Let us introduce what is new in the spreadsheet.
Have been thinking about this and made an example with matrices on a jupyter notebook, so that those interested might better follow WHENCE the numbers come from. Using matricial product, the explanation is rather elegant. However, in the end, it seems best to provide the numbers with a layman's explanation, and leave the thorough one for those interested, later.
So, novelties: first, to the right of the first table, the Controversy colum. It is simply the variance of the opinions about someone. The less Controversy, the more homogeneous are opinions about someone.
Then, the big novelties: in the second image, see the IDAHO matrix, and below, the Trust matrix. The Trust matrix is simply how much you trust someone, meaning 1-p, with p being what everyone is submitting, the estimated probability of someone being hostile to Town.
What is the IDAHO matrix? Let us go step by step: Imagine that we make an average of all opinions on someone, but it is not a regular average. Instead, we make a weighted average. Weighted with what? Good question: weighted with the degree of trust that you have for each one posing their opinion. That means that confirmed scum get zero input, and an innocent child gets full input, relative to the rest, and a 0,5 gets 0,5 input, relative to an IC.
Remember the movie
My own private Idaho? what you see under the columns in pink in the upper matrix of the second image is YOUR OWN PRIVATE IDAHO. Meaning, the average on the opinions of people, with more weight for the people you trust the most to be Town. You search the column and what appears is an average of the readings, customized for your. The serious name of this matrix is the
Trust-Weighted Matrix, if you prefer that. All the private Idahos, together, make the Idaho matrix.
Then you can get an average of all that, which appears to the right, plus another Controversy column. Ta-da.
The examples attached are from the last data from D2, so that we can compare. In the next post, the same data, BUT with the suspicion matrix UPDATED with what we do know in D4: Meaning, two scum and an innocent child. This is so that you can compare.
Will be compiling data and hopefully Caesar and Joppo can provide more input as well, so that we have a proper D4 spreadsheet. Please bump if you can...
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T H E O RY (for those willing to enter the rabbit hole; read at your own peril)
Let us call
M the matrix with the players's input, with the elements being pij, meaning the opinion of player j about the suspect i
T the matrix whose elements are 1-pij
n the number of suspects
m the number of players providing input with their estimations
(if all players are being considered, the matrix M is squared and n,m = 14)
The average suspicion is obtained post-multiplying the suspicion matrix (M) by an equal-weight column vector which has all elements equal to 1/n, with n being the total number of suspects (as you know, in order to make a weighted average, you can build any weight vector assigning relative weights manually and then dividing by the sum of all the relative weights)
But lets us use the Trust matrix:
The Trust-Weighted (Mtw) Idaho matrix is M post-multiplied by T
M*T = Mtw
But M can also be weighted with the Skill that you think that each player has for detecting scum and clearing townies. All those Skill weights, put together, make the Skill matrix (S).
And you would obtain a Skill-weighted suspicion matrix (Msw) customized for each participant in the same way:
M*S = Msw
Actually, nothing prevents us from combining the two weights, if we wish to do that: taking into account the Skill of each player and the degree of Trust that we have for them. Thence we obtain the Trust and Skill weighted Suspicion Matrix (Mtsw).
M*T*S = Mtsw
Why stop here? Probably you might have reflected that some people can read this or that player better than they can read others, or the opposite (when someone is someone else's kryptonite). This means that we might assign to each player a different degree of skill when reading each particular player.
M*T*Sj
And if we are to consider that each player might have a different opinion of each other's skill when evaluating each of the other players, we end up with a three-dimensional array, that, if all players are being considered, will bear
the shape of a CUBE. Not using the Skill matrices nor the Cube, currently, but it could be done. At least you have been entertained, hopefully (and your sanity checked only slightly ;-)