The Dynamics of Social Influence - ETH Z · Hamilton (1971) analyses herding behavior of a group of...

The Dynamics of Social Influence

ETH Zurich

Controversies in Game Theory III

31st May 2016

Bary S. R. Pradelski

Bary Pradelski |The Dynamics of Social Influence 1

Technology facilitates social influence and resurfaces old questionsIntroduction


Phenomena of ‘collective action’ are explained by social influence models

Social influence Social influence describes how

individuals’ adjust their opinions, beliefs, and actions

in light of information about others

Definition

Examples

▪ Opinion dynamics, e.g., financial markets, political opinions/riots, voting

▪ Innovation diffusion, e.g., technological advances, fashion trends

▪ Behavioral trades, e.g., smoking, obesity

Introduction


The study of social influence has a long tradition in the Social Sciences

… economists started

to recognize socially

driven behavior …

▪ Keynes (1930, 1936)

explains financial

instability by the

sociological forces of

uncertainty

▪ Shiller (1984, 2000)

argues that individuals

take decisions based

on beliefs of uncertain

events

… which is supported

by more recent

experimental work

▪ Cukiermann (1991)

finds evidence for

influence of opinion

polls in voting

▪ Salganik, Dodds and

Watts (2006) show

how ‘social influence’

can change hit songs

▪ Lorenz et al. (2011)

show how ‘social

influence’ can

undermine the wisdom

of crowds

Initially discussed in

psychology …

▪ Trotter (1916) identifies

herd instinct

▪ Psychologists identify

group identity (LeBon,

1895; Freud, 1921)

▪ Asch (1955) conducts

simple experiment

supporting social

influence

Introduction


The dynamics of social influence have also been studied

▪ Schelling (1971) studies a dynamic model of social influence for

neighborhood segregation

▪ Hamilton (1971) analyses herding behavior of a group of animals fleeing

from a predator

▪ Schelling (1978) and Granovetter (1978) develop threshold models for

heterogeneous populations and analyze their equilibria

▪ Banerjee (1992) and Bikchandani et al. (1992) study social influence under

the assumption of Bayesian learning

Early dynamic models of social influence

Most related to our

research, detailed on

following page

Introduction


Threshold models are characterized by a heterogeneous populationIntroduction

▪ Actors have two alternatives (e.g., adopting an innovation or not, voting

Democrats vs. Republican, smoking vs. non-smoking)

▪ The costs/benefits of each depend on how many other actors choose which

alternative

▪ Heterogeneous population, i.e., each player may react differently to social

influence:

Threshold: the proportion of others taking one action

in order for a given actor to take the same action

▪ No underlying network, i.e., a player is influenced by any other player with the

same probability

Model assumptions (based on Granovetter, 1978)

Theorem (Granovetter, 1978)

The equilibria of a social influence model are the fixpoints of the frequency

distribution of thresholds.


We compare the classic model of social influence and an alternative modelIntroduction

▪ Players respond to the current

number of adopters

▪ Examples

– Voting: each player is only counted

once

Adoption – the classic model

▪ Players respond to the cumulative

usage in the recent history

▪ Examples

– Media coverage

– Stock market participation: total

order volume

Usage – new model


We compare the classic model of social influence and an alternative modelIntroduction

▪ Players respond to the current

number of adopters

▪ Examples

– Voting: each player is only counted

once

Adoption – the classic model

▪ Players respond to the cumulative

usage in the recent history

▪ Examples

– Media coverage

– Stock market participation: total

order volume

Usage – new model

▪ Providing incentives to purchase a

bicycle

▪ Providing incentives to use a bicycle

Our guiding example


We identify the stable states of Adoption and Usage and find that they are

different

Our contribution

▪ We show selection of long-run stable equilibria for

perturbed processes of the two models

▪ We compare the classical model of observing the

number of adopters (Adoption) with the model where

players observe the cumulative usage (Usage) and find

different stable states

▪ We formulate tests to empirically discriminate between

the processes

Introduction


Definition – static game

▪

The model

is of type 1

types 1,2,...,n

players 1,..., , where each player is associated with one type

1ratio of players of type :

actions , for all

utility function : 0,1 , when resp

R

p

i j ij

i

i

N

P p

i qp

A A m d i P

u A

2

onding to the signal 0,1

about society:

if ( )

(1 ) if

with p and 0 constant

response function : , specifying 's probability to take

an action given

R

i ii

i

i i

i

s

p s a du a

s a m

f d m i

his utilities ( , ), ( , ) :

if ( , ) ( ,m)

f , (50 - 50) if ( , ) ( ,m)

else

i i

i i

i i i

u s d u s m

d u s d u s

d m u s d u s

m

▪

▪

▪

▪

▪


Definition – dynamic gameThe model

▪

the game is played in continous time where each player is

activated by iid Poisson arrival processes - a time step is

defined by the activation of one (!) player: t=1,2,3,...

accordingly let ( ) be ages t

0

act not

nt 's observation and ( ) his

action at time

let ( ) be the state at the end of period ; let be

any permissible initial configuration, then

( ( ( ), ), ( ( ), ))

a a

1 1

i

t ti i P

ti i i i

i a t

t

a t

a f u s t d u s t m 1act

for all 1.

tia

t

▪

▪


A crucial variable is the signal about society to which players respond toThe model

Definition

Adoption Usage

▪ An active player responds to the

current number of adopters

▪ An active player responds to the

cumulative usage in the past k

periods

for constant

1

1

( )

1 ti

adoption

p

a di

s t

p0

1

, i act in 1

( )

1 vi

usage

pt

a d vv t i

s t

k

0 , t t k k


The signal about society differs dependent on the model of social

influence

Illustration

Suppose we are at time step t=7 and play unfolded as shown below:

Suppose k=5. Then: (7) 50%

(7) 80%

adoption

usage

s

s


Each player has a threshold which determines his preferenceAnalysis

Threshold

Definition

Let each player i’s threshold be such that he wants to

play d if and only if

is indifferent if and wants to play m otherwise.

That is is the (unique) zero of the function

Ri

( ), i s t

( ( ), ) ( ( ),m)i iu s t d u s t

i

( ) i s t


The Aggregate Dynamic gives the share of players preferring the

innovation given the signal

Analysis

Definition

Aggregate

Dynamics

Let Agg be the Aggregate Dynamic. That is given the true

average action

Agg gives the share of players who would play d, given they

observe this state:

1

1

1i

p

a di

ap

a1

1 2: [0,1] 0, , ,...,1

1 (a)

1i

p

i

Aggp p

Aggp


Results for AdoptionAnalysis

Theorem: Adoption

Suppose the model is Adoption and uniform action tremble. The stochastic

potential of a recurrence class associated with fixpoint

is given by

The stochastically stable states are those states associated with fixpoints of Agg that

minimize stochastic potential. For generic games there exists a unique long-term

stable state.

Suppose players have a uniform action tremble. That is there exists a small probability

such that a player picks an action uniformly at random. 0

*ai

* * *1,...,i kx x x

* * * *1 1

1

*, ,1 1

max ( ) max ( )

a

i l

x x x x x xi i

x Agg x x Agg x

* * *1,...,a a ai k


Proof sketch Adoption – stochastic stability

▪ Suffices to analyze the change of

▪ Absorbing states of unperturbed dynamic are singleton recurrent classes

▪ Definition: A state is stochastically stable if the limit of its invariant

measure is positive

▪ Young (1993) shows that the computation of stochastically stable states

can be reduced to an analysis of rooted trees on the set of recurrent

classes

▪ Now note that that the process governing has a linear transition

structure: in order to go from one recurrent class to

another one has to go through all states in between

Analysis

Stochastic stability analysis (Foster/Young ‘90, Kandori et al. ‘93, Young ‘93)

1

1

1i

p

a di

ap

* * *1,...,a a ai k

a


Results for UsageAnalysis

Suppose the model is Usage and players have finite sampling with sample size k, let

For let

Let be such that for :

then

is a subset of the fixpoints of Agg. For generic games .

Theorem: Usage

Players have finite sampling. An active player samples the most recent k (const.) actions.

1

111

12

1if

: 1(1 )

0 else

j j

i i

ij

i i ii jji i

qq

qq q

1

:

1,...,

i

i jj

q q

i n

* * *

, active in period 0lim lim

1 vi

t

a d i vv

i i Ik tq

t

1,...,i n

* 1,...,I n * *i I

*1,...,n

max

ii i

* * *i i Iq * 1I


Proof sketch Usage – reinforced random walksAnalysis

▪ Suffices to analyze the change of

▪ We can transform the stochastic process governing into a random walk

on which was recently studied by Pinsky (2013)

▪ He studies a random walk reinforced by its recent history. It goes left or

right with fixed probabilities. If in the last k steps it went right more than m

times these probabilities change. He gives a closed formula for the

expected average limit

▪ The proof defines an auxiliary Markov chain and computes its invariant

measure

Reinforced random walks (Pinsky 2013)

1

1

1i

p

a di

ap

a

Ζ


The two models yield different outcomesAnalysis

Theorem

The two models Adoption and Usage yield different outcomes. In

particular, the set of games where the outcomes differ is generic.

Proof by example on following pages.


Innovation diffusion with a population à la Rogers (1962)Example

Given a population of …

2.5% innovators:

13.5% early adopters:

34% early majority:

34% late majority:

13.5% laggards:

2.5% non-adopters:

…who…

always play the innovation action d, that is, they play the

innovation independent of social influence and hence their

threshold is `negative‘

play the innovation if at least `few‘ play the innovation

play the innovation if at least an `intermediate proportion‘

play the innovation

play the innovation if at least `many' play the innovation

play the innovation if at least `almost everybody' play the

innovation

never play the innovation

< 0

9.25%

32.00%

68.00%

90.75%

> 1

…. have threshold

q


Fixpoints of AggExample

Agg


Long-run stable states under Adoption and Usage differExample

Agg


Intuition for different results for Adoption and Usage

▪ Shift from one fixpoint to another is

governed by erroneous behavior of players

currently not playing the innovation

Probability is independent on the

number of current adopters

Analysis

Adoption

▪ Shift from one fixpoint to another is

governed by higher usage intensity of

players currently playing the innovation

Probability is dependent on the number

of current adopters

Usage

* *,i jx x

* *0.5 0.5 i jx x

*ix

*jx

▪ The stability of a fixpoint is independent

of whether it is more or less mixed than

another fixpoint, when all else is equal

▪ The stability of a fixpoint is lower for more

mixed states, when all else is equal

Definition: For two fixpoints, , say that is more mixed than if


Empirically discriminating between Adoption and Usage

▪ The behavior of a player is characterized

by a Bernoulli trial (with different

parameters)

▪ is thus the sum of

independent non-identical Bernoulli trials

▪ The resulting distribution is not Binomial,

and in particular its variance depends on

the error rate 𝜀

Analysis

Adoption

▪ The behavior of a player is characterized

by a Bernoulli trial (with different

parameters)

▪ By considering non-overlapping sequences

of the resulting distribution is

Binomial; in particular its variance is

independent of 𝜀

Usage

Identifying which process is at work allows us to inform decisions on, for example,

policy interventions or marketing campaigns.

𝑠𝑎𝑑𝑜𝑝𝑡𝑖𝑜𝑛

𝑠𝑢𝑠𝑎𝑔𝑒


Conclusion and future workConclusion

We study social influence for

• Adoption: players respond to the current number of adopters

• Usage: players respond to the cumulative usage

The long-run outcome of the two models is – in general – different

We provide empirically testable predictions allowing to discriminate between

the two models

Future work:

We ran field experiments to test social influence on opinions

We provide subjects with different information about prior players’ decisions


There are many settings where you can test social influence

▪ Online news pages enable user

engagement provide natural testing

ground

▪ For example, a user sees previous

users’ opinions and opinions of different

opinion leaders (e.g., other newspapers)

▪ This can create interesting dynamics

and potentially polarization,

extremization, or consensus

Outlook


▪ Social Influence

– Does social influence exist? How strong are the effects?

– Does social influence drive opinion polarization or do we observe

convergence?

▪ Opinion Leadership

– Does leader influence exist?

– Do we observe persuasion bias?

▪ Self-selection and the spiral of silence

– Does opting in/out affect distribution of observed opinions

We are currently working on experiments regardingOutlook


Selected references

Bass, F. M. (1969), “A new product growth model for consumer durables."

Management Science, 15, 215-227.

Granovetter, M. (1978), “Threshold models of collective behavior." The American

Journal of Sociology, 83, 1420-1443.

Kandori, M., G. J. Mailath, and R. Rob (1993), “Learning, mutation, and long run

equilibria in games." Econometrica, 61, 29-56.

Lopez-Pintado, D. and D. J. Watts (2008), “Social influence, binary decisions and

collective dynamics." Rationality and Society, 20, 399-443.

Pinsky, R. G. (2013), “The speed of a random walk excited by its recent history.“

Working Paper, arXiv: 1305.7242.

Rogers, E. M. (1962), Diffusion of Innovations. The Free Press.

Schelling, T. C. (1978), Micromotives and Macrobehavior. Norton & Company.

Young, H. P. (1993), “The evolution of conventions." Econometrica, 61, 57-84.

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