  In digital markets comprised of many competing services, each user chooses
between multiple service providers according to their preferences, and the
chosen service makes use of the user data to incrementally improve its model.
The service providers' models influence which service the user will choose at
the next time step, and the user's choice, in return, influences the model
update, leading to a feedback loop. In this paper, we formalize the above
dynamics and develop a simple and efficient decentralized algorithm to locally
minimize the overall user loss. Theoretically, we show that our algorithm
asymptotically converges to stationary points of of the overall loss almost
surely. We also experimentally demonstrate the utility of our algorithm with
real world data.
