There are m players guessing numbers. The answer is an integer randomly chosen from 1 to n. They guess in this order: player 1, 2, …, m, 1, …
Every turn, they can guess an integer also from 1 to n. Everyone will know whether the guessed number is equal, less or greater than the answer. Who guesses out the answer will win, and they all want to win.
Is it possible that this game will never stop? If not so, what's the winning probability for every player?
Upd: It seems that the game will never stop. So I add this: Everyone should guess a number that hasn't been guessed.
Upd 2: If there are two or more ways to maximize the winning probability, the player will randomly choose one.
I don't think there is an answer because we don't know enough to define how people will play. Take the case $n=m$ for example. Presumably the first player must guess something in range. This wins with probability $1/m$, so it is fair for the first player. The number will be guessed before it comes around to the first player again. S/he can guess $1$, giving minimal information to the rest and leaving the game fair for the second person, or guess $n/2$ (or something in between) and giving the second player twice the chance to win while eliminating any chance for all the players after $n/2$. Very often when there are more than two players it is difficult to define the strategies.