[EM] Highly-expressive preference voting

[Kristofer Munsterhjelm]

On 08/27/2015 02:02 PM, Sebastiaan Snoeckx wrote:
> Hello
>
> This may sound like an insanely strange question, but I was wondering
> whether there were specific election algorithms and ballot designs that
> would allow a voter to express preferences between specific candidates,
> without having to specify their preference between the expressions of
> preferences themselves.
>
> Don't worry if this sounds inconsistent, I'll explain by example:
>   1. The voter prefers A over B (A>B)
>   2. The voter prefers C over D (C>D)
>   3. The voter prefers E over F and G (E>F=G)
>   4. The voter prefers their own preference of A>B over their preference
> C>D, but could care less whether E>F=G is preferred over the others
>
> Notationally, it would be a bit like this: ((A>B)>(C>D))=(E>F=G)

A suitably altered version of Condorcet could do something like this. 
Here's a simple (but probably impractical) example:

1. For each pair of candidates X, Y:

1.1. If the voter has already answered whether he prefers Y to X or X to 
Y, skip.

1.2. Otherwise, ask the voter if he prefers X to Y, Y to X, or considers 
them equal.

1.3. Then ask the voter his strength of preference from 0 to 1. Call 
this preference intensity p.

1.4. If the voter preferred X to Y, let c[X, Y] = p, c[Y, X] = 0; 
otherwise, let c[X, Y] = 0, c[Y, X] = p.

Once all voters have constructed their c-matrices, add the matrices up 
and run the result through your Condorcet method of choice.

If the meta-preferences are ranked rather than rated, it gets trickier. 
I imagine variants of say, Ranked Pairs would still work, though. 
Something like:

when you determine whether to lock A>B, only count the voters whose 
preferences for A>B are stronger than preferences that have already been 
locked in or excluded.

Another option is to only consider stronger preferences and use weaker 
preferences as tiebreaks. The reasoning would be: say that you're voter 
X and have a strong preference for A>B and a weak for C>D, and say some 
other voter Y has a strong preference for B>A and a weak for D>C. Then 
it would be strange if his weak preference counts the same as your 
strong one. Hence, only strong preferences should be considered at 
first; then, if there are any ties, use the weak preferences. But 
strategy might complicate the picture.

> Ow! I can imagine any voting system choking over this (and imagine this
> happening with loops allowed!), but it is an incredibly common thing in
> real life: people prefer burgers over pizza and prefer coke over sprite
> (YMMV!), but when you ask them wether this mean that they prefer burgers
> over coke or pizza over sprite, they'll shrug and say these are not
> comparable: (burgers>pizza)=(coke>sprite).
>
> In real-life elections, candidates are rarely comparable to each other
> (ie. one-issue candidates or mutually-complementary ideologies), and
> forcing voters to rank (or score, in a cardinal system) incomparable
> candidates or ideologies seems to me like a lot of information is lost.
>
> Did this make any sense at all?

I think that's more like having a partial order, which is to say that 
you know (about your own preferences):

A>B, B>C, but you don't know whether A>C.

This is very similar to the problem ranked voting tries to solve in the 
first place, because you have something like:

voter X: A>B
voter Y: B>C

but you can't say anything *directly* about society's preference about 
A>C. Society's preference might even be cyclical, as Condorcet cycle 
examples show.

If the analogy is accurate, then the easiest way to deal with 
burgers>pizza, coke>sprite would be to pretend that you're two 
"mini-voters". These "mini-voters" would have weight 1/2 each (since 
there's no reason the coke preference should override the burger 
preference or vice versa), and they would vote:

mini-voter A: burgers>pizza
mini-voter B: coke>sprite

Note that they say nothing about the combined preference of burgers>coke 
or pizza>sprite.
Similarly, say you were evaluating candidates on three different axes 
(say defense, financial policy, and resistance to corruption), then you 
could have three mini-voters who rank like this:

defense: A>B>C
economics: B>A>C
incorruptibility: C>A>B

That will leave it to the system how to extrapolate the social 
preferences, which is the very purpose *of* that system :-) The system 
would thus have more information to go on.

Note that if everybody does this, then you don't even have to scale the 
power of the mini-voters, since each voter will have three different 
mini-voters and thus equal power. In practice, though, you'd want to 
scale the weight of each mini-voter so that different people with 
different number of mini-voters each get the same power.