Dictatorship mechanism explained

In social choice theory, a dictatorship mechanism is a degenerate voting rule or mechanism where the result depends on only one person's preferences, without considering any other voters. 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.^[1] In Social Choice and Individual Values, Kenneth Arrow defines non-dictatorship as:

There is no voter

i in such that, for every set of orderings in the domain of the constitution, and every pair of social states x and y, x \succeq _{i} y implies x \succeq y .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. 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^[2] 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 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 or exceptions to theorems, e.g. Arrow's theorem. If there are at least three alternatives, dictatorship is the only ranked voting rule that satisfies unrestricted domain, Pareto efficiency, and independence of irrelevant alternatives. Similarly, by Gibbard's theorem, 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: there is never any advantage to tactical voting.

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

1. Game Theory Second Edition Guillermo Owen Ch 6 pp124-5 Axiom 5 Academic Press, 1982
2. Book: Felix Brandt. https://books.google.com/books?id=0qY8DwAAQBAJ&dq=multiwinner++voting+a+new+challenge&pg=PA27. Trends in Computational Social Choice. 2017-10-26. Lulu.com. 978-1-326-91209-3. Endriss. Ulle. en. Probabilistic Social Choice.