# American Institute of Mathematical Sciences

July  2017, 4(3): 255-284. doi: 10.3934/jdg.2017015

## Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations

 Economics Department, University of Essex, Wivenhoe Park, Colchester, UK

* Corresponding author: Sheri M. Markose

Received  April 2016 Revised  April 2017 Published  July 2017

Fund Project: I'm grateful for constructive comments from an anonymous referee, which have improved the quality and structure of the paper. I have benefitted from encouragement from Noam Chomsky and from recent discussions with Jeffrey Johnson and Thanos Yannacopoulos, respectively, at the Global Systems Science Conference in Genoa in November 2015 and at the 2015 AUEB 12th Annual Summer School, where this paper was given. At the 2014 ESRC funded Diversity in Macroeconomics Conference, I had the chance to assemble Vittorio Gallese, Scott Kelso and Eshel Ben-Jacob, who helped me take this field to a new frontier. Over the years, there have been discussions with Steve Spear, Peyton Young, Aldo Rustichini, Ken Binmore, Arthur Robson, Kevin McCabe, Steven Durlauf, Shyam Sunder, James Foster and Vela Velupillai. I appreciate discussions with the students who attend my Complexity Economics lectures at the University of Essex, and those who have done dissertations on this such as Alexander Thierschmidt

The new digital economy has renewed interest in how digital agents can innovate. This follows the legacy of John von Neumann dynamical systems theory on complex biological systems as computation. The Gödel-Turing-Post (GTP) logic is shown to be necessary to generate innovation based structure changing Type 4 dynamics of the Wolfram-Chomsky schema. Two syntactic procedures of GTP logic permit digital agents to exit from listable sets of digital technologies to produce novelty and surprises. The first is meta-analyses or offline simulations. The second is a fixed point with a two place encoding of negation or opposition, referred to as the Gödel sentence. It is postulated that in phenomena ranging from the genome to human proteanism, the Gödel sentence is a ubiquitous syntactic construction without which escape from hostile agents qua the Liar is impossible and digital agents become entrained within fixed repertoires. The only recursive best response function of a 2-person adversarial game that can implement strategic innovation in lock-step formation of an arms race is the productive function of the Emil Post [58] set theoretic proof of the Gödel incompleteness result. This overturns the view of game theorists that surprise and innovation cannot be a Nash equilibrium of a game.

Citation: Sheri M. Markose. Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations. Journal of Dynamics & Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015
##### References:
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##### References:
 [1] P. Albin, The metalogic of economic predictions, calculations and propositions, Mathematical Social Sciences, 3 (1982), 329-358. doi: 10.1016/0165-4896(82)90016-6. [2] P. Albin, Barriers and Bounds to Rationality, Essays on Economic Complexity and Dynamics in Interactive Systems, (Edited) [with an Introduction by Duncan Foley], Princeton University Press, 1998. [3] L. Anderlini, Some notes on Church's thesis and the theory of games, Theory and Decision, 29 (1990), 19-52. doi: 10.1007/BF00134103. [4] L. Anderlini and H. Sabourian, Cooperation and effective computability, Econometrica, 63 (1995), 1337-1369. doi: 10.2307/2171773. [5] M. Arbib and A. Fagg, Modeling parietal-premotor interactions in primate control of grasping, Neural Networks, 11 (1998), 1277-1303. [6] W. B. Arthur, On The evolution of complexity, Santa Fe Institute Working Paper, 93-11-070 (1993). [7] R. Axelrod, Risk in Networked Information Systems, Mimeo, Gerald R. Ford School of Public Policy, University of Michigan, 2003. [8] W. Baumol, The Free Market Innovation Machine, Princeton University Press, 2002. [9] W. Baumol, Red-Queen games: Arms races, rule of law and market economies, Journal of Evolutionary Economics, 14 (2004), 237-247. doi: 10.1007/s00191-004-0207-y. [10] E. Beinhocker, Evolution as computation: Integrating self-organization with generalized Darwinism, Journal of Institutional Economics, 7 (2011), 393-423. doi: 10.1017/S1744137411000257. [11] E. Ben-Jacob, Bacterial wisdom, Gödel's theorem and creative genomic webs, Physica A: Statistical Mechanics and Its Applications, 248 (1998), 57-76. doi: 10.1016/S0378-4371(97)00529-3. [12] M. Bhatt and C. Camerer, Self-referential thinking and equilibrium as states of mind in games: FMRI evidence, Games and Economic Behaviour, 52 (2005), 424-459. doi: 10.1016/j.geb.2005.03.007. [13] K. Binmore, Modelling rational players: Part 1, Economics and Philosophy, 3 (1987), 179-214. [14] E. Brynjolfsson and A. McAfee, The Second Machine Age: Work, Progress and Prosperity in a Time of Brilliant Technologies, W. W. Norton, New York, 2014. [15] R. Byrne and A. Whiten, Machiavellian Intelligence: Social Expertise and the Evolution of Intellect in Monkeys, Apes, and Humans, Oxford University Press, Oxford, 1988. [16] D. Canning, Rationality, computability and Nash equilibrium, Econometrica, 60 (1992), 877-888. doi: 10.2307/2951570. [17] J. Casti, Complexification: Explaining A Paradoxical World Through the Science of Surprises, London Harper Collins, London, 1994. [18] N. Chomsky, On certain formal properties of grammars, Information and Control, 2 (1959), 137-167. doi: 10.1016/S0019-9958(59)90362-6. [19] N. Chomsky, Three models for the description of language, IRE Transactions on Information Theory, 2 (1956), 113-124. doi: 10.1109/TIT.1956.1056813. [20] M. Corballis, The Recursive Mind: The Origins of Human Language, Thought, and Civilization, Princeton University Press, 2014. doi: 10.1515/9781400851492. [21] N. J. Cutland, Computability: An Introduction to Recursive Function Theory, Cambridge University Press, 1980. [22] R. Dawkins, The Extended Phenotype, Oxford University Press, Oxford, 1989. [23] P. M. Driver and D. A. Humphries, Protean behaviour: The biology of unpredictability, Ethology and Sociobiology, 10 (1989), 393-394. doi: 10.1016/0162-3095(89)90041-1. [24] H. Gaifman, Naming, Diagonalization, from Cantor, to Gödel to Kleene, Logic Journal of the IGPL, 14 (2006), 709-728. doi: 10.1093/jigpal/jzl006. [25] V. Gallese, Mirror neurons, embodied simulation, and the neural basis of social identification, Psychoanalytic Dialogues, 19 (2009), 519-536. doi: 10.1080/10481880903231910. [26] V. Gallese, L. Fadiga, L. Fogassi and G. Rizzolatti, Action recognition in the premotor cortex, Brain, 119 (1996), 593-609. doi: 10.1093/brain/119.2.593. [27] V. Gallese and C. Sinigaglia, What is so special about embodied simulation, Trends in Cognitive Sciences, 15 (2011), 512-519. doi: 10.1016/j.tics.2011.09.003. [28] K. Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, (English) [Gödel's Theorem in Focus, (Ed. S. G Shanker)], Publishers, New York, 1963. [29] D. E. Goldberg, The existential pleasures of genetic algorithms, in Genetic Algorithms in Engineering and Computer Science (eds. G. Winter, J. Perioux, M. Galan and P. Cuesta), John Wiley and Sons (1995), 23-31. [30] A. Gunnthorsdottir, K. McCabe and V. Smith, Using the Machiavellianism instrument to predict trustworthiness in a bargaining game, Journal of Economic Psychology, 23 (2002), 49-66. doi: 10.1016/S0167-4870(01)00067-8. [31] A. Haldane, Financial Arms Races, Speech delivered at the Institute for New Economic Thinking, Berlin, 14 April 2012. [32] M. Hauser, N. Chomsky and W. Fitch, The faculty of language: What is it, who has it, and how did it evolve?, Science, 298 (2002), 1569-1579. doi: 10.1017/CBO9780511817755.002. [33] F. A. Hayek, The Sensory Order, The University of Chicago Press, Chicago, 1999. doi: 10.7208/chicago/9780226321301.001.0001. [34] F. A. Hayek, The Constitution of Liberty, The University of Chicago Press, Chicago, 2011. doi: 10.7208/chicago/9780226320519.001.0001. [35] F. A. Hayek, The Theory of Complex Phenomena, In Studies In Philosophy, Politics, and Economics, The University of Chicago Press, Chicago, 1967. [36] J. Holland, Adaptation in Natural and Artificial Systems, MIT Press, New York, 1975. [37] A. Kashiwagi and T. Yomo, Ongoing Phenotypic and Genomic Changes in Experimental Coevolution of RNA Bacteriophage Qβ and Escherichia coli, PLoS Genet, 7 (2011), e1002188. doi: 10.1371/journal.pgen.1002188. [38] R. Koppl and B. Rosser, Everything I Might Say Will Already Have Passed Through Your Mind, Metroeconomica, 53 (2002), 339-360. [39] C. Langton, Life at the edge of chaos, (Ed. ) [C. Langton, C. Taylor, D. Farmer, and S, Rasmussen (Eds. ) Artificial Life Ⅱ], Santa Fe Institute Studies in the Sciences of Complexity 10 (1992). [40] E. Lieberman-Aiden, N. van Berkum, L. Williams, M. Imakaev, T. Ragoczy, A. Telling, I. Amit, B. Lajoie, P. Sabo, M. Dorschner, R. Sandstrom, B. Bernstein, M. Bender, M. Groudine, A. Gnirke, J. Stamatoyannopoulos, L. Mirny, E. Lander and J. Dekker, Comprehensive mapping of long-range interactions reveals folding principles of the human genome, Science, 326 (2009), 289-293. doi: 10.1126/science.1181369. [41] J. MacInnes, Modelling the Enemy: Recursive Cognitive Models in Dynamic Environments, Division of Life Sciences University of Toronto (Scarborough), Paper presented at 2006 Cognitive Science Conference, 2006. [42] B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1982. [43] S. M. Markose, Mirroring, Offline Simulation and Complex Strategic Interactions: Coordination, Anti-Coordination and Innovation, University of Essex Mimeo, 2015. [44] S. M. Markose, Computability and evolutionary complexity: Markets as complex adaptive systems (CAS), Economic Journal, 115 (2005), F159-F192. doi: 10.1111/j.1468-0297.2005.01000.x. [45] S. M. Markose, Novelty in complex adaptive systems (CAS): A computational theory of actor innovation, Physica A: Statistical Mechanics and Its Applications, 344 (2004), 41-49. doi: 10.1016/j.physa.2004.06.085. [46] S. M. Markose, The new evolutionary computational paradigm of complex adaptive systems: Challenges and prospects for economics and finance, In Genetic Algorithms and Genetic Programming in Computational Finance, (Ed. Shu-Heng Chen), Kluwer Academic Publishers, (2002), 443-484. doi: 10.1007/978-1-4615-0835-9_21. [47] B. McClintock, The significance of responses of the genome to challenge, Science 226(4676) (1984), 792{801. [48] G. F. Miller, Protean primates: The evolution of adaptive unpredictability in competition and courtship, in Machiavellian Intelligence: Ⅱ. Extensions and Evaluations, Cambridge University Press, (1997), 312{340. [49] P. Mirowski, Machine Dreams: How Economics Became a Cyborg Science, Cambridge University Press, New York, 2012. doi: 10.1017/CBO9780511613364. [50] Y. N. Moschovakis, Kleene's amazing second recursion theorem, Bulletin of Symbolic Logic, 16 (2010), 189-239. doi: 10.2178/bsl/1286889124. [51] G. B. Müller and S. A. Newman, The innovation triad: An EvoDevo agenda, Journal of Experimental Zoology Part B Molecular and Developmental Evolution, 304 (2005), 487-503. [52] J. H. Nachbar and W. R. Zame, Non-computable strategies and discounted repeated games, Economic Theory, 8 (1996), 103-122. doi: 10.1007/s001990050079. [53] M. Naeem, G. Prasad, D. R. Watson and J. A. S. Kelso, Electrophysiological signatures of intentional social coordination in the 10-12 Hz range, Neuroimage(2), 59 (), 1795-1803. [54] New Scientist, Feature, You Are Junk: Why It's Not Your Genes that Make You Human, 2016. Available from: https://www.newscientist.com/article/mg23130840-400-the-junk-that-makes-you-human/?cmpid=NLC. [55] S. Ohno, So much 'junk' DNA in our genome, Evolution of Genetic Systems, Brookhaven Symposia in Biology, 23 (1972), 366-370. [56] M. Osborne and A. Rubinstein, A Course in Game Theory, MIT Press, 1994. [57] A. J. 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Prediction Function, Meta–Information on Outcomes and Dynamics of 2-person Turing Machine Game
Mirror Neurons As Offline Simulations for Mutual Predictions with Self and Other as Gödel 2-Place Substitution Function for Meta-Analysis (Rogers [61,p.202-204])
The Incompleteness of $p$'s Nash Equilibrium Strategy Set $\mathbf{B}_p$. Note that the arrow denotes the many-one recursive reduction of Lemma 3.8 using the second subroutine $f_p \sigma(b_a^{\neg},b_a^{\neg})=b^2$ for the surprise strategy function in (20) from the recursively enumerable subset $\mathbf{W}_{\sigma_n^{\neg}}$ of the archetypical productive set $\mathbf{\tilde{C}}$ in Lemma 6.1 to the Surprise Strategy set $\mathbf{W}_{\sigma_n^{!}}$, of Theorem 6.2 yielding the productive surprise strategy function $f_p^{E!}$ with $g.n(b^2(g(\sigma_n^{\neg})))$
Arms Race in Surprises/Innovations: Productive Function Growth of the Surprise Strategy Set (see equation (25))(NB g.n: Gödel number)
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