[EM] feedback loops in elections and election methods

[Forest W Simmons]

Designated Strategy Voting (DSV) methods relieve the voter of repeated 
returns to the polls for each iteration of the feedback loop, and also 
solve the anonimity requirement, but as has been noted, methods that 
are supposed to iterate unto an equilibrium may not converge.

[What follows requires a little pre-calculus math, or good graphical 
intuition to understand.]

In that regard, suppose that you are solving a scalar equation of the 
form  x=f(x),  i.e. you are searching for an equilibrium point (i.e. 
fixed point) x for the function f.  You might try starting with a guess 
 x0, and then iterate the function f to see if you get a convergent 
sequence x0, x1, x2, ...
where each number of the sequence is obtained by applying the function 
f to the previous number, for example  x5=f(x4).

Sometimes this works, and sometimes not.  Even if it does work, it may 
converge slowly.  In fact, for a generic equilibrium, if it does 
converge, it will converge linearly.  

[This is because the graph of f will generally cross the line y=x with 
some slope other than zero.  If the absolute value of that slope is 
less than one, then the sequence will converge, provided the starting 
point x0 is not too far off.  If that slope is zero, then convergence 
will supralinear.  If the absolute value of the slope is greater than 
one, then the sequence will not converge.]

But linear order convergence can be speeded up by making use of a 
convergence accelerator.

If  a, b, and c are successive numbers in a sequence of iterates where 
the convergence is linear (as per the usual case) then  the quantity

Q=(a*c - b^2)/(a+c-2*b)

will be a superlinear improvement over any of a, b, or c, as an 
estimate of the equilibrium value.

Note that if we interchange a and c in this formula, then Q is not 
changed.  This fact will help you understand why a convergence 
accelerator can also help bring convergence to divergent sequences.

In other words, if you were to run the sequence backwards, so that you 
are iterating the inverse of f instead of f itself, then the sequence 
would converge, since if the graph of f has a slope with abs value 
greater than one, the graph of the iinverse of f will have a slope with 
abs value less than one. [Their slopes are reciprocals at the 
equilibrium point.]

If f and g are mutual inverse functions, then they share equilibria, 
and a unstable equilibrium for one will be stable for the other. The 
situation is more complicated in higher dimensions, but convergence 
accelerators still have the same salutory effect on divergent 
iterations near unstable equilibria.

So convergence accelerators can speed up convergence of slowly 
converging iterations towards a stable equilibrium, and can reverse 
divergence from an unstable equilibrium.

Bottom line:  somebody should try convergence accelerators on iterative 
DSV methods.

Forest