An impossibility theorem in economics and social choice theory is a result that demonstrates that a seemingly desirable set of conditions for a system, such as a voting method or social welfare function, are mutually contradictory.
In other words, no system can satisfy all of the specified conditions simultaneously.
These theorems reveal the inherent limits of collective decision-making.
1. Arrow’s Impossibility Theorem
The most famous impossibility theorem is Arrow’s Impossibility Theorem, also known as the General Possibility Theorem. It was proven by economist Kenneth Arrow in 1950. The theorem states that when voters have at least three distinct options, no ranked-choice voting system can convert individual preferences into a community-wide ranking while also satisfying a set of “fairness” criteria.
The conditions for the theorem are:
- Unrestricted Domain (Universality): The voting system must be able to handle any possible combination of individual preference rankings.
- Non-Dictatorship: The system’s outcome cannot be determined solely by the preferences of a single individual.
- Pareto Efficiency (Unanimity): If every voter prefers option A over option B, then the social ranking must also prefer A over B.
- Independence of Irrelevant Alternatives (IIA): The social ranking between two options, A and B, should only depend on how individuals rank A and B, not on how they rank a third, “irrelevant” option C.
Arrow’s theorem proves that it’s impossible for any voting system to satisfy all four of these conditions simultaneously. This means that any voting system that is not a dictatorship will either fail to respect unanimous preferences, be unable to handle certain preference orderings, or be susceptible to the “spoiler effect” where a third candidate changes the outcome between two others.
While Arrow’s is the most well-known, other impossibility theorems exist that deal with similar issues in social choice theory.
2. The Gibbard–Satterthwaite Impossibility Theorem
This theorem addresses strategic voting. It states that for any voting method with at least three possible outcomes, the method is either dictatorial (one person’s vote determines the outcome) or manipulable (voters have an incentive to misrepresent their true preferences to achieve a more favorable outcome).
This is a foundational result in mechanism design and highlights the challenge of creating a system where voters are always incentivized to be truthful.
3. The Muller–Satterthwaite Impossibility Theorem
This is a related result that demonstrates the conditions under which a social choice function becomes dictatorial.
It states that any social choice function that is monotonic (raising a candidate’s rank in a voter’s ballot can’t make them lose) and unanimous (if everyone ranks a candidate first, they win) is also dictatorial.