50 research outputs found
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