Automated Reasoning Tightens Bounds on Condorcet Winning Set Size
A new arXiv preprint (2604.19851) investigates the minimum number of candidates needed to guarantee a Condorcet winning set in ranked elections. Condorcet's paradox shows that no single candidate can be a Condorcet winner in some elections, and the same holds for committees of size two. Recent work proves that a set of five candidates always exists. This leaves a gap between the lower bound (k≥3) and upper bound (k≤5). The authors employ automated reasoning via a mixed-integer linear program (MILP) to search for counterexamples to conjectured bounds, aiming to tighten the gap between existence guarantees and impossibility results.
Key facts
- arXiv:2604.19851v1 is a cross-type announcement.
- Condorcet winning set: a committee of k candidates such that for any outside candidate, a majority of voters prefer some committee member.
- Condorcet's paradox shows some elections have no Condorcet winner (k=1).
- No Condorcet winning set of size k=2 exists for some elections.
- A set of size k=5 exists for every election.
- Theoretical gap: lower bound k≥3, upper bound k≤5.
- Authors use a mixed-integer linear program (MILP) to search for counterexamples.
- Goal: tighten bounds between existence guarantees and impossibility results.
Entities
Institutions
- arXiv