Wisdom of Crowds in Social Networks with Structural Shocks

Wisdom of Crowds in Social Networkswith Structural Shocks

Ines LindnerJia-Ping Huang and Bernd Heidergott

VU University AmsterdamEconometrics & OR

Bremen, July 6th 2017

Lindner Wisdom of Crowds in Social Networks with Structural Shocks

Information Sharing and Opinion Formation

Social networksI shape opinions and beliefs

I subsequently shape behavior

Critical to understand how structure affects learning and informationdiffusion



Fundamental Questions:I Common consensus or divided opinions?

I most influential agents

I speed of learning

I whether initially diverse information scattered throughout societycan be aggregated in accurate manner


Literature

Two kinds of models:

(1) Bayesian models:I very cumbersome and complex analysis

I nevertheless provide insight in conditions for consensus


Literature


(2) Naıve learning modelI DeGroot (1974): Convergence to consensus

I Golub and Jackson (2010): wisdom of crowds (WoC)- Extend DeGroot model to growing networks

- Conditions of converging to “truth”

I This research: allow network variability→ Consensus is achieved

→ WoC may be lost


Social learning: DeGroot model

I Set of agents: N = 1, . . . , n

I Social influence matrix: P = [Pij ]i,j∈NPij : the weight or trust that agent i places on the current belief ofagent j in forming i’s belief for the next period.

I Initial beliefs: fi ∈ [−1, 1] for all i

I Updating: f (t)i =∑

j Pijf(t−1)j



Updating

1

2

3

1/3

1/31/30

1

01

2

3

1/3

1/31/31/3



Updating

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

01

01

2

3

1/3

1/3

1/21/21/3

1/2

1/2

1/3 1/2

0



Updating

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

01

01

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3

1/3

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1/3 1/2

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1/2

5/185/12

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Updating

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

2/7 2/7

2/7…

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

01

01

2

3

1/3

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1

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3

1/3

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1/21/21/3

1/2

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5/185/12

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I Updating beliefs:

f (t) = Pf (t−1) = P t−1f (0) where f (0) = f

I Convergence: if the limit limt→∞

f (t) exists for all initial beliefs,


Convergence of belief

P =

1/3 1/3 1/3

1/2 1/2 0

1/2 0 1/2

, . . . , P∞ →3/7 2/7 2/7

3/7 2/7 2/7

3/7 2/7 2/7

, π = (3/7, 2/7, 2/7),

f (∞) =3

7f1 +

2

7f2 +

2

7f3

TheoremSuppose P is strongly connected.

P is convergent if and only if it is aperiodic.

P is convergent if and only if limt Pt = Π. Each row of Π equals the

stationary distribution vector π> which is the unique solution of

π>P = π>, π>1 = 1.


Wisdom of Crowds

Extension of Golub and Jackson (2010): wisdom of crowds

Assume µ ∈ [−1, 1], the true state of nature.

Assume each agent has some imperfect information about µ. Belief fiis the signal received by agent i.Each agent i’s belief fi is a random variable with mean µ and varianceσ2i ≥ σ2 > 0.


Wisdom of Crowds




Question: For what networks is limt P tf closer to the truth than f?

Golub and Jackson (2010): answer for n → ∞.


Wisdom of crowds: the double limit in detail

……

1 2 3 4 t t+1

1

2

3

n

n+1

time

network size

π (1)

π (2)

π (3)

π (n)

π (n+1)

……


Wisdom of crowds: Golub and Jackson (2010)

Denote P (n) a stochastic matrix with size n× n.

DefinitionThe sequence P (n)∞n=1 is wise if

limn→∞

Pr[

maxi≤n

∣∣f (∞)i (n)− µ

∣∣ > ε]

= 0

for any ε > 0.

In words, a sequence of networks is wise if the limiting beliefsconverge to the true state µ in probability.



PropositionIf P (n)∞n=1 is a sequence of strongly connected and aperiodicstochastic matrices, then it is wise if and only if the associatedstationary vectors π(n)∞n=1 are such that

maxi≤n

πi(n)→ 0.

Intuitively, this is to say that no individual is more influential than othersin the limit.


Wisdom of crowds: examples

Wise: growing ring

π(n) =

1/n

1/n...

1/n

Non-wise: growing star

π(1) > 0 for n→∞



Non-wise network: a line

1

1-a

2 3 n

a

……a

1-a

a

1-a

P (n) =

1− a a 0 . . . 0

1− a 0 a . . . 0

0 1− a 0 . . . 0...

......

. . ....

0 0 0 . . . a

For a < 1/2, it holds that

πi(n) =( a

1− a

)i−1·

1− a1−a

1−(

a1−a)n .

Hence,limn→∞

π1(n) =(

1− a

1− a

).


Our interests

Drawbacks Golub and Jackson (2010):Network is constant for t→∞

Research questions:I Will wisdom be preserved under structural shocks?

I Does wisdom hold for more realistic updating?


Our interests




Examples of shocks:I financial crisis, natural disaster, war, etc.

I natural communication


Random shock: superposition

Let P (n) be wise and and Q(n) a series of arbitrary Markovchains. For θ ∈ (0, 1), consider

P (θ, n) = (1− θ)P (n) + θQ(n).

P (n) wiseQ(n) series of arbitrary Markov chains


Random shock: randomization

We consider a random sequence P (t)t>0 whose elements areindependently drawn according to a Bernoulli trial.

1 2 3 4 ttime

P P Q P……

……

P……

Q

Ɵ1- Ɵ

……

In expectation, superposition and randomization are the same.

E[P (t)] = θP + (1− θ)Q.


Wisdom of crowds

What is the insight of the randomization case?

1 2 3 4 ttime

P P Q P……

……

P……

Q

Ɵ1- Ɵ

……

TheoremAssume P and Q are strongly connected and aperiodic.For fixed n consensus exists if every agent influences at least twoother agents per update. This consensus is path dependent.


An illustrating example

I Consider the ring network with self links(omitted in the picture).

I Assume

P (θ, n) =

12

14

14

14

12

14

14

12

14

. . . . . . . . .14

12

14

14

12

14

14

14

12

1

2 3

4 5

6 7

2m 2m+1



P =

12

14

14

13

12

16

13

12

16

. . .. . .

. . .13

12

16

13

12

16

13

16

12

,

Q =

12

14

14

c 12

d

c 12

d

. . .. . .

. . .c 1

2d

c 12

d

c d 12

,

c = (3 − 4θ)/(12 − 12θ),d = 1/2 − c = (3 − 2θ)/(12 − 12θ).

1

2 3

4 5

6 7

2m 2m+1

P (θ, n) = θP (n) + (1− θ)Q(n) for θ ∈ (0, 34).Lindner Wisdom of Crowds in Social Networks with Structural Shocks

Convergence under fixed networks

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by matrix P

Network P (θ), θ = 0.3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by matrix X

Network P

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by matrix Y

Network Q

Horizontal axis: time.Vertical axis: individual belief.n = 25. The initial belief f (0) = (1, 1112 , . . . ,−

1112 ,−1)>.


Convergence under randomized networks

Taking θ = 0.3, we generate 5 sample paths of f(t)t>0, all startingfrom the same initial belief vector.

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by randomly chosen matrix from X and Y: sample path 1

Belief dist of sample path 1

0 100 200 300 400 500−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time

Ave

rag

e b

elie

f

Updated by randomly chosen matrix from X and Y: 5 sample paths

Sample path 1

Sample path 2

Sample path 3

Sample path 4

Sample path 5

Consensus levels


Distribution of consensus under randomized networks

Histograms of consensus level of 5000 samples.

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.10

50

100

150

200

250

300

350

Consensus level

Fre

quency

Histogram of consensus: n = 25, sample size = 5000

Mean = −0.000273

Std = 0.018190

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

50

100

150

200

250

Consensus level

Fre

quency


Mean = −0.004247

Std = 0.215243

θ = 0.02 θ = 0.73

Note that consensus level under θ = 1 is around 0.8 and that under θ = 0 is around −0.5.


Pathwise convergence

Let Ω be the space of all infinite sequences drawn from A.

For any ω ∈ Ω, let B(t)ω = P

(t)ω P

(t−1)ω · · · P (1)

ω be the partial backwardproduct up to time t of ω.

Under what conditions is B(t)ω convergent as t→∞?





(t)ω P

(t−1)ω · · · P (1)



I We look at sample paths because we have only one history.

I Primitivity of A ∈ A is not sufficient for convergence.0 × 0

0 0 ×× × ×

·× × ×× 0 0

0 × 0

=

× 0 0

0 × 0

× × ×

primitive primitive reducible



Definition (Scrambling matrix)A non-negative square matrix T is said to be scrambling if for anyi, j ∈ N there exists at least one k ∈ N such that tik > 0 and tjk > 0.

[C] There is an integer k ≥ 1 such that for each t ≥ k the matrix B(t)ω is

scrambling.

Theorem (Anthonisse & Tijms (1977))Suppose that condition [C] holds. Then for any ω ∈ Ω, there is a vectorvω such that v>ω 1 = 1 and lim

t→∞B(t)

ω = 1v>ω , where the convergence isexponentially fast.

Note that if A contains only scrambling matrices, then [C] holds for anyω ∈ Ω with k = 1.



Corollary

If [C] holds, then for any ω ∈ Ω, there is a vector vω such that v>ω 1 = 1

andlimt→∞

(f (t)ω )i = v>ω f(0)

for all i ∈ N .

The consensus is reached but its value depends on the sample path ω.


Summary

Main insight:I Carefully distinguish between superposition and randomization.

I Collecting data on social networks can be misleading as itassumes superposition.

I Randomization is more natural for information updating.


Summary

Result for superposition:Wisdom will be preserved if influence of non-wise network vanishesquick enough.

Results for randomization:I Consensus can be reached under general conditions.

I Consensus 6= Truth as consensus is path dependent.

Variability in the communication structure can explain irrationalconsensus of a network of rational agents.


Thank you for your attention!


Wisdom of Crowds in Social Networkswith Structural Shocks

Ines LindnerJia-Ping Huang and Bernd Heidergott

VU University AmsterdamEconometrics & OR

Bremen, July 17th 2017



Social Networks have central role

Examples:I choice of electronic devices

I relative merits of politician

I information about scientific research and results

I coming to a public verdict



Social networksI shape opinions and beliefs

I subsequently shape behavior

Critical to understand how structure affects learning and informationdiffusion



Fundamental Questions:I Common consensus or divided opinions?

I most influential agents

I speed of learning

I whether initially diverse information scattered throughout societycan be aggregated in accurate manner


Literature


(1) Bayesian models:I individuals observe actions and results experienced by neighbors

I process information in sophisticated manner

I firm normative foundations

I leads to very cumbersome and complex analysis

I not necessarily descriptive


Literature

(1) Bayesian models:

Bala and Goyal (1998):I limited form of Bayesian updating

I agents process information from actions and outcomes

I ignore indirect information

See e.g. on three-link networksGale and Kariv (2003), Choi, Gale and Kariv (2005),Celen, Kariv, Schotter (2004)

Lobel and Sadler (2015)


Literature


(1) Bayesian models:I very cumbersome and complex analysis

I nevertheless provide insight in conditions for consensus


Literature


(2) Naıve learning modelI DeGroot (1974): Convergence to consensus

I Golub and Jackson (2010): wisdom of crowds (WoC)- Extend DeGroot model to growing networks

- Conditions of converging to “truth”

I This research: allow network variability→ Consensus is achieved

→ WoC may be lost



I Set of agents: N = 1, . . . , n

I Social influence matrix: P = [Pij ]i,j∈NPij : the weight or trust that agent i places on the current belief ofagent j in forming i’s belief for the next period.

I Initial beliefs: fi ∈ [−1, 1] for all i

I Updating: f (t)i =∑

j Pijf(t−1)j



Updating

1

2

3

1/3

1/31/30

1

01

2

3

1/3

1/31/31/3



Updating

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

01

01

2

3

1/3

1/3

1/21/21/3

1/2

1/2

1/3 1/2

0



Updating

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

01

01

2

3

1/3

1/3

1/21/21/3

1/2

1/2

1/3 1/2

0

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

5/185/12

1/6



Updating

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

2/7 2/7

2/7…

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

01

01

2

3

1/3

1/3

1/21/21/3

1/2

1/2

1/31/2

0

1

2

3

1/3

1/3

1/21/21/3

1/2

1/2

5/185/12

1/6



I Updating beliefs:

f (t) = Pf (t−1) = P t−1f (0) where f (0) = f

I Convergence: if the limit limt→∞

f (t) exists for all initial beliefs,



Example of non-convergence:

2

1

3

1

1/2

1/2

1

P =

0 1/2 1/2

1 0 0

1 0 0

, P 2 =

1 0 0

0 1/2 1/2

0 1/2 1/2

,P 3 =

1/2 1/2 0

1 0 0

1 0 0

, P 4 →

1 0 0

0 1/2 1/2

0 1/2 1/2

, . . .




2

1

3

1

1/2

1/2

1

P =

0 1/2 1/2

1 0 0

1 0 0

, P 2 =

1 0 0

0 1/2 1/2

0 1/2 1/2

,P 3 =

1/2 1/2 0

1 0 0

1 0 0

, P 4 →

1 0 0

0 1/2 1/2

0 1/2 1/2

, . . .The matrix oscillates and there is no convergence. If we start withf = (1, 0, 0),




2

1

3

1

1/2

1/2

1

P =

0 1/2 1/2

1 0 0

1 0 0

, P 2 =

1 0 0

0 1/2 1/2

0 1/2 1/2

,P 3 =

1/2 1/2 0

1 0 0

1 0 0

, P 4 →

1 0 0

0 1/2 1/2

0 1/2 1/2

, . . .The matrix oscillates and there is no convergence. If we start withf = (1, 0, 0),

f (1) =

0

1

1

, f (2) =

1

0

0

, f (3) =

0

1

1

, f (4) =

1

0

0

, . . .


Convergence of belief

P =

1/3 1/3 1/3

1/2 1/2 0

1/2 0 1/2

, . . . , P∞ →3/7 2/7 2/7

3/7 2/7 2/7

3/7 2/7 2/7

, π = (3/7, 2/7, 2/7),

f (∞) =3

7f1 +

2

7f2 +

2

7f3

TheoremSuppose P is strongly connected.

P is convergent if and only if it is aperiodic.

P is convergent if and only if limt Pt = Π. Each row of Π equals the

stationary distribution vector π> which is the unique solution of

π>P = π>, π>1 = 1.


Wisdom of Crowds





Wisdom of Crowds




Question: Is limt Ptf closer to the truth than individual signals?


Wisdom of Crowds




Question: Is limt Ptf closer to the truth than individual signals?

Golub and Jackson (2010): answer for n→∞.


Wisdom of crowds: the double limit in detail

……

1 2 3 4 t t+1

1

2

3

n

n+1

time

network size

π (1)

π (2)

π (3)

π (n)

π (n+1)

……



Denote P (n) a stochastic matrix with size n× n.

DefinitionThe sequence P (n)∞n=1 is wise if

limn→∞

Pr[

maxi≤n

∣∣f (∞)i (n)− µ

∣∣ > ε]

= 0

for any ε > 0.

In words, a sequence of networks is wise if the limiting beliefsconverge to the true state µ in probability.



PropositionIf P (n)∞n=1 is a sequence of strongly connected and aperiodicstochastic matrices, then it is wise if and only if the associatedstationary vectors π(n)∞n=1 are such that

maxi≤n

πi(n)→ 0.

Intuitively, this is to say that no individual is more influential than othersin the limit.



Wise: growing ring

π(n) =

1/n

1/n...

1/n

Non-wise: growing star

π(1) > 0 for n→∞



Non-wise network: a line

1

1-a

2 3 n

a

……a

1-a

a

1-a

P (n) =

1− a a 0 . . . 0

1− a 0 a . . . 0

0 1− a 0 . . . 0...

......

. . ....

0 0 0 . . . a

For a < 1/2, it holds that

πi(n) =( a

1− a

)i−1·

1− a1−a

1−(

a1−a)n .

Hence,limn→∞

π1(n) =(

1− a

1− a

).


Our interests





Our interests




Examples of shocks:I financial crisis, natural disaster, war, etc.

I natural communication


Random shock: superposition

Let P (n) be wise and and Q(n) a series of arbitrary Markovchains. For θ ∈ (0, 1), consider

P (θ, n) = (1− θ)P (n) + θQ(n).

P (n) wiseQ(n) series of arbitrary Markov chains


Random shock: randomization

We consider a random sequence P (t)t>0 whose elements areindependently drawn according to a Bernoulli trial.

1 2 3 4 ttime

P P Q P……

……

P……

Q

Ɵ1- Ɵ

……

In expectation, superposition and randomization are the same.

E[P (t)] = θP + (1− θ)Q.


Wisdom under superposition

TheoremAssume the superposition case

P (θ, n) = (1− θ)P (n) + θQ(n).

P (n)∞n=n0Q(n)∞n=n0

P (θ, n)∞n=n0

wise wise wisewise non-wise non-wise

non-wise non-wise can be wise or non-wise



TheoremAssume the superposition case

P (θ, n) = (1− θ)P (n) + θQ(n).

P (n)∞n=n0Q(n)∞n=n0

P (θ, n)∞n=n0

wise wise wisewise non-wise non-wise

non-wise non-wise can be wise or non-wise

Possible: Let θ(n)→ 0 quick enough to preserve wisdom.



Letδ(n) :=

1∥∥(Q(n)− P (n))DP (n)

∥∥1

where DP =∑∞

t=0(Pt −ΠP ) = FP −ΠP .

TheoremIf P (n) is wise, then P (θ(n), n) is wise provided that

limn→∞

θ(n)

δ(n)= 0.

δ(n): perturbation resilience,typically: δ(n)→ 0.


Wisdom of crowds

What is the insight of the randomization case?

1 2 3 4 ttime

P P Q P……

……

P……

Q

Ɵ1- Ɵ

……

TheoremAssume P and Q are strongly connected and aperiodic.For fixed n consensus exists if every agent influences at least twoother agents per update. This consensus is path dependent.



I Consider the ring network with self links(omitted in the picture).

I Assume

P (θ, n) =

12

14

14

14

12

14

14

12

14

. . . . . . . . .14

12

14

14

12

14

14

14

12

1

2 3

4 5

6 7

2m 2m+1



P =

12

14

14

13

12

16

13

12

16

. . .. . .

. . .13

12

16

13

12

16

13

16

12

,

Q =

12

14

14

c 12

d

c 12

d

. . .. . .

. . .c 1

2d

c 12

d

c d 12

,

c = (3 − 4θ)/(12 − 12θ),d = 1/2 − c = (3 − 2θ)/(12 − 12θ).

1

2 3

4 5

6 7

2m 2m+1



P =

12

14

14

13

12

16

13

12

16

. . .. . .

. . .13

12

16

13

12

16

13

16

12

,

Q =

12

14

14

c 12

d

c 12

d

. . .. . .

. . .c 1

2d

c 12

d

c d 12

,

c = (3 − 4θ)/(12 − 12θ),d = 1/2 − c = (3 − 2θ)/(12 − 12θ).

1

2 3

4 5

6 7

2m 2m+1


Note that c = 1/4 for θ = 0, c'(θ) < 0 and c = 0 for θ =3/4.

An illustrating example (again)

Both P (n)∞n=n0and Q(n)∞n=n0

are non-wise.

πP :1(2m+ 1) =2m

2m+2 − 3

m→∞→ 1

4,

πQ:2m(2m+ 1) = πQ:2m+1(2m+ 1) =(1− 4c)(1− 2c)m−1/2

(1− 2c)m − 2(2c)m+1

m→∞→ 1− 4c

2− 4c=

θ3 − 2θ

for i = 1, . . . ,m,

Note that the superposition P (θ, n)∞n=n0is wise.


Convergence under fixed networks

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by matrix P

Network P (θ), θ = 0.3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by matrix X

Network P

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by matrix Y

Network Q

Horizontal axis: time.Vertical axis: individual belief.n = 25. The initial belief f (0) = (1, 1112 , . . . ,−

1112 ,−1)>.


Convergence under randomized networks

Taking α = 0.3, we generate 5 sample paths of f (t)t>0, all startingfrom the same initial belief vector.

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Time

Be

lief

dis

trib

utio

n

Updated by randomly chosen matrix from X and Y: sample path 1

Belief dist of sample path 1

0 100 200 300 400 500−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time

Ave

rag

e b

elie

f

Updated by randomly chosen matrix from X and Y: 5 sample paths

Sample path 1

Sample path 2

Sample path 3

Sample path 4

Sample path 5

Consensus levels



Histograms of consensus level of 5000 samples.

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.10

50

100

150

200

250

300

350

Consensus level

Fre

quency


Mean = −0.000273

Std = 0.018190

α = 0.02

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

50

100

150

200

250

Consensus level

Fre

quency


Mean = −0.004247

Std = 0.215243

α = 0.73

Note that consensus level under α = 1 is around 0.8 and that underα = 0 is around −0.5.


Randomization: more general

I We consider a random sequence P (t) : t = 1, 2, . . . whoseelements are independently drawn from a finite set A of stochasticmatrices according to a probability distribution R over A, where

P =∑A∈A

R(A) ·A = E[P (t)] for all t ≥ 1.

I The corresponding belief process

f (t) = P (t)f (t−1), f(0)i = µ+ ei.





(t)ω P

(t−1)ω · · · P (1)





Theorem: Probability that sequence converges tends to one.





(t)ω P

(t−1)ω · · · P (1)



I We look at sample paths because we have only one history.

I Primitivity of A ∈ A is not sufficient for convergence.0 × 0

0 0 ×× × ×

·× × ×× 0 0

0 × 0

=

× 0 0

0 × 0

× × ×

primitive primitive reducible



Definition (Scrambling matrix)A non-negative square matrix T is said to be scrambling if for anyi, j ∈ N there exists at least one k ∈ N such that tik > 0 and tjk > 0.

[C] There is an integer k ≥ 1 such that for each t ≥ k the matrix B(t)ω is

scrambling.

Theorem (Anthonisse & Tijms (1977))Suppose that condition [C] holds. Then for any ω ∈ Ω, there is a vectorvω such that v>ω 1 = 1 and lim

t→∞B(t)

ω = 1v>ω , where the convergence isexponentially fast.

Note that if A contains only scrambling matrices, then [C] holds for anyω ∈ Ω with k = 1.



Corollary

If [C] holds, then for any ω ∈ Ω, there is a vector vω such that v>ω 1 = 1

andlimt→∞

(f (t)ω )i = v>ω f(0)

for all i ∈ N .

The consensus is reached but its value depends on the sample path ω.


Conclusion: Two ways of modeling

I We say P (t) : 1, 2, . . . and P are the randomization andsuperposition of A, respectively.

I Randomization:

Imagine that you estimate communication structures for each dayin [1, . . . , T ] and end up with a finite set of estimates together withtheir frequencies.E.g., X for weekdays and Y for weekends with R(X) = 5/7 andR(Y ) = 2/7.

I Superposition:

Imagine that you consider [1, . . . , T ] as one long period and use allthe data to obtain one averaged communication structure and youassume that agents use this structure everyday.


Summary

Main insight:I Carefully distinguish between superposition and randomization.

I Collecting data on social networks can be misleading as itassumes superposition.

I Randomization is more natural for information updating.


Summary

Result for superposition:Wisdom will be preserved if influence of non-wise network vanishesquick enough.

Results for randomization:I Consensus can be reached under general conditions.

I Consensus 6= Truth as consensus is path dependent.

Variability in the communication structure can explain irrationalconsensus of a network of rational agents.


Thank you for your attention!


Wisdom of Crowds in Social Networks with Structural Shocks

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