In a single round-robin tournament, every player faces every other player exactly once. Call a player a “boss” if every other player in the tournament either lost to the boss or lost to someone who lost to the boss (or both). Curiously, every round-robin tournament has at least one boss. Why? (Assume games cannot end in a tie.)
We’ll argue that the person who won the most games is always a boss. If multiple people share this achievement, then they’re all bosses. (In fact, other bosses with fewer wins may also exist, but we only need to establish the existence of one boss in every single round-robin tournament.)
Call the person who won the most games Serena. For Serena to be a boss, every player must have either:
1. Lost to Serena
or
2. Lost to someone who lost to Serena
We’ll show that it’s impossible for Serena to not be a boss. If a player satisfies either of the above conditions, then they don’t contradict Serena’s boss status. So for Serena to not be a boss, there would have to be another player, let’s call her Venus, who violates both of the above conditions. In other words, Venus must have both beaten Serena (violating condition 1) and beaten everybody who Serena beat (violating condition 2). If there is even a single person Serena beat that Venus didn’t, then condition 2 isn’t violated. But if Venus beat Serena and everybody who Serena beat, then Venus won more games than Serena. This is impossible because we stipulated at the start that Serena won the most games (or at least tied for most games).
To recap: Serena won the most games in the tournament, and if she weren’t a boss, then there would be some other player who won even more games, which is clearly nonsense. So Serena must be a boss.
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