 One way to avoid the dubious conclusion of the backward induction argument without delving too deeply into conditions of knowledge and rationality is to consider infinitely repeated PDs. No human agents can actually play an infinitely repeated game, of course, but the infinite IPD has been considered an appropriate way to model a series of interactions in which the participants never have reason to think the current interaction is their last. In this setting a pair of strategies determines an infinite path through of the game tree. If the payoffs of the one-shot game are positive, their total along any such path is infinite. This makes it somewhat awkward to compare strategies. In many cases, the average payoff per round approaches a limit as the number of rounds increases, and so that limit can conveniently serve as the payoff. (See Binmore 1992, page 365 for further justification.) For example, if we confine ourselves to those strategies that can be implemented by mechanical devices (with finite memories and speeds of computation), then the sequence of payoffs to each player will always, after a finite number of rounds, cycle repeatedly through a particular finite sequence of payoffs. The limit of the average payoff per round will then be the average payoff in the cycle. In recent years, Press and Dyson have shown that for many purposes, investigation of the infinite IPD can be confined to the “memory-one” strategies, in which the probability of cooperating in any round depends only on what happened in the previous meeting between the strategies. The average payoff per round is again always well-defined in the limit. The ideas of Press and Dyson have inspired much new work on the infinite IPD. (See  Zero-Determinant Strategies  below.) Since there is no last round, it is obvious that backward induction does not apply to the infinite IPD.