15. There are 10 players $A_{1}, A_{2}, \cdots, A_{10}$, whose points are $9,8,7,6,5,4,3,2,1,0$, and their rankings are 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th. Now a round-robin tournament (i.e., each pair of players plays exactly one match) is held, and each match must have a winner. If the player with a higher ranking beats the player with a lower ranking, the winner gets 1 point, and the loser gets 0 points; if the player with a lower ranking beats the player with a higher ranking, the winner gets 2 points, and the loser gets 0 points. After all the matches, the total points of each player (i.e., the sum of the points obtained in this round-robin tournament and the previous points) are calculated, and the players are re-ranked based on the total points. Find the minimum value of the new champion's total points (ties in ranking are allowed).
15. The minimum cumulative score of the new champion is 12.

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 by at most 6 points, but there are only 5 players with fewer points than him at the start. Therefore, if he increases 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 by at most 7 points, but there are only 4 players with fewer points than him at the start. Therefore, if he increases 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 by at most 8 points, but there are only 3 players with fewer points than him at the start. Therefore, if he increases 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 by at most 9 points, but there are only 2 players with fewer points than him at the start. Therefore, if he increases 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 by at most 10 points, but there is only 1 player with fewer points than him at the start. Therefore, if he increases 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 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 $\mathrm{C}_{10}^{2}=45$ wins, 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 $0+2 \times 5=10$.