Abstract

Prisoner’s Dilemma, Chicken, Stag Hunts, and other two-person two-move (2 × 2) models of strategic situations have played a central role in the development of game theory. The Robinson–Goforth topology of payoff swaps reveals a natural order in the payoff space of 2 × 2 games, visualized in their four-layer “periodic table” format that elegantly organizes the diversity of 2 × 2 games, showing relationships and potential transformations between neighboring games. This article presents additional visualizations of the topology, and a naming system for locating all 2 × 2 games as combinations of game payoff patterns from the symmetric ordinal 2 × 2 games. The symmetric ordinal games act as coordinates locating games in maps of the payoff space of 2 × 2 games, including not only asymmetric ordinal games and the complete set of games with ties, but also ordinal and normalized equivalents of all games with ratio or real-value payoffs. An efficient nomenclature can contribute to a systematic understanding of the diversity of elementary social situations; clarify relationships between social dilemmas and other joint preference structures; identify interesting games; show potential solutions available through transforming incentives; catalog the variety of models of 2 × 2 strategic situations available for experimentation, simulation, and analysis; and facilitate cumulative and comparative research in game theory.

Highlights

  • Two-person, two-move games form the simplest possible models of strategic situations, where the outcomes of each person’s action depend on choices by another

  • Robinson and Goforth’s topology of payoff swaps reveals an elegant structure in the payoff space of 2 × 2 games, which can be mapped onto a flat four-layer display

  • Names for games based on symmetric games provide coordinates for locating 2 × 2 ordinal games within this payoff space

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Summary

Introduction

Two-person, two-move games form the simplest possible models of strategic situations, where the outcomes of each person’s action depend on choices by another. Prisoner’s Dilemma, Chicken ( known as Hawk-Dove), and other 2 × 2 preference structures have played a central role in the development and application of game theory in economics, political science, evolutionary biology, and other fields. Most attention has focused on a small subset of strict symmetric games, with less attention to the much larger numbers of asymmetric games, where players do not face the same incentive structure, and to non-strict games, those with indifference (ties) between outcomes. Taxonomies and associated naming systems have played a significant role in organizing knowledge in many fields of science, such as the Periodic Table of the Elements, molecular names in chemistry, and Linnaean classification of species, but have not seen much application in game theory. Rapoport and Guyer [6]

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