Is there a name for the game with extensive form given in the image below? Are there any references that study it? I am not well versed in game theory so I would appreciate any pointers. I am sure it isn't new.
It seems like a simple example of a game where only one player has access to some information (here just a single bit) that the second player needs to make the right move. The two players must then coordinate so that the second player knows how to interpret the first player's move.
More specifically, if nature (here denoted $W$) plays $x$ then player two (denoted $A_2$) has to play $x$ as well for both players to get payoff $1$ else both players get payoff $0$. Only $A_1$ sees the move by nature and $A_2$ only sees the move by $A_1$. So the players must coordinate either so that when $A_1$ plays $x$ it means that $W$ played $x$ or when $A_1$ plays $x$ it means that $W$ played $(1-x) \mod 2$.
