About approval "equilibria" I suppose I should give my proof that the following voting system will yield the Voronoi diagram [same as Condorcet systems] in the large #voters limit.
system: 1. do approval election - call the winner "X." 2. each voter now approves Y if Y>X, flips coin to approve Y if Y=X, and disapproves Y if Y<X. 3. back to step 1 until reach a steady state where some X keeps winning. remark: This only is true in the geometric-probabilistic settings used in the Yee voting sim pictures. [It is possible abstractly to set up a situation where a steady state never is reached, e.g. the approval winner just keeps "walking round a Condorcet cycle."] proof: In those pictures, by a proof based on properties of convolutions under the assumption the voter distribution is centrosymmetric and the utility function is a decreasing function of distance, you can prove there is always a transitive social ordering of all candidates, i.e. condorcet cycles are impossible. We shall take that as known (see *Plott below). Now in that case a condorcet winner W exists. We shall show W is the steady state winner. Note if X is W, then next approval election, W gets 50% approval due to the coin-toss rule in large#voters limit; others get below 50% approval due to W being a Condorcet winner and the Y>X and Y<X approval rules; hence W wins, proving the steady state continues. So all we need to prove is that after a finite number of iterations, W will manage to win an approval election. Each approval election, W>X in the view of over 50% of the voters (if X and W are not identical) and indeed for any Y with Y>X in the social ordering, Y will get >50% approval - whereas X gets only 50% by the cointoss rule and each Z with Z<X in the social ordering gets below 50%. Hence the next approval election, the winner will be somebody ABOVE the previous winner in the social ordering. So we keep walking upward in the transitive social order and such a walk must end with W after at most N-1 walk-steps (if there are N candidates). QED. remark: Gaussians, whether spherical or elliptical, are centrosymmetric. An elliptical gaussian with an elliptical "hole" in it (the two ellipses can have unrelated shapes, so long as they have same center) is centrosymmetric. *Plott remark: It has been suggested that perhaps some or all of my theorems were proven long ago by Charles R. Plott: A Notion of Equilibrium and its Possibility Under Majority Rule, The American Economic Review 57,4 (Sep., 1967) 787-806. I do not know that because I have not read Plott's paper yet (it is available on JSTOR but I do not currently have access). Warren D Smith http://rangevoting.org ---- election-methods mailing list - see http://electorama.com/em for list info
