Overview
Arrow's Impossibility Theorem demonstrates that no procedure exists for deriving rational collective preferences from individual preferences when certain reasonable conditions are met. The theorem, developed by Kenneth Arrow while a graduate student, revolutionized social choice theory and welfare economics, earning him the 1972 Nobel Prize in Economics.
The Paradox of Voting
- Condorcet discovered (1785) that majority voting can produce cycles rather than rational social orderings.
- Example: Three voters rank A, B, C differently; majority prefers A to B, B to C, but C to A—creating a cycle.
- Individual preferences can be maximized separately, but aggregated social preferences may lack a maximum.
- This demonstrates how rational choice possibilities vanish when individual preferences are aggregated using pairwise majority decision.
- Arrow investigated whether other aggregation procedures avoid such shortcomings or have problems of their own.
Arrow's Technical Framework
- Arrow studied procedures for deriving social orderings from individual preferences among fixed alternatives X and people 1,…,n.
- The problem arises before preferences are known; procedures must handle any possible preferences.
- Individual preferences are represented as weak orderings Rᵢ (connected and transitive binary relations).
- A preference profile lists all individual orderings; multiple profiles represent different possible worlds.
- Social welfare functions map profiles to social orderings, aggregating individual preferences into collective rankings.
The Five Conditions
| Condition | Symbol | Meaning |
|---|
| Unrestricted Domain | U | Function handles every logically possible profile of individual preferences |
| Social Ordering | SO | Output is always a complete, transitive ranking (no cycles) |
| Weak Pareto | WP | Unanimous strict preferences must be respected in social ordering |
| Non-Dictatorship | D | No individual's strict preferences always determine social preferences |
| Independence of Irrelevant Alternatives | I | Social preference between x,y depends only on individual preferences between x,y |
The Impossibility Theorem
- Arrow proved that with more than two alternatives, no social welfare function satisfies all five conditions simultaneously.
- The theorem conflicts with Enlightenment ideals of a common will of the people.
- It suggests democratic government by popular will may be incoherent, though interpretations vary.
- Arrow demonstrated this impossibility while answering whether procedures exist for rational collective decision-making.
Critical Examination of Conditions
- Unrestricted Domain requires handling any preferences, not any alternatives; weakening this enables some possibility results.
- Social Ordering ensures path-independent choices; relaxing transitivity to quasi-transitivity permits only oligarchies.
- Weak Pareto conflicts with liberalism (Sen's Impossibility of a Paretian Liberal) despite appearing harmless.
- Non-Dictatorship can exclude democratic procedures in restricted domains; dictator doesn't imply power or control.
- Independence of Irrelevant Alternatives excludes positional information; most controversial condition; supports non-manipulability.
Domain Restrictions
- Single-peaked preferences arise when everyone cares about same attribute (temperature, cost, political orientation).
- Black showed pairwise majority decision yields orderings on single-peaked domains with odd numbers of voters.
- The median voter's preference becomes the social maximum under single-peakedness.
- Phantom voters enable Arrow-consistency with even numbers of voters on single-peaked domains.
- Restricting domains to profiles sharing preference similarities enables satisfaction of Arrow's other conditions.
Alternative Information Structures
- Positional voting (Borda counting) uses rank information but violates Independence; susceptible to strategic manipulation.
- Fairness considerations (envy-freeness) require information about preferences for irrelevant alternatives.
- Grading systems allow individuals to input ordinal scores; median grading satisfies reformulated Arrow conditions.
- Cardinal utilities with interpersonal comparability enable classical utilitarian and Rawlsian approaches.
- Sen's framework coordinates social orderings based on measurability and comparability assumptions about utilities.
Applications and Reinterpretations
- Judgment aggregation applies Arrow framework to collective belief formation; similar impossibilities arise.
- Multi-criterial decision-making in engineering, ethics, and epistemology faces Arrow-like constraints.
- Theory choice in science involves weighing criteria like simplicity, fit to data, and scope.
- Overall similarity aggregation relevant to Lewis's metaphysics faces impossibility results.
- Verisimilitude combines truth-likeness dimensions; Arrow-inspired results limit combination possibilities.
Key Terms and Definitions
- Weak ordering: Binary relation that is both connected (comparable) and transitive (consistent).
- Preference profile: List of individual weak orderings, one per person, representing preferences.
- Social welfare function: Maps preference profiles to social orderings according to specified rules.
- Welfarism: Doctrine that social goodness depends exclusively on individual utilities or preferences.
- Single-peaked preferences: Each person has a bliss point; preferences decline moving away from it.
- Arrow-consistent domain: Restricted domain permitting social welfare functions satisfying all non-domain conditions.