15 If the new champion's score does not exceed 11 points, then $A_{1}$ can win at most 2 games; $A_{2}$ can win at most 3 games; $A_{3}$ can win at most 4 games; $A_{4}$ can win at most 5 games; $A_{5}$ can increase his score by at most 6 points, but there are only 5 players with fewer points than him at the start. Therefore, if he increases his score by 6 points, he must win at least 1 game against players ranked higher than him, meaning he can win at most 4 games against players ranked lower, thus he can win at most 5 games; $A_{6}$ can increase his score by at most 7 points, but there are only 4 players with fewer points than him at the start. Therefore, if he increases his score by 7 points, he must win at least 2 games against players ranked higher than him, meaning he can win at most 3 games against players ranked lower, thus he can win at most 5 games; $A_{7}$ can increase his score by at most 8 points, but there are only 3 players with fewer points than him at the start. Therefore, if he increases his score by 8 points, he must win at least 3 games against players ranked higher than him, meaning he can win at most 2 games against players ranked lower, thus he can win at most 5 games; $A_{8}$ can increase his score by at most 9 points, but there are only 2 players with fewer points than him at the start. Therefore, if he increases his score by 9 points, he must win at least 4 games against players ranked higher than him, meaning he can win at most 1 game against players ranked lower, thus he can win at most 5 games; $A_{9}$ can increase his score by at most 10 points, but there is only 1 player with fewer points than him at the start. Therefore, if he increases his score by 10 points, he must win at least 5 games against players ranked higher than him, thus he can win at most 5 games; $A_{10}$ can increase his score by at most 11 points, and he can win at most 5 games against players ranked higher than him, thus he can win at most 5 games.
In summary, the maximum number of games won by all players is $2+3+4+5 \times 7=44$, but each match between two players results in one win, totaling $\mathrm{C}_{10}^{2}=45$ (games), which is a contradiction.
The following example shows that the new champion's cumulative score can be 12 points.
$A_{1}$ wins against $A_{2}, A_{3}, A_{4}$, loses to $A_{5}, A_{6}, A_{7}, A_{8}, A_{9}, A_{10}$, with a cumulative score of $9+3=12$;
$A_{2}$ wins against $A_{3}, A_{4}, A_{5}, A_{6}$, loses to $A_{7}, A_{8}, A_{9}, A_{10}$, with a cumulative score of $8+4=12$;
$A_{3}$ wins against $A_{4}, A_{5}, A_{6}, A_{7}$, loses to $A_{8}, A_{9}, A_{10}$, with a cumulative score of $7+4=11$;
$A_{4}$ wins against $A_{5}, A_{6}, A_{7}, A_{8}$, loses to $A_{9}, A_{10}$, with a cumulative score of $6+4=10$;
$A_{5}$ wins against $A_{6}, A_{7}, A_{8}, A_{9}$, loses to $A_{10}$, with a cumulative score of $5+2+4=11$;
$A_{6}$ wins against $A_{7}, A_{8}, A_{9}, A_{10}$, with a cumulative score of $4+2+4=10$;
$A_{7}$ wins against $A_{8}, A_{9}, A_{10}$, with a cumulative score of $3+2 \times 2+3=10$;
$A_{8}$ wins against $A_{9}, A_{10}$, with a cumulative score of $2+2 \times 3+2=10$;
$A_{9}$ wins against $A_{10}$, with a cumulative score of $1+2 \times 4+1=10$;
$A_{10}$ has a cumulative score of $0+2 \times 5=10$.
15 There are 10 players $A_{1}, A_{2}, \cdots, A_{10}$, their points are $9, 8, 7, 6, 5, 4, 3, 2, 1, 0$, and their ranks are 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th. Now a round-robin tournament is held, that is, each pair of players will play exactly one match, and each match must have a winner. If the player with a higher rank wins the player with a lower rank, the winner gets 1 point, and the loser gets 0 points; if the player with a lower rank wins the player with a higher rank, the winner gets 2 points, and the loser gets 0 points. After all the matches, the total points of each player (the sum of the points obtained in this round-robin tournament and the previous points) are calculated, and the players are re-ranked according to the total points. Find the minimum value of the new champion's total points (ties are allowed).