 In a game like this, the notion of nash equilibrium loses some of its privileged status. Recall that a pair of moves is a nash equilibrium if each is a best reply to the other. Let us extend the notation used in the discussion of the asynchronous PD and let \(\bDu\) be the strategy that calls for defection at every node of an IPD. It is easy to see that \(\bDu\) and \(\bDu\) form a nash equilibrium. But against \(\bDu\), a strategy that calls for defection unless the other player cooperated at, say, the fifteenth node, would determine the same play (and therefore the same payoffs) as \(\bDu\) itself does. The components that call for cooperation never come into play, because the other player does not cooperate on the fifteenth (or any other) move. Similarly, a strategy calling for cooperation only after the second cooperation by itself does equally well. Thus these strategies and many others form nash equilibria with \(\bDu\). There is a sense in which these strategies are clearly not equally rational. Although they yield the same payoffs at the nodes along the path representing the actual play, they would not yield the same payoffs if other nodes had been reached. If Player One had cooperated in the past, that would still provide no good reason for him to cooperate now. A nash equilibrium requires only that the two strategies are best replies to each other as the game actually develops. A stronger solution concept for extensive-form games requires that the two strategies would still be best replies to each other no matter what node on the game tree were reached. This notion of subgame-perfect equilibrium is defined and defended in Selten 1975. It can be expressed by saying that the strategy-pair is a nash equilibrium for every subgame of the original game, where a subgame is the result of taking a node of the original game tree as the root, pruning away everything that does not descend from it.