Towards an economic theory of meaning and language

1. 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

2. 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

3. Saussurean language game Based on F. de Saussure 1916 Assuming that communication is beneficial coherent language can emerge by rules of evolution. What if meanings are not pre-existing? M A Nowak & N L Komarova, Trends Cogn. Sci. 5, 288 (2001)

4. 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 i agent j 3/10 9/10 7/10 0/10

5. 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

6. Valuation accuracy / utility In the case of language: utility = valuation accuracy average over alternatives decision contexts approximate payoff exact payoff Measures the quality of the agent’s mental representation (the extent of bounded rationality)

7. Mental representation „Human mind is a feature detector. It only perceives the part of reality which it has a concept for.” • 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 approximate valuations mental weights concept vectors attributes of decision alternative

8. Meaning - Language Meaning of concept  = The role it plays in the mental rep. hierarchy Language = The collection of meanings

9. Valuation utility For the given mental rep.: Assumptions: 1, 2, i.e., concepts are independent 3, are fast variables

10. Valuation utility Maximization for gives: Now the accuracy is a function of only: fixed by subjective reality trace over concepts World matrix: fixed by subjective reality

11. Language as a social process „Meanings are deformed by social interactions. Language gets determined in a social process.” Decision contexts = INDividual contexts + SOCial contexts We have seen but ?

12. Social interaction - COM Communication • 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 agent i agent j If benefit is only on i’s side: If benefit is symmetric:

13. Social interaction - TOM TOM (Theory Of Mind) • i benefits from predicting j’s valuation • j-related contexts with j-related reality, i observes: i’s benefit: This is symmetric

14. Explicitly: COM-AS: COM-S: TOM: SPL: + constraint: Mean field Fully connected, uniform social network

15. 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 matrixW. Practically any learning mechanism finds this solution….

16. 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)

17. There may be many equilibria! Dynamic equilibrium selection Bifurcations, phase transition 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

18. Spontaneous emergence Can language (coherent meaning) appear spontaneously in a heterogeneous population? Assume unbiased random preferences: Wi are Wishart distributed random matrices For all model versions in equilibrium: g < gc: disordered g > gc1, gc2 … ordered Spontaneous ordering in a series of 1st order transitions COM-AS model I=120 D=X=10 K=3 gc1 gc2 gc3

19. Analytic results for TOM Disordered solution loses stability at gc gc can be calculated using 1st order perturbationtheory and RMT (Wishart) For K<<D=X : complexity of world capacity of agents critical social coupling strength

20. Unbiased random population Ordered Collective meanings Coherent Language Strength of social interactions g Disordered Individual meanings No Language Agent intelligence K/D TOM phase diagram Cultural explosion ~50,000 years ago ?

21. 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 fromwww.szfki.hu/~fath)