 As Becker and Cudd astutely observe, we don't need an upper bound on the number of possible iterations to make a backward induction argument for defection possible. If the players know all the values of \(p_i\) from the outset, then, as long as the value of \(p_i\) becomes and remains sufficiently small, they (and we) can compute a stage \(k\) at which the risk of future punishment and the chance of future reward no longer outweighs the benefit of immediate defection. So they know their opponent will defect at stage \(k\), and the induction begins. This modification of the Kavka/Carroll argument, however, only further exposes the implausibility of its assumptions. Not only are Smith and Jones expected to believe that there is non-zero probability that they will be interacting in a thousand years, each is expected to be able to compute the precise day on which future interactions will become and remain so unlikely that their expected future return is outweighed by that day's payoff. Furthermore each is expected to believe that the other has made this computation, and that the other expects him to have made it, and so on.