[EM] Criteria linking elections with the same number of candidates?

[Kevin Venzke]

Hi Kristofer,

> Le dimanche 19 décembre 2021, 05:02:25 UTC−6, Kristofer Munsterhjelm <km_elmet at t-online.de> a écrit :
> 
> I got to thinking about my minimally strategically vulnerable method
> finder again. While it can find results for small numbers of candidates
> and voters - basically a lookup table - I realized that I don't have a
> way to connect the optimum for, say, 3 candidates and 4 voters, to 3
> candidates and 5 voters.
> 
> Since models tend to overfit unless they're kept in check somehow, this
> raises the chance that the lookup table it finds from elections to
> results for 3 candidates and 4 voters is optimized only locally, while
> any method that would actually behave this way would be very strange
> away from the direction of the optimizer's spotlight. As it were.
> 
> So in order to constrain the method, it would be a good idea to optimize
> over many number-of-voters settings simultaneously (e.g. 3 voters, 4
> voters, 7 voters). But I don't any good "tethering" properties to do so:
> to relate behavior with fewer voters to behavior with more, that are
> also passed by most useful methods to begin with.

I guess I don't, either. I usually simulate like-minded factions rather
than individual voters. And even when I've done a dozen or so individual
voters, the results were still random simulations, not an exhaustive
calculation.

> So how about this? "Restricted mono-add-top": Suppose that the social
> order is A>B>C. Then adding another ballot of the form A>B>C should not
> change the winner (or perhaps more strongly, the social order).
> 
> Do any methods commonly considered to be good fail this criterion? Is it
> incompatible with Condorcet?

I think it may be compatible with Condorcet, but I definitely see good
reason to suspect that it wouldn't be: When you repeatedly add ballots
matching the social order, you could gradually reverse defeats, boding
poorly especially for Smith methods which have heavy reliance on the
directions of defeats.

> And can you think of other such criteria that relate elections with
> (v+k) voters to elections with v voters? The multiplicative scale
> invariance:
> 
> v voters to kv voters, k integer: if you duplicate each ballot k times,
> the outcome should be the same
> 
> is simple enough, but kv gets very large very quickly, as my solver has
> trouble dealing with anything beyond 10 voters.
> 
> Is this v -> v+2 any good?
> 
> Suppose A is the winner. Adding a ballot and its reverse should not make
> someone else win.

I would say no, it's not. Imagine any arbitrary ballot and its reverse
getting duplicated indefinitely. Of the original scenario you will preserve
only the direction and margin of the pairwise wins. The reversed ballots
could dramatically change the overall character of the ballots. I guess
that will severely limit the scope of your search.

Kevin