The undecidability of the spatialized prisoner's dilemma
 Patrick Grim
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
In the spatialized Prisoner's Dilemma, players compete against their immediate neighbors and adopt a neighbor's strategy should it prove locally superior. Fields of strategies evolve in the manner of cellular automata (Nowak and May, 1993; Mar and St. Denis, 1993a,b; Grim 1995, 1996). Often a question arises as to what the eventual outcome of an initial spatial configuration of strategies will be: Will a single strategy prove triumphant in the sense of progressively conquering more and more territory without opposition, or will an equilibrium of some small number of strategies emerge? Here it is shown, for finite configurations of Prisoner's Dilemma strategies embedded in a given infinite background, that such questions are formally undecidable: there is no algorithm or effective procedure which, given a specification of a finite configuration, will in all cases tell us whether that configuration will or will not result in progressive conquest by a single strategy when embedded in the given field. The proof introduces undecidability into decision theory in three steps: by (1) outlining a class of abstract machines with familiar undecidability results, by (2) modelling these machines within a particular family of cellular automata, carrying over undecidability results for these, and finally by (3) showing that spatial configurations of Prisoner's Dilemma strategies will take the form of such cellular automata.
 Axelrod, R. (1984) The Evolution of Cooperation. Basic Books, New York
 Berlekamp, E., Conway, J., Guy, R. (1982) Winning Ways for your Mathematical Plays. Academic Press, London
 Boolos, G., Jeffrey, R. (1989) Computability and Logic. Cambridge Univ. Press, New York.
 Demongeot, J., Golès, E., Tchuente, M. eds. (1985) Dynamical Systems and Cellular Automata. Academic Press, New York
 Dewdney, A.K. (1993) The (New) Turing Omnibus. Computer Science Press, New York.
 Grim, P. (1994) Computation and Undecidability in the Spatialized Prisoner's Dilemma. Group for Logic and Formal Semantics. Dept. of Philosophy, SUNY at Stony Brook
 Grim, P. (1994) An NPComplete Question Regarding the Spatialized Prisoner's Dilemma. Group for Logic and Formal Semantics. Dept. of Philosophy, SUNY at Stony Brook
 Grim, P. (1995) The Greater Generosity of the Spatialized Prisoner's Dilemma. Journal of Theoretical Biology 173: pp. 353359
 Grim, P. (1996) Spatialization and greater generosity in the stochastic Prisoner's Dilemma. BioSystems 37: pp. 317
 Gutowitz, H. (Ed.): 1990, Cellular Automata: Theory and Experiment, NorthHolland, New York.
 Mar, G., and St. Denis, P.: 1993a, 'The Evolution of Dynamical MetaStrategies in the Prisoner's Dilemma', International Conference on Game Theory, SUNY at Stony Brook, July 1993, and research report No. 9301, Group for Logic and Formal Semantics, Dept. of Philosophy, SUNY at Stony Brook.
 Mar, G., St. Denis, P. (1993) Chaos in Cooperation: TwoDimensional Prisoner's Dilemmas in InfiniteValued Logic. International Journal of Bifurcation and Chaos. Dept. of Philosophy, SUNY at Stony Brook, pp. 943958
 Minsky, M. (1967) Computation: Finite and Infinite Machines. PrenticeHall, Englewood Cliffs, N.J.
 Nowak, M. (1990) Stochastic Strategies in the Prisoner's Dilemma. Theoretical Population Biology 38: pp. 93112
 Nowak, M., May, R. (1992) Evolutionary games and spatial chaos. Nature 359: pp. 826829
 Nowak, M., May, R. (1993) The Spatial Dimensions of Evolution. International Journal of Bifurcation and Chaos 3: pp. 3578
 Nowak, M., Sigmund, K. (1989) GameDynamical Aspects of the Prisoner's Dilemma. Applied Mathematics and Computation 30: pp. 191213
 Nowak, M., Sigmund, K. (1992) Tit for tat in heterogeneous populations. Nature 355: pp. 250252
 Nowak, M., Sigmund, K. (1993) A strategy of winstay, loseshift that outperforms titfortat in the Prisoner's Dilemma game. Nature 364: pp. 5658
 Silverman, B. (1987) The Phantom Fish Tank: An Ecology of Mind. Logo Computer Systems, Montreal
 Toffoli, T., Margolus, N. (1987) Cellular Automata Machines: A New Environment for Modelling. MIT Press, Cambridge, Mass.
 Wolfram, S. (1984) Cellular automata as models of complexity. Nature 311: pp. 419424
 Wolfram, S. (1986) Theory and Applications of Cellular Automata. World Scientific, Philadelphia
 Title
 The undecidability of the spatialized prisoner's dilemma
 Journal

Theory and Decision
Volume 42, Issue 1 , pp 5380
 Cover Date
 19970101
 DOI
 10.1023/A:1004959623042
 Print ISSN
 00405833
 Online ISSN
 15737187
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Undecidability
 Prisoner's Dilemma
 cellular automata
 game theory
 decision theory
 computability
 Industry Sectors
 Authors

 Patrick Grim ^{(1)}
 Author Affiliations

 1. Department of Philosophy, SUNY at Stony Brook, Group for Logic & Formal Semantics, Stony Brook, NY, 11794, U.S.A.