Robust Competitive Ratio for Deterministic Monopoly Pricing

Abstract

We consider deterministic monopoly pricing when only partial market knowledge is available, specifically when the monopolist only knows summary statistics of the valuation distribution such as mean, dispersion, and maximum valuation. Employing distributionally robust optimization and max-min analysis, we use the competitive ratio (CR) to evaluate pricing performance and compare it with traditional expected revenue metrics. We provide a comprehensive solution to the minimization problem for CR by identifying the worst-case market scenario given the limited information. Additionally, we derive closed-form solutions for optimal pricing under various dispersion measures, including variance and other power moments. Our findings reveal that the worst-case market for CR aligns with that for expected revenue, which challenges the conventional belief that CR restricts the adversary’s ability to create extreme scenarios. Through novel proof techniques tailored to CR, we also show how dispersion and maximum valuation impact optimal deterministic pricing. Our results offer valuable insights for adjusting prices effectively and avoiding unrealistic worst-case scenarios, thus leading to more robust and resilient pricing strategies. In particular, setting a maximum valuation introduces a new high-pricing strategy and proves a key factor in unlocking the potential for more profitable (not overly conservative) pricing schemes.