Arrow's Impossibility Theorem

ID: 915ec704-db4e-463b-a1a6-7a1a7d4542fe
MTIME: [2025-01-25 Sat 18:38]

Result in economics. Solving economics is just a matter of allocating resources to people in the most efficient possible way, right? What happens if you try to do that?

This result proves that at least one reasonable-seeming way to model that situation is a dead-end.

Properties of the model:

• Universality - allocation function is complete of the space of resources and people' preferences
• Pareto-optimality - all possible Pareto improvements are made: that is, allocations A where there exists no allocations A' that are preferred by at least one person and equivelently good per everybody else.
• Independence of Irrelevant Alternatives or i.i.d) - people's preferences are consistent across the subset relation - if \(R = P_Alice({A,B}) = A >= B\), then for all \(\phi\), \(R' = P_Alice({A,B} \bigcup \phi)\) must be such that \(A R_2 B\) is \(A >= B\).
• (No) Dictatorship - No single person in the allocation function is empowered to "choose among allocations" - that is, there should be no person whose preferences completely determine the entire allocation function.

There exist no allocation functions with all four of these properties.