There are no realizable 15_4- and 16_4-configurations

There exist a finite number of natural numbers n for which we do not know whether a realizable n_4-configuration does exist. We settle the two smallest unknown cases n=15 and n=16. In these cases realizable n_4-configurations cannot exist even in the more general setting of pseudoline-arrangements. The proof in the case n=15 can be generalized to n_k-configurations. We show that a necessary condition for the existence of a realizable n_k-configuration is that n > k^2+k-5 holds.Comment: 11 pages, 8 figures, added pseudoline realizations by Branko Gr{\"u}nbau

Robust Market Equilibria under Uncertain Cost

This work studies equilibrium problems under uncertainty where firms maximize their profits in a robust way when selling their output. Robust optimization plays an increasingly important role when best guaranteed objective values are to be determined, independently of the specific distributional assumptions regarding uncertainty. In particular, solutions are to be determined that are feasible regardless of how the uncertainty manifests itself within some predefined uncertainty set. Our mathematical analysis adopts the robust optimization perspective in the context of equilibrium problems. First, we present structural insights for a single-stage, nonadjustable robust setting. We then go one step further and study the more complex two-stage or adjustable case where a part of the variables can adjust to the realization of the uncertainty. We compare equilibrium outcomes with the corresponding centralized robust optimization problem where thesum of all profits are maximized. As we find, the market equilibrium for the perfectly competitive firms differs from the solution of the robust central planner, which is in stark contrast to classical results regarding the efficiency of market equilibria with perfectly competitive firms. For the different scenarios considered, we furthermore are able to determine the resulting price of anarchy. In the case of non-adjustable robustness, for fixed demand in every time step the price of anarchy is bounded whereas it is unbounded if the buyers are modeled 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.Comment: 26 page