In
social choice theory
Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...
, a dictatorship mechanism is a
degenerate voting rule or
mechanism
Mechanism may refer to:
*Mechanism (economics), a set of rules for a game designed to achieve a certain outcome
**Mechanism design, the study of such mechanisms
*Mechanism (engineering), rigid bodies connected by joints in order to accomplish a ...
where the result depends on one person's. A serial dictatorship is similar, but also designates a series of "backup dictators", who break ties in the original dictator's choices when the dictator is indifferent.
Formal definition
Non-dictatorship is one of the necessary conditions in
Arrow's impossibility theorem
Arrow's impossibility theorem is a key result in social choice theory showing that no ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Arrow showed no such rule can satisfy the ind ...
.
[''Game Theory'' Second Edition Guillermo Owen Ch 6 pp124-5 Axiom 5 Academic Press, 1982 ] In ''
Social Choice and Individual Values
Kenneth Arrow's monograph ''Social Choice and Individual Values'' (1951; revised in 1963 and 2012) and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. Somew ...
'',
Kenneth Arrow
Kenneth Joseph Arrow (August 23, 1921 – February 21, 2017) was an American economist, mathematician and political theorist. He received the John Bates Clark Medal in 1957, and the Nobel Memorial Prize in Economic Sciences in 1972, along with ...
defines non-dictatorship as:
:There is no voter
in such that, for every set of orderings in the domain of the constitution, and every pair of social states ''x'' and ''y'', ''
'' implies
.
Unsurprisingly, a dictatorship is a rule that does not satisfy non-dictatorship.
Anonymous voting rules automatically satisfy non-dictatorship (so long as there is more than one voter).
Serial dictatorship
When the dictator is indifferent between two or more best-preferred options, it is possible to choose one of them arbitrarily or randomly, but this will not be strictly
Pareto efficient
In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse ...
. A more efficient solution is to appoint a secondary dictator, who has a right to choose, from among all the first dictator's best options, the one that they most prefer. If the second dictator is also indifferent between two or more options, then a third dictator chooses among them, and so on; in other words, ties are broken
lexicographically. This rule is called serial dictatorship
{{Rp, 6 or the priority mechanism.
The priority mechanism is sometimes used in problems of
house allocation. For example, when allocating dormitory rooms to students, it is common for
academic administrators
Academic administration is a branch of university or college employees responsible for the maintenance and supervision of the institution and separate from the faculty or academics, although some personnel may have joint responsibilities. Some ...
to care more about avoiding effort than about the students' well-being or fairness. Thus, students are often assigned a pre-specified priority order (e.g. by age, grades, distance, etc.) and is allowed to choose their most preferred room from the available ones.
Properties
Dictatorships often crop up as
degenerate cases
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descri ...
or exceptions to theorems, e.g.
Arrow's theorem
Arrow's impossibility theorem is a key result in social choice theory showing that no Ordinal utility, ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Kenneth Arrow, Arrow showed no ...
. If there are at least three alternatives, dictatorship is the only
ranked voting rule that satisfies ''
unrestricted domain'', ''
Pareto efficiency
In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse ...
'', and ''
independence of irrelevant alternatives
Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between A and B (which are both related) should not depend on the quality of a third, unrelated outcome C. There are several dif ...
''. Similarly, by
Gibbard's theorem
In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It states that for any deterministic process of collective decision, at least one of the following three properti ...
, when there are at least three candidates, dictatorship is the only
strategyproof rule.
Single-winner
Satisfied criteria include:
*
Perfect decisiveness: there is no possibility of a tied vote, assuming some selected voter has expressed a preference.
*
Strategyproofness
In mechanism design, a strategyproof (SP) mechanism is a game form in which each player has a weakly- dominant strategy, so that no player can gain by "spying" over the other players to know what they are going to play. When the players have privat ...
: there is never any advantage to
tactical voting
Strategic or tactical voting is voting in consideration of possible ballots cast by other voters in order to maximize one's satisfaction with the election's results.
Gibbard's theorem shows that no voting system has a single "always-best" stra ...
.
Failed criteria include:
*
Determinism: the results depend on chance.
*
Majority-rule: even if a single candidate has support from a majority in every subelection, that candidate may lose.
References
Voting theory