PSPACE-completeness for unrestricted-ballot Plurality SPNE Winner

Establish PSPACE-completeness, or at least PSPACE-hardness, for deciding whether a designated candidate belongs to the winner set in a subgame-perfect Nash equilibrium of the sequential Plurality voting game with abstention when every voter may vote for every candidate.

Background

The paper proves PSPACE-completeness only for a restricted-ballot variant in which each voter may vote for a candidate from a specified prefix of her ranking or abstain. The cited conjecture concerns the corresponding unrestricted model, where every voter can vote for every candidate.

Because the restriction to prefix ballots is essential to the presented reduction, the result does not settle whether the unrestricted-ballot problem is PSPACE-complete. The authors explicitly identify PSPACE-hardness of this unrestricted problem as an unresolved issue and note that a proof may require a less elegant construction.

References

We note that our result does not fully resolve the conjecture of \citet{DE10}, as their conjecture applies to unrestricted ballots (i.e., when every voter can vote for every candidate). We strongly believe that this problem is PSPACE-hard as well; however, we expect the proof to be less elegant than the one presented here.

— Subgame-Perfect Nash Equilibria of Plurality Voting with Abstention: a PSPACE-Completeness Result for Restricted Ballots  (2609.24292 - Elkind, 21 Sep 2026) in Final paragraph following the proof of Theorem 1