Arrow’s Impossibility Theorem

Question: Is there a public decision making process, voting method, or “Social Welfare Function” (SWF) that will tell us if people are better off?

Arrow’s Impossibility Theorem Question: Is there a public decision making process, voting method, or “Social Welfare Function” (SWF) that will tell us if people are better off? Arrow argued that a reasonable SWF should meet 5 minimal conditions:

Arrow’s Impossibility Theorem: Conditions Condition #1: Given any three alternative states, the SWF must provide a complete, transitive, and reflexive ranking of states for any set of individual preferences over states.

Arrow’s Impossibility Theorem: Conditions Condition # 2: (Responsiveness) Let R 1, …, R n be rankings of two alternatives (Z and Z’) by individuals 1, 2, …, n, and let R’ 1, …, R’ n be an alternative ranking of Z and Z’ such that Z is ranked higher by some individuals and no lower by others. If the SWF selects Z under rankings R 1, …, R n then it must also select Z under rankings R’ 1, …, R’ n

Arrow’s Impossibility Theorem: Conditions Condition # 3: (Independence of Irrelevant Alternatives) Let R 1, …, R m and R’ 1, …, R’ m be individual rankings of alternative social states where Z and Z’ are the socially preferred alternatives under R j and R’ j (j = 1, …, m). Assume Z R j Z’ if and only if Z R’ j Z‘. Then Z = Z’.

Arrow’s Impossibility Theorem: Conditions Condition # 3: (Independence of Irrelevant Alternatives) – Explanation Suppose the rankings of three voters are: –V1 – A > B > C > D –V2 – B > C > A > D –V3 – C > A > B > D D is an irrelevant alternative and should not affect a pairwise comparison between other alternatives, but they do, Nader, e.g.

Arrow’s Impossibility Theorem: Conditions Condition # 4: (Pareto Condition) If an alternative Z is preferred to Z’ by all individual rankings R j (j = 1, …, m) then Z must be socially preferred to Z’.

Arrow’s Impossibility Theorem: Conditions Condition # 5: (Non-Dictatorship) There does not exist an individual k such that the SWF chooses Z as socially preferred if and only if Z R k Z’.

Arrow’s Impossibility Theorem: Summary of Conditions Condition # 1 – Completeness, Transitivity Condition # 2 – Responsiveness Condition # 3 – Independence of Irrelevant Alternatives Condition # 4 – Pareto Optimality Condition # 5 – Non-Dictatorship

Arrow’s Impossibility Theorem No SWF satisfies conditions 1 – 5 simultaneously.