  Memory-one strategies are a set of Iterated Prisoner's Dilemma strategies
that have been praised for their mathematical tractability and performance
against single opponents. This manuscript investigates best response memory-one
strategies with a theory of mind for their opponents. The results add to the
literature that has shown that extortionate play is not always optimal by
showing that optimal play is often not extortionate. They also provide evidence
that memory-one strategies suffer from their limited memory in multi agent
interactions and can be out performed by optimised strategies with longer
memory. We have developed a theory that has allowed to explore the entire space
of memory-one strategies. The framework presented is suitable to study
memory-one strategies in the Prisoner's Dilemma, but also in evolutionary
processes such as the Moran process, Furthermore, results on the stability of
defection in populations of memory-one strategies are also obtained.
