Discovery and Diffusion Discovery and Diffusion of Knowledge in an of Knowledge in an Endogenous Social Endogenous Social

NetworkNetwork

Myong-Hun Chang and Joe Myong-Hun Chang and Joe HarringtonHarrington

Social Accumulation of Social Accumulation of KnowledgeKnowledge

►Essential elements of the processEssential elements of the process Generation of Ideas by IndividualsGeneration of Ideas by Individuals Adoption of Ideas by IndividualsAdoption of Ideas by Individuals Diffusion of Ideas through Social Diffusion of Ideas through Social

NetworksNetworks

SocialSocial

NetworkNetwork

DiffusionDiffusion

ProcessProcess

►Generation of KnowledgeGeneration of Knowledge Research: process of Research: process of searchingsearching for better for better

ways of doing things (finding better solution ways of doing things (finding better solution to a problem?)to a problem?)

►Decentralized Research Decentralized Research Parallel Search Parallel Search►How is this search carried out by an How is this search carried out by an

individual agent?individual agent? Innovation (individual learning)Innovation (individual learning)

►Production of “new” ideasProduction of “new” ideas

Imitation (social learning)Imitation (social learning)►Adoption of “existing” ideas of others (Adoption of “existing” ideas of others (

diffusion)diffusion)

►Underlying social system: A Underlying social system: A population of autonomous agents (= population of autonomous agents (= problem-solvers), each endowed with problem-solvers), each endowed with a a goalgoal unknown to her unknown to her ex anteex ante Goal: “Optimal” solution to the given Goal: “Optimal” solution to the given

problemproblem Goals can differ: Diversity in solution Goals can differ: Diversity in solution

due to diversity in local environmentsdue to diversity in local environments Goals can change: Local environments Goals can change: Local environments

may be subject to inter-temporal may be subject to inter-temporal fluctuations fluctuations Problems may change Problems may change

Decision-making by individual Decision-making by individual agentsagents

► Should we model them as being hyper-rational Should we model them as being hyper-rational with perfect foresight, etc.?with perfect foresight, etc.? No. The decision environment is too complex.No. The decision environment is too complex. Full rationality – too demandingFull rationality – too demanding

► Boundedly rational and engage in myopic Boundedly rational and engage in myopic searchsearch for the unknown goal (local optimum) for the unknown goal (local optimum) Adaptive and capable of learning from past Adaptive and capable of learning from past

experiencesexperiences Search alone (innovation) or search by learning Search alone (innovation) or search by learning

from others (imitation)from others (imitation)

Research QuestionsResearch Questions► Individual learning versus social learning via Individual learning versus social learning via

networknetwork How do individuals choose among the two How do individuals choose among the two

learning mechanisms?learning mechanisms?► Network as an outcome of interactive Network as an outcome of interactive

choices among individuals as to whom to choices among individuals as to whom to observe and whom to ignoreobserve and whom to ignore What are the determinants of their emergent What are the determinants of their emergent

structure?structure?► Performance at the individual and Performance at the individual and

community levelcommunity level How does the reliability of communication How does the reliability of communication

technology affect the performance?technology affect the performance? How does the innovativeness of the population How does the innovativeness of the population

affect the performance?affect the performance?

The ModelThe Model

►Social system: Social system: LL individualsindividuals►Each individual engages in an operation Each individual engages in an operation

with with HH separate tasksseparate tasks►For each task, there is a fixed number For each task, there is a fixed number

of possible methods that can be used to of possible methods that can be used to perform the task.perform the task. A given method is a string of A given method is a string of dd bits – 0’s bits – 0’s

and 1’sand 1’s 22dd possible methods per task possible methods per task

►An individual An individual ii in period in period tt characterized characterized by a binary vector of by a binary vector of HdHd dimensions: dimensions: zzii((tt))

►Example: Example: HH = 4 and = 4 and dd = 4= 4 ZZ11= 1101 0100 1111 1001= 1101 0100 1111 1001

ZZ22= 0101 1101 0011 1000= 0101 1101 0011 1000

►Distance between two vectorsDistance between two vectors Hamming distance: Hamming distance: DD((zz11, , zz22)) No. positions for which the corresponding No. positions for which the corresponding

bits differ bits differ DD((zz11, , zz22)) = 6 for the above = 6 for the above vectorsvectors

►There exists a goal vector (binary) of There exists a goal vector (binary) of HdHd dimensions for each agent in dimensions for each agent in tt: : ggii((tt)) Unique optimal solution to the problem Unique optimal solution to the problem

agent i is facingagent i is facing►Inter-agent diversityInter-agent diversity

It is possible that It is possible that ggii((tt))≠ ≠ ggjj((tt) ) for i for i ≠ j.≠ j.

►Inter-temporal variabilityInter-temporal variability It is possible that It is possible that ggii((tt))≠ ≠ ggii((t’t’) ) for t for t ≠ t’.≠ t’.

Individuals uninformed about Individuals uninformed about ggii((tt)) ex anteex ante, , but engage in “search” to get as close to it but engage in “search” to get as close to it as possible.as possible.

g3(t)

z1(t)

z2(t)

z3(t)

z4(t)

D(z1(t), g1(t))

g1(t)g2(t)

g4(t)

D(z2(t), g2(t))

D(z3(t), g3(t))

D(z4(t), g4(t))

►Period-Period-tt performance of agent performance of agent i i How close is agent How close is agent ii to his current to his current

optimum?optimum? ππii((tt) = ) = HdHd – – DD((zzii((tt), ), ggii((tt))))

►Period-Period-tt performance of the social performance of the social systemsystem Sum of all agent’s performances.Sum of all agent’s performances.

► Decision-Making Sequence (in Decision-Making Sequence (in tt) for an ) for an AgentAgent

Choose to Innovate

Choose to Imitate

Innovate

Idle

Idle

Observe

Observe j = 1

Observe j = 2

Observe j = 3

Observe j = L

Observe j ≠ i

i

qi(t)

1 - qi(t)

μiin

1 - μiin

μiim

1 - μiim

pi1(t)

pi2(t)

piL(t)

pij(t)

► If fail to generate an idea or fail to connect to If fail to generate an idea or fail to connect to the network, the network, zzii((tt+1) = +1) = zzii((tt).).

► Otherwise, there exists an idea, Otherwise, there exists an idea, zzii’(’(tt)), , proposed under innovation or imitation such proposed under innovation or imitation such that that zzii’(’(tt)) is adopted iff it gets is adopted iff it gets i i closer to closer to ggii((tt)).. Innovation: Randomly chosen method for a Innovation: Randomly chosen method for a

randomly chosen taskrandomly chosen task

ZZ11= 1101 0100 = 1101 0100 11111111 1001 1001Z’Z’11= 1101 0100 = 1101 0100 10111011 1001 1001

Imitation: Method used by another agent for a Imitation: Method used by another agent for a randomly chosen taskrandomly chosen task

ZZ11= 1101 0100 1111 1001= 1101 0100 1111 1001ZZ22= 0101 1101 = 0101 1101 00110011 1000 1000Z’Z’11= 1101 0100 = 1101 0100 00110011 1001 1001

Evolving ChoicesEvolving Choices

►Exogenous Probabilities: (Exogenous Probabilities: (μμiiinin, , μμii

imim))

►Endogenous Probabilities: (Endogenous Probabilities: (qqii((tt), {), {ppiijj((tt)})}j≠ij≠i))

►((qqii((tt), {), {ppiijj((tt)})}j≠ij≠i) evolve via ) evolve via reinforcement reinforcement

learninglearning A positive outcome realized from a course of A positive outcome realized from a course of

action reinforces the likelihood of that same action reinforces the likelihood of that same action being chosen again [action being chosen again [Experience-Experience-Weighted Attraction Learning – Camerer & Weighted Attraction Learning – Camerer & Ho, Ho, EconometricaEconometrica, 1999], 1999]

Evolving Evolving qqii((tt))

► Attractions for the available courses of Attractions for the available courses of actionaction BBii

inin((tt): i’s attraction measure for innovation): i’s attraction measure for innovation

BBiiimim((tt): i’s attraction measure for imitation): i’s attraction measure for imitation

► Evolution of the attractionsEvolution of the attractions BBii

inin((tt+1)= B+1)= Biiinin((tt) + 1, if innovation successful in ) + 1, if innovation successful in tt

BBiiinin((tt+1)= B+1)= Bii

inin((tt), otherwise), otherwise

BBiiimim((tt+1)= B+1)= Bii

imim((tt) + 1, if imitation successful in ) + 1, if imitation successful in ttBBii

imim((tt+1)= B+1)= Biiimim((tt), otherwise), otherwise

BBiiinin((tt))

BBiiinin((tt) + B) + Bii

imim((tt))qqii((tt) =) =

BBiiimim((tt))

BBiiinin((tt) + B) + Bii

imim((tt))1 - q1 - qii((tt) =) =

Evolving Evolving ppi i jj((tt))

► ppi i jj((tt): Probability with which ): Probability with which ii observes observes jj in in tt

► ∑∑j≠ij≠ippi i jj((tt) = 1 for all ) = 1 for all ii

► AAiijj((tt): agent i): agent i’’s attraction to another agent s attraction to another agent jj

► Evolution of the attractionsEvolution of the attractions AAii

jj((tt+1) = A+1) = Aiijj((tt) + 1, if ) + 1, if ii successfully imitated successfully imitated jj in in tt

AAiijj((tt+1) = A+1) = Aii

jj((tt), otherwise), otherwisefor all j ≠ i and for all i.for all j ≠ i and for all i.

ppii jj((tt) =) =AAii

jj((tt))

∑∑i≠ji≠jAAii jj((tt))

for all j ≠ i and for all i.for all j ≠ i and for all i.

social network of agent social network of agent ii

{{ppiijj((tt)})}j≠ij≠i

Agents with Uniform Learning Agents with Uniform Learning CapabilitiesCapabilities

► μμiiinin = = μμin in for allfor all i i (innovativeness)(innovativeness)

►μμiiimim = = μμim im for allfor all i i (reliability of the (reliability of the

network)network)► Improved productivity in innovation: a Improved productivity in innovation: a

rise in rise in μμinin

► Improved communication: a rise in Improved communication: a rise in μμimim

Initial Conditions at Initial Conditions at t t = 0= 0

►All agents have equal probabilities of All agents have equal probabilities of innovation vs. imitation: innovation vs. imitation: qqii(0) = 1- (0) = 1- qqii(0) = .5 for all (0) = .5 for all ii

►All agents have equal probabilities of All agents have equal probabilities of observing all other agents in the observing all other agents in the population: population: ppii

jj((tt) = 1/() = 1/(LL-1) for all -1) for all j≠ij≠i, , for all for all ii

Goals, Groups, and NetworksGoals, Groups, and Networks

► Similarity or diversity in the objectives drive Similarity or diversity in the objectives drive the formation of networksthe formation of networks

► Partition the population into a fixed number Partition the population into a fixed number of groupsof groups Agents belonging to the same group tend to Agents belonging to the same group tend to

have more similar goals – i.e., working on similar have more similar goals – i.e., working on similar problems – than those belonging to different problems – than those belonging to different groups.groups.

Two sources for social learningTwo sources for social learning► Another agent in the same groupAnother agent in the same group► Another agent in a different groupAnother agent in a different group

► Efficacy of social learning dependent on the Efficacy of social learning dependent on the tightness of goals within a given group tightness of goals within a given group relative to the tightness of the goals relative to the tightness of the goals between different groups.between different groups.

How are the goals specified?How are the goals specified?

Goal DiversityGoal Diversity

► Intra-group diversity: Intra-group diversity: κκ► Inter-group diversity: XInter-group diversity: X

Shift Dynamics for the GoalsShift Dynamics for the Goals

►ggii(t) stays the same with probability (t) stays the same with probability σσ and changes with probability and changes with probability (1-(1-σσ))..

► If gIf gii(t+1) is different from g(t+1) is different from gii(t), it is (t), it is chosen from the set of points that lie chosen from the set of points that lie both within the Hamming distance both within the Hamming distance ρρ of of ggii(t) and within Hamming distance (t) and within Hamming distance κκ of of the original group seed vector gthe original group seed vector gkk..

Parameters – {Parameters – {σσ, , ρρ, , κκ, X}, X}

► σσ The greater is The greater is σσ, the more stable is an agent’s , the more stable is an agent’s

goal vector.goal vector.► ρρ

The greater is The greater is ρρ, the more variable is the change , the more variable is the change in an agent’s goal vector.in an agent’s goal vector.

► κκ The higher is The higher is κκ, the lower is the intra-group goal , the lower is the intra-group goal

congruence.congruence.► XX

The higher is The higher is XX, the greater is the inter-group , the greater is the inter-group goal diversity.goal diversity.

Specs for Computational Specs for Computational ExperimentsExperiments

► 20 agents20 agents► 4 groups4 groups► 24 tasks, each with 4 bits24 tasks, each with 4 bits

16 methods in each task16 methods in each task Search space contains over 7.9 x 10Search space contains over 7.9 x 1028 28

possibilitiespossibilities

► BBiiinin(0)=B(0)=Bii

imim(0)=1 for all i(0)=1 for all i

► AAiijj(0)=1 for all i and all j (0)=1 for all i and all j ≠i≠i

► Horizon: 20,000 periodsHorizon: 20,000 periods

Baseline Parameter ValuesBaseline Parameter Values

►μμimim = = μμinin = 0.5 = 0.5►σσ = 0.75 = 0.75►ρρ = 4 = 4►κκ = 16 = 16►X = 16X = 16

Performance Time Series Performance Time Series (Baseline)(Baseline)

Steady-State Values of Steady-State Values of Endogenous VariablesEndogenous Variables

►Average over the 5,000 periods from Average over the 5,000 periods from t=15,001 to t=20,000t=15,001 to t=20,000

►Let’s look at the learning patterns: Let’s look at the learning patterns: ppi i jj((tt))

Do agents choose their learning partners Do agents choose their learning partners randomly?randomly?

Or do they learn from a small subset of other Or do they learn from a small subset of other agents?agents?

Do agents go to those who also come to Do agents go to those who also come to them?them?

►MUTUAL LEARNING? Are MUTUAL LEARNING? Are ppi i jj((tt) and ) and ppjj

ii((tt) correlated?) correlated?

ppi i jj((tt) and ) and ppjj

ii((tt))

Mutuality in LearningMutuality in Learning

►Correlation positive for all parameter Correlation positive for all parameter valuesvalues

►Correlation increases in XCorrelation increases in X The greater the The greater the interinter-group diversity in -group diversity in

goals goals The stronger the mutuality in The stronger the mutuality in learninglearning

►Correlation decreases in Correlation decreases in κκ The lower the The lower the intraintra-group diversity -group diversity The The

stronger the mutuality in learningstronger the mutuality in learning

Okay, but who is learning from whom?Okay, but who is learning from whom?

What drives the tendency for mutual What drives the tendency for mutual learning?learning?

Can we be more specific? How about Can we be more specific? How about tracking the evolution of the imitation tracking the evolution of the imitation probabilities?probabilities?

- Very complex. 20 agents, each with probabilities of - Very complex. 20 agents, each with probabilities of observing 19 other agents in each period (380 observing 19 other agents in each period (380 probabilities per period over the horizon of 20,000 probabilities per period over the horizon of 20,000 periods).periods).

Visualization of pVisualization of piijj((tt)s)s

ii

(observe(observer)r)

j (observed)j (observed)

11

11

2020

2020

1313

99

99

1818

►Four groupsFour groups Group 1 = {1, 2, 3, 4, 5}Group 1 = {1, 2, 3, 4, 5} Group 2 = {6, 7, 8, 9, 10}Group 2 = {6, 7, 8, 9, 10} Group 3 = {11, 12, 13, 14, 15}Group 3 = {11, 12, 13, 14, 15} Group 4 = {16, 17, 18, 19, 20}Group 4 = {16, 17, 18, 19, 20}

►At t=0, everyone observes everyone At t=0, everyone observes everyone else with an equal probability.else with an equal probability.

Steady-state Steady-state ppiijj((tt)s averaged )s averaged

over 20 replicationsover 20 replications

►Learning is mutual Learning is mutual andand more active more active among agents sharing similar goals.among agents sharing similar goals.

► Intra-group mutual learning is more Intra-group mutual learning is more intensive when the groups are more intensive when the groups are more segregated and isolated from one segregated and isolated from one another.another.

What about the aggregate What about the aggregate performance of the social performance of the social

system?system?

How is it affected by various How is it affected by various parameters?parameters?

Comparative DynamicsComparative Dynamics

► How is performance influenced byHow is performance influenced by Reliability of the communication technology Reliability of the communication technology

supporting the network – supporting the network – μμimim?? Productivity of agents engaging in innovation – Productivity of agents engaging in innovation – μμinin??

► How is the above relationship affected by How is the above relationship affected by features of the environment?features of the environment? Turbulence in the task environments – Turbulence in the task environments – σσ and and ρρ Inter-agent goal diversity – X and Inter-agent goal diversity – X and κκ

► Focus on steady-state (t=15,000 to t=20,000) Focus on steady-state (t=15,000 to t=20,000) – Long run properties of the system– Long run properties of the system

The rate of innovation declines in The rate of innovation declines in μμimim

σσ = 0.75; = 0.75; ρρ = 4; = 4; κκ = 16; X = 16 = 16; X = 16

Performance of the social system rises in Performance of the social system rises in μμimim

σσ = 0.75; = 0.75; ρρ = 4; = 4; κκ = 16; X = 16 = 16; X = 16

► Is this universal?Is this universal?►Does the steady-state performance Does the steady-state performance

always rise in the reliability of always rise in the reliability of communication technology?communication technology?

►Try:Try: Lower Lower μμinin to .25 to .25

►Agents less productive at innovationAgents less productive at innovation

Lower Lower κκ to 4 to 4►Degree of intra-group goal congruence higherDegree of intra-group goal congruence higher

μμinin = .25 = .25σσ = 0.75; = 0.75; ρρ = 4; = 4; κκ = 4 = 4; X = ; X =

1616

►An improvement in communication An improvement in communication technology can lead to a deterioration technology can lead to a deterioration in performance.in performance.

► Is this a general property? Do we Is this a general property? Do we observe this for different parameter observe this for different parameter configurations?configurations? YESYES

Property 1Property 1

► When the reliability of the network is When the reliability of the network is sufficiently low, steady-state performance is sufficiently low, steady-state performance is increasing in reliability.increasing in reliability.

► When the reliability of the network is When the reliability of the network is sufficiently high ANDsufficiently high AND

- task environment is sufficiently volatile (- task environment is sufficiently volatile (σσ low, low, ρρ high) high)- goal diversity among groups is sufficiently great (X high)- goal diversity among groups is sufficiently great (X high)- intra-group goal diversity is sufficiently low (- intra-group goal diversity is sufficiently low (κκ low) low)

performance decreases in reliability.performance decreases in reliability.

More reliable network (higher More reliable network (higher μμimim))

Imitation substituted for innovationImitation substituted for innovation

Faster diffusion of ideasFaster diffusion of ideas

Formation of more structured networkFormation of more structured network

Agents in a network (and in a group) become more Agents in a network (and in a group) become more alike (homogenization of the networks)alike (homogenization of the networks)

LACK OF DIVERSITYLACK OF DIVERSITY IN THE SYSTEM!!!IN THE SYSTEM!!!

►More stable environmentMore stable environment Lack of diversity is not a problemLack of diversity is not a problem Faster social learning is more critical Faster social learning is more critical

(speeding up convergence to local (speeding up convergence to local optima)optima)

►Volatile environmentVolatile environment Agents must modify their tasks Agents must modify their tasks

continually to solve new problems.continually to solve new problems. As network becomes more reliable, As network becomes more reliable,

imitation crowds out innovation and the imitation crowds out innovation and the ensuing lack of diversity makes it less ensuing lack of diversity makes it less likely that there will be useful ideas that likely that there will be useful ideas that serve the new environment.serve the new environment.

►What about the role of inter-group What about the role of inter-group diversity (X)?diversity (X)? Low X Low X the optima of agents from the optima of agents from

different groups overlap to a greater different groups overlap to a greater extent extent social learning is more social learning is more globalglobal (agents observe other agents in different (agents observe other agents in different groups more frequently) groups more frequently) inter-agent inter-agent diversity survives over time diversity survives over time superior superior adaptability to changing environmentadaptability to changing environment

SummarySummary

►Simple improvement in the reliability of Simple improvement in the reliability of the network may harm long-run the network may harm long-run performance.performance.

►Enhanced communication technology Enhanced communication technology can induce too much imitation can induce too much imitation Loss of Loss of diversity in responding to future diversity in responding to future environmentsenvironments

►An individual’s capacity to carry on An individual’s capacity to carry on independent innovationindependent innovation is crucial in is crucial in supplying the necessary fuel for the supplying the necessary fuel for the effective operation of the social network.effective operation of the social network.

Individual Choices and Social Individual Choices and Social OutcomesOutcomes

► At the individual level, innovation and imitation At the individual level, innovation and imitation are substitutes.are substitutes. Agents can choose to allocate effort to discovering Agents can choose to allocate effort to discovering

new ideas or to observing the ideas of others.new ideas or to observing the ideas of others.

► At the social level, innovation and imitation are At the social level, innovation and imitation are complements.complements. Innovation provides a pool of ideas that imitation can Innovation provides a pool of ideas that imitation can

then spread.then spread.

► Reliability of communication can reduce overall Reliability of communication can reduce overall performance by generating excessive performance by generating excessive uniformity.uniformity.

What if agents have heterogeneous What if agents have heterogeneous learning capabilities?learning capabilities?

QuestionsQuestions

► What if agents are heterogeneous in their What if agents are heterogeneous in their capacity for innovating and imitating?capacity for innovating and imitating? Some agents more creative and more productive Some agents more creative and more productive

in generating ideasin generating ideas Some agents more sociable or more capable to Some agents more sociable or more capable to

understanding the ideas of othersunderstanding the ideas of others

► To what extent does this heterogeneity lead To what extent does this heterogeneity lead to more order?to more order? Do agents form links with those who are most Do agents form links with those who are most

innovative or those who are most imitative?innovative or those who are most imitative?

►Underlying social system: A Underlying social system: A population of autonomous agents with population of autonomous agents with heterogeneous skills and capabilitiesheterogeneous skills and capabilities Some more skilled as innovatorsSome more skilled as innovators Some more skilled as imitatorsSome more skilled as imitators

►Example: The Scientific RevolutionExample: The Scientific Revolution Inventors and Innovators: Copernicus, Inventors and Innovators: Copernicus,

Kepler, Galileo, …, Newton (sources of Kepler, Galileo, …, Newton (sources of original ideas)original ideas)

Imitators/Connectors: Mersenne, Hartlib, Imitators/Connectors: Mersenne, Hartlib, …, Oldenburg (hubs of scientific …, Oldenburg (hubs of scientific correspondence networks)correspondence networks)

Research ObjectivesResearch Objectives

►Observe the network architecture that Observe the network architecture that evolves in the search processevolves in the search process Who learns from whom?Who learns from whom? What parameters affect the network What parameters affect the network

architecture and how?architecture and how?

►Socially optimal combination of Socially optimal combination of innovators and imitators?innovators and imitators?

The ModelThe Model

► Social system: Social system: LL individualsindividuals► There exists a common goal vector (binary) There exists a common goal vector (binary)

of of HdHd dimensions for the population in dimensions for the population in tt: : gg((tt))

► gg((tt)) may fluctuate over time may fluctuate over time Each period, Each period, gg((tt)) stays the same with probability stays the same with probability

σσ and changes with probability and changes with probability 1-1-σσ.. If it changes, the new goal vector is an If it changes, the new goal vector is an iidiid

selection from the set of points that lie within the selection from the set of points that lie within the Hamming distance Hamming distance ρρ of the old goal vector. of the old goal vector.

g(t)z1(t)

z2(t)z3(t)

z4(t)

D(z1(t), g(t))

Agent HeterogeneityAgent Heterogeneity

►Baseline Case: Baseline Case: LL = 50 = 50►Three types of agentsThree types of agents

Type Type NN: : Super-Innovators: Super-Innovators: ((μμiiinin, , μμii

imim) = (1, 0)) = (1, 0)

Type Type MM: : Super-Imitators: Super-Imitators: ((μμiiinin, , μμii

imim) = (0, 1)) = (0, 1)

Type Type RR: : Regular Agents: Regular Agents: ((μμiiinin, , μμii

imim) = ) = (.25, .25)(.25, .25)

►Assume |Assume |RR| = 40| = 40►||NN|+||+|MM| = 10 | = 10

Initial Conditions at Initial Conditions at t t = 0= 0

►All agents start with the same methods All agents start with the same methods vector: vector: zz11(0) = (0) = zz22(0) = … = (0) = … = zzLL(0)(0)

►All agents have equal probabilities of All agents have equal probabilities of innovation vs. imitation: innovation vs. imitation: qqii(0) = 1- (0) = 1- qqii(0) (0) = .5 for all = .5 for all ii

►All agents have equal probabilities of All agents have equal probabilities of observing all other agents in the observing all other agents in the population: population: ppii

jj((tt) = 1/() = 1/(LL-1) for all -1) for all j≠ij≠i, for , for all all ii

Steady-State MeasuresSteady-State Measures

►TT = 15,000 = 15,000►Steady-state performance of agent Steady-state performance of agent ii

Average of Average of ππii((tt)s over the last 5,000 periods)s over the last 5,000 periods

►Steady-state imitation probabilitiesSteady-state imitation probabilities Average of {Average of {ppii

jj((tt)})}j≠ij≠i over the last 5,000 over the last 5,000 periodsperiods

►100 replications100 replications►All measures are the averages over the All measures are the averages over the

100 replications100 replications

►How do these measures respond to How do these measures respond to changes in the values of:changes in the values of: ||NN|:||:|MM| : Composition of the super-types | : Composition of the super-types

in the populationin the population σσ: Frequency of goal change: Frequency of goal change ρρ: Magnitude of goal change: Magnitude of goal change

Steady-State MeasuresSteady-State Measures

►TT = 15,000 = 15,000►Steady-state performance of agent Steady-state performance of agent ii

Average of Average of ππii((tt)s over the last 5,000 periods)s over the last 5,000 periods

►Steady-state imitation probabilitiesSteady-state imitation probabilities Average of {Average of {ppii

jj((tt)})}j≠ij≠i over the last 5,000 over the last 5,000 periodsperiods

►100 replications100 replications►All measures are the averages over the All measures are the averages over the

100 replications100 replications

►How do these measures respond to How do these measures respond to changes in the values of:changes in the values of: ||NN|:||:|MM| : Composition of the super-types | : Composition of the super-types

in the populationin the population σσ: Frequency of goal change: Frequency of goal change ρρ: Magnitude of goal change: Magnitude of goal change

ppiijj((tt): Prob. Of ): Prob. Of ii observing observing jj

►Let agents be distributed as follows:Let agents be distributed as follows: N = {1, 2, 3, 4}N = {1, 2, 3, 4} M = {5, 6, 7, 8, 9, 10}M = {5, 6, 7, 8, 9, 10} R = {11, 12, 13, …… , 48, 49, 50}R = {11, 12, 13, …… , 48, 49, 50}

t = 1t = 1

t = 501t = 501

t = 1001t = 1001

t = 3001t = 3001

t = 6001t = 6001

t = 10001t = 10001

t = 14901t = 14901

Steady-State pSteady-State piij j with with σσ = .8, = .8, ρρ = 1, and | = 1, and |NN|:||:|MM|=4:6|=4:6

(average over 100 replications)(average over 100 replications)

Network StructureNetwork Structure Chain learningChain learning

InnovatorsInnovators generate ideas generate ideas

ImitatorsImitators learn from learn from InnovatorsInnovators

Regular AgentsRegular Agents learn from learn from ImitatorsImitators..

Group-Mean Learning Group-Mean Learning ProbabilitiesProbabilities

► ffgg’gg’ = The probability with = The probability with which an average agent which an average agent in group in group gg learns from an learns from an average agent in group average agent in group g’g’

► Network structure Network structure dictated by (dictated by (ffMNMN, f, fMMMM, f, fMRMR) ) and (and (ffRNRN, f, fRMRM, f, fRRRR) )

ffNNNN ffNMNM ffNRNR

ffMNMN ffMMMM ffMRMR

ffRNRN ffRMRM ffRRRR

Property 1Property 1

►Chain LearningChain Learning ffMNMN >> f fMMMM >> >> ffMRMR

ffRMRM >> f fRNRN >>>> f fRRRR

►A A Regular AgentRegular Agent learns learns throughthrough ImitatorsImitators rather than directly from rather than directly from InnovatorsInnovators.. ImitatorsImitators as the integrators of knowledgeas the integrators of knowledge More productive for an More productive for an Regular AgentRegular Agent to to

observe an observe an ImitatorImitator than multiple than multiple InnovatorsInnovators

Property 2Property 2► ||NN| plentiful relative to || plentiful relative to |MM||

An increase in environmental volatility (lower An increase in environmental volatility (lower σ and higher ρ) induces Imitators and Regular Agents to connect more with Innovators rather than Imitators.

►More stability Imitators as the central hub

► ||NN| scarce relative to || scarce relative to |MM|| An increase in volatility (lower An increase in volatility (lower σ and higher ρ)

induces Regular Agents to connect more with Innovators but induces Imitators to connect less with Innovators.

► Regular AgentsRegular Agents engage in more innovation (useful source engage in more innovation (useful source of ideas for of ideas for ImitatorsImitators).).

► More stability More stability Chain-learning Chain-learning

fMN – fMM as a function of the |N|:|M| ratio

for varying values of σ

fRM – fRN as a function of the |N|:|M| ratio

for varying values of σ

Flow of KnowledgeFlow of Knowledge- The Baseline Case -- The Baseline Case -

NN MM RR

Connected Connected InnovatorsInnovators and and Innovative Innovative ImitatorsImitators

►Assume:Assume: ((μμii

inin, , μμiiimim) = (.75, .25) for all ) = (.75, .25) for all ii in in NN

((μμiiinin, , μμii

imim) = (.25, .75) for all ) = (.25, .75) for all ii in in MM

((μμiiinin, , μμii

imim) = (.25, .25) for all ) = (.25, .25) for all ii in in RR

►Do Do InnovatorsInnovators learn from each other? learn from each other?►What does the flow of knowledge look What does the flow of knowledge look

like?like?

Flow of KnowledgeFlow of Knowledge

NN ↔↔ MM RR

Socially Optimal Mix of Agent Socially Optimal Mix of Agent TypesTypes

►When is the social performance at its When is the social performance at its maximum?maximum?

►Optimal Mix of Super-typesOptimal Mix of Super-types All innovators? (10 Newtons?)All innovators? (10 Newtons?) All imitators? (10 copycats?)All imitators? (10 copycats?) A Mixture?A Mixture?

Aggregate Performance as a Aggregate Performance as a function of Super-type function of Super-type

CompositionComposition

Heterogeneous Mixture is Heterogeneous Mixture is Optimal!Optimal!

5 5 InnovatorsInnovators and 5 and 5 ImitatorsImitators preferable to 10 preferable to 10 InnovatorsInnovators!!!!!!