Hardness with only one model generalization

Determine whether equilibrium computation remains computationally hard in discrete multiplayer Colonel Blotto games when only one of the two generalizations—multiple players or player-specific battlefield values—is present.

Background

The paper establishes PPAD-completeness for approximate equilibrium computation in discrete multiplayer Colonel Blotto games with both multiple players and player-specific battlefield values under uniform tie-breaking. It also shows tractability in the two-player constant-sum setting and in the one-resource uniform-tie-breaking setting. The authors explicitly leave unresolved whether hardness persists when only one of the two generalizations is allowed: either multiple players with otherwise common values or player-specific values in an otherwise restricted player setting.

References

It remains open whether the problem remains hard with only one of these generalizations, either with multiple players or with player-specific values.

— The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values  (2609.30019 - Bichler et al., 24 Sep 2026) in Section Conclusion and Open Problems

The case with player-specific values but a constant number of players also remains open.

— The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values  (2609.30019 - Bichler et al., 24 Sep 2026) in Section Conclusion and Open Problems

We currently do not know how to implement this logic using only two resources for the primary player. We leave the complexity of computing approximate equilibria in Blotto games with uniform tie-breaking, player-specific values, and at most two resources per player as an open problem.

— The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values  (2609.30019 - Bichler et al., 24 Sep 2026) in Section 4, paragraph “Number of resources”; reiterated in Section Conclusion and Open Problems