Battle Royale

Chef invented a new game named Quick Battle Royale. $N$ people play the game. One or more rounds are played until all players except one are eliminated. In each round: - Let there be $x$ non-eliminated players at the start of the round. Let's number them $1$ through $x$. - First, each of these players chooses an integer between $1$ and $x$ inclusive. For each valid $i$, let's denote the integer ch