Incomparability of PMA and PTW

Prove that the syntactic classes PMA (Polynomial Majority Argument) and PTW (Polynomial Tournament Winner) are incomparable, i.e., establish both PMA ⊄ PTW and PTW ⊄ PMA by constructing explicit languages L1 ∈ PMA \ PTW and L2 ∈ PTW \ PMA or by otherwise demonstrating non-containment in each direction.

Background

The paper introduces two unambiguous subclasses of UΣ2: PTW, capturing tournament-style winner existence problems with polynomial-time pairwise comparisons, and PMA, capturing problems whose uniqueness arises from a majority-of-edges argument in a bounded-label bipartite structure.

While the authors prove various inclusions and separations for these classes relative to other complexity classes, the relationship between PMA and PTW themselves is unsettled. They note non-triviality of establishing containment either way and explicitly conjecture that the two are incomparable due to fundamentally different combinatorial bases for unambiguity.