 One clever implementation of the idea that a strategies in an imperfect environment should pay attention to their previous interactions is the family of “Pavlovian” strategies investigated by Kraines and Kraines. For each natural number \(n\), \(n\)-Pavlov, or \(\bP_n\), adjusts its probability of cooperation in units of \(\tfrac{1}{n}\), according to how well it fared on the previous round. More precisely, if \(\bP_n\) was cooperating with probability \(p\) on the last round, then on this round it will cooperate with probability \(p\,[+]\tfrac{1}{n}\) if it received the reward payoff on the previous round, \(p\,[-]\tfrac{1}{n}\) if it received the punishment payoff, \(p\,[+]\tfrac{2}{n}\) if it received the temptation payoff, and \(p\,[-]\tfrac{2}{n}\) if it received the sucker payoff. \([+]\) and \([-]\) are bounded addition and subtraction, i.e., \(x\,[+]\,y\) is the sum \(x+y\) unless that number exceeds one, in which case it is one (or as close to one as the possibility of error allows), and \(x\,[-]\,y\) is similarly either \(x-y\) or close to zero. Strictly speaking, \(\bP_n\) is not fully specified until an initial probability of cooperation is given, but for most purposes the value of that parameter becomes insignificant in sufficiently long games and can be safely ignored. It may appear that \(\bP_n\) requires far more computational resources to implement than, say, TFT. Each move for the latter depends on only on its opponent's last move, whereas each move for \(\bP_n\) is a function of the entire history of previous moves of both players. \(\bP_n\), however, can always calculate its next move by tracking only its current probability of cooperation and its last payoff. As its authors maintain, this seems like “a natural strategy in the animal world.” One can calculate that for \(n \gt1\), \(\bP_n\) does better against the random strategy than does TFT. More generally, \(\bP_n\) does as well or better than TFT against the generous unresponsive strategies Cp that always cooperate with fixed probability \(p \ge \tfrac{1}{2}\) (because an occasional temptation payoff can teach it to exploit the unresponsive strategies.) In these cases the “slow learner” versions of Pavlov with higher values of \(n\) do slightly better than the “fast learners” with low values. Against responsive strategies, like other Pavlovian strategies and TFT, \(\bP_n\) and its opponent eventually reach a state of (almost) constant cooperation. The total payoff is then inversely related to the “training time,” i.e., the number of rounds required to reach that state. Since training time of \(\bP_n\) varies exponentially with \(n\), Kraines and Kraines maintain that \(\bP_3\) or \(\bP_4\) are to be preferred to other Pavlovian strategies, and are close to “ideal” IPD strategies. It should be noted, however, that when (deterministic) TFT plays itself, no training time at all is required, whereas when a Pavlovian strategy plays TFT or another Pavlov, the training time can be large. Thus the cogency of the argument for the superiority of Pavlov over TFT depends on the observation that its performance shows less degradation when subject to imperfections. It is also worth remembering that no strategy is best in every environment, and the criteria used in defense of various strategies in the IPD are vague and heterogeneous. One advantage of the evolutionary versions of the IPD discussed in the next section is that they permit more careful formulation and evaluation of success criteria.