Emergence of Collective Decision Behavior in Bounded Rational Agents: Modeling Two-Phase Market Dynamics

1. Emergence of two-phase behavior in markets through interaction and learning in agents with bounded rationality Sitabhra Sinha The Institute of Mathematical Sciences, Chennai, India in collaboration with: S. Raghavendra Madras School of Economics, Chennai, India

2. Market Behavior : The Problem of Collective Decision • Process of emergence of collective decision • in a society of agents free to choose…. • but constrained by limited information and having heterogeneous beliefs. • Example: Movie popularity. • Movie rankings according to votes by IMDB users.

3. Collective Decision: A Naive Approach • Each agent chooses randomly - independent of all other agents. • Collective decision: sum of all individual choices. • Example: YES/NO voting on an issue • For binary choice Individual agent: S = 0 or 1 Collective decision: M = Σ S • Result: Normal distribution. NO YES 0 % Collective Decision M 100%

4. But… • Prevalence of bimodal distributions across social domains: Movies Elections Financial Markets Plerou, Gopikrishna, Stanley (2003)

5. Collective Choice: Interaction among Agents • Modeling social phenomena : Emergence of collective properties from agent-level interactions. • Approach : Agent Interaction Dynamics • Assumption: Bounded Rationality of Agents • Limited perception: information about choice behavior of the entire system is limited to agent’s immediate neighborhood. • Perfect rationality: Neighborhood ≡ entire system → complete information. The agents quickly synchronize their decisions.

6. Background Physica A 323 (2003) • Weisbuch-Stauffer Binary Choice Model • Agents interact with their ‘social neighbors’ [e.g., in square lattice with 4 nearest neighbors] … • …and their own belief. • Belief changes over time as a function of previous decisions. • Result: • Very small connected groups of similar choice behavior. • On average, equal number of agents with opposite choice preferences.

7. 100 x 100 lattice of agents in the Weisbuch-Stauffer model. No long-range order : Unimodal distribution

8. So what’s missing ? • 2 factors affect the evolution of an agent’s belief • Adaptation (to previous choice): Belief increases on making a positive choice and decreases on making a negative choice • Global Feedback (by learning): The agent will also be affected by how her previous choice accorded with the collective choice (M). • Influence of mass media ?

9. The Model:‘Adaptive Field’ Ising Model • Binary choice :2 possible choice states (S = ± 1). • Choice dynamics of the ith agent at time t: • Belief dynamics of the ith agent at time t: is the collective decision where • μ: Adaptation timescale • λ: Global feedback timescale

10. Results • Long-range order for λ > 0

11. Initial state of the S field: 1000 × 1000 agents

12. μ =0.1 λ = 0: No long-range order N = 1000, T = 10000 itrns Square Lattice (4 neighbors)

13. μ =0.1 λ > 0: clustering λ = 0.05 N = 1000, T = 200 itrns Square Lattice (4 neighbors)

14. Results • Long-range order for λ > 0 • Self-organized pattern formation

15. μ =0.1 Ordered patterns emerge asymptotically λ = 0.05

16. Results • Long-range order for λ > 0 • Self-organized pattern formation • Multiple ordered domains • Behavior of agents belonging to each such domain is highly correlated – • Distinct ‘cultural groups’ (Axelrod). • These domains eventually cover the entire system. [dislocation lines at the boundary of two domains]

17. μ =0.1 Pattern formation even for randomly distributed λ λ = uniform distribution [0,0.1]

18. μ =0.1 Pattern formation in higher dimensions λ = 0.05 3-D 100 × 100 ×100 : 50000 iterations

19. Results • Long-range order for λ > 0 • Self-organized pattern formation • Multiple ordered domains • Behavior of agents belonging to each such domain is highly correlated – • Distinct ‘cultural groups’ (Axelrod). • These domains eventually cover the entire system. [dislocation lines at the boundary of two domains] • Phase transition • Unimodal to bimodal distribution as λ increases.

20. Behavior of collective decision M with increasing λ λ=0.0 λ=0.05 μ =0.1 λ=0.1 λ=0.2 • As λ increases the system gets locked into either positive or negative M • Reminiscent of lock-in due to positive feedbacks in economies (Arthur 1989).

21. Phase transition with increasing λ

22. OK… but does it explain reality ? Rank distribution: Compare real data with model US Movie Opening Gross Model: randomly distributed λ Model

23. Outlook • Two-phase behavior of financial markets • Efficiency of marketing strategies: Mass media campaign blitz vs targeted distribution of free sample • The Mayhew Effect: Bimodality in electoral behavior • Evolution of co-operation and defection: Each individual is rational and cooperates some of the time; But society as a whole gets trapped into non-cooperative mode and vice versa • How does a paper become a "citation classic" ? S. Redner, "How popular is your paper?", E P J B 4 (1998) 131. The role of citation indices in making a paper a citation classic.