Economics Department and Centre for Computational Finance and Economic Agents (CCFEA)

39

Meta-Representation, Mirroring and the Liar: Complex Strategic Behaviour and Arms Race In Novelty and Surprises

Sheri M. Markose[1]

Economics Department and Centre for Computational Finance and Economic Agents (CCFEA)

Email:,

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 meta-representation 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

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 Queen[3] 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 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 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 Gödel-Turing-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], 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 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 1-1 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 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 others beliefs of one’s beliefs. Oberman and Ramachandran (2004) suggest that a system of neurons in the medial prefrontal cortex[9] 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 interact[10] is still in its infancy and possibly neuro-economic game theorists are leading the way here.