As power to gas (P2G) technology gradually matures, the coupling between electricity networks and natural gas networks should ideally evolve synergistically. With the intent of characterizing market behaviors of integrated electric power and natural gas networks (IPGNs) with P2G facilities, this paper establishes a steady-state model of P2G and constructs optimal dispatch models of an electricity network and a natural gas network separately. In addition, a concept of slack energy flow (SEF) is proposed as a tool for coordinated optimal dispatch between the two networks. To study how the market pricing mechanism affects coordinated optimal dispatch in an IPGN, a market equilibrium-solving model for an IPGN is constructed according to game theory, with a solution based on the Nikaido-Isoda function. Case studies are conducted on a joint model that combines the modified IEEE 118-node electricity network and the Belgian 20-node gas network. The results show that if the game between an electric power company and a natural gas company reaches market equilibrium, not only can both companies maximize their profits, but also the coordinated operation of the coupling units, i.e., gas turbines and P2G facilities, will contribute more to renewable energy utilization and carbon emission reduction.