Arrow's Impossibility Theorem

Game Theory · Axiom Academy

LESSON Arrow's Impossibility Theorem Understanding why no perfect voting system can exist: the fundamental impossibility of social choice 1. The Four Fairness Conditions Arrow identified four seemingly reasonable conditions that any "fair" voting system should satisfy. Let's explore each condition and what it means for social choice. Unrestricted Domain (U): The voting system must work for any possible set of individual preference orderings Pareto Efficiency (P): If every voter prefers option A over option B, then A should be ranked higher than B in the social ordering Independence of Irrelevant Alternatives (IIA): The social preference between A and B should depend only on individual preferences between A and B, not on preferences involving other options Non-Dictatorship (ND): No single voter's preferences should always determine the social preference, regardless of other voters' preferences These conditions seem eminently reasonable. Surely a democratic voting system should satisfy all four, right? Arrow's theorem shows this is impossible. 2. Independence of Irrelevant Alternatives The IIA condition is often the most controversial and counterintuitive. It states that introducing or removing a third alternative should not change the relative ranking between two options. This is why "spoiler candidates" can affect elections. The IIA condition would prevent this, but Arrow showed we cannot have IIA along with the other three conditions. 3. Proof Sketch: The Impossibility

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