Recently, Cheryche et al. (2011) proved the important negative result that deciding the strong feasibility of the Marshallian equilibrium inequalities, introduced by Brown and Matzkin (1996), is NP-complete. Here, I show that the weak feasibility of the equivalent Hicksian equilibrium inequalities, introduced by Brown and Shannon (2000), can be decided in oracle-polynomial time.
Brown, Donald J., "Notes on Computational Complexity of GE Inequalities" (2012). Cowles Foundation Discussion Papers. 2226.