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 ---- election-methods mailing list - see http://electorama.com/em for list info
