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An RG theory of cultural evolution Gábor Fáth Hungarian Academy of Sciences Budapest, Hungary in collaboration with Miklos Sarvary - INSEAD, Fontainebleau, France

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What is Culture? Sociology defines: „Culture is the sum of knowledge, beliefs, values, norms and behavioral patterns represented by a group of human beings and transmitted from one generation to the next.” mental representation shared evolving

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Cultural identity as a clustering problem unique individuals humanity similarity threshold culturessubcultures

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Two possible approaches How can cultural diversity emerge despite our fundamental biological similarities? How can cultural coherence emerge despite our fundamental biological/economic/ environmental differences? Axelrod’s theory of culture (1997)

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Outline 1.Mental representation: RG approach for bounded rationality 2.Heterogeneous interacting agents: Spontaneous ordering of mental reps.

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Mental representation Objective (physical) reality Representation in the mind Bounded rationality = representation error

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Mental representation (2) COMPLEXITY ACCURACY Optimal representation for given complexity Sub-optimal representations - - - - - - - - - - B o u n d e d r a t i o n a l i t y - - - - - - - - Super rationality

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Mental representation (3) COMPLEXITY UTILITY Agents maximize: UTILITY = ACCURACY – COMPLEXITY COST accuracy complexity cost

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Concepts Cognitive science: Human mind is a feature detector („pattern recognizer”, „filter”) We only perceive the part of reality which we have a concept for. e.g.: chess concepts

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Model of bounded rationality Choice among decision alternatives Mental model on concepts + Concepts as feature detectors Microscopic variables describing decision alternatives Behavior Perception Mental representation in MIND

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Bounded rationality in chess Objective payoff (super rationality) Estimated payoff (bounded rationality) How to evaluate a move ? Microscopic attributes of alternative (board configuration) Concepts Value of alternative in decision context (value of move) Weak pawn Pinned piece Positional advantage decision context (adversary) possible move payoff of move decision context-2 (adversary-2) decision context-3 (adversary-3)

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Bounded rationality in general Objective payoff (super rationality) Estimated payoff (bounded rationality) How to evaluate a decision alternative ? Microscopic attributes of alternative a = {a 1,…,a D } Concepts Concept decision alternative payoff of alternative payoff of alternative in decision context - 1 payoff of alternative in decision context - X 1D 1K Fixed Linear

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Linear world approximation Microscopic attributes of alternative a = {a 1,…,a D } Concepts Concept payoff of alternative in decision context - 1 payoff of alternative in decision context - X 1D 1K Linear context dependent preference vectors mental weights concept vectors

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Representation error Assume that attributes are -correlated: How to choose the mental representation (v (x) and d ) to minimize the error? (we assume K and d (x) fixed, | |=1)

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Representation error (2) Assuming v (x) (mental weights) are fast variables we get How to choose the concepts { , 2,…, K } to maximize utility? Def: Agent’s utility with and World matrix

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Principal Component Analysis Theorem (Principle Component Analysis) Error is minimal if are the K most significant eigenvectors of W. This is the PCA problem:

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Cultural profile Def: Agent’s cultural profile = PCA-defined K dim -subspace of D dim World Determines what/how the agent canunderstand predict communicate mean behave …

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Connection with DMRG Superblock target state Renormalized superblock ground state Block {d}Environment {x} Superblock Renormalization

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Heterogeneous agents Heterogeneous preferences World matrices differ W i ≠ W j Cultural profiles ( -subspaces) differ

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Tensor order parameter (à la de Gennes): Eigenvalue structure measures cultural ordering! Perfect order: Perfect disorder: Order parameter How to measure cultural (subspace) coherence?

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Interactions Understanding/predicting other agents is advantageous Reality = Individual + Social strength of social interactions S j projects onto agent j’s -subspace Mean field: Agent’s utility:

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Dynamics Best response dynamics co-adaptation to natural & social environment PCA Population average RG iteration cycle RG fixed point: Nash equilibrium (no incentive to deviate)

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Phase transition Fixed point properties for heterogeneous agents with unbiased random preferences W i 0 = Wishart distributed h < h c : disordered h > h c : ordered Spontaneous ordering in 1st order transition

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Analytic results Disordered solution loses stability at h c h c can be calculated using 1st order perturbation theory and RMT (Wishart) For K << D=X complexity of world capacity of agents critical social coupling strength

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Phase diagram Agent intelligence K Strength of social interactions h Unbiased random population D = fixed Disordered No Culture No Language Ordered Coherent Culture, Language Cultural explosion ~50,000 years ago

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Summary Culture: Ordering of mental representations (Concepts) Bounded rationality: Mental rep. should be accurate and simple RG agents: Sorting / truncating the degrees of freedom Iteratively Fixed points Spontaneous ordering with jumps as mental abilities improve as interactions strengthen Archeological evidence: „Cultural explosion” G. Fath and M. Sarvary, nlin.AO/0312070

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