# 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

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Spatial models of voting |

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Limitations of spatial models of voting |

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.

Mapping voter opinion to multi-dimensional space is tricky. When theorists choose four or more dimensions for their models, the result is difficult to visualize in a three-dimensional space. Moreover, the 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 involving many candidates, with the vast majority of election scenarios being impossible to arise. For example, for a 2-dimensional Euclidean spatial model with 6 candidates, there are possible rankings, but it is geometrically impossible to construct a voter and candidate distribution which produces more than 101 distinct ballots, meaning at least 86% of ballots will never emerge in any voter or candidate distribution. Elections involving more than 101 unique ballots are impossible in such a scenario, and these correspond to the vast majority of possible elections. See space of possible elections and limitations of spatial models of voting for more details.

Conversely, notwithstanding any assumptions of a spatial model, certain voting methods with arbitrary ballot restrictions may be fundamentally unable to capture the information available in an electorate.

## Ranking

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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

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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 other criteria he defined 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 are unable to simultaneously satisfy some seemingly sensible criteria that one would hope all election methods would comply.

## Examples of restricted domains

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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 they like best along that line. Their ordering of the choices is determined by the distances from the spot defined by the voter's preference. 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 (see "Median voter theory") 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 "independence of irrelevant alternatives" criterion.

## References

- 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.