Transforming 2x2 Games

A map of relationships between the 2x2 games, based on how changes in payoffs transform one game into another
Prisoner's Dilemma, Chicken, and other 2x2 games can be transformed by swaps in adjoining payoff ranks:
  • 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

2x2 games mini display

2x2 Games Mini-Display. A small schematic visualization of the topology of 2x2 games, with payoff icons, strategy incentives, game and tile borderlines, payoff family colors, layer icons, and abbreviations for the twelve strict symmetric ordinal games.
   Robinson and Goforth's (2005) topology of swaps in adjoining payoffs maps the pathways for transforming one strict ordinal 2x2 game into another, and thereby elegantly arranges the games according to symmetries, alignment of highest payoffs, alignment of interests, number of dominant strategies and Nash Equilibria and other properties. In their "periodic table" display, the twelve strict symmetric ordinal games form an axis from southwest to northeast, surrounded by the more numerous but less-studied asymmetric games. Win-win games, where both players can get their best result, make up one fourth of the games, while Prisoner's Dilemmas and Alibi games with Pareto-deficient outcomes are only about five percent of the possible games.

This enhanced visualization of the topology of 2x2 games offers a tool for understanding the relationships between 2x2 games, particularly potential transformations between games. Arranged with the layer of simpler, win-win games in the lower left, it includes the order graphs and structures shown in Robinson and Goforth's original visualization, and also shows:
  • 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. 
The topology maps routes for escaping from Prisoner's Dilemma and other conflicts, ways to transform social dilemmas into win-win games.

This is a beta version, a working draft prototype. Comments welcome. These visualizations may be freely printed and used under the Creative Commons Attribution-ShareAlike License.


2x2 table thumbnail with link

Periodic Table of 2x2 Games

Names for Games:

A Binomial Nomenclature for 2x2 Ordinal Games

Presented at the International Conference on Game Theory, Stony Brook, NY, July 07 - 11, 2014

A binomial nomenclature identifies any two-person, two-move (2x2) ordinal game as a combination of symmetric game payoffs, based on the topology of payoff swaps that arranges 2x2 ordinal games in a natural order. Preference orderings categorize 2x2 ordinal games according to type of ties formed by transformations of strict games. Location of best payoffs defines orientations for games equivalent by interchanging rows or columns. Two-letter abbreviations for symmetric game names provide a compact notation. A systematic and efficient nomenclature identifying equivalent and similar 2x2 games helps locate interesting games; aids in understanding the diversity of elementary models of strategic situations available for experimentation, simulation, and analysis; and facilitates comparative and cumulative research in game theory.

Changing Games:

An Atlas of Conflict and Cooperation in 2x2 Games

 This atlas presents a series of diagrams mapping the the 2x2 games and their relationships, based on the Robinson-Goforth topology of 2x2 games, which shows how payoff swaps change one strategic situation into another. As a tool for researchers, students, and anyone interested in game theory, this atlas offers a visual introduction to the diversity of 2x2 games, illustrating the relationships among elementary models of strategic interaction, and pathways for transforming conflict into cooperation. Under construction. This working draft is composed of a series of pdf files. Comments welcome.

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.

Brams, S. J. 1994. Theory of Moves. Cambridge University Press.

Brams, S. J, and D. M Kilgour. 2009. How Democracy Resolves Conflict in Difficult Games. Games, Groups, and the Global Good: 229.

Bruns, B. R. Transmuting Samaritan's Dilemmas in Irrigation Aid: An Application of the Topology of 2x2 Games. International Association for the Study of Commons North American Meeting. Tempe AZ, September 30-October 2, 2010.

———.  2010. Navigating the Topology of 2x2 Games: An Introductory Note on Payoff Families, Normalization, and Natural Order. Arxiv preprint arXiv:1010.4727.

———.  Switching Games: Visualizing the Adjacent Possible in the Topology of Two-person Two-strategy Games. 2x2 Working Group Session 2, Canadian Economics Association, Ottawa, June 4, 2011

———. 2011. Visualizing the Topology of 2x2 Games: From Prisoner’s Dilemma to Win-win. In Stony Brook, NY: Game Theory Center, July 15, 2011

DeCanio, S.J., and A. Fremstad. 2011. Game Theory and Climate Diplomacy. Ecological Economics.

Dragicevic, Pierre 2011 Game Theory Icons. http://www.lri.fr/~dragice/gameicons/

Goforth, D. J. and Robinson, D. R., 2004. Periodic Table of the 2x2 Games Poster.

———.  2004. The Ecology of the Space of 2x2 Social Dilemmas. Presented at the Meetings of the Canadian Economics Association. Toronto, June 4-6, 2004.

———.  2009. Complex Behavior in Challenging Social Situations. Presented at the Meetings of the Canadian Economics Association.  University of Toronto, May 29-31, 2009.

———.  2009. The Interactive Applet of the Periodic Table,

———.  2009. Dynamic Periodic Table of the 2x2 Games: User’s Reference and Manual.

Greenberg, J. 1990. The Theory of Social Situations: An Alternative Game-theoretic Approach. New York: Cambridge Univ Pr.

Hopkins, Brian. 2011. Between Neighboring Strict Ordinal Games. Presented at the Meetings of the Canadian Economics Association. June 2-5, 2011.

Irwin, T. 2009. Implications for Climate-Change Policy of Research on Cooperation in Social Dilemmas.World Bank Policy Research Working Paper 5006. Washington, D.C.

Pasha, S. T. 2010. 2x2games.pdf (Graphic with Perl + Tex source code). Wikipedia

Perlo-Freeman, S. 2006. The Topology of Conflict and Co-operation. University of the West of England, Dept of Economics, Discussion Paper 609.

Rapoport, A. 1967. Exploiter, leader, hero, and martyr: the four archetypes of the 2 times 2 game. Behavioral science 12 (2): 81.

Rapoport, A., and M. Guyer. 1966. A taxonomy of 2 x 2 games. General Systems 11 (1-3): 203–214.

Rapoport, A., M. Guyer, and D. G Gordon. 1976. The 2 x 2 Game. University of Michigan Press.

Robinson, D. R., and D.J., Goforth. 2003. A Topologically-based Classification of the 2x2 Ordinal games. Presented at the Meetings of the Canadian Economics Association. Carlton University.

———.  Alibi games: the Asymmetric Prisoner’s Dilemmas. Presented at the Meetings of the Canadian Economics Association. Toronto, June 4, 2004.

———.  2004. Graphs and Groups for the Ordinal 2x2 Games.  Presented at the Canadian Theory Conference, Montreal, May 7-9 2004

———. 2005 The Topology of 2x2 Games: A New Periodic Table (Routledge Advances in Game Theory) London: Routledge.

———.  2005. Conflict, No Conflict, Common Interests, and Mixed Interests in 2x2 Games. Presented at the Meetings of the Canadian Economics Association. Hamilton, Ontario, May 27-29, 2005.

———.  2011. Teaching Economics by Teaching 2x2 Game Theory. In Presented at the Meetings of the Canadian Economics Association. June 2-6, 2011.

Robinson, David, David Goforth and Matt Cargill. 2007.  Toward a Topological Treatment of the Non-strictly Ordered 2x2 Games.

Schelling, T.C. 1980. The Strategy of Conflict. Harvard University Press.

Sen, A. K. 1967. Isolation, Assurance and the Social Rate of Discount. The Quarterly Journal of Economics 81 (1): 112–124.

Simpson, J. 2010. Simulating Strategic Rationality. Ph.D. Dissertation, Edmonton: University of Alberta.

Simpson, J. 2011. Overcoming the (near) Fetishization of a Small Group of 2x2 Games. Presented at the Meetings of the Canadian Economics Association. Toronto, June 2-5, 2011.