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 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.

When Kenneth Arrow proved his theorem in 1950, it inaugurated the modern field of social choice theory, a branch of welfare economics studying mechanisms to aggregate preferences and beliefs across a society. Such a mechanism of study can be a market, voting system, constitution, or even a moral or ethical framework.

In the context of Arrow's theorem, citizens are assumed to have ordinal preferences, i.e. orderings of candidates. If A and B are different candidates or alternatives, then ${\displaystyle A\succ B}$ means A is preferred to B. Individual preferences (or ballots) are required to satisfy intuitive properties of orderings, e.g. they must be transitive—if ${\displaystyle A\succeq B}$ and ${\displaystyle B\succeq C}$, then ${\displaystyle A\succeq C}$. The social choice function is then a mathematical function that maps the individual orderings to a new ordering that represents the preferences of all of society.

Arrow's theorem assumes as background that any non-degenerate social choice rule will satisfy:

Arrow's original statement of the theorem included non-negative responsiveness as a condition, i.e., that increasing the rank of an outcome should not make them lose—in other words, that a voting rule shouldn't penalize a candidate for being more popular. However, this assumption is not needed or used in his proof (except to derive the weaker condition of Pareto efficiency), and Arrow later corrected his statement of the theorem to remove the inclusion of this condition.