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For the iterated prisoners dilemma with random ending time, is it the case that "both players defects each round" is a Nash equilibrium?

I understood a Nash equilibrium as a set of strategies for which it holds that no player benefits from switching strategy if the other player(s) keep their strategy. This indeed seems to be the case here since when everyone is defecting, there is no benefit to cooperating.

Did I understand it correctly?

(I am aware that it is not necessarily the best strategy in practice but I am specifically asking if it is a Nash equilibrium)

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Yes, both players defecting in every round regardless of the history of the game is a Nash equilibrium. Moreover, it is also a subgame perfect equilibrium, although there are also other subgame perfect equilibria of the in(de)finitely repeated prisoner's dilemma game (see folk theorem for detail).

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