  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.
