# [EM] Better cardinal methods?

Forest Simmons forest.simmons21 at gmail.com
Thu Sep 30 21:51:42 PDT 2021

```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

El dom., 26 de sep. de 2021 11:44 p. m., Andy Jennings <
elections at jenningsstory.com> escribió:

> Kristofer,
>
> I, too, find myself going back to the risk-neutral lottery-based
> definition of utility. I feel like it goes so naturally with "random
> ballot".
>
> Suppose V1 has a ranking of A > B > C and V2 has a ranking of C > B > A.
> In random ballot, V1's vote becomes a lottery ticket that causes A to win
> and V2's vote becomes a lottery ticket that causes C to win. Let us ask
> when would V1 and V2 both agree to trade their "one chance of A winning and
> one chance of C winning" for "two chances of B winning".
>
>
> We know we can say little about the "absolute" or "interpersonal"
> utilities of how U_V1(A), U_V1(B), and U_V1(C) compare to U_V2(A), U_V2(B),
> and U_V2(C).
>
> But asking each voter to quantify exactly where B lies on the spectrum
> between A and C, as a number between 0 and 1, is completely meaningful (in
>
> Let b_1 = (U_V1(B) - U_V1(C)) / (U_V1(A) - U_V1(C))
>
> Let b_2 = (U_V2(B) - U_V2(A)) / (U_V2(C) - U_V2(A))
>
> (In other words, rescale each voter's utility so their favorite candidate
> is at 1.0 and their least favorite is at 0.0 and examine their utility
> estimations of B.)
>
>
>
> If b_1 = 0.5 and b_2 = 0.5, we propose they trade "one chance of A winning
> and one chance of C winning" for "two chances of B winning". The voters are
> actually completely neutral toward this trade, though as an outsider I much
> prefer the lowered entropy. b_1 and b_2 would have to be strictly greater
> than 0.5 for both voters to be excited about the transaction.
>
> It works for other fractions, too. If b_1 = 0.6 and b_2 = 0.4, the
> utility-neutral trade is "one chance of A winning and one chance of C
> winning" for "1.666 chances of B winning and 0.333 chance of C winning".
> (If b_1 > 0.6 and b_2 > 0.4, then the trade is positive-sum.)
>
>
>
> Can we actually set up this market (declared-strategy style), let all
> voters submit their three-candidate ranking and a utility (between 0 and 1)
> for their middle candidate, then we simulate all the trades and come up
> with a final, optimal lottery?
>
> A > B > C voters and C > B > A voters would trade with each other. A > C >
> B voters and B > C > A voters would trade with each other. B > A > C voters
> and C > A > B voters would trade with each other.
>
> It seems obvious to me that in an election where 50% of the voters want A
> > B > C and 50% want C > B > A, if you can get a number from each voter on
> where B is on their scale from 0 to 1, that information is useful AND
> meaningful. I mean, if all the voters say 0.9 then clearly we should just
> elect B as the compromise candidate. And if all the voters say 0.1, then
> giving them a 50/50 lottery between A and C is probably the best we can do.
> Why should we decline to collect and use this "utility of the middle
> candidate" information?
>
> How can we simulate those trades? Line up all the A > B > C voters in
> order of decreasing "B" utility and line up all the C > B > A voters in
> order of increasing "B" utility and match up the two lines somehow? What
> about the mismatch in length?
>
>
>
> One problem I see is that whenever a transaction is perfectly fair, it is
> utility-neutral, and the two parties are indifferent to whether the trade
> actually happens. A trade that is positive-sum, on the other hand, has some
> surplus utility and we could be unfair about which voter captures it.
>
> If b_1 = b_2 = 0.6, then trading "one chance of A and one chance of C" for
> any of the following would be utility-neutral or -positive for both voters:
>
> - 1.666 chances of B winning and 0.333 chance of C winning
> - 2 chances of B winning
> - 1.666 chances of B winning and 0.333 chance of A winning
>
> Obviously, as neutral election administrators, we should choose the middle
> option. But I think this illustrates the opportunity for strategic voting
> in this system. If you, as a voter, have perfect information about the
> other voters, maybe your utility for B is 0.6 and you see that you can
> decrease your declared utility for B to 0.400001 and still get a trade. It
> will be a trade the other person barely agrees to, and you'll maximize your
> utility, capturing all the surplus from the transaction.
>
> Is there something else we could do as election administrators to make
> dishonesty less profitable? Does it depend on the way we line up and match
> up the opposing voters? If we always try to make sure that we match up
> voters with a "sum of compromise utility" that is greater than one but as
> small as possible, does that help somehow?
>
> Perhaps in a large election, it will be difficult to know enough
> information about the other voters and the benefits will be small enough
> that voters will just be honest?
>
> Or maybe we just discard the concept of matching up individual voters,
> look at all the data, and come up with a "market-clearing price" for
> turning A and C chances into B chances? Does that fix anything, or just
> leave a lot of positive-sum transactions unfulfilled?
>
> Can it be generalized to more than three candidates?
>
> ~ Andy
>
> On Sun, Aug 15, 2021 at 3:02 PM Kristofer Munsterhjelm <
> km_elmet at t-online.de> wrote:
>
>> Suppose we take the risk neutral lottery-based definition of utility as
>> a basis for honesty. (That is, if you're indifferent between a 100%
>> chance of choice X and a 70% chance of Y, 30% chance of Z, then your
>> utility for X is equal to 0.7 * u(Y) + 0.3 * u(Z).)
>>
>> What kind of cardinal system could incentivize voters to report this
>> kind of information? It seems very hard to do it with any of the broad
>> IIA class (score for X is just a function of ratings for X, highest
>> score wins, and increasing your rating for X never decreases X's chance
>> of winning), because those methods encourage minmax strategy.
>>
>> Perhaps some kind of cumulative voting? There's Hay, but it sucks.
>>
>> The lottery definition above can't determine both a natural zero and
>> unit value, because if you scale all utilities by some constant, the
>> lottery equations remain the same. So any method that takes this kind of
>> input should pass a kind of "irrelevance of constant scaling" property,
>> which says that if every voter v scales his ballot by some (private)
>> constant factor v_F, then the outcome remains the same.
>>
>> In a Range-type method, that means the system should scale every ballot
>> so that one candidate is max rated and another is min rated (I think).
>> This would probably lead to IIA because in a two-candidate election,
>> you'd get majority rule (whichever candidate voter v prefers gets max
>> rating, and the other one gets min rating).
>>
>> Maybe it's possible to preserve IIA, but I doubt it.
>>
>> Any thoughts on how a method with better "honesty" incentives could be
>> designed for lottery-type cardinal ballots?
>>
>> -km
>> ----
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>>
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