  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.
