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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show all references

References:
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[30]

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[31]

A. Haldane, Financial Arms Races, Speech delivered at the Institute for New Economic Thinking, Berlin, 14 April 2012. Google Scholar

[32]

M. HauserN. 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.  Google Scholar

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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.  Google Scholar

[38]

R. Koppl and B. Rosser, Everything I Might Say Will Already Have Passed Through Your Mind, Metroeconomica, 53 (2002), 339-360.   Google Scholar

[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). Google Scholar

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[42]

B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1982.  Google Scholar

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S. M. Markose, Mirroring, Offline Simulation and Complex Strategic Interactions: Coordination, Anti-Coordination and Innovation, University of Essex Mimeo, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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[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. Google Scholar

[49]

P. Mirowski, Machine Dreams: How Economics Became a Cyborg Science, Cambridge University Press, New York, 2012. doi: 10.1017/CBO9780511613364.  Google Scholar

[50]

Y. N. Moschovakis, Kleene's amazing second recursion theorem, Bulletin of Symbolic Logic, 16 (2010), 189-239.  doi: 10.2178/bsl/1286889124.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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Figure 1.  Prediction Function, Meta–Information on Outcomes and Dynamics of 2-person Turing Machine Game
Figure 2.  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])
Figure 3.  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 $g.n(b^2(g(\sigma_n^{\neg})))$
Figure 4.  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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