An Interactive Exploration — 3 Voters, 3 Alternatives
The Axioms
Unrestricted Domain (U)
Every logically possible profile of strict preference orderings is in the domain of the social welfare function.
Weak Pareto (WP)
If every voter prefers x to y, then the social ordering must rank x above y.
Independence of Irrelevant Alternatives (IIA)
The social ranking of x vs. y depends only on how each voter ranks x vs. y — not on how they rank any third alternative z.
Non-Dictatorship (ND)
No single voter always determines the social outcome: there is no voter i such that for all profiles, the social ordering equals i's ordering.
Collective Rationality (CR)
The output of the SWF must be a complete, transitive, antisymmetric ordering — a strict linear order. This rules out cycles and incomparability. In this tool, the dropdown enforces CR by construction, but note that bare (unmodified) majority rule can violate it via Condorcet cycles.
How to use: Rows where Weak Pareto forces a unique outcome are pre-filled automatically.
When you assign an SWF output to a row, IIA constrains every other row sharing the same
pairwise voter-pattern — impossible options are removed from the dropdowns, and rows that
become fully determined are filled in automatically. Choose a preset rule below, or build
your own SWF and watch the constraints propagate.
Choose a Rule
Social Welfare Function Table
Pareto-forced rows have been pre-filled. Make selections to continue.
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Voter 1
Voter 2
Voter 3
SWF Output
Status
Constraint Propagation Trace
This panel shows how your choices propagate through the constraint system. Each assignment creates pairwise constraints via IIA (the social ranking of {x, y} must be the same across all profiles where voters rank x vs y identically). When two constrained pairs within a single profile force the third via transitivity (Collective Rationality), that creates a new IIA constraint — which can cascade further. This interaction between IIA and transitivity is the engine that drives Arrow's impossibility.