<div dir="auto">Great!<div dir="auto"><br></div><div dir="auto">One way to extend the reals to allow comparison of quantities that are "incommensurate" with standard real ratings ... is to allow polynomials in epsilon as ratings.</div><div dir="auto"><br></div><div dir="auto">Another thought ... the Ultimate Lottery method allows ballots to be arbitrary positively homogeneous functions of the lottery probability variables ...</div><div dir="auto"><br></div><div dir="auto">f(p1, p2, ... p_n)</div><div dir="auto"><br></div><div dir="auto">The Ultimate Lottery is the point P of real n-space that maximizes the product of the ballots, subject to the non-negativity constraints p_k >=0, and the normalization to unity of the Sum p_k .</div><div dir="auto"><br></div><div dir="auto">The one person, one vote condition is that all of the ballots have the same degree d of homogeneity.</div><div dir="auto"><br></div><div dir="auto">f(lambda*p)=f(p)*lambda^d</div><div dir="auto"><br></div><div dir="auto">-Forest</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Oct 31, 2022, 5:41 PM Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de">km_elmet@t-online.de</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Good news - I'm almost done with my non-EM big project, and so I should <br>
be able to create some strategy simulations soon to investigate which <br>
Friendly variants are good :-)<br>
<br>
But here's a thought that I've talked about earlier, that came up in a <br>
private mail response.<br>
<br>
Suppose that von Neumann-Morgenstern utilities are the best we can do as <br>
far as interpersonal comparisons go.<br>
<br>
Then we can imagine that every cardinal method has an idealized <br>
(Platonic, so to speak) form that operates not on ballots with discrete <br>
ratings, but on real ratings. E.g. the idealized version of Range <br>
involves giving each candidate a rating on [0...1), and the candidate <br>
with the highest mean wins.<br>
<br>
Then vNM utilities are unique up to some affine scaling. Let u(v, X) be <br>
voter v's utility should X be elected, on an objective scale that we <br>
can't access due to interpersonal comparison problems. Then if v is <br>
asked to rate X, he would rate X as r(v, X) = a_v * u(v, X) + b_v, where <br>
a_v and b_v are the positive scaling constants for that voter.<br>
<br>
For something like Range, suppose that when v goes to vote, v chooses <br>
some a and b so that every rating fits on the scale, then submits that vote.<br>
<br>
For an idealized method, we can then propose the following invariance <br>
property:<br>
<br>
- Affine invariance: The outcome of the election should not depend on <br>
the particular a and b constants chosen by the voter.<br>
<br>
Methods that pass affine invariance plus anonymity and symmetry satisfy <br>
a type of one-man-one-vote because expanding your scale is equivalent to <br>
making the a constant larger. Hence such a method, if it accepts <br>
ratings, doesn't have to limit itself to a bounded scale, though the <br>
user interface is probably more pleasant if it does.<br>
<br>
Furthermore, these methods have another property that there exists one <br>
honest ballot, which solves rb-j's objection of how much a voter should <br>
vote his second favorite if he's honest. Every such method that I know <br>
of fails IIA, but that's better than the sort of "I pass IIA but the <br>
outcome still depends on who's in the running" thing that Range does.<br>
<br>
What methods pass this criterion?<br>
<br>
Well, ranked methods clearly do. No amount of positive affine scaling <br>
can turn r(v, X) > r(v, Y) into r(v, X) < r(v, Y).<br>
<br>
In addition, there's this "double normalized Range": the voter has to <br>
fill in a ballot where at least one candidate is rated max and at least <br>
one is rated min. The rules are otherwise as Range.<br>
<br>
Each forced rating gets rid of one degree of freedom, so forcing two <br>
candidates fixes both a and b. The intermediate candidates' ratings can <br>
be found as follows:<br>
<br>
Suppose Best is the best candidate, and Worst is the worst, and the <br>
scale is from 1 being best to 0 being worst. Then for any other <br>
candidate X, rate X at p if you're indifferent between getting Best with <br>
probability p and Worst with probability 1-p, and getting X for sure.<br>
<br>
Incentivizing honesty here would be hard, though.<br>
<br>
(Perhaps a Ranked Pairs-like algorithm is possible for triples rather <br>
than pairs.)<br>
<br>
<br>
I also had an idea that perhaps we could find the best VSE affine <br>
invariant method by something like my "optimum strategy" search, but <br>
over the space of cardinal ballots instead of ordinal ones, for a small <br>
number of candidates. In particular, the results for three candidates <br>
could be interesting.<br>
<br>
We couldn't use the idealized method because there are infinite reals on <br>
a bounded interval. But the observation about double normalized Range <br>
above suggests that we could replace the ballot format for three <br>
candidates with:<br>
(Who's min), (Who's max), (the p value for the third candidate),<br>
<br>
or really, one of:<br>
(A: X, B: 0, C: 10)<br>
(A: X, B: 10, C: 0)<br>
(A: 0, B: X, C: 10)<br>
(A: 10, B: X, C: 0)<br>
(A: 0, B: 10, C: X)<br>
(A: 10, B: 0, C: X)<br>
<br>
for a scale of 0..10, where X is an integer between 0 and 10 inclusive. <br>
This gives each voter 60 different ballots to choose between (contrasted <br>
with 6 per voter for ordinal).<br>
<br>
So for e.g. 10 voters, 3 candidates, there are about (60+10-1) choose 10 <br>
= 3e11 elections. There are 3003 ordinal ones. Perhaps a rating scale of <br>
0..10 is too high for an integer programming search. The trick would be <br>
to find a quantization level that's high enough that the search is <br>
feasible, yet low enough that it's possible to interpolate to an <br>
idealized method, somewhat like I did with fpA-fpC.<br>
<br>
<br>
Finally, I'd say it is perhaps possible to go beyond vNM, but I'm not <br>
entirely sure how to do it, much less how to do it in a <br>
strategy-resistant manner. On the other hand, there may be limitations <br>
beyond vNM itself: e.g. it might be hard to tell who would be a better <br>
winner between Stalin and Ivan the Terrible, because they're both so <br>
thoroughly awful as to saturate the scale.<br>
----<br>
Election-Methods mailing list - see <a href="https://electorama.com/em" rel="noreferrer noreferrer" target="_blank">https://electorama.com/em</a> for list info<br>
</blockquote></div>