Mirroring and the Liar Qua Rule-breaker: Complex Strategic Behaviour and Arms Race in Novelty and

Surprises  

7 July 2015 Athens University 12 Stochastic Finance Summer School

New Foundations for Social Cognition and Strategic Interaction: Coordination, Anti-Coordination and

Innovation

Sheri M. Markose

Economics Department University of Essex, UK. scher@essex.ac.uk 1

ADAPTIVE NOVELTY :INNOVATION, PROTEANISM , HYPER INTELLIGENCE

 First modern study of capacity of systems to produce ‘new’ objects : von Neumann in the 1940s with his self-reproducing machines. Provenance of Langston/Casti/Wolfram/Chomsky agents of

computational powers of Turing Machines are needed to produce Type 4 dynamics with structure changing undecidable dynamics based on novelty and surprises they call the sine qua non of complex adaptive systems (see, Economist , Peter Albin, 1998)

Capacity to produce novelty, ‘new’ objects is rampant in market systems, evolutionary immune systems with host parasite relationships and other evolutionary systems.

In extant economic and game theory, the mathematical framework used is inadequate to model innovation as arising from strategic necessity in a Nash equilibrium of a game. Mutation is taken to be random and the ESS equilibria study how systems can be locked in and be resistant to mutants who arise in an exogenous way. Herbert Simon said to me ( re game theorists) : Randomize too much, don’t they ?

NEO-DARWINIAN TRADITION POSTULATES INNOVATION ARISES ONLY FROM RANDOM MUTATION ON WHICH NATURAL SELECTION OPERATES

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  “Bacterial wisdom, Gödel's theorem and creative genomic webs” Eshel Ben Jacob (1998); Markose (2005) Economic Journal paper on Markets as Complex Adaptive Systems

• Ben-Jacob : The emergence of new forms or entities not previously there “ this is not the result of successful accumulation of mistakes ( or random mutations) in the replication of the genetic order”. The genome even in most early life forms such as in bacteria already has endogenous ‘creative’ capabilities and does not have to rely only on random mutations enhanced by natural selection to evolve.  

 Ben-Jacob gives a number of capabilities that the ‘genome units’ should possess (p59) to be able to engineer, possibly as emergent outcomes, novel artifacts and behaviours: • (i)Capacity to recognize difficulties in the environment or hostile conditions

that is a threat to some functionality , • (ii) The need for advanced language that includes self-referential mapping.  • Finally and significantly, • (iii) Even with  very earliest form of life as with bacteria, they evolved social

capabilities of recognizing signals between self and the other (bottom of p 63) which requires a rich (chemical) language.

• (ii) and (iii) we will call meta-analysis. 3

In addition to fight or flight: Get smart and produce novel behaviours, artifacts and technologies

This produces primordial fear from anti-coordination of the most radical kind

Necessary and sufficient condition for the generation of novel objects contextual to a code or functionality that is under threat is the intelligent system must be able to syntactically embed the Gödel sentence related to this.Often regarded as funky and esoteric construction It is ubiquitous and must be there from the word get go with life itself as it embodies recognition of hostile agency in relation to self and specific to a code. Why ?  The Godel sentence simply encodes in a self-referential way the presence of a hostile agent in the form of negation to a given code. In pair of similarly mentalizing hostile agents, it symbolizes the mutual recognition of a mortal embrace of opposition/hostility. In logic, these will signal points of exit from any known listable set of actions.

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To Date Game Theory Eschews Possibility of an Agent Considering a ‘Surprise’ or ‘Novel’ Action and in turn that he will be ‘Surprised’ and Face a

Non-Listed Action The main point of departure between extant game theory and the above Gödelian logic of agents capable of meta-analysis has been succinctly stated by Bhatt and Camerer (2005) as follows: “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 characterization of a Nash equilibrium in which players mutually and logically expect that they will need to surprise and be surprised in that novel actions follow.

Markose (2013) paper is concerned with extending game theory to incorporate social proteanism : A Nash equilibrium in which agents are involved in a arms race of competitive innovation.

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In Gödel Logic Liar like structures (the agency to negate what can be predicted) has Great Significance

Robert Lucas ( 1972,1976) only economist to talk about the strategic use of surpriseGödel Logic

• Archetypical agents called the Liar who negate /falsify what can be predicted/computed

• Fixed point of a negation produces mechanized exit points from Listable Sets, viz surprises

• Nash equilibrium is mutual 'surprise'- expect to surprise and to be surprise in an arms race of innovations

• This generates innovation based undecidable structure changing Type 4 dynamics. No meta model can identify this point of structure changes. This has highest degree of non-computability a la solving Hilbert 10 problem.

 

Lucas Postulates: Policy Ineffectiveness and Surprise

• Policy is rendered ineffective by regulatees if it can be rationally expected (Why?)

• Hence policy surprises need to be used : Lucas couched it in ‘surprise’ inflation

y = y* + b( p - p e ) + e ,

 This says that output, y, will not increase beyond the natural rate , y*, unless there is ‘surprise’ inflation, (p - p e

) which is the prediction error from expected inflation, p e.

• Lucas Critique : No econometric model can identify innovation based structure changes

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New Neurophysiological Foundations of Social Cognition: Evidence (1)

• Gallese’s theory of the Mirror Neuron System (MNS) as means of offline simulation for action prediction

MNS so called as there is synchronous firing of mirror neurons when canonical neurons fire with actual action execution by self and also when observing another • MNS provides a cellular depiction of meta-analysis with

self as both actor and observer of others in the mapping. • The modelling of and neuronal evidence for how the MNS

system integrates self and the other in complex social interaction has only just begun

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MNS as a 2 place (self and other) Offline Simulation vs Gödel Representational System ( Hartley Rogers, 1967, p. 202-204 )

MNS : LHS off line simulation vs RHS Action Execution

LHS Godel meta system off line simulation RHS Action Execution /Online Calculation

)s()y,x( ,)()( qsyx iff xy() (3)

Mirror Neurons Fire (yellow) MNS (self, other)

Canonical Neurons Firing during action execution of self LHS ‘signifies’ or records off line

with 2 place (g.n) of Gödel Substitution function (x,y) a 1-1 mapping of the online execution on RHS a la canonical neurons In (x,x) , first place x (lhs) is action of self and second place x is self ‘s belief that other has correctly predicted self : coordination

Self as observer of other

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Anti-Coordination Part of MNS : Evidence (2) We will use symbol from

logic x negation of x ; f will denote the Liar strategy or

negation/anticoordination strategy

• Neuro-physiological marker for MNS activity is mu wave suppression. Kelso et al (2007,2012) have done real time social interaction experiments using dual EEG and were able to identify phi 1 and phi 2 waves that coincide, respectively, with intentional anti coordination and coordination behaviours.

• The significance of this is that both coordination and anti coordination behaviours are part of the MNS system as identified by the mu wave suppression which Kelso et al find irrespective of the type of social interaction.

• MNS shows that we are hardwired to coordinate and perform mimetic activities . If anti-coordination was not part of MNS – entrainment in the species will follow.

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Players of Matching Pennies automatically imitate opponents’ gestures against strong

incentives not to do so , (2012) Michele Belot,  Vincent P. Crawford and Cecilia Heyes

• In a paper just out, I think for the first time it has been suggested by economists Belot and Crawford (with neuro-psychologist Cecilia Heyes)  that the role of mirror neurons to coordinate actions is great despite strategic incentives to not coordinate. 

•  http://www.pnas.org/content/110/8/2763.abstract

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The Logic of Anti-Coordination -The Liar: Self Reflexive Statement of Negation

• The modelling of and neuronal evidence for how the MNS system integrates self and the other in complex social interaction is still in early days.

• Any system with MNS capacity to do offline simulations viz operations on encoded information is already in possession of powers of Turing Machines.

• Mimetic behaviours and action prediction when no conflicting objectives exist can be shown to be automated and instantaneous deduction. Such fixed point mappings are stock in trade of Gödel representational systems. We need fixed point theorem called 2nd Recursion

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Computation Theory 101

MECHANISM, ALGORITHM, COMPUTATION

 

The Church Turing Thesis states that models of computation considered so far for implementing finitely encoded instructions, prominent among these being that of the Turing machine (T.M for short), have all been shown to be equivalent to the class of general recursive functions.

Operations or mappings on encoded information, referred to as Gödel numbers (g.n) are number theoretic functions ; the operation itself is encoded and gives instructions for execution .

G.Ns of total computable functions are blueprints or executable descriptions of a technology Eg A 3 D printer ! 12

As computable functions operate on encoded information they are number theoretic functions, f : N N where N is the set of all integers. f(x) a(x) =q . (1.a) 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) }. (1.b) Range of function is denoted by set E. Definition 1: The number theoretic functions that are defined on the full domain of N are called total functions. Partial functions are those functions that are defined only on some subset of N.

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Two Person Game and MNS Style Social Cognition: What will constitute novelty and surprise strategy ?

G= {(p,q), (Ap, Ag), s Î S}. This information is in the public domain.

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 by Ai with A= Ai, are finitely countable with ail ÎAi , iÎ (g, p) being the g.n of an action rule of player i and l=0,1,2,.....,L. g.n of action  An element s Î S denotes a finite vector of state variables S is a finitely countable set.  The strategy functions denoted by (bg , bp )  The strategy sets containing the g.ns of computable strategies denoted by (Bp, Bg). Lower case b are g.n for strategies and b^ g.n of beliefs by self of other players’s action prediction for self.

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Definition 5: The best response functions fi, i (p,g) that are total

computable functions can belong to one of the following classes –

such that the g.ns of fi are contained in set , 

= { m | f i = f m , fm is total computable}. (5.b)

Remark 4: The set which is the set of all total computable functions is not recursively enumerable. The proof of this is standard, see, Cutland (1980, p.7). As will be clear, (5.b) draws attention to issues on how innovative

actions/institutions can be constructed from existing action sets.  

fi =

Surprisef

BreakingRulef

BendingRulef

AbidingRuleFunctionIdentity

i

i

i

!

)(1

(5.a)

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Set is not recursively listable A surprise strategy is an

innovation fi = fi ! = fm , m -A,

Set of Total Computable Functions representing all possible technologies

Action set A Subset of (A contains known technologies)

Surprise is encoding of a technology that lies in ( -A)

Definition 7 : The objective functions of players are computable functions Pi , i (p,g) defined over the partial

recursive payoff/outcome functions specified in state variables in (3).

Arg ii Bb

max

i )s()̂jb,ib( , i,j (p,g)

The Nash equilibrium strategies with g.ns denoted by (bpE, bg E) entail two subroutines or iterations, to be specified later. Note bj ^ = bi is i’s assignment of perfect action prediction of j of i’s action

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LIKE CHESS NOTATION: MNS Style META ANALYSIS OF GAME

All meta-information on the outcomes of the game for any given set of state variables, s S and state of

play can be effectively organized by the so called

prediction function f s (x,y) (s) in an infinite matrix X of the enumeration of all

computable functions, given in Figure 2.Fixed Point of Game of a Strategy function f ,

starting from some desired base point from perspective of g (x=ba , ) f f p s (x,x) (s) = f s(x,x) (s)

In s (x,x) on LHS , first x describes where g desires the system and 2nd element that g knows p has predicted this , the s(x,x) on RHS is g’s action prediction of fp

= 1 viz. perfect coordination 18

FIGURE 2 PREDICTABLE PAYOFFS : Only Diagonal Elements Nash Equilibrium X0 fs(0,0) (0,1) (0,2) (0,3) .... fs fs fsfs(0,y) .... X1 fs(1,0) (1,1) (1,2) (1,3) .... fs fs fsfs(1,y) .... X2 fs(2,0) (2,1) (2,2) (1,3) .... fs fs fsfs(2,y) ..... .. Xx fs(x,0) fs (x,1) fs (x,2) fs (x,3) .... fs(x,y) .... fs(x,x)

The best response function fi can dynamically move the system from row to row. f s (x,y) (s) = q .

 q in some code, is the vector of state variables determining the outcome of the game.Nash Equilibria are DIAGONAL ELEMENTS  s(x,y) is the index of the program for prediction function f that produces the output of the game when self plays strategy x and the self predicts other player plays a strategy that is consistent with other’s belief that self has used strategy y: out of equilibrium

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Second Recursion Theorem: Fixed Point Result : First Used by Steve Spear 1989. "Learning Rational Expectations under Computability

Constraints," Econometrica

X0 fs(0,0) f (0,1)s f (0,2)s f (0,3)s .... fs(0,y) ....

X1 fs(1,0) f (1,1)s f (1,2)s f (1,3)s .... fs(1,y) ....

X2 fs(2,0) f (2,1)s f (2,2)s f (1,3)s .... fs(2,y) .....

.

.

Xx fs(x,0) f s (x,1) f s (x,2) f s (x,3) .... fs(x,x ) .....

Xm f

f(s(0,0)) ff(s(1,1)) f

f(s(2,2)) ff(s(3,3)) ...

ff(s(m,m))

f'

Xm ff(s(0,0)) ff(s(1,1)) ff(s(2,2)) ff(s(3,3)) ... ff(s(m,m))

= fs(m,m)

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THE STRUCTURE OF OPPOSITION: THE LIAR STRATEGY 

For any state s when the rule a applies,THE LIAR STRATEGY fp¬ :

(i)ab

E abE he outcomes (q~ , q ) can be zero sum but in

general we refer to property q~ ab

E in (14.a) as being oppositional

or subversive. (ii)The Liar can subvert/destroy only from a computable fixed point. From latter he can destroy with certainty if a total computable function

fp¬exists.

For all s when policy rule a does not apply,

fp¬ = 0 . (14.b)Implications of the Liar Strategy

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Proposition 3: The outcome of the game at this out of equilibrium point

s(ba ¬,ba ) is predictable with

)ab,ab( q~

The no-win for g is recursively ascertainable and rule a cannot be a Nash equilibrium strategy for g. From the perspective of g fp¬ on ba to give

ba is a direct negation of g’s code a and will be directly experienced by g . Player g thus obtains the action prediction for p’s Liar strategy in a fixed

point on the RHS of (15)

On RHS, the second ba

is that g knows that p knows that g has predicted that p is the Liar.

)ab,ab(pf (s)

)ab,ab( (s). (15)

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3.3 The Non-computable Fixed Point : Gödel Sentence

Once the identity of the Liar has been acknowledged, g must rationally abandon the transparent rule a in (14.a) as per Proposition 3. Prediction leads to negation by the Liar !

Theorem 3: The prediction function indexed by the fixed point of the

Liar/rule breaker best response function fp¬ in (15) is not computable.

)ab,ab(pf (s)

)ab,ab( (s). (15)

Here, the fixed point which signals mutual knowledge that p will falsify predicted outcomes of g’s rule will lack structural invariance relative to the

best response function fp¬ whose fixed point it is.  

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3.4 Surprise Nash Equilibria and The Productive Function , Emil Post (1947) A total computable function operating on a Gödel sentence ,the fixed

point of which is s (ba¬ , ba¬ ), has to map outside two listable disjoints sets (see definition 5 in end

notes) 

g’s Nash equilibrium strategy bgE

with g.n bgE

implemented by the total computable function b1 in

(11.a) must be such that bg

E (fgs (ba¬ , ba¬ ), z, s, A) - A and

fg = fg! = fm , m -A. (16.a)

 That is, fg! implements an innovation and bg

E ! is the

g.n of the surprise strategy function in (16.a). 24

Theorem 5

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 4 with

the g.n of the surprise strategy:

bpE ! ßp+c - ßp¬.

Construct a witness for why ßp+c is not recursively enumerable.

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FIGURE 3

THE INCOMPLETENESS OF p’s STRATEGY SET Bp

bpE ! = b2 ( zp , b1 ( zg , ba

¬ )):SURPRISE STRATEGY

bpE ! = b2 ( zp , b1 ( zg , ba

¬ ))

ßp+

ßp¬

ßp+c

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b0¬ b1

¬ …. bn-1¬

g.n (fp¬(σn))= bn

¬

ARMS RACE IN SURPRISES/INNOVATIONS : Like a record of antibodies of past ‘attacks’

Bp+c

Wσn+1

g.n: Godel Number

Wσn

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CONCLUDING REMARKS

A Mirror Neuron Type System is needed which has a 1-1 synchrony with online executions and off line ‘records’ of this which can be processed in a two place formulation to integrate

action prediction of self and the other

Work of Ben Jacob, Vittorio Gallese and Scott Kelso is vital for further multi-disciplinary work and some neuro experiments a la

Kelso experiments on anti-coordination

THE STRUCTURE OF OPPOSITION IS A LOGICAL NECESSARY CONDITION FOR INNOVATION TO BE A STRATEGIC RATIONAL

OUTCOME AND A NASH EQUILBRIUM OF A GAME.THIS I BELIEVE IS THE FIRST DEMONSTRATION OF THIS.

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Binmore (1987) : Modelling Rational Players (Journal of Economics and Philosophy) Seminal work that introduced to game theory the requisite dose of mechanism with players with powers of Turing machines, and along with it ‘the spectre of Gödel’.

Binmore’s critique of traditional game theory is that it cannot accommodate a generic model of a rule breaker that comes in the form of Gödel’ s Liar which formalizes the structure of opposition.

 

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II. Agents must have oppositional interests for novel behaviours : Why?

. Axelrod (1987) in his classic study on cooperative and non-

cooperative behaviour in governing design principles behind evolution had raised this crucial question on the necessity of hostile agents :“ we can begin asking about whether parasites are inherent to all complex systems, or merely the outcome of the way biological systems have happened to evolve” (ibid. p. 41).

Hyperintelligence , getting smart and innovation which involves encodable technologies is simply a way to escape from the enemy, in a coevolutionary framework

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Recognition of deception or contrarian and Liar like structures have special significance in Gödelian logic as it first triggers chaotic dynamics : Do we

need more than syntactic encoding which operates at a sub-personal level ?• Skarda and Freeman (1987) conclude that

“chaos confers the neural system with a deterministic “I don’t know state” from within which new patterns can emerge ”

• Encoding the Gödel sentence needs an MNS system to signals mutual recognition of hostility between oneself and another is considered to be basis of all creative and protean behaviours which produce novelty and surprises.

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So Where Did Economists Go Wrong With Lucas Surprise Inflation Supply Function ?

• Surprise inflation sounds like a bad thing; precommitment to fixed simple rules was recommended. First, with currency pegs which got destroyed by Soros in 1992 when he said he was using the Liar strategy and then inflation targeting to the exclusion of monitoring rampant financial innovation which upended the economy

• In a game of co-evolution no party has the luxury of withdrawing from the game unilaterally without courting full wipe out

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Some end notes

See Markose (2013, 2005, 2003)

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What do non-computable emergent equilibria look like?

It corresponds to the famous Langton thesis on “life at the edge of chaos” and is formally identical to recursively inseparable sets first discovered in the context of formally undecidable propositions and algorithmically unsolvable problems by Post (1944).

Figure 1 gives the set theoretic representation of the

Wolfram-Chomsky schema of complexity classes for dynamical systems which formally corresponds to Post’s set theoretic proof of Gödel Incompleteness Result. 34

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C o r r e s p o n d i n g t o t h o s e ( a g l , s ) t u p l e s , a g l A g o f g ’ s b a s e p o i n t o p t i m a l s t r a t e g y f o r w h i c h p ’ s b e s t r e s p o n s e f p i s t o b e r u l e a b i d i n g o r r u l e

b e n d i n g , v i z . f p { 1 , f p + } , t h e g . n s o f t h e s e o p t i m a l s t r a t e g i e s f o r p , b p

*

B p r e s u l t i n c o m p u t a b l e f i x e d p o i n t s

ß p + = { *)b(|*b p*bp p f o r a l l ( a g l , s ) , a g l A g , f p { 1 , f p

+ } } . ( 1 7 . a )

S e t ß p ¬ t h a t c o n t a i n s t h e g . n s o f p ’ s s t r a t e g i e s f o r w h e n i t i s o p t i m a l f o r p

t o u s e t h e L i a r b e s t r e s p o n s e f u n c t i o n f p¬ t o t h o s e ( a g l , s ) t u p l e s , a g l

A g o f g ’ s b a s e p o i n t o p t i m a l s t r a t e g y .

T h i s s e t o f p ’ s s t r a t e g i e s t h a t r e s u l t s i n n o n - c o m p u t a b l e f i x e d p o i n t s .

ß p ¬ = { *)b(|*b p*bp p f o r a l l ( a g l , s ) , a g l A g , f p = f p

¬ } . ( 1 7 . b )

ß p + ß p ¬ = a n d t h e s e a r e d i s j o i n t s e t s . N o w , d e f i n e t h e c o m p l i m e n t s e t o f ß p + d e n o t e d b y ß p + c a s

ß p + c = { x | x ( x ) , x B p } . ( 1 8 )

A s ß p + ß p ¬ = , t h e t w o s e t s a r e r e c u r s i v e l y e n u m e r a b l e d i s j o i n t

s e t s w i t h ß p ¬ ß p + c b y d e f i n i t i o n i n ( 1 7 . b ) . 36

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 Ì 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).

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