[EM] Good As L
Forest Simmons
forest.simmons21 at gmail.com
Thu Oct 7 13:51:49 PDT 2021
Before the promised continuation .. first a correction on the "weakly
better " relation on lotteries in the rankings context... I got the
containment backwards ... in general a voter V prefers L' over L if L'
gives more probability (than does L) to V's higher ranked alternatives than
to V's lower ranked alternatives , which tends to raise the GAL' cutoff
above the GAL cutoff ... so the set of alternatives marked GAL will contain
the set marked GAL', the reverse of what I mistakenly said in my previous
message.
El mar., 5 de oct. de 2021 1:12 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:
> With ratings we judged "Good As L" by the ballot expectation under the
> lottery L.
>
> With rankings we might be tempted to base GAL on expected rank numbers...
> but that would be a huge mistake ... a repeat of the classic mistake of
> Borda ... treating rankings as interchangeable with ratings ... thereby
> losing clone independence.
>
> The way that has the best chance of yielding positive results is to mark
> alternative X as GAL ("Good as L") iff it is more likely that L will pick
> an alternative ranked below (behind) X than one ranked above (before) X.
>
> 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 one or 1/2, respectively depending on
> whether or not X is ranked (Equal)Top.
>
> Finally, let L'(X) be the percentage of ballots beta such that c(beta) = X.
>
> How about the "weakly better" relation on lotteries in this ranking
> context?
>
> In the ratings context we made use of dot products to define this
> relation, an expedient that is unavailable in the context of mere
> rankings...
>
> Let's say that L' is weakly better than L as far as ballot beta is
> concerned if the set of alternatives marked as GAL' on ballot beta contains
> the set marked as GAL.
>
*** CORRECTION ... Should be...
the set of alternatives marked as GAL on ballot beta contains the set
marked as GAL'.
>
> If this is true for all beta, and strict containment holds for at least
> one beta, then L' is weakly better than L.
>
> In our next message we talk more about entropy.... (TO BE CONT'D)
>
> Forest
>
> El lun., 4 de oct. de 2021 8:09 p. m., Forest Simmons <
> forest.simmons21 at gmail.com> escribió:
>
>> 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 at 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 = p*x + q*z .
>>>
>>> Having done this for each interior Y, we now have a system of equations
>>>
>>> {y = p*x+q*a | 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
>>>
>>
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