Using a theory of mind to find best responses to memory-one strategies

Nikoleta E. Glynatsi,Vincent A. Knight

doi:10.1038/s41598-020-74181-y

Nikoleta E. Glynatsi, Vincent A. Knight

Open Access

https://doi.org/10.1038/s41598-020-74181-y

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Abstract

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.

Highlights

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
Memory-one strategies are a set of Iterated Prisoner’s Dilemma (IPD) strategies that have been studied thoroughly in the literature[3,4], they have gained most of their attention when a certain subset of memory-one strategies was introduced in[5], the zero-determinant strategies (ZDs)
One specific advantage of memory-one strategies is their mathematical tractability. They can be represented completely as an element of R4[0,1]. This originates from[20] where it is stated that if a strategy is concerned with only the outcome of a single turn there are four possible ‘states’ the strategy could be in; both players cooperated (C C ), the first player cooperated whilst the second player defected (C D ), the first player defected whilst the second player cooperated ( DC ) and both players defected ( DD )

Summary

Introduction

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. There are several works in the literature that have considered strategies with a theory of mind[5,6,13,14,15,16] These works defined “theory of mind” as intention recognition[13,14,15,16] and as the ability of a strategy to realise that their actions can influence o pponents[6]. Theory of mind is defined here as the ability of a strategy to know the behaviour of their opponents and alter their own behaviour in response to that

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