I see. Yeah, it seems easy to prove that my game can’t be made symmetric by your definition. (If player 1 and player 2′s mutual pairing must happen independently from their identities and also from the actions of everyone else, there’s nothing stopping player 3 from submitting the same action as player 2.) I wonder if the definition I was implicitly using is studied anywhere...
I see. Yeah, it seems easy to prove that my game can’t be made symmetric by your definition. (If player 1 and player 2′s mutual pairing must happen independently from their identities and also from the actions of everyone else, there’s nothing stopping player 3 from submitting the same action as player 2.) I wonder if the definition I was implicitly using is studied anywhere...