Transcript Novelty And Surprises In Complex Adaptive System (CAS

Gödelian Foundations of Non-Computability and
Heterogeneity In Economic Forecasting and Strategic
Innovation
Sheri M. Markose
Lecture 2
Economics Department and Centre For
Computational Finance and Economic Agents
(CCFEA)
University of Essex, UK. [email protected]
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 metamodels for forecasting and strategizing by agents
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
ROAD MAP: III Ubiquity of contrarian self-reflexive
calculations 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.
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 selforganizes
Lack of a unique effective decision procedure :endogenous to
the 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 Pt+1.
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 Rice’s 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.
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 selfreferential 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.
Example: II Design of Market Games:Should 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
Part II: Main ingredients of a Nash
Equilibrium With Surprise or Innovation
I.
II.
III.
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
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.
II.
Agents must have oppositional interests.
Why?
Axelrod (1987) in his classic study on cooperative and noncooperative 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.
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
Post’s set theoretic proof of Gödel Incompleteness
Result.
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.
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.
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).
(a,x) = u(a)(x)  a(x)
(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.
.
C = { x | x(x) ) ; TMx(x) halts ; x  Wx }
(3.a)
The complement of C
C~ = { x | TMx (x) does not halt; x(x) not defined; 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. [2]
[2] 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~ .
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).
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), s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 i ,
i (g, p) being the g.n of an action rule 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 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.
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 s (x,y) (s)
in an infinite matrix X of the enumeration of all
computable functions, given in Figure 2.
FIGURE 2 PREDICTABLE PAYOFFS
X0 s(0,0)
s(0,1)s(0,2)s(0,3)....s(0,y)
....
X1 s(1,0)
s(1,1)s(1,2)s(1,3)....s(1,y)
....
X2 s(2,0)
.
.
s(2,1)s(2,2)s(1,3)....s(2,y)
.....
Xx s(x,0)
s(x,1)
s(x,2) s(x,3) ....s(x,y) ....s(x,x)
The best response function fi can dynamically move the system from row
to row.
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  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.
Second Recursion Theorem: Fixed Point Result
f'
X0 s(0,0)
s(0,1)
s(0,2)
s(0,3) ....s(0,y)
....
X1 s(1,0)
s(1,1)
s(1,2)
s(1,3) ....s(1,y)
....
X2 s(2,0)
s(2,1)
s(2,2)
.
.
s(1,3) ....s(2,y)
.....
Xx s(x,0)
s(x,1)
s(x,2) s(x,3) ....s(x,x )
.....
Xm f(s(0,0)) f(s(1,1)) f(s(2,2)) f(s(3,3)) ...f(s(m,m)) =s(m,m)
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:
s ( x ,y ) ( s )
  x ( y ) ( s) =q,
iff  x ( y)  (3)
Here the total computable function s(x,y) modelled along the lines of
Gödel’s substitution function (see, Rogers, 1967,p.202-204) has the
feature that it names or ‘signifies’ in the meta system X the points in
the game that correspond to the different internal calculations on the
right- hand side of (3) as we substitute different values for (x,y). The
g.ns implemented by s(x,y) can always be obtained whether or not the
partial recursive function  x ( y) on the right-hand side of (3) which
executes internal calculations halts or not.
Definition 5: The best response functions fi, i  (p,g) that are total
computable functions can belong to one of the following classes –
1( IdentityFunction) 

fi =
fi

fi
fi
!


Rule
Abiding
Rule
Rule
Bending
Breaking
(5.a)
 Surprise
such that the g.ns of fi are contained in set ,
 = { m | f j = m , m 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.
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 max
Pi (s ( b , b ^ )( s ) ) ,
b B
i j
i
i,j (p,g).
i
The Nash equilibrium strategies with g.ns denoted by
(bpE, bg E) entail two subroutines or iterations, to be specified
later.
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= 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 player’s 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 ! = m , m  -A,
i  (p,g).
(7.b)
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.
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
Implications of the Liar Strategy
(i) Es b
a

(14.b)
Es b  =.he outcomes (q~ , q ) can be zero sum but in
a
general we refer to property q~ Es b in (14.a) as being oppositional
a
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.
Proposition 3: The outcome of the game at this out of
equilibrium point
s(ba ¬,ba ) is predictable with

s ( ba  ,ba )
= q~
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.
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.

(s)=
(s).





s
(
b
,
b
)
f p s ( ba ,ba )
a
a
(15)
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 g’s rule will lack structural invariance relative to the
best response function fp¬ whose fixed point it is.
Herbert Simon calls this the outguessing problem
3.4 Surprise Nash Equilibria and The Productive
Function
g’s 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! = m , m  -A. (16.a)
That is, fg! implements an innovation and bgE ! is the
g.n of the surprise strategy function in (16.a).
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 ! = m , 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.
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.
FIGURE 3
THE INCOMPLETENESS OF p’s STRATEGY SET Bp
¬
bpE ! = b2 ( zp , b1 ( zg , ba )):SURPRISE STRATEGY
E
bp
ßp+
! = b2 ( zp , b1 ( zg , ba¬ ))

ßp¬
ßp+c
ARMS RACE IN SURPRISES/INNOVATIONS
Bp+c
b0¬ b1¬ …. bn-1¬
g.n (fp¬(σn))= bn¬
Wσn
Wσn+1
g.n: Godel Number
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.
• 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 Ray’s 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 selforganized 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.
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.