Right, game theory models how cooperation can emerge from strategic self-interest when the theorists have the temerity to treat an indefinite, Groundhog-Day style of Purgatory as a “game.”
Even in repeated prisoner’s dilemmas, the players betray each other when they know when the repetitions will end and even when they don’t know exactly when they’ll end but they know the repetitions will end at some point. Only when the players have reason to think the repetitions will never end do they adopt the tit-for-tat strategy since in that case there’s no benefit to double-crossing the opponent when the opponent has no chance of retaliating (since the repetitions won’t end to allow for an unretaliated betrayal).
This is what the Bill Murray movie Groundhog Day dramatized: a mean guy turns nice eventually with potentially infinite repetitions of the same day. He learns to be his best self, thanks to infinite trials and errors.
Alas, an indefinitely iterated prisoner’s dilemma, as in a potentially infinite series of repetitions is no longer a game at all. This becomes theology, then, rather than scientific modeling of behaviour. A game has a beginning, a middle, and an end. In an infinite series of prisoner's dilemmas, the resolution of each dilemma is only part of the series which isn’t itself a game. The players learn to cooperate when they realize they just might be in a supernatural setting in which their contests could be repeated forever. In such a setting, shit gets real, and you no longer play mere games, as it were. You face your Maker with Heaven or Hell on the line, and with your soul to be purified in Purgatory.
The reason the infinite series of prisoner's dilemmas--or the imagination of such a series in the minds of the players who, for some artificial reason, suspect their interactions could go on forever--isn’t a game is that a game must have a result, as in an end that’s determined by the players’ actions. An infinite series has no end, so it’s not a game. Therefore, it’s not in the province of game theory unless economists are okay with their discipline being interpreted as theological rather than scientific.
Consequently, game theorists reconciled classical liberalism with experiments on strategic thinking only by positing the religious scenario in which players learn to cooperate by being transported to the supernatural realm of Purgatory.
In the real world, of course, when we know our interactions are finite and when we usually know when they'll end (as in a fixed contract situation), there’s no such clear emergence of the morally optimal scenario from purely strategic self-interest. Hence, the upshot of nontheological game theory conflicts with classical liberalism, in the way I’ve outlined.
