[ExI] EvilTriplets.xlsm

[Anders Sandberg]

Ah, this is fun! Some mathematical notes:

The system is essentially x(n+1)=A*x(n),where x(n) is the vector of the 
triplets' states in generation n, and A is a matrix with their weightings.

The sum of each row of A is 0. This means that we can view the system as 
defined by three vectors lying on the plane defined by x*[1 1 1]=0. So 
there are 6 degrees of freedom to play with, like Spike did with his 
scrollbars.

If A has eigenvalues lambda_i and eigenvectors Lambda_i (i=1,2,3) then 
it is easy to see that they are invariant: A*Lambda_i=lambda_i*Lambda_i. 
So there are potentially three invariant directions, but eigenvalues can 
coincide of course. If you start at some random state x it can be 
expressed as a sum of eigenvectors: x = 
c1*Lambda_1+c2*Lambda_2+c3*Lambda_3. As the system updates, the vector 
that has the biggest real part of the eigenvalue dominates: if this is 
positive you get exponential growth, if it is negative you get 
exponential decay. The imaginary part determines how much things 
oscillate: if it is nonzero there will be jumping. If it is small you 
get sinusoidal oscillations. A pure imaginary one makes a cyclic pattern.

So by this argument we can tell what the eventual distribution will be: 
it will be a multiple of the largest eigenvector (whether a positive or 
negative multiple depends on starting value), possibly with some 
oscillation going on.

The next question is what conditions lead to different behavior. We can 
rotate our coordinate system so Alice is [+1,0,-1] , giving us four 
degrees of freedom. Still one too much to visualize.


-- 
Anders Sandberg
Future of Humanity Institute
Oxford Martin School
Oxford University