Emergence of two-phase Emergence of two-phase behavior in markets behavior in markets through interaction and through interaction and

learning learning in agents with bounded in agents with bounded

rationality rationality Sitabhra SinhaSitabhra Sinha

The Institute of Mathematical Sciences, Chennai, The Institute of Mathematical Sciences, Chennai, IndiaIndia

in collaboration with:in collaboration with:

S. RaghavendraS. RaghavendraMadras School of Economics, Chennai, IndiaMadras School of Economics, Chennai, India

Market Behavior : The Market Behavior : The Problem of Collective Problem of Collective

DecisionDecision Process of emergence of collective decisionProcess 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 but constrained by limited information and

having heterogeneous beliefs. having heterogeneous beliefs.

Example: Example:

Movie popularity.Movie popularity. Movie rankings Movie rankings according to votes by according to votes by IMDB users.IMDB users.

Collective Decision: A Naive Collective Decision: A Naive ApproachApproach

Each agent chooses randomly - Each agent chooses randomly - independent of all other agents.independent of all other agents.

Collective decision: sum of all individual Collective decision: sum of all individual choices. choices.

Example: YES/NO Example: YES/NO voting on an issuevoting on an issue For binary choiceFor binary choice

Individual agent: S = Individual agent: S = 0 or 1 0 or 1

Collective decision: M Collective decision: M = = ΣΣ S S Result: Normal Result: Normal distribution.distribution.

NO YES

0 % Collective Decision M 100%

But…But… Prevalence of bimodal distributions Prevalence of bimodal distributions

across social domains:across social domains:MoviesMovies

ElectionsElections

Financial MarketsFinancial Markets

Plerou, Gopikrishna, Stanley (2003)Plerou, Gopikrishna, Stanley (2003)

Collective Choice: Collective Choice: Interaction among AgentsInteraction among Agents

Modeling social phenomena : Emergence of Modeling social phenomena : Emergence of collective properties from agent-level collective properties from agent-level interactions.interactions.

Approach : Agent Interaction Dynamics Approach : Agent Interaction Dynamics Assumption: Bounded Rationality of Agents Assumption: Bounded Rationality of Agents

Limited perception: information about choice Limited perception: information about choice behavior of the entire system is limited to agent’s behavior of the entire system is limited to agent’s immediate neighborhood.immediate neighborhood.

Perfect rationality: Perfect rationality:

Neighborhood ≡ entire system → complete information.Neighborhood ≡ entire system → complete information.

The agents quickly synchronize their decisions. The agents quickly synchronize their decisions.

BackgroundBackground Weisbuch-Stauffer Binary Choice Model Weisbuch-Stauffer Binary Choice Model Agents interact with their ‘social neighbors’ Agents interact with their ‘social neighbors’

[e.g., in square lattice with 4 nearest [e.g., in square lattice with 4 nearest neighbors] …neighbors] …

……and their own belief. and their own belief. Belief changes over time as a function of Belief changes over time as a function of

previous decisions. previous decisions. Result: Result:

Very small connected groups of similar choice Very small connected groups of similar choice behavior. behavior.

On average, equal number of agents with opposite On average, equal number of agents with opposite choice preferences. choice preferences.

Physica A 323 (2003)Physica A 323 (2003)

100 x 100 lattice of agents in the Weisbuch-Stauffer model. 100 x 100 lattice of agents in the Weisbuch-Stauffer model.

No long-range order : Unimodal distributionNo long-range order : Unimodal distribution

So what’s missing ?So what’s missing ? 2 factors affect the evolution of an 2 factors affect the evolution of an

agent’s belief agent’s belief Adaptation (to previous choice): Adaptation (to previous choice): Belief increases on making a Belief increases on making a

positive choice and decreases on positive choice and decreases on making a negative choicemaking a negative choice

Global Feedback (by learning): Global Feedback (by learning): The agent will also be affected by The agent will also be affected by

how her previous choice accorded how her previous choice accorded with the collective choice (M).with the collective choice (M).

Influence of mass media ?Influence of mass media ?

The Model:The Model:‘Adaptive Field’ Ising 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 iBelief dynamics of the ithth agent at time t: agent at time t:

wherewhere is the collective is the collective decisiondecision μμ: Adaptation timescale: Adaptation timescale

λλ: Global feedback timescale: Global feedback timescale

Choice dynamics of the ith agent at time t:Choice dynamics of the ith agent at time t:

ResultsResults

Long-range order for Long-range order for λλ > 0 > 0

Initial state of the S field: 1000 × 1000 agentsInitial state of the S field: 1000 × 1000 agents

λλ = 0: No long-range order = 0: No long-range orderμμ =0.1 =0.1

N = 1000, T = 10000 N = 1000, T = 10000 itrnsitrnsSquare Lattice (4 Square Lattice (4 neighbors)neighbors)

μμ =0.1 =0.1 λλ > 0: clustering > 0: clusteringλλ = 0.05 = 0.05

N = 1000, T = 200 N = 1000, T = 200 itrnsitrnsSquare Lattice (4 Square Lattice (4 neighbors)neighbors)

ResultsResults

Long-range order for Long-range order for λλ > 0 > 0 Self-organized pattern formation Self-organized pattern formation

μμ =0.1 =0.1

λλ = 0.05 = 0.05

Ordered patterns emerge Ordered patterns emerge asymptoticallyasymptotically

ResultsResults

Long-range order for Long-range order for λλ > 0 > 0 Self-organized pattern formation Self-organized pattern formation

Multiple ordered domainsMultiple ordered domains Behavior of agents belonging to each Behavior of agents belonging to each

such domain is highly correlated – such domain is highly correlated – Distinct ‘cultural groups’ (Axelrod). Distinct ‘cultural groups’ (Axelrod). These domains eventually cover the These domains eventually cover the

entire system. [dislocation lines at the entire system. [dislocation lines at the boundary of two domains] boundary of two domains]

μμ =0.1 =0.1

λλ = uniform distribution [0,0.1] = uniform distribution [0,0.1]

Pattern formation even for Pattern formation even for randomly distributed randomly distributed λλ

Pattern formation in higher Pattern formation in higher dimensionsdimensions

μμ =0.1 =0.1

λλ = 0.05 = 0.053-D3-D

100 × 100 100 × 100 ×× 100 : 50000 iterations100 : 50000 iterations

ResultsResults

Long-range order for Long-range order for λλ > 0 > 0 Self-organized pattern formation Self-organized pattern formation

Multiple ordered domainsMultiple ordered domains Behavior of agents belonging to each such Behavior of agents belonging to each such

domain is highly correlated – domain is highly correlated – Distinct ‘cultural groups’ (Axelrod). Distinct ‘cultural groups’ (Axelrod). These domains eventually cover the entire These domains eventually cover the entire

system. [dislocation lines at the boundary system. [dislocation lines at the boundary of two domains] of two domains]

Phase transitionPhase transition Unimodal to bimodal distribution as Unimodal to bimodal distribution as λλ

increases.increases.

Behavior of collective decision M with increasing Behavior of collective decision M with increasing λλ

μμ =0.1 =0.1λ=0.0 λ=0.05

λ=0.1 λ=0.2

• As As λλ increases the system gets locked into either positive or increases the system gets locked into either positive or negative Mnegative M• Reminiscent of lock-in due to positive feedbacks in economies Reminiscent of lock-in due to positive feedbacks in economies (Arthur 1989). (Arthur 1989).

Phase transition with increasing Phase transition with increasing λλ

OK… but does it explain OK… but does it explain reality ?reality ?

Rank distribution:Rank distribution:Compare real data with modelCompare real data with model

US Movie Opening Gross

Model

Model: randomly distributed λ

OutlookOutlook Two-phase behavior of financial marketsTwo-phase behavior of financial markets Efficiency of marketing strategies:Efficiency of marketing strategies: Mass media campaign blitz vs targeted distribution of free Mass media campaign blitz vs targeted distribution of free

sample sample

The Mayhew Effect: The Mayhew Effect: Bimodality in electoral behaviorBimodality 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 But society as a whole gets trapped into non-cooperative mode and vice versa mode and vice versa

How does a paper become a "citation How does a paper become a "citation classic" ? classic" ?

S. Redner, "How popular is your paper?", E P J B 4 (1998) S. Redner, "How popular is your paper?", E P J B 4 (1998) 131. The role of citation indices in making a paper a 131. The role of citation indices in making a paper a citation classic. citation classic.