A plays B.  call the winner A by symmetry.
A plays C 
IF A WINS, A plays D
   IF A WINS, A plays E 4 times (A wins ties)
   IF D WINS, A plays D twice more
      IF SAME WINNER (say A) BOTH TIMES, 
         A plays E twice (E loses ties)
      IF DIFFERENT WINNERS, 
         A plays E and winner plays D
IF C WINS, A plays C again
   IF A WINS, A plays C again
      IF A WINS, A plays D, A plays E, winners play
      IF C WINS, A plays D
         IF A WINS, A plays E and winner plays C
         IF D WINS, C plays E and winner plays D
   IF C WINS, D plays E, and winner plays C best-of-3