Price Competition in Linear Fisher Markets: Stability, Equilibrium and Personalization
Juncheng Li, Pingzhong Tang
TL;DR
This work studies price competition among sellers in linear Fisher markets by introducing a model where sellers own item subsets and choose personalized prices and total preorders (priorities) over buyers. Stability is defined so that no buyer–seller deviation strictly improves either party's outcome, and a stable allocation exists and is computable in strongly poly-time; however, stability is not generally unique, reflecting market uncertainty. The authors distinguish regimes by personalization level: with no buyer personalization, the competitive equilibrium forms a maximin steady state and revenue-optimal allocations are tractable (APX-hard for minimization), while personalization dramatically increases complexity, enabling revenue improvements but also APX-hardness and potential absence of static equilibria. The paper connects these economic insights to algorithmic structures from Gale-Shapley, network flow, and stable matching, providing a rich computational-structural view of how personalization affects efficiency, predictability, and market dynamics in Fisher markets.
Abstract
Linear Fisher market is one of the most fundamental economic models. The market is traditionally examined on the basis of individual's price-taking behavior. However, this assumption breaks in markets such as online advertising and e-commerce, where several oligopolists dominate the market and are able to compete with each other via strategic actions. Motivated by this, we study the price competition among sellers in linear Fisher markets. From an algorithmic game-theoretic perspective, we establish a model to analyze behaviors of buyers and sellers that are driven by utility-maximizing purposes and also constrained by computational tractability. The main economic observation is the role played by personalization: the classic benchmark market outcome, namely competitive equilibrium, remains to be a steady-state if every buyer must be treated "equally"; however, sellers have the incentive to personalize, and as a result the market would become more unpredictable and less efficient. In addition, we build a series of algorithmic and complexity results along the road to justify our modeling choices and reveal market structures. We find interesting connections between our model and other computational problems such as stable matching, network flow, etc. We believe these results and techniques are of independent interest.