# [EM] More thoughts about cardinal methods

Kristofer Munsterhjelm km_elmet at t-online.de
Tue Oct 12 06:43:45 PDT 2021

```A possible criterion for cardinal methods that take lottery information
is this:

"If there exists a single lottery that, no matter what affine scaling we
apply to a voter, every voter prefers to every other lottery, then that
lottery should win."

For a deterministic voting method, only consider deterministic lotteries
(i.e. 100% election probability for some candidate, 0% for the rest).
(There's also a Smith set analog: it should elect from the smallest
group for which some lottery inside is preferred to every such outside.)

But this gives rise to a utilitarian/OMOV tradeoff problem that I've
mentioned earlier: it might be the case that one of the voters feels
*very* strongly about the outcome, so that by the intensity of his
preference, he would have dictatorial powers over the outcome, if we
were aiming to maximize utility.

So even if we assume complete honesty, there might exist an inherent
dictator by the logic of utilitarianism itself. Thus there's a tension
between OMOV (which limits the relative power of one voter over another)
and utilitarian maximization.

A cardinal voting method based on utilitarian reasoning has to set that
limit somewhere. One reason that it's so hard to construct cardinal
methods based on lottery information might be that we haven't decided
just where that limit should be, or even thought about how it factors
into the design of the method itself.

To set the limit, there seem to be three alternatives:

- The method can directly set the tradeoff as a tunable parameter, or
- The method can be designed not to be directly utilitarian, just be
better at distinguishing "weak centrist" scenarios from "true consensus"
scenarios, or
- The method can allocate some maximum power to each ballot and then let
the voters voluntarily claim only a portion of this power if they feel
that their opinions are not sufficiently strong to warrant full intensity.

I *think* MJ is in the second category and Range in the third. Applying
DSV to the third type of method would result in something in the second,
because the voters who deliberately choose not to use the full range of
their ballots could strategize depending on who's in the running. But
this normalization doesn't have to look like Range - for instance, a
type three method could be cumulative voting with a maximum on the
Euclidean norm of the ballot; and then the corresponding strategic
method chooses a ballot with Euclidean norm at that exact maximum.

Methods of type one (I don't know of any) would try to resolve the
problem by saying "suppose each voter's utilities are within some
interval; then with honest voting with lottery information, if there
exists a dominating lottery in the sense above, then that's sure to be
the one that maximizes social utility".

Another thought: suppose that the method normalizes lottery information
to get the maximum power out of any comparison. This scale has to
involve more than two candidates - otherwise the normalization is just
"100% power to whichever I prefer", which turns into Condorcet. Would it
be possible to make a three-candidate variant with its own analogs of a
Condorcet winner and Smith set? Such a method might end up majoritarian,
but it's possible to be majoritarian and cardinal -- at least more
cardinal than ranked -- as e.g. shown by MJ.
```