# [EM] Better cardinal methods?

Kristofer Munsterhjelm km_elmet at t-online.de
Sun Aug 15 14:54:11 PDT 2021

```Suppose we take the risk neutral lottery-based definition of utility as
a basis for honesty. (That is, if you're indifferent between a 100%
chance of choice X and a 70% chance of Y, 30% chance of Z, then your
utility for X is equal to 0.7 * u(Y) + 0.3 * u(Z).)

What kind of cardinal system could incentivize voters to report this
kind of information? It seems very hard to do it with any of the broad
IIA class (score for X is just a function of ratings for X, highest
score wins, and increasing your rating for X never decreases X's chance
of winning), because those methods encourage minmax strategy.

Perhaps some kind of cumulative voting? There's Hay, but it sucks.

The lottery definition above can't determine both a natural zero and
unit value, because if you scale all utilities by some constant, the
lottery equations remain the same. So any method that takes this kind of
input should pass a kind of "irrelevance of constant scaling" property,
which says that if every voter v scales his ballot by some (private)
constant factor v_F, then the outcome remains the same.

In a Range-type method, that means the system should scale every ballot
so that one candidate is max rated and another is min rated (I think).
This would probably lead to IIA because in a two-candidate election,
you'd get majority rule (whichever candidate voter v prefers gets max
rating, and the other one gets min rating).

Maybe it's possible to preserve IIA, but I doubt it.

Any thoughts on how a method with better "honesty" incentives could be
designed for lottery-type cardinal ballots?

-km
```