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 \displaystyle\frac12 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?
Answer Choices
A. 8
B. 8\frac12
C. 9
D. 9\frac12
E. 10