 Another noteworthy feature of the 2IPD, proved rigorously in Press and Dyson (Appendix A), is that a long memory is unnecessary to play well. Suppose I adopt a memory-one strategy i.e., I condition each move only on our last interaction. Then Press and Dyson show that you can't benefit by using a longer memory: whatever strategy you adopt, there is an equivalent memory-one strategy you could have adopted that would net us both the same scores. By adopting a memory-one strategy myself, I ensure that a longer memory will be of no benefit to you. Thus we can, without loss of generality, take the 2IPD game to be a game between agents with memory-one strategies.