A plays B, say A wins by symmetry.  C plays D, say C wins by symmetry.  A plays C, say A wins by symmetry.
A plays C again.
IF A WINS, D plays E
  IF D WINS, A plays D again
    IF A WINS, A plays E
      IF A WINS, B plays D, and the winner plays A 3 times and needs to win them all
      IF E WINS, E plays A 4 times (A wins ties)
    IF D WINS, A plays D twice more
      IF A WINS BOTH, E plays A 3 times and needs to win them all
      IF THEY SPLIT, A plays D best-of-3
      IF D WINS BOTH, B plays C, and winner plays D twice (D wins ties)
  IF E WINS, A plays E twice
    IF A WINS BOTH, B plays D, and winner plays A 3 times and needs to win them all
    IF THEY SPLIT, E plays A 4 times (A wins ties)
    IF E WINS BOTH, A plays E again
      IF A WINS, A plays E best-of-3
      IF E WINS, B plays E 3 times and needs to win them all
IF C WINS, A plays C again
  IF A WINS, C plays E
    IF C WINS, A plays C twice more
      IF A WINS BOTH, D plays E, and winner plays A twice (A wins ties)
      IF THEY SPLIT, A plays C best-of-3
      IF C WINS BOTH, A plays B, and winner plays C twice (C wins ties)
    IF E WINS, A plays E twice
      IF A WINS BOTH, D plays A 3 times and needs to win them all
      IF THEY SPLIT, A plays E best-of-3
      IF E WINS BOTH, A plays D, and winner plays E twice (E wins ties)
  IF C WINS, A plays E
    IF A WINS, A plays C twice
      IF A WINS BOTH, C plays D, and winner plays A twice (A wins ties)
      IF THEY SPLIT, A plays C best-of-3
      IF C WINS BOTH, B plays E, and winner plays C twice (C wins ties)
    IF E WINS, C plays E twice
      IF C WINS BOTH, B plays C 3 times and needs to win them all
      IF THEY SPLIT, C plays E best-of-3
      IF E WINS BOTH, B plays C, and winner plays E (E wins ties)