Math Artifacts

The MathMatters competition consists of $10$ players $P_1$, $P_2$, $\dots$, $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$: the loser is ranked $10^{\text{th}}$, while the winner plays $P_8$. The loser of that game is ranked $9^{\text{th}}$, while the winner plays $P_7$.

They keep repeating this process until someone plays $P_1$: the loser of that final game is ranked $2^{\text{nd}}$, while the winner is ranked $1^{\text{st}}$.

(Source: Math Prize for Girls 2013)