An RG theory ofcultural evolution

Gábor FáthHungarian Academy of Sciences

Budapest, Hungary

in collaboration with

Miklos Sarvary - INSEAD, Fontainebleau, France

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

Cultural identity as a clustering problem

unique individuals

humanity

similarity threshold

culturessubcultures

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)

Outline

1. Mental representation:

RG approach for bounded rationality

2. Heterogeneous interacting agents:

Spontaneous ordering of mental reps.

Mental representation

Objective (physical) reality Representation in the mind

Bounded rationality = representation error

Mental representation (2)

COMPLEXITY

AC

CU

RA

CY

Optimal representationfor given complexity

Sub-optimal representations

- - - - - - - - - - B o u n d e d r a t i o n a l i t y - - - - - - - - Superrationality

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

Model of bounded rationality

Choice among decision alternatives

Mental model on concepts+

Concepts as feature detectors

Microscopic variables describing decision alternatives

Behavior

Perception

Mental representationin MIND

Bounded rationality in chess

xa x

a

Objectivepayoff

(super rationality)

Estimatedpayoff

(bounded rationality)

How to evaluate a move ?

Microscopicattributes

of alternative(board configuration)

Concepts

Value of alternative in

decision context(value of move)

Weak pawn

Pinned piece

Positionaladvantage

decisioncontext

(adversary) possiblemove

payoffof move

decisioncontext-2

(adversary-2)

decisioncontext-3

(adversary-3)

Bounded rationality in general

xa x

a

Objectivepayoff

(super rationality)

Estimatedpayoff

(bounded rationality)

How to evaluate a decision alternative ?

Microscopic attributesof alternative a = {a1,…,aD}

ConceptsConcept ConceptConcept

decisionalternative

payoffof alternative

payoff of alternative in decisioncontext - 1

payoff of alternative in decisioncontext - X

1 D

1 KFixed

Fixed

Fixed

Linear

Linear

Linear world approximation

Microscopic attributesof alternative a = {a1,…,aD}

ConceptsConcept ConceptConcept

payoff of alternative in decisioncontext - 1

payoff of alternative in decisioncontext - X

1 D

1 K

Linear

Linear

context dependentpreference vectors

mental weights

concept vectors

Representation error

Assume that attributes are -correlated:

How to choose the mental representation (v(x) and d)

to minimize the error?

(we assumeK andd(x) fixed, ||=1)

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

Principal Component Analysis

Theorem (Principle Component Analysis)

Error is minimal if are the K most significant eigenvectors of W.

This is the PCA problem:

Cultural profile

Def:

Agent’s cultural profile

=

PCA-defined K dim -subspace of D dim

World

Determines what/how the agent can understandpredict communicatemeanbehave…

Connection with DMRG

xd

d,x

xd

,x d

xd

d,x

22

x xd d

d,x

x xd d 'dd ' dd '

x

d x

v x with d

v d x

Trunc. Error v Re p. Error

Density matrix World matrix

Superblock target state

Renormalized superblock ground state

Block {d} Environment {x}

Superblock

Renormalization

Heterogeneous agents

Heterogeneous preferences

World matrices differ Wi ≠ Wj

Cultural profiles (-subspaces) differ

Tensor order parameter (à la de Gennes):

Eigenvalue structure measures cultural ordering!

Perfect order:

Perfect disorder:

Order parameter

How to measure cultural (subspace) coherence?

Interactions

Understanding/predicting other agents is advantageous

Reality = Individual + Social

strength of social interactions

Sj projects onto agent j’s -subspace

Mean field:

Agent’s utility:

DynamicsBest response dynamics

co-adaptation to natural & social environment

0i iW (t) W h O(t 1)

i (t)

O(t)

PCA

Population average

RG

iter

atio

n cy

cle

RG fixed point: Nash equilibrium (no incentive to deviate)

Phase transition

Fixed point properties for

heterogeneous agents with unbiased random preferences

Wi0 = Wishart distributed

h < hc: disorderedh > hc: ordered

Spontaneous ordering in1st order transition

Analytic results

Disordered solution loses stability at hc

hc can be calculated using

1st order perturbation theory and RMT (Wishart)

For K << D=Xcomplexity of world

capacity of agents

critical social coupling strength

Phase diagram

Agent intelligence K

Str

engt

h of

soc

ial i

nter

actio

ns h

Unbiased random population D = fixed

DisorderedNo Culture No Language

OrderedCoherent Culture, Language

Cultural explosion ~50,000 years ago

Summary

Culture: Ordering of mental representations (Concepts)

Bounded rationality: Mental rep. should be accurate and simpleRG agents: Sorting / truncating the degrees of freedom

IterativelyFixed points

Spontaneous ordering with jumpsas mental abilities improveas interactions strengthen

Archeological evidence: „Cultural explosion”

G. F

ath

and

M. S

arva

ry,

nlin

.AO

/031

2070