Arrow's Impossibility Theorem

What Is Arrow's Impossibility Theorem?

Arrow's Impossibility Theorem is a result in social choice theory, published by economist Kenneth Arrow in 1951 in his book Social Choice and Individual Values. It addresses how a group can combine the rankings of its members into one group ranking. Arrow showed that when there are three or more options, no ranking method can satisfy a set of reasonable conditions at the same time. He later received the Nobel Memorial Prize in Economic Sciences, shared with John Hicks in 1972.

The Conditions

Arrow's conditions are usually described as follows:

• Unrestricted domain: any combination of individual rankings is allowed as input.
• Pareto efficiency: if every member prefers option A to option B, the group ranking must also place A above B.
• Independence of irrelevant alternatives: the group's ranking of two options depends only on how members rank those two, not on a third option.
• Non-dictatorship: no single member's preferences automatically determine the group ranking.

Arrow proved that no method meets all four conditions once there are at least three options.

Why It Matters

Simple majority voting shows the problem. Suppose three voters rank three options. Their preferences can form a cycle: a majority may prefer A to B, a majority may prefer B to C, and a majority may prefer C to A. This is known as the Condorcet paradox, and it means majority rule can fail to produce a coherent group ranking. Arrow's result extends this difficulty to every ranking system that meets his conditions. It is best read as a limit on what any fair aggregation method can achieve, not as a claim that group decisions are meaningless.

Relevance to Investors

The theorem matters in finance wherever groups combine rankings or preferences, such as investment committees choosing among strategies or shareholders voting on several competing proposals. Corporate governance and proxy voting are areas where collective decisions can depend heavily on the voting rule used. Understanding the theorem helps readers see why the outcome of a group ranking can change with the method chosen, and why such outcomes should be interpreted with that in mind.