Those of you who have read through the Top 100 SNES list, and possibly the Top 100 NES list, might be wondering how we came to the final ordering for these games. Well, each game received a score from 1 to 1000 based on it's position on each individual list, and was then ranked against all other games with the highest scoring game getting the top spot.
Now you may be wondering how each games score was calculated. Well, to put it simply:
Where n is the number of individual lists the game appeared on,
x is the game's position on an individual list,
and l is the total number of individual lists.
For those of you who haven't taken university mathematics, I shall go into a more detailed explaination of each step along the way. (For those of you who have, I apologize for using 1.n, it just looks nicer than 1 + n/10).
Alright, let's start from the inside and work our way out. For clarity, from here out when I use the term 'rank', I'm refering to a games rank on an indivdual list.
The first thing we do is take the reciprocal of the games rank. If I rated one game 15th on my list, and another game 30th, it wouldn't make sense to give them 15 points and 30 points respectively, as we wanted the 15th ranked game to be higher on the final list. So we take the reciprocal of each number to achieve the desired result. 1/15 = 0.067, 1/30 = 0.033.
Before we take the reciprocal, we want to add what I call a 'limiter' to the games rank. The limiter is the total number of individual lists, less 1. It might not be immediately clear why this is necessary, so let's move forward.
The next step is to add up all those reciprocals from the individual lists. The more lists a game appears on, and the higher its rank, the higher this number will be. This is also where the limiter becomes important. Let's say there were 5 lists, and everybody voted a certain game #1. Then we'd have this happening:
So the highest possible total a game can have a this point is 1. Without the limiter, a game could have a score higher than 1 at this point, which would cause some wonky things in the next step.
Finally we have a booster. When a game appears on multiple lists, it gets a boost for the number of lists it appears on. So again, if there are 5 lists, a game could receive a low boost of 1.1 for being on 1 list, up to 1.5 for appearing on 5 lists. If you want to see why I had to use the limiters, take the 1.5th root of 2, and the 1.1th root of 2. For this system we would want the former to be higher, but you'll see that this is not the case. When the numbers are between 0 and 1, this achieves the desired effect.
Well, that was fairly long-winded. Let's walk through an example to see how A Link to the Past scored. There were 7 lists in total, and it was ranked 6, 1, 1, 4, 2, 2, and 4. Then we'll get this:
Multiplying the number at the end by 1000 is purely for aesthetic reasons. It is easier to rank numbers that range from 1 to 1000 than it is to rank numbers ranging from 0 to 1. To show that the booster does indeed work the way it is supposed to, let's take the 1.1th root of 0.8190476. This results in 0.834 or a score of 834.
So that's how the ranking system was designed. I hope someone out there enjoyed reading this.