Scale invariance: Difference between revisions
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(Added in "additive" scale invariance an addition to multiplicative scale invariance.) |
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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 voting systems#Scale%20invariance|cardinal method criterion]]
{{Definition|Multiplying every ballot's score of every candidate by a constant <math>\alpha>0</math> should not change the outcome.}}The additive version:
{{Definition|Adding a constant <math>\alpha</math> to every ballot's score of every candidate should not change the outcome.}}
with a stronger variant being (multiplicative):
{{Definition|Multiplying one or more ballot's score of every candidate by a constant <math>\alpha>0</math> should not change the outcome.}}Additive:
The ordinal method criterion is also called the [[homogeneity criterion]]. It is:▼
{{Definition|Adding a constant <math>\alpha</math> 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):
{{Definition|For every way of ranking the candidates, multiplying the number of voters who express this preference by a constant <math>\alpha>0</math> should not change the outcome.}}
Additive:
{{Definition|
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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* [[Single distributed vote]]
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Latest revision as of 21:42, 9 October 2021
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 should not change the outcome.
The additive version:
Adding a constant 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 should not change the outcome.
Additive:
Adding a constant 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 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 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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