Full-dimensional equilibrium separation

Determine whether the strict separation between correlated equilibrium and conservative/proximal correlated equilibrium can be realized by a full-dimensional absolutely continuous joint distribution, rather than only by a distribution with absolutely continuous marginals supported on a lower-dimensional diagonal.

Background

The paper constructs a two-player convex game and an uncountably supported distribution satisfying conservative and proximal correlated equilibrium but not unrestricted correlated equilibrium. The distribution has absolutely continuous two-dimensional marginals, but its joint law is supported on the diagonal and is therefore singular with respect to full-dimensional Lebesgue measure on the product action space.

The authors explicitly leave unresolved whether the same strict separation persists for a genuinely diffuse outcome distribution whose joint law has a full-dimensional density.

References

Second, can the strict continuous-support equilibrium separation be realized by a full-dimensional absolutely continuous joint distribution? The present construction has absolutely continuous marginals but is supported on a lower-dimensional diagonal, so resolving this question would determine whether the separation persists for genuinely diffuse outcome distributions.

— Exact-Form Regret for Gradient Descent, Mirror Descent and Follow-the-Regularized-Leader  (2609.09466 - Soleymani et al., 8 Sep 2026) in Section Conclusion; related formulation in Section 10, subsection “Strict Separation from CE on Continuous Support”