I don’t think so. The prisoner’s dilemma is just a payoff matrix. Adding communication doesn’t change it as long as each player can’t see the other’s choice before making their own. And whether the payoff matrix arises because of a mutual adversary or some other way doesn’t seem like it should make a difference to me.
Make it an iterated prisoner’s dilemma, or add some other way for players to punish each other after seeing their choice, and that does change the game.
The relationship between a boss and an employee is so far removed from a classical single-round prisoner's dilemma that I don't think you'll draw reliably accurate conclusions from it. As it sounds like you know, constraints like iteration change the game significantly, as would making it a game with many actors and reputational effects.
All models are wrong, some models are useful, and in this case it seems more wrong than useful.
>Adding communication doesn’t change it as long as each player can’t see the other’s choice before making their own.
Saying that communication doesn't change the prisoner's dilemma so long as each player can't communicate is a contradiction. The only difference is you used the word "see" instead of the word "communicate", but "see"ing is a form of communication.
The prisoners' dilemma is one of a whole class of thought experiment "games" that are defined by the structure of the payoff matrix. These games are generally framed in terms of two or more "players" making simultaneous choices.
In that context, not being able to objectively see the choice made by other players making their own choice is functionally equivalent to making simultaneous choices. You can re-order when the players make choices and communicate at will without changing the payoff matrix and mathematical model.
They're saying there's a distinction of being able to communicate freely (and possibly lie) while still not showing your final action to defect or cooperate. Which yes, leaves the prisoners dilemma much the same as the 0 forms of comms version.
Make it an iterated prisoner’s dilemma, or add some other way for players to punish each other after seeing their choice, and that does change the game.