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7 "dominant strategy" in Question 2 modified to "fixed optimal strategy".; added 4 characters in body

n players numbered 1~n play a shooting game. Their accuracy rates p1~pn are strictly between 0 and 1, and strictly increases from p1 to pn. This is common knowledge.

Before the game starts, the referee arranges the n players in some order. When game starts, players take turns to fire at one another according to that order. (For example, if n=4, and the referee arranges the players in order (3,4,1,2), when game starts, 3 fires first, then 4 fires, then 1, then 2, then 3 again, so on and so forth, as long as they are alive) The last person left is the winner of the game.

To be more specific, define $S_{i}$={1,2,3,...,i-1,i+1,i+2,...,n}. Let $S_{i}^{k}$ be any subset of $S_{i}$ that contains k elements(n>k>0). Then the strategy of player i is a function that maps $S_{i}^{k}$ to one of its element, for any $S_{i}^{k}$. What this definition means is that, given any k players (excluding i himself) left, player i's strategy tells him whom to shoot first. Notice that we rule out the possibility that any player can hold fire in his turn: he must choose someone to shoot.

After the referee arranges the firing order, all players must announce their strategies simultaneously. A player's payoff is his winning probability.

Question 1: Is there always an Nash equilibrium in this game, for any firing order?

Question 2: Suppose firing order is (1,2,3,...,n). Which players have dominant fixed optimal strategies for with respect to changes in (0,0,...,0)<(p1,p2,...,pn)<(1,1,...,1)? p1,p2,...,pn)? (When n=3, all have dominant fixed optimal strategies; when n=4, player 1,2 and 4 have dominant fixed optimal strategies) Furthermore, these fixed optimal strategies are intuitive and simple in the sense that they always instruct the player to fire at the most accurate person alive. I guess there could be regularities as n gets larger? At least can we say for player 1 this strategy is always has optimal?

EDIT: "dominant strategy? )" in Question 2 is changed to "fixed optimal strategy with respect to changes in probabilities", which is more appropriate.

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n players numbered 1~n play a shooting game. Their accuracy rates p1~pn are strictly between 0 and 1, and strictly increases from player 1 p1 to npn. This is common knowledge.

Before the game starts, the referee arranges the n players in some order. When game starts, players take turns to fire at one another according to that order. (For example, if n=4, and the referee arranges the players in order (3,4,1,2), when game starts, 3 fires first, then 4 fires, then 1, then 2, then 3 again, so on and so forth, as long as they are alive) The last person left is the winner of the game.

To be more specific, define $S_{i}$={1,2,3,...,i-1,i+1,i+2,...,n}. Let $S_{i}^{k}$ be any subset of $S_{i}$ that contains k elements(n>k>0). Then the strategy of player i is a function that maps $S_{i}^{k}$ to one of its elmentelement, for any $S_{i}^{k}$. What this definition means is that, given any k players (excluding i himself) left, player i's strategy tells him whom to shoot first. Notice that we rule out the possibility that any player can hold fire in his turn: he must choose someone to shoot.

After the referee arranges the firing order, all players must announce their strategies simultaneously. A player's payoff is his winning probability.

Question 1: Is there always an Nash equilibrium in this game, for any firing order?

Question 2: Suppose firing order is (1,2,3,...,n). Which players have dominant strategies for (0,0,...,0)<(p1,p2,...,pn)<(1,1,...,1)? (When n=3, all have dominant strategies; when n=4, player 1,2 and 4 have dominant strategies. I guess there could be regularities as n gets larger? At least can we say 1 always has dominant strategy? )

5 all symbols behind "<" sign are invisible. I have to avoid using "<". Don't know what's wrong.

n players numbered 1~n play a shooting game. Their accuracy rates are strictly between 0 and 1, and strictly increases from player 1 to n. This is common knowledge.

Before the game starts, the referee arranges the n players in some order. When game starts, players take turns to fire at one another according to that order. (For example, if n=4, and the referee arranges the players in order (3,4,1,2), when game starts, 3 fires first, then 4 fires, then 1, then 2, then 3 again, so on and so forth, as long as they are alive) The last person left is the winner of the game.

To be more specific, define $S_{i}$={1,2,3,...,i-1,i+1,i+2,...,n}. Let $S_{i}^{k}$ be any subset of $S_{i}$ that contains k elements(n>k>0). Then the strategy of player i is a function that maps $S_{i}^{k}$ to one of its elment, for any $S_{i}^{k}$. What this definition means is that, given any k players (excluding i himself) left, player i's strategy tells him whom to shoot first. Notice that we rule out the possibility that any player can hold fire in his turn: he must choose someone to shoot.

After the referee arranges the firing order, all players must announce their strategies simultaneously. A player's payoff is his winning probability.

Question 1: Is there always an Nash equilibrium in this game, for any firing order?

Question 2: Suppose firing order is (1,2,3,...,n). Which players have dominant strategies for (0,0,...,0)<(p1,p2,...,pn)<(1,1,...,1)? (When n=3, all have dominant strategies; when n=4, player 1,2 and 4 have dominant strategies. I guess there could be regularities as n gets larger? At least can we say 1 always has dominant strategy? )

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