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$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or lower than the average of the remaining players. All who guess wrongly are eliminated before the next round starts.

Assumptions:

• Players only know their own number, and not anyone else’s.

• Players are myopic and play only to optimise their survival probability in the present round.

• Players all follow an optimal strategy.

• The game goes on until only one player remains.

Question: What is the expected average of the remaining players after $n$ rounds if

• The players are not given any information on the actions of other players, nor how many players remain?

• The players are given full information on the actions of other players in previous rounds and subsequent eliminations?

Note: If all players are eliminated, then the expected value of the average defaults to the last round in which there were surviving players.

$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or lower than the average of the remaining players. All who guess wrongly are eliminated before the next round starts.

Assumptions:

• Players only know their own number, and not anyone else’s.

• Players are myopic and play only to optimise their survival probability in the present round.

• Players all follow an optimal strategy.

• The players are given full information on the actions of other players in previous rounds and subsequent eliminations.

Without any analysis, we know that the optimal strategy is to guess "higher" if one's number exceeds a certain value depending on the information available to the player so far.

Question: What is the optimal strategy?

$N \geq 2$ players play a game - they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or lower than the average of the remaining players. All who guess wrongly are eliminated before the next round starts.

Assumptions:

• Players only know their own number, and not anyone else’s.

• Players are myopic and play only to optimise their survival probability in the present round.

• Players all follow an optimal strategy.

• The game goes on until only one player remains.

Question: What is the expected average of the remaining players after $n$ rounds if

• The players are not given any information on the actions of other players, nor how many players remain?

• The players are given full information on the actions of other players in previous rounds and subsequent eliminations?

Note: If all players are eliminated, then the expected value of the average defaults to the last round in which there were surviving players.

$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or lower than the average of the remaining players. All who guess wrongly are eliminated before the next round starts.

Assumptions:

• Players only know their own number, and not anyone else’s.

• Players are myopic and play only to optimise their survival probability in the present round.

• Players all follow an optimal strategy.

• The game goes on until only one player remains.

Question: What is the expected average of the remaining players after $n$ rounds if

• The players are not given any information on the actions of other players, nor how many players remain?

• The players are given full information on the actions of other players in previous rounds and subsequent eliminations?

Note: If all players are eliminated, then the expected value of the average defaults to the last round in which there were surviving players.

$N \geq 2$ players play a game - they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or lower than the average of the remaining players. All who guess wrongly are eliminated before the next round starts.

Assumptions:

• Players only know their own number, and not anyone else’s.

• Players are myopic and play only to optimise their survival probability in the present round.

• Players all follow an optimal strategy.

• The game goes on until only one player remains.

Question: What is the expected average of the remaining players after $n$ rounds if

• The players are not given any information on the actions of other players, nor how many players remain?

• The players are given full information on the actions of other players in previous rounds and subsequent eliminations?

$N \geq 2$ players play a game - they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or lower than the average of the remaining players. All who guess wrongly are eliminated before the next round starts.

Assumptions:

• Players only know their own number, and not anyone else’s.

• Players are myopic and play only to optimise their survival probability in the present round.

• Players all follow an optimal strategy.

• The game goes on until only one player remains.

Question: What is the expected average of the remaining players after $n$ rounds if

• The players are not given any information on the actions of other players, nor how many players remain?

• The players are given full information on the actions of other players in previous rounds and subsequent eliminations?

Note: If all players are eliminated, then the expected value of the average defaults to the last round in which there were surviving players.