Arrow's Impossibility Theorem

Arrow's Impossibility Theorem, proposed by economist Kenneth Arrow in 1951, is a fundamental result in social choice theory that highlights the challenges of aggregating individual preferences into a collective or societal preference ranking. The theorem demonstrates that it's impossible to design a voting system that satisfies a set of seemingly reasonable criteria simultaneously.

Here's a simplified explanation of Arrow's Impossibility Theorem:

• Individual Preferences: Arrow's theorem starts with the assumption that individuals have preferences over a set of alternatives. These preferences are ranked, meaning individuals can express which alternatives they prefer over others, but there's no assumption of cardinal utility (i.e., no numerical measurement of the intensity of preferences).
• Social Choice Functions: A social choice function is a method for aggregating individual preferences into a societal preference ranking or outcome. The goal is to design a fair and consistent method that reflects the collective will of the individuals in the society.
• Arrow's Criteria: Arrow identified several criteria that a desirable social choice function should ideally satisfy. These include:
  • Universal Domain: The social choice function should be applicable to all possible profiles of individual preferences.
  • Pareto Efficiency: If every individual prefers one alternative to another, the societal preference should reflect this.
  • Independence of Irrelevant Alternatives: The societal preference between two alternatives should not be affected by the presence or absence of other irrelevant alternatives.
  • Non-Dictatorship: No single individual's preferences should always determine the societal preference ranking. In other words, the social choice function should not be equivalent to the preferences of a single individual.
• Impossibility Result: Arrow's theorem demonstrates that it's impossible to design a social choice function that satisfies all of these criteria simultaneously. In other words, no voting system can consistently and fairly aggregate individual preferences into a societal preference ranking without violating at least one of Arrow's criteria.

The theorem has profound implications for democratic decision-making and the design of voting systems. It suggests that no voting method can perfectly capture the will of the people in all situations without encountering some form of paradox or inconsistency. This has led to ongoing debates in political science and economics about the best methods for collective decision-making and the trade-offs involved in different voting systems.