Arrow’s Impossibility Theorem was proved by Kenneth J. Arrow in his 1950 paper “A Difficulty in the Concept of Social Welfare” and subsequently published in expanded form in his book Social Choice and Individual Values in 1951.
With this theorem, he aimed to aggregate individual preferences into a single social preference, making use of the following components to set up the framework:
- The set, ‘N’ of agents.
- The alternatives, ‘A’, which must be three at minimum for the theory to apply.
- Individual preferences that must be complete (for any two alternatives A and B, either A is weakly preferred over B or B is weakly preferred over A, or both, meaning indifference is allowed) and transitive (if A is preferred to B and B is preferred to C, then A is preferred to C).
The SWF or Social Welfare Function, ‘F’ maps the individual preference profiles to a single social order.
The axioms are as follows:
- Unrestricted Domain (UD): Any logically possible combination of individual preferences will be accepted as input. Constraints like single-peakedness, where each agent has a most-preferred alternative and preferences decline on either side of it, do not exist.
- Weak Pareto (WP): This means that if A is preferred over B by all individuals, this implies that A is ranked over B by that society as a whole.
- Independence of Irrelevant Alternatives (IIA): The social ranking of A over B must depend only on how individuals rank A versus B, and on nothing else, including how individuals rank any third alternative C.
- Non-dictatorship: There must not be one single agent who dictates the social order of preference.
The Impossibility that Arrow states is that if UD, WP and IIA are all satisfied, then the system must be dictatorial in nature.
The Decisive Sets proof is used in proving the Propagation Lemma, by making use of the IIA property.
A subset ‘S’ of N is decisive for a pair (A, B) if whenever every member of S prefers A over B, society prefers A over B, regardless of the preferences of everyone outside S. Once the Propagation Lemma forces the proof to narrow down to a single agent, a dictatorship exists.
By definition, an ultrafilter is a mathematical structure that picks out “winning” subsets. The collection of decisive sets forms an ultrafilter: it is closed under supersets and finite intersections, and for any subset of agents, either that subset or its complement must be decisive. Since we are working with a finite set of agents, the only ultrafilters that can exist are principal, meaning the smallest decisive set must be a single agent i.e. the dictator.
If a subset ‘S’ is decisive for one pair of alternatives, it is fully decisive for all pairs. The subsequent ultrafilter logic systematically proves that the only remaining, smallest fully decisive set must be a single agent or singleton, which is the Dictator.
In short, a decisive set is a group of individuals whose unanimous preference for A over B forces society’s preference for A over B, regardless of the preferences of those outside the subset S. According to the Propagation Lemma, such a set, if it exists, becomes decisive for all pairs, and therefore leads to a dictator.
An escape hatch exists in the form of discarding the UD axiom and restricting preferences to be single-peaked. When all preferences are single-peaked, Condorcet’s Paradox, the problem of cyclical social preferences cannot arise. Under majority rule, a single alternative exists that defeats every other alternative in pairwise comparison. This is called the Condorcet Winner. This outcome is rational and non-dictatorial in nature.
IIA is the key aspect of this Theorem. The requirement that pairwise social rankings depend only on pairwise individual rankings is often considered unrealistic in real-world elections. The introduction or removal of a third candidate can dramatically change individual assessments of A and B, and IIA rules out any such influence.
This breaks the system.
There are modern relaxations to the theorem as follows:
- Borda Count – This method assigns points to the placement of an alternative in an agent’s preference profile (3 points for 1st preference, 2 for 2nd, 1 for 3rd). The preference with the highest total points across agents is the outcome. This violates IIA because removing an alternative changes the point totals for the remaining alternatives and an alternative that ranked second above the removed option now ranks second above a different option, altering the scores of A and B even if no individual changed their relative ranking of A versus B.
- Copeland – This method compares alternatives based on pairwise wins and losses. The score for each alternative is its total wins minus its total losses. This also violates IIA.
- Cardinal Utilities Framework – Instead of only using ordinal rankings, cardinal numerical scores are used to rank the preferences, pushing the calculation beyond the scope of Arrow’s ordinal approach to the theorem.
Arrow’s theorem is not just a mathematical proof; it’s a warning.
From a policy perspective, the theorem shows that no aggregation procedure can simultaneously satisfy all four axioms when there are three or more alternatives. Rather than searching for a perfect system that converts individual preferences into a unified social preference, the more useful approach is to decide which axioms to relax and which trade-offs to accept, so that the outcome does not collapse into effective dictatorship.
Preparing for MA Economics Entrances?
Social Choice Theory including Arrow’s Theorem appears in CUET PG, IIT JAM, GATE and ISI MSQE. It is the kind of topic where knowing the proof structure matters as much as knowing the result.
At Haq Se Economics, the MA Economics Entrance 2027 program covers topics like this from first principles, with daily practice problems, weekly tests, and dedicated ISI MSQE subjective sessions. The full syllabus for CUET PG, IIT JAM, GATE and ISI MSQE is covered in a single structured program- live lectures, recordings available throughout, and access until all exams are done.
If you want to test yourself before enrolling, the Test Series is available separately — topic-wise and full mocks across CUET PG, IIT JAM, and ISI MSQE.
