Good As L (was "Better cardinal methods?")
This is the promised continuation that suggests how one can put to good use
the cardinal ratings we defined in our previous message.
The following steps constitute a way of transforming a non-proportional
lottery L into a proportional one L', given sincere cardinal ratings as
described in our previous message.
If we are not mistaken, optimal rational strategy for certain methods (as
suggested at the end of this message) that incorporate this transformation
will elicit from rational voters sincere ratings of that kind.
Let L be the input lottery. On each ballot mark GAL for "Good As L" next to
each alternative whose ballot rating is not less than that ballot's lottery
expectation.
[If the ballot ratings and lottery probabilities are given in vector form,
then the ballot's lottery expectation is the dot product of the two
vectors.]
For each alternative X, let GAL(X) be the percentage of ballots on which X
is marked GAL. Then for each ballot beta, let c(beta) be the "chosen"
alternative X, among those marked GAL on beta, with the greatest value of
GAL(X)*beta(X), where beta(X) is ballot beta's rating of X.
So each ballot beta invests its share of the probability in a candidate who
has support from other ballots (as attested by the factor GAL(X)) and is
also rated relatively high by beta's voter (as attested by the factor
beta(X)). Thus the cooperative viability and the individual voter's
estimate of desirability have equal weight in this product.
Finally, let L'(X) be the percentage of ballots beta such that c(beta) = X.
Thus we see how the output lottery L' is determined by the input lottery L.
If a subset S of the ballots rate only candidate X above zero, then each of
those ballots will choose X, i.e. for each beta in S, the choice c(beta)
will be X, which entails that L'(X) is at least #S/N, where N is the total
number of ballots and #S is the cardinality of the subset S.
Therefore L' is a fair (proportional) lottery even if L is not.
Suppose that for every ballot beta, the dot product beta•L is no greater
than the dot product beta•L', and that for at least one ballot beta (L' -
L)•beta>0. Then we can say that L' is at least weakly better than L.
One method based on this lottery improvement transformation is to
initialize L as the random favorite "benchmark" lottery. Then iterate the
transformation L ---> L' until L' is no longer weakly better than L. At
that point the last improved L is the winning lottery ... i.e. the one used
to pick a single winner, or the one used to apportion seats among parties
in a multi winner party list context ... as the case may be.
Another way to use this L ---> L' transformation is to solicit nominations
for L from all voters, candidates, and other interested (non-bot) parties.
Then choose by random ballot from among the corresponding L' lotteries that
are tied for minimum entropy.
In our next message let's see how well we can mimic these results using
only ordinal ballots ... to be continued...
FWS
El jue., 30 de sep. de 2021 9:51 p. m., Forest Simmons <
forest.simmons21@gmail.com> escribió:
Here are some of my thoughts about determining sincere ratiings with the
help of sincere rankings ... ratings adequate for use in lottery methods:
We set up a system of equations (to be solved iteratively) whose solutions
are the desired ratings.
First assign Top and Bottom ranked (or truncated) candidates the
respective boundary values of 100 and zero percent.
Each remaining candidate Y is interior to the ranks, i.e. ranked between
two neighbors X and Z. We use the lower case variables x, y, and z to
represent the ratings (whether given or to be determined) of the respective
candidates X, Y, and Z.
For each interior Y adjust parameters p and q (while keeping p + q =
100%) interactively until the user is indifferent between the lotteries p*X
- q*Z and 100%Y, where X and Z are adjacent to Y in the ranking.
Then set y = px + qz .
Having done this for each interior Y, we now have a system of equations
{y = px+qa | Y is ranked consecutively between X and Z}
which together with the previously mentioned boundary conditions are
sufficient to uniquely determine the desired ratings.
In fact, an approximate solution set for this system can be obtained by
initializing all of the interior variables randomly and then iterating the
set of equations (always respecting boundary conditions) until the
variables converge (e.g.) to the accuracy of the math coprocessor, ... as
long as you realize the accuracy of the actual ratings cannot exceed the
accuracy of the p and q estimates provided by the user ... GIGO.
The main purpose of the above verbiage is to show that there is a
conceptually rigorous way to define meaningful ratings adequate for use in
lottery methods without mention of "utilities."
That said, forty plus years of assigning partial credit to student work
has taught me some useful shortcuts.
A problem that can be solved in n sinificant steps gets fraction k/n
partial credit if the student successfully completes k steps before getting
derailed.
Similarly, a candidate gets rating k/n if she meets k out of your n
equally important criteria. If not equally important, then includes weights.
Sometimes the easiest way to assign partial credit is to ask yourself the
question, "What is the probability that this student would successfully
solve a typical problem of this kind on another similar test?"
Similarly, you can ask what is the probability that this candidate would
faithfully represent your position on issues of importance to you (weighted
by importance)?
List the candidates in order of these weighted probabilities, then
subtract the smallest from all of them .... finally divide the resulting
values by the largest of these. Note, however, that these normalization
steps form an affine transformation so they are not necessary if your
lottery method is invariant under affine transformations of the ballot
ratings ... an indispensable requirement for a decent lottery method.
I promise to show how to use these ratings ballots to make a lottery
based, but completely deterministic, party list proportional representation
method.
How can that be?
Here's the trick: the alternatives of the lottery method are the party
lists themselves. Voters rate the lists rather than the separate candidates
within the lists. Then the number of candidates contributed by a list is N
times p, where p is the lottery probability of that list and where N is the
number of seats to be filled by the election.
If the lottery method is "random favorite party," then you get a basic
party list method depending on how you round the N*p values to whole
numbers. Note that this method is absolutely deterministic despite its use
of lottery language to describe the distribution of winning candidates
among the various party lists.
But other proportional lottery methods (besides the benchmark
random-favorite lottery) with significantly lower entropy can lead to less
fragmentation and more potential for cooperation, without sacrificing
proportional representation of minority groups.
To be continued ...
FWS