I hate it when I read a sentence that says “Game theory tells us _______.” For example, “game theory tells us prisoners should confess to crimes.” If you ever see something to that effect, know that the author doesn’t really understand what game theory is about.
The example is a reference to the prisoner’s dilemma. In that model, the prisoners only want to minimize jail time and defect accordingly, which leads them to an outcome that is strictly worse for both players than had they cooperated with each other.
But game theory does not generally tell us that the prisoners should rat each other out. Whenever we solve a model, we make some assumptions, formulate those assumptions into equations, do some game theory, and produce conclusions. Game theory is nothing more than overly glorified math. (Sorry to burst your bubble.) The nice part is that game theory maps assumptions to logically valid conclusions. So, really, game theory only tells us how to solve the game.
The assumptions do all of the dirty work. In the prisoner’s dilemma example, the critical assumption is that players only want to minimize jail time. Alternatively, they might want to minimize jail time and maintain their friendship. In my textbook, I formulate this as a stag hunt. The players want to keep quiet if and only if the other keeps quiet as well, while they players want to confess if and only if the other confesses as well. Consequently, in this game, the prisoners might stay silent. Changing the assumptions changes the prediction.
Bottom line: the assumptions tell us everything.