Transforming 2x2 Games

• Swapping one person's highest-ranked payoffs changes Prisoner's Dilemma into an asymmetric Prisoner's Dilemma, while high swaps for both players lead to a precarious Stag Hunt, with one win-win equilibrium and another inferior but safer (risk-dominant) equilibrium.
• Swapping middle-ranked payoffs for one player converts Prisoner's Dilemma into a zero-sum game of Total Conflict, but swapping middle payoffs for both forms a game of Deadlock, where both players can get their second-best outcome.
• Swapping the two lowest payoffs turns Prisoner's Dilemma into Chicken, while setting the two lowest payoffs equal to each other, making ties, creates a game in between, Low Dilemma, where incentives may lead both players to get their worst-ranked outcome.
• Swapping the highest payoffs for a single player shifts from Chicken to Hegemony, where one player always wins and the other loses, but high swaps for both move to a win-win game of No Conflict.
• Swapping the two highest payoffs in a Battle of the Sexes game produces a Coordination game.
• Such swaps in how outcomes are ranked, and the resulting transformations in game structures and outcomes, could arise from changes in information, technology, perception, rules, monitoring, enforcement, or side payment

• Numeric payoffs, indicating Pareto optimal, Pareto-deficicient, and maximin outcomes
• Names for the twelve symmetric games provide coordinates for locating the asymmetric games.
  
• Common names for asymmetric games and tiles.
  
• Categorization by payoffs at Nash Equilibria not only distinguishes Cyclic Games, Stag Hunts, and Battles of the Sexes, but also adds families of Second Best, Biased, and Unfair games, and extends the Prisoner's Dilemma family to include a Tragic subfamily.
  
• Borderlines outline tiles of games linked by swaps in the two lowest payoffs, and help to locate the games with ties and other normalized games that lie between the strict ordinal games. 
• A second page shows structures in the topology: symmetric games on the diagonal, payoff patterns combine to form games, dominant strategies lead to equilibria, high swaps link tiles across layers, distances to win-win, mixtures of interests, borderlines of games with ties, and game numbers for the Robinson and Goforth topology, Rapoport and Guyer taxonomy, and Brams typology of 2x2 games. 

Escaping Prisoner's Dilemmas: From Discord to Harmony
in the Landscape of 2x2 Games

Bryan Bruns

Working Paper

Changes in payoffs can transform Prisoner’s Dilemma and other social dilemmas into harmonious win-win games. Using the Robinson-Goforth topology of 2x2 games, this paper analyzes how payoff swaps turn Prisoner’s Dilemma into other games, compares Prisoner’s Dilemmas with other families of games, traces paths that affect the difficulty of transforming Prisoner’s Dilemma and other social dilemmas into win-win games, and shows how ties connect simpler and more complex games. Charts illustrate the relationships between the 144 strict ordinal 2x2 games, the 38 symmetric 2x2 ordinal games with and without ties, and the complete set of 1,413 2x2 ordinal games. Payoffs from the symmetric ordinal 2x2 games combine to form asymmetric games, generating coordinates for a simple labeling scheme to uniquely identify and locate all asymmetric ordinal 2x2 games. The expanded topology elegantly maps relationships between 2x2 games with and without ties, enables a systematic understanding of the potential for transformations in social dilemmas and other strategic interactions, offers a tool for institutional analysis and design, and locates a variety of interesting games for further research. 

A Brief Bibliography on the Topology of 2x2 Games

For those with some background in game theory, the 2007 article, Toward a Topological Treatment of the Non-strictly Ordered 2x2 Games, by Robinson, Goforth, and Cargill, offers a quick introduction to the topology and discusses its extension to include games with ties. Robinson and Goforth's 2005 book, The Topology of 2x2 Games provides a systematic introduction, including an accessible formal presentation using concepts from topology and group theory.