[EM] Score DSV

[Forest Simmons]

I have never been completely satisfied with my previous attempts at
Designated Strategy Voting, where the strategy converts a set of Score
ballots into a set of approval ballots and elects the approval winner, but
hope springs eternal.

My new idea combines insights from Michael Ossipoff and Martin Harper among
others.

Among Ossipoff's rules of thumb for Approval voting was, "Approve the
candidate you would have voted for under one mark Plurality, as well as
everybody you prefer over her.

This sensible advice was a constructive, mild rebuke to anybody who claimed
that traditional Plurality strategy was easier than Approval strategy.

Now, how can we use this insight to automatically convert ratings into
approvals?

How do we figure out whom to vote for under single shot plurality?

We want to spend our one vote on the decent candidate who has the most
support among other voters, or on the most decent candidate who has a
chance of winning.

We judge "decency" by the debates, public record, hair style, etc. But how
do we judge support among other voters?

It would help if we had the results of an opinion poll. Then we could use
Martin Harper's insight ... vote for the decent guy with the most support.

How to quantify this?

Since popular support and decency are both important considerations for how
to spend our vote, why not vote for the candidate who maximizes the product
of viability and suitability?

Now shifting to the point of view of the DSV decision maker ... the one
responsible for converting Score ballots into Approval ballots ... we take
the ballot scores to be quantified measures of voter estimates of
suitability.  But what about estimates of viability?

If we had an Approval style opinion poll, that would serve, but we can only
use the information on the ballots ... no external polls.

It's starting to sound circular ... in order to figure out the ballot
approval cutoffs (the candidates that maximize the products of ballot
ratings and viability), we need an estimate of approval from the ballots.

A simpler problem of this kind is that of solving an equation like x=1+1/x
.  It looks like in order to find x you need to know x, and of course that
would help.

But with algebra, you can find a positive solution x=(1+5^.5)/2.

Another more general technique is by a fixed point iteration of an initial
guess:
1-->2-->3/2-->5/3-->8/5-->13/8-->21/13 ... 144/89 ... etc.

The approval problem we are dealing with is too complex for the algebraic
approach, but is well suited to the iterative approach.

For the initial approval estimates a0(X) we use the average ratings of the
respective candidates:

a0(X)=(1/N)Sum (over ballots B) of B(X),
where B(X) is the ballot B rating of X, and N is the number of ballots.

Our next approvals a1(X) are based on ballot approval cutoffs given for
each ballot B, by C(B)=argmax(B(X)a0(X)).

In general, the next approval estimates a'(X) are determined from the
current approval estimates a(X) by use of the ballot cutoffs
C(B)=argmax(B(X)a(X)).

To be definite ...

a'(X)=(1/N) #{B| B(X)>=B(C(B))}

Note that C(B) is the candidate with the greatest product of suitability
and support according to ballot B and the current estimates of viability.
In other words, according to my interpretation of Martin Harper's idea,
C(B) is the candidate deserving the Plurality vote from ballot B.

It follows from Ossipoff's rule of thumb, that if X is rated above or equal
to B(C(X)), then X should be approved on/by ballot B.

So if a_n(X) is a(X), then a_(n+1)(X)=a'(X).

If the sequence a0(X), a1(X), ...a_n(X) ...
converges, then the limit A(X) must satisfy A'(X)=A(X). ****

Our DSV method not only provides a list of approvals A(X) for the
candidates, but also an allocation of the equivalent Plurality votes by
ballot X=C(B), such that Approval winner AW is also the Plurality winner PW:

AW=argmax A(X).

PW=argmax P(X),

where P(X) = #{B| X=C(B)}.

In fact, if we normalize P to P' by dividing it by Sum over X of P(X), we
see that P' is a proportional lottery whose probabilities are in precise
proportion to the ideal  Plurality votes according to our Martin Harper
heuristic.

This fact can be seen most clearly from the cutoff formula in the limit ..

C(B)=argmax(B(X)A(X)),

and the fact that for every ballot B, the unique Plurality choice C(B) for
ballot B is also approved on B in its capacity of (the inclusive) cutoff
candidate for ballot B, as seen from the formula ...

a'(X)=(1/N) #{B| B(X)>=B(C(B))},

where in the limit, the left hand side a'(X) becomes A(X).

The first DSV paper I ever saw was a masters thesis by Lori???

The output was a set of Plurality ballots. I remember thinking ... since
Approval is much better than Plurality, why not have a set of Approval
ballots as output?

Later Rob Legrand did his thesis converting Score ballots to approvals
among other things.

This new DSV converts Score ballots to both Plurality ballots and Approval
ballots. The Plurality choice on each ballot is the inclusive approval
cutoff.

Thus, you can answer both questions: whom did I give full support to? And
... whose lottery probability increased by 1/N because of my ballot?

That's your one person one vote candidate if you want to boil it down that
way.

****[The Brouwer fixed point theorem proves the existence of such an
approval equilibrium function A'=A,  irrespective of the convergence of the
sequence a0, a1, a2, ... --> A.

The proof depends on the assumption that the transformation a --> a' is
continuous, and that the Score ballots are high resolution. If the ballots
are low resolution, a generalization of the Brouwer theorem (the Kakutani
fixed-point theorem) saves the day.

If the transformation a-->a is a contraction mapping, then we don't need
Kakutani or Brouwer. In fact, even if it is not a contraction mapping, it
can be converted into one by an averaging process, if I am not mistaken.

Numerical experiments are needed to see if averaging is needed.]
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