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)