An RG theory of cultural evolution

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

2. mental representation shared evolving 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.” What is Culture?

3. Cultural identity as a clustering problem similarity threshold unique individuals subcultures cultures humanity

4. 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)

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

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

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

8. 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

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

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

11. Bounded rationality in general Linear Fixed Fixed Fixed Linear Objective payoff (super rationality) Estimated payoff (bounded rationality) How to evaluate a decision alternative ? decision alternative payoff of alternative payoff of alternative in decision context - 1 payoff of alternative in decision context - X Concept Concept Concept Concepts 1 K 1 D Microscopicattributes of alternative a = {a1,…,aD}

12. Linear world approximation Linear Linear context dependent preference vectors payoff of alternative in decision context - 1 payoff of alternative in decision context - X Concept Concept Concept Concepts 1 K 1 D mental weights Microscopicattributes of alternative a = {a1,…,aD} concept vectors

13. Representation error Assume that attributes are d-correlated: How to choose the mental representation (vm(x) and gmd) to minimize the error? (we assumeK andwd(x) fixed, |gm|=1)

14. Representation error (2) Assuming v(x) (mental weights) are fast variables we get with World matrix and Def:Agent’s utility How to choose the concepts {g1,g2,…,gK} to maximize utility?

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

16. Cultural profile Def: Agent’scultural profile = PCA-defined K dimg-subspaceof D dim World Determines what/how the agent can understand predict communicate mean behave …

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

18. Heterogeneous agents Heterogeneous preferences World matrices differ Wi ≠ Wj Cultural profiles (g-subspaces) differ

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

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

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

22. Phase transition Fixed point properties for heterogeneous agents with unbiased random preferences Wi0 = Wishart distributed h<hc: disordered h>hc: ordered Spontaneous ordering in 1st order transition

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

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

25. 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