Unrestricted domain
Encyclopedia
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
.
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.
, independence of irrelevant alternatives
, and non-dictatorship
. However, the conditions of the theorem can be satisfied if unrestricted domain is removed.
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 she likes best along that line. Her 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
.
Social choice theory
Social choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
, 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
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
.
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 Arrow's impossibility theorem
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 efficiencyPareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
, independence of irrelevant alternatives
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
, and non-dictatorship
Non-dictatorship
In voting theory, non-dictatorship is a property of social choice functions, where the results cannot simply mirror that of any ONE single person's preferences without consideration of the other voters. Fairness requires that the social welfare function take into account the desires of more than...
. However, the conditions of the theorem can be satisfied if unrestricted domain is removed.
Examples of restricted domains
Duncan BlackDuncan Black
Duncan Black was a Scottish economist who laid the foundations of social choice theory. In particular he was responsible for unearthing the work of many early political scientists, including Charles Dodgson, and was responsible for the Black electoral system, a Condorcet method whereby, in the...
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 she likes best along that line. Her 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
Median voter theory
The median voter theory, also known as the median voter theorem or Black's theorem, is a famous voting theorem. It posits that in a majority election, if voter policy preferences can be represented as a point along a single dimension, if all voters vote deterministically for the politician who...
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
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
.