Arrow’s impossibility theorem

Arrow’s Impossibility Theorem, also known as Arrow’s Paradox or the General Possibility Theorem, is a fundamental result in social choice theory.
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Updated on May 29, 2024
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3 key takeaways

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  • Arrow’s Impossibility Theorem demonstrates the inherent limitations of voting systems in translating individual preferences into a fair collective decision.
  • The theorem highlights that no voting system can satisfy all fairness criteria simultaneously, leading to potential conflicts in social choice.
  • It has profound implications for economics, political science, and decision theory, questioning the feasibility of perfectly fair and rational collective decision-making.

What is Arrow’s Impossibility Theorem?

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Arrow’s Impossibility Theorem, formulated by economist Kenneth Arrow in his 1951 book “Social Choice and Individual Values,” asserts that it is impossible to design a voting system that meets a set of seemingly reasonable criteria for fair decision-making. These criteria are intended to ensure that the voting system accurately reflects the preferences of individuals in a group decision.

Importance of Arrow’s Impossibility Theorem

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The theorem is crucial because it reveals the challenges and limitations inherent in collective decision-making processes. It underscores the complexity of aggregating individual preferences into a social welfare function that is fair and rational. Arrow’s theorem has significant implications for the design of voting systems, policy-making, and the evaluation of democratic processes.

Criteria in Arrow’s Impossibility Theorem

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  1. Non-dictatorship: No single individual should possess the power to determine the outcome of the decision regardless of the preferences of others.
  2. Unrestricted domain (universality): The voting system should work for any possible set of individual preferences.
  3. Pareto efficiency (unanimity): If every individual prefers one option over another, then the group preference should reflect the same ranking.
  4. Independence of irrelevant alternatives (IIA): The group’s preference between any two alternatives should not be affected by changes in the ranking of other irrelevant alternatives.
  5. Transitivity: If the group prefers option A over B and B over C, then the group should also prefer A over C.

How Arrow’s Impossibility Theorem works

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Arrow’s theorem demonstrates that no voting system can simultaneously satisfy all five criteria. The proof involves showing that any system attempting to meet these conditions will inevitably violate at least one of them. This result implies that all voting systems must compromise on one or more fairness criteria, leading to potential issues like dictatorship, inconsistency, or vulnerability to irrelevant alternatives.

Examples of Arrow’s Impossibility Theorem

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  • Voting systems: Common voting methods like plurality voting, ranked-choice voting, and Borda count all fail to satisfy one or more of Arrow’s criteria. For instance, ranked-choice voting may fail the IIA criterion.
  • Policy decisions: In public policy-making, Arrow’s theorem suggests that it is challenging to create a decision-making process that perfectly reflects the diverse preferences of all stakeholders while being fair and consistent.

Real-world application

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Consider a small committee tasked with choosing a new project. The members rank the potential projects based on their preferences. Using Arrow’s criteria, the committee attempts to find a voting system that can aggregate these preferences into a collective decision. They soon realize that every method they consider either gives one member undue influence (violating non-dictatorship), fails to handle all possible preference sets (violating unrestricted domain), or is inconsistent with other criteria. This scenario illustrates the practical challenges posed by Arrow’s theorem in designing fair decision-making processes.

Understanding Arrow’s Impossibility Theorem is essential for economists, political scientists, and policymakers. It provides critical insights into the limitations of collective decision-making and the need for compromise and trade-offs in designing fair and effective voting systems.

Related topics you might want to learn about include social choice theory, voting systems, and public choice theory. These areas offer further exploration into the complexities of aggregating individual preferences and the implications for democratic decision-making.


Sources & references

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