I am investigating a setting where one agent, $P_1$, may not know if the other agent, $P_2$, is present in the system. Thus, $P_1$ is unsure whether it is actually playing a game or simply taking actions and receiving rewards in absence of the other agent.
Have such games been studied? If so, do they have a name? Perhaps such situations can simply be modeled as $P_2$ having a type (exists, doesn't exist) where $P_1$ doesn't know the type.