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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: [email protected]
Kae Nemoto
Quantum Information Science
National Institute of Informatics, Japan
Email: [email protected]
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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
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 physicsRosen: “Correlated” Components in biology, rather than “uncorrelated” partsMattick: 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 networksProkaryotic 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)
= Ft (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 Dilemma Iterated 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
Px
Py
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 nth stage
Myopic Agent Optimization
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Myopic Agent Optimization
Myopic agents ( probability distributions) uncorrelated no additional constraints
Backwards Induction & Minimax
x1 y1
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
CooperationDefect
CKR: Common Knowledge of Rationality
Payoff Matrix
Py
Px
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 PNx,HN-1(1) = 1
=1 > 0 PN-1,x,HN-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 P1x(1) = 0, so Px cooperates
< 0 P1y(1) = 0, so Py 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: [email protected]
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)
Kae Nemoto
Email: [email protected]
URL: http://www.qis.ex.nii.ac.jp/knemoto.html
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