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.
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]] is:
The [[Cardinal voting systems#Scale%20invariance|cardinal method criterion]] (multiplicative version):


{{Definition|Multiplying every ballot's score of every candidate by a constant <math>\alpha>0</math> should not change the outcome.}}
{{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.}}
{{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:
Strong variant, for [[Cardinal voting systems#Scale%20invariance|cardinal method]] is:


{{Definition|Multiplying one or more ballot's score of every candidate by a constant <math>\alpha>0</math> should not change the outcome.}}
{{Definition|For every way of ranking the candidates, adding to the number of voters who express this preference by a constant <math>\alpha>0</math> 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.
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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==Relevant Pages==
==See also==


* [[Single distributed vote]]
* [[Single distributed vote]]

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.

This page is a stub - please add to it.

See also