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)