Theoretical background

The score system is inspired by

(1) Pythagorean Expectations => http://en.wikipedia.org/wiki/Pythagorean_expectation" onclick="window.open(this.href);return false;

(2) Random Walk => http://rankings.amath.unc.edu/old/monkeys.htm" onclick="window.open(this.href);return false;

Calculation

(1) The base assumption is that there exist an Pythagorean Expectation formula that can connect votes for and votes against to the Win% of a character. Furthermore, this win% may be close to what people expect from a character in an ISML Match. After many trial and error, it was decided that the following formula is the simplest formula that adequately fits ISML results

Expected Win% = 1 / [ 1 + ( VA/VF) ^ 5 ]

The Root Mean Square Error = sqrt [ sum of all of (Expected Win% - Actual Win% ) ^2 ] = 0.066326

The RMSE value corresponds to average error of being off by about two wins in typical ISML season, but any formula that reduced error involved much more factors, which would have made this part very cumbersome. The form above was chosen for its simplicity, and also because the number 5 played important role in the next part.

(2) The next assumption is how much of an expectation that a character carries into the match is at the stake for a match. It is like how much of the "Pot" the character has will be "Bet" into the match pool. Stronger characters are likely to carry a bigger "Pot" thus have more expectation at stake for a match. How much of the "Pot" was "bet" on each match was of great importance, as "betting" too little would make the "Expectation Value" change too slowly, while "betting" too much would make it very volatile and not mirror the reality.

Shmion and I tested the "bet" amount between 0.01 = 1% of the whole pot to 0.5=50% of the whole pot on 2014 Nova and Stella Data. The calculation becomes as follow

- i) All characters start with 100 points as their Pot

ii) In each match character bets N% of the Pot into the Match Pot. We explored N%=0.01=1% to N%=0.5=50%

iii) After the match between Character A and B, they will divide up the match pot by the following equation

Amount of Match Pot earned by character A = Match Pot Value / [ 1 + ( Vote for character B / Vote for character A ) ^5 ]

iv) The character's pot value will be readjusted and they will go to the next match

i) Character A and B would have pot value of 100 at the start of the season

ii) In the first match of the season they are against each other. Each put up Pot * N% = 100*N% into the match pot. The Match Pot Value is (100 ( = Pot_A ) +100( = Pot_B) )*N%

iii) In the match Character A got Vf_A votes and Character B got Vf_B votes

iv) The Character A's Pot after 1st match would be New Pot value = 100 ( = initial pot ) - 100*N% ( Amount placed into the match pot ) + (100 ( = Pot_A ) +100( = Pot_B) )*N% / [ 1 + ( Vf_B / Vf_A ) ^5 ]

Shmion and I ranked the characters by their expectation value at the end of the season and compared it to our traditional point ranking. We wanted to find a case where average rank difference between expectation value and traditional point ranking was bigger than 1, but less than 5. We also didn't want more than two cases where rank difference would be more than 10. Another constraint I looked at was that maximum amount of "Pot" change that occurred from a match be near 50, where this 50 was chosen as value that is about half the initial pot value.

What we found was that when N% is greater than 0.2, we get the average rank difference to be greater than 1, but things somewhat stabilize after that, thus we couldn't exceed average rank difference of 2 even for N% = 0.5 in case of Nova ( 2.333 for Stella which is still near 2 ) . However, maximum amount of Pot change was as great as 174 for Nova and 232 for Stella, which means there was potential for it to be overly volatile. After much simulation, N% = 1/5 = 0.2 was chosen to be optimal. The value of 0.2 may not be mere coincidence as the exponent in our Pythagorean Expectation formula also was 5.

You can see the actual numbers at https://docs.google.com/spreadsheets/d/ ... 1236600324" onclick="window.open(this.href);return false; which was made by Shmion .

3) discussions

i) Illya at the end of 2014 Stella showed significant rise in Expectation value ranks. I believe the increase to be close to apparent level of interest people gave Illya due to her Topaz wins.

ii) Depending on how much characters placed into match pot, it is possible for a character to gain expectation value even when she lost. This is akin to a real world sports case where a team that used to struggle starts doing suddenly well, and in an important match, nearly beat an opponent that was considered to be one of the best team in the league. Many people would start paying more attention to the team that exceeded expectations.

iii) Unlike 1%-35% point stealing scheme, nor our traditional point system, every vote really matters, WIN OR LOSS . While this is great for voter participation, it also makes this "Value" system be vulnerable to manipulations by small yet consistent faction voting.

iv) The value rank should move faster than traditional point rank when a character receives a late season push. This can be very useful for ********* .

v) The Value and Value Rank can replace the SVAO and SWVO column in the stat table. VP and SVDO column should be replaced by the 1%-35% Score column and the Score Rank.

vi) It should be noted that the Value tracks expectation, not the actual strength level of a character. There already are many Actual Strength Level calculation systems that beats this Value system in predicting the results.

----- Conclusion ----

I believe we have found a nice formula that somewhat matches amount of hype a character may have. This hype amount can be used in various ways. It is recommended that the columns for the Expectation Value and the Value rank replace two of the columns in current stat table so its evolution can be watched and analyzed by the public.