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

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

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

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

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

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

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

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

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

Results Long-range order for λ > 0 Long-range order for λ > 0

Initial state of the S field: 1000 × 1000 agents

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

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

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

μ =0.1 λ = 0.05 Ordered patterns emerge asymptotically

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

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

Pattern formation in higher dimensions μ =0.1 λ = D 100 × 100 × 100 : iterations

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

Behavior of collective decision M with increasing λ μ =0.1 λ=0.0λ=0.05 λ=0.1λ=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). Reminiscent of lock-in due to positive feedbacks in economies (Arthur 1989).

Phase transition with increasing λ

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

Outlook Two-phase behavior of financial markets Two-phase behavior of financial markets Efficiency of marketing strategies: Efficiency of marketing strategies: Mass media campaign blitz vs targeted distribution of free sample Mass media campaign blitz vs targeted distribution of free sample The Mayhew Effect: Bimodality in electoral behavior The Mayhew Effect: Bimodality in electoral behavior Evolution of co-operation and defection: Evolution of co-operation and defection: Each individual is rational and cooperates some of the time; Each individual is rational and cooperates some of the time; But society as a whole gets trapped into non-cooperative mode and vice versa But society as a whole gets trapped into non-cooperative mode and vice versa How does a paper become a "citation classic" ? 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. 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.

One-dimensional lattice: Phase transition with increasing λ

The Mayhew Effect: Vanishing marginals in US Congressional elections David Mayhew (1974) : Apparent decline in electoral competition for US Congressional seats ( )  Increase in the proportion of ‘safe seats’  Attributed to incumbency advantage …has prompted a wave of new electoral laws - from campaign finance regulation to term limits. The bimodal distribution of percentage of votes polled by Democratic Party candidates in US federal elections.

Challenged by Garand & Gross (1984) both incumbent and nonincumbent winners have significantly increased their share of the two-party vote. Advantage Winner’s Advantage

State elections do not show vanishing marginals … … yet the incumbency advantage is similar to federal elections (Ansolabehere and Snyder, 2002) Clear Unimodal Distribution

Note that the two types of distributions (i.e., for state and federal elections) correspond to λ = 0 and λ > 0 (respectively) in our model. This implies that the influence of global feedback (in the form of mass media and opinion polls) play a very significant role in determining voting behavior in federal elections. This is often termed as ‘bandwagon effect’ where the information about majority opinion, widely dispersed in the community, causes people to alter their opinion to accord with the majority view. This conclusion is supported by:  Goidel & Shields (1994): Individuals have become more reliant on media coverage for individual-level voting cues.  McAllister & Studlar (1991): Similar conclusion for British general elections during