We probably should also take into account intended quads that are popped into doubles, etc., that do not show up on the protocols as quads.
No incident of "no score" and singling a quad was found in men's LPs during the 2011-2012 season. The frequency of doubling a quad was 5, with an average of 1.346 points earned, and the frequency of tripling a planned quad was also 5, with an average of 4.78 points earned. So we have:
(1) Greater than opportunity cost (74%):
Full rotation without fall ==> 61% (based on the judgment of the technical panel, therefore including the border-line "non-fall")
Full rotation with fall ==> 10%
Underrotation without fall ==> 3%
(2) Lower than opportunity cost (26%):
Underrotation with fall ==> 13%
Downgrade without fall ==> 4%
Downgrade with fall ==> 4%
Popping/doubling ==> 5%
Percentage for full rotation = 71%.
Percentage for underrotation = 16%
Percentage for downgrade = 8%
Let Y = total deduction for a fall, and assume that the average GOE deduction for an underrotated quad without fall is -2 and for a downgraded quad without fall is -2.5.
(71% x 25% x 10.30) + (71% x 75% x (10.30 - Y)) + (16% x 25% x (7.20 - 2) + (16% x 75% x (7.20 -Y)) + (8% x 25% x (4.1 - 2.5 x 0.7)) + (8% x 75% x (4.1 - 0.7Y)) + 5% x 1.346 = 4.1
1.8283 + 5.4848 -0.5325Y + 0.208 + 0.864 -0.12Y + 0.047 + 0.246 - 0.042Y + 0.0673 = 4.1
4.6454 = 0.6945Y
Y = 6.6888
To be really conservative, let's have a 6-point deduction for a quad fall.
One difficulty here is that under-rotating and falling are not independent...A skater who avoids a fall only 25% of the time is not going to hit the average of 74.74% for full rotations.
A skater who can rotate the jump only 25% of the time and land upright only 25% of the time should not be encouraged to try it in a competition under any scoring system anyway. Shouldn't he be practicing at home till he can reach a satisfactory rate for rotation? The proper interpretation of my calculation should be: What should be the deduction for a fall in order to ensure a minimum of 25% success rate for landing even if a skater can hit the average rate for rotation? The answer is 6 approximately (or 6.6888 to be specific). And the proposed deduction will not discourage a skater that has a lower rotation rate as long as he can compensate it with a higher landing rate. For instance,
Let total deduction for a quad fall = 6, full rotation rate = 50%, non-fall rate = 50%, underrotation = 28%, downgrade rate = 14%, and popping/doubling rate = 8%.
(50% x 50% x 10.30) + (50% x 50% x (10.30 - 6)) + (28% x 50% x (7.20 - 2) + (28% x 50% x (7.20 -6)) + (14% x 50% x (4.1 - 2.5 x 0.7)) + (14% x 50% x (4.1 - 0.7x5-1)) + 8% x 1.346 = 2.575 + 1.075 + 0.728 + 0.168 + 0.1645 -0.028 + 0.1077 = 4.7902 > 4.1 (base value of 3T).
That is to say, a skater who can fully rotate the quad 50% of the time and manage not to fall 50% of the time of his quad attempts can yield an expected value higher than the base mark of a 3T or 3S, even if the total deduction for a quad fall increases from the current -4 (-3 GOE and -1 mandatory deduction) to -6.
How to educate the public to accept the fact that an element on which the skater falls ought to be awarded some points.
The more education is needed for the public, the more we take the fun out of the sport.
As a "casual" fan, I trust whatever the scores/levels the adjudicators assigned and don't care about the "black box" in TES because I acknowledge the limit of my technical knowledge. The presentation aspect (PE, CH, IN), however, is where the spectators feel they can understand and be involved emotionally and intellectually. In my opinion, there is no urgent need to "educate" the public. What is urgent for the ISU is to close the gap between the judges and the fans in the assessment of presentation.