 There is an observation, apparently originating in Kavka 1983, and given more mathematical form in Carroll, that the backward induction argument applies as long as an upper bound to the length of the game is common knowledge. For if \(b\) is such an upper bound, then, if the players were to get to stage \(b\), they would know that it was the last round and they would defect; if they were to get to stage \(b-1\), they would know that their behavior on this round cannot affect the decision to defect on the next, and so they would defect; and so on. It seems an easy matter to compute upper bounds on the number of interactions in real-life situations. For example, since shopkeeper Jones cannot make more than one sale a second and since he will live less than a thousand years, he and customer Smith can calculate (conservatively) that they cannot possibly conduct more than \(10^{12}\) transactions. It is instructive to examine this argument more closely in order to dramatize the assumptions made in standard treatments of the indefinite IPD and other indefinitely repeated games. Note first that, in an indefinite IPD as described above, there can be no upper bound on the length of the game. There is, instead, some fixed probability \(p\) that, at any time in which the game is still being played, it will continue to be played with probability \(p\). If the interaction of Smith and Jones were modeled as an indefinite IPD, therefore, the probability of their interacting in a thousand years would not be zero, but rather some number greater than \(p^k\) where \(p\) is the probability of their interacting again now and \(k\) is the number of seconds in a thousand years. A more realistic way to model the interaction might be to allow the value of \(p\) to decrease as the game progressed. As long as \(p\) always remains greater than zero, however, it remains true that there can be no upper bound on the number of possible interactions, i.e., no time at which the probability of future interactions becomes zero. Suppose, on the other hand, that there was a number \(n\) such that that there was zero probability of the game's continuing to stage \(n\). Let \(p_1, \ldots, p_n\), be the probabilities that game continues after \(\text{stage } 1, \ldots, \text{stage } n\). Then there must be a smallest \(i\) such that \(p_i\) becomes \(0\). (It would happen at \(i=n\) if not sooner.) Given the standard common knowledge assumptions that we have been making, the players would know this value of \(i\), and the IPD would be one of fixed length, and not an indefinite IPD at all. In the case of the shopkeeper and his customer, we are to suppose that both know today that their last interaction will occur, let's say, at noon on June 10th, 2020. The very plausible idea that we began with, viz., that some upper bounds on the number of interactions are common knowledge, even though the smallest upper bound is not, is incompatible with the assumption that we know all the continuation probabilities \(p_i\) from the start.