2015 Cayley Problem 22

Six players compete in a chess tournament. Each player plays exactly two games against every other player. In each game, the winning player earns 1 point and the losing player earns 0 points; if the game results in a draw (tie), each player earns \frac{1}{2} point. What is the minimum possible number of points that a player needs to earn in order to guarantee that he has more points than every other player?

\textbf{(A)}\ 8\quad \textbf{(B)}\ 8\frac{1}{2}\quad \textbf{(C)}\ 9\quad \textbf{(D)}\ 9\frac{1}{2}\quad \textbf{(E)}\ 10

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.