1 / 41

AGENT-BASED COMPUTATIONAL ECONOMICS: Complexity Economics and the Red Queen

AGENT-BASED COMPUTATIONAL ECONOMICS: Complexity Economics and the Red Queen. Sheri M. Markose Economics Department University of Essex, UK . scher@essex.ac.uk. ABSTRACT.

tosca
Download Presentation

AGENT-BASED COMPUTATIONAL ECONOMICS: Complexity Economics and the Red Queen

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. AGENT-BASED COMPUTATIONAL ECONOMICS:Complexity Economics and the Red Queen Sheri M. Markose Economics Department University of Essex, UK. scher@essex.ac.uk

  2. ABSTRACT  The nexus between evolutionary complexity and innovation:Based on Complex Adaptive Systems (CAS) Theory of Wolfram- Langton-Kauffman thesis:    The sine qua non of a CAS is its capacity to produce innovation or surprises with irregular structure changing dynamics I develop the full formal foundations of the famous phase transition that physicists call ‘life at the edge of chaos’, viz. the domain on which novel objects emerge. I give a formalization of the Arms Race type Red Queen Dynamic in strategic innovation

  3. ADAPTIVE NOVELTY :“INNOVATION, INNOVATION, INNOVATION”INTRODUCTION • First modern study of capacity of systems to produce ‘new’ objects : von Neumann in the 1940s with his self-reproducing machines. (COMPUTATIONAL LEGACY OF VON NEUMANN TO CAS THEORY) • The capacity of the system to produce novelty, ‘new’ objects is rampant in market systems, evolutionary immune systems with host parasite relationships and other evolutionary systems. • In extant economic 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. • DARWINIAN TRADITION POSTULATES INNOVATION ARISES FROM RANDOM MUTATION ON WHICH NATURAL SELECTION OPERATES

  4. But Where from a Theory of Innovation ? A Computational Theory of Actor Innovation Goldberg (1995) claims that the mystery shrouding innovation can be dispelled .. “by a heavy dose of mechanism. Many of the difficulties in the social sciences comes from a lack of a computational theory of actor innovation …. . population oriented systems are dominated by what economists call the law of unintended consequences (which is itself largely the result of the innovative capability of the actors ) and interacting with GAs provides hands-on experience in understanding what for most people is counterintuitive behaviour”, (Ibid. p.28, italics added).

  5. 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. The computational theory of actor innovation did not follow, till I hit on these ideas!

  6. Three main ingredients of a Computational Theory of Actor 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

  7. 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.

  8. Agents must have oppositional interests. Why? The evolution of cooperation (Axelrod, 1984)has received a lot of attention. 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 developed in this paper, 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.

  9. The Liar: Self Reflexive Statement of Negation The ubiquitous structure of opposition that necessitates secrecy and emergence of innovation though intuitively familiar has not received formal attention in economic models. In oppositional zero sum games, randomized strategies are Nash equilibrium (eg. Matching Pennies). Innovation does not feature. In the classic Santa Fe Artificial Life simulations (Ray;’s Tierra , Hillis, Langton et.al.1992) when some agents recognize that they are going to be hosts to parasitic behaviour, they start hiding their whereabouts and mutating /innovating so as not to be identified ! Recent research on RNA virus (see, Solé et. al. 2001) has likewise identified a ‘phenotype for mutation’, viz. a behavioural or strategic response favouring novelty, which is a far cry from the notion of random mutation. Quite simply, one mutates as a computationally rational strategy to escape from the enemy. Remarkably, despite the deep mathematical foundations of CAS on the ubiquitous structure of opposition formalized in the Liar and the capacity for self-referential calculations by agents of hostile behaviour of other agents, systems capable of adaptive novelty are commonplace and by and large only involve the intuitively familiar notion of the need to evade hostile agents. Markose (2003) refers to this as the Red Queen dynamics that exists among coevolving species. In Ray’s classic artificial life simulation called Tierra, Ray (1992), when some agents perceive that others are parasitic on them, they start hiding their whereabouts and also mutate to evade the parasite.   (Note, this is not obvious that this is the structure of the game that will give innovative structure changing dynamics. Peter Albin in his book attempted to get it out of a Prisoner’s Dilemma Structure). MATHEMATICALLY, TO MODEL THIS WE NEED INCOMPLETENESS AND UNLIKE EXTANT GAME THEORY WHERE ACTION SET IS FIXED AND GIVEN GEORGE SOROS USED ‘LIAR STRATEGY ‘ TO CLEAN OUR BOE IN 1992

  10. CANTOR DIAGONALIZATION LEMMA f¬ • 0 1 2 3 … m … • 1 0 1 0 1 0 … • 0 1 0 1 0 0 … • 0 1 0 1 0 … • 0 1 0 1 0 … • 0 1 0 1 0 0 1 … W0 W1 W2 W3 … Wm Set D D ¬ = {0 0 0 1 … 1 } D ¬ E { WN for any N}

  11. 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.

  12. Red Queen : To maintain Status Quo in Market Shares firms innovate continuously (see Markose 2005, 4 oligopolistic firms competing in margarine market from 1975,1995)

  13. Van Valen (1973) Introduced the Notion of the Arms Race • Lewis Carol has the following passage in his book Alice Through the Looking Glass which has lend itself to the so called Red Queen principle : “Well in our country ” said Alice, still panting a little “you’d generally get to somewhere else if you ran very fast for a long time as we been doing.” “A slow kind of country!” said the Red Queen. “Now here, you see, it takes all the running you can do, to be in the same place”.

  14. SECTION 2 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.

  15. 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.

  16. . 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, Cantor’s 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 ÎC~ as we have assumed C~ = Wy . But by the definition of C~ in (3.b) if y Î Wy , then y ÎC 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~ .

  17. Definition 5: A creative set Q is a recursively enumerable set whose compliment, Q~, is a productiveset. The set Q~ is productive if there exists a recursively enumerable set Wx disjoint from Q (viz. WxÌ 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).

  18. 2.2 REGULATORY ARBITRAGE/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 l=0,1,2,.....,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.

  19. LIKE CHESS NOTATION: META ANALYSIS OF GAMEAll meta-information with regard to 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.

  20. 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 ELEMENTSs(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.

  21. Second Recursion Theorem: Fixed Point ResultX0fs(0,0) fs (0,1) fs (0,2) fs (0,3) .... fs(0,y) .... X1fs(1,0) fs (1,1) fs (1,2) fs (1,3) .... fs(1,y) .... X2fs(2,0) fs (2,1) fs (2,2) fs (1,3) .... fs(2,y) ..... ..Xxfs(x,0) fs (x,1) fs (x,2) fs (x,3) .... fs(x,x ) ..... f' Xmff(s(0,0)) ff(s(1,1))ff(s(2,2))ff(s(3,3)) ... ff(s(m,m)) Xmff(s(0,0)) ff(s(1,1))ff(s(2,2))ff(s(3,3)) ... ff(s(m,m)) = fs(m,m)

  22. 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:

  23. 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.

  24. 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.

  25. 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 setsbi ( 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 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 ! = fm , m -A, i  (p,g). (7.b)

  26. Anderlini and Sabourian (1995, p.1351) have noted from Holland (1975), that heterogeneity in forms do not arise primarily by random mutation but by algorithmic recombinations that operate on existing patterns. Preconceptions from traditional game theory such as the ‘givenness’ of actions sets prevent Anderlini and Sabourian(1995) from positing that players, equipped with the wherewithal for algorithmic recombinations of existing actions, do indeed innovate from strategic necessity rather than by random mutation. The innovation per se is emergent phenomena, but the strategic necessity for it is fully deducible.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.

  27. THE STRUCTURE OF OPPOSITION: THE LIAR STRATEGYFor any state s when the rule a applies,THE LIAR STRATEGY fp¬ : For all s when policy rule a does not apply, fp¬ = 0 . (14.b)Implications of the Liar Strategy

  28. Proposition 3: The outcome of the game at this out of equilibrium points(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.

  29. 3.3 TheNon-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. This underpins Ray’s Tierra simulation when agents recognize the need for secrecy. 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

  30. 3.4 Surprise Nash Equilibria and The Productive Function g’s Nash equilibrium strategy bgEwith 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).

  31. Likewise for player p, fp! implements an innovation in (16.b) and bpE ! is the g.n of the surprise strategy function. Thus,bpE(fps (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. Thus, it is clear from (16.a,b) that what we call surprise strategies implement institutional innovations that signal structural breaks in the system.

  32. Theorem 5The 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.

  33. ARMS RACE IN SURPRISES/INNOVATIONS Bp+c b0¬ b1¬ …. bn-1¬ g.n (fp¬(σn))= bn¬ Wσn Wσn+1 g.n: Godel Number

  34. LEMMA 1For any function kwhich is total computable then k(x)  ßp+(k(x)) fx (k(x))  or k(x)  Wx = Dom fx . (A.1) However, if Wx is disjoint from ßp+ such that Wx ßp+c asfx (x)  , then we must have k(x) ßp+c - Wx . Assume the opposite viz. k(x)  Wx = Dom fx , then by the L.H.S of (A.1) as k is total , k(x)  ßp+ and is enumerable with fx (x) . The latter contradicts our assumption that Wx ßp+c asfx (x) . Such a k (x) which cannot be in the recursively enumerable sets ßp+ and Wx is the productive function of the set ßp+c .

  35. CONCLUDING REMARKS INNOVATION FAR FROM BEING A RANDOM OUTCOME, AS IS POPULARLY HELD, IS THE RESULT PRIMARILY OF COMPUTATONAL INTELLIGENCEWolfram (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 INSIGHT FROM HILLIS/LANGTON ET AL (1992) ARTIFICIAL LIFE SIMULATIONS IS THAT COEVOLVING RIVALROUS SPECIES ENCOURAGES INNOVATION AND ESCAPE FROM ENTRAPMENT AT LOCAL OPTIMA. 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.

  36. 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 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.

More Related