Arrow’s Theorem (SEP)
Kenneth Arrow’s “impossibility” theorem—the “general possibility” theorem, as he called it—answers a very basic question in the theory of collective decision-making. Say there are some alternatives to choose among. They could be policies, public projects, candidates in an election, distributions of income and labour requirements or just about anything else. There are some people whose preferences among these alternatives will inform this choice, and the question is: which procedures are there for deriving, from what is known or can be found out about their preferences, a collective or “social” ordering of the alternatives from better to worse? The answer is startling. Arrow’s theorem says there are no such procedures at all—none, anyway, that meet certain conditions concerning the autonomy of the people and the rationality of their preferences. The technical framework in which Arrow gave the question of social orderings a precise sense and its rigorous answer is now widely used for studying problems in welfare economics. The impossibility theorem itself set the agenda for contemporary social choice theory. Arrow accomplished this while still a graduate student. In 1972, he received the Nobel Prize in economics for his contributions.
Read the rest here: https://plato.stanford.edu/entries/arrows-theorem/
Chatbot
Arrow’s Impossibility Theorem is a result in social choice theory about voting systems.
It states that no voting system can convert individual preference rankings into a collective ranking while satisfying a small set of seemingly reasonable conditions, as long as there are at least three options.
The conditions are usually given as:
- Unrestricted domain: voters may have any preference orderings.Pareto efficiency: if everyone prefers A over B, society should prefer A over B.
- Independence of irrelevant alternatives: the social ranking of A vs. B depends only on individual rankings of A vs. B, not on other options.
- Non-dictatorship: no single voter always determines the outcome.
Arrow proved that these conditions cannot all be met simultaneously. If a system satisfies the first three, it must be dictatorial. The point is not that democracy is impossible, but that every voting system involves trade-offs. Any real system violates at least one of these conditions, often in hidden or non-obvious ways. Philosophically, the theorem shows that “the will of the people” is not a well-defined object. Collective rationality does not follow automatically from individual rationality.
Wikipedia
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, American economist Kenneth Arrow showed 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.
The result is often cited in discussions of voting rules, where it shows no ranked voting rule can eliminate the spoiler effect.This result was first shown by the Marquis de Condorcet, whose voting paradox showed the impossibility of logically-consistent majority rule; Arrow’s theorem generalizes Condorcet’s findings to include non-majoritarian rules like collective leadership or consensus decision-making.
While the impossibility theorem shows all ranked voting rules must have spoilers, the frequency of spoilers differs dramatically by rule. Plurality-rule methods like choose-one and ranked-choice (instant-runoff) voting are highly sensitive to spoilers, creating them even in some situations where they are not mathematically necessary (e.g. in center squeezes). In contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections by restricting them to voting cycles, which are rare in ideologically-driven elections. Under some models of voter preferences (like the left-right spectrum assumed in the median voter theorem), spoilers disappear entirely for these methods.
Rated voting rules, where voters assign a separate grade to each candidate, are not affected by Arrow’s theorem. Arrow initially asserted the information provided by these systems was meaningless and therefore could not be used to prevent paradoxes, leading him to overlook them. However, Arrow would later describe this as a mistake, admitting rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.
Read the rest here: https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem
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