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

Kristofer Munsterhjelm km_elmet at t-online.de
Sun Dec 19 03:02:14 PST 2021

```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 do have a way of connecting 3 cddts, k voters to 4 cddts, k voters:
clone independence. But the optimization program turns out to be very
slow since the number of elections grows much more rapidly in the number
of candidates than voters.)

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?

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.

-km
```

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