Real Dictionary

play at
If someone, especially a child, plays at being someone or doing something, they pretend to be that person or do that thing as a game.
If you say that someone is playing at something, you disapprove of the fact that they are doing it casually and not very seriously.
to participate in
to pretend interest in
phrasal verb: If you do not know what someone is playing at, you do not understand what they are doing or what they are trying to achieve.