A tournament is defined as a contest among $n$ players, all of whom compete against each other except for one pair. In each contest between a pair, one individual wins. A tournament winner is defined as the player who, for each other player, either won his game against that player, or won a game against a player who in turn won his game against that player (or both).
Prove that for any $n\geq 2$, there is a tournament with n players and no tournament winners.
My thoughts: I believe that if we consider the two players who don't play each other and they each beat everyone else and everyone else's matchups are in any legal fashion, an arrangement like this qualifies for any sized tournament. Is my intuition correct? I don't need help with the proof but merely would like to know whether this is a valid scenario that I describe.