1
Meta-Representation, Mirroring and the Liar: Complex Strategic
Behaviour and Arms Race In Novelty and Surprises
Sheri M. Markose1
Economics Department and Centre for Computational
Finance and Economic Agents (CCFEA)
Email:scher@essex.ac.uk,
University of Essex, Wivenhoe Park
Colchester C04 3SQ, UK.
June 2012
Abstract:
Oppositional or contrarian structures, deception, self-reference and the necessity to
innovate to out-smart hostile agents in an arms race are ubiquitous in socio-economic
systems, immunology and evolutionary biology. However, such phenomena with
strategic innovation are outside the ambit of extant game theory. How can strategic
innovation with novel actions be a Nash equilibrium of a game? Having reviewed
evidence from cognitive social neuroscience and neuro-economics, this paper shows
how the only known Gödel-Turing-Post (GTP) axiomatic framework on metarepresentation of procedures singles this out as a necessary condition for self-referential
simulations. Logical implications of such ‘mirroring’ include the detection of negation or
deception and the capacity of such agents to innovate, “think outside the box” and embark
on an arms race in novelty or surprises. The only recursive best response function of the
game that can implement strategic innovation in a lock-step formation of an arms race
is the productive function of the Emil Post (1944) set theoretic proof of the Gödel
incompleteness result.
Keywords: Meta-representation; Self-reference; Contrarian; Simulation; Strategic
innovation; Novelty; Surprises; Red Queen type arms race; Creative and Productive
Sets; Productive function ; Surprise Nash Equilibrium
I’m grateful for recent discussions with Peyton Young, Aldo Rustichini, Kevin Mc Cabe, Steve Spear
and Michael Arbib. Over the years, there have been discussions with Ken Binmore, Arthur Robson
and Vela Velupillai. The paper has benefitted from feedback from participants at invited lectures given
at the Kiel Institute of the World Economy in 26-27 June 2012, the Ruhr Bochum Economics
Department on Markets as Complex Adaptive Systems in 2010, the Institute for Advanced Studies at
Glasgow workshops on “Limits to Rationality in Economics and Financial Markets“ in June/July 2009
and from 2002 at the Centre for Computational Finance and Economic Agents at the University of
Essex.
1
2
1. Introduction
In a recent paper on strategic behaviour, Crawford (2003) begins with the elaborate
subterfuge involved in the D-Day Allied landings of World War II in order to surprise
and wrong foot the enemy. There is a long standing tradition, albeit an informal one,
in the macro-economic policy literature of Lucas (1972) on the strategic use of
‘surprises’ by policy makers against a private sector which may render policy
ineffective if policy can be predicted. 2 Binmore (1987) seminally indicated that any
strategist who upholds deterministic strategies as being optimal must answer the
question “what of the Liar ?” He had raised the alarm for game theory regarding the
logical and strategic necessity for novelty that arises from the archetype of the rule
breaker or contrarian who controverts or falsifies what can be computed / predicted.
Baumol (2002, 2004), in keeping with the Schumpeter (1934) vision of ‘creative
destruction’, has extensively discussed and documented the role of the relentless Red
Queen3 type strategic arms race in innovation by firms of products and processes in
capitalism which he claims is not addressed in mainstream economics. In social
neuro- science, the role of social proteanism has been discussed at length when
predictability is punished by hostile animals capable of prediction (see, Driver and
Humphries, 1988, Miller,1988, Bryne and Whiten, 1988,1997). Markose (2004,
2005), in the context of markets as complex adaptive systems, and Robson (2003,
2005) in regard to of strategic rationality and how the human neo-cortex grew big,
have argued that much of game theory overlooks the arms race in complexity and the
endogenous emergence of innovations from the strategic interaction of intelligent
agents.
The contrarian need not only appear in the agency of a player. It can arise
from the structure of the payoffs of a game where a player wins only if his actions
2
Despite, the seminal inclusion of the necessity of surprises by Lucas (1972) in a policy context in response to
regulatees who can contravene policy, it was the exact opposite that was followed. The notion of a surprise
strategy in the macro-economics literature appears in the so called Lucas surprise supply function which entails a
term for surprise inflation. The idea here is that the private sector contravenes the effects of anticipated inflation,
viz. the neutrality result. Hence, it is intuitively asserted that authorities who seek to expand output beyond the
natural rate need to use surprise inflation. As surprise inflation sounds like a ‘bad’ thing to do – the objective of
mainstream monetary policy for over two decades became one of pre-committing authorities to a fixed currency
peg till it got destroyed by regulatees in 1992, and thereafter a fixed quantitative rule for inflation control became
the sole macro-economic policy tool for stability (see, Markose, 2005 Sections 3 and 4). Remarkably as Axelrod
(2003) said that system failure occurs because “coevolution is not anticipated”, the 2007 financial crisis is
acknowledged by many to be the consequence of a system left vulnerable to the proteanism of regulatees by
flawed micro-financial and macro-economic policy doctrines.
3 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. Baumol (2002) shows how Red Queen
type arms race in product or process innovation is undertaken by firms simply to retain status quo in market share.
3
diverge from that of co-players. The classic zero-sum 2-person game of matching
pennies is an example of this. The earliest discussion of this is by Morgenstern
(1935) in terms of the mortal combat between Moriarty and Holmes. Moriarty who
seeks the demise of Holmes has to be in proximity with him while Holmes needs to
elude Moriaty. Arthur (1994) noted that asset markets have a contrarian pay off
structure, rewards tend to accrue to those agents who are contrarian or in the minority.
That is, if it is most profitable to buy when the majority are selling and sell when the
majority are buying, then if all agents act in an identical homogenous fashion having
made predictions from the same meta-model, they will fail in their objective to be
profitable and any trend movements in prices will be broken down by contrarians who
will arise endogenously from untagged agents, Arthur et. al. (1997).4 The lack of
effective procedures to determine winning strategies in games with contrarian payoff
structures and the impossibility of homogenous rational expectations, cleverly
identified by Arthur (1994) in the above informal statement of this problem in stock
markets is typically called the Minority or El Farol game. So unlike the traditional no
trade results of Milgrom and Stokey (1982) and a cessation of trade under conditions
of homogenous rational expectations, there is instead heterogeneity of beliefs and
myriad technical trading strategies that will endogenously bring about the boom and
bust dynamics seen in asset markets.
Few economists have acknowledged that the 1987 Binmore critique of game
theory based on the Liar, the Baumol type characterization of technology races, the
Lucas ‘surprise’ based postulates for policy design and Arthur’s (1994) challenge to
the possibility of homogenous rational expectations in contrarian/minority stock
market games, signify the need for a new set of logico-axiomatic foundations for
handling self-referential mappings of meta-representational systems and the role of
protean strategic behaviour that involves novel objects not previously there. One is
confronted with the disjunction between game theory on the one hand and strategic
behaviour on the other in which outsmarting others using deceit, novelty and surprises
are key ingredients. As partly indicated by Binmore (1987) extant mathematics of
game theory is closed and complete. It is impossible to produce novelty or surprises in
4
A lucid statement of the problem of self-reference in asset markets can be found in Arthur et. al
(1997) : “In asset markets, agents’ forecasts create the world that agents are trying to forecast. Thus,
asset markets have a reflexive nature in that prices are generated by traders’ expectations, but these
expectations are formed on the basis of anticipation of others’ expectations. This reflexivity, or self-
4
a Nash equilibrium, let alone the structure of an arms race in strategic innovation. In
game theory there are strategy mappings to a fixed action set and indeterminism
extends only to randomizations between given actions. Regarding contrarian
behaviour, the Liar/deceit and strategic innovations or surprises to escape from hostile
agents, as pointed out by Crawford (2003) –to date , economic “theory lags behind the
public’s intuition”... and “we are left with no systematic way to think about such
ubiquitous phenomena”. While surprise is an affective state elicited in response to an
unexpected event or in the detection of a contradiction or conflict between a new
discovery and a previously held theory, the significance of surprise and its associated
cognitive processes in humans in the context of survival in a complex and dynamic
environment is far from well formulated.
The objective of this paper is to provide the only known logical and axiomatic
foundations of meta-representational systems (MRS, here after), viz. the GödelTuring-Post (GTP) theory of computation or recursion function theory. It will be
shown that mathematical logic naturally provides the framework for a game which
involves a contrarian or oppositional structure and also on what constitutes a surprise
in mutual meta-analysis involving two players. Meta-analysis in the GTP framework
involves offline operations on encoded information. In preparation for this, in Section
2, a brief survey will be made of the two areas that have to date contributed to
conjectures and hypotheses relating to the capacity for meta-representation and
mirroring, the simulation theory of mind reading, the identification of deceit or the
contrarian and the necessity to surprise and resort to protean innovative behaviour in
an interactive setting. The first of these covers recent developments in cognitive and
social neuroscience which includes neuro-economics of strategic behaviour. The
second area is that of the applications, to date, of recursion function theory to game
theory.
Recent advances in the neurophysiology of the brain on mirror neurons, hailed
as one of the great discoveries in science, provide a cellular explanation for why the
capacity of meta-representation with self as both an actor and observer in the mapping
is essential for social and strategic behaviour. In the macaque monkeys, where this
phenomena was first found by the so called Parma group (Rizzolatti et. al., 1996)5,
referential character of expectations, precludes expectations being formed by deductive means (italics
added), so that perfect rationality ceases to be well defined”.
5
While single-cell recordings of F5 neurons have thus far been limited to monkeys, a variety of
5
mirror neurons discharge when the subject observes goal-directed action in another
individual and also when performing the action herself. Hence, they serve to
internally "represent" an action, by self or another, within the pre-motor cortex of the
observer in the form of an offline simulation. The neurons that fire to execute the
action by the agent herself, in contrast, have been called canonical neurons, Fagg and
Arbib (1998).6 The ‘meta’ or virtual status attributed to mirror neurons distinguishes
them from the functioning of the canonical neurons. This has led to the mirror system
hypothesis (Arbib and Rizzolatti,1997, Rizzolatti and Arbib,1998) and also to the
simulation theory of mind reading (see, Gallesse and Goldman,1998)7. The hypothesis
here is that an individual can recognize goal oriented actions of others because the
neural pattern elicited in the observational meta system of the individual is similar to
what is internally generated by the canonical neurons to physically produce a similar
action by the individual. Initially, the proponents of the mirror system hypothesis
were careful only to attribute syntactic machine learning qualities to the activation of
mirror neurons per se in the case of biological motion rather than affective
understanding in humans which requires a larger repertoire of interconnections that
trigger internal states linked to memory and emotions emanating from the limbic
system which are all further mediated by chemical/hormonal neurotransmitters (see,
Cohen, 2005). Now there is evidence that the emotional centers of the brain also have
mirror-like systems such that observation of emotions in others activates the same
neuron systems as in the case when the person actually experiences the emotion
(Singer et. al., 2004, and Wicker et. al., 2003).8
This paper aims to use the GTP framework to construct a Nash equilibrium of
a two person game with an oppositional structure in which the only logically
consistent and strategically rational thing to do is to innovate or surprise in a sequence
evidence derived from EEG and fMRI exists to support the claim of the homologous brain region
in humans serving similar functions (see,Fadiga et al., 1995, Rizzolatti and Craighero, Iacoboni et. al.,
2005).
6
Grézes et. al (2004) have mapped the respective areas of activation for some canonical functionalities
and their corresponding mirror neurons in the human brain. Mulamel et. al. (2010) have obtained
direct evidence from individual neurons of mirror activity and they find that mirror neurons are
widespread in the brain accompanying the myriad canonical activities of humans.
7
For the most recent set of reviews on where this field is at, see
http://www.psychologicalscience.org/index.php/publications/observer/2011/july-august-11/reflecting-onbehavior-giacomo-rizzolatti-takes-us-on-a-tour-of-the-mirror-mechanism.html
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Same regions of the insula and anterior cingulate cortex are activated both with the experience of
disgust and when observing others
6
which also entails an arms race. Section 3 sets up the mathematical preliminaries for
a recursive function approach to analyse decision problems within a 2-person finite
game. Actions involve encodable procedures that can always be executed in finite
time and these will be modelled as total computable functions. The productive
function of Post (1944) will form the basis of novelty production and hence the
surprise strategy will be defined in terms of this. In Section 3, it will be shown how
meta-representation of the full gamut of procedures is done and how meta-analyses
which involves offline simulations for the prediction of the outputs of the game uses a
2 place notation based representation of players’ actions. For this a framework well
known as Gödel meta mathematics (see, Rogers,1967) is used which implements a 11 mapping between executable calculations made by players and their respective meta
representations. The analogy with the canonical neurons and the mirror neurons can
be made here. The diagonal alignment in the meta system is shown to correspond to
potential Nash equilibria of a game.
Section 4 proceeds with the specification of a two person game with the
classic oppositional structure of the Moriarty-Holmes game which also characterize
the parasite-host or regulator-regulatee games. As already noted, the second of these
twosomes have to conduct ‘deceit’ to evade the first and will apply what will be
called the Liar strategy. The first significant point is that the Liar can win only out of
equilibrium when the identity of the Liar is not known or the formal structure of the
game involving the Liar is not acknowledged by the other player. From the
perspective of the Liar, the success of his strategy requires that the first player has a
false belief about the Liar. This has the self-referential second order belief structure
that underpins deceit, Bhatt and Carmerer (2005). The Second Recursion Theorem is
used to determine the Nash equilibria of the game as fixed points of recursive
functions. When there is mutual or common knowledge of the Liar, the point at which
this occurs is the famous non-computable fixed point at which a hostile agent qua Liar
“knows that the other knows that he is the Liar”. This also has the self-referential
second order belief structure discussed in Bhatt and Carmerer (2005) except that it
entails mutual beliefs on the need for deceit. This is a major point of departure from
standard game theory. It will be shown that the only best response function, within
the class of recursive functions, from the mutually deducible non-computable fixed
point is the Emil Post productive function which implements novel objects. This also
provides proof of a fully deducible fixed point at which agents mutually infer that
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they must surprise the other. This is a result that cannot be obtained without the GTP
framework. As recursive functions are used in recombining encoded information, novelty
refers to new blueprints for technologies/actions not previously there. This results in the
Type IV novelty based structure changing dynamics of complex adaptive systems which
is distinct from chaotic dynamics. A brief concluding section will summarize the results
and directions of future work.
2. Review of Meta-representation and Protean Strategic Behaviour in
Cognitive and Social Neuro-Science and in Mathematical Logic
2.1 Meta-representation and the mirror neuron system
Unlike, neural competences that enhance individual functionality regarding
vision, memory and even reward systems to food - the mirror neuron system (MNS)
is solely oriented to equip an individual for social interaction. It is the fact that the
MNS system with encoded goal related action information which exists separately
from the machinery for its physical execution has a two place structure involving
self and others which warrants study by game theorists. Game theory and strategic
behaviour whether cooperative or non-cooperative presuppose mutual mentalizing of
others’ intentions, beliefs and ‘types’. On the MNS basis of mentalizing about
others, many cognitive and social neuroscientists have subscribed to the simulation
theory of understanding goal related actions of others. On the role of mirror
neurons, Ramachandran (2006) reiterates the Gallesse and Goldman (1998)
hypothesis on the simulation theory of the mind: “It's as if anytime you want to make
a judgement about someone else's movements you have to run a VR (virtual reality)
simulation of the corresponding movements in your own brain and without mirror
neurons you cannot do this.” The narratives of those espousing the MNS hypothesis
is that understanding others involve self-referential meta-calculations arising from
encoded neural imprints emanating from agent’s own execution of procedures via
the canonical neurons. Further, Ramachandran (2006) views Machiavellian
behaviour, über intelligence, deceit and creativity as being part and parcel of this
capacity for meta-representation.
In addition to MNS associated with biological movement (especially of
con-specifics), the question is whether the latter only informs judgements or
mentalizing (see, Grézes et. al, 2004, Centelles et. al. 2011) about others beliefs or
whether there is a separate MNS for drawing a congruence between own beliefs and
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others beliefs of one’s beliefs. Oberman and Ramachandran (2004) suggest that a
system of neurons in the medial prefrontal cortex9 may serve as the mirror-like
shared representation for the experience and perception of mental states (Ochsner et.
al. 2004). The neuroscience of how a mental simulation framework operates when
two people directly interact10 is still in its infancy and possibly neuro-economic
game theorists are leading the way here.
2.2 Protean strategic behaviour and the role of deceit in social neuro-science and
neuro-economics
There is a growing social neuroscience literature (see, Grammar et. al. 2002)
which hypothesizes that MNS provide the neurophysiological basis of ‘the shared
manifold’ for inferential communication in society. Many (see, Sperber, 2000) hold
the capacity for meta-representations as the prime faculty in humans and adduce from
this much credence for the hypothesis of an evolutionary arms race in higher order
meta-representational abilities that has been called ‘Machiavellian intelligence’ by
Bryne and Whiten (1988), Whiten and Bryne (1997). The evolution of deception in
animals and primates in environments with conflicting goals and the detection of
falsity have been identified as an important landmark of meta-representational
competence in humans (see, Baron-Cohen 1995).11 Miller (1997) has catalogued
deceitful behaviour to combat situations with the potential for conflict as follows:
deceit takes the form of hiding intentions, the deliberate spreading of misinformation
and finally the development of protean strategies based on unpredictable adaptive
behaviour to escape from hostile agents or rivalrous conspecifics. Miller (1997) and
Grammar et. al. (2002) cite a co-evolutionary arms race in foundational social
interactions such as human courtship where deception and proteanism feature.
The regions of the brain is associated with ‘mind reading’ or forming beliefs of the intentions of
others are the following: the medial prefrontal cortex (MPFC), Dorsolateral Prefrontal Cortex (DLPC)
or Paracingulate Frontal Cortex, Rostral Prefrontal Cortex or Brodman Area( BA10), the temporoparietal junction (TPJ), the anterior superior temporal sulci (aSTS) the posterior superior temporal
sulcus pSTS and the amygdala. The latter three are meant to be involved in making judgements about
trustworthiness.
10
In Tognoli et. al. (2007) the neuro-physiology of the simplest mutual two person direct interaction is
studied. They confirm that there is suppression of so called mu rhythm which has been identified as a
marker of the activation of the MNS. This is observed irrespective of whether or not coordinated
behaviour is needed. In addition they also find a pair of oscillatory components, phi1 and phi2, with the
first of these featuring with independent behaviour and the second with coordinated behaviour.
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There is evidence that autistic individuals have difficulty in passing the so called Sally-Ann test on
ascribing false beliefs to others. It has been found that this group has dysfunction in their MNS and
,irrespective of high IQ, they have trouble with mind reading or with making out intentions of others
and hence social and strategic skills.
9
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The field of empirical and experimentally based investigations for the
neurophysiological correlates of strategic behaviour, though in its infancy, is
burgeoning. Economists, within the subfield of neuroeconomics, are actively
contributing to both in the context of individual decision making and in terms of
strategic interaction (see, Smith,2002, Mc Cabe et. al. 2000, Rustichini, 2005, 2009,
and Cameron et. al., 2005 and Caplin and Dean, 2008). Bhatt and Camerer (2005) give
one of the most explicit discussions of how to figure out the other player which is a
necessary condition of strategic behaviour : “One way agents might form 2nd-order
beliefs is to use general circuitry for forming beliefs, but apply that circuitry as if they
were the other player (put themselves in the “other player’s brain”). Another method
is self-referential: Think about what they would like to choose, and ask themselves if
the other player will guess their choice or not.” Bhatt and Camerer (2005) refer to
the figuring out by player 1 of what another player 2 believes that player 1 will do
him-self, as 2nd- order self-referential beliefs. They note that such 2nd-order selfreferential mappings feature when intentional deception is involved.
Gréze et. al. (2004) conduct experiments aimed at identifying the neural
mechanisms in humans that are involved in making judgements about mental states
from non-verbal but visual interactions with own and others’ actions. They find that
there is activation of the parietal and the pre-motor cortex associated with assessing
action prediction role of MNS occurs along with areas that have been associated with
so called ‘mind reading’ when judgements of others’ mental states are made. The
interesting finding in Gréze et. al. (2004) that is common with the results in Bhatt and
Carmerer (2005) is that when there was a mismatch between the player/observer’s
prediction for the actions of the other and the actual actions, brain activity is far more
widespread and enhanced.12 In contrast, in a ‘prediction equilibrium’, brain activation
is much attenuated. Bhatt and Camerer (2005) make an important observation that in a
Nash equilibrium when there is a consistent alignment of beliefs between a player’s
belief of the other player’s action corresponds to the condition of the brain when
prediction equilibrium is observed. Gréze et. al. (2004) conclude that in cases of
prediction failure regarding others, individuals must update their own representation
of the mental state of the target. In Bhatt et.al. (2010) the conditions of the game
12
Those brain areas that have been observed to be active and remain so when there are violations of
‘prediction error’ are the Temporal Parietal Junction (TPJ), Orbital Frontal Cortex and the
neighbouring anterior insula.
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require players to mutually maintain others in a state of false belief in order to
maximize returns in the game. However, in that paper the neuro-physiological
conditions for the mental identification of a mutual belief state by a player that the
other player ‘knows’ that he intends to lie/mislead has not been investigated.
Despite above developments, in contrast to the Baumol/Schumpeter
legacy on the role of innovative behaviour in capitalist growth and the large literature
in social neuroscience following the provenance of Byrne and Whiten (1988), where
the ubiquity of strategic deception and arms race in protean strategies and novel
products is well established, this phenomena is missing in the models of Nash
equilibria of games studied to date by economists. Bhatt and Camerer (2005)
succinctly state this as “in a Nash equilibrium nobody is surprised about what others
actually do, or what others believe, because strategies and beliefs are synchronized,
presumably due to introspection, communication or learning.” What is missing in this
statement is the category of mutual belief and expectation of surprise and the
characterization of a Nash equilibrium in which players mutually and logically expect
that they will need to surprise and be surprised. Finally, though Arbib (2006) states :
“mirror neurons are not restricted to recognition of an innate set of actions but can
be recruited to recognize and encode an expanding repertoire of novel actions”, we
do not yet have the neurophysiological correlates of novel actions arising in the
context of interaction between the self and others.
2.3 Gödel-Turing-Post (GTP) theory of meta-representation, simulation and novelty
production
So what light can the Gödel-Turing-Post (GTP) theory of computation
throw on the above conjectures and hypotheses on what amounts to the simulation
theory of mind reading ? The GTP theory now also called recursive function theory,
provides the only known axiomatic foundations for meta-representation of an
underlying system in terms of encoding using integers (also known as Gödel
numbers) to represent the instructions utilizing strings of symbols to achieve encoded
outputs from inputs in a finite number of steps in terms of an algorithm or program.
Both the encoding of information and the offline execution of the encoded
instructions which one can regard as a simulation can be run on ‘mechanisms’
involving any substrata ranging from intra-cellular neuronal biology to silicon chips.
This capacity for meta-representation yields the notion of a universal Turing machine
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(UTM) which can take encoded information of other machines and replicate their
behaviour. The combination of operations on encoded information and the finite
mechanisms for execution will collectively be called the GTP meta-representational
system (MRS). Remarkably, UTMs can run codes involving themselves, which is the
basis of self-reference. Self-referential mappings of UTMs involve fixed points of
recursive functions. As extensively discussed in Smullyan (1961), in formal systems,
a large number of the self-referential computations that can be recursively enumerated
relate to provable propositions. The remarkable point is that in a consistent MRS
system where two naturally disjoint sets arise from provable and refutable
propositions,13 the latter being negations of the former, Gödel (1931) provided the
constructive proof that the self-referential computations that cannot be recursively
listed, uniquely arise from the operation of negation.14 Hence, the centre piece of
Gödel (1931, p.19) is a formal analogue of the Liar15 or hostile agent which led to the
proof on the limits of deduction or calculation. The Liar, as the agent or function that
falsifies or contravenes, can be formally seen to embody the logic of opposition. This
is what gives the GTP framework a natural structure of a two person game with
contrarian payoffs. The self-refuting fixed points of contrarian mappings of the Liar
‘produce’, in a manner that will be shown in detail, the famous non-computable fixed
points which involve epistemic uncertainty referred to as undecidability. As noted
above, while it is not yet clear what neuro-physiological mechanisms are in place for
actor-actor interactions involving a mutual belief state on the necessity to propagate
false beliefs, the Second Recursion Theorem sets out the conditions needed for fixed
13
If sentences in a formal system 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. Refutable
sentences are those that have no proof and hence Turing machines will not halt when attempting their
proofs. If a formal system is complete then the set of all sentences denoted as FS satisfies the
condition that FS = T U R, where Tand R , respectively, are the set of provable and refutable
sentences. FS is said to be incomplete if T U R  FS and consistent if T and R are disjoint. The
Gödel (1931) incompleteness result and the set theoretic proof of this by Post (1944) provides a
constructive proof of a sentence denoted as u such that u  FS and u T U R. The sentence u is the
‘witness’ that FS is incomplete. To date, there is only one known way for the construction of such
sentences.
14
See, Smullyan (1961) Chapter III , Section 3Theorem 2.
15
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 ( ~ |-). Here ‘~’ is the negation sign and ‘|-‘ signifies proof. 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
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points involving recursive functions.
The significance of the MRS is that the agent within whom this is
embedded will be able to deduce and encode this non-computable fixed point
involving the Liar or the operation of negation. Thus, the first major breakthrough
here is how non-computability of certain self-referential mappings instead of being a
paradox are theorems in the formal system such that agents with MRS can identify
such non-computable fixed points qua undecidable propositions by deduction. Now
the question is why is the capacity to make inferences involving the Liar or contrarian
is (logically) critical to creative behaviour (see, Markose 2004, 2005) ? Emil Post
(1944) developed a set theoretic proof of the Gödel-Turing incompleteness result
utilising the notions of creative and productive sets that represent the logic behind the
so called productive function which by mapping outside of given disjoint
enumerable/listable sets encodes the innovation. The productive set grows by
incorporating the encoding for the innovation after it has occurred. Smullyan (1961,
Chapter 5) and Cutland (1980) give diagrams that respectively show how Post’s
creative and productive sets generate the ‘witness’ for such novel objects and the
growth of the productive set manifests an arms race in such novel objects.
Remarkably, these are generated from encounters with the Liar or hostile agent.
Intuitively, this says expect surprises from the opposition and if innovation is not
adopted then expect to be ‘negated’ by the hostile agent, viz. innovate or die. The
significance of the mathematical logic of GTP can be summarized as follows: (i)Meta
representational systems (MRS) and self referential or reflexive mappings involve
computational intelligence of a Universal Turing machine (ii) Contrarian or selfnegating structures like the Liar (as in ‘this is false’) are given recursive analogues in
terms of non-computability and these are fully deducible or provable in the MRS as a
non-computable state, and (iii) The consequences of (i) and (ii) can be represented by
so called creative and productive sets with the latter depicting an arms race in novelty
production or 'surprises'.
It is Binmore (1987) who first raised the “spectre of Gödel” (ibid) in the
context of game theory which attempts to restrict best responses to what can be
formally deduced, ie restrict the scope of strategic behaviour to a system that is
logically closed and complete. The question that is pertinent here is what of the
lemma (which was used to prove that the power set of a set has greater cardinality than a set) we begin
13
hostile agent who will falsify or negate one’s actions if he could deduce what they
are? When faced by hostile agents, can one rationally play an action that is known or
can be formally deduced, both of which will be called ‘transparent’, or does one
innovate and ‘surprise’ the enemy? Following a provenance of complex adaptive
systems (CAS) that defines the sine qua non of CAS as its capacity to produce
novelty and surprises, Albin (1988) is perhaps the first economist16 to have discussed
the necessity for agents to have powers of Turing machines to produce the WolframChomsky Type IV novelty producing structure changing dynamics. As noted by
Markose (2005) and Durlauf (2005), almost all mainstream accounts of dynamics in
economic interactions eschews this provenance of proteanism with complexity and
mostly confines economic dynamics to Type I and Type II dynamical outcomes in the
Wolfram-Chomsky schema, viz. limit points and limit cycles.
A number of game theory papers such as Anderlini (1990), Anderlini and
Sabourian (1995), Canning(1992), Nachbar and Zame (1996), which use recursion
function theory confine their analysis to defining the problem of indeterminacy
associated with self-refuting decision structures. Indeed, it is interesting to note that
these game theory papers discuss neither the significance of nor the possibility for
innovation and surprise strategies arranged in a structure of an arms race. The
problems here arise for two main reasons. These papers appear not to utilize the
major methodological triumph of Gödel (1931) which is the meta analysis that
produces fully definable meta propositions, in an ever extendable sequence, in terms
of what Post (1944) calls productive functions that map from self-refuting fixed
points, that are both true, non-computable and avoids paradox by being verifiably so.
Secondly, the characterization of Nash equilibria as fixed points of recursive functions
seems not to be specified as such.
It was in the seminal paper of Spear (1989) that the Second Recursion
Theorem was introduced to formalize the problem of computing rational expectations
equilibria as fixed points. Though Spear’s paper is not one that explicitly depicts a
game, it will be shown that without the proper formalism of defining fixed points of
to have so called “good” uses of self-refuting structures that result in theorems rather than paradoxes.
16
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). Much as it was seminal in the
way that Hayek redirected the discussion on the limits of deductive inference from Humean scepticism
to the Gödelian logic of incompleteness (Markose, 2004, 2005), Hayeks’s own account of this did not
go beyond the Cantor diagonal lemma (see, footnote 15).
14
recursive functions, the Nash equilibria of a game which requires the identification of
the meta-representations of mutual best response functions in a two place diagonal
alignment, one could be forced into different ‘resolutions’ of the classic problems
such as the one between Holmes and Moriarty. Koppl and Rosser (2002) attempt to
characterize the Nash equilibrium of the zero sum game that depicts the machinations
of Holmes and Moriarty using recursive function theory. They conclude as follows:
“We can see that there are best-reply functions, f(x), such that f(x)x for all 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 is fully deducible from within the MRS
of the players. Thereafter, any total computable function that is a mapping from the
non-computable fixed point, which defines the Emil Post production function and
which will be called the surprise strategy function can only map outside given
recursively enumerable sets. 17 This is so, in order for the MRS to avoid
inconsistency. Hence, the aphorism that sufficiently rich formal systems cannot be
both consistent and complete. The Emil Post productive function which is a recursive
function provides a ‘witness’ (epithet used in Post, 1944) for the incompleteness of
the formal system in which such novel objects are produced by a fully mechanized
exit route. Finally, in the Wolfram-Chomsky schema on Type IV novelty producing
dynamics, the logical necessity to ‘step out’ can only arise in the context of agents
with the full powers of Universal Turing Machines.
Likewise, consider the Nachbar and Zame (1996) conclusion that “for a large class of discounted
repeated games (including the repeated Prisoner's Dilemma) there exist strategies implementable by a
Turing machine for which no best response is implementable by a Turing machine” . The
implementation of the Gödel incompleteness result shows that that from fully deducible noncomputable fixed points of a game, the only (italics added) strategies that can be implemented by
recursive functions, viz. Turing machines, are those that satisfy the property of productive functions
producing surprises or innovations that lie outside given recursively enumerable sets. The point here is
that Type IV dynamics arise from the logical necessity to ‘step out’ of recursively enumerable sets of
the MRS and can be produced only by Universal Turing Machines though such activity cannot be
enumerated by them in advance.
17
15
3. Gödel-Turing-Post (GTP) Meta-Mathematics and the Logic of
Novelty Production
The main purpose of the formal analysis is to show the relevance of the GTP
mathematics of incompleteness for the characterization of systems capable of novelty
based complex Type IV dynamics. Gödel (1931) pioneered the framework of
analysis called meta mathematics pertinent to self-referential structures where he
obtains epochal results on the sort of statements an internal observer can make as a
meta-theorist if he is constrained to be very precise in what he can know and how he
can make inferences. As highlighted by Binmore (1987), the theoretical significance
of the analogue of the Gödel type incompleteness or indeterminacy result for
formalized game theory stems precisely because this can be proven to arise not from
incorrect or inconsistent reasoning or calculation but rather to avoid strategic
irrationality and logical contradiction. To this end instrumentally rational players are
accorded the full powers of an idealised computation machine in the calculation of
Nash equilibrium strategies and all information has to be in a codifiable form.
Following from the Church-Turing thesis, the computability constraint on the decision
procedures implies that these are computable functions that can only entail finitely
specified set of instructions in the computation. Again by a method introduced by
Gödel (1931) called Gödel numbering, all objects of a formalisable system
describable on the basis of a countable alphabet are put into 1-1 mapping with the set
of natural numbers referred to as their Gödel numbers (g.ns, for short). Thus,
computable functions can be indexed by the g.n of their finitely encoded program.
Impossibility results on computation, therefore, become the only constraints on what
rational/optimizing players cannot calculate given the same information on the
encoded primitives on the game.
3.1 Some preliminaries on computable functions
By the Church-Turing thesis computable functions are number theoretic functions,
f : N N where N is the set of all integers.18 Each computable function is identified
by the index or g.n of the program that computes it when operating on an input and
18
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  being denoted by 0, while from Cantor we know

that the set of all number theoretic functions have cardinality of 2 0. Hence, not all number theoretic
functions are computable (see,Cutland,1980 ).
16
producing an output if the function is defined or the calculation terminates at this
point. Following a well known notational convention, we state this for a single valued
computable function as follows
f(x)  a(x) =q .
(1.a)
That is, the value of a computable function f(x) when computed using the
program/TM with index a is equal to an integer a(x) = q, if a(x) is defined or halts
(denoted as a(x) ) or the function f(x) is undefined (~) when a(x) does not halt
(denoted as a(x) ). The domain of the function f(x) denoted by Dom a or Wa is
such that,
Dom a = Wa ={ x | a(x)  : TMa(x) halts}.
(1.b)
The range of a computable function is defined by the set Ea,
Range a = Ea ={ q | a(x)  : TMa(x) halts}.
(1.b)
Definition 1: Computable functions that are defined on the full domain of N are
called total computable functions. Partial computable functions are those functions
that are defined only on some subset of N.
Related to (1.b) is the notion of sets whose members can be enumerated by an
algorithm or a TM.
Definition 2: A set which is the null set or the domain or the range of a
recursive/computable function is a recursively enumerable (r.e) set. Sets that cannot
be enumerated by T.Ms are not r.e .
The one feature of computation theory that is crucial to game theory where
players have to simulate the decision procedure of other players, is the notion of the
Universal Turing Machine (UTM).
Definition 3: The UTM is a partial computable function, defined as (a,x), which
uses the index a of the TM whose behaviour it has to simulate. By what is called the
Parameter or Iteration Theorem, there is a total computable function u(a) which
determines the index of the UTM such that
(a,x) = u(a)(x)  a(x) .
(2)
17
Equation (2) says that the UTM, on the left-hand side of (2) on input x will halt and
output what the TMa on the right-hand side does when the latter halts and otherwise
both are undefined.
Of particular significance are Turing Machines that use their own code/g.n as
inputs in their calculation. We will refer to these as self-referential or diagonal
calculations.
Definition 4: The set denoted by C is the set of g.ns of all TMs that halt when
operating on their own g.ns or alternatively C contains the g.ns of those recursively
enumerable sets that contain their own codes (see, Cutland , 1980, p.123, Rogers,
1967, p.62).
C = { x | x(x) )  ; TMx(x) halts ; x  Wx }
(3.a)
The complement of C
C~ = { x | x(x) ; TMx (x) does not halt; x  Wx}
(3.b).
Theorem 1: The set C~ is not recursively enumerable.
In the proof that C~ is not recursively enumerable, viz there is no computable function
that will enumerate it, Cantor’s diagonalization method is used. 19
3.2 Post (1944) set theoretic characterization of Gödel Incompleteness
As indicated in Section 2, we will now state the formal character of systems
capable of the endogenous production of novelty or surprises in terms of the notion of
creative and productive sets first defined by Emil Post (1944).
Definition 5: A creative set Q is a recursively enumerable set whose compliment, Q~,
is a productive set. The set Q~ is productive if there exists a recursively enumerable
set Wx disjoint from Q (viz. Wx  Q~) and there is a total computable function f !(x)
which belongs to Q~ - Wx. f !(x)  Q~ – Wx is referred to as the productive function
and is a ‘witness’ to the fact that Q~ is not recursively enumerable. Any effective
enumeration of Q~ will fail to list f !(x), Cutland (1980, p. 134-136).
Assume that there is a computable function f = y , whose domain Wy = C~ . Now, if y  Wy , then y
 C~ as we have assumed C~ = Wy . But by the definition of C~ in (3.b) if y  Wy , then y  C and not
to C~ . Alternatively, if yWy , y C~ , given the assumption that C~ = Wy . Then, again we have a
contradiction, as since from (3.b) when yWy , yC~ . Thus, we have to reject the assumption that
for some computable function f = y , its domain Wy= C~ .
19
18
Lemma 1: Set C in (3.a) is a simple example of a creative set. The productive
function f(i)= 1 is the identity function for set C.
By the definition of C if any number i  C
 i  Wi by the definition of C.
Hence, for f(i)= 1 if f(i)  C  i  Wi . If Wi is disjoint from C, then f(i)  C 
Wi.. If i  Wi , then i  C and Wi will not be disjoint from C.
For any generic r.e set Q and the productive function f(i) which is not an
identity function that we have by Lemma 1 for set C in (3.a), we need some
‘reduction’ of set C to Q. Lemma 2 below is analogous to Proposition 2 in
Smullyan (1961, p 96, Chapter IV).
Lemma 2: Let the recursive function f(i) define the following reduction of set C to Q
f: i  C then f(i)  Q. Hence, C =f -1 (Q). Let Wi¬ be disjoint to Q . Then there is
a recursive function t(.) s.t it is the index of set, Wt(i`) = f -1(Wi¬ ) , viz
Wt(i`) 
C~ . Hence, Wt(i`)  C~ , t(i`) is the productive function of C with t(i¬)  C 
Wt(i`) . Likewise, f (t(i¬)) is the productive function for Q~ and f (t(i¬))  Q 
Wi¬ .
The notation f!(x) for the productive function given in Definition (5) will be
justified as it will shown that f!(x) the productive function which implements the
proof of the incompleteness of the formal system also corresponds to the best
response surprise strategy function in the Nash equilibrium of a game that produces
the innovation based structure changing Type IV undecidable dynamics.
The set Q is reducible to the prototypical creative set C in equation (3.a)
which contains self-referential calculations that converge. They will be shown to
correspond to computable fixed points and equivalent to provable theorems in a
formal system. The creative set on which Turing Machines halt is associated with
Type 1 and Type II dynamics which can be called (computable) order associated
with limit points and limit cycles. In a formal system all negations of theorems in the
system can be listed as being refutable and hence we have a recursively enumerable
subset Wx of C~ in the domain on which Turing machines do not halt. Such mappings
represent Type III deterministic chaotic dynamics. In the context of a game, the noncomputable fixed involving the Liar strategy when there is 2nd order self-referential
recognition that the other player has wised up to the deceit or negation qua Liar,
should correspond to this chaotic mapping.
19
Figure 1 Post (1944) Set Theoretic Representation of Gödel Incompleteness in the
Domain Outside Disjoint Recursively Enumerable Sets (See Definition 5)
diuu
Type I and Type II
Dynamics (Limit Points
Or Homogeneity and
Limit Cycles)
Type IV
Novelty
based
structure
changing
undecidable
dynamics
Type III
Chaotic
Dynamics
Recursive
Enumerable
subset Wx on
which TMs
which TMs can
halt
be deduced not

to halt
Remarkably, Korn and Faure (2003) who investigate the role of chaotic
Creative
Set C on
Productive
set C~ on
which TMs
do not halt
dynamics in the neuro-physiology of the brain, review the work of Freeman and
collaborators (Skarda and Freeman, 1987) and conclude that “chaos confers the
(neural) system with a deterministic ‘I don’t know state’ from within which new
activity patterns can emerge… chaotic states… are well designed for preventing
convergence and for easy ‘destabilization’ of their activity by a novel input .. . they
are ideally fit for accommodating the neural networks with a new and still unlearned
stimulus”. Precisely on cue, it is indeed from this self-negating non-computable fixed
point where calculations cannot converge that the recursive productive function f !(x)
maps outside the equivalent reductions (using Lemma 2 above) of the r.e disjoint sets
C and Wx. Hence, the domain for novelty producing Type IV dynamics in the
proposed game lies outside such recursively enumerable disjoint sets.
20
3.3 Meta-representational System (MTS) and a simulation theory for a two person
game
This section sets out how a MTS organizes encoded information involving self and
other. This interactive situation is best characterized by a two person game.
The primitives of the game, best interpreted as one in which both cooperation and
opposition arise such as in a regulatory/policy game or a parasite-host game, is
codified as follows.
G= {(p,g), (Ap, Ag), sS}.
Here,(p,g) denote the respective g.ns of the objective functions, to be specified, of
players, p, the private sector/regulatee and g, government/regulator. The action sets
denoted by Ai are finite and countable with ail i , i (g, p) being the g.n of an
action of player i and l=0,1,2,.....,L. An element sS denotes a finite vector of state
variables and other archival information and S is a finite and countable set. The
action set A = A1 U A1 represents the known technologies. In order to highlight the
fundamental recursive nature of actions as technologies and the potential for new
technologies, the class of best response strategy functions will be defined as a set of
total computable functions.
Definition 6: The best response strategy functions fi, i  (p,g) that are total
computable functions can belong to one of the following classes –
f i1  1( Identity Function) 
fi =
fi
Rule
Abiding
Rule
Breaking / Liar


!
 Surprise
fi
(4.a)
such that the g.ns of fi are contained in set ,
 = { m | fi = m ,m is total computable}.
(4.b)
The set  which is the set of all total computable functions is not recursively
enumerable. The proof of this is standard, see, Cutland (1980).
The total computability of best response functions fi = m, m  in (4.a,b)
yields the notion of constructible/effective action rules such that a finitely codifiable
description of some (institutional) procedure which is defined for all mutually
exclusive states of the world is obtained. As will be clear, (4.b) draws attention to
issues on how innovative actions/institutions can be constructed from existing action
21
sets. The remarkable nature of the set  is that potentially there is an uncountable
infinite number of ways in which ‘new’ institutions can be constructed from extant
action set A. The task is to show the conditions under which it is mutually deducible
that the best response function fi, i  (p,g) satisfies Post’s productive function and is a
surprise strategy, fi = fi!= m , such that m  -A. Only such innovations will be
accorded with the status of strategic innovations.
A major implication of imposing computability constraints on all aspects of
the game is that all meta-information with regard to the outcomes of the game for any
given set of state variables, s S, can be effectively organized by the so called
prediction function  (x,y) (s) in an infinite matrix  of the enumeration of all partial
computable functions. This is given in Figure 2 (see, Cutland, 1980, p.208). The
tuple (x,y ) identifies the row and column of this matrix  whose rows are denoted as
j, i= 0,1,2,...... .
Figure 2 : Meta –Information on Outcomes of Decision Problem for a 2-person
Games
0
(0,0) (0,y) 
1
(1,0) (1,y) 
2
(2,0) (2,y) 
.
x (x,0) (x,1) (x,2) (x,3) (x,x) 

(5).
.
The function  (x,y) (s) if defined at a given state s and (x,y) yields
 (x,y) (s) = q .
Here, q in some code, determines the outcome of the decision problem of the game
and q E x . Note, (x,y) is the index of the program for this function  that
produces the output of the strategic decision problem of the 2-person game. The tuple
also identifies a point on the matrix  in Figure 2. The conditions under which the
output of the prediction function for each (x,y) point in the above matrix is defined is
given in the following Theorem.
22
Theorem 2: The representational system is a 1-1 mapping between meta information
in matrix  in Figure 2 and executable calculations such that the conditions under
which the prediction function which determines the output of the game for each (x,y)
point is defined are as follows:
 ( x ,y ) ( s)   
x
( y)
(s)  q , iff  x ( y)  . (6)
Here, the total computable function (x,y) modelled along the lines of Gödel’s
substitution function20 (see, Rogers, 1967,p.202-204) has the feature that it names or
‘signifies’ in the meta system  the points in the game that correspond to the different
executed calculations on the right-hand-side of (6) as we substitute different values
for (x,y) for a given state s. The g.ns representing (x,y) can always be obtained
whether or not the partial recursive function  x ( y ) on the right-hand side of (6)
which executes internal programs halts or not.
Proof : See Rogers (1967).
By the necessary condition in (6) if the function x (y) on the right-hand side (RHS)
executing the internal calculation is defined, we say the prediction function  (x,y) in
the meta system on the left-hand side producing the output of the game is computable
and the outcome q of the game at that point is predictable. Likewise, the ‘only if ’
condition in (6) implies that meta statements that are valid on the predictability of the
outcomes of the game at any (x,y) must give the correct inference on whether
program executions on the right-hand side terminate.
Definition 6: The two place notation of the meta-system (x,y) can be used to define
two second-order self-referential encodings of the following kind:
(a) When player i has to determine her own best response function, the first place
entry x in (x,y) refers to what the player i does (viz the g.n of best response
function fi) given that player j plays a strategy that is consistent with player i’s
belief denoted by y of what player j believes player i has done.21
(b) When player j determines the best response of the other player i then the first
place entry x in (x,y) refers to j’s simulation of what i does (viz the g.n of best
20
This approach economizes on formalism and enables us to high light and exploit the Fixed Point
Theorems of recursive function theory to determine Nash equilibrium outcomes more readily than has
been the case in for instance in Anderlini(1991),Canning(1992) and Albin(1982).
21
Note this is what has been referred to as self-referential 2nd order beliefs in Bhatt and Carmerer
(2005).
23
response function fi ) and the second place y denotes j’s belief of what i’s
believes regarding j’s simulation of i.
(c) All Nash equilibria and other relevant fixed points of the game satisfying what
has been referred to as consistent alignment of beliefs (CAB, for short, Osborne
and Rubinstein,1994) have to be elements, (x,x), along the diagonal array of
this matrix. Note, (x,y) which are off diagonal entries in matrix violate the
CAB condition.
The set up in (6) formalizes the relationship between a mirror/meta system on
the LHS of (6) which records all ‘successful’ machine executions on the RHS of (6).
The latter relates to the canonical system involving online activity. The case (b) in
Definition (6) given in the form (x,x) can be seen as an example when a trigger such
as direct observation of the action of another occurs and ‘fires’ up off line simulation
in the meta system of j’s prediction of i’s action with the full prediction of the two
person interaction following automatically with great economy from past recorded
points such as (x,x). GTP meta-analyses are operations on Godel numbers and
bypass the online calculations involved. In other words, all permissible inferences are
obtained in short hand from encoded information. Likewise, on account of the ‘only
if’ condition in Theorem 2, many interesting aspects of the Nash equilibria of
computable games can be established only with reference to the meta analyses and
information in the matrix  with no explicit reference to physical executions of
programs such as the optimization algorithm, to be discussed, being made by the
players.
Two out- of-equilibrium belief states will be defined. The case when player i
simply attributes a different belief to player j of i’s own action x is denoted by (x,y).
The case when player i deliberately acts in such a way he believes player j is in a state
of false belief about i will be typically denoted as follows.
Definition 7 Deliberate Deceit and False Belief: Denoting by x¬ the negation of x
brought about by best response function fi¬ defined in (4.a), we have (x¬ ,x) in the
two place meta representation of the game by i. This is the case when player i knows
that he has negated action with g.n x and believes that player j believes that he, viz.
player i, is playing x.
Both logically and nuero-physiologically as noted in the introduction, this
out-of-equilibrium situation involving false beliefs has great significance.
24
It will be shown how total computable functions for the best response function
fi , i=p, g in a 2 person game when applied to the diagonal array of the matrix  can
dynamically move it to a specific row in matrix  . The Fixed Point or Second
Recursion Theorem states that there exists an index n of a program/set of instructions
that computes f(n) and then applies f(n) so that both n and f(n) are instructions for the
computation of the same recursive function and if the latter is computable at this point
the same outcome q is predicted by the operation of the two programs.
Theorem 3: Fixed Point or Second Recursion Theorem (Cutland, 1980 p. 200)
Let f be a total unary computable function then there exists a number n such that
such that
f (n) = n.
(7)
Note, f(n)  n being codes for different programs, but they identify the same function
and both sides of equation will yield an identical output if f has a computable fixed
point.
The proof that any computable function f has a fixed point follows from the fact
that a function representing an encoded set of instructions when applied to the
diagonal array of matrix  belongs to some row of the matrix  say v, such that the
v+1th element in the vth row ,f((v,v)) , and the v+1th element in the diagonal array of
 coincide, yielding
f((v,v))= (v,v).
(8)
Thus, the vth row of matrix satisfies:
vf((0,0)) f((1,1)) f((2,2)) f((3,3)).... f((v,v))= (v,v) …. f((x,x))..
.
A major advantage of this framework is that the determination of Nash equilibrium
strategies involves the use of total computable best response functions
(fp , fg) which can be shown to operate directly on points such as (x,x) to effect
computable transformations of the system from one row to another of matrix  with
special reference to its diagonal array, see, Figure 2. Theorem 3 is used in the
determination of the fixed points for the total computable functions best response
function fi , i=p,g. When one player applies his best response fi , the condition that
25
both players identify the same prediction function as producing the output of the game
at that point is called a rational expectations.
 fi  ( v , v ) (s) =   ( v , v ) (s) ,
i  (p,g).
(9)
How player j≠i, identifies v,v) as the fixed point of i’s best response function fi
will be derived in the next section
4. Nash Equilibria : When Does One Surprise the Opposition ?
4.1 Total computable best response functions and optimal strategy functions
The optimization algorithms entailed in achieving best responses in the game arise
from the objective functions of players.
Definition 8 : The objective functions of players are computable functions i , i
(p,g) defined over the partial recursive payoff/outcome functions specified as in (4).
Arg max i   ( bi , bi / j ) (s)  ,
bi  Bi
i (p,g)
The choice set Bi contains the g.ns of strategy functions. The Nash equilibrium
strategies (gE , pE ) with g.ns denoted by (bpE, bgE) entail up to two subroutines or
iterations, to be specified below. In principle, the strategy functions (g , p ) are
Universal Turing Machines that simulate optimal strategies of the players that satisfy
(10) and involve the total computable best response functions (fp , fg) which
incorporate elements from the respective action sets A= (Ap ,Ag) and given mutual 2nd
order self-referential beliefs of one another’s optimal strategy. Note, ail , l= 1,2,… L,
denote the elements of the sets Ai, i=p,g. In the two place notation given in (6), bi is
the g.n of i’s optimal strategy given that i’s belief that j has optimally chosen its
strategy on the basis of j’s belief, bi/j , of i’s strategy . Note that we will use g.ns zi, i
(p,g) to represent encoding of the optimization calculus with respect to respective
objective functions. In the Nash equilibrium best response calculus, the first
subroutine denoted by g.n b1 simulates the other player’s optimization calculus to
determine optimal action. The problem is that actions can in general be implemented
by any total computable best response function, fi = m , m , i  (p,g) in (4.b).
In standard rational choice models of game theory, the optimization calculus
in the choice of best response restricts choice to given actions sets. Hence, starting
26
from some point x,x), the strategy functions map from a relevant tuple that encodes
meta information of the game into given action sets
i ( fix,x), z, s, A)  Ai and fi= m , mA, i  (p,g) .
(11.a)
Unless this is the case, as the set  is not recursively enumerable there is in general
no computable decision procedure that enables a player to determine the other
player’s best response functions. However, in principle, a strategic decision
procedure (g , p ) for choice of best response, fi= m , m , i  (p,g), can map into
-A , implying that an innovative action not previously in given action sets is used.

i (fi(x,x)), z, s, A)  - A and fi = fi ! = m , m  -A, i  (p,g). (11.b)
The question is which fixed point (x,x), fully deducible in the meta-mathematics,
will trigger such Nash equilibrium surprise strategies, (gE! , pE! ), with g.ns denoted
by (bpE!, bgE!) ? It has been noted in passing by Anderlini and Sabourian (1995,
p.1351), based on the work of Holland (1975), that heterogeneity in forms does not
arise primarily by random mutation but by algorithmic recombinations that operate on
existing patterns. However, a number of preconceptions from traditional game theory
such as the ‘givenness’ of actions sets prevent Anderlini and Sabourian(1995) from
positing that players who as in (11.b), equipped with the wherewithal for algorithmic
recombinations of existing actions, do indeed innovate from strategic necessity rather
than by random mutation. Indeed, it is the very function of the Gödel meta
framework to ensure that no move in the game made by rational and calculating
players can entail an unpredictable/surprise response function from set - A unless
players can mutually infer by strictly codifiable deductive means from (x,x) that
(11.b) is a logical implication of the optimal strategy at the point in the game. In
other words, the necessity of an innovative/surprise strategy as a best response and
that an algorithmic decision procedure is impossible at this point are fully codifiable
propositions in the meta analysis of the game. While it will be shown what specific
structure of opposition logically and strategically necessitates surprise strategies in the
Nash equilibrium of the game, in keeping with the set theoretic formulation of novelty
27
production in Figure 1, the corresponding creative and productive disjoint subsets of
the strategy sets have also to be developed.
4.2 Fixed Point/Second Recursion Theorem: The base-point
The meta analysis in the determination of Nash equilibrium strategies (pE , gE) with
g.ns (bpE, bgE ) will be undertaken here. In the classic matching pennies game format,
the optimal outcomes for the government/regulator arise when the regulatee/private
sector is rule abiding or coordinating. Calculations start at this so called base-point
which is the fixed point of fg which has to be arrived at by player p on the RHS on
(12) :
 f g  ( ba , ba ) (s) =   ( b , b ) (s)  q
a
a
.
(12)
Here, ba is the g.n of the strategy fg that selects the optimal action a from set A in
(11.a) when g is put in for the index i. In the two place notation in ( ba ba ) on the
RHS of (12), the first ba is the code of the program from (11.a) as adopted by p to
simulate the optimal policy rule a and the second place ba denotes that p believes that
g believes and acts on the basis that the p is rule abiding and has left the policy rule a
unchanged. The prediction functions in (12)  ba ,ba )  s is computable and outcomes
of the policy rule a is predictable and q is the desired outcome that g wants in state
variables when applying this policy rule a . It is convenient to assume that policy rule
a is optimal for g if the private sector is rule abiding. By rule abiding is meant that p
will leave the system unchanged in terms of the row ba of matrix .
4.2 The Liar/Rule Breaker Strategy :The logic of opposition
For player p, for the given (a,s) it may be optimal for p to apply the Liar strategy, fp¬
( ba , ba ), with code ba¬ . Formally, the Liar strategy has the following generic
structure.
For any state s when the rule a applies,
  f

p
 ( ba ,ba )
(s)  q ~ q~  E  b a  ( ba ,ba ) (s)  q  q  E  b a .(13.a)
For all s when policy rule a does not apply,
fp¬ = 0 : Do Nothing .
(13.b)
28
The Liar can successfully subvert with certainty in (13.a) if and only if () the
policy rule a has predictable outcomes (LHS of (13.a)) and fp¬ itself is total
computable. Thus, fp¬ = m , mp, must include a codified description of an action
rule if undertaken by the Liar can subvert the predictable outcomes of the policy rule
a. Formally, if q is predicted then the application of fp¬ to ( ba , ba ) is equivalent to
the condition of deliberate deceit in Definition 7 and the g.n of this strategy is ( ba¬,
ba ). That is, p has negated ba and he knows that g harbours a false belief about him,
that p is rule abiding with ba. This out- of- equilibrium ( ba¬, ba ) point in the game is
off diagonal in terms of the matrix andwill bring about an outcome q
~
 E b which belongs to a set disjoint from the set that contains the desired output of
a
rule a for all s for which rule a applies, viz. E b  E b   E b  he outcomes (q~
a
,q
a
a
) can be zero sum but in general we refer to property q~  E b in (13.a) as being
a
oppositional or subversive.
Thus, we come to the point as to why agents who precipitate the WolframChomsky Type IV dynamics with innovation have to have powers of self-referential
calculation. Firstly, g acknowledges the identity of the Liar in (13.a) and understands
that transparent rule a cannot be implemented rationally as the outcome now defined
by   ( b  , b ) (s) = q~ is the opposite of what is optimal for g. Player g has to identify
a
a
the fixed point of fp which is formally given as  (ba , ba ) on the RHS of (14)
¬
¬
¬
where ba¬ is the code for the Liar strategy in (13.a). Likewise, both g and p in their
respective second place entry  (ba¬ , ba¬ ) on te RHS and LHS of equation (14)
remove any attribution of false belief to the other. Thus, the Liar, p, knows that g
knows that p is the Liar on the LHS of (14) and likewise, g knows that p knows that g
has identified him.
Theorem 4: The prediction function indexed by the fixed point of the Liar/rule
breaker best response function fp¬ in (14) is not computable and corresponds to the
famous Gödel non-computable fixed point.
 f  ( b  ,b  ) (s)  ( b  , b  ) (s).
p
a
a
a
a
(14)
29
The proof is standard. Assume it is computable and the R.H.S of (14) produces the
output q~ and the L.H.S by the definition of the Liar strategy produces output q.
Hence, if (14) is computable then we have q=q~ which is a contradiction.
Though the conditions of the out-of-equilibrium success of the Liar are spelt out in
(13.a , 13.b), in many fast moving co-evolutionary systems, predictable strategies
such as ba or ba¬ may not be observed, and instead only the arms race in novelty
given in the next section is what persists such that both players survive. On the other
hand, in environments suitable for neuro-physiological 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 kept in a
state of false belief ( ba¬, ba ) given in Definition 7.
4.3 Surprise Nash Equilibria
There is no paradox in stating that as both players can prove the noncomputability of (14) they will be able to mutually deduce that that the only Nash
equilibrium strategies for both players that is consistent with meta information in the
fixed point in (14), is one that involves strategies that elude prediction from within the
system. On substituting the fixed point  (ba¬ , ba¬ ) in (14) for (x,x) in (11.b), g’s
Nash equilibrium strategy gE with g.n bgE implemented by an appropriate total
computable function such as (12.a) must be such that
gE (fg (ba¬ , ba¬ ), z, s, A)  - A and fg = fgE! = m , m  -A. (15.a)
That is, fg! implements an innovation and bgE ! is the g.n of the surprise strategy
function in (14.a) hence  (bgE !, bgE !) is the fixed point of fg !.
Likewise, for player p, fp! implements an innovation in (15.b) and bpE ! is the g.n of
the surprise strategy function viz.  (bpE !, bpE !) the fixed point of fpE!. Thus,

pE (fp  ( ba¬, ba¬ ), z, s, A)  - A and fp = fp E! = m , m  -A. (15.b)
30
The intuition here is that from the non-computable fixed point with the Liar, the total
computable best response function implementing the Nash equilibrium strategies can
only map as above into domains of the action and strategy sets of the players that
cannot be algorithmically enumerated in advance.
Using Theorem 4, Definition 5 and Lemma 2, we will now prove the
incompleteness results for the strategy sets of the players from the Liar/rule breaking
strategy. Analysis will be done for p’s strategy set Bp as the strategy functions fp and
fg, respectively, can be shown to implement a reduction, as in Lemma 2, of the
prototypical creative set C in ( 3.a).
Corresponding to those (agl , s) tuples, agl  Ag of g’s base point optimal
strategy for which p’s best response fp is to be rule abiding viz. fp =1, the g.ns of
these optimal strategies for p, bp 1  Bp result in computable fixed points . Here b1
indicates the subroutine 1 in the determination of the Nash equilibrium strategy. This
set denoted by ßp+ can be generated by eductive/recursive methodology entailed in
the proof of Theorem 3. Thus,
 ßp+ = { b p |b p 1 (b p )  for all (agl , s), agl  Ag , fp =1 }. (16.a)
1
1
Using logic in (13.a,b), a set ßp¬ can be recursively be generated that contains the g.ns
of p’s strategies for when it is optimal for p to use the Liar best response function fp¬
to those (agl , s) tuples, agl  Ag of g’s action set. By Theorem 4, this is a set of p’s
strategies that can be proven to result in non-computable fixed points. Hence,
ßp¬ = { b p |b p 1 (b p )  for all (agl , s), agl  Ag , fp = fp¬ }.
1
1
(16.b)
For the same (agl , s) tuple, agl  Ag constituting g’s base point optimal strategy, p’s
optimal strategy bp* cannot belong to both ßp+ and ßp¬ . Thus, logical consistency of
the meta analyis requires ßp+  ßp¬ = and these are disjoint sets.
Now, define the compliment set of ßp+ denoted by ßp+c as
ßp+c = { x | x (x) , x  Bp }.
(17)
31
As ßp+  ßp¬ = , the two sets are recursively enumerable disjoint sets with ßp¬ 
ßp+c by definition in (16.b). Hence, the incompleteness of p’s strategy set Bp that
arises from the agency of the Liar strategy requires the proof that ßp+c is productive as
in Definition 5 with the g.n of the surprise strategy bpE !=b2(fp , bpE !  ßp+c - ßp¬.
FIGURE 3
The Incompleteness of p’s Strategy Set Bp
bpE ! = b2 ( fp  ( ba¬,ba¬ ) ) :SURPRISE STRATEGY
bpE ! = b2 ( fp  ( ba ,ba ) )
¬
ßp+
¬


W !
n
ßp+c
ßp+c is defined in (17)
Theorem 5: The g.n of player p’s Nash equilibrium surprise strategy is defined as
bpE ! = b2 ( fp  ( ba¬,ba¬ ) ) from (12.b) having substituted in
 ( ba ¬ , ba ¬ ) from the non-computable fixed point in (16). Then, by construction
bpE ! is a ‘witness’ for the productivity of the set ßp+c such that bpE ! ßp+c –b2( ßp¬ )
and p’s optimal strategy set Bp is incomplete . (i) As bpE ! is the g.n of the total
computable best response function fp! implementing the surprise or the innovation in
the system as defined in (15.b), fp! is the productive function for the set ßp+c. This is
shown in Figure 3. (ii) Once the surprise Nash equilibrium strategy has been
implemented by p which has g.n b2 (fp ( n¬)), the growth of the strategy set can be
proven to take the following form:
W
!
n 1
= W 1  { b2 (fp ( n¬)) }
n
32
This is shown in Figure 4.
Proof: See Appendix.
FIGURE 4
Arms Race in Surprises/Innovations: Growth of the Productive Strategy Set
Bp+c
bpbE0!!
, b1! …. bn-1!
g.n (fp(σn¬)= bn!
W σn !
W σn+1!
g.n: Godel Number
The significance of Theorem 5 is that the surprise strategy is fully definable as a
meta–proposition and is paradox free as the surprise strategy is indeed a pure
innovation in the strategy set Bp and outside of sets ßp+  ßp¬ that can be
enumerated by eductive calculation and information in G, see Figure 3. It is
precisely the absence of logical inconsistency and strategic irrationality in the meta
proposition on the surprise strategy that sustains the consistent alignment of beliefs
condition of a Nash equilibrium with surprises. Thus, as already observed, for human
players utilizing ideal reasoning provided by Gödel meta analysis, the set  of best
response functions in (6.b) should provide an inexhaustible source of surprise or
innovative strategies. However, by the same token, by Theorem 5, there is no
algorithmic way by which the prediction function with the index ( bpE!, bpE!) at the
surprise equilibrium can produce an output q though both players can mutually
identify that ( bpE!, bpE!) is the fixed point of the surprise Nash equilibrium best
33
response function fpE. Indeed, ( bpE!, bpE!) says that this is so self-referentially. In a
nutshell ‘innovate or die’ describes this Nash equilibrium in which neither party can
unilaterally deviate without drastically impairing their prospects.
Theorem 5 and Figure 3 on the surprise strategy in a Nash equilibrium of a
game formally corresponds to the set theoretic proof of Gödel’s undecidable
proposition in miniature, Cutland (1980). We have succeeded in showing the formal
equivalence between the Nash equilibrium with surprise or novelty in Figure 3 and
the phase transition in dynamical systems theory that characterizes the endogenous
production of novelty as in Figure 1. Following from part (ii) of Theorem 5 and as
seen in Figure 4, once players are locked in an oppositional structure, the strategy set
of each player will grow utilizing the formalism of an arms race in novelty.
5. Conclusion
This paper has brought together the logical, neurophysiological and social neuroscience literature to throw light on strategic protean behaviour. Based on extensive
studies, the picture that Baumol (2002) paints of technology races in capitalism is not
of isolated individuals making random discoveries, but of a concerted and
institutionalized strategy of innovation to stay ahead of the technology race.
Following the legacy of Driver and Humpheries (1988) and Bryne and Whiten
(1988), social proteanism is hypothesized to be the consequence of predictability
which can be punished by agents capable of prediction. The cognitive basis of the
recurring pursuit-evasion type contests that entail arms-races in new behaviours that
are diverse as they are spectacular, is still not fully understood.
It is only in the meta-mathematics of Gödel that negation of what is
predictable by agents plays a central role in the production of novel objects in the
system to bear ‘witness’ to the incompleteness of the system. The self-negating
diagonal or self-referential Gödel sentences in their productive function variant, often
regarded as dubious and artificial mathematical constructs, are shown to be Nash
equilibrium fixed points of the best response functions of a ubiquitous game where
players are locked in a hostile structure. Indeed, despite the invention of the epithets
‘creative’ and ‘productive’ sets by Emil Post(1944) to underscore these characteristics
of the logical procedures involved in Gödel incomplete systems, almost no mention
or use of such constructs have been made to date even by game theorists who have
34
used recursive function theory in the context of strategic behaviour. Albin (1998)
and Markose ( 2004,2005) are the only exceptions here as they draw on a provenance
of complexity adaptive systems Wolfram-Chomsky schema (see, Casti
and
Wolfram,1984) to highlight the role of agents with powers of universal Turing
Machines to generate Type IV novelty producing structure changing dynamics.
The ingredients of the Gödel meta-representational system (MRS) as set out in
Theorem 2 and equation (6) capable of offline operations on encoded information
which while related to the physical execution of same, has been shown to be the basis
of powerful meta-analysis. The parallels between canonical and mirror neurons and
the Gödel MRS have been noted. Just as undecidability and incompleteness are
theorems rather than paradoxes in a rich enough meta-representational system (MRS),
in this theory, players with the specified MRS will infer the logical necessity to
mutually surprise. It still remains to be seen if the human mirror neuron system has
the capacity for meta-analysis that extends to the 2nd order self-referential recursions
that are needed for deceit. Tognoli et. al. (2007) have made some advances in the
experimental procedure to detect the mirror neuron activity in a mutual two person
direct interactive setting.
The mutual recognition of hostility, negation or deceit - places the metarepresentational system of agents in a state of chaos corresponding to non-converging
calculations neuronal mappings. Such implications for novelty recognition and
production have been cited in Korn and Faure (2003). The GTP theory says that the
only recursive mappings from such fixed points involving the logical persona of the
Liar are those that map out side of recursively enumerable sets. This in layman’s
terms corresponds to thinking ‘outside the box’. The deeply contextual points of exit
and innovation which follow in lock step have to be noted. The exit routes are guided
by the encoded information on specific hostile interactions. Thus spelling out of the
logical foundations of novelty production in a strategic setting suggests many rich
investigative lines for empirical neuro-physiological experiments. The urgency for
these lines of investigation arises from the fact that extant mathematical models of
strategic behaviour cannot account for protean behaviour which is ubiquitous in
socio-economic systems.
35
APPENDIX
PROOF OF THEOREM 5: The proof entails showing that the best response
function fp in (17.b) is the productive function denoted as fp! with the ‘!’ intended to
focus on the feature that an innovation outside given action sets is involved, viz.
fp! = m , m  -A. We will use the two following Lemmas in the proof as well as
the property of the set ßp+c given in (17) and recall that b1 and b2 are the indexes of
the first and second subroutines of the program for the determination of Nash
equilibrium strategies..
Part (i): Consider fpE , the best response function for player p, at the second subroutine denoted by b2 of the Nash equilibrium strategy, to be the recursive reduction
function in Lemma 2. Here with no loss of generality, let
b1n¬   ( ba¬ , ba¬)
(A.1)
signify the non-computable fixed point from p’s Liar/rule breaking strategy in (14)
and Theorem 4 and b1 indicates the index of the first subroutine of the Nash
equilibrium. Consider
W ! to be the recursively enumerable subset W ! 
n
n
ßp+c in
Lemma 2. Then as fp E is the reduction of the prototypical creative set C such that
there exists an index , here b1, which yields Wb1 

n
= fpE-1 ( W n! ). Note Wb1   =
n
ßp¬ and ßp¬ is defined in (16.a) . It then follows from Lemma 2 that fpE( b1n¬ )
is the productive function for the set
W !
n
implementing the surprise Nash
equilibrium and b2 ( fp  ( ba¬,ba¬ ) )= bpE ! is the Gödel number for it.
Part (ii) : Generically, let any Wx ,Wx  ßp+c of be constructed as
W ! of Part (i) to
n
yield a non-repeating, recursive enumeration of surprise strategies thereof with
 !n(.) denoting this total computable enumerating function such that
and, W
In other words
W !
n
!
n 1
= W 1  { b2 (fp ( n¬)) } .
(A.2)
n
contains the reduction facilitated by subroutine 2 on the best
response function fpE of the members of ßp¬ defined in (16.b) and this yields the
surprise strategies thereof as {bo! , b 1! ,. . . . . . .,b n! }. In (A.2) W
!
n 1
is obtained by
36
adding on the index of the productive function or the novel object produced at this
stage. Note in this construction at no time can the g.n of fp(n¬ )  W ! . If the g.n of
n
fp(n¬ )  W ! , then due to the reduction argument in Lemma 2 we will have n¬ 
n

ßp+ and hence ba  (ba ) . However, this leads to a contradiction as from (15.a)
b  (ba  ) and hence b2( fp(n¬ ))  W .
a
!
n
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