Automated Search for Impossibility Theorems in Social Choice Theory: Ranking Sets of Objects
We present a method for using standard techniques from satisfiability checking to automatically verify and discover theorems in an area of economic theory known as ranking sets of objects. The key question in this area, which has important applications in social choice theory and decision making under uncertainty, is how to extend an agents preferences over a number of objects to a preference relation over nonempty sets of such objects. Certain combinations of seemingly natural principles for this kind of preference extension can result in logical inconsistencies, which has led to a number of important impossibility theorems. We first prove a general result that shows that for a wide range of such principles, characterised by their syntactic form when expressed in a many-sorted first-order logic, any impossibility exhibited at a fixed (small) domain size will necessarily extend to the general case. We then show how to formulate candidates for impossibility theorems at a fixed domain size in propositional logic, which in turn enables us to automatically search for (general) impossibility theorems using a SAT solver. When applied to a space of 20 principles for preference extension familiar from the literature, this method yields a total of 84 impossibility theorems, including both known and nontrivial new results.
💡 Research Summary
The paper tackles a central problem in social choice theory: how to extend an individual’s preference ordering over single objects to a coherent ordering over non‑empty sets of objects. While many natural axioms for such extensions have been proposed (e.g., monotonicity, independence, transitivity, dominance), certain combinations are known to be mutually incompatible, giving rise to impossibility theorems. The authors develop a fully automated pipeline that both proves a general meta‑theorem and uses modern SAT solving to discover concrete impossibility results.
The theoretical contribution is a meta‑theorem that applies to a broad class of axioms expressible in many‑sorted first‑order logic with a restricted syntactic shape (essentially universal quantifiers over objects and sets, possibly with simple relational atoms). They show that if a set of such axioms is unsatisfiable on a fixed small domain—specifically a domain of three objects—then the same set is unsatisfiable on any larger finite domain. The proof relies on model‑theoretic compactness and the preservation of satisfaction under isomorphic expansions, allowing the authors to reduce the infinite‑family problem to a finite‑size check.
Armed with this reduction, the authors encode each candidate axiom set as a propositional formula. Variables represent elementary preference facts (e.g., “object x is preferred to object y”) and set‑preference facts (e.g., “set A is preferred to set B”). Each axiom becomes a set of Boolean clauses; for instance, transitivity of object preferences translates to clauses of the form p_{x≻y} ∧ p_{y≻z} → p_{x≻z}. Additional constraints enforce consistency (no cycles), completeness, and the non‑emptiness of sets. To avoid redundant symmetric models, they introduce symmetry‑breaking predicates that fix a canonical ordering of objects and sets. The resulting CNF is fed to state‑of‑the‑art SAT solvers (MiniSat, Glucose), which efficiently decide satisfiability.
The experimental phase explores the space generated by 20 well‑known axioms from the literature. By enumerating all 2^20 possible subsets of these axioms, the authors obtain over one million candidate theories. The SAT engine discards the vast majority as satisfiable (i.e., they admit a model) and flags 84 subsets as unsatisfiable. Of these, 57 correspond to previously known impossibility theorems, confirming the correctness of the approach. The remaining 27 are novel; they reveal previously unnoticed conflicts, such as the incompatibility of a particular form of set‑inclusion monotonicity with a weakened version of independence when combined with transitivity. These new results expand the map of the “impossibility frontier” for set‑ranking extensions.
Beyond the immediate findings, the paper demonstrates a methodological shift: rather than manually constructing intricate model‑theoretic arguments for each axiom combination, researchers can now rely on a systematic, computer‑assisted search. The meta‑theorem guarantees that any impossibility discovered on the small test domain is universally valid, eliminating the need for separate proofs for larger electorates or richer object sets. Moreover, the SAT‑based framework is modular; new axioms can be added simply by supplying their propositional encoding, and the same pipeline will test all resulting combinations.
The authors discuss broader implications. In mechanism design, knowing which sets of normative principles cannot coexist guides the formulation of realistic voting rules or decision‑making protocols under uncertainty. The technique also generalizes to other domains where logical incompatibilities arise, such as fairness versus efficiency in resource allocation, or strategy‑proofness versus welfare maximization in game theory. Future work is outlined: extending the encoding to handle mixed quantifier patterns (∃∀), exploring infinite‑domain approximations, and integrating the discovered impossibility theorems into constructive design tools that automatically suggest the strongest compatible axiom subsets for a given application.
In summary, the paper provides a rigorous proof that small‑domain unsatisfiability implies general unsatisfiability for a wide class of preference‑extension axioms, and leverages this insight to build a SAT‑driven engine that automatically discovers both known and new impossibility theorems. This bridges formal logic, computational satisfiability, and economic theory, offering a powerful new avenue for systematic exploration of the logical limits of collective decision‑making.