  We consider a periodical equilibrium pricing problem for multiple firms over
a planning horizon of T periods. At each period, firms set their selling prices
and receive stochastic demand from consumers. Firms do not know their
underlying demand curve, but they wish to determine the selling prices to
maximize total revenue under competition. Hence, they have to do some price
experiments such that the observed demand data are informative to make price
decisions. However, uncoordinated price updating can render the demand
information gathered by price experimentation less informative or inaccurate.
We design a nonparametric learning algorithm to facilitate coordinated dynamic
pricing, in which competitive firms estimate their demand functions based on
observations and adjust their pricing strategies in a prescribed manner. We
show that the pricing decisions, determined by estimated demand functions,
converge to underlying equilibrium as time progresses. We obtain a bound of the
revenue difference that has an order of O(F^2 T^3/4) and a regret bound that
has an order of O(F T^1/2) with respect to the number of the competitive firms
F and T . We also develop a modified algorithm to handle the situation where
some firms may have the knowledge of the demand curve.
