The Dynamics of Social Influence
Bary S. R. Pradelski
ETH Zurich
Controversies in Game Theory III
31st May 2016
Introduction
Technology facilitates social influence and resurfaces old questions
The Dynamics of Social Influence
Bary Pradelski
| 1
Introduction
Phenomena of ‘collective action’ are explained by social influence models
Definition
Social influence Social influence describes how
individuals’ adjust their opinions, beliefs, and actions
in light of information about others
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
The Dynamics of Social Influence
Bary Pradelski
| 2
Introduction
The study of social influence has a long tradition in the Social Sciences
Initially discussed in
psychology …
… economists started
to recognize socially
driven behavior …
… which is supported
by more recent
experimental work
▪
Trotter (1916) identifies
herd instinct
▪
▪
▪
Psychologists identify
group identity (LeBon,
1895; Freud, 1921)
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
▪
Asch (1955) conducts
simple experiment
supporting social
influence
The Dynamics of Social Influence
▪
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
Bary Pradelski
| 3
Introduction
The dynamics of social influence have also been studied
Most related to our
research, detailed on
following page
Early dynamic models of social influence
▪
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
The Dynamics of Social Influence
Bary Pradelski
| 4
Introduction
Threshold models are characterized by a heterogeneous population
Model assumptions (based on Granovetter, 1978)
▪
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
Theorem (Granovetter, 1978)
The equilibria of a social influence model are the fixpoints of the frequency
distribution of thresholds.
The Dynamics of Social Influence
Bary Pradelski
| 5
Introduction
We compare the classic model of social influence and an alternative model
Adoption – the classic model
Usage – new model
▪
▪
▪
Players respond to the current
number of adopters
Examples
– Voting: each player is only counted
once
The Dynamics of Social Influence
▪
Players respond to the cumulative
usage in the recent history
Examples
– Media coverage
– Stock market participation: total
order volume
Bary Pradelski
| 6
Introduction
We compare the classic model of social influence and an alternative model
Adoption – the classic model
Usage – new model
▪
▪
▪
Players respond to the current
number of adopters
Examples
– Voting: each player is only counted
once
▪
Players respond to the cumulative
usage in the recent history
Examples
– Media coverage
– Stock market participation: total
order volume
Our guiding example
▪
Providing incentives to purchase a
bicycle
The Dynamics of Social Influence
▪
Providing incentives to use a bicycle
Bary Pradelski
| 7
Introduction
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
The Dynamics of Social Influence
Bary Pradelski
| 8
The model
Definition – static game
▪ types N  1,2,...,n
▪ players P  1,..., p, where each player is associated with one type
1 p
▪ ratio of players of type i: qi  p   j
j 1
is of type i
▪ actions A  Ai  m, d  for all i  P
▪ utility function ui : 0,1  A  R, when responding to the signal s  0,1
about society:
 p  i s if a  d
ui ( a )   i
 i (1  s ) if a  m
with pi and i  0 constant
▪ response function fi : R 2  d, m specifying i's probability to take
an action given his utilities ui (s, d ), ui (s, m ) :
d


fi  d , m (50 - 50)

m

The Dynamics of Social Influence
if ui (s, d )  ui (s,m)
if ui (s, d )  ui (s,m)
else
Bary Pradelski
| 9
The model
Definition – dynamic game
▪ 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 s(t ) be agent i's observation and ai (t ) his
action at time t
▪ let at  (ait )i P be the state at the end of period t ; let a0 be
any permissible initial configuration, then
ait  1act fi (ui (s(t ), d ), ui (s(t ), m ))  1notact ait 1
for all t  1.
The Dynamics of Social Influence
Bary Pradelski
| 10
The model
A crucial variable is the signal about society to which players respond to
Adoption
Usage
▪
▪
An active player responds to the
current number of adopters
s adoption (t ) 
Definition
p
 1a
i 1
t 1
d
i
p
An active player responds to the
cumulative usage in the past k
periods
s usage (t ) 
t 1
p
  1a
v  t0 i 1
v
i
 d , i act in v
k
for t0  t  k , k constant
The Dynamics of Social Influence
Bary Pradelski
| 11
Illustration
The signal about society differs dependent on the model of social
influence
Suppose we are at time step t=7 and play unfolded as shown below:
Suppose k=5. Then:
s adoption (7)  50%
s usage (7)
The Dynamics of Social Influence
 80%
Bary Pradelski
| 12
Analysis
Each player has a threshold which determines his preference
Definition
Let each player i’s threshold be i  R such that he wants to
play d if and only if
i  s(t ),
Threshold
is indifferent if i  s(t ) and wants to play m otherwise.
That is i is the (unique) zero of the function
ui (s(t ), d )  ui (s(t ),m)
The Dynamics of Social Influence
Bary Pradelski
| 13
Analysis
The Aggregate Dynamic gives the share of players preferring the
innovation given the signal
Definition
Let Agg be the Aggregate Dynamic. That is given the true
average action
1 p
a   1ai  d
p i 1
Aggregate
Dynamics
Agg gives the share of players who would play d, given they
observe this state:
 1 2

Agg : [0,1]  0, , ,...,1
 p p

1 p
Agg (a)   1i  a
p i 1
The Dynamics of Social Influence
Bary Pradelski
| 14
Analysis
Results for Adoption
Suppose players have a uniform action tremble. That is there exists a small probability
  0 such that a player picks an action uniformly at random.
Theorem: Adoption
Suppose the model is Adoption and uniform action tremble. The stochastic
potential  a*i of a recurrence class a*i  a1* ,..., a*k
associated with fixpoint
*
*
* is given by
xi  x1 ,..., xk




a*i

i 1
 xmax
x ,x
 1

*

*

 1 
x  Agg ( x ) 

l

max
  i 1 x x 1, x 
*
*
x  Agg ( x )
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.
The Dynamics of Social Influence
Bary Pradelski
| 15
Analysis
Proof sketch Adoption – stochastic stability
Stochastic stability analysis (Foster/Young ‘90, Kandori et al. ‘93, Young ‘93)
▪
Suffices to analyze the change of
▪
▪
Absorbing states of unperturbed dynamic are singleton recurrent classes
1 p
a   1ai  d
p i 1
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 a has a linear transition
*
*
*
structure: in order to go from one recurrent class a i  a1,..., a k to
another one has to go through all states in between
The Dynamics of Social Influence


Bary Pradelski
| 16
Analysis
Results for Usage
Players have finite sampling. An active player samples the most recent k (const.) actions.
Theorem: Usage
Suppose the model is Usage and players have finite sampling with sample size k, let


qi :


For i  1,..., n let

1

 
 i :  q i (1  q )1 i
i
i




qj 
i  1,..., n
j 1

i
 q j 1 
  1  q 
j 1 
j 2 
0
i
Let I *  1,..., n be such that for i *  I * :
if i  qi  i 1
else
 i *  max  i
i 1,...,n
t
then
 1a
lim lim
k  t 
qi i I
*
 j   j 1
*
*
v 0
v
i
 d ,i active in period v
t
 i I
 qi *
*
*
*
is a subset of the fixpoints of Agg. For generic games I  1 .
The Dynamics of Social Influence
Bary Pradelski
| 17
Analysis
Proof sketch Usage – reinforced random walks
Reinforced random walks (Pinsky 2013)
▪
Suffices to analyze the change of
▪
We can transform the stochastic process governing a into a random walk
on Ζ 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
The Dynamics of Social Influence
1 p
a   1ai  d
p i 1
Bary Pradelski
| 18
Analysis
The two models yield different outcomes
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.
The Dynamics of Social Influence
Bary Pradelski
| 19
Example
Innovation diffusion with a population à la Rogers (1962)
Given a population of …
…who…

2.5%
always play the innovation action d, that is, they play the
innovation independent of social influence and hence their
threshold is `negative‘

13.5% early adopters:
play the innovation if at least `few‘ play the innovation

34%
early majority:
play the innovation if at least an `intermediate proportion‘
play the innovation
32.00%

34%
late majority:
play the innovation if at least `many' play the innovation
68.00%

13.5% laggards:
play the innovation if at least `almost everybody' play the
innovation
90.75%

2.5%
never play the innovation
innovators:
non-adopters:
…. have threshold

<0
9.25%
>1
q

The Dynamics of Social Influence
Bary Pradelski
| 20
Example
Fixpoints of Agg
Agg

The Dynamics of Social Influence
Bary Pradelski
| 21
Example
Long-run stable states under Adoption and Usage differ
Agg

The Dynamics of Social Influence
Bary Pradelski
| 22
Analysis
Intuition for different results for Adoption and Usage
Adoption
Usage
▪
▪
Shift from one fixpoint to another is
governed by erroneous behavior of players
currently not playing the innovation
Probability is independent on the
Probability is dependent on the number
number of current adopters
▪
of current adopters
The stability of a fixpoint is independent
of whether it is more or less mixed than
another fixpoint, when all else is equal
Definition:
Shift from one fixpoint to another is
governed by higher usage intensity of
players currently playing the innovation
▪
The stability of a fixpoint is lower for more
mixed states, when all else is equal
*
*
*
*
For two fixpoints, xi , x j , say that xi is more mixed than x j if
xi*  0.5  x *j  0.5
The Dynamics of Social Influence
Bary Pradelski
| 23
Analysis
Empirically discriminating between Adoption and Usage
Adoption
Usage
▪
The behavior of a player is characterized
by a Bernoulli trial (with different
parameters)
▪
The behavior of a player is characterized
by a Bernoulli trial (with different
parameters)
▪
𝑠 𝑎𝑑𝑜𝑝𝑡𝑖𝑜𝑛
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
By considering non-overlapping sequences
of 𝑠 𝑢𝑠𝑎𝑔𝑒
the resulting distribution is
Binomial; in particular its variance is
independent of 𝜀
Identifying which process is at work allows us to inform decisions on, for example,
policy interventions or marketing campaigns.
The Dynamics of Social Influence
Bary Pradelski
| 24
Conclusion
Conclusion and future work
 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
The Dynamics of Social Influence
Bary Pradelski
| 25
Outlook
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
The Dynamics of Social Influence
Bary Pradelski
| 26
Outlook
We are currently working on experiments regarding
▪
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
The Dynamics of Social Influence
Bary Pradelski
| 27
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.
The Dynamics of Social Influence
Bary Pradelski
| 28