# Unrestricted domain

In social choice theory, **unrestricted domain**, or **universality**, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed. Intuitively, unrestricted domain is a common requirement for social choice functions, and is a condition for Arrow's impossibility theorem.

The **universality criterion** requires that a system give unique results for a given set of ranked ballots, i.e., that any set of ranked ballots should be viable as a valid set in the domain of the social choice function.

## Spatial modeling[edit | edit source]

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Spatial model of voting |

In spatial models of voting, the choice of dimension for the latent space of voter opinions imposes 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.

## Ranking[edit | edit source]

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Ranked ballot |

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[edit | edit source]

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Arrow's impossibility theorem |

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Gibbard–Satterthwaite theorem |

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Social welfare function |

It was stated by Kenneth Arrow as part of his 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-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.

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 have problems.

## Examples of restricted domains[edit | edit source]

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 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 IIA.

## References[edit | edit source]

- Arrow, K.J. (August 1950), "A Difficulty in the Concept of Social Welfare" (PDF),
*Journal of Political Economy*,**58**(4): 328–346, doi:10.1086/256963.