Transforming 2x2 Games
A map of relationships between the 2x2 games, based on how changes in payoffs transform one game into another
Dilemma, Chicken, and other 2x2 games can be transformed by
swaps in adjoining payoff ranks:
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:
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 Games Chart. An explanatory chart for the topology of 2x2 games, with order graphs, numeric bimatrices, visualizations of payoff and game properties, brief explanations, and instructions for locating games.
Prisoner’s Dilemma turns into an Asymmetric Prisoner’s Dilemma, transforms into a precarious Stag Hunt, becomes a dominance-solvable Privileged Hunt, and then changes into No Conflict.
A Brief Bibliography on the Topology of 2x2 Games
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
———. 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
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.
The Ecology of the Space of 2x2 Social Dilemmas.
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):
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.
2 x 2 Game. University of Michigan
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.
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.
Robinson, David, David Goforth and Matt Cargill. 2007.
a Topological Treatment of the
Non-strictly Ordered 2x2 Games.
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.
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.