A dominant strategy is the best strategy for the play no matter what other plays' strategies are, i.e., it always maximizes his surviving probability, regardless of what strategies other people choose)

Question 2: Is there a Nash equilibrium for the order in Q1?

Question 3: Suppose the referee arranges the shooters in some arbitary order, is there always a dominant strategy for the first person in that order? Is there always a Nash equilibrium?

Question 4: Above is a simplistic way to specify players' strategies. When there are fewer players left, a shooter may very well want to change his shooting priorities. Thus it is more reasonable to place no restriction on who the shooter must shoot first (of course, he must shoot some person other than himself. But there's no restrictions on who, in any case). Alternatively, in this case we can think of his strategy as specifying the shooting priority list for any k people (k=2,3,4,...,n-1). How can we rethink Q1~Q3 in this case?

--------------------------------------------------------------------------------

It is a puzzle in many books when n=3. But i've been considering the general case: whether the problem remains simple or becomes intractably complex. Any suggestions or opinions are welcome.