Online Price Competition under Generalized Linear Demands

We study a sequential price competition among N sellers, each influenced by the pricing decisions of their rivals. Specifically, the demand function for each seller i follows the single index model λi(mathbf p) = μi(langle boldsymbol θi,0, mathbf p rangle), with known increasing link μi and unknown parameter boldsymbol θi,0, where the vector p denotes the vector of prices offered by all the sellers simultaneously at a given instant. Each seller observes only their own realized demand - unobservable to competitors - and the prices set by rivals. We propose a novel decentralized policy, PML-GLUCB, that combines penalized MLE with an upper-confidence pricing rule. Our approach (i) removes the need for coordinated front-loaded exploration phases across sellers - which is integral to previous models - making our method aligned with realistic market conditions; (ii) generalizes existing approaches that focus solely on linear demand models; (iii) accommodates both binary and real-valued demand observations. Relative to a dynamic benchmark policy, each seller achieves widetildeO(√T) regret, which matches the optimal rate known in the linear setting.