CWMO 2010 Problem 7

There are $n$ $(n \ge 3)$ players in a table tennis tournament, in which any two players have a match. Player $A$ is called not out-performed by player $B$ , if at least one of player $A$ 's losers is not a $B$ 's loser.

Determine, with proof, all possible values of $n$ , such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.