towards an economic theory of meaning and language gábor fáth research institute for solid state...

Towards an economic theory of

meaning and language

Gábor FáthResearch Institute for Solid State Physics and Optics

Budapest, Hungary

in collaboration with Miklos Sarvary - INSEAD, Fontainebleau, France

Agenda

• Saussurean language game

• Meaning formation in economic decisions

• Optimal concepts (meanings) for a single agent

• Language as a social process: co-evolution of concepts

• Spontaneous emergence of language

Saussurean language game

M A Nowak & N L Komarova, Trends Cogn. Sci. 5, 288 (2001)

Assuming that communication is beneficialcoherent language can emerge by rules of evolution.

What if meanings are not pre-existing?

Based on F. de Saussure 1916

Meanings are not well-defined on the social level

They can vary from agent to agent:

Is this shirt „trendy”?

How about eating „dogs”?

Personal tastes/preferences/cultural background modify meaning!Dispersion of meaning is especially large for abstract concept.

Trade-off: Concepts should serve

1, personal decision making (individual meaning)

2, communication (collective meaning)

agent jagent i

3/10 9/10

7/10 0/10

Economic decision problem

Discrete choice problem

Alternatives to choose from:

Payoff (profit) function:

Ordering: Best choice =

Valuation problem

Estimating under bounded rationality (complexity) is a problem

Exact payoff: under perfect rationality

Estimated payoff: under bounded rationality

using the agent’s mental representation

(simplified model of reality)

valuation error

Valuation accuracy / utility

Measures the quality of the agent’s mental representation

(the extent of bounded rationality)

decision contexts

exact payoff

approximate payoff

In the case of language: utility = valuation accuracy

average over alternatives

Mental representation

• Concepts are coarse-grained degrees of freedom.

• Multi-level hierarchy of concepts

• Lowest (perceptual) layer is common for everybody

• Highest (payoff) layer is preference dependent (agent heterogeneity)

• Simplest model is linear with one concept layer

• K<<D,X dimension reduction

mental weights

concept vectors

„Human mind is a feature detector. It only perceives the part of reality which it has a concept for.”

attributes of decision alternative

approximate valuations

Meaning - Language

Meaning of concept = The role it plays in the mental rep. hierarchy

Language = The collection of meanings

Valuation utility

Assumptions:

1,

2, i.e., concepts are independent

3, are fast variables

For the given mental rep.:

Maximization for gives:

Now the accuracy is a function of only:

Valuation utility

fixed by subjective reality

trace over concepts

World matrix:

fixed by subjective reality

Decision contexts = INDividual contexts + SOCial contexts

Language as a social process

We have seen

but?

„Meanings are deformed by social interactions. Language gets determined in a social process.”

• Assume a (Saussurean) matching between concepts of agents i and j• j has direct observation of reality along j’s concepts • i uses j’s concept scores and i’s mental weights in valuation

Social interaction - COM

If benefit is only on i’s side:

If benefit is symmetric:

Communication

agent i agent j

• i benefits from predicting j’s valuation• j-related contexts with j-related reality, i observes:

Social interaction - TOM

i’s benefit:

This is symmetric

TOM (Theory Of Mind)

Mean field

Fully connected, uniform social network

Explicitly:

COM-AS:

COM-S:

TOM:

SPL:+ constraint:

Optimal concepts for a single agent

Adding the constraint as a Lagrange multiplicator:

Varying with respect to yields:

The optimal concepts span the K-dimensional PCA subspace of the world matrix W.

Practically any learning mechanism finds this solution….

Interacting agents:Social dynamics / learning

Asynchronous update of concepts depending on valuation/prediction success:

• Continuous local optimization

Gradient dynamics

Global optimization (e.g., Best Response) is inadequate due to complexity

• Discrete relabeling of concepts to handle the Saussurean matching problem

REGA dynamics(Rematching Enabled Gradient Adjustment)

REGA equilibria

Easy to prove existence if interaction utility is symmetric:

Game has a potential V:

argmax(V) is a dynamically stable equilibrium (local, multi-agent stability)

It is also a Nash equilibrium (global, single-agent stability)

Existence can also be proven for the non-symmetric COM-AS version

There may be many equilibria!

Dynamic equilibrium selectionBifurcations, phase transition

0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

social coupling g

con

cep

t co

he

ren

ce

concept 1

concept 2

concept 3

Can language (coherent meaning) appear spontaneously

in a heterogeneous population?

Assume unbiased random preferences:

Wi are Wishart distributed random matrices

Spontaneous emergence

For all model versions in equilibrium:

g < gc: disordered g > gc1, gc2 … ordered

Spontaneous ordering in aseries of 1st order transitions

COM-AS modelI=120 D=X=10 K=3

gc1 gc2 gc3

Analytic results for TOM

Disordered solution loses stability at gc

gc 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

TOM phase diagram

Cultural explosion ~50,000 years ago ?

Agent intelligence K/D

Str

engt

h of

soc

ial i

nter

actio

ns g

Unbiased random population

DisorderedIndividual meaningsNo Language

OrderedCollective meaningsCoherent Language

Summary

• Concepts are coarse-grained degrees of freedom

• Meaning manifests itself in (economic) decision making

• Meaning is defined by the couplings of the hierarchical mental representation

• Utility for language is valuation/prediction accuracy

• Optimal language for a single agent is a PCA problem

• Language gets determined in a social process

• Co-evolution of meanings under COM and TOM interactions

• Rematching Enabled Gradient Adjustment (REGA) dynamics

• Spontaneous emergence of collective meaning in random population

• Cultural explosion 50,000 years ago as a phase transition

• G. Fath and M. Sarvary, A renormalization group theory of cultural evolution Physica A 348: 611-629, 2005

• G. Fath and M. Sarvary, An economic theory of language Working paper, 2005 (downloadable from www.szfki.hu/~fath)