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Atlas of 2x2 Games: Transforming Conflict and Cooperation

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Type: Working Paper
Author: Bruns, Bryan
Date: 2015
Agency:
Series:
URI: http://hdl.handle.net/10535/9629
Sector: Theory
Region:
Subject(s): game theory
social dilemmas
cooperation
prisoner's dilemma
chicken game
coordination game
Abstract: "In social dilemmas and other strategic situations, the outcome of each person's choice depends on what the other decides. Payoff structures may lead to win-win results, or to tragic failure when individual incentives clash with cooperation that could benefit both. Prisoner's Dilemma, Chicken, Stag Hunt, and other two-person two-move (2x2) games form the foundation for analysis of cooperation and conflict in game theory and its application in economics, political science, ecology, evolutionary biology, and other fields. Changes in payoffs can turn one game into another, for example swaps in the highest payoffs turn Prisoner's Dilemma into a Stag Hunt. Robinson and Goforth showed how a topology of payoff swaps elegantly arranges 2x2 games into a periodic table, organized in a natural order according to swap neighbors, alignment of best outcomes, symmetry, number of dominant strategies and equilibria, and other properties. This atlas further visualizes the the topology of 2x2 games, displaying relationships between games and pathways for transforming strategic situations. The diagrams in this atlas: show how payoff swaps transform games and so open opportunities to achieve different outcomes, mapping the adjacent possible for institutional design; display the diversity of 2x2 games, the variety of strategic situations where the outcome of each person's action depends on what the other decides, the range of possible incentive structures where two people have two interdependent choices; illustrate how interests may be aligned, opposed, or mixed, combinations of incentives and externalities when payoff patterns induce help or harm, kindness or cruelty; present enhanced visualizations of the periodic table of 2x2 games, including payoff families, game graphs and payoff values, simpler games with ties, and the structure of bands, hotspots, and pipes; reveal robustness or sensitivity in how changes in payoffs shift equilibrium outcomes; indicate likely frequencies of different kinds of games, if payoffs occur randomly; and specify names for games, a binomial nomenclature that forms part of a convenient framework for identifying and naming the complete set of all 2x2 ordinal games. The periodic table of 2x2 games is organized around the symmetric 2x2 games on a diagonal axis, showing the twelve strict ordinal games where each person has four distinctly ranked payoffs. Payoff patterns from symmetric games combine to form asymmetric games, and so provide a convenient basis for naming games. Neighboring games are linked by swaps in adjoining payoffs. The games with ties lie between the strict symmetric games, linked by half-swaps that make or break ties, as illustrated by additional diagrams. Normalized versions of games with real or ratio value payoffs also map onto the periodic table of 2x2 games. As a tool for researchers, students, and anyone interested in game theory and understanding behavior in social situations, this atlas offers a visual introduction to the diversity of 2x2 games, illustrating the relationships among elementary models of strategic interaction, and mapping pathways for transforming conflict into cooperation."

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