Optimizing Free-to-Play Multiplayer Games with Premium Subscription

We consider the optimal operating policies of a free-to-play online multiplayer game with premium sub- scription to maximize its lifetime operating profit. Accounting for social-comparison effects between free and premium players, we model the game attracting or losing players with a hybrid of the Bass diffusion model and the replicator equation in evolutionary game theory. Leveraging optimal control theory, we characterize optimal dynamic pricing and advertising policies and show that the developer should prioritize initial growth by aggressively advertising while postponing the introduction of premium subscription, and surprisingly the subscription price may start high and gradually decrease. We further show that in general the developer should foster stronger social-comparison effects, that player matching can be an efficient and effective operational lever to facilitate monetization without inhibiting growth, and that our main findings remain robust when either the developer or players are financially constrained. These findings are potentially instructive for game developers adopting the premium subscription model