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 $\alpha >0$ should not change the outcome.

The additive version:

Adding a constant $\alpha$ 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 $\alpha >0$ should not change the outcome.

Additive:

Adding a constant $\alpha$ 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 $\alpha >0$ should not change the outcome.

Additive:

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

These criteria represent a desideratum that the method should not rely on absolute numbers when selecting a winner, just on the candidates' or factions' relative support.

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See also