Robust Equilibrium Problems

We consider equilibrium problems under uncertainty where firmsmaximize their profits in a robust way when selling their output. Robustoptimization plays an increasingly important role when best guaranteed objec-tive values are to be determined, independently of the specific distributionalassumptions regarding uncertainty. In particular, solutions are to be deter-mined that are feasible regardless of how the uncertainty manifests itself withinsome predefined uncertainty set. Our analysis adopts the robust optimizationperspective in the context of equilibrium problems. First, we consider a single-stage, nonadjustable robust setting. We then go one step further and study themore complex two-stage or adjustable case where a part of the variables canadjust to the realization of the uncertainty. We compare equilibrium outcomeswith the corresponding centralized robust optimization problem where thesum of all profits are maximized. As we find, the market equilibrium forthe perfectly competitive firms differs from the solution of the robust centralplanner, which is in stark contrast to classical results regarding the efficiency ofmarket equilibria with perfectly competitive firms. For the different scenariosconsidered, we furthermore are able to determine the resulting price of this http URL the case of non-adjustable robustness, for fixed demand in every time stepthe price of anarchy is bounded whereas it is unbounded if the buyers aremodeled by elastic demand functions. For the two-stage adjustable setting, we show how to compute subsidies for the firms that lead to robust welfareoptimal equilibria.