Proportional Dynamics in Linear Fisher Markets with Auto-bidding: Convergence, Incentives and Fairness
Juncheng Li, Pingzhong Tang
TL;DR
This work extends proportional dynamics to auto-bidding markets where a single seller can own multiple items, and buyers allocate budgets across sellers. It proves convergence to the competitive equilibrium with rate $O(1/T)$ via a KL-divergence based potential and clarifies the connection to pacing equilibria in auto-bidding. The authors show that buyers have strong incentives to follow the proportional rule (with a $2$-approximation guarantee), while sellers can profitably deviate, leading to a unique pure Nash equilibrium in the seller competition game and a Nash social welfare guarantee of at least $(1-\Delta)$ of the optimum, where $\Delta$ captures market monopolization. These results offer practical insights for price formation in online advertising markets and provide a meaningful generalization of proportional dynamics to the auto-bidding setting.
Abstract
Proportional dynamics, originated from peer-to-peer file sharing systems, models a decentralized price-learning process in Fisher markets. Previously, items in the dynamics operate independently of one another, and each is assumed to belong to a different seller. In this paper, we show how it can be generalized to the setting where each seller brings multiple items and buyers allocate budgets at the granularity of sellers rather than individual items. The generalized dynamics consistently converges to the competitive equilibrium, and interestingly relates to the auto-bidding paradigm currently popular in online advertising auction markets. In contrast to peer-to-peer networks, the proportional rule is not imposed as a protocol in auto-bidding markets. Regarding this incentive concern, we show that buyers have a strong tendency to follow the rule, but it is easy for sellers to profitably deviate (given buyers' commitment to the rule). Based on this observation, we further study the seller-side deviation game and show that it admits a unique pure Nash equilibrium. Though it is generally different from the competitive equilibrium, we show that it attains a good fairness guarantee as long as the market is competitive enough and not severely monopolized.