[EM] STAR

[Kristofer Munsterhjelm]

On 8/16/23 18:52, fdpk69p6uq at snkmail.com wrote:
> 
> Ranked ballots can't capture strength of preference. It's possible for a 
> majority-preferred candidate to be very polarizing (loved by 51% and 
> hated by 49%), while the minority-preferred candidate is broadly-liked 
> and has a much higher overall approval/favorability rating.  Which 
> candidate is the rightful winner?
> 
> https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html <https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html>
> 
> "Suppose you and a pair of friends are looking to order a pizza. You, 
> and one friend, really like mushrooms, and prefer them over all other 
> vegetable options, but you both also really, /really/ like pepperoni. 
> Your other friend also really likes mushrooms, and prefers them over all 
> other options, but they're also vegetarian. What one topping should you 
> get?
> 
> Clearly the answer is mushrooms, and there is no group of friends worth 
> calling themselves such who would conclude otherwise. It's so obvious 
> that it hardly seems worth calling attention to. So why is it, that if 
> we put this decision up to a vote, do so many election methods, which 
> are otherwise seen as perfectly reasonable methods, fail? Plurality, 
> top-two runoffs <http://en.wikipedia.org/wiki/Two-round_system>, instant 
> runoff voting <http://en.wikipedia.org/wiki/Instant-runoff_voting>, all 
> variations of Condorcet's method 
> <http://en.wikipedia.org/wiki/Condorcet_method>, even Bucklin voting 
> <http://en.wikipedia.org/wiki/Bucklin_voting>; all of them, incorrectly, 
> choose pepperoni."

As a ranked voting guy, I'd say because incommensurability is a pain and 
because it invites strategy by honest voters.

(Both of these incidentally are motivations for my attempt to make a 
rated method that "takes von Neumann-Morgenstern utilities seriously". 
But it's very difficult, mainly because strategic equilibria for 
Approval result in the ordinal Condorcet winner being elected. See my 
quick and dirty STAR ideas earlier.)

Point one: friends usually know what they mean when they say "that's 
good!" or "that sucks". But suppose you have an electorate of 200 
million. Some of these only rate totalitarian dictators a zero, while 
others rate mildly unpleasant candidates a zero. Some of these only rate 
true angels a ten, while others always rate their favorite frontrunner ten.

Now suppose the vote is so that A is the majority winner but B is ever 
so slightly ahead by ratings. How do you know that B is truly the best 
winner, and doesn't just have voters who tend to vote the whole range?

Point two: Now suppose you're an honest voter in this situation. The 
ballot asks you for how much you like a candidate. What's the "right" 
way to vote? How far do you lower the candidate you like the least? This 
kind of ambiguity invites strategy and min/maxing, and in the case of 
Range, further to voting Approval style (going "all the way" adapting 
your scores to the situation).

My generalized STAR attempts thus reduce to majority rule in the 
two-candidate case, because there's no way to absolutely calibrate the 
different voters' scales. And I've been playing with lp norm 
normalization among candidate triples, since that discourages 
Approval-style voting. (IIRC, there are some arguments that l2 
normalization incentivizes ratings whose intervals are proportional to 
the utility differences, but I wouldn't be able to prove it.)

> (And strength of preference is clearly a real thing in our brains.  If 
> you prefer A > B > C, and are given the choice between Box 1, which 
> contains B, and Box 2, which has a 50/50 chance of containing A or C, 
> which do you choose?  What if the probability were 1 in a million of Box 
> 2 containing C?  By varying the probability until it's impossible to 
> decide, you can measure the relative strength of preference for B > C vs 
> A > C.)

That's von Neumann-Morgenstern utilities, as I understand it. (From my 
experience, sometimes it gets very hard to decide when you're close to 
indifferent, though.)

The lottery information provides strength of preference up to an affine 
scaling. So for each voter, there are two free parameters:

R(v, x) = a_v * U(v, x) + b_v

where R(v, x) is the voter's rating, a and b are constants, and U(v, x) 
is voter v's (perceived) utility of getting x elected. The lottery 
method gives us the ratio between the utilities (assuming risk 
neutrality). But not a_v or b_v!

We can design the method so that it assumes every voter's scaling 
constants are the same. That gives something like Range. Or we can 
design it so that each voter's power is similar (OMOV), which gives 
something that reduces to majority rule in the two-candidate case.

-km