Proportional Dynamics in Linear Fisher Markets with Auto-bidding: Convergence, Incentives and Fairness

TL;DR

This paper extends proportional dynamics to multi-item sellers in Fisher markets, demonstrating convergence to equilibrium, analyzing incentive issues in auto-bidding, and showing a fair outcome under certain market conditions.

Contribution

It generalizes proportional dynamics to multi-item sellers, connects it with auto-bidding, and analyzes the incentive and fairness properties in this setting.

Findings

Generalized dynamics converge to the competitive equilibrium.

Buyers tend to follow the proportional rule, but sellers can profitably deviate.

Seller deviation game has a unique pure Nash equilibrium with good fairness under certain market conditions.

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.