What is the classic name of the game "The N-th player to respond wins"? What are general strategies for the case when the number of total players is not known (or maybe there's only a prior on the number of players)? What theory exists on this topic?
Assume that there is only a single simultaneous round: each player $p$ chooses a time $t_p$ to respond, with no knowledge (other than a prior, perhaps) of when the other players respond.
If necessary to make the problem well-defined, let's also assume a maximum time $T_\max$, beyond which no response will win, as well as $P$ total players, discrete time steps. The hardest, most interesting case, is when no player can observed when other players have responded. Another case is if the game is repeated, and all players can observe all responses after the game is over.