Solution

The problem has been characterized as a quickie: once you hit on the right idea, the solution takes just a few lines.

There may be several players with the stated property: for any other player p, this one either beat p or beat a player who has beaten p. The nice idea that makes the problem a quickie is that the player who won most of the games has this property. Let P be such a player and G_P a group of players beaten by P. Assume there is a player q not in G_P who, in addition, has not lost to any p in G_P. Since, it's a round robin tournament, that is a tournament in which every player meets any other player, q met with P and any player from G_P. Moreover, since he has not lost to either P or any member of the G_P group, he won all those meets implying that he won more meets than P. Contradiction with the selection of P. In other words, any q that is not in G_P has been beaten by a member of the G_P group.