Price competition under linear demand and finite inventories: Contraction and approximate equilibria

We consider a competitive pricing problem where there are multiple firms with limited inventories of substitutable products. Each firm chooses the prices that it charges for its product over a finite selling horizon. The demand of each firm jointly depends on the prices of all firms in a deterministic fashion through a linear demand function. The goal of each firm is to choose its prices to maximize its total revenue. We study two types of equilibrium. In equilibrium without recourse, each firm chooses its entire price trajectory at the beginning of the selling horizon. In equilibrium with recourse, each firm adjusts its price at each time period as a function of the current inventories of all of the firms. Although the demand of each firm is a deterministic function of the prices so that there is no uncertainty in the responses of the firms, there is a stark difference between equilibria with and without recourse. Considering the commonly studied diagonally dominant regime, where the demand of a firm is affected more by its price than the prices of the other firms, we show that the equilibrium without recourse exists and it is unique. In contrast, we demonstrate that the equilibrium with recourse may not exist or may not be unique. Motivated by this result, we look for approximate equilibrium with recourse. Considering a low influence regime, where the effect of the price of each firm on the demand of the others is diminishing, we show that the equilibria without recourse is an approximate equilibrium even when we allow the firms not to commit to a price trajectory at the beginning of the selling horizon.