Towards an economic theory of meaning and language Gábor Fáth Research Institute for Solid State Physics and Optics Budapest, Hungary in collaboration with Miklos Sarvary - INSEAD, Fontainebleau, France

Language research Language is studied in many disciplines: Cognitive science, Linguistics Economics Rational game theory (signaling, cheap talk) Biology Evolutionary game theory Computer science Artificial Intelligence Physics Complexity: networks, heterogeneity, dynamics, transitions

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 beneficial coherent 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 iagent 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 selection Bifurcations, phase transition

Can language (coherent meaning) appear spontaneously in a heterogeneous population? Assume unbiased random preferences: W i are Wishart distributed random matrices Spontaneous emergence For all model versions in equilibrium: g < g c : disordered g > g c1, g c2 … ordered Spontaneous ordering in a series of 1st order transitions COM-AS model I=120 D=X=10 K=3 g c1 g c2 g c3

Analytic results for TOM Disordered solution loses stability at g c g c 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 Strength of social interactions g Unbiased random population Disordered Individual meanings No Language Ordered Collective meanings Coherent 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: , 2005 G. Fath and M. Sarvary, An economic theory of language Working paper, 2005 (downloadable from