Evolutionary games are studied here with two types of players located on a chessboard or on a bipartite random regular graph. Each player's income comes from matching-pennies games played with the four neighbors. The players can modify their own strategies according to a myopic strategy update resembling the Glauber dynamics for the kinetic Ising model. This dynamical rule drives the system into a stationary state where the two strategies are present with the same probability without correlations between the nearest neighbors while a weak correlation is induced between the second and the third neighbors. In stationary states, the deviation from the detailed balance is quantified by the evaluation of entropy production. Finally, our analysis is extended to evolutionary games where the uniform pair interactions are composed of an anticoordination game and a weak matching-pennies game. This system preserves the Ising type order-disorder transitions at a critical noise level decreasing with the strength of the matching-pennies component for both networks.