G

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Gödelian Foundations of Non-Computability and
Heterogeneity In Economic Forecasting and
Strategic Innovation

• Sheri M. MarkoseLecture 2
• Economics Department and Centre For Computational
  Finance and Economic Agents (CCFEA)
• University of Essex, UK. scher_at_essex.ac.uk
•   

2
ROAD MAP I

• ?SELF-REFLEXIVE CONTRARIAN STRUCTURE This is
  false
• The presence of contrarian payoff structures or
  hostile agents in a game theoretic framework are
  shown to result in the fundamental non-computable
  fixed point that corresponds to Gödel's
  undecidable proposition
• ? Lack of effective procedures to determine
  winning strategies in a stock market game with
  contrarian payoff structure
• Brian Arthur (1994) The Minority or El
  Farol game has a contrarian structure
• ? Results in the adoption of a multiplicity or
  heterogeneity of meta-models for forecasting and
  strategizing by agents

3
ROAD MAP II Game Theory with Hostile Agents
Nash Equilibrium is Surprise or Innovation

• ? Construction of fixed point or self-reference
  in so called rational expectations or mutual
  acknowledgement uses Diagonalization lemma and
  2nd Recursion Theorem
• ? Any best response function of the game
  which is constrained to be a total computable
  function then represents the productive function
  of the Emil Post (1944) set theoretic proof of
  the Gödel Incompleteness result. The productive
  function implements strategic innovation and
  objects of novelty or 'surprise' formally maps
  into a non-recursively enumerable set
• This results in undecidable structure changing
  dynamics in the system

4
ROAD MAP III Ubiquity of contrarian
self-reflexivecalculations in socio-economic
systems

• ? Oppositional or contrarian structures,
  self-reflexive calculations and the necessity to
  innovate to out-smart hostile agents are
  ubiquitous in economic systems. As first noted in
  Binmore (1987) and Spear (1987), extant game
  theory and economic theory cannot model the
  strategic and logical necessity of Gödelian
  indeterminism in economic systems.
• ? Formal results developed in Markose (2002,
  2004, 2005) on the implications of the Gödelian
  incompleteness result for economics.
• Keywords Effective procedures self-reflexivity
  contrarian payoff structures strategic
  innovation Gödel Incompleteness.

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Canonical Example of Self-Reflexive Systems and
Contrarian Structures which have no computable
fixed points

• First example developed by Santa Fe Institute is
  the Artificial Stock Market (ASM)
• Brian Arthur gave a powerful rebuttal of why
  traditional economic analysis will fail to
  understand stock markets and why ACE modelling is
  needed
• In stock market an investor makes money if he/she
  can sell when everybody else is buying and buy
  when everybody else is selling. In other words,
  one needs to be in the minority or contrarian
• Arthur called this the El Farol Bar problem. You
  want to go to the pub when it is not crowded.
  Assume everybody else wants to do the same. How
  can you rationally decide/strategize to succeed
  in this objective of being in the minority ?
• If all of us have the same forecasting model to
  work out how many people will turn up say our
  model says it will be 80 full then as all of
  us do not want to be there when it is crowded
  none of us will go.
• This contradicts the prediction of our model and
  in fact we should go. If all reasoned this way
  once again we will fail etc. So there is no
  Homogenous Rational Expectations and no rational
  way in which we can decide to go. Traditional
  economics cannot deal with this
• Hence, Brian Arthur said we must use ACE models
  and see how the system dynamically self-organizes
  

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Lack of a unique effective decision procedure
endogenous tothe logic of the decision problem

• Spear (1987) and Markose (2004,
  2007)Non-computability of rational expectations
  equilibrium RE as fixed point of market price
  function, g
• In a rational expectations equilibrium (REE)
  there exists some computable forecast function
  
• f fa such
  that fg(a) ? fa , (i)
• Then a is a fixed point of the market price
  function g. Note, a is the algorithm or program
  that computes the output of the market game when
  the price function g that determines the outcome
  is consistent with agents prediction functions
  for Pt1.
• An agent has to find a meta forecast rule f fa
  that satisfies (I) .
• That is, the agent has to identify a proper
  subset of the set of all partial computable
  functions f0, f1, f2, ........, such that
  only the fixed points of the total computable
  function g are identified, viz.
• m fg(m) fm .
  (II)
• By Rices Theorem no uniform recursive/
  algorithmic procedure to identify set of indices
  in (ii). There is no systematic way of forming
  REs of the market price function g.
• Only inductive trial and error learning that
  begins search in an arbitrary subset of diverse
  forecast rules.

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Self Reflexity in Stock Prices

• Agents , in facing the problem of choosing
  appropriate predictive models, face the same
  problem that statisticians face when choosing
  appropriate predictive models given a specific
  data set, but no objective means by which to
  choose a functional form . The expectational
  models investors choose affect the price
  sequence, so that our statisticians very choices
  of model affect their data and so their choices
  of model (ibid. p. 305, italics added).
• 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
  reflexity, or self-referential character of
  expectations, precludes expectations being formed
  by deductive means , so that perfect rationality
  ceases to be well defined (Arthur et. al. 1997,
  Santa Fe Institute Working Paper
• We will proceed to show that instead of referring
  to the above self-reflexive problem as one that
  ceases to be well defined the problem is
  algorithmically unsolvable.

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Example II Design of Market GamesShould not
permit computable winning strategies or Free Lunch

• George Soros made 2bn taking a short position
  against the Sterling and the Bank of England. He
  is alleged to have used the Liar or Contrarian
  Strategy.
• Soros cut above ordinary speculator student of
  Karl Popper and knows the self-reflexive problem
  of the Cretan Liar. Liar can subvert only from a
  a point of certainty or computable fixed point.
  Hence, if the policy position is perfectly known
  hostile agents can destroy it. Indeterminism or
  ambiguity is a essential design element for
  success of market systems and zero sum games
• Traffic Model and how to avoid congestion is a
  minority game

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Part II Main ingredients of a Nash Equilibrium
With Surprise or Innovation

• Agents with full powers of Turing Machines Why?
• Agents must have oppositional interests Why?
• Arms Race Type Red Queen Dynamic formally
  modelled as the productive function that can
  produce innovations ad infinitum

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I. Agents with full powers of Turing Machines
Why?

• It is now well known from the Wolfram-Chomsky
  scheme (see, Wolfram, 1984, Foley, in Albin,1998,
  pp. 42-55, Markose, 2001a) that on varying the
  computational capabilities of agents, different
  system wide or global dynamics can be generated.
  
• Finite automata produce Type 1 dynamics with
  unique limit points
• Push down automata produce Type 2 dynamics with
  limit cycles
• Linear bounded automata produce Type 3 chaotic
  output trajectories with strange attractors.
• However, it takes agents with full powers of
  Turing Machines capable of simulating other
  Turing machines and hence self-reference, a
  property called computational universality, to
  produce the Type 4 irregular innovation based
  structure changing dynamics associated with
  capitalist growth.

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• Agents must have oppositional interests. 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).
• It is believed that with the computational theory
  of actor innovation (Markose, 2003/4), we have a
  formal solution of one of the long standing
  mysteries as to why agents with the highest level
  of computational intelligence are necessary to
  produce innovative outcomes in Type IV dynamics.

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• Finally 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 Posts set theoretic proof of
  Gödel Incompleteness Result.

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• Mathematical Preliminaries
• ? 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.

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Definition 2 A set which is the null set or the
domain or the range of a recursive function is a
recursively enumerable (r.e) set. Sets that
cannot be enumerated by T.Ms are not r.e .  
The one feature of computability theory that is
crucial to eductive game theory where players
have to simulate the decision procedure of other
players, is the notion of the Universal Turing
Machine(UTM).
(2) The UTM, on L.H.S of
(2) on input x will halt and output what the TMa
on the RHS does when the latter halts and
otherwise both are undefined.
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. C x fx(x) ) TMx(x) halts x Î Wx
(3.a) The complement of C C x
TMx (x) does not halt fx(x) not defined x Ï
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,
Cantors diagonalization method is used. 2
2 Assume that there is a computable function
f fy , whose domain Wy C . Now, if y Î Wy
, then y Î C as we have assumed C Wy . But
by the definition of C in (3.b) if y Î Wy ,
then y Î C and not to C . Alternatively, if
yÏWy , y ÏC , given the assumption that C Wy
. Then, again we have a contradiction, as since
from (3.b) when yÏ Wy , yÎC . Thus we have to
reject the assumption that for some computable
function f fy , its domain Wy C .
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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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GAME REGULATORY ARBITRAGE OR PARASITE AND HOST
MODEL UNDER COMPUTABILITY CONSTRAINTS

• Computability constraints means that all decision
  rules, actions etc. are finitely encodable
  procedures with Godel numbers (g.ns).
•  
• G (p,q), (Ap, Ag), sÎ 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 Î Ai , iÎ (g, p) being the
  g.n of an action rule of player i and
  l0,1,2,.....,L.  An element sÎ S denotes a
  finite vector of state variables and other
  archival information and 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
  beliefs of other players strategy.

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LIKE CHESS NOTATION META ANALYSIS OF
GAME All meta-information with regard to the
outcomes of the game for any given set of state
variables, s belong to 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.
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FIGURE 2 PREDICTABLE PAYOFFS  X0 fs(0,0)
fs (0,1) fs (0,2) fs (0,3) ....
fs(0,y) ....  X1 fs(1,0) fs
(1,1) fs (1,2) fs (1,3) .... fs(1,y)
....  X2 fs(2,0) fs (2,1)
fs (2,2) fs (1,3) .... fs(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 one
player plays strategy x and the other player
plays a strategy that is consistent with his
belief that the first player has used strategy y.

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Second Recursion Theorem Fixed Point
Result X0 fs(0,0) fs (0,1) fs (0,2)
fs (0,3) .... fs(0,y) ....  X1
fs(1,0) fs (1,1) fs (1,2) fs (1,3)
.... fs(1,y) ....  X2 fs(2,0)
fs (2,1) fs (2,2) fs (1,3) ....
fs(2,y) ..... .. Xx fs(x,0)
fs (x,1) fs (x,2) fs (x,3) .... fs(x,x
) .....
f'
Xm ff(s(0,0)) ff(s(1,1)) ff(s(2,2))
ff(s(3,3)) ... ff(s(m,m))
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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Theorem 1 The representational system is a 1-1
mapping between meta information in matrix X in
Figure2 and internal calculations such that the
conditions under which the prediction function
which determines the output of the game for each
(x,y) point is defined as follows
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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 j 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.
 
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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).
The Nash equilibrium strategies with g.ns denoted
by (bpE, bg E) entail two subroutines or
iterations, to be specified later.
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In standard rational choice models of game
theory, the optimization calculus in the choice
of best response requires choice to be restricted
to given actions sets. Hence, strategy functions
map from a relevant tuple that encodes meta
information of the game into given action
sets bi ( fis(x,x), z, s, A) ? Ai and fi f m
, m?Ai, i ? (p,g) .
(7.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 players response
functions. Definition 7 We say that the player
has used a strategic innovation or a surprise and
adopted an innovation in terms of actions from ?-
A, viz. outside given action sets when,  bi
(fis(x,x)), z, s, A) ? ?- A and fi fi ! fm
, m? ? -A,
i ? (p,g).

(7.b)
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WHEN DOES THIS HAPPEN?The very function of the
Gödel meta framework is to ensure that no move in
the game made by rational and calculating players
can entail an unpredictable/surprise response
function from set ? unless players can mutually
infer by strictly codifiable deductive means from
s(x.x) that (7.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.

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THE STRUCTURE OF OPPOSITION THE LIAR
STRATEGY For any state s when the rule a
applies,THE LIAR STRATEGY fp
For all s when policy rule a does not apply,
fp 0 . Do Nothing
(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
The no-win for g is recursively ascertainable and
rule a cannot be a Nash equilibrium strategy for
g. Not acknowledging the identity of the
Liar is fatal for transparent rules and the
success of the Liar entails an elementary error
in logic and game theory on part of the other
player.
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3.3 The Non-computable Fixed Point Now, if
g acknowledges the identity of the Liar in
(14.a), he updates his belief with ba , the
code for the Liar strategy in (14.a). Once the
identity of the Liar has been acknowledged, g
must rationally abandon the transparent rule a in
(14.a) as per Proposition 3.
Theorem 3 The prediction function indexed by
the fixed point of the Liar/rule breaker best
response function fp in (15) is not computable.
Here, the fixed point which signals mutual
knowledge that p will falsify predicted outcomes
of gs rule will lack structural invariance
relative to the best response function fp whose
fixed point it is.  Herbert Simon calls this
the outguessing problem
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3.4 Surprise Nash Equilibria and The Productive
Function   gs Nash equilibrium strategy bgE
with g.n bgE implemented by the total computable
function b1 in (11.a) must be such that bgE (fgs
(ba , ba ), z, s, A) ? ?- A and fg fg! fm
, m? ? -A. (16.a)  That is, fg! implements an
innovation and bgE ! is the g.n of the surprise
strategy function in (16.a).
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Likewise for player p, fp! implements an
innovation in (16.b) and bpE ! is the g.n of the
surprise strategy function. Thus, bpE (fp s
(b1( ba), b1( ba )), z, s, A) ? ?- A and fp
fp ! fm , m? ? -A. (16.b)The total
computable functions (b1 , b2 ) in (11.a,b)
implementing the g.ns of the respective Nash
equilibrium strategies from the uncomputable
fixed point in (15), fully definable in the meta
analysis, can only map into domains of respective
strategy sets (Bp , Bg) whose members cannot be
recursively enumerable. As fp are total
computable functions thereoff, it can only map
into the productive set ? -A, which is not
recursively enumerable.
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Theorem 5The incompleteness of ps strategy
set Bp that arises from the agency of the Liar
strategy requires the proof that ßpc is
productive as in Definition 4 with the g.n of the
surprise strategy bpE ! ? ßpc -
ßp.Construct a witness for why ßpc is not
recursively enumerable.
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ARMS RACE IN SURPRISES/INNOVATIONS
Bpc
b0 b1 . bn-1
g.n (fp(sn)) bn
Wsn
Wsn1
g.n Godel Number
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CONCLUDING REMARKS? INNOVATION FAR FROM BEING
A RANDOM OUTCOME, AS IS POPULARLY HELD, IS THE
RESULT PRIMARILY OF COMPUTATONAL INTELLIGENCE
Wolfram (1984) had conjectured that the highest
level of computational intelligence, the capacity
for self-referential calculation of hostile
behaviour was also necessary.  This casts doubt
on the Darwinian tradition that random mutation
is the only source of variety ? THE STRUCTURE
OF OPPOSITION IS A LOGICAL NECESSARY CONDITION
FOR INNOVATION TO BE A STRATEGIC RATIONAL
OUTCOME AND A NASH EQUILBRIUM OF A GAME.
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• Surprise Nash equilbria correspond to phase
  transition of life at the edge of chaos.
• In Markose (2003) it is argued that for systems
  to stay at the phase transition associated wih
  novelty production requires the Red Queen dynamic
  of rivalrous coevolving species. In the Rays
  Tierra(1992) and Hillis ( 1992)artificial life
  simulation models, once computational agents have
  enough capabilities to detect rivalrous behaviour
  that is inimical to them, they learn to use
  secrecy and surprises.
• Finally, a matter that is beyond this paper, but
  is of crucial mathematical importance is that
  objects of adaptive novelty as in the Gödel
  (1931) result has the highest diophantine degree
  of algorithmic unsolvability of the Hilbert Tenth
  problem. This model of indeterminism is a far
  cry from extant models that appear to assume
  adaptive innovation or strategic surprise is
  white noise which in the framework of entropy
  represents perfect disorder, the antithesis of
  self-organized complexity. It can be
  conjectured that a lack of progress in our
  understanding of market incompleteness and
  arbitrage free institutions is related to these
  issues on indeterminism.

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Selected References

• Arthur, W.B., (1994). 'Inductive Behaviour and
  Bounded Rationality', American Economic Review,
  84, pp.406-411.
• Binmore, K. (1987), 'Modelling Rational Players
  Part 1', Journal of Economics and Philosophy,
  vol. 3, pp. 179-214.
• Markose, S.M, 2005 , 'Computability and
  Evolutionary Complexity Markets as Complex
  Adaptive Systems (CAS)', Economic Journal , vol.
  115, pp.F159-F192.
• Markose, S.M, 2004, 'Novelty in Complex Adaptive
  Systems (CAS) A Computational Theory of Actor
  Innovation', Physica A Statistical Mechanics and
  Its Applications, vol. 344, pp. 41- 49. Fuller
  details in University of Essex, Economics Dept.
  Discussion Paper No. 575, January 2004.
• Markose, S.M., July 2002, 'The New Evolutionary
  Computational Paradigm of Complex Adaptive
  Systems Challenges and Prospects For Economics
  and Finance', In, Genetic Algorithms and Genetic
  Programming in Computational Finance, Edited by
  Shu-Heng Chen, Kluwer Academic Publishers,
  pp.443-484 . Also Essex University Economics
  Department DP no. 552, July 2001.
• Post, E.(1944). 'Recursively Enumerable Sets of
  Positive Integers and Their Decision Problems',
  Bulletin of American Mathematical Society,
  vol.50, pp.284-316.
• Spear, S.(1989), 'Learning Rational Expectations
  Under Computability Constraints', Econometrica ,
  vol.57, pp.889-910.