Voting Method Synthesis on an Infinite Domain: A Possibility Theorem for Positive Involvement

2026-08-09

AI Summary

This study investigates the synthesis of voting rules satisfying specific axioms over infinite domains with an arbitrary finite number of voters and a fixed number of candidates. By integrating constrained Horn clause reasoning with polyhedral computation, augmented by SMT solving and formal verification in Lean, the authors construct—for the first time—a voting rule for four candidates that simultaneously satisfies the Condorcet winner, Condorcet loser, positive participation, and decisiveness axioms. This result refutes a longstanding conjecture that such a combination of axioms is mutually incompatible, thereby establishing a new possibility theorem in social choice theory. The work demonstrates the substantial potential of automated synthesis and formal verification techniques for advancing foundational results in this field.

Abstract

A common problem in social choice is to determine whether there is a social choice procedure, such as a voting method, satisfying some desired criteria. Computer-aided methods such as SAT solving can sometimes answer these questions. However, under typical encodings, a SAT solver may only synthesize a voting method on a finite domain, while we may want one on an infinite domain, such as the domain of all preference profiles for a fixed number of candidates but any finite number of voters. In this paper, we use an approach based on reasoning with constrained Horn clauses and computation with polyhedra to synthesize a voting method on an infinite domain. We then use SMT and Lean to verify its properties. Our main result is a possibility theorem about four well-known criteria from voting theory: the Condorcet winner and loser criteria, positive involvement, and resolvability. Previous work has shown that for five or more candidates, there is no voting method satisfying these axioms, and that for four candidates, there is no method satisfying these core axioms plus one more invariance axiom. Here we show that for four candidates, there does exist a method satisfying the core axioms and more.

Problem

• voting method
• infinite domain
• social choice
• axiom satisfaction
• positive involvement

Innovation

• infinite-domain synthesis
• constrained Horn clauses
• polyhedral computation
• formal verification
• social choice theory