11. (10 points) Two people are playing a table tennis match, a best-of-three games, where the first to 11 points and the opponent has fewer than 10 points wins the game, or after a 10-all tie, the first to gain a 2-point lead wins. The total score of both players is 31 points, and one player won the first game and the match. How many possible scores are there for the second game?
【Analysis】For the sake of explanation, let's set the winner as Party A and the loser as Party B. Party A wins the first round. If Party A wins the second round again, then Party A will definitely have a higher total score, which does not meet the problem's requirement, so Party A loses the second round and wins the third round. Since the maximum score difference to win each round is 2 points, Party A winning the first and third rounds means that Party A will have at least $2+2=4$ points more than Party B in the first and third rounds. Therefore, Party B must win the second round by at least 4 points, so Party B can only win with 11 points, otherwise it would not comply with the game rules. Then we can discuss the possible scores of the second round.

【Solution】Let the winner be Party A and the loser be Party B.
Party A wins the first round. If Party A wins the second round again, then Party A will definitely have a higher total score, which does not meet the problem's requirement, so Party A loses the second round and wins the third round.

Party A winning the first and third rounds means that Party A will have at least $2+2=4$ points more than Party B in the first and third rounds, so Party B must win the second round by at least 4 points, so Party B can only win with 11 points.

Therefore, the score of the second round can be: $0: 11, 1: 11, 2: 11, \cdots 7: 11$; a total of 8 possibilities. (No matter how many points Party B wins in the second round, Party A can make up the points by winning the first and third rounds to make the total scores equal, so Party A can score from $0 \sim 7$ in the second round; for example, the scores of the three rounds can be 20: 18, 0: 11, 11: 2). Answer: There are 8 possible scores for the second round.