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The game $G(N,M)$ is played:

$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$. In their move, a player announces an integer $C$ and toss a fair coin: if head, $C$ more chips are added to the pot; if tail, $C$ chips are removed from the pot, where $1\leq C\leq $ the current number of chips in the pot.

The game ends on two conditions:

1. The pot is empty after a player's move, in which case that player loses the game, and everyone else wins.
2. There're $M$ or more chips in the pot, in which case everyone wins.

Communication is not allowed, and we assume a player's choice of integer $C$ is a function only of the current number of chips in the pot. Formally, a player's strategy is a function $f: \{1,2,...,M-1\} \longmapsto \{1,2,...,M-1\}$.

Question: Is there always an equilibrium for $G(N,M)$? If so, what can we say about the equilibria? Is it feasible to search for an equilibrium of, say $G(3,1000)$?

Edit: Notice that the strategy of always betting all but one chips can't be an equilibrium for many games. For example in $G(3,5)$, if the other 2 players stick to that strategy and there are 4 chips, you're better off betting 1 rather than 3.

The game $G(N,M)$ is played:

$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$. In their move, a player announces an integer $C$ and toss a fair coin: if head, $C$ more chips are added to the pot; if tail, $C$ chips are removed from the pot, where $1\leq C\leq $ the current number of chips in the pot.

The game ends on two conditions:

1. The pot is empty after a player's move, in which case that player loses the game, and everyone else wins.
2. There're $M$ or more chips in the pot, in which case everyone wins.

Communication is not allowed, and we assume a player's choice of integer $C$ is a function only of the current number of chips in the pot. Formally, in game $G(N,M)$ a player's strategy is a function $f: \{1,2,...,M-1\} \longmapsto \{1,2,...,M-1\}$, with the restriction $f(x)\leq x, \forall x$.

Question: Is there always an equilibrium for $G(N,M)$? If so, what can we say about the equilibria? Is it feasible to search for an equilibrium of, say $G(3,1000)$?

Edit: Notice that the strategy of always betting all but one chips can't be an equilibrium for many games. For example in $G(3,5)$, if the other 2 players stick to that strategy and there are 4 chips, you're better off betting 1 rather than 3.

The game $G(N,M)$ is played:

$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$. In their move, a player announces an integer $C$ and toss a fair coin: if head, $C$ more chips are added to the pot; if tail, $C$ chips are removed from the pot, where $1\leq C\leq $ the current number of chips in the pot.

The game ends on two conditions:

1. The pot is empty after a player's move, in which case that player loses the game, and everyone else wins.
2. There're $M$ or more chips in the pot, in which case everyone wins.

Communication is not allowed, so a player's choice of integer $C$ is a function only of the current number of chips in the pot. Formally, a player's strategy is a function that maps every integer $i\in [1,M-1]$ to an integer $j\in[1,i]$.

Question: Is there always an equilibrium for $G(N,M)$? If so, what can we say about the equilibria? Is it feasible to search for an equilibrium of, say $G(3,1000)$?

Edit: Notice that the strategy of always betting all but one chips can't be an equilibrium for many games. For example in $G(3,5)$, if the other 2 players stick to that strategy and there are 4 chips, you're better off betting 1 rather than 3.

The game $G(N,M)$ is played:

$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$. In their move, a player announces an integer $C$ and toss a fair coin: if head, $C$ more chips are added to the pot; if tail, $C$ chips are removed from the pot, where $1\leq C\leq $ the current number of chips in the pot.

The game ends on two conditions:

1. The pot is empty after a player's move, in which case that player loses the game, and everyone else wins.
2. There're $M$ or more chips in the pot, in which case everyone wins.

Communication is not allowed, and we assume a player's choice of integer $C$ is a function only of the current number of chips in the pot. Formally, a player's strategy is a function $f: \{1,2,...,M-1\} \longmapsto \{1,2,...,M-1\}$.

Question: Is there always an equilibrium for $G(N,M)$? If so, what can we say about the equilibria? Is it feasible to search for an equilibrium of, say $G(3,1000)$?

Edit: Notice that the strategy of always betting all but one chips can't be an equilibrium for many games. For example in $G(3,5)$, if the other 2 players stick to that strategy and there are 4 chips, you're better off betting 1 rather than 3.

The game $G(N,M)$ is played:

$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$. In their move, a player announces an integer $C$ and toss a fair coin: if head, $C$ more chips are added to the pot; if tail, $C$ chips are removed from the pot, where $1\leq C\leq $ the current number of chips in the pot.

The game ends on two conditions:

1. The pot is empty after a player's move, in which case that player loses the game, and everyone else wins.
2. There're $M$ or more chips in the pot, in which case everyone wins.

Communication is not allowed, so a player's choice of integer $C$ is a function only of the current number of chips in the pot. Formally, a player's strategy is a function that maps every integer $i\in [1,M-1]$ to an integer $j\in[1,i]$.

Question: Given $N$ and $M$, is there always an equilibrium? If so, what can we say about the equilibria?

Edit: Notice that the strategy of always betting all but one chips can't be an equilibrium for many games. For example in the game of $G(3,5)$, if the other 2 players stick to that strategy and there are 4 chips, you're better off betting 1 rather than 3.

The game $G(N,M)$ is played:

$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$. In their move, a player announces an integer $C$ and toss a fair coin: if head, $C$ more chips are added to the pot; if tail, $C$ chips are removed from the pot, where $1\leq C\leq $ the current number of chips in the pot.

The game ends on two conditions:

1. The pot is empty after a player's move, in which case that player loses the game, and everyone else wins.
2. There're $M$ or more chips in the pot, in which case everyone wins.

Communication is not allowed, so a player's choice of integer $C$ is a function only of the current number of chips in the pot. Formally, a player's strategy is a function that maps every integer $i\in [1,M-1]$ to an integer $j\in[1,i]$.

Question: Is there always an equilibrium for $G(N,M)$? If so, what can we say about the equilibria? Is it feasible to search for an equilibrium of, say $G(3,1000)$?

Edit: Notice that the strategy of always betting all but one chips can't be an equilibrium for many games. For example in $G(3,5)$, if the other 2 players stick to that strategy and there are 4 chips, you're better off betting 1 rather than 3.