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Myopic and non-myopic agent optimization in game theory, economics, biology and artificial intelligence Michael J Gagen Institute of Molecular Bioscience University of Queensland Email: m.gagen@imb.uq.edu.au Kae Nemoto Quantum Information Science National Institute of Informatics, Japan Email: nemoto@nii.ac.jp

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Overview: Functional Optimization in Strategic Economics (and AI) Mathematics / Physics (minimize action) Formalized by von Neumann and Morgenstern, Theory of Games and Economic Behavior (1944)

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Overview: Functional Optimization in Strategic Economics (and AI) Strategic Economics (maximize expected payoff) Formalized by von Neumann and Morgenstern, Theory of Games and Economic Behavior (1944) Functionals: Fully general Not necessarily continuous Not necessarily differentiable Nb: Implicit Assumption of Continuity !!

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von Neumann’s “myopic” assumption Overview: Functional Optimization in Strategic Economics (and AI) Strategic Economics (maximize expected payoff) Evidence: von Neumann & Nash used fixed point theorems in probability simplex equivalent to a convex subset of a real vector space von Neumann and Morgenstern, Theory of Games and Economic Behavior (1944) J. F. Nash, Equilibrium points in n-person games. PNAS, 36(1):48–49 (1950)

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Non-myopic Optimization Correlations Constraints and forbidden regions Overview: Functional Optimization in Strategic Economics (and AI) No communications between players

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∞ correlations & ∞ different trees constraint sets Non-myopic Optimization Overview: Functional Optimization in Strategic Economics (and AI) Myopic “The” Game Tree lists All play options X Myopic One Constraint = One Tree “Myopic” Economics (= Physics)

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Myopic = Missing Information! Correlation = Information Chess: “Chunking” or pattern recognition in human chess play Experts: · Performance in speed chess doesn’t degrade much · Rapidly direct attention to good moves · Assess less than 100 board positions per move · Eye movements fixate only on important positions · Re-produce game positions after brief exposure better than novices, but random positions only as well as novices Learning Strategy = Learning information to help win game Nemoto: “It is not what they are doing, its what they are thinking!” What Information?

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Optimization and Correlations are Non-Commuting! Complex Systems Theory Emergence of Complexity via correlated signals higher order structure

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Optimization and Correlations are Non-Commuting! Life Sciences (Evolutionary Optimization) Selfish Gene Theory Mayr: Incompatibility between biology and physics Rosen: “Correlated” Components in biology, rather than “uncorrelated” parts Mattick: Biology informs information science 6 Gbit DNA program more complex than any human program, implicating RNA as correlating signals allowing multi-tasking and developmental control of complex organisms. Mattick: RNA signals in molecular networks Prokaryotic gene mRNA protein Eukaryotic gene mRNA & eRNA protein networking functions Hidden layer

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Optimization and Correlations are Non-Commuting! Economics Selfish independent agents: “homo economicus” Challenges: Japanese Development Economics, Toyota “Just-In-Time” Production System

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Optimization and Correlations are Non-Commuting! o = F(i) = F(t,d) = F t (d) {F(x,y,z), …,F(x,x,z),…} Functional Programming, Dataflow computation, re-write architectures, … o i 1 Player Evolving / Learning Machines (neural and molecular networks) endogenously exploit correlations to alter own decision tree, dynamics and optima

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Discrepancies: Myopic Agent Optimization and Observation Heuristic statistics Iterated Prisoner’s DilemmaIterated Ultimatum Game Chain Store Paradox (Incumbent never fights new market entrants)

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Sum-Over-Histories or Path Integral formulation Myopic Agent Optimization Normal FormStrategic Form PxPx PyPy von Neumann and Morgenstern (1944): All possible information = All possible move combinations for all histories and all futures ??

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Optimization Sum over all stages Probability of each path Payoff from each stage for each path Sum over all paths to n th stage Myopic Agent Optimization

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Myopic agents ( probability distributions) · uncorrelated · no additional constraints Backwards Induction & Minimax x1x1 y1y1 1-p p 0 ≤ p ≤ 1/2

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Non-Myopic Agent Optimization Fully general, notationally emphasized by: Optimization Sum over all correlation strategies Constraint set of each strategy Payoff for each path Sum over all paths given strategy Probability of each strategy Conditioned path probability

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Non-Myopic Agent Optimization in the Iterated Prisoner’s Dilemma In 1950 Melvin Dresher and Merrill Flood devised a game later called the Prisoner’s Dilemma Two prisoners are in separate cells and must decide to cooperate or defect Cooperation Defect CKR: Common Knowledge of Rationality Payoff Matrix P y P x C D C (2, 2) (0, 3) D (3,0) (1,1)

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Non-Myopic Agent Optimization in the Iterated Prisoner’s Dilemma Myopic agent assumption max

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Non-Myopic Agent Optimization in the Iterated Prisoner’s Dilemma Myopic agents: N max constraints = 0 > 0 P Nx,H N-1 (1) = 1 =1 > 0 P N-1,x,H N-2 (1) = 1 Simultaneous solution Backwards Induction myopic agents always defect

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Non-Myopic Agent Optimization in the Iterated Prisoner’s Dilemma Correlated Constraints: (deriving Tit For Tat) < 0 P 1x (1) = 0, so P x cooperates < 0 P 1y (1) = 0, so P y cooperates 2 max constraints

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Non-Myopic Agent Optimization in the Iterated Prisoner’s Dilemma Families of correlation constraints: k, j index Change of notation: “dot N” = N, “dot dot N” = 2N, “dot dot N -2 ” = 2N-2, etc Optimize via game theory techniques Many constrained equilibria involving cooperation Cooperation is rational in IPD

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Further Reading and Contacts Michael J Gagen Email: m.gagen@imb.uq.edu.au URL: http://research.imb.uq.edu.au/~m.gagen/ See: Cooperative equilibria in the finite iterated prisoner's dilemma, K. Nemoto and M. J. Gagen, EconPapers:wpawuwpga/0404001 (http://econpapers.hhs.se/paper/wpawuwpga/0404001.htm)0404001http://econpapers.hhs.se/paper/wpawuwpga/0404001.htm Kae Nemoto Email: nemoto@nii.ac.jp URL:http://www.qis.ex.nii.ac.jp/knemoto.html

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