Platform Competition With Cash-Back Rebates Under No Surcharge Rules

We analyze competing strategic platforms setting fees to a local monopolist merchant and cash-back rebates to end users, when the merchant may not surcharge platforms' customers, a rule imposed by some credit card networks. Each platform has an incentive to gain transactions by increasing the spread between its merchant fee and user rebate above its rival's spread. This incentive yields non-existence of a pure strategy equilibrium in many natural environments. In some circumstances, a mixed strategy equilibrium exists where platforms choose fee structures that induce merchants to accept only one platform with equal probability, a form of monopolistic market allocation.