Scale invariance

Scale invariance can refer to one of two criteria: a cardinal voting method criterion and an ordinal one, and these can be multiplicative or additive.

The cardinal method criterion (multiplicative version):

Multiplying every ballot's score of every candidate by a constant ${\displaystyle \alpha>0}$ should not change the outcome.
Adding a constant ${\displaystyle \alpha}$ to every ballot's score of every candidate should not change the outcome.

with a stronger variant being (multiplicative):

Multiplying one or more ballot's score of every candidate by a constant ${\displaystyle \alpha>0}$ should not change the outcome.
Adding a constant ${\displaystyle \alpha}$ to one or more ballot's score of every candidate should not change the outcome.

The ordinal method criterion is also called the homogeneity criterion. It is (multiplicative):

For every way of ranking the candidates, multiplying the number of voters who express this preference by a constant ${\displaystyle \alpha>0}$ should not change the outcome.

For every way of ranking the candidates, adding to the number of voters who express this preference by a constant ${\displaystyle \alpha>0}$ should not change the outcome.