  In two-sided marketplaces such as online flea markets, recommender systems
for providing consumers with personalized item rankings play a key role in
promoting transactions between providers and consumers. Meanwhile, two-sided
marketplaces face the problem of balancing consumer satisfaction and fairness
among items to stimulate activity of item providers. Saito and Joachims (2022)
devised an impact-based fair ranking method for maximizing the Nash social
welfare based on fair division; however, this method, which requires solving a
large-scale constrained nonlinear optimization problem, is very difficult to
apply to practical-scale recommender systems. We thus propose a fast solution
to the impact-based fair ranking problem. We first transform the fair ranking
problem into an unconstrained optimization problem and then design a gradient
ascent method that repeatedly executes the Sinkhorn algorithm. Experimental
results demonstrate that our algorithm provides fair rankings of high quality
and is about 1000 times faster than application of commercial optimization
software.
