This paper examines how competition among payment card networks three-party scheme networks and four-party scheme networks affects pricing as well as the welfare of various parties. A competing network has an incentive to provide rewards to its card users. By providing more generous rewards than its rival networks, the network can increase its own card transactions because multihoming cardholders who hold multiple networks cards choose to use its card instead of using its rivals. Although a monopoly network does not have such an incentive, in a monopoly four-party scheme network, competition among card issuers likely makes issuers provide rewards. Due to rewards, the merchant fees under competition can be higher than the merchant fees set by a monopoly network, unless the majority of cardholders are multihoming. Generally, cardholding consumers are better off under network competition. In contrast, non-cardholding consumers are better off only when network competition reduces merchant fees lower than those under monopoly. The results suggest that policies that simply encourage network competition will likely increase cardholder rewards but will not necessarily lower merchant fees in the U.S. payment card market. Several empirical indicators may possibly tell which direction the U.S. payments system needs to go.