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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.
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:
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).
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] : 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 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.
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.
Satisfied criteria include:
Failed criteria include:
In social choice theory, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently self-contradictory. The result implies that it is logically impossible for any voting system to guarantee a winner will have support from a majority of voters: in some situations, a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called Condorcet cycles or cyclic ties.
Arrow's impossibility theorem is a key result in social choice theory, showing that no ranking-based decision rule can satisfy the requirements of rational choice theory. Most notably, Arrow showed that no such rule can satisfy independence of irrelevant alternatives, the principle that a choice between two alternatives A and B should not depend on the quality of some third, unrelated option C.
In welfare economics and social choice theory, a social welfare function—also called a socialordering, ranking, utility, or choicefunction—is a function that ranks a set of social states by their desirability. Each person's preferences are combined in some way to determine which outcome is considered better by society as a whole. It can be seen as mathematically formalizing Rousseau's idea of a general will.
Independence of irrelevant alternatives (IIA) is a major axiom of decision theory which codifies the intuition that a choice between and should not depend on the quality of a third, unrelated outcome . There are several different variations of this axiom, which are generally equivalent under mild conditions. As a result of its importance, the axiom has been independently rediscovered in various forms across a wide variety of fields, including economics, cognitive science, social choice, fair division, rational choice, artificial intelligence, probability, and game theory. It is closely tied to many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski-Young theorem, and the money pump arguments.
The Gibbard–Satterthwaite theorem is a theorem in voting theory. It was first conjectured by the philosopher Michael Dummett and the mathematician Robin Farquharson in 1961 and then proved independently by the philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner, and shows that for every voting rule of this form, at least one of the following three things must hold:
A random ballot or random dictatorship is a randomized electoral system where the election is decided on the basis of a single randomly-selected ballot. A closely-related variant is called random serialdictatorship, which repeats the procedure and draws another ballot if multiple candidates are tied on the first ballot.
In social choice theory, May's theorem, also called the general possibility theorem, says that majority vote is the unique ranked social choice function between two candidates that satisfies the following criteria:
The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox proposed by Amartya Sen which shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:
Social choice theory is a branch of welfare economics that analyzes methods of combining individual opinions, beliefs, or preferences to reach a collective decision or create measures of social well-being. It contrasts with political science in that it is a normative field that studies how societies should make decisions, whereas political science is descriptive. Social choice incorporates insights from economics, mathematics, philosophy, political science, and game theory to find the best ways to combine individual preferences into a coherent whole, called a social welfare function.
Kenneth Arrow's monograph Social Choice and Individual Values and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. Somewhat formally, the "social choice" in the title refers to Arrow's representation of how social values from the set of individual orderings would be implemented under the constitution. Less formally, each social choice corresponds to the feasible set of laws passed by a "vote" under the constitution even if not every individual voted in favor of all the laws.
In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationality of preference aggregation rules, such as voting rules. It is an indicator of the extent to which an aggregation rule can yield well-defined choices.
Arunava Sen is a professor of economics at the Indian Statistical Institute. He works on Game Theory, Social Choice Theory, Mechanism Design, Voting and Auctions.
Maximal lotteries refers to a probabilistic voting rule. The method uses preferential ballots and returns a probability distribution of candidates that a majority of voters would weakly prefer to any other.
In economics, dichotomous preferences (DP) are preference relations that divide the set of alternatives to two subsets, "Good" and "Bad".
Random priority (RP), also called Random serial dictatorship (RSD), is a procedure for fair random assignment - dividing indivisible items fairly among people.
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 properties must hold:
Fractional, stochastic, or weighted social choice is a branch of social choice theory in which the collective decision is not a single alternative, but rather a weighted sum of two or more alternatives. For example, if society has to choose between three candidates, then in standard social choice exactly one of these candidates is chosen. By contrast, in fractional social choice it is possible to choose any linear combination of these, e.g. "2/3 of A and 1/3 of B".
Budget-proposal aggregation (BPA) is a problem in social choice theory. A group has to decide on how to distribute its budget among several issues. Each group-member has a different idea about what the ideal budget-distribution should be. The problem is how to aggregate the different opinions into a single budget-distribution program.
The median voting rule or median mechanism is a rule for group decision-making along a one-dimensional domain. Each person votes by writing down his/her ideal value, and the rule selects a single value which is the median of all votes.
In mechanism design, a regret-free truth-telling mechanism is a mechanism in which each player who reveals his true private information does not feel regret after seeing the mechanism outcome. A regret-free mechanism incentivizes agents who want to avoid regret to report their preferences truthfully.
