$N$ ($N\geq 2$) players numbered $1$~$N$ are betting chips. 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. Naturally, a player's payoff is their expected winning probability.

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