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

• * Corresponding author: Sheri M. Markose
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.

Mathematics Subject Classification: Primary: 03D20, 68Q05, 91A40; Secondary: 0300, 92D15.

 Citation:

• 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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