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Fundamental Limitations of Networked Decision
Systems
Munther A. Dahleh
Laboratory for Information and Decision Systems
MIT
AFOSR-MURI Kick-off meeting, Sept, 2009
Smart Grid
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Drug Prescription: Marketing
The drugs your physician prescribes may well depend on the
behavior of an opinion leader in his or her social network in
addition to your doctor’s own knowledge of or familiarity with those
products.
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Smoking
Whether a person quits smoking is largely shaped by social
pressures, and people tend to quit smoking in groups. If a spouse
quits smoking, the other spouse is 67% less likely to smoke. If a
friend quits, a person is 36% less likely to still light up. Siblings who
quit made it 25% less likely that their brothers and sisters would
still smoke.
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Social Networks and Politics
Network structure of
political blogs prior to 2004
presidential elections
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Outline
Human/
Selfish
Engineered
Networks: Connectivity/capacity
Nature of Interaction:
Cyclic/Sequential
Computation
Learning
Decisions
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Learning Over Complex Networks
In Collaboration:
Daron Acemoglu
Ilan Lobel
Asuman Ozdaglar
The Tipping Point: M. Gladwell
The Tipping Point is that magic moment when an idea, trend, or
social behavior crosses a threshold, tips, and spreads like
wildfire. Just as a single sick person can start an epidemic of
the flu, so too can a small but precisely targeted push cause
fashion trend, the popularity of a new product, or a drop of
crime rate.
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Objective
Develop Models that can capture the impact of a
social network on learning and decision making
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Who's Buying the Newest Phone and Why?
I read positive
reviews, and
Lisa got it.
This phone
has great
functionality.
Got it!
1
Got it!
2
Everyone has it,
but 3G speeds are
rather lacking.
3
5
Got it!
Got it!
Didn’t.
It looks good, but
Jane didn’t get it
despite all her
friends having it.
Didn’t.
6
Before I asked
around, I thought the
phone was perfect.
But now I’m getting
mixed opinions.
4
7
Should I get it?
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Formulation: Two States
I read positive
reviews, and
Lisa got it.
This phone
has great
functionality.
Got it!
1
Got it!
2
Everyone has it,
but 3G speeds are
rather lacking.
3
5
Got it!
Got it!
Didn’t.
It looks good, but
Jane didn’t get it
despite all her
friends having it.
Didn’t.
6
Before I asked
around, I thought the
phone was perfect.
But now I’m getting
mixed opinions.
4
7
Should I get it?
11
General Setup



Two possible states of the world µ 2 f 0; 1g both equally likely.
A sequence of agents (n = 1; 2; :::) making binary decisions x n .
Agent n obtains utility 1 if x n = µ and utility 0 otherwise.

Each agent has an iid private signal sn 2 [0; 1]. The signal is
sampled from a cumulative density F µ .

The neighborhood: -

The neighborhoods - n ½ f 1; : : : n ¡ 1g is generated according to
arbitrary independent distributions f Q n ; n 2 N g.

Information: I n = f sn ; - n ; x k for all k 2 - n g
n
½ f 1; : : : n ¡ 1g
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The World According to Agent 7
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Rationality

Rational Choice: Given information set I n agent n chooses
¾n (I n ) 2 arg max P (µ = yjI n ) :
y2 f 0;1g
¾= f ¾n g

Strategy profile:

Asymptotic Learning: Under what conditions does
limn !
1
P ¾(x n = µ) = 1
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Equilibrium Decision Rule

The belief about the state decomposes into two parts
µ
¶¡
dF0 (sn )
p
(s
)
=
P
(µ
=
1js
)
=
1
+
n n
¾
n
 Private Belief,
dF1 (sn )


1
Social belief, P ¾(µ = 1j- n ; x k for all k 2 - n )
Strategy profile ¾ is a perfect Bayesian equilibrium if and only if:
P ¾(µ = 1jsn ) + P ¾(µ = 1j- n ; x k for all k 2 - n ) > 1 =) ¾n (I n ) = 1;
P ¾(µ = 1jsn ) + P ¾(µ = 1j- n ; x k for all k 2 - n ) < 1 =) ¾n (I n ) = 0:
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Selfish vs. Engineered Response: Star Topology (Cover)

Hypothesis testing:

If nodes communicate their observations:
L(¢) = Likelihood Ratio
1X
Pf
L (si ) ¡ EL (S) > dg · en g( d)
n i

What if nodes communicate only their decisions: x i = f (si )
1X
Pf
L (x i ) ¡ EL (x) > dg · en g1 ( d)
n i
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Selfishness and Herding Phenomenon: [Banerjee (92), BHW 92]

Setup:
 Full network: - n = f 1; 2; :::; n ¡ 1g

with probability 0.8
Absence of Collective Wisdom
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Private Beliefs
µ
dF 0 (sn )
1+
dF 1 (sn )
¶¡
1

Private Beliefs pn (sn ) = P ¾(µ = 1jsn ) =

Definition: The private beliefs are called unbounded if
sup
s2 S
dF0 (s)
= 1
dF1 (s)
and
inf
s2 S
:
dF 0 (s)
= 0
dF 1 (s)

If the private beliefs are unbounded, then there exist some agents
with beliefs arbitrarily close to 0 and other agents with beliefs
arbitrarily close to 1.

Discrete example:
with probability 0.8?
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Expanding Observations

Definition: A network topology f Q n gn 2 N is said to have expanding
observations if for all ² > 0, and all K 2 R , there exists some N
such that for all n ¸ N
µ
Qn

¶
max b < K
b2 -
< ²
n
Conversely
Absence of Excessively Influential Agents
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Influential References
Vidyasagar
Astrom/Murray
New Book
New Book
Doyle, Francis
Tannenbaum
Kailath
New Book
Ljung
New Book
No “learning”!
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Summary
Expanding
Observat ions
Ot her Topologies
Unbounded Beliefs
YES
NO
Bounded Beliefs
USUALLY NO,
SOMET IMES YES
NO

If j- n j · M then learning is impossible for signals with bounded
beliefs
 Line network

It is possible to learn with expanding observations and bounded
beliefs
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Deterministic Networks: Examples

Full topology: -

Line topology: -
n
n
= f 1; 2; :::; n ¡ 1g
= f n ¡ 1g
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Examples: A Random Sample

Suppose each agent observes a sample C > 0 of randomly drawn
(uniformly) decisions from the past. If the private beliefs are
unbounded, then asymptotic learning occurs.
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Examples: Binomial Sample (Erdős–Rényi)

Suppose all links in the network are independent, and
for two constants A and B we have Q n (m 2 - n ) = AB ;
n


If the beliefs are unbounded and B < 1, asymptotic learning
occurs
If B ¸ 1 , asymptotic learning does not occur
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Active Research

Just scratching the surface….

Presented a simple model of information aggregation
 Private signal
 Network topology

More complex models for sequential decision making
 Dependent neighbors
 Heterogeneous preferences
 Multi-class agents
 Cyclic decisions

Rationality

Topology measures: depth, diameter, conductance
 Expanding observations
 Learning Rate

Robustness
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