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Unrestricted domain: Difference between revisions
Clarifying the prose a bit (especially the #Spatial modeling section), and making some of the links a bit clearer
(Organizing the article a bit, and noting that Gibbard–Satterthwaite theorem involves different criteria, and shows that even rated methods have problems.) |
(Clarifying the prose a bit (especially the #Spatial modeling section), and making some of the links a bit clearer) |
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== Spatial modeling ==
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{{seealso|Spatial models of voting
In [[spatial model of voting|spatial models of voting]], the choice of dimension for the latent space of voter opinions imposes [[dimensional limitations of the spatial model|fundamental limitations on the set of allowed elections]], depending on the number of candidates, as there may be insufficient room in the space for all ranked ballots to occur. This geometric result implies that violations of unrestricted domain are common in low-dimensional simulations, with the vast majority of election scenarios being impossible, and that certain voting methods with arbitrary ballot restrictions may be fundamentally unable to capture the information available in an electorate. ▼
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{{seealso|Limitations of spatial models of voting
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When voting theorists create [[spatial models of voting]], they map voter opinions regarding different topics onto different dimensions in space. The most readily understood models have one, two, or three dimensions, but more dimensions are possible.
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== Ranking ==▼
{{seealso|Ranked ballot}}▼
{{main|Ranked ballot}}
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With unrestricted domain, the social welfare function accounts for all preferences among all voters to yield a unique and complete ranking of societal choices. Thus, the voting mechanism must account for all individual preferences, it must do so in a manner that results in a complete ranking of preferences for society, and it must deterministically provide the same ranking each time voters' preferences are presented the same way.
== Relation to other social choice theorems==
{{seealso|Arrow's impossibility theorem
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{{seealso|
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It was stated by Kenneth Arrow as part of his [[Arrow's impossibility theorem|impossibility theorem]], and it is such a basic criterion that it's satisfied by all non-random ranked systems. However, since it was defined by Kenneth Arrow before there had been theoretical analysis of rated voting systems, it does not apply to rated ballots, and so all rated systems technically violate universality. This is why some rated systems, such as [[MCA|MCA-P]], can appear to violate Arrow's theorem by satisfying all of his more-interesting criteria such as [[monotonicity]] and [[independence of irrelevant alternatives]]. When not combined with (ranked) universality, those other criteria are not incompatible.▼
{{seealso|Social welfare function
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▲It was stated by Kenneth Arrow as part of his [[Arrow's impossibility theorem|impossibility theorem]], and it is such a basic criterion that it's satisfied by all non-random ranked systems. However, since it was defined by Kenneth Arrow before there had been theoretical analysis of rated voting systems, it does not apply to rated ballots, and so all rated systems technically violate universality. This is why some rated systems, such as [[MCA|MCA-P]], can appear to violate Arrow's theorem by satisfying
Unrestricted domain is one of the conditions for Arrow's impossibility theorem. Under that theorem, it is impossible to have a social choice function that satisfies ''unrestricted domain'', ''[[Pareto efficiency]]'', ''[[independence of irrelevant alternatives]]'', and ''[[non-dictatorship]]''. However, the conditions of the theorem can be satisfied if unrestricted domain is removed.
Note that [[Gibbard–Satterthwaite theorem]] involves different criteria, and shows that even rated methods
==Examples of restricted domains==
{{Seealso|Median voter theory
[[Duncan Black]] defined a restriction to domains of social choice functions called ''"single-peaked preferences"''. Under this principle, all of the choices have a predetermined position along a line, giving them a linear ordering. Every voter has some special place he likes best along that line. His ordering of the choices is determined by their distances from that spot. For example, if voting on where to set the volume for music, it would be reasonable to assume that each voter had their own ideal volume preference and that as the volume got progressively too loud or too quiet they would be increasingly dissatisfied. Black [[Median voter theory|proved]] that by replacing unrestricted domain with single-peaked preferences in Arrow's theorem removes the impossibility: there are Pareto-efficient non-dictatorships that satisfy [[Independence of irrelevant alternatives|IIA]].▼
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▲[[Duncan Black]] defined a restriction to domains of social choice functions called ''"single-peaked preferences"''. Under this principle, all of the choices have a predetermined position along a line, giving them a linear ordering. Every voter has some special place he likes best along that line. His ordering of the choices is determined by their distances from that spot. For example, if voting on where to set the volume for music, it would be reasonable to assume that each voter had their own ideal volume preference and that as the volume got progressively too loud or too quiet they would be increasingly dissatisfied. Black [[Median voter theory|proved]] that by replacing unrestricted domain with single-peaked preferences in Arrow's theorem removes the impossibility: there are Pareto-efficient non-dictatorships that satisfy the "[[
==References==
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[[Category:Social choice theory]]
[[Category:Voting system criteria]]
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