Rebates in a Bertrand game

We study a price competition game in which customers are heterogeneous in the rebates they get from either of two firms. We characterize the transition between competitive pricing (without rebates), mixed strategy equilibrium (for intermediate rebates), and monopoly pricing (for larger rebates). In the mixed equilibrium, a firm's support consists of two parts: (i) aggressive prices that can steal away customers from the other firm, and (ii) defensive prices that can only attract customers who get the rebate. Both firms earn positive expected profits. We show that, counter-intuitively, for intermediate rebates, an increase in rebates leads to a lower market segmentation. (C) 2013 Elsevier B.V. All rights reserved.