We study a system of self-propelled, proliferating finite-size disks with game-theoretical interactions, where growth rates depend on local population composition. We analyze how these interactions influence spatial distribution, coexistence, and extinction. Three scenarios emerge: (i) stable coexistence with well-mixed distributions, (ii) bistability, where the outcome depends on initial conditions, and (iii) dominance, where one species always outcompetes the other. We further show that activity enhances spatial mixing in coexistence and reduces extinction times in both dominance and bistability. Additionally, stronger interactions promote coexistence while accelerating extinction in the latter two regimes. By varying diffusivity, activity, and interaction strength, we show how movement and spatial constraints affect population dynamics.