15. There are 10 players $A_{1}, A_{2}, \cdots, A_{10}$, whose initial points are $9,8,7,6,5,4,3,2,1,0$, and their initial rankings are 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th. Now a round-robin tournament is held, meaning that every two players will play exactly one match, and each match must have a winner. If a higher-ranked player beats a lower-ranked player, the winner gets 1 point and the loser gets 0 points; if a lower-ranked player beats a higher-ranked player, the winner gets 2 points and the loser gets 0 points. After all matches are completed, the cumulative points of each player (the sum of the points from this round-robin tournament and their initial points) are calculated, and the players are re-ranked based on their cumulative points. Find the minimum possible cumulative points of the new champion (ties are allowed).
Solve 12.
.5 points
Assume 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, so if he increases by 6 points, he must win at least 1 game against players ranked higher than him, thus he can win at most 4 games against players ranked lower, making his maximum wins 5; $A_{6}$ can increase by at most 7 points, but there are only 4 players with fewer points than him at the start, so if he increases by 7 points, he must win at least 2 games against players ranked higher than him, thus he can win at most 3 games against players ranked lower, making his maximum wins 5; $A_{7}$ can increase by at most 8 points, but there are only 3 players with fewer points than him at the start, so if he increases by 8 points, he must win at least 3 games against players ranked higher than him, thus he can win at most 2 games against players ranked lower, making his maximum wins 5; $A_{8}$ can increase by at most 9 points, but there are only 2 players with fewer points than him at the start, so if he increases by 9 points, he must win at least 4 games against players ranked higher than him, thus he can win at most 1 game against players ranked lower, making his maximum wins 5; $A_{9}$ can increase by at most 10 points, but there is only 1 player with fewer points than him at the start, so if he increases by 10 points, he must win at least 5 games against players ranked higher than him, making his maximum wins 5; $A_{10}$ can increase by at most 11 points, and he can win at most 5 games against players ranked higher than him, making his maximum wins 5. In summary, the maximum number of games won by all players is $2+3+4+5 \times 7=44$, but each two points
The following example illustrates that the new champion's cumulative score can be 12 points.
$A_{1}$ wins against $A_{2}, A_{3}, A_{4}$, loses to $A_{5}, A_{6}, A_{7}, A_{8}, A_{9}, A_{10}$, cumulative score is $9+3=12$;
$A_{2}$ wins against $A_{3}, A_{4}, A_{5}, A_{6}$, loses to $A_{7}, A_{8}, A_{9}, A_{10}$, cumulative score is $8+4=12$;
$A_{3}$ wins against $A_{4}, A_{5}, A_{6}, A_{7}$, loses to $A_{8}, A_{9}, A_{10}$, cumulative score is $7+4=11$;
$A_{4}$ wins against $A_{5}, A_{6}, A_{7}, A_{8}$, loses to $A_{9}, A_{10}$, cumulative score is $6+4=10$;
$A_{5}$ wins against $A_{6}, A_{7}, A_{8}, A_{9}$, loses to $A_{10}$, cumulative score is $5+2+4=11$;
$A_{6}$ wins against $A_{7}, A_{8}, A_{9}, A_{10}$, cumulative score is $4+2+4=10$;
$A_{7}$ wins against $A_{8}, A_{9}, A_{10}$, cumulative score is $3+2 \times 2+3=10$;
$A_{8}$ wins against $A_{9}, A_{10}$, cumulative score is $2+2 \times 3+2=10$;
$A_{9}$ wins against $A_{10}$, cumulative score is $1+2 \times 4+1=10$;
$A_{10}$ cumulative score is $0+2 \times 5=10$. $\qquad$