Robust Market Equilibria under Uncertain Cost

TL;DR

This paper investigates robust market equilibria under uncertainty, analyzing how firms and market outcomes behave when firms optimize profits in a worst-case scenario, revealing differences from classical models and proposing subsidy mechanisms.

Contribution

It introduces a robust optimization framework for market equilibria, including single-stage and adjustable cases, and compares these with centralized solutions, highlighting efficiency and welfare implications.

Findings

Market equilibrium differs from robust central planner solutions.

Price of anarchy is bounded with fixed demand, unbounded with elastic demand.

Subsidies can achieve robust welfare-optimal equilibria in adjustable settings.

Abstract

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.