Abstract
Self-referential calculations of oppositional or contrarian structures and the necessity to innovate to outsmart hostile agents in an arms race are ubiquitous in socio-economic systems, immunology and evolutionary biology. However, such phenomena with strategic innovation, which entails novel actions beyond listable sets, are outside the ambit of extant game theory. How can strategic innovation with novel actions be a Nash equilibrium of a game? Based on the only known Gödel-Turing-Post (GTP) axiomatic framework on meta-analyses of offline simulations that involve recursive operations on encoded information, we show that mutually mentalising agents capable of such offline simulations can “think outside the box” and embark on an arms race in novelty or surprises. A key logical ingredient of this is the self-referential encoding of a proposition on mutual negation or opposition, often referred to as the Gödel sentence. The only recursive best response function of a two-person game with an oppositional structure that can implement strategic innovation in a lock-step formation of an arms race is the productive function of the Emil Post set theoretic proof of the Gödel incompleteness result.
The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-32543-9_20
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-32543-9_20
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Notes
- 1.
These include the EC FP 7 calls, the Horizon 20–20 Global Systems Science initiatives and the UK Economic and Social Research Council (ESRC) endeavours at the Oxford Symposium of 2012 and the 2014 Essex Diversity in Macroeconomics conference. The latter aimed at breaking up the monoculture in mainstream economics by bringing together developments from at least three new branches of economics. These include, agent-based computational, complexity and behavioural economics. The aim of the highly interdisciplinary studies of computational and digital technologies, complexity sciences and neurophysiology of the brain is to address erstwhile gaps and controversies in the micro- and macroscopic aspects of socio-economic interactions.
- 2.
The Red Queen, the character in Lewis Carol’s Alice Through the Looking Glass, who signifies the need “to run faster and faster to stay in the same square” has become emblematic of the outcome of competitive co-evolution for evolutionary biologists in that no competitor gains absolute ground; see Markose (2005).
- 3.
Even though Durlauf (2012) notes that economics as a science is still evolving, he does not think it needs any substantive ‘paradigm’ shifts or new foundational studies to understand and model economic systems as complex adaptive systems. Of the issues on non-linearity, heterogeneity and interconnectedness that Durlauf (2012) considers, the latter two, along with the use of agent-based simulation models, are relatively new approaches in the literature (see Tesfatsion and Judd 2006). In the UK, there are no more than a dozen economists who pursue these nonmainstream approaches.
- 4.
One of the more prescient of macroeconomists, William White, has recently stated “it seems to me that nobody on the regulatory (and macroeconomics) side has really got to grips with the reality of this constant innovation” (Financial Times, June 25, 2014).
- 5.
Arthur (1993) argues for the need to take a computational perspective to model complexity.
- 6.
Sayama (2008) suggests that routine programs called quines appended to the end of other programs to read, copy and ‘print’ are important building blocks in self-replication algorithms.
- 7.
The neurons that fire with actual action execution are called canonical neurons, Arbib and Fagg (1998), and represent online machine executions in the GTP logic.
- 8.
Since antiquity, it has been known that self-refuting statements generate paradoxes as in the Cretan Liar proposition: this is false. Gödel’s analogue of the Liar proposition is the undecidable proposition. The latter, denoted as A, has the following structure: A ↔ ∼ | − (A). That is, A says of itself that it is not provable ( ∼ | −). However, unlike the Cretan Liar, there is no paradox in Gödel’s undecidable proposition as it can be proved that this is so. Any attempt to prove the proposition A results in a contradiction with both A and ∼ A, its negation being provable in the system. Simmons (1993, p. 29) has noted how with the Cantor diagonal lemma (which was used to prove that the power set of a set has greater cardinality than a set) we begin to have so-called ‘good’ uses of self-refuting structures that result in theorems rather than paradoxes.
- 9.
It is beyond the scope of this paper to discuss certain anti-machine views of Ben-Jacob (1998) which makes claims for semantic knowledge that goes beyond sub-personal syntactic expression of the Gödel sentence.
- 10.
F.A. Hayek is the first economist to have discussed the implications for economics that arise from the problems of non-computability that he called the limits of constructive reason and on the possibility that the brain manifests Gödel incompleteness (Hayek 1952, 1967). Hayek seminally redirected the discussion on the limits of deductive inference from Humean scepticism to the Gödel logic of incompleteness (Markose 2002, 2005) and indeed brought this to my attention by instructing me to read his book on the Sensory Order (1952). However, Hayek’s own account of this did not go beyond the Cantor diagonal lemma (see footnote 9), which led him to a view on cognitive incompleteness that there is much knowledge that cannot be formally enumerated.
- 11.
Byrne and Whiten (1999) have presented extensive evidence on the development of the Machiavellian Brain.
- 12.
Koppl and Rosser (2002) attempt to characterise the Nash equilibrium of the zero sum game that depicts the machinations of a well-known oppositional game involving Holmes and Moriarty using recursive function theory. Moriarty, who seeks the demise of Holmes, has to be in proximity with him, while Holmes needs to elude Moriarty. They conclude as follows: ‘We can see that there are best-reply functions, f(x), such that \(f(x)\neq x,\ \forall \ x\). That is, there are best-reply functions without a fixed point. (A fixed point is defined by the condition that f(x) = x.)’. It will be shown that Gödel meta-representational system has no problem ‘referring’ to the fixed point of the best response function that seeks to negate or deceive as in the Holmes-Moriarty game. The important point here, therefore, is not that one or the other player has to find a best response function that does not have a fixed point, but that the fixed point of an important class of best response functions is not computable, and this undecidability is fully encoded from within the meta-representational system of the players.
- 13.
The first limitative result on functions computable by T.Ms is that at most there can only be a countable number of these with the cardinality of \(\aleph \) being denoted by \(\aleph _{0}\), while from Cantor we know that the set of all number theoretic functions have cardinality of \(2^{\aleph _{0}}\). Hence, not all number theoretic functions are computable (see Cutland 1980).
- 14.
Assume that there is a computable function f = ϕ y , whose domain W y = C ∼ . Now, if y ∈ W y , then y ∈ C ∼ as we have assumed C ∼ = W y . But, by the definition of C ∼ in (11.5) if y ∈ W y , then y ∈ C and not to C ∼ . Alternatively, if y ∉ W y , y ∉ C ∼ , given the assumption that C ∼ = W y . Then, again we have a contradiction, as since from (11.5) when y ∉ W y , y ∈ C ∼ . Thus, we have to reject the assumption that for some computable function f = ϕ y and that its domain W y = C ∼ .
- 15.
This is analogous to the proof in Smullyan (1961, p. 96, Chap. V, Proposition 2).
- 16.
If sentences in a formal system, denoted as FS, are provable and have the status of being theorems (proof being defined as the operation of a Turing machine that halts), then their negations are refutable in that it is known that they belong to the domain on which Turing machines will not halt when attempting their proofs. If a FS is complete, then the set of all sentences satisfies the condition that FS = TUR, where T and R, respectively, are the set of provable and refutable sentences. FS is consistent if T and R are disjoint. The subset of R that is recursively enumerable are negations of those propositions that are known to be provable. The set FS is said to be incomplete if TUR ⊂ FS. The Gödel (1931) incompleteness result and the set theoretic proof of this by Post (1944) provide a constructive proof of a sentence denoted as u such that u ∈ FS and u does not belong to TUR. The sentence u is undecidable and is the ‘witness’ that FS is incomplete.
- 17.
- 18.
- 19.
In Markose (2015), it is suggested that using environments suitable for neurophysiological experiments of such a game, it is interesting to identify the juncture at which a player p knows his best payoffs come from the out-of-equilibrium configuration wherein the other player g has to be kept in a state of false belief σ(b a ⇁ , b a ) given in Definition 7.
- 20.
It is beyond the scope of this paper to discuss why a complexity perspective on radical uncertainty has a bearing on the long tradition of liberal jurisprudence associated with Kant (1781) and Hayek (1960) that universalisable rules of just society are end neutral and have no predictable consequences that are person, place or time specific. It can be conjectured, along the lines of the famous dictum of Hirschman (1991) that certain rules that aim to achieve specific ends in society may lead to futility, perversity and jeopardy that culminates in system failure as a result of gaming and protean agents. Likewise, absenting predictability of outcomes is the basis of a no arbitrage equilibrium in many economic models.
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Acknowledgements
I’m grateful for the interest at the 2015 AUEB 12th Annual Summer School in ideas relating to computability and complexity for economics. Discussions with AUEB Summer School participants in July 2015 and encouragement from Thanos Yannacouplos have contributed to this paper. Over the years, there have been discussions with Steve Spear, Peyton Young, Aldo Rustichini, Ken Binmore, Arthur Robson, Kevin McCabe, Steven Durlauf, Shyam Sunder and Vela Velupillai. The 2014 ESRC-funded Diversity in Macroeconomics Conference where I had the chance to assemble Vittorio Gallese, Scot Kelso and Eshel Ben Jacob has helped me to take this field to a new frontier. I have also benefitted from feedback from participants at invited Graduate lectures at the following institutions/workshops: 2012 Kiel Institute of the World Economy Summer School, 2010 Ruhr Bochum Economics Department invited Graduate lectures, 2009 Institute for Advanced Studies at Glasgow Workshop, 2002–2009 Lectures at the Centre for Computational Finance and Economic Agents and presently the new MSc Computational Economics, Financial Markets and Policy at the University of Essex.
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Markose, S. (2016). The Gödelian Foundations of Self-Reference,the Liar and Incompleteness: Arms Racein Complex Strategic Innovation. In: Pinto, A., Accinelli Gamba, E., Yannacopoulos, A., Hervés-Beloso, C. (eds) Trends in Mathematical Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32543-9_11
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