We introduce a stochastic influence game modeling strategic interactions between two competing decision makers (DMs) promoting their projects on social network platforms. The network has many agents, each DM wants to influence as many as possible. Influence propagates over time as new active agents also influence neighbors, but may switch to the competing project due to global influence (from reviewers, comments, etc.), yielding coupled spreading dynamics. We develop a recursive method to obtain closed-form expressions for the cumulative expected number of activated agents for each DM. The existence of Nash Equilibrium (NE) is proved since the game is convex. Moreover, due to the strict convexity of the utility functions, the explicit determination of the NE is possible for specific parameters. Finally, simulation results on two network models illustrate a comparison with the theoretical expression of the expected cumulative number of activated agents.