Warren Smith wds at math.temple.edu
Wed Dec 27 11:36:33 PST 2006

```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

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